Contents

TitlePageContentsCopyrightIntroductionInfinityTheManWhoHarnessedInfinityDiscoveringtheLawsofMotionTheDawnofDifferentialCalculusTheCrossroadsTheVocabularyofChangeTheSecretFountainFictionsoftheMindTheLogicalUniverseMakingWavesTheFutureofCalculusConclusionAcknowledgmentsIllustrationCreditsNotesBibliographyIndexSampleChapterfromJOYOFXBuytheBookAbouttheAuthorConnectwithHMH

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LibraryofCongressCataloging-in-PublicationDataNames:Strogatz,StevenH.(StevenHenry),author.

Title:Infinitepowers:howcalculusrevealsthesecretsoftheuniverse/StevenStrogatz.Description:Boston:HoughtonMifflinHarcourt,2019.|

Includesbibliographicalreferencesandindex.Identifiers:LCCN2018042561(print)|LCCN2018049721(ebook)|ISBN9781328880017(ebook)|

ISBN9781328879981(hardcover)Subjects:LCSH:Calculus.|Calculus—History.|Archimedes.|Differentialcalculus.

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v1.0319

Introduction

Withoutcalculus,wewouldn’thavecellphones,computers,ormicrowaveovens.Wewouldn’thaveradio.Ortelevision.Orultrasoundforexpectantmothers,orGPSforlosttravelers.Wewouldn’thavesplittheatom,unraveledthehumangenome,orputastronautsonthemoon.WemightnotevenhavetheDeclarationofIndependence.It’sacuriosityofhistorythattheworldwaschangedforeverbyanarcane

branchofmathematics.Howcoulditbethatatheoryoriginallyaboutshapesultimatelyreshapedcivilization?TheessenceoftheanswerliesinaquipthatthephysicistRichardFeynman

madetothenovelistHermanWoukwhentheywerediscussingtheManhattanProject.WoukwasdoingresearchforabignovelhehopedtowriteaboutWorldWarII,andhewenttoCaltechtointerviewphysicistswhohadworkedonthebomb,oneofwhomwasFeynman.Aftertheinterview,astheywereparting,FeynmanaskedWoukifheknewcalculus.No,Woukadmitted,hedidn’t.“Youhadbetterlearnit,”saidFeynman.“It’sthelanguageGodtalks.”Forreasonsnobodyunderstands,theuniverseisdeeplymathematical.Maybe

Godmadeitthatway.Ormaybeit’stheonlywayauniversewithusinitcouldbe,becausenonmathematicaluniversescan’tharborlifeintelligentenoughtoaskthequestion.Inanycase,it’samysteriousandmarvelousfactthatouruniverseobeyslawsofnaturethatalwaysturnouttobeexpressibleinthelanguageofcalculusassentencescalleddifferentialequations.Suchequations

describethedifferencebetweensomethingrightnowandthesamethinganinstantlaterorbetweensomethingrighthereandthesamethinginfinitesimallycloseby.Thedetailsdifferdependingonwhatpartofnaturewe’retalkingabout,butthestructureofthelawsisalwaysthesame.Toputthisawesomeassertionanotherway,thereseemstobesomethinglikeacodetotheuniverse,anoperatingsystemthatanimateseverythingfrommomenttomomentandplacetoplace.Calculustapsintothisorderandexpressesit.IsaacNewtonwasthefirsttoglimpsethissecretoftheuniverse.Hefound

thattheorbitsoftheplanets,therhythmofthetides,andthetrajectoriesofcannonballscouldallbedescribed,explained,andpredictedbyasmallsetofdifferentialequations.TodaywecallthemNewton’slawsofmotionandgravity.EversinceNewton,wehavefoundthatthesamepatternholdswheneverweuncoveranewpartoftheuniverse.Fromtheoldelementsofearth,air,fire,andwatertothelatestinelectrons,quarks,blackholes,andsuperstrings,everyinanimatethingintheuniversebendstotheruleofdifferentialequations.IbetthisiswhatFeynmanmeantwhenhesaidthatcalculusisthelanguageGodtalks.Ifanythingdeservestobecalledthesecretoftheuniverse,calculusisit.Byinadvertentlydiscoveringthisstrangelanguage,firstinacornerof

geometryandlaterinthecodeoftheuniverse,thenbylearningtospeakitfluentlyanddecipheritsidiomsandnuances,andfinallybyharnessingitsforecastingpowers,humanshaveusedcalculustoremaketheworld.That’sthecentralargumentofthisbook.Ifit’sright,itmeanstheanswertotheultimatequestionoflife,theuniverse,

andeverythingisnot42,withapologiestofansofDouglasAdamsandTheHitchhiker’sGuidetotheGalaxy.ButDeepThoughtwasontherighttrack:thesecretoftheuniverseisindeedmathematical.

CalculusforEveryone

Feynman’squipaboutGod’slanguageraisesmanyprofoundquestions.Whatiscalculus?HowdidhumansfigureoutthatGodspeaksit(or,ifyouprefer,thattheuniverserunsonit)?Whataredifferentialequationsandwhathavetheydonefortheworld,notjustinNewton’stimebutinourown?Finally,howcananyofthesestoriesandideasbeconveyedenjoyablyandintelligiblytoreadersofgoodwilllikeHermanWouk,averythoughtful,curious,knowledgeablepersonwithlittlebackgroundinadvancedmath?

InacodatothestoryofhisencounterwithFeynman,Woukwrotethathedidn’tgetaroundtoeventryingtolearncalculusforfourteenyears.Hisbignovelballoonedintotwobignovels—WindsofWarandWarandRemembrance,eachaboutathousandpages.Oncethosewerefinallydone,hetriedtoteachhimselfbyreadingbookswithtitleslikeCalculusMadeEasy—butnoluckthere.Hepokedaroundinafewtextbooks,hoping,asheputit,“tocomeacrossonethatmighthelpamathematicalignoramuslikeme,whohadspenthiscollegeyearsinthehumanities—i.e.,literatureandphilosophy—inanadolescentquestforthemeaningofexistence,littleknowingthatcalculus,whichIhadheardofasadifficultboreleadingnowhere,wasthelanguageGodtalks.”Afterthetextbooksprovedimpenetrable,hehiredanIsraelimathtutor,hopingtopickupalittlecalculusandimprovehisspokenHebrewontheside,butbothhopesranaground.Finally,indesperation,heauditedahigh-schoolcalculusclass,buthefelltoofarbehindandhadtogiveupafteracoupleofmonths.Thekidsclappedforhimonhiswayout.Hesaiditwaslikesympathyapplauseforapitifulshowbizact.I’vewrittenInfinitePowersinanattempttomakethegreatestideasand

storiesofcalculusaccessibletoeveryone.Itshouldn’tbenecessarytoendurewhatHermanWoukdidtolearnaboutthislandmarkinhumanhistory.Calculusisoneofhumankind’smostinspiringcollectiveachievements.Itisn’tnecessarytolearnhowtodocalculustoappreciateit,justasitisn’tnecessarytolearnhowtopreparefinecuisinetoenjoyeatingit.I’mgoingtotrytoexplaineverythingwe’llneedwiththehelpofpictures,metaphors,andanecdotes.I’llalsowalkusthroughsomeofthefinestequationsandproofsevercreated,becausehowcouldwevisitagallerywithoutseeingitsmasterpieces?AsforHermanWouk,heis103yearsoldasofthiswriting.Idon’tknowifhe’slearnedcalculusyet,butifnot,thisone’sforyou,Mr.Wouk.

TheWorldAccordingtoCalculus

Asshouldbeobviousbynow,I’llbegivinganappliedmathematician’stakeonthestoryandsignificanceofcalculus.Ahistorianofmathematicswouldtellitdifferently.Sowouldapuremathematician.Whatfascinatesmeasanappliedmathematicianisthepushandpullbetweentherealworldaroundusandtheidealworldinourheads.Phenomenaoutthereguidethemathematicalquestionsweask;conversely,themathweimaginesometimesforeshadowswhatactuallyhappensoutthereinreality.Whenitdoes,theeffectisuncanny.

Tobeanappliedmathematicianistobeoutward-lookingandintellectuallypromiscuous.Tothoseinmyfield,mathisnotapristine,hermeticallysealedworldoftheoremsandproofsechoingbackonthemselves.Weembraceallkindsofsubjects:philosophy,politics,science,history,medicine,allofit.That’sthestoryIwanttotell—theworldaccordingtocalculus.Thisisamuchbroaderviewofcalculusthanusual.Itencompassesthemany

cousinsandspinoffsofcalculus,bothwithinmathematicsandintheadjacentdisciplines.Sincethisbig-tentviewisunconventional,Iwanttomakesureitdoesn’tcauseanyconfusion.Forexample,whenIsaidearlierthatwithoutcalculuswewouldn’thavecomputersandcellphonesandsoon,Icertainlydidn’tmeantosuggestthatcalculusproducedallthesewondersbyitself.Farfromit.Scienceandtechnologywereessentialpartners—andarguablythestarsoftheshow.Mypointismerelythatcalculushasalsoplayedacrucialrole,albeitoftenasupportingone,ingivingustheworldweknowtoday.Takethestoryofwirelesscommunication.Itbeganwiththediscoveryofthe

lawsofelectricityandmagnetismbyscientistslikeMichaelFaradayandAndré-MarieAmpère.Withouttheirobservationsandtinkering,thecrucialfactsaboutmagnets,electricalcurrents,andtheirinvisibleforcefieldswouldhaveremainedunknown,andthepossibilityofwirelesscommunicationwouldneverhavebeenrealized.So,obviously,experimentalphysicswasindispensablehere.Butsowascalculus.Inthe1860s,aScottishmathematicalphysicistnamed

JamesClerkMaxwellrecasttheexperimentallawsofelectricityandmagnetismintoasymbolicformthatcouldbefedintothemawofcalculus.Aftersomechurning,themawdisgorgedanequationthatdidn’tmakesense.Apparentlysomethingwasmissinginthephysics.MaxwellsuspectedthatAmpère’slawwastheculprit.Hetriedpatchingitupbyincludinganewterminhisequation—ahypotheticalcurrentthatwouldresolvethecontradiction—andthenletcalculuschurnagain.Thistimeitspatoutasensibleresult,asimple,elegantwaveequationmuchliketheequationthatdescribesthespreadofripplesonapond.ExceptMaxwell’sresultwaspredictinganewkindofwave,withelectricandmagneticfieldsdancingtogetherinapasdedeux.Achangingelectricfieldwouldgenerateachangingmagneticfield,whichinturnwouldregeneratetheelectricfield,andsoon,eachfieldbootstrappingtheotherforward,propagatingtogetherasawaveoftravelingenergy.AndwhenMaxwellcalculatedthespeedofthiswave,hefound—inwhatmusthavebeenoneofthegreatestAha!momentsinhistory—thatitmovedatthespeedoflight.Soheusedcalculusnotonlytopredicttheexistenceofelectromagneticwavesbutalsotosolveanage-oldmystery:Whatwasthenatureoflight?Light,herealized,wasanelectromagneticwave.

Maxwell’spredictionofelectromagneticwavespromptedanexperimentbyHeinrichHertzin1887thatprovedtheirexistence.Adecadelater,NikolaTeslabuiltthefirstradiocommunicationsystem,andfiveyearsafterthat,GuglielmoMarconitransmittedthefirstwirelessmessagesacrosstheAtlantic.Sooncametelevision,cellphones,andalltherest.Clearly,calculuscouldnothavedonethisalone.Butequallyclearly,noneof

itwouldhavehappenedwithoutcalculus.Or,perhapsmoreaccurately,itmighthavehappened,butonlymuchlater,ifatall.

CalculusIsMorethanaLanguage

ThestoryofMaxwellillustratesathemewe’llbeseeingagainandagain.It’softensaidthatmathematicsisthelanguageofscience.There’sagreatdealoftruthtothat.Inthecaseofelectromagneticwaves,itwasakeyfirststepforMaxwelltotranslatethelawsthathadbeendiscoveredexperimentallyintoequationsphrasedinthelanguageofcalculus.Butthelanguageanalogyisincomplete.Calculus,likeotherformsof

mathematics,ismuchmorethanalanguage;it’salsoanincrediblypowerfulsystemofreasoning.Itletsustransformoneequationintoanotherbyperformingvarioussymbolicoperationsonthem,operationssubjecttocertainrules.Thoserulesaredeeplyrootedinlogic,soeventhoughitmightseemlikewe’rejustshufflingsymbolsaround,we’reactuallyconstructinglongchainsoflogicalinference.Thesymbolshufflingisusefulshorthand,aconvenientwaytobuildargumentstoointricatetoholdinourheads.Ifwe’reluckyandskillfulenough—ifwetransformtheequationsinjustthe

rightway—wecangetthemtorevealtheirhiddenimplications.Toamathematician,theprocessfeelsalmostpalpable.It’sasifwe’remanipulatingtheequations,massagingthem,tryingtorelaxthemenoughsothatthey’llspilltheirsecrets.Wewantthemtoopenupandtalktous.Creativityisrequired,becauseitoftenisn’tclearwhichmanipulationsto

perform.InMaxwell’scase,therewerecountlesswaystotransformhisequations,allofwhichwouldhavebeenlogicallyacceptablebutonlysomeofwhichwouldhavebeenscientificallyrevealing.Giventhathedidn’tevenknowwhathewassearchingfor,hemighteasilyhavegottennothingoutofhisequationsbutincoherentmumblings(orthesymbolicequivalentthereof).Fortunately,however,theydidhaveasecrettoreveal.Withjusttherightprodding,theygaveupthewaveequation.

Atthatpointthelinguisticfunctionofcalculustookoveragain.WhenMaxwelltranslatedhisabstractsymbolsbackintoreality,theypredictedthatelectricityandmagnetismcouldpropagatetogetherasawaveofinvisibleenergymovingatthespeedoflight.Inamatterofdecades,thisrevelationwouldchangetheworld.

UnreasonablyEffective

It’seeriethatcalculuscanmimicnaturesowell,givenhowdifferentthetwodomainsare.Calculusisanimaginaryrealmofsymbolsandlogic;natureisanactualrealmofforcesandphenomena.Yetsomehow,ifthetranslationfromrealityintosymbolsisdoneartfullyenough,thelogicofcalculuscanuseonereal-worldtruthtogenerateanother.Truthin,truthout.Startwithsomethingthatisempiricallytrueandsymbolicallyformulated(asMaxwelldidwiththelawsofelectricityandmagnetism),applytherightlogicalmanipulations,andoutcomesanotherempiricaltruth,possiblyanewone,afactabouttheuniversethatnobodyknewbefore(liketheexistenceofelectromagneticwaves).Inthisway,calculusletsuspeerintothefutureandpredicttheunknown.That’swhatmakesitsuchapowerfultoolforscienceandtechnology.Butwhyshouldtheuniverserespecttheworkingsofanykindoflogic,let

alonethekindoflogicthatwepunyhumanscanmuster?ThisiswhatEinsteinmarveledatwhenhewrote,“Theeternalmysteryoftheworldisitscomprehensibility.”Andit’swhatEugeneWignermeantinhisessay“OntheUnreasonableEffectivenessofMathematicsintheNaturalSciences”whenhewrote,“Themiracleoftheappropriatenessofthelanguageofmathematicsfortheformulationofthelawsofphysicsisawonderfulgiftwhichweneitherunderstandnordeserve.”Thissenseofawegoeswaybackinthehistoryofmathematics.Accordingto

legend,Pythagorasfeltitaround550BCEwhenheandhisdisciplesdiscoveredthatmusicwasgovernedbytheratiosofwholenumbers.Forinstance,imaginepluckingaguitarstring.Asthestringvibrates,itemitsacertainnote.Nowputyourfingeronafretexactlyhalfwayupthestringandpluckitagain.Thevibratingpartofthestringisnowhalfaslongasitusedtobe—aratioof1to2—anditsoundspreciselyanoctavehigherthantheoriginalnote(themusicaldistancefromonedotothenextinthedo-re-mi-fa-sol-la-ti-doscale).Ifinsteadthevibratingstringis⅔ofitsoriginallength,thenoteitmakesgoesupbyafifth(theintervalfromdotosol;thinkofthefirsttwonotesoftheStarsWarstheme).

Andifthevibratingpartis¾aslongasitwasbefore,thenotegoesupbyafourth(theintervalbetweenthefirsttwonotesof“HereComestheBride”).TheancientGreekmusiciansknewaboutthemelodicconceptsofoctaves,fourths,andfifthsandconsideredthembeautiful.Thisunexpectedlinkbetweenmusic(theharmonyofthisworld)andnumbers(theharmonyofanimaginedworld)ledthePythagoreanstothemysticalbeliefthatallisnumber.Theyaresaidtohavebelievedthateventheplanetsintheirorbitsmademusic,themusicofthespheres.Eversincethen,manyofhistory’sgreatestmathematiciansandscientistshave

comedownwithcasesofPythagoreanfever.TheastronomerJohannesKeplerhaditbad.SodidthephysicistPaulDirac.Aswe’llsee,itdrovethemtoseek,andtodream,andtolongfortheharmoniesoftheuniverse.Intheenditpushedthemtomaketheirowndiscoveriesthatchangedtheworld.

TheInfinityPrinciple

Tohelpyouunderstandwherewe’reheaded,letmesayafewwordsaboutwhatcalculusis,whatitwants(metaphoricallyspeaking),andwhatdistinguishesitfromtherestofmathematics.Fortunately,asinglebig,beautifulidearunsthroughthesubjectfrombeginningtoend.Oncewebecomeawareofthisidea,thestructureofcalculusfallsintoplaceasvariationsonaunifyingtheme.Alas,mostcalculuscoursesburythethemeunderanavalancheofformulas,

procedures,andcomputationaltricks.Cometothinkofit,I’veneverseenitspelledoutanywhereeventhoughit’spartofcalculuscultureandeveryexpertknowsitimplicitly.Let’scallittheInfinityPrinciple.Itwillguideusonourjourneyjustasitguidedthedevelopmentofcalculusitself,conceptuallyaswellashistorically.I’mtemptedtostateitrightnow,butatthispointitwouldsoundlikemumbojumbo.Itwillbeeasiertoappreciateifweinchourwayuptoitbyaskingwhatcalculuswants...andhowitgetswhatitwants.Inanutshell,calculuswantstomakehardproblemssimpler.Itisutterly

obsessedwithsimplicity.Thatmightcomeasasurprisetoyou,giventhatcalculushasareputationforbeingcomplicated.Andthere’snodenyingthatsomeofitsleadingtextbooksexceedathousandpagesandweighasmuchasbricks.Butlet’snotbejudgmental.Calculuscan’thelphowitlooks.Itsbulkinessisunavoidable.Itlookscomplicatedbecauseit’stryingtotacklecomplicatedproblems.Infact,ithastackledandsolvedsomeofthemostdifficultandimportantproblemsourspecieshaseverfaced.

Calculussucceedsbybreakingcomplicatedproblemsdownintosimplerparts.Thatstrategy,ofcourse,isnotuniquetocalculus.Allgoodproblem-solversknowthathardproblemsbecomeeasierwhenthey’resplitintochunks.Thetrulyradicalanddistinctivemoveofcalculusisthatittakesthisdivide-and-conquerstrategytoitsutmostextreme—allthewayouttoinfinity.Insteadofcuttingabigproblemintoahandfulofbite-sizepieces,itkeepscuttingandcuttingrelentlesslyuntiltheproblemhasbeenchoppedandpulverizedintoitstiniestconceivableparts,leavinginfinitelymanyofthem.Oncethat’sdone,itsolvestheoriginalproblemforallthetinyparts,whichisusuallyamucheasiertaskthansolvingtheinitialgiantproblem.Theremainingchallengeatthatpointistoputallthetinyanswersbacktogetheragain.Thattendstobeamuchharderstep,butatleastit’snotasdifficultastheoriginalproblemwas.Thus,calculusproceedsintwophases:cuttingandrebuilding.In

mathematicalterms,thecuttingprocessalwaysinvolvesinfinitelyfinesubtraction,whichisusedtoquantifythedifferencesbetweentheparts.Accordingly,thishalfofthesubjectiscalleddifferentialcalculus.Thereassemblyprocessalwaysinvolvesinfiniteaddition,whichintegratesthepartsbackintotheoriginalwhole.Thishalfofthesubjectiscalledintegralcalculus.Thisstrategycanbeusedonanythingthatwecanimagineslicingendlessly.

Suchinfinitelydivisiblethingsarecalledcontinuaandaresaidtobecontinuous,fromtheLatinrootscon(togetherwith)andtenere(hold),meaninguninterruptedorholdingtogether.Thinkoftherimofaperfectcircle,asteelgirderinasuspensionbridge,abowlofsoupcoolingoffonthekitchentable,theparabolictrajectoryofajavelininflight,orthelengthoftimeyouhavebeenalive.Ashape,anobject,aliquid,amotion,atimeinterval—allofthemaregristforthecalculusmill.They’reallcontinuous,ornearlyso.Noticetheactofcreativefantasyhere.Soupandsteelarenotreally

continuous.Atthescaleofeverydaylife,theyappeartobe,butatthescaleofatomsorsuperstrings,they’renot.Calculusignorestheinconvenienceposedbyatomsandotheruncuttableentities,notbecausetheydon’texistbutbecauseit’susefultopretendthattheydon’t.Aswe’llsee,calculushasapenchantforusefulfictions.Moregenerally,thekindsofentitiesmodeledascontinuabycalculusinclude

almostanythingonecanthinkof.Calculushasbeenusedtodescribehowaballrollscontinuouslydownaramp,howasunbeamtravelscontinuouslythroughwater,howthecontinuousflowofairaroundawingkeepsahummingbirdoranairplanealoft,andhowtheconcentrationofHIVvirusparticlesinapatient’sbloodstreamplummetscontinuouslyinthedaysafterheorshestartscombination-drugtherapy.Ineverycasethestrategyremainsthesame:splita

complicatedbutcontinuousproblemintoinfinitelymanysimplerpieces,thensolvethemseparatelyandputthembacktogether.Nowwe’refinallyreadytostatethebigidea.

TheInfinityPrinciple

Toshedlightonanycontinuousshape,object,motion,process,orphenomenon—nomatterhowwildandcomplicateditmayappear—reimagineitasaninfiniteseriesofsimplerparts,analyzethose,andthenaddtheresultsbacktogethertomakesenseoftheoriginalwhole.

TheGolemofInfinity

Therubinallofthisistheneedtocopewithinfinity.That’seasiersaidthandone.Althoughthecarefullycontrolleduseofinfinityisthesecrettocalculusandthesourceofitsenormouspredictivepower,itisalsocalculus’sbiggestheadache.LikeFrankenstein’smonsterorthegoleminJewishfolklore,infinitytendstoslipoutofitsmaster’scontrol.Asinanytaleofhubris,themonsterinevitablyturnsonitsmaker.Thecreatorsofcalculuswereawareofthedangerbutstillfoundinfinity

irresistible.Sure,occasionallyitranamok,leavingparadox,confusion,andphilosophicalhavocinitswake.Yetaftereachoftheseepisodes,mathematiciansalwaysmanagedtosubduethemonster,rationalizeitsbehavior,andputitbacktowork.Intheend,everythingalwaysturnedoutfine.Calculusgavetherightanswers,evenwhenitscreatorscouldn’texplainwhy.Thedesiretoharnessinfinityandexploititspowerisanarrativethreadthatrunsthroughthewholetwenty-five-hundred-yearstoryofcalculus.Allthistalkofdesireandconfusionmightseemoutofplace,giventhat

mathematicsisusuallyportrayedasexactandimpeccablyrational.Itisrational,butnotalwaysinitially.Creationisintuitive;reasoncomeslater.Inthestoryofcalculus,morethaninotherpartsofmathematics,logichasalwayslagged

behindintuition.Thismakesthesubjectfeelespeciallyhumanandapproachable,anditsgeniusesmoreliketherestofus.

Curves,Motion,andChange

TheInfinityPrincipleorganizesthestoryofcalculusaroundamethodologicaltheme.Butcalculusisasmuchaboutmysteriesasitisaboutmethodology.Threemysteriesaboveallhavespurreditsdevelopment:themysteryofcurves,themysteryofmotion,andthemysteryofchange.Thefruitfulnessofthesemysterieshasbeenatestamenttothevalueofpure

curiosity.Puzzlesaboutcurves,motion,andchangemightseemunimportantatfirstglance,maybeevenhopelesslyesoteric.Butbecausetheytouchonsuchrichconceptualissuesandbecausemathematicsissodeeplywovenintothefabricoftheuniverse,thesolutiontothesemysterieshashadfar-reachingimpactsonthecourseofcivilizationandonoureverydaylives.Aswe’llseeinthechaptersahead,wereapthebenefitsoftheseinvestigationswheneverwelistentomusiconourphones,breezethroughthelineatthesupermarketthankstoalasercheckoutscanner,orfindourwayhomewithaGPSgadget.Itallstartedwiththemysteryofcurves.HereI’musingthetermcurvesina

veryloosesensetomeananysortofcurvedline,curvedsurface,orcurvedsolid—thinkofarubberband,aweddingring,afloatingbubble,thecontoursofavase,orasolidtubeofsalami.Tokeepthingsassimpleaspossible,theearlygeometerstypicallyconcentratedonabstract,idealizedversionsofcurvedshapesandignoredthickness,roughness,andtexture.Thesurfaceofamathematicalsphere,forinstance,wasimaginedtobeaninfinitesimallythin,smooth,perfectlyroundmembranewithnoneofthethickness,bumpiness,orhairinessofacoconutshell.Evenundertheseidealizedassumptions,curvedshapesposedbafflingconceptualdifficultiesbecausetheyweren’tmadeofstraightpieces.Trianglesandsquareswereeasy.Sowerecubes.Theywerecomposedofstraightlinesandflatpiecesofplanesjoinedtogetheratasmallnumberofcorners.Itwasn’thardtofigureouttheirperimetersorsurfaceareasorvolumes.Geometersallovertheworld—inancientBabylonandEgypt,ChinaandIndia,GreeceandJapan—knewhowtosolveproblemslikethese.Butroundthingswerebrutal.Noonecouldfigureouthowmuchsurfaceareaaspherehadorhowmuchvolumeitcouldhold.Evenfindingthecircumferenceandareaofacirclewasaninsurmountableproblemintheolddays.Therewasnowaytogetstarted.

Therewerenostraightpiecestolatchonto.Anythingthatwascurvedwasinscrutable.Sothisishowcalculusbegan.Itgrewoutofgeometers’curiosityand

frustrationwithroundness.CirclesandspheresandothercurvedshapesweretheHimalayasoftheirera.Itwasn’tthattheyposedimportantpracticalissues,atleastnotatfirst.Itwassimplyamatterofthehumanspirit’sthirstforadventure.LikeexplorersclimbingMountEverest,geometerswantedtosolvecurvesbecausetheywerethere.Thebreakthroughcamefrominsistingthatcurveswereactuallymadeof

straightpieces.Itwasn’ttrue,butonecouldpretendthatitwas.Theonlyhitchwasthatthosepieceswouldthenhavetobeinfinitesimallysmallandinfinitelynumerous.Throughthisfantasticconception,integralcalculuswasborn.ThiswastheearliestuseoftheInfinityPrinciple.Thestoryofhowitdevelopedwilloccupyusforseveralchapters,butitsessenceisalreadythere,inembryonicform,inasimple,intuitiveinsight:Ifwezoomincloselyenoughonacircle(oranythingelsethatiscurvedandsmooth),theportionofitunderthemicroscopebeginstolookstraightandflat.Soinprinciple,atleast,itshouldbepossibletocalculatewhateverwewantaboutacurvedshapebyaddingupallthestraightlittlepieces.Figuringoutexactlyhowtodothis—noeasyfeat—tooktheeffortsoftheworld’sgreatestmathematiciansovermanycenturies.Collectively,however,andsometimesthroughbitterrivalries,theyeventuallybegantomakeheadwayontheriddleofcurves.Spinoffstoday,aswe’llseeinchapter2,includethemathneededtodrawrealistic-lookinghair,clothing,andfacesofcharactersincomputer-animatedmoviesandthecalculationsrequiredfordoctorstoperformfacialsurgeryonavirtualpatientbeforetheyoperateontherealone.Thequesttosolvethemysteryofcurvesreachedafeverpitchwhenitbecame

clearthatcurvesweremuchmorethangeometricdiversions.Theywereakeytounlockingthesecretsofnature.Theyarosenaturallyintheparabolicarcofaballinflight,intheellipticalorbitofMarsasitmovedaroundthesun,andintheconvexshapeofalensthatcouldbendandfocuslightwhereitwasneeded,aswasrequiredfortheburgeoningdevelopmentofmicroscopesandtelescopesinlateRenaissanceEurope.Andsobeganthesecondgreatobsession:afascinationwiththemysteriesof

motiononEarthandinthesolarsystem.Throughobservationandingeniousexperiments,scientistsdiscoveredtantalizingnumericalpatternsinthesimplestmovingthings.Theymeasuredtheswingingofapendulum,clockedtheacceleratingdescentofaballrollingdownaramp,andchartedthestatelyprocessionofplanetsacrossthesky.Thepatternstheyfoundenrapturedthem—

indeed,JohannesKeplerfellintoastateofself-described“sacredfrenzy”whenhefoundhislawsofplanetarymotion—becausethosepatternsseemedtobesignsofGod’shandiwork.Fromamoresecularperspective,thepatternsreinforcedtheclaimthatnaturewasdeeplymathematical,justasthePythagoreanshadmaintained.Theonlycatchwasthatnobodycouldexplainthemarvelousnewpatterns,atleastnotwiththeexistingformsofmath.Arithmeticandgeometrywerenotuptothetask,eveninthehandsofthegreatestmathematicians.Thetroublewasthatthemotionsweren’tsteady.Aballrollingdownaramp

keptchangingitsspeed,andaplanetrevolvingaroundthesunkeptchangingitsdirectionoftravel.Worseyet,theplanetsmovedfasterwhentheygotclosetothesunandsloweddownastheyrecededfromit.Therewasnoknownwaytodealwithmotionthatkeptchanginginever-changingways.Earliermathematicianshadworkedoutthemathematicsofthemosttrivialkindofmotion,namely,motionataconstantspeedwheredistanceequalsratetimestime.Butwhenspeedchangedandkeptonchangingcontinuously,allbetswereoff.MotionwasprovingtobeasmuchofaconceptualMountEverestascurveswere.Aswe’llseeinthemiddlechaptersofthisbook,thenextgreatadvancesin

calculusgrewoutofthequesttosolvethemysteryofmotion.TheInfinityPrinciplecametotherescue,justasithadforcurves.Thistimetheactofwishfulfantasywastopretendthatmotionatachangingspeedwasmadeupofinfinitelymany,infinitesimallybriefmotionsataconstantspeed.Tovisualizewhatthiswouldmean,imaginebeinginacarwithajerkydriveratthewheel.Asyouanxiouslywatchthespeedometer,itmovesupanddownwitheveryjerk.Butoveramillisecond,eventhejerkiestdrivercan’tmakethespeedometerneedlemovebymuch.Andoveranintervalmuchshorterthanthat—aninfinitesimaltimeinterval—theneedlewon’tmoveatall.Nobodycantapthegaspedalthatfast.Theseideascoalescedintheyoungerhalfofcalculus,differentialcalculus.It

waspreciselywhatwasneededtoworkwiththeinfinitesimallysmallchangesoftimeanddistancethataroseinthestudyofever-changingmotionaswellaswiththeinfinitesimalstraightpiecesofcurvesthataroseinanalyticgeometry,thenewfangledstudyofcurvesdefinedbyalgebraicequationsthatwasalltherageinthefirsthalfofthe1600s.Yes,atonetime,algebrawasacraze,aswe’llsee.Itspopularitywasaboonforallfieldsofmathematics,includinggeometry,butitalsocreatedanunrulyjungleofnewcurvestoexplore.Thus,themysteriesofcurvesandmotioncollided.Theywerenowbothatthecenterstageofcalculusinthemid-1600s,bangingintoeachother,creatingmathematicalmayhemand

confusion.Outofthetumult,differentialcalculusbegantoflower,butnotwithoutcontroversy.Somemathematicianswerecriticizedforplayingfastandloosewithinfinity.Othersderidedalgebraasascabofsymbols.Withallthebickering,progresswasfitfulandslow.AndthenachildwasbornonChristmasDay.Thisyoungmessiahofcalculus

wasanunlikelyhero.Bornprematureandfatherlessandabandonedbyhismotheratagethree,hewasalonesomeboywithdarkthoughtswhogrewintoasecretive,suspiciousyoungman.YetIsaacNewtonwouldmakeamarkontheworldlikenoonebeforeorsince.First,hesolvedtheholygrailofcalculus:hediscoveredhowtoputthepieces

ofacurvebacktogetheragain—andhowtodoiteasily,quickly,andsystematically.Bycombiningthesymbolsofalgebrawiththepowerofinfinity,hefoundawaytorepresentanycurveasasumofinfinitelymanysimplercurvesdescribedbypowersofavariablex,likex2,x3,x4,andsoon.Withtheseingredientsalone,hecouldcookupanycurvehewantedbyputtinginapinchofxandadashofx2andaheapingtablespoonofx3.Itwaslikeamasterrecipeandauniversalspicerack,butchershop,andvegetablegarden,allrolledintoone.Withithecouldsolveanyproblemaboutshapesormotionsthathadeverbeenconsidered.Thenhecrackedthecodeoftheuniverse.Newtondiscoveredthatmotionof

anykindalwaysunfoldsoneinfinitesimalstepatatime,steeredfrommomenttomomentbymathematicallawswritteninthelanguageofcalculus.Withjustahandfulofdifferentialequations(hislawsofmotionandgravity),hecouldexplaineverythingfromthearcofacannonballtotheorbitsoftheplanets.Hisastonishing“systemoftheworld”unifiedheavenandearth,launchedtheEnlightenment,andchangedWesternculture.ItsimpactonthephilosophersandpoetsofEuropewasimmense.HeeveninfluencedThomasJeffersonandthewritingoftheDeclarationofIndependence,aswe’llsee.Inourowntime,Newton’sideasunderpinnedthespaceprogrambyprovidingthemathematicsnecessaryfortrajectorydesign,theworkdoneatNASAbyAfrican-AmericanmathematicianKatherineJohnsonandhercolleagues(theheroinesofthebookandhitmovieHiddenFigures).Withthemysteriesofcurvesandmotionnowsettled,calculusmovedontoits

thirdlifelongobsession:themysteryofchange.It’sacliché,butit’strueallthesame—nothingisconstantbutchange.It’srainyonedayandsunnythenext.Thestockmarketrisesandfalls.EmboldenedbytheNewtonianparadigm,thelaterpractitionersofcalculusasked:AretherelawsofchangesimilartoNewton’slawsofmotion?Aretherelawsforpopulationgrowth,thespreadof

epidemics,andtheflowofbloodinanartery?Cancalculusbeusedtodescribehowelectricalsignalspropagatealongnervesortopredicttheflowoftrafficonahighway?Bypursuingthisambitiousagenda,alwaysincooperationwithotherpartsof

scienceandtechnology,calculushashelpedmaketheworldmodern.Usingobservationandexperiment,scientistsworkedoutthelawsofchangeandthenusedcalculustosolvethemandmakepredictions.Forexample,in1917AlbertEinsteinappliedcalculustoasimplemodelofatomictransitionstopredictaremarkableeffectcalledstimulatedemission(whichiswhatthesandestandforinlaser,anacronymforlightamplificationbystimulatedemissionofradiation).Hetheorizedthatundercertaincircumstances,lightpassingthroughmattercouldstimulatetheproductionofmorelightatthesamewavelengthandmovinginthesamedirection,creatingacascadeoflightthroughakindofchainreactionthatwouldresultinanintense,coherentbeam.Afewdecadeslater,thepredictionprovedtobeaccurate.Thefirstworkinglaserswerebuiltintheearly1960s.Sincethen,theyhavebeenusedineverythingfromcompact-discplayersandlaser-guidedweaponrytosupermarketbar-codescannersandmedicallasers.Thelawsofchangeinmedicinearenotaswellunderstoodasthoseinphysics.

Yetevenwhenappliedtorudimentarymodels,calculushasbeenabletomakelifesavingcontributions.Forexample,inchapter8we’llseehowadifferential-equationmodeldevelopedbyanimmunologistandanAIDSresearcherplayedapartinshapingthemodernthree-drugcombinationtherapyforpatientsinfectedwithHIV.Theinsightsprovidedbythemodeloverturnedtheprevailingviewthattheviruswaslyingdormantinthebody;infact,itwasinaragingbattlewiththeimmunesystemeveryminuteofeveryday.Withthenewunderstandingthatcalculushelpedprovide,HIVinfectionhasbeentransformedfromanear-certaindeathsentencetoamanageablechronicdisease—atleastforthosewithaccesstocombination-drugtherapy.Admittedly,someaspectsofourever-changingworldliebeyondthe

approximationsandwishfulthinkinginherentintheInfinityPrinciple.Inthesubatomicrealm,forexample,physicistscannolongerthinkofanelectronasaclassicalparticlefollowingasmoothpathinthesamewaythataplanetoracannonballdoes.Accordingtoquantummechanics,trajectoriesbecomejittery,blurry,andpoorlydefinedatthemicroscopicscale,soweneedtodescribethebehaviorofelectronsasprobabilitywavesinsteadofNewtoniantrajectories.Assoonaswedothat,however,calculusreturnstriumphantly.ItgovernstheevolutionofprobabilitywavesthroughsomethingcalledtheSchrödingerequation.

It’sincrediblebuttrue:EveninthesubatomicrealmwhereNewtonianphysicsbreaksdown,Newtoniancalculusstillworks.Infact,itworksspectacularlywell.Aswe’llseeinthepagesahead,ithasteamedupwithquantummechanicstopredicttheremarkableeffectsthatunderliemedicalimaging,fromMRIandCTscanstothemoreexoticpositronemissiontomography.It’stimeforustotakeacloserlookatthelanguageoftheuniverse.Naturally,

theplacetostartisatinfinity.

1

Infinity

THEBEGINNINGSOFmathematicsweregroundedineverydayconcerns.Shepherdsneededtokeeptrackoftheirflocks.Farmersneededtoweighthegrainreapedintheharvest.Taxcollectorshadtodecidehowmanycowsorchickenseachpeasantowedtheking.Outofsuchpracticaldemandscametheinventionofnumbers.Atfirsttheyweretalliedonfingersandtoes.Latertheywerescratchedonanimalbones.Astheirrepresentationevolvedfromscratchestosymbols,numbersfacilitatedeverythingfromtaxationandtradetoaccountingandcensustaking.WeseeevidenceofallthisinMesopotamianclaytabletswrittenmorethanfivethousandyearsago:rowafterrowofentriesrecordedwiththewedge-shapedsymbolscalledcuneiform.Alongwithnumbers,shapesmatteredtoo.InancientEgypt,themeasurement

oflinesandangleswasofparamountimportance.Eachyearsurveyorshadtoredrawtheboundariesoffarmers’fieldsafterthesummerfloodingoftheNilewashedtheborderlinesaway.Thatactivitylatergaveitsnametothestudyofshapeingeneral:geometry,fromtheGreekgē,“earth,”andmetrēs,“measurer.”Atthestart,geometrywashard-edgedandsharp-cornered.Itspredilectionfor

straightlines,planes,andanglesreflecteditsutilitarianorigins—triangleswereusefulasramps,pyramidsasmonumentsandtombs,andrectanglesastabletops,altars,andplotsofland.Buildersandcarpentersusedrightanglesforplumblines.Forsailors,architects,andpriests,knowledgeofstraight-linegeometrywasessentialforsurveying,navigating,keepingthecalendar,predictingeclipses,anderectingtemplesandshrines.

Yetevenwhengeometrywasfixatedonstraightness,onecurvealwaysstoodout,themostperfectofall:thecircle.Weseecirclesintreerings,intheripplesonapond,intheshapeofthesunandthemoon.Circlessurroundusinnature.Andaswegazeatcircles,theygazebackatus,literally.Theretheyareintheeyesofourlovedones,inthecircularoutlinesoftheirpupilsandirises.Circlesspanthepracticalandtheemotional,aswheelsandweddingrings,andtheyaremysticaltoo.Theireternalreturnsuggeststhecycleoftheseasons,reincarnation,eternallife,andnever-endinglove.Nowondercircleshavecommandedattentionforaslongashumanityhasstudiedshapes.Mathematically,circlesembodychangewithoutchange.Apointmoving

aroundthecircumferenceofacirclechangesdirectionwithouteverchangingitsdistancefromacenter.It’saminimalformofchange,awaytochangeandcurveintheslightestwaypossible.And,ofcourse,circlesaresymmetrical.Ifyourotateacircleaboutitscenter,itlooksunchanged.Thatrotationalsymmetrymaybewhycirclesaresoubiquitous.Wheneversomeaspectofnaturedoesn’tcareaboutdirection,circlesareboundtoappear.Considerwhathappenswhenaraindrophitsapuddle:tinyripplesexpandoutwardfromthepointofimpact.Becausetheyspreadequallyfastinalldirectionsandbecausetheystartedatasinglepoint,therippleshavetobecircles.Symmetrydemandsit.Circlescanalsogivebirthtoothercurvedshapes.Ifweimagineskeweringa

circleonitsdiameterandspinningitaroundthataxisinthree-dimensionalspace,therotatingcirclemakesasphere,theshapeofaglobeoraball.Whenacircleismovedverticallyintothethirddimensionalongastraightlineatrightanglestoitsplane,itmakesacylinder,theshapeofacanorahatbox.Ifitshrinksatthesametimeasit’smovingvertically,itmakesacone;ifitexpandsasitmovesvertically,itmakesatruncatedcone(theshapeofalampshade).

Circles,spheres,cylinders,andconesfascinatedtheearlygeometers,butthey

foundthemmuchhardertoanalyzethantriangles,rectangles,squares,cubes,andotherrectilinearshapesmadeofstraightlinesandflatplanes.Theywonderedabouttheareasofcurvedsurfacesandthevolumesofcurvedsolidsbuthadnocluehowtosolvesuchproblems.Roundnessdefeatedthem.

InfinityasaBridgeBuilder

Calculusbeganasanoutgrowthofgeometry.Backaround250BCEinancientGreece,itwasahotlittlemathematicalstartupdevotedtothemysteryofcurves.Theambitiousplanofitsdevoteeswastouseinfinitytobuildabridgebetweenthecurvedandthestraight.Thehopewasthatoncethatlinkwasestablished,themethodsandtechniquesofstraight-linegeometrycouldbeshuttledacrossthebridgeandbroughttobearonthemysteryofcurves.Withinfinity’shelp,alltheoldproblemscouldbesolved.Atleast,thatwasthepitch.Atthetime,thatplanmusthaveseemedprettyfar-fetched.Infinityhada

dubiousreputation.Itwasknownforbeingscary,notuseful.Worseyet,itwasnebulousandbewildering.Whatwasitexactly?Anumber?Aplace?Aconcept?

Nevertheless,aswe’llseesoonandinthechapterstocome,infinityturnedouttobeagodsend.Givenallthediscoveriesandtechnologiesthatultimatelyflowedfromcalculus,theideaofusinginfinitytosolvedifficultgeometryproblemshastorankasoneofthebestideasanyoneeverhad.Ofcourse,noneofthatcouldhavebeenforeseenin250BCE.Still,infinitydid

putsomeimpressivenotchesinitsbeltrightaway.Oneofitsfirstandfinestwasthesolutionofalong-standingenigma:howtofindtheareaofacircle.

APizzaProof

BeforeIgointothedetails,letmesketchtheargument.Thestrategyistoreimaginethecircleasapizza.Thenwe’llslicethatpizzaintoinfinitelymanypiecesandmagicallyrearrangethemtomakearectangle.Thatwillgiveustheanswerwe’relookingfor,sincemovingslicesaroundobviouslydoesn’tchangetheirareafromwhattheywereoriginally,andweknowhowtofindtheareaofarectangle:wejustmultiplyitswidthtimesitsheight.Theresultisaformulafortheareaofacircle.Forthesakeofthisargument,thepizzaneedstobeanidealizedmathematical

pizza,perfectlyflatandround,withaninfinitesimallythincrust.Itscircumference,abbreviatedbytheletterC,isthedistancearoundthepizza,measuredbytracingaroundthecrust.Circumferenceisn’tsomethingthatpizzaloversordinarilycareabout,butifwewantedto,wecouldmeasureCwithatapemeasure.

Anotherquantityofinterestisthepizza’sradius,r,definedasthedistancefromitscentertoeverypointonitscrust.Inparticular,ralsomeasureshowlongthestraightsideofasliceis,assumingthatalltheslicesareequalandcutfromthecenterouttothecrust.

Supposewestartbydividingthepieintofourquarters.Here’sonewayto

rearrangethem,butitdoesn’tlooktoopromising.

Thenewshapelooksbulbousandstrangewithitsscallopedtopandbottom.It’scertainlynotarectangle,soitsareaisnoteasytoguess.Weseemtobegoingbackward.Butasinanydrama,theheroneedstogetintotroublebeforetriumphing.Thedramatictensionisbuilding.

Whilewe’restuckhere,though,weshouldnoticetwothings,becausetheyaregoingtoholdtruethroughouttheproof,andtheywillultimatelygiveusthedimensionsoftherectanglewe’reseeking.Thefirstobservationisthathalfofthecrustbecamethecurvytopofthenewshape,andtheotherhalfbecamethebottom.Sothecurvytophasalengthequaltohalfthecircumference,C/2,andsodoesthebottom,asshowninthediagram.Thatlengthiseventuallygoingtoturnintothelongsideoftherectangle,aswe’llsee.Theotherthingtonoticeisthatthetiltedstraightsidesofthebulbousshapearejustthesidesoftheoriginalpizzaslices,sotheystillhavelengthr.Thatlengthiseventuallygoingtoturnintotheshortsideoftherectangle.Thereasonwearen’tseeinganysignsofthedesiredrectangleyetisthatwe

haven’tcutenoughslices.Ifwemakeeightslicesandrearrangethemlikeso,ourpicturestartstolookmorenearlyrectangular.

Infact,thepizzastartstolooklikeaparallelogram.Notbad—atleastit’salmostrectilinear.Andthescallopsonthetopandbottomarealotlessbulbousthantheywere.Theyflattenedoutwhenweusedmoreslices.Asbefore,theyhavecurvylengthC/2onthetopandbottomandaslanted-sidelengthr.Tospruceupthepictureevenmore,supposewecutoneoftheslantedend

piecesinhalflengthwiseandshiftthathalftotheotherside.

Nowtheshapelooksverymuchlikearectangle.Admittedly,it’sstillnotperfectbecauseofthescallopedtopandbottomcausedbythecurvatureofthecrust,butatleastwe’remakingprogress.Sincemakingmorepiecesseemstobehelping,let’skeepslicing.With

sixteenslicesandthecosmeticsprucing-upoftheendpiece,aswedidbefore,wegetthisresult:

Themoresliceswetake,themoreweflattenoutthescallopsproducedbythe

crust.Ourmaneuversareproducingasequenceofshapesthataremagicallyhominginonacertainrectangle.Becausetheshapeskeepgettingcloserandclosertothatrectangle,we’llcallitthelimitingrectangle.

Thepointofallthisisthatwecaneasilyfindtheareaofthislimiting

rectanglebymultiplyingitswidthbyitsheight.Allthatremainsistofindthatheightandwidthintermsofthecircle’sdimensions.Well,sincetheslicesarestandingupright,theheightisjusttheradiusroftheoriginalcircle.Andthewidthishalfthecircumferenceofthecircle;that’sbecausehalfofthecircumference(thecrustofthepizza)wentintomakingthetopoftherectangleandtheotherhalfgotusedonthebottom,justasitdidateveryintermediatestageofworkingwiththebulbousshapes.Thusthewidthishalfthecircumference,C/2.Puttingeverythingtogether,theareaofthelimitingrectangleisgivenbyitsheighttimesitswidth,namely,A=r×C/2=rC/2.And

sincemovingthepizzaslicesarounddidnotchangetheirarea,thismustalsobetheareaoftheoriginalcircle!Thisresultfortheareaofacircle,A=rC/2,wasfirstproved(usingasimilar

butmuchmorecarefulargument)bytheancientGreekmathematicianArchimedes(287–212BCE)inhisessay“MeasurementofaCircle.”Themostinnovativeaspectoftheproofisthewayinfinitycametotherescue.

Whenwehadonlyfourslices,oreight,orsixteen,thebestwecoulddowasrearrangethepizzaintoanimperfectscallopedshape.Afteranunpromisingstart,themoresliceswetook,themorerectangulartheshapebecame.Butitwasonlyinthelimitofinfinitelymanyslicesthatitbecametrulyrectangular.That’sthebigideabehindcalculus.Everythingbecomessimpleratinfinity.

LimitsandtheRiddleoftheWall

Alimitislikeanunattainablegoal.Youcangetcloserandclosertoit,butyoucannevergetallthewaythere.Forexample,inthepizzaproofwewereabletomakethescallopedshapes

moreandmorenearlyrectangularbycuttingenoughslicesandrearrangingthem.Butwecouldnevermakethemgenuinelyrectangular.Wecouldonlyapproachthatstateofperfection.Fortunately,incalculus,theunattainabilityofthelimitusuallydoesn’tmatter.Wecanoftensolvetheproblemswe’reworkingonbyfantasizingthatwecanactuallyreachthelimitandthenseeingwhatthatfantasyimplies.Infact,manyofthegreatestpioneersofthesubjectdidpreciselythatandmadegreatdiscoveriesbydoingso.Logical,no.Imaginative,yes.Successful,very.Alimitisasubtleconceptbutacentraloneincalculus.It’selusivebecause

it’snotacommonideaindailylife.PerhapstheclosestanalogyistheRiddleoftheWall.Ifyouwalkhalfwaytothewall,andthenyouwalkhalftheremainingdistance,andthenyouwalkhalfofthat,andonandon,willthereeverbeastepwhenyoufinallygettothewall?

Theanswerisclearlyno,becausetheRiddleoftheWallstipulatesthatateach

step,youwalkhalfwaytothewall,notalltheway.Afteryoutaketenstepsoramillionoranyothernumberofsteps,therewillalwaysbeagapbetweenyouandthewall.Butequallyclearly,youcangetarbitrarilyclosetothewall.Whatthismeansisthatbytakingenoughsteps,youcangettowithinacentimeterofit,oramillimeter,orananometer,oranyothertinybutnonzerodistance,butyoucannevergetallthewaythere.Here,thewallplaystheroleofthelimit.Ittookabouttwothousandyearsforthelimitconcepttoberigorouslydefined.Untilthen,thepioneersofcalculusgotbyjustfinewithintuition.Sodon’tworryiflimitsfeelhazyfornow.We’llgettoknowthembetterbywatchingtheminaction.Fromamodernperspective,theymatterbecausetheyarethebedrockonwhichallofcalculusisbuilt.Ifthemetaphorofthewallseemstoobleakandinhuman(whowantsto

approachawall?),trythisanalogy:Anythingthatapproachesalimitislikeaheroengagedinanendlessquest.It’snotanexerciseintotalfutility,likethehopelesstaskfacedbySisyphus,whowascondemnedtorollaboulderupahillonlytoseeitrollbackdownagainoverandoverforeternity.Rather,whenamathematicalprocessadvancestowardalimit(likethescallopedshapeshominginonthelimitingrectangle),it’sasifaprotagonistisstrivingforsomethingheknowsisimpossiblebutforwhichhestillholdsoutthehopeofsuccess,encouragedbythesteadyprogresshe’smakingwhiletryingtoreachanunreachablestar.

TheParableof.333...

Toreinforcethebigideasthateverythingbecomessimpleratinfinityandthatlimitsarelikeunattainablegoals,considerthefollowingexamplefromarithmetic.It’stheproblemofconvertingafraction—forexample,⅓—intoan

equivalentdecimal(inthiscase,⅓=0.333...).Ivividlyrememberwhenmyeighth-grademathteacher,Ms.Stanton,taughtushowtodothis.Itwasmemorablebecauseshesuddenlystartedtalkingaboutinfinity.Untilthatmoment,I’dneverheardagrownupmentioninfinity.Myparents

certainlyhadnouseforit.Itseemedlikeasecretthatonlykidsknewabout.Ontheplayground,itcameupallthetimeintauntsandone-upmanship.

“You’reajerk!”“Yeah,well,you’reajerktimestwo!”“Andyou’reajerktimesinfinity!”“Andyou’reajerktimesinfinityplusone!”“That’sthesameasinfinity,youidiot!”

Thoseedifyingsessionshadconvincedmethatinfinitydidnotbehavelikean

ordinarynumber.Itdidn’tgetbiggerwhenyouaddedonetoit.Evenaddinginfinitytoitdidn’thelp.Itsinvinciblepropertiesmadeitgreatforfinishingargumentsintheschoolyard.Whoeverdeployeditfirstwouldwin.ButnoteacherhadevertalkedaboutinfinityuntilMs.Stantonbroughtitup

thatday.Everyoneinourclassalreadyknewaboutfinitedecimals,thefamiliarkindusedforamountsofmoney,like$10.28,withitstwodigitsafterthedecimalpoint.Bycomparison,infinitedecimals,whichhadinfinitelymanydigitsafterthedecimalpoint,seemedstrangeatfirstbutappearednaturalassoonaswestartedtodiscussfractions.Welearnedthatthefraction⅓couldbewrittenas0.333...wherethedot-

dot-dotsmeantthatthethreesrepeatedindefinitely.Thatmadesensetome,becausewhenItriedtocalculate⅓bydoingthelong-divisionalgorithmonit,Ifoundmyselfstuckinanendlessloop:threedoesn’tgointoone,sopretendtheoneisaten;thenthreegoesintotenthreetimes,whichleavesaremainderofone;andnowI’mbackwhereIstarted,stilltryingtodividethreeintoone.Therewasnowayoutoftheloop.That’swhythethreeskeptrepeatingin0.333....Thethreedotsattheendof0.333...havetwointerpretations.Thenaive

interpretationisthatthereareliterallyinfinitelymany3spackedsidebysidetotherightofthedecimalpoint.Wecan’twritethemalldown,ofcourse,sincethereareinfinitelymanyofthem,butbywritingthethreedotswesignifythattheyareallthere,atleastinourminds.I’llcallthisthecompletedinfinityinterpretation.Theadvantageofthisinterpretationisthatitseemseasyandcommonsensical,aslongaswearewillingnottothinktoohardaboutwhatinfinitymeans.

Themoresophisticatedinterpretationisthat0.333...representsalimit,justlikethelimitingrectangledoesforthescallopedshapesinthepizzaprooforlikethewalldoesforthehaplesswalker.Excepthere,0.333...representsthelimitofthesuccessivedecimalswegeneratebydoinglongdivisiononthefraction⅓.Asthedivisionprocesscontinuesformoreandmoresteps,itgeneratesmoreandmore3sinthedecimalexpansionof⅓.Bygrindingaway,wecanproduceanapproximationascloseto⅓aswelike.Ifwe’renothappywith⅓≈0.3,wecanalwaysgoastepfurtherto⅓≈0.33,andsoon.I’llcallthisthepotentialinfinityinterpretation.It’s“potential”inthesensethattheapproximationscanpotentiallygoonforaslongasdesired.There’snothingtostopusfromcontinuingforamillionorabillionoranyothernumberofsteps.Theadvantageofthisinterpretationisthatweneverhavetoinvokewoolly-headednotionslikeinfinity.Wecansticktothefinite.Forworkingwithequationslike⅓=0.333...,itdoesn’treallymatterwhich

viewwetake.They’reequallytenableandyieldthesamemathematicalresultsinanycalculationwecaretoperform.Butthereareothersituationsinmathematicswherethecompletedinfinityinterpretationcancauselogicalmayhem.ThisiswhatImeantintheintroductionwhenIraisedthespecterofthegolemofinfinity.Sometimesitreallydoesmakeadifferencehowwethinkabouttheresultsofaprocessthatapproachesalimit.Pretendingthattheprocessactuallyterminatesandthatitsomehowreachesthenirvanaofinfinitycanoccasionallygetusintotrouble.

TheParableoftheInfinitePolygon

Asachasteningexample,supposeweputacertainnumberofdotsonacircle,spacethemevenly,andconnectthemtooneanotherwithstraightlines.Withthreedots,wegetanequilateraltriangle;withfour,asquare;withfive,apentagon;andsoon,runningthroughasequenceofrectilinearshapescalledregularpolygons.

Noticethatthemoredotsweuse,therounderthepolygonsbecomeandthe

closertheygettothecircle.Meanwhile,theirsidesgetshorterandmorenumerous.Aswemoveprogressivelyfurtherthroughthesequence,thepolygonsapproachtheoriginalcircleasalimit.Inthisway,infinityisbridgingtwoworldsagain.Thistimeit’stakingusfrom

therectilineartotheround,fromsharp-corneredpolygonstosilky-smoothcircles,whereasinthepizzaproof,infinitybroughtusfromroundtorectilinearasittransformedacircleintoarectangle.Ofcourse,atanyfinitestage,apolygonisstilljustapolygon.It’snotyeta

circleanditneverbecomesone.Itgetscloserandclosertobeingacircle,butitnevertrulygetsthere.Wearedealingherewithpotentialinfinity,notcompletedinfinity.Soeverythingisairtightfromthestandpointoflogicalrigor.Butwhatifwecouldgoallthewaytocompletedinfinity?Wouldthe

resultinginfinitepolygonwithinfinitesimallyshortsidesactuallybeacircle?It’stemptingtothinkso,becausethenthepolygonwouldbesmooth.Allitscornerswouldbesandedoff.Everythingwouldbecomeperfectandbeautiful.

TheAllureandPerilofInfinity

There’sagenerallessonhere:Limitsareoftensimplerthantheapproximationsleadinguptothem.Acircleissimplerandmoregracefulthananyofthethornypolygonsthatapproachit.Sotooforthepizzaproof,wherethelimitingrectanglewassimplerandmoreelegantthanthescallopedshapes,withtheirunsightlybulgesandcusps.Andlikewiseforthefraction⅓.Itwassimplerand

morehandsomethananyoftheungainlyfractionscreepinguponit,withtheirbiguglynumeratorsanddenominators,like3/10and33/100and333/1000.Inallthesecases,thelimitingshapeornumberwassimplerandmoresymmetricalthanitsfiniteapproximators.Thisistheallureofinfinity.Everythingbecomesbetterthere.Withthatlessoninmind,let’sreturntotheparableoftheinfinitepolygon.

Shouldwetaketheplungeandsaythatacircletrulyisapolygonwithinfinitelymanyinfinitesimalsides?No.Wemustn’tdothat,mustn’tyieldtothattemptation.Doingsowouldbetocommitthesinofcompletedinfinity.Itwouldcondemnustologicalhell.Toseewhy,supposeweentertainthethought,justforamoment,thatacircle

isindeedaninfinitepolygonwithinfinitesimalsides.Howlong,exactly,arethosesides?Zerolength?Ifso,theninfinitytimeszero—thecombinedlengthofallthosesides—mustequalthecircumferenceofthecircle.Butnowimagineacircleofdoublethecircumference.Infinitytimeszerowouldalsohavetoequalthatlargercircumferenceaswell.Soinfinitytimeszerowouldhavetobeboththecircumferenceanddoublethecircumference.Whatnonsense!Theresimplyisnoconsistentwaytodefineinfinitytimeszero,andsothereisnosensiblewaytoregardacircleasaninfinitepolygon.Nevertheless,thereissomethingsoenticingaboutthisintuition.Likethe

biblicaloriginalsin,theoriginalsinofcalculus—thetemptationtotreatacircleasaninfinitepolygonwithinfinitesimallyshortsides—isveryhardtoresist,andforthesamereason.Ittemptsuswiththeprospectofforbiddenknowledge,withinsightsunavailablebyordinarymeans.Forthousandsofyears,geometersstruggledtofigureoutthecircumferenceofacircle.Ifonlyacirclecouldbereplacedbyapolygonmadeofmanytinystraightsides,theproblemwouldbesomucheasier.Bylisteningtothehissofthisserpent—butholdingbackjustenough,by

usingpotentialinfinityinsteadofthemoretemptingcompletedinfinity—mathematicianslearnedhowtosolvethecircumferenceproblemandothermysteriesofcurves.Inthecomingchapters,we’llseehowtheydidit.Butfirst,weneedtogainanevendeeperappreciationofjusthowdangerouscompletedinfinitycanbe.It’sagatewaysintomanyothers,includingthesinourteacherswarnedusaboutfirst.

TheSinofDividingbyZero

Allacrosstheworld,studentsarebeingtaughtthatdivisionbyzeroisforbidden.Theyshouldfeelshockedthatsuchatabooexists.Numbersaresupposedtobeorderlyandwellbehaved.Mathclassisaplaceforlogicandreasoning.Andyetit’spossibletoasksimplethingsofnumbersthatjustdon’tworkormakesense.Dividingbyzeroisoneofthem.Therootoftheproblemisinfinity.Dividingbyzerosummonsinfinityin

muchthesamewaythataOuijaboardsupposedlysummonsspiritsfromanotherrealm.It’srisky.Don’tgothere.Forthosewhocan’tresistandwanttounderstandwhyinfinitylurksinthe

shadows,imaginedividing6byanumberthat’ssmallandgettingclosetozero,butthatisn’tquitezero,saysomethinglike0.1.There’snothingtabooaboutthat.Theanswerto6dividedby0.1is60,afairlysizablenumber.Divide6byanevensmallernumber,say0.01,andtheanswergrowsbigger;nowit’s600.Ifwedaretodivide6byanumbermuchclosertozero,say0.0000001,theanswergetsmuchbigger;insteadof60or600,nowit’s60,000,000.Thetrendisclear.Thesmallerthedivisor,thebiggertheanswer.Inthelimitasthedivisorapproacheszero,theanswerapproachesinfinity.That’stherealreasonwhywecan’tdividebyzero.Thefaintofheartsaytheanswerisundefined,butthetruthisit’sinfinite.Allofthiscanbevisualizedasfollows.Imaginedividinga6-centimeterline

intopiecesthatareeach0.1centimeterlong.Those60pieceslaidendtoendmakeuptheoriginal.

Likewise(butIwon’tattempttosketchit),thatsamelinecanbechoppedinto

600piecesthatareeach0.01centimeteror60,000,000piecesthatareeach0.0000001centimeter.Ifwekeepgoingandtakethischoppingfrenzytothelimit,weareledtothe

bizarreconclusionthata6-centimeterlineismadeupofinfinitelymanypieces

oflengthzero.Maybethatsoundsplausible.Afterall,thelineismadeupofinfinitelymanypoints,andeachpointhaszerolength.Butwhat’ssophilosophicallyunnervingisthatthesameargumentappliestoa

lineofanylength.Indeed,there’snothingspecialaboutthenumber6.Wecouldjustaswellhaveclaimedthatalineoflength3centimeters,or49.57,or2,000,000,000ismadeupofinfinitelymanypointsofzerolength.Evidently,multiplyingzerobyinfinitycangiveusanyandeveryconceivableresult—6or3or49.57or2,000,000,000.That’shorrifying,mathematicallyspeaking.

TheSinofCompletedInfinity

Thetransgressionthatdraggedusintothismesswaspretendingthatwecouldactuallyreachthelimit,thatwecouldtreatinfinitylikeanattainablenumber.BackinthefourthcenturyBCE,theGreekphilosopherAristotlewarnedthatsinningwithinfinityinthiswaycouldleadtoallsortsoflogicaltrouble.Herailedagainstwhathecalledcompletedinfinityandarguedthatonlypotentialinfinitymadesense.Inthecontextofchoppingalineintopieces,potentialinfinitywouldmean

thatthelinecouldbecutintomoreandmorepieces,asmanyasdesiredbutstillalwaysafinitenumberandallofnonzerolength.That’sperfectlypermissibleandleadstonologicaldifficulties.What’sverbotenistoimaginegoingallthewaytoacompletedinfinityof

piecesofzerolength.That,Aristotlefelt,wouldleadtononsense—asitdoeshere,inrevealingthatzerotimesinfinitycangiveanyanswer.Andsoheforbadetheuseofcompletedinfinityinmathematicsandphilosophy.Hisedictwasupheldbymathematiciansforthenexttwenty-twohundredyears.Somewhereinthedarkrecessesofprehistory,somebodyrealizedthat

numbersneverend.Andwiththatthought,infinitywasborn.It’sthenumericalcounterpartofsomethingdeepinourpsyches,inournightmaresofbottomlesspits,andinourhopesforeternallife.Infinityliesattheheartofsomanyofourdreamsandfearsandunanswerablequestions:Howbigistheuniverse?Howlongisforever?HowpowerfulisGod?Ineverybranchofhumanthought,fromreligionandphilosophytoscienceandmathematics,infinityhasbefuddledtheworld’sfinestmindsforthousandsofyears.Ithasbeenbanished,outlawed,andshunned.It’salwaysbeenadangerousidea.DuringtheInquisition,therenegademonkGiordanoBrunowasburnedaliveatthestakeforsuggestingthatGod,inHisinfinitepower,createdinnumerableworlds.

Zeno’sParadoxes

AbouttwomillenniabeforetheexecutionofGiordanoBruno,anotherbravephilosopherdaredtocontemplateinfinity.ZenoofElea(c.490–430BCE)posedaseriesofparadoxesaboutspace,time,andmotioninwhichinfinityplayedastarringandperplexingrole.Theseconundrumsanticipatedideasattheheartofcalculusandarestillbeingdebatedtoday.BertrandRussellcalledthemimmeasurablysubtleandprofound.Wearen’tsurewhatZenowastryingtoprovewithhisparadoxesbecause

noneofhiswritingshavesurvived,ifanyexistedtobeginwith.HisargumentshavecomedowntousthroughPlatoandAristotle,whosummarizedthemmainlytodemolishthem.Intheirtelling,Zenowastryingtoprovethatchangeisimpossible.Oursensestellusotherwise,butoursensesdeceiveus.Change,accordingtoZeno,isanillusion.ThreeofZeno’sparadoxesareparticularlyfamousandstrong.Thefirstof

them,theParadoxoftheDichotomy,issimilartotheRiddleoftheWallbutvastlymorefrustrating.Itholdsthatyoucan’tevermovebecausebeforeyoucantakeasinglestep,youneedtotakeahalfastep.Andbeforeyoucandothat,youneedtotakeaquarterofastep,andsoon.Sonotonlycan’tyougettothewall—youcan’tevenstartwalking.It’sabrilliantparadox.Whowouldhavethoughtthattakingasteprequired

completinginfinitelymanysubtasks?Worsestill,thereisnofirsttasktocomplete.Thefirsttaskcannotbetakinghalfastepbecausebeforethatyou’dhavetocompleteaquarterofastep,andbeforethat,aneighthofastep,andsoon.Ifyouthoughtyouhadalottodobeforebreakfast,imaginehavingtofinishaninfinitenumberoftasksjusttogettothekitchen.Anotherparadox,calledAchillesandtheTortoise,maintainsthataswift

runner(Achilles)cannevercatchuptoaslowrunner(atortoise)iftheslowrunnerhasbeengivenaheadstartinarace.

ForbythetimeAchillesreachesthespotwherethetortoisestarted,thetortoisewillhavemovedalittlebitfartherdownthetrack.AndbythetimeAchillesreachesthatnewlocation,thetortoisewillhavecreptslightlyfartherahead.Sinceweallbelievethatafastrunnercanovertakeaslowrunner,eitheroursensesaredeceivingusorthereissomethingwronginthewaythatwereasonaboutmotion,space,andtime.Inthesefirsttwoparadoxes,Zenoseemedtobearguingagainstspaceand

timebeingfundamentallycontinuous,meaningthattheycanbedividedendlessly.Hiscleverrhetoricalstrategy(somesayheinventedit)wasproofbycontradiction,knowntolawyersandlogiciansasreductioadabsurdum,reductiontoanabsurdity.Inbothparadoxes,Zenoassumedthecontinuityofspaceandtimeandthendeducedacontradictionfromthatassumption;therefore,theassumptionofcontinuitymustbefalse.Calculusisfoundedonthatveryassumptionandsohasalotatstakeinthisfight.ItrebutsZenobyshowingwherehisreasoningwentwrong.Forexample,here’showcalculustakescareofAchillesandthetortoise.

Supposethetortoisestarts10metersaheadofAchillesbutAchillesruns10timesfaster,sayataspeedof10meterspersecondcomparedtothetortoise’s1meterpersecond.ThenittakesAchilles1secondtomakeupthetortoise’s10-meterheadstart.Duringthattimethetortoisewillhavemoved1meterfartherahead.IttakesAchillesanother0.1secondtomakeupthatdifference,bywhichtimethetortoisewillhavemovedanother0.1meterahead.Continuingthisreasoning,weseethatAchilles’sconsecutivecatch-uptimesadduptotheinfiniteseries

1+0.1+0.01+0.001+···=1.111...seconds.

Rewrittenasanequivalentfraction,thisamountoftimeisequalto10/9seconds.That’showlongittakesAchillestocatchuptothetortoiseandovertakehim.AndalthoughZenowasrightthatAchilleshasinfinitelymanytaskstocomplete,there’snothingparadoxicalaboutthat.Asthemathshows,hecandothemallinafiniteamountoftime.Thislineofreasoningqualifiesasacalculusargument.Wejustsummedan

infiniteseriesandcalculatedalimit,aswedidearlierwhenwediscussedwhy0.333...=⅓.Wheneverweworkwithinfinitedecimals,wearedoingcalculus(eventhoughmostpeoplewouldpooh-poohitasmiddle-schoolarithmetic).Incidentally,calculusisn’ttheonlywaytosolvethisproblem.Wecoulduse

algebrainstead.Todoso,wefirstneedtofigureoutwhereeachrunnerisonthetrackatanarbitrarytimetsecondsaftertheracebegins.SinceAchillesrunsataspeedof10meterspersecondandsincedistanceequalsratetimestime,hisdistancedownthetrackis10t.Asforthetortoise,hehadaheadstartof10metersandherunswithaspeedof1meterpersecond,sohisdistancedownthetrackis10+t.ToascertainthetimewhenAchillesovertakesthetortoise,wehavetosetthosetwoexpressionsequaltooneanother,becausethat’sthealgebraicwayofaskingwhenAchillesandthetortoiseareatthesameplaceatthesametime.Theresultingequationis

10t=10+t.Tosolvethisequation,subtracttfrombothsides.Thatgives9t=10.Thendividebothsidesby9.Theresult,t=10⁄9seconds,isthesameaswefoundwithinfinitedecimals.Sofromtheperspectiveofcalculus,therereallyisnoparadoxaboutAchilles

andthetortoise.Ifspaceandtimearecontinuous,everythingworksoutnicely.

ZenoGoesDigital

Inathirdparadox,theParadoxoftheArrow,Zenoarguedagainstanalternativepossibility—thatspaceandtimearefundamentallydiscrete,meaningthattheyarecomposedoftinyindivisibleunits,somethinglikepixelsofspaceandtime.Theparadoxgoeslikethis.Ifspaceandtimearediscrete,anarrowinflightcannevermove,becauseateachinstant(apixeloftime)thearrowisatsome

definiteplace(aspecificsetofpixelsinspace).Hence,atanygiveninstant,thearrowisnotmoving.Itisalsonotmovingbetweeninstantsbecause,byassumption,thereisnotimebetweeninstants.Therefore,atnotimeisthearrowevermoving.Tomymind,thisisthemostsubtleandinterestingoftheparadoxes.

Philosophersarestilldebatingitsstatus,butitseemstomethatZenogotittwo-thirdsright.Inaworldwherespaceandtimearediscrete,anarrowinflightwouldbehaveasZenosaid.Itwouldstrangelymaterializeatoneplaceafteranotherastimeclicksforwardindiscretesteps.Andhewasalsorightthatoursensestellusthattherealworldisnotlikethat,atleastnotasweordinarilyperceiveit.ButZenowaswrongthatmotionwouldbeimpossibleinsuchaworld.Weall

knowthisfromourexperienceofwatchingmoviesandvideosonourdigitaldevices.OurcellphonesandDVRsandcomputerscreenschopeverythingintodiscretepixels,andyet,contrarytoZeno’sassertion,motioncantakeplaceperfectlywellinthesediscretizedlandscapes.Aslongaseverythingisdicedfineenough,wecan’ttellthedifferencebetweenasmoothmotionanditsdigitalrepresentation.Ifweweretowatchahigh-resolutionvideoofanarrowinflight,we’dactuallybeseeingapixelatedarrowmaterializinginonediscreteframeafteranother.Butbecauseofourperceptuallimitations,itwouldlooklikeasmoothtrajectory.Sometimesoursensesreallydodeceiveus.Ofcourse,ifthechoppingistooblocky,wecantellthedifferencebetween

thecontinuousandthediscrete,andweoftenfinditbothersome.Considerhowanold-fashionedanalogclockdiffersfromamodern-daydigital/mechanicalmonstrosity.Ontheanalogclock,thesecondhandsweepsaroundinabeautifullyuniformmotion.Itdepictstimeasflowing.Whereasonthedigitalclock,thesecondhandjerksforwardindiscretesteps,thwack,thwack,thwack.Itdepictstimeasjumping.Infinitycanbuildabridgebetweenthesetwoverydifferentconceptionsof

time.Imagineadigitalclockthatadvancesthroughtrillionsoflittleclickspersecondinsteadofoneloudthwack.Wewouldnolongerbeabletotellthedifferencebetweenthatkindofdigitalclockandatrueanalogclock.Likewisewithmoviesandvideos;aslongastheframesflashbyfastenough,sayatthirtyframesasecond,theygivetheimpressionofseamlessflow.Andiftherewereinfinitelymanyframespersecond,theflowtrulywouldbeseamless.Considerhowmusicisrecordedandplayedback.Myyoungerdaughter

recentlyreceivedanold-fashionedVictrolarecordplayerforherfifteenthbirthday.She’snowabletolistentoEllaFitzgeraldonvinyl.Thisisaquintessentialanalogexperience.AllofElla’snotesandscatsglidejustas

smoothlyastheydidwhenshesangthem;hervolumegoescontinuouslyfromsofttoloudandeverywhereinbetween,andherpitchclimbsjustasgracefullyfromlowtohigh.Whereaswhenyoulistentoherondigital,everyaspectofhermusicismincedintotiny,discretestepsandconvertedintostringsof0sand1s.Althoughconceptuallythedifferencesaregigantic,ourearscan’thearthem.Soineverydaylife,thegulfbetweenthediscreteandthecontinuouscanoften

bebridged,atleasttoagoodapproximation.Formanypracticalpurposes,thediscretecanstandinforthecontinuous,aslongasweslicethingsthinlyenough.Intheidealworldofcalculus,wecangoonebetter.Anythingthat’scontinuouscanbeslicedexactly(notjustapproximately)intoinfinitelymanyinfinitesimalpieces.That’stheInfinityPrinciple.Withlimitsandinfinity,thediscreteandthecontinuousbecomeone.

ZenoMeetstheQuantum

TheInfinityPrincipleasksustopretendthateverythingcanbeslicedanddicedendlessly.We’vealreadyseenhowusefulsuchconceptscanbe.Imaginingpizzasthatcanbecutintoarbitrarilythinpiecesenabledustofindtheareaofacircleexactly.Thequestionnaturallyarises:Dosuchinfinitesimallysmallthingsexistintherealworld?Quantummechanicshassomethingtosayaboutthat.It’sthebranchof

modernphysicsthatdescribeshownaturebehavesatitssmallestscales.It’sthemostaccuratephysicaltheoryeverdevised,anditislegendaryforitsweirdness.Itsterminology,withitszooofleptons,quarks,andneutrinos,soundslikesomethingoutofLewisCarroll.Thebehavioritdescribesisoftenweirdaswell.Attheatomicscale,thingscanhappenthatwouldneveroccurinthemacroscopicworld.Forinstance,considertheRiddleoftheWallfromaquantumperspective.If

thewalkerwereanelectron,there’sachanceitmightwalkrightthroughthewall.Thiseffectisknownasquantumtunneling.Itactuallyoccurs.It’shardtomakesenseofthisinclassicalterms,butthequantumexplanationisthatelectronsaredescribedbyprobabilitywaves.Thosewavesobeyanequationformulatedin1925bytheAustrianphysicistErwinSchrödinger.ThesolutiontoSchrödinger’sequationshowsthatasmallportionoftheelectronprobabilitywaveexistsonthefarsideofanimpenetrablebarrier.Thismeansthereissomesmallbutnonzeroprobabilitythattheelectronwillbedetectedonthefarsideofthebarrier,asifithadtunneledthroughthewall.Withthehelpofcalculus,we

cancalculatetherateatwhichsuchtunnelingeventsoccur,andexperimentshaveconfirmedthepredictions.Tunnelingisreal.Alphaparticlestunneloutofuraniumnucleiatthepredictedratetoproducetheeffectknownasradioactivity.Tunnelingalsoplaysanimportantroleinthenuclear-fusionprocessesthatmakethesunshine,solifeonEarthdependspartiallyontunneling.Andithasmanytechnologicaluses;scanningtunnelingmicroscopy,whichallowsscientiststoimageandmanipulateindividualatoms,isbasedontheconcept.Wehavenointuitionforsucheventsattheatomicscale,beingthegargantuan

creaturescomposedoftrillionsupontrillionsofatomsthatweare.Fortunately,calculuscantaketheplaceofintuition.Byapplyingcalculusandquantummechanics,physicistshaveopenedatheoreticalwindowonthemicroworld.Thefruitsoftheirinsightsincludelasersandtransistors,thechipsinourcomputers,andtheLEDsinourflat-screenTVs.Althoughquantummechanicsisconceptuallyradicalinmanyrespects,in

Schrödinger’sformulation,itretainsthetraditionalassumptionthatspaceandtimearecontinuous.Maxwellmadethesameassumptioninhistheoryofelectricityandmagnetism;sodidNewtoninhistheoryofgravityandEinsteininhistheoryofrelativity.Allofcalculus,andhencealloftheoreticalphysics,hingesonthisassumptionofcontinuousspaceandtime.Thatassumptionofcontinuityhasbeenresoundinglysuccessfulsofar.Butthereisreasontobelievethatatmuch,muchsmallerscalesofthe

universe,farbelowtheatomicscale,spaceandtimemayultimatelylosetheircontinuouscharacter.Wedon’tknowforsurewhatit’slikedownthere,butwecanguess.SpaceandtimemightbecomeasneatlypixelatedasZenoimaginedinhisParadoxoftheArrow,butmorelikelythey’ddegenerateintoadisorderlymessbecauseofquantumuncertainty.Atsuchsmallscales,spaceandtimemightseetheandroilatrandom.Theymightfluctuatelikebubblingfoam.Althoughthereisnoconsensusabouthowtovisualizespaceandtimeatthese

ultimatescales,thereisuniversalagreementabouthowsmallthosescalesarelikelytobe.Theyareforceduponusbythreefundamentalconstantsofnature.Oneofthemisthegravitationalconstant,G.Itmeasuresthestrengthofgravityintheuniverse.ItappearedfirstinNewton’stheoryofgravityandagaininEinstein’sgeneraltheoryofrelativity.Itisboundtooccurinanyfuturetheorythatsupersedesthem.Thesecondconstant,ħ(pronounced“hbar”),reflectsthestrengthofquantumeffects.Itappears,forexample,inHeisenberg’suncertaintyprincipleandinSchrödinger’swaveequationofquantummechanics.Thethirdconstantisthespeedoflight,c.Itisthespeedlimitfortheuniverse.Nosignalofanykindcantravelfasterthanc.Thisspeedmustnecessarilyenteranytheoryof

spaceandtimebecauseittiesthetwoofthemtogetherviatheprinciplethatdistanceequalsratetimestime,wherecplaystheroleoftherateorspeed.In1899,thefatherofquantumtheory,aGermanphysicistnamedMaxPlanck,

realizedthattherewasoneandonlyonewaytocombinethesefundamentalconstantstoproduceascaleoflength.Thatuniquelength,heconcluded,wasanaturalyardstickfortheuniverse.Inhishonor,itisnowcalledthePlancklength.Itisgivenbythealgebraiccombination

Plancklength=√ħG/c3.

WhenwepluginthemeasuredvaluesofG,ħ,andc,thePlancklengthcomesouttobeabout10–35meters,astupendouslysmalldistancethat’saboutahundredmilliontrilliontimessmallerthanthediameterofaproton.ThecorrespondingPlancktimeisthetimeitwouldtakelighttotraversethisdistance,whichisabout10–43seconds.Spaceandtimewouldnolongermakesensebelowthesescales.They’retheendoftheline.Thesenumbersputaboundonhowfinewecouldeverslicespaceortime.To

getafeelforthelevelofprecisionwe’retalkingabouthere,considerhowmanydigitswewouldneedtomakeoneofthemostextremecomparisonsimaginable.Takethelargestpossibledistance,theestimateddiameteroftheknownuniverse,anddivideitbythesmallestpossibledistance,thePlancklength.Thatunfathomablyextremeratioofdistancesisanumberwithonlysixtydigitsinit.Iwanttostressthat—onlysixtydigits.That’sthemostwewouldeverneedtoexpressonedistanceintermsofanother.Usingmoredigitsthanthat—sayahundreddigits,letaloneinfinitelymany—wouldbecolossaloverkill,waymorethanwewouldeverneedtodescribeanyrealdistancesoutthereinthematerialworld.Andyetincalculus,weuseinfinitelymanydigitsallthetime.Asearlyas

middleschool,studentsareaskedtothinkaboutnumberslike0.333...whosedecimalexpansiongoesonforever.Wecalltheserealnumbers,butthereisnothingrealaboutthem.Therequirementtospecifyarealnumberbyaninfinitenumberofdigitsafterthedecimalpointisexactlywhatitmeanstobenotreal,atleastasfarasweunderstandrealitythroughphysicstoday.Ifrealnumbersarenotreal,whydomathematicianslovethemsomuch?And

whyareschoolchildrenforcedtolearnaboutthem?Becausecalculusneedsthem.Fromthebeginning,calculushasstubbornlyinsistedthateverything—spaceandtime,matterandenergy,allobjectsthateverhavebeenorwillbe—shouldberegardedascontinuous.Accordingly,everythingcanandshouldbe

quantifiedbyrealnumbers.Inthisidealized,imaginaryworld,wepretendthateverythingcanbesplitfinerandfinerwithoutend.Thewholetheoryofcalculusisbuiltonthatassumption.Withoutit,wecouldn’tcomputelimits,andwithoutlimits,calculuswouldcometoaclankinghalt.Ifallweeverusedweredecimalswithonlysixtydigitsofprecision,thenumberlinewouldbepockmarkedandcratered.Therewouldbeholeswherepi,thesquarerootoftwo,andanyothernumbersthatneedinfinitelymanydigitsafterthedecimalpointshouldexist.Evenasimplefractionsuchas⅓wouldbemissing,becauseittoorequiresaninfinitenumberofdigits(0.333...)topinpointitslocationonthenumberline.Ifwewanttothinkofthetotalityofallnumbersasformingacontinuousline,thosenumbershavetoberealnumbers.Theymaybeanapproximationofreality,buttheyworkamazinglywell.Realityistoohardtomodelanyotherway.Withinfinitedecimals,aswiththerestofcalculus,infinitymakeseverythingsimpler.

2

TheManWhoHarnessedInfinity

ABOUTTWOHUNDREDyearsafterZenoponderedthenatureofspace,time,motion,andinfinity,anotherthinkerfoundinfinityirresistible.HisnamewasArchimedes.We’vemethimalreadyinconnectionwiththeareaofacircle,butheislegendaryformanyotherreasons.Foronething,therearealotoffunnystoriesabouthim.Severalportrayhim

astheoriginalmathgeek.Forexample,thehistorianPlutarchtellsusthatArchimedescouldbecomesoengrossedingeometrythatit“madehimforgethisfoodandneglecthisperson.”(Thatcertainlyringstrue.Formanyofusmathematicians,mealsandpersonalhygienearen’ttoppriorities.)PlutarchgoesontosaythatwhenArchimedeswaslostinhismathematics,hewouldhavetobe“carriedbyabsoluteviolencetobathe.”It’sinterestingthathewassuchareluctantbather,giventhatabathisthesettingfortheonestoryabouthimthateverybodyknows.AccordingtotheRomanarchitectVitruvius,Archimedesbecamesoexcitedbyasuddeninsighthehadinthebaththatheleapedoutofthetubandrandownthestreetnakedshouting,“Eureka!”(“Ihavefoundit!”)Otherstoriescasthimasamilitarymagician,awarrior-scientist/one-man

deathsquad.Accordingtotheselegends,whenhishomecityofSyracusewasundersiegebytheRomansin212BCE,Archimedes—bythenanoldman,aroundseventy—helpeddefendthecitybyusinghisknowledgeofpulleysandleverstomakefantasticalweapons,“warengines”suchasgrapplinghooksand

giantcranesthatcouldlifttheRomanshipsoutoftheseaandshakethesailorsfromthemlikesandbeingshakenoutofashoe.AsPlutarchdescribedtheterrifyingscene,“Ashipwasfrequentlylifteduptoagreatheightintheair(adreadfulthingtobehold),andwasrolledtoandfro,andkeptswinging,untilthemarinerswereallthrownout,whenatlengthitwasdashedagainsttherocks,orletfall.”Inamoreseriousvein,allstudentsofscienceandengineeringremember

Archimedesforhisprincipleofbuoyancy(abodyimmersedinafluidisbuoyedupbyaforceequaltotheweightofthefluiddisplaced)andhislawofthelever(heavyobjectsplacedonoppositesidesofaleverwillbalanceifandonlyiftheirweightsareininverseproportiontotheirdistancesfromthefulcrum).Bothoftheseideashavecountlesspracticalapplications.Archimedes’sprincipleofbuoyancyexplainswhysomeobjectsfloatandothersdonot.Italsounderliesallofnavalarchitecture,thetheoryofshipstability,andthedesignofoil-drillingplatformsatsea.Andyourelyonhislawofthelever,evenifunknowingly,everytimeyouuseanailclipperoracrowbar.Archimedesmighthavebeenaformidablemakerofwarmachines,andhe

undoubtedlywasabrilliantscientistandengineer,butwhatreallyputshiminthepantheoniswhathedidformathematics.Hepavedthewayforintegralcalculus.Itsdeepestideasareplainlyvisibleinhiswork,butthentheyaren’tseenagainforalmosttwomillennia.Tosayhewasaheadofhistimewouldbeputtingitmildly.Hasanyoneeverbeenmoreaheadofhistime?Twostrategiesappearagainandagaininhiswork.Thefirstwashisardent

useoftheInfinityPrinciple.Toprobethemysteriesofcircles,spheres,andothercurvedshapes,healwaysapproximatedthemwithrectilinearshapesmadeoflotsofstraight,flatpieces,facetedlikejewels.Byimaginingmoreandmorepiecesandmakingthemsmallerandsmaller,hepushedhisapproximationseverclosertothetruth,approachingexactitudeinthelimitofinfinitelymanypieces.Thisstrategydemandedthathebeawizardwithsumsandpuzzles,sinceheendeduphavingtoaddmanynumbersorpiecesbacktogethertoarriveathisconclusions.Hisotherdistinguishingstratagemwasblendingmathematicswithphysics,

theidealwiththereal.Specifically,hemingledgeometry,thestudyofshapes,withmechanics,thestudyofmotionandforce.Sometimesheusedgeometrytoilluminatemechanics;sometimestheflowwentintheotherdirection,withmechanicalargumentsprovidinginsightintopureform.ItwasbyusingbothstrategieswithconsummateskillthatArchimedeswasabletopenetratesodeeplyintothemysteryofcurves.

SqueezingPi

WhenIwalktomyofficeorgooutwithmydogforaneveningstroll,thepedometeronmyiPhonekeepstrackofhowfarIwalk.Thecalculationissimple:TheappestimatesthelengthofmystridebasedonmyheightandcountshowmanystepsI’vetaken,thenitmultipliesthosetwonumberstogether.Thedistancetraveledequalsstridelengthtimesthenumberofstepstaken.Archimedesusedasimilarideatocalculatethecircumferenceofacircleand

toestimatepi.Thinkofthecircleasatrack.Ittakesalotofstepstowalkallthewayaround.Thepathwouldlooksomethinglikethis.

Eachstepisrepresentedbyatinystraightline.Bymultiplyingthenumberofstepsbythelengthofeachone,wecanestimatethelengthofthetrack.It’sonlyanestimate,ofcourse,becausethecircleisnotactuallymadeupofstraightlines.It’smadeupofcurvedarcs.Whenwereplaceeacharcbyastraightline,we’retakingaslightshortcut.Andsotheapproximationissuretounderestimatethetruelengthofthecirculartrack.But,atleastintheory,bytakingenoughstepsandmakingthemsmallenough,wecanapproximatethelengthofthetrackasaccuratelyaswewish.Archimedesdidaseriesofcalculationslikethis,startingwithapathmadeup

ofsixstraightsteps.

Hebeganwithahexagonbecauseitwasaconvenientbasecampfromwhich

toembarkonthemorearduouscalculationsahead.Theadvantageofthehexagonwasthathecouldeasilycalculateitsperimeter,thetotallengtharoundthehexagon.It’ssixtimestheradiusofthecircle.Whysix?Becausethehexagoncontainssixequilateraltriangles,eachsideofwhichequalsthecircle’sradius.Sixofthetriangle’ssidesmakeuptheperimeterofthehexagon.

Sotheperimeterequalssixtimestheradius;insymbols,p=6r.Then,sincethecircle’scircumferenceCislongerthanthehexagon’sperimeterp,wemusthaveC>6r.ThisargumentgaveArchimedesalowerboundonwhatwewouldcallpi,

writtenastheGreekletterπanddefinedastheratioofthecircumferencetothediameterofthecircle.Sincethediameterdequals2r,theinequalityC>6rimplies

π=C/d=C/2r>6r/2r=3.

Thusthehexagonargumentdemonstratesπ>3.Ofcourse,sixisaridiculouslysmallnumberofsteps,andtheresulting

hexagonisobviouslyaverycrudecaricatureofacircle,butArchimedeswasjustgettingstarted.Oncehefiguredoutwhatthehexagonwastellinghim,heshortenedthestepsandtooktwiceasmanyofthem.Hedidthatbydetouringtothemidpointofeacharc,takingtwobabystepsinsteadofstridingacrossthearcinonebigstep.

Thenhekeptdoingthat,overandoveragain.Amanobsessed,hewentfrom

sixstepstotwelve,thentwenty-four,forty-eight,and,ultimately,ninety-sixsteps,workingouttheirever-shrinkinglengthstomigraine-inducingprecision.

Unfortunately,itgotprogressivelyhardertocalculatethesteplengthsasthey

shrank,becausehehadtokeepinvokingthePythagoreantheoremtofindthem.Thatrequiredhimtocalculatesquareroots,anastychoretodobyhand.Furthermore,toensurethathewasalwaysunderestimatingthecircumference,hehadtomakesurethathisapproximatingfractionsboundedthebothersome

squarerootsfrombelowwhenheneededthemtobeunderestimatesandfromabovewhenheneededthemtobeoverestimates.WhatI’mtryingtosayisthathiscalculationofπwasheroic,bothlogically

andarithmetically.Byusinga96-goninsidethecircleanda96-gonoutsidethecircle,heultimatelyprovedthatπisgreaterthan3+10/71andlessthan3+10/70.Forgetaboutmathforaminute.Justsavorthisresultatavisuallevel:

3+10/71<π<3+10/70.

Theunknown,andforeverunknowable,valueofπistrappedinanumericalvise,squeezedbetweentwonumbersthatlookalmostidenticalexceptthattheformerhasadenominatorof71andthelatterof70.Thatlatterresult,3+10/70,reducesto22/7,thefamousapproximationtoπthatallstudentsstilllearntodayandthatsomeunfortunatelymistakeforπitself.ThesqueezetechniquethatArchimedesused(buildingonearlierworkbythe

GreekmathematicianEudoxus)isnowknownasthemethodofexhaustionbecauseofthewayittrapstheunknownnumberpibetweentwoknownnumbers.Theboundstightenwitheachdoubling,thusexhaustingthewiggleroomforpi.Circlesarethesimplestcurvesingeometry.Yet,surprisingly,measuringthem

—quantifyingtheirpropertieswithnumbers—transcendsgeometry.Forexample,youwillfindnomentionofπinEuclid’sElements,writtenagenerationortwobeforeArchimedes.Youwillfindaproofbyexhaustionthattheratioofacircle’sareatothesquareofitsradiusisthesameforallcirclesbutnohintthattheuniversalratioiscloseto3.14.Euclid’somissionwasasignalthatsomethingdeeperwasneeded.Tocometogripswithπ’snumericalvaluerequiredanewkindofmathematics,onethatcouldcopewithcurvedshapes.Howtomeasurethelengthofacurvedlineortheareaofacurvedsurfaceorthevolumeofacurvedsolid—thesewerethecutting-edgequestionsthatconsumedArchimedesandledhimtotakethefirststepstowardwhatwenowcallintegralcalculus.Piwasitsfirsttriumph.

TheTaoofPi

Itmayseemstrangetomodernmindsthatpidoesn’tappearinArchimedes’sformulafortheareaofacircle,A=rC/2,andthatheneverwrotedownanequationlikeC=πdtorelatethecircumferenceofacircletoitsdiameter.Heavoideddoingallthatbecausepiwasnotanumbertohim.Itwassimplyaratiooftwolengths,aproportionbetweenacircle’scircumferenceanditsdiameter.Itwasamagnitude,notanumber.Wenolongermakethisdistinctionbetweenmagnitudeandnumber,butitwas

importantinancientGreekmathematics.Itseemstohavearisenfromthetensionbetweenthediscrete(asrepresentedbywholenumbers)andthecontinuous(asrepresentedbyshapes).Thehistoricaldetailsaremurky,butitappearsthatsometimebetweenPythagorasandEudoxus,betweenthesixthandthefourthcenturiesBCE,somebodyprovedthatthediagonalofasquarewasincommensurablewithitsside,meaningthattheratioofthosetwolengthscouldnotbeexpressedastheratiooftwowholenumbers.Inmodernlanguage,someonediscoveredtheexistenceofirrationalnumbers.ThesuspicionisthatthisdiscoveryshockedanddisappointedtheGreeks,sinceitbeliedthePythagoreancredo.Ifwholenumbersandtheirratioscouldn’tevenmeasuresomethingasbasicasthediagonalofaperfectsquare,thenallwasnotnumber.ThisdeflatingletdownmayexplainwhylaterGreekmathematiciansalwayselevatedgeometryoverarithmetic.Numberscouldn’tbetrustedanymore.Theywereinadequateasafoundationformathematics.Todescribecontinuousquantitiesandreasonaboutthem,theancientGreek

mathematiciansrealizedtheyneededtoinventsomethingmorepowerfulthanwholenumbers.Sotheydevelopedasystembasedonshapesandtheirproportions.Itreliedonmeasuresofgeometricalobjects:lengthsoflines,areasofsquares,volumesofcubes.Allofthesetheycalledmagnitudes.Theythoughtofthemasdistinctfromnumbersandsuperiortothem.This,Ibelieve,iswhyArchimedesheldpiatarm’slength.Hedidn’tknow

whattomakeofit.Itwasastrange,transcendentcreature,moreexoticthananynumber.Todayweacceptpiasanumber—arealnumber,aninfinitedecimal—anda

fascinatingoneatthat.Mychildrencertainlywereintriguedbyit.Theyusedtostareatapieplatehanginginourkitchenthathadthedigitsofpirunningaroundtherimandspiralingintowardthecenter,shrinkinginsizeastheyswirledintotheabyss.Forthem,thefascinationhadtodowiththerandom-lookingsequenceofdigits,neverrepeating,nevershowinganypatternatall,goingonforever,infinityonaplatter.Thefirstfewdigitsinpi’sinfinitedecimalexpansionare

3.1415926535897932384626433832795028841971693993751058209749...

Wewillneverknowallthedigitsofpi.Nevertheless,thosedigitsareout

there,waitingtobediscovered.Asofthiswriting,twenty-twotrilliondigitshavebeencomputedbytheworld’sfastestcomputers.Yettwenty-twotrillionisnothingcomparedtotheinfinitudeofdigitsthatdefinetheactualpi.Thinkofhowphilosophicallydisturbingthisis.Isaidthatthedigitsofpiareoutthere,butwherearetheyexactly?Theydon’texistinthematerialworld.TheyexistinsomePlatonicrealm,alongwithabstractconceptsliketruthandjustice.There’ssomethingsoparadoxicalaboutpi.Ontheonehand,itrepresents

order,asembodiedbytheshapeofacircle,longheldtobeasymbolofperfectionandeternity.Ontheotherhand,piisunruly,disheveledinappearance,itsdigitsobeyingnoobviousrule,oratleastnonethatwecanperceive.Piiselusiveandmysterious,foreverbeyondreach.Itsmixoforderanddisorderiswhatmakesitsobewitching.Piisfundamentallyachildofcalculus.Itisdefinedastheunattainablelimit

ofanever-endingprocess.Butunlikeasequenceofpolygonssteadfastlyapproachingacircleorahaplesswalkersteppinghalfwaytoawall,thereisnoendinsightforpi,nolimitwecaneverknow.Andyetpiexists.Thereitis,definedsocrisplyastheratiooftwolengthswecanseerightbeforeus,thecircumferenceofacircleanditsdiameter.Thatratiodefinespi,pinpointsitasclearlyascanbe,andyetthenumberitselfslipsthroughourfingers.Withitsyinandyangbinaries,piislikeallofcalculusinminiature.Piisa

portalbetweentheroundandthestraight,asinglenumberyetinfinitelycomplex,abalanceoforderandchaos.Calculus,foritspart,usestheinfinitetostudythefinite,theunlimitedtostudythelimited,andthestraighttostudythecurved.TheInfinityPrincipleisthekeytounlockingthemysteryofcurves,anditaroseherefirst,inthemysteryofpi.

CubismMeetsCalculus

Archimedeswentdeeperintothemysteryofcurves,againguidedbytheInfinityPrinciple,inhistreatiseTheQuadratureoftheParabola.Aparaboladescribesthefamiliararcofathree-pointshotinbasketballorwatercomingoutofadrinkingfountain.Actually,thosearcsintherealworldareonlyapproximatelyparabolic.Atrueparabola,toArchimedes,wouldhavemeantacurveobtained

byslicingthroughaconewithaplane.Imagineameatcleaverslicingthroughaduncecaporaconicalpapercup;thecleavercanmakedifferentkindsofcurvesdependingonhowsteeplyitcutsthroughthecone.Asliceparalleltothebaseofaconemakesacircle.

Aslightlysteepercutproducesanellipse.

Acutthathasthesameslopeastheconeitselfproducesaparabola.

Viewedintheplaneoftheslice,theparabolaappearsasagraceful,

symmetricalcurvewithalineofsymmetrydownitsmiddle.Thislineiscalledtheparabola’saxis.

Inhistreatise,Archimedessethimselfthechallengeofworkingoutthe

quadratureofaparabolicsegment.Inmoremodernlanguage,asegmentofaparabolameansthecurvedregionlyingbetweentheparabolaandalinethatcutsacrossitobliquely.

Findingitsquadraturemeansexpressingitsunknownareaintermsoftheknownareaofasimplershapelikeasquare,rectangle,triangle,orotherrectilinearfigure.ThestrategyusedbyArchimedeswasastonishing.Hereimaginedthe

parabolicsegmentasinfinitelymanytriangularshardsgluedtogetherlikepiecesofbrokenpottery.

Theshardscameinanendlesshierarchyofsizes:onebigtriangle,twosmaller

ones,foursmallerstill,andsoon.Hisplanwastofindalltheirareasandthenaddthembacktogethertocalculatethecurvedareahewaswonderingabout.Ittookakaleidoscopicleapofartisticimaginationtoseeasmooth,gentlycurvingparabolicsegmentasamosaicofjaggedshapes.Ifhehadbeenapainter,Archimedeswouldhavebeenthefirstcubist.Tocarryouthisstrategy,Archimedesfirsthadtofindtheareasofallthe

shards.Buthow,precisely,werethoseshardstobedefined?Afterall,therearecountlesswaystopiecetrianglestogethertoformaparabolicsegment,justastherearecountlesswaystosmashaplateintojaggedbits.Thebiggesttrianglecouldlooklikethis,orthis,orthis:

Hecameupwithabrilliantidea—brilliantbecauseitestablishedarule,a

consistentpatternthatheldfromonelevelofthehierarchytothenext.Heimaginedslidingtheobliquelineatthebaseofthesegmentupwardwhile

keepingitparalleltoitselfuntilitjustbarelytouchedtheparabolaatasinglepointnearthetop.

Thatspecialpointofgrazingcontactiscalledapointoftangency(fromthe

Latinroottangere,meaning“touching”).Itdefinedthethirdcornerofthebigtriangle,theothertwobeingthepointswheretheobliquelinecuttheparabola.Archimedesusedthesameruletodefinethetrianglesateverystageinthe

hierarchy.Atthesecondstage,forexample,thetriangleslookedlikethis.

Noticethatthesidesofthebigtrianglenowplaytheroleoftheobliquelineusedearlier.Next,Archimedesinvokedknowngeometricalfactsaboutparabolasand

trianglestorelateonelevelofthehierarchytothenext.Heprovedthateachnewlycreatedtrianglehadone-eighthasmuchareaasitsparenttriangle.Thus,ifwesaythatthefirst,biggesttriangleoccupies1unitofarea—thattrianglewill

serveasourareastandard—thenitstwodaughtertrianglestogetheroccupy⅛+⅛=¼asmucharea.

Ateachsubsequentstagethesameruleapplies:thedaughtertrianglesalways

contributeatotalofaquarterasmuchareaastheirparentdoes.Sothetotalareaoftheparabolicsegment,reassembledfromthewholeinfinitehierarchyofshards,mustbe

Area=1+1⁄4+1⁄16+1⁄64+···,

aninfiniteseriesinwhicheachtermisone-quarterofthetermprecedingit.There’sashortcuttosumthiskindofinfiniteseries,whichisknowninthe

tradeasageometricseries.ThetrickistocancelallbutoneofitsinfinitelymanytermsbymultiplyingbothsidesoftheequationforAreaby4andsubtractingtheoriginalsumfromit.Watch:Multiplyingeachtermby4intheinfiniteseriesabovegives

4×Area=4(1+1⁄4+1⁄16+1⁄64+···)=4+4⁄4+4⁄16+4⁄64+···=4+1+1⁄4+1⁄16+···=4+Area

Themagichappensbetweenthenext-to-lastlineandthelastlineabove.Theright-handsideofthelastlineequals4+Area,becausetheoriginalsum,Area=1+1⁄4+1⁄16+···,has,likeaphoenix,beenreborninthetermsfollowingthe4inthenext-to-lastline.So

4×Area=4+Area.

SubtractoneAreafrombothsidestoget3×Area=4.Thus

Area=4⁄3.

Inotherwords,theparabolicsegmenthas4/3theareaofthebigtriangle.

ACheesyArgument

Archimedeswouldnothaveapprovedofthelegerdemainabove.Hearrivedatthesameresultbyadifferentroute.Heresortedtoasubtlestyleofargumentationoftendescribedasdoublereductioadabsurdum,adoubleproofbycontradiction.Heprovedthattheareaoftheparabolicsegmentcouldnotbelessthan4/3orgreaterthan4/3,soitmustequal4/3.AsSherlockHolmeslaterputit,“Whenyouhaveeliminatedtheimpossible,whateverremains,howeverimprobable,mustbethetruth.”What’sconceptuallycrucialhereisthatArchimedeseliminatedtheimpossible

withargumentsbasedonafinitenumberofshards.Heshowedthattheircombinedareacouldbemadeascloseto4/3asdesired,closerthananyprescribedtolerance,simplybytakingenoughofthem.Heneverhadtosummoninfinity.Soeverythingabouthisproofwasironclad.Itstillmeetsthehigheststandardsofrigortoday.Thegistofhisargumentbecomeseasytounderstandifweputitineveryday

terms.Supposethreepeoplewanttosharefouridenticalslicesofcheese.

Thecommonsensesolutionwouldbetogiveeachpersonaslice,thencuttheremainingsliceintothirdsandhandthemout.That’sfair.Intotal,everyonewouldget1+⅓=4/3ofaslice.Butsupposethethreepeoplehappentobemathematicianswhoaremilling

aroundthefoodtablebeforetheseminar,eyeingthelastfourslicesofcheese.Thecleverestofthethree,coincidentallynamedArchimedes,mightsuggestthefollowingsolution:“I’lltakeasliceandyouguystakeyours,whichleavesonemoreforustoshare.Euclid,cutthatleftoversliceintoquarters,notthirds,andeveryone,takeaquarterofthatleftoverslice.We’regoingtokeepdoingthis,alwayscuttingwhat’sleftoverintofourequalportions,untiltheremainingcrumbisofnointeresttoanyone.Okay?Eudoxus,stopwhining.”

Howmanyslicesofcheese,total,wouldeachofthemgettoeatifthiswereto

goonindefinitely?Onewaytolookatitistokeeparunningtallyofhowmanysliceseachpersongets.Afterroundone,eachgetsoneslice.Afterroundtwo,whenthequarterslicesarepassedout,eachpersonhasaccumulated1+¼slices.Afterroundthree,whenthequartersarethemselvesquarteredintosixteenths,therunningtotalforeachis1+¼+1/16slices.Andsoon.Looselyspeaking,eachofthethreepeoplewouldeventuallygettoeat1+¼+1/16+...slicesintotalifthecuttingwentonforever.Andsincethisamountmustrepresentathirdoftheoriginalfourslices,itmustbethat1+¼+1/16+...equalsone-thirdof4,whichis4/3.InTheQuadratureoftheParabola,Archimedesgaveanargumentveryclose

tothis,includingadiagramwithsquaresofdifferentsizes,butheneverinvokedinfinityorusedthecounterpartofthethreedots[...]abovetosignifythatthesumwentonendlessly.Rather,hephrasedhisargumentintermsoffinitesumssothatitwasunimpeachablyrigorous.Hiskeyobservationwasthatthetinysquareintheupperrightcorner—thecurrentleftoverremainingtobeshared—couldbemadesmallerthananygivenamountbyconsideringasufficientlylargebutfinitenumberofrounds.Andbysimilarreasoning,thefinitesum1+¼+1/16+...+¼n(thetotalamountofcheesethateachpersongets)couldbemadeascloseto4/3asdesiredbymakingnlargeenough.Sotheonlypossibleanswerwas4/3.

TheMethodIt’satthispointthatIbegintofeelrealaffectionforArchimedes,becausehedoessomethinginoneofhisessaysthatfewgeniuseseverdo:Heinvitesusinandrevealshowhethinks.(I’musingthepresenttenseherebecausetheessayissointimate,itfeelslikehe’sspeakingtoustoday.)Heshareshisprivateintuition,avulnerable,soft-belliedthing,andsayshehopesthatfuturemathematicianswilluseittosolveproblemsthateludedhim.TodaythissecretisknownastheMethod.Ineverheardofitincalculusclass.Wedon’tteachitanymore.ButIfoundthestoryofitandtheideabehinditenthrallingandastounding.HewritesaboutitinalettertohisfriendEratosthenes,thelibrarianat

Alexandriaandtheonlymathematicianofhiserawhocouldunderstandhim.HeconfessesthateventhoughhisMethod“doesnotfurnishanactualdemonstration”oftheresultshe’sinterestedin,ithelpshimfigureoutwhat’strue.Itgiveshimintuition.Ashesays,“Itiseasiertosupplytheproofwhenwehavepreviouslyacquired,bythemethod,someknowledgeofthequestionsthanitistofinditwithoutanypreviousknowledge.”Inotherwords,bynoodlingaround,playingwiththeMethod,hegetsafeelfortheterritory.Andthatguideshimtoawatertightproof.Thisissuchanhonestaccountofwhatit’sliketodocreativemathematics.

Mathematiciansdon’tcomeupwiththeproofsfirst.Firstcomesintuition.Rigorcomeslater.Thisessentialroleofintuitionandimaginationisoftenleftoutofhigh-schoolgeometrycourses,butitisessentialtoallcreativemathematics.Archimedesconcludeswiththehopethat“therewillbesomeamongthe

presentaswellasfuturegenerationswhobymeansofthemethodhereexplainedwillbeenabledtofindothertheoremswhichhavenotyetfallentoourshare.”Thatalmostbringsateartomyeye.Thisunsurpassedgenius,feelingthefinitenessofhislifeagainsttheinfinitudeofmathematics,recognizesthatthereissomuchlefttobedone,thatthereare“othertheoremswhichhavenotyetfallentoourshare.”Weallfeelthat,allofusmathematicians.Oursubjectisendless.IthumbledevenArchimedeshimself.ThefirstmentionoftheMethodappearsatthebeginningoftheessayonthe

quadratureoftheparabola,beforethecubistproofwiththeshards.ArchimedesconfessesthattheMethodledhimtothatproofandtothenumber4/3inthefirstplace.WhatistheMethod,andwhatissopersonal,brilliant,andtransgressiveabout

it?TheMethodismechanical;Archimedesfindstheareaoftheparabolic

segmentbyweighingitinhismind.Hethinksofthecurvedparabolicregionasamaterialobject—I’mpicturingitasathinsheetofmetalcarefullytrimmedintothedesiredparabolicshape—andthenheplacesitatoneendofanimaginarybalancescale.Or,ifyouprefer,thinkofitasbeingseatedatoneendofanimaginaryseesaw.Nexthefiguresouthowtocounterbalanceitagainstashapehealreadyknowshowtoweigh:atriangle.Fromthishededucestheareaoftheoriginalparabolicsegment.It’sanevenmoreimaginativeapproachthanthecubist/geometric/shards-and-

trianglestechniqueofhisthatwediscussedearlier,becauseinthiscase,he’sgoingtobuildtheimaginaryseesawaspartofthecalculationanddesignittocomportwiththeparabola’sdimensions.Together,theywillproducetheanswerheseeks.Hestartswiththeparabolicsegmentandtiltsittoensurethattheparabola’s

symmetryaxisisvertical.

Thenhebuildstheseesawaroundit.Theinstructionmanualreadsasfollows:

Drawthebigtriangleinsidetheparabolicsegment,asbefore,andlabelitABC.Asinthecubistproof,thistriangleisagaingoingtoserveasanareastandard.Theparabolicsegmentwillbecomparedtoitandwillturnouttohavefour-thirdsitsarea.

Nextenclosetheparabolicsegmentinamuchbiggertriangle,ACD.

Thetriangle’stopsideischosentobealinetangenttotheparabolaatthe

pointC.ItsbaseisthelineAC.AnditsleftsideisaverticallinethatextendsupwardfromAuntilitmeetsthetopsideatpointD.UsingstandardEuclideangeometry,ArchimedesprovesthatthishugeoutertriangleACDhasfourtimestheareaoftheinnertriangleABC.(Thatfactwillbecomeimportantlater.Setitasidefornow.)Thenextstepistobuildtherestoftheseesaw—itslever,itstwoseats,andits

fulcrum.Theleveristhelinethatjoinsthetwoseats.ThatlinestartsatC,goesthroughB,emergesfromthehugeoutertriangleatF(thefulcrum),andcontinuestotheleftuntilithitsapointS(theseat).TheconditionthatdefinesSisthatit’sasfarfromFasCis.Inotherwords,FisthemidpointofthelineSC.

Nowcomesthestunninginsightthatunderliesthewholeconception.Using

knownfactsaboutparabolasandtriangles,Archimedesprovesthathecanbalancethehugeoutertriangleagainsttheparabolicsegmentifhethinksaboutthemoneverticallineatatime.Heregardsthembothasbeingcomposedofinfinitelymanyparallellines.Thoselinesarelikeinfinitesimallythinslatsorribs.Here’satypicalpairofthem,definedbyasingleverticallinethroughbothshapes.Onthatline,ashortribconnectsthebasetotheparabola,

andatallribconnectsthebasetothetopsideofthehugeoutertriangle.

Hisamazinginsightisthattheseribsbalanceeachotherperfectly,likekids

playingonaseesaw,aslongastheysitintherightplaces.HeprovesthatifheslidestheshortribovertothepointSandleavesthetallribinplace,theybalance.

Thesameistrueforeveryverticalslice.Nomatterwhichverticalsliceyou

take,theshortribalwaysbalancesthetallribifyouslidetheshortribtoSandleavethetallribinplace.Sothetwoshapesbalanceeachother,ribbyrib.Alltheribsfromtheparabola

endupatS.TogethertheybalancealltheribsfromthehugeoutertriangleACD.

Andsincethoseribshaven’tmoved,thatmeansalltheparabolicmassshiftedtoSbalancesthehugetrianglerightwhereitis.Next,Archimedesreplacestheinfinitelymanyribsofthehugeoutertriangle

withanequivalentpointoftheirown,calledthetriangle’scenterofgravity.Itservesasaproxy.Asfarasseesawsareconcerned,thehugetriangleactsasifitsentiremasswereconcentratedatthatsinglecenterofgravity.Thatlocation,Archimedeshasalreadyshowninotherwork,liesonthelineFCatapointpreciselythreetimesclosertothefulcrumFthanSis.So,sincetheentiremassofthetrianglesitsthreetimesclosertothepivot

point,theparabolicsegmentmustweighathirdasmuchasthehugetriangleinorderforthemtobalance;that’sthelawofthelever.Thereforetheareaoftheparabolicsegmentmustbeone-thirdthatofthehugeoutertriangleACD.AndsincethatoutertrianglehasfourtimestheareaoftheinnertriangleABC(thefactwesetasideearlier),Archimedesdeducesthattheparabolicsegmentmusthave4/3theareaofthetriangleABCinsideit...justaswefoundearlierbysummingtheinfiniteseriesoftriangularshards!IhopeI’vemanagedtoconveywhatanacidtripofanargumentthisis.

Insteadofapotterreassemblingshards,hereArchimedesismorelikeabutcher.Hetakesthetissueoftheparabolicregionapart,oneverticalstripatatime,andhangsalltheseinfinitesimallythinstripsoffleshfromahookatS.Thetotalweightofallthefleshstaysthesameasitwasbackwhenitwasanintactparabolicsegment.It’sjustthathehasshreddedtheoriginalshapeintolotsofvertical,stringystrips,allhangingfromthesamemeathook.(It’ssuchaweirdimage.Maybeweshouldstickwithseesaws.)WhydidIcallthisargumenttransgressive?Becauseittrafficswithcompleted

infinity.Atonestage,Archimedesopenlydescribestheoutertriangleasbeing“madeupofalltheparallellines”insideitself.That,ofcourse,istabooinGreekmathematics;there’sacontinuousinfinityoftheseparallellines,theseverticalribs.He’sopenlythinkingofthetriangleasacompletedinfinityofribs.Indoingso,he’sunleashingthegolem.Likewisehedescribestheparabolicsegmentasbeing“madeupofallthe

parallellinesdrawninsidethecurve.”Dallyingwithcompletedinfinitylowersthestatusofthisreasoning,inhisestimation,toaheuristic—ameansoffindingananswer,notaproofofitscorrectness.InhislettertoEratosthenes,hedownplaystheMethodasgivingnothingmorethan“asortofindication”thattheconclusionistrue.Whateveritslogicalstatus,Archimedes’sMethodhasanepluribusunum

qualitytoit.ThisLatinphrase,themottooftheUnitedStates,means“outofmany,one.”Outoftheinfinitelymanystraightlinesmakinguptheparabola,

oneareaemerges.Thinkingofthatareaasamass,Archimedesshiftsit,linebyline,tothefarleftseatontheseesaw.Theinfinitudeoflinesistherebyrepresentedbyasinglemassseatedatasinglepoint.Theonereplacesthemanyandstandsforit,representingitperfectlyandfaithfully.Thesameistrueforthecounterbalancingoutertriangleontherightofthe

seesaw.Outofitscontinuumofverticallines,onepointischosen—itscenterofgravity.Ittoostandsforthewhole.Infinitycollapsestounity;epluribusunum.Exceptthisisnotpoetryorpolitics.Thisisthebeginningofintegralcalculus.Trianglesandparabolicregionsareapparentlyandmysteriouslyequivalent,insomesensethatArchimedescouldnotquitemakerigorous,toinfinitudesofverticallines.AlthoughArchimedesseemsembarrassedbyhisdalliancewithinfinity,heis

braveenoughtoownuptoit.Anyonetryingtomeasureacurvedshape—tofindthelengthofitsboundaryortheareaorthevolumeinsideit—hastograpplewiththelimitofaninfinitesumofinfinitesimallysmallpieces.Carefulsoulsmaytrytosidestepthatnecessity,finessingitwiththemethodofexhaustion.Butatbottom,thereisnoescapingit.Copingwithcurvedshapesmeanscopingwithinfinity,onewayoranother.Archimedesisopenaboutthis.Whenheneedsto,hecandressuphisproofsinrespectablegarb,sportingfinitesumsandthemethodofexhaustion.Butinprivate,he’sdirty.Headmitstoweighingshapesinhismind,dreamingofleversandcentersofgravity,balancingregionsandsolidslinebyline,oneinfinitesimalpieceatatime.ArchimedeswentontoapplytheMethodtomanyotherproblemsabout

curvedshapes.Forexample,heusedittodiscoverthecenterofgravityofasolidhemisphere,aparaboloid,andsegmentsofellipsoidsandhyperboloids.Hisfavoriteresult,whichhelovedsomuchthatheaskedthatitbecarvedonhistombstone,concernedthesurfaceareaandvolumeofasphere.Pictureaspheresittingsnuglyinacylindricalhatbox.

UsingtheMethod,Archimedesdiscoveredthatthespherehas⅔thevolumeoftheenclosinghatbox,aswellas⅔ofitssurfacearea(assumingthetopandbottomlidsarealsocountedinthehatbox’ssurfacearea).Noticethathedidn’tgiveformulasforthevolumeorthesurfaceareaofthesphere,aswewouldtoday.Rather,hephrasedhisresultsasproportions.That’sclassicGreekstyle.Everythingwasexpressedasaproportion.Anareawascomparedtoanotherarea,avolumetoanothervolume.Andwhentheirratioinvolvedsmallwholenumbers,astheydoherewith3and2andastheydidwith4and3inthequadratureoftheparabola,thatmusthavebeenasourceofparticularpleasuretohim.Afterall,thosesameratios,3:2and4:3,heldspecialsignificancetotheancientGreeksbecauseoftheircentralroleinthePythagoreantheoryofmusicalharmony.Recallthatwhentwootherwiseidenticalstringswithlengthsintheratio3:2arestruck,theyharmonizebeautifully,separatedinpitchbyanintervalknownasafifth.Similarly,stringsina4:3ratioproduceafourth.ThesenumericalcoincidencesbetweenharmonyandgeometrymusthavedelightedArchimedes.Hiswordsinhisessay“OntheSphereandCylinder”suggestjusthowtickled

hewas:“Nowthesepropertieswereallalongnaturallyinherentinthefigures,butremainedunknowntothosewhowerebeforemytimeengagedinthestudyofgeometry.”Ignorehowproudhesoundsandfocusinsteadonhisclaimthatthepropertieshediscovered“wereallalongnaturallyinherentinthefigures,butremainedunknown.”Hereheisexpressingaparticularphilosophyof

mathematicsdeartotheheartsofallworkingmathematicians.Wefeelwearediscoveringmathematics.Theresultsarethere,waitingforus.Theyhavebeeninherentinthefiguresallalong.Wearenotinventingthem.UnlikeBobDylanorToniMorrison,wearenotcreatingmusicornovelsthatneverexistedbefore;wearediscoveringfactsthatalreadyexist,thatareinherentintheobjectswestudy.Althoughwehavecreativefreedomtoinventtheobjectsthemselves—tocreateidealizationslikeperfectspheresandcirclesandcylinders—oncewedo,theytakeonlivesoftheirown.WhenIreadthewayArchimedesexpresseshispleasureatunveilingthe

surfaceareaandvolumeofthesphere,IfeellikeI’mfeelingthesamethingshefelt.Or,rather,thathewasfeelingthesamethingsIfeelandthatallofmycolleaguesfeelwhenwedomathematics.Althoughwearetoldthatthepastisaforeigncountry,itmaynotbeforeignineveryrespect.PeoplewereadaboutinHomerandtheBibleseemalotlikeus.Andthesameappearstobetrueofancientmathematicians,oratleastofArchimedes,theonlyonewholetusintohisheart.Twenty-twocenturiesago,Archimedeswrotealettertohisfriend

Eratosthenes,thelibrarianatAlexandria,essentiallysendinghimamathematicalmessageinabottlethatvirtuallynoonecouldappreciatebutthathehopedmightsomehowsailsafelyacrosstheseasoftime.Hehadsharedhisprivateintuition,hisMethod,inthewishthatitmightenablefuturegenerationsofmathematicians“tofindothertheoremswhichhavenotyetfallentoourshare.”Theoddswereagainsthim.Asalways,theravagesoftimewerecruel.Kingdomsfellandlibrarieswereburned.Manuscriptsdecayed.NotasinglecopyoftheMethodwasknowntohavesurvivedtheMiddleAges.AlthoughLeonardodaVinci,Galileo,Newton,andothergeniusesoftheRenaissanceandthescientificrevolutionporedoverwhatwasleftofArchimedes’streatises,theyneverhadachancetoreadtheMethod.Itwasthoughttobeirretrievablylost.Andthen,miraculously,itwasfound.InOctober1998abatteredmedievalprayerbookcameupforauctionat

Christie’sandsoldtoananonymousprivatecollectorfor$2.2million.BarelyvisibleunderitsLatinprayerslayfaintgeometricaldiagramsandmathematicaltextwrittenintenth-centuryGreek.Thebookisapalimpsest;inthethirteenthcentury,itsparchmentfolioshadbeenwashedandscrapedcleanoftheoriginalGreekandoverwrittenwithLatinliturgicaltext.Fortunately,theGreekwasnotcompletelyobliterated.ItcontainstheonlysurvivingcopyofArchimedes’sMethod.TheArchimedesPalimpsest,asitisnowknown,firstcametolightin1899in

aGreekOrthodoxlibraryinConstantinople.ItspenttheRenaissanceandthe

scientificrevolutionundetectedinaprayerbookinthemonasteryofSt.SabasnearBethlehem.ItnowlivesintheWaltersArtMuseuminBaltimore,whereithasbeenlovinglyrestoredandexaminedusingthelatestimagingtechnology.

ArchimedesToday:FromComputerAnimationtoFacialSurgery

Archimedes’slegacylivesontoday.Considerthecomputer-animatedmoviesthatourkidslovetowatch.ThecharactersinfilmslikeShrek,FindingNemo,andToyStoryseemlifelikeandreal,inpartbecausetheyembodyanArchimedeaninsight:Anysmoothsurfacecanbeconvincinglyapproximatedbytriangles.Forexample,herearethreetriangulationsofamannequin’shead.

Themoretrianglesweuseandthesmallerwemakethem,thebettertheapproximationbecomes.What’strueformannequinsisequallytrueforogres,clownfish,andtoy

cowboys.JustasArchimedesusedamosaicofinfinitelymanytriangularshardstorepresentasegmentofasmoothlycurvedparabola,modern-dayanimatorsatDreamWorkscreatedShrek’sroundbellyandhiscutelittletrumpet-likeearsoutoftensofthousandsofpolygons.EvenmorewererequiredforatournamentsceneinwhichShrekbattledlocalthugs;eachframeofthatscenetookoverforty-fivemillionpolygons.Buttherewasnotraceofthemanywhereinthefinishedmovie.AstheInfinityPrincipleteachesus,thestraightandthejaggedcanimpersonatethecurvedandthesmooth.

WhenAvatarwasreleasednearlyadecadelater,in2009,thelevelofpolygonaldetailbecamemoreextravagant.AtdirectorJamesCameron’sinsistence,animatorsusedaboutamillionpolygonstorendereachplantontheimaginaryworldofPandora.Giventhatthemovietookplaceinalushvirtualjungle,thatamountedtoalotofplants...andalotofpolygons.NowonderAvatarcostthreehundredmilliondollarstoproduce.Itwasthefirstmovietousepolygonsbythebillions.Theearliestcomputer-generatedmoviesusedfarfewerpolygons.

Nonetheless,thecomputationsseemedstaggeringatthetime.ConsiderToyStory,releasedin1995.Backthen,ittookasingleanimatoraweektosyncaneight-secondshot.Thewholefilmtookfouryearsandeighthundredthousandhoursofcomputertimetocomplete.AsPixarco-founderSteveJobstoldWired,“TherearemorePhDsworkingonthisfilmthananyotherinmoviehistory.”SoonafterToyStorycameGeri’sGame,thefirstcomputer-animatedfilm

withahumanmaincharacter.Thisfunny/sadstoryofalonesomeoldmanwhoplayschesswithhimselfintheparkwonthe1998AcademyAwardforBestAnimatedShortFilm.

Likeothercharactersgeneratedbyacomputer,Geriwasbuiltfromangular

shapes.Atthebeginningofthissection,Ishowedacomputergraphicofafacemadefromevermoretriangles.Inmuchthesameway,theanimatorsatPixarfashionedGeri’sheadfromacomplexpolyhedron,athree-dimensionalgem-likeshapethatconsistedofaboutforty-fivehundredcornerswithflatfacetsinbetweenthem.Theanimatorssubdividedthosefacetsrepeatedlytocreatean

increasinglydetaileddepiction.Thissubdivisionprocesstookupmuchlessmemoryinthecomputerthanearliermethodshad,anditallowedformuchfasteranimations.Itwasarevolutionaryadvanceincomputeranimationatthetime.Butinspirit,itchanneledArchimedes.Recallthattoestimatepi,Archimedesstartedwithahexagon,thensubdividedeachofitssidesandpushedtheirmidpointsouttothecircletogeneratea12-gon.Afteranothersubdivision,the12-gonbecamea24-gon,thena48-gon,andfinallya96-gon,eachencroachingevermorecloselyonitstarget,alimitingcircle.Likewise,Geri’sanimatorsapproximatedthecharacter’swrinklyforehead,hisprotuberantnose,andthefoldsofskininhisneckbyrepeatedlysubdividingapolyhedron.Byrepeatingthatprocessenoughtimes,theycouldmakeGerilooklikewhathewasintendedtobe,apuppet-likecharacterwhoconveyedawiderangeofhumanfeeling.Afewyearslater,aPixarrival,DreamWorks,tookthenextstepsforwardin

realismandemotionalexpressivenessintheirstoryofasmelly,grouchy,heroicogrenamedShrek.

Althoughheneverexistedoutsideacomputer,Shrekseemedpractically

human.Thatwaspartlybecausetheanimatorstooksuchgreatcaretoreproducehumananatomy.Underneathhisvirtualskin,theybuiltvirtualmuscle,fat,bones,andjoints.ItwasdonesofaithfullythatwhenShrekopenedhismouthtospeak,theskinonhisneckformedadoublechin.WhichbringsustoanotherfieldwhereArchimedes’sideaofpolygonal

approximationhasproveduseful:facialsurgeryforpatientswithsevereoverbites,misalignedjaws,orothercongenitalmalformations.In2006,the

GermanappliedmathematiciansPeterDeuflhard,MartinWeiser,andStefanZachowreportedtheresultsoftheirworkusingcalculusandcomputermodelingtopredicttheoutcomesofcomplexfacialsurgeries.Theteam’sfirststepwastobuildanaccuratemapofapatient’sfacial-bone

structure.Todoso,theyscannedthepatientswithcomputerizedtomography(CT)ormagneticresonanceimaging(MRI).Theresultsgaveinformationaboutthethree-dimensionalconfigurationoffacialbonesintheskull,fromwhichtheresearcherscreatedacomputermodelofthepatient’sface.Themodelwasnotjustgeometricallyaccurate;itwasbiomechanicallyaccurate.Itincorporatedrealisticestimatesofthematerialpropertiesofskinandsofttissuessuchasfat,muscle,tendons,ligaments,andbloodvessels.Withthehelpofthecomputermodel,surgeonscouldthenperformoperationsonvirtualpatients,similartohowfighterpilotssharpentheirskillsinflightsimulators.Virtualbonesintheface,jaw,andskullcouldbecut,relocated,augmented,orremovedentirely.Thecomputercalculatedhowthevirtualsofttissuebehindthefacewouldmoveandreconfigureitselfinresponsetostressesproducedbytheface’snewbonestructure.Theresultsofsuchsimulationswerehelpfulinseveralways.Theyalertedthe

surgeonstopossibleadverseeffectstheprocedurescouldhaveonvulnerablestructureslikenerves,bloodvessels,andtherootsofteeth.Theyalsorevealedwhatthepatient’sfacewouldlooklikepostoperatively,sincethemodelpredictedhowthesofttissueswouldrepositionthemselvesafterthepatienthealed.Anotheradvantagewasthatthesurgeonscouldpreparebetterfortheactualoperationsinlightofthesimulatedresults.Andthepatientscouldmakebetterdecisionsaboutwhethertohavetheoperations.Archimedescameinwhentheresearchersmodeledthesmoothtwo-

dimensionalsurfaceoftheskullwithanenormousnumberoftriangles.Thesofttissueposeditsowngeometricalchallenges.Unliketheskull,softtissueformsafullythree-dimensionalvolume.Itfillsthecomplicatedspaceinfrontoftheskullandbehindtheskinoftheface.Theteamrepresenteditbyhundredsofthousandsoftetrahedrons,thethree-dimensionalcounterpartsoftriangles.Intheimagebelow,theskullsurfaceisapproximatedby250,000triangles(they’retoosmalltobeseen)andthevolumeofsofttissueconsistsof650,000tetrahedrons.

Thearrayoftetrahedronsallowedtheresearcherstopredicthowthepatient’s

softtissueswoulddeformaftersurgery.Roughlyspeaking,softtissueisadeformableyetspringymaterial,abitlikerubberorspandex.Ifyoupinchyourcheek,itchangesshape;whenyouletgo,itreturnstonormal.Eversincethe1800s,mathematiciansandengineershaveusedcalculustomodelhowdifferentmaterialsstretch,bend,andtwistwhentheyaoutcomesofaboutthirtysurgicalrepushed,pulled,orshearedinvariousways.Thetheoryismosthighlydevelopedinthemoretraditionalpartsofengineering,whereit’susedtoanalyzethestressesandstrainsinbridges,buildings,airplanewings,andmanyotherstructuresmadeofsteel,concrete,aluminum,andotherhardmaterials.TheGermanresearchersadaptedthetraditionalapproachtosofttissuesandfoundthatitworkedwellenoughtobevaluabletosurgeonsandpatientsalike.Theirbasicideawasthis.Thinkofthesofttissueasameshworkof

tetrahedronsconnectedtooneanotherlikebeadsconnectedbyelasticthreads.Thebeadsrepresenttinyportionsoftissue.Theyaretiedtogetherelasticallybecause,inreality,atomsandmoleculesinthetissuearelinkedbychemicalbonds.Thosebondsresiststretchingandcompression,whichiswhatendowsthemwithelasticity.Duringavirtualoperation,asurgeoncutsbonesinthevirtualfaceandrelocatessomeofthebonesegments.Whenapieceofboneis

movedtoanewplace,itpullsonthetissuesit’sconnectedto,whichinturnpullontheirneighboringtissues.Themeshworkreconfiguresitselfduetotheeffectofcascadingforces.Aspiecesoftissuemove,theychangetheforcestheyexertontheirneighborsbystretchingorcompressingthebondsbetweenthem.Thoseaffectedneighborsthemselvesreadjust,andsoon.Keepingtrackofalltheresultingforcesanddisplacementsisamassivecalculationthatcanbedoneonlybycomputer.Stepbystep,analgorithmupdatesthemyriadofforcesandmovesthetinytetrahedronsaccordingly.Ultimatelyalltheforcesbalanceandthetissuesettlesintoitsnewequilibriumstate.That’sthenewshapeofthepatient’sfacethatthemodelpredicts.In2006,Deuflhard,Weiser,andZachowtestedtheirmodel’spredictions

againsttheclinicaloutcomesofaboutthirtysurgicalcases.Theyfoundthatthemodelworkedremarkablywell.Asonemeasureofitssuccess,itcorrectlyforecast—towithinonemillimeter—thepositionof70percentofthepatient’sfacialskin.Only5to10percentoftheskinsurfacedeviatedbymorethanthreemillimetersfromitspredictedpostoperativelocation.Inotherwords,themodelcouldbetrusted.Anditwascertainlybetterthanguesswork.Here’sanexampleofonepatientbeforeandaftersurgery.Thefourpanels

showhisprofilebeforetheoperation(farleft),thecomputermodelofhisfaceatthattime(mid-left),thepredictedoutcomeofthesurgery(mid-right),andtheactualoutcome(farright).Lookatthepositionofhisjawbeforeandafter.Theresultsspeakforthemselves.

OnwardtotheMysteryofMotion

Iamwritingthesewordsthedayafterablizzard.YesterdaywasMarch14,PiDay,andwegotoverafootofsnow.Thismorning,whileIwasshovelingmydrivewayforthefourthtime,Iwatchedjealouslyasasmalltractorwithafront-mountedsnowthrowermadeitswayeasilydownthesidewalkacrossthestreet.Itusedarotatingscrewbladetopullsnowintothemachineandthenejecteditontomyneighbor’syard.Thisuseofarotatingscrewforpropellingsomethinggoesbackto

Archimedes,atleastaccordingtolegend.Inhishonor,todaywecallitanArchimedeanscrew.HeissaidtohavecomeupwiththeinventionduringatriptoEgypt(althoughitmayhavebeenusedmuchearlierbytheAssyrians);itwasdevelopedtoliftwaterfromalow-lyingareauptoanirrigationditch.Today,cardiac-assistdevicesuseArchimedean-screwpumpstosupportcirculationwhentheheart’sleftventricleisimpaired.Butapparently,Archimedesdidnotwanttoberememberedforhisscrewsor

hiswarenginesoranyotherpracticalinventions;heneverleftusanywritingsaboutthem.Hewasproudestofhisinventionsinmathematics.WhichalsogetsmethinkingthatitisfittingtobereflectingonhislegacyonPiDay.Inthetwenty-twohundredyearssinceArchimedestrappedpi,numericalapproximationstopihavebeenimprovedmanytimes,butalwaysbyusingmathematicaltechniquesthatArchimedeshimselfintroduced:approximationsbypolygonsorbyinfiniteseries.Morebroadly,hislegacywasthefirstprincipleduseofinfiniteprocessestoquantifythegeometryofcurvedshapes.Atthishewasunrivaled,andheremainssotothisday.Yetthegeometryofcurvedshapestakesusonlysofar.Wealsoneedtoknow

howthingsmoveinthisworld—howhumantissueshiftsaftersurgery,howbloodflowsthroughanartery,howaballfliesthroughtheair.Onthis,Archimedeswassilent.Hegaveusthescienceofstatics,ofbodiesbalancingonleversandfloatingstablyinwater.Hewasamasterofequilibrium.Theterritoryaheadconcernedthemysteriesofmotion.

3

DiscoveringtheLawsofMotion

WHENARCHIMEDESDIED,themathematicalstudyofnaturenearlydiedalongwithhim.EighteenhundredyearspassedbeforeanewArchimedesappeared.InRenaissanceItaly,ayoungmathematiciannamedGalileoGalileipickedupwhereArchimedeshadleftoff.Hewatchedhowthingsmovedwhentheyflewthroughtheairorfelltotheground,andhelookedfornumericalrulesintheirmovements.Hedidcarefulexperimentsandmadecleveranalyses.Hetimedpendulumsswingingbackandforthandrolledballsdowngentlerampsandfoundmarvelousregularitiesinboth.Meanwhile,ayoungGermanmathematiciannamedJohannesKeplerstudiedhowtheplanetswanderedacrossthesky.Bothmenwerefascinatedbypatternsintheirdataandsensedthepresenceofsomethingfardeeper.Theyknewtheywereontosomething,butcouldn’tquitemakeoutitsmeaning.Thelawsofmotiontheywerediscoveringwerewritteninanalienlanguage.Thatlanguage,asyetunknown,wasdifferentialcalculus.Thesewerehumanity’sfirsthintsofit.BeforetheworkofGalileoandKepler,naturalphenomenahadrarelybeen

understoodinmathematicalterms.Archimedeshadrevealedthemathematicalprinciplesofbalanceandbuoyancyinhislawsoftheleverandhydrostaticequilibrium,butthoselawswerelimitedtostatic,motionlesssituations.GalileoandKeplerventuredbeyondthestaticworldofArchimedesandexploredhowthingsmoved.Theirstrugglestomakesenseofwhattheysawspurredthe

inventionofanewkindofmathematicsthatcouldhandlemotionatavariablerate.Itaddressedthetypeofchangethatkeepschanging,likeaballgainingspeedasitrollsdownaramportheplanetsspeedingupastheymoveclosertothesunandslowingdownastheyrecedefromit.In1623,Galileodescribedtheuniverseas“thisgrandbook...whichstands

continuallyopentoourgaze,”butcautionedthat“thebookcannotbeunderstoodunlessonefirstlearnstocomprehendthelanguageandreadthelettersinwhichitiscomposed.Itiswritteninthelanguageofmathematics,anditscharactersaretriangles,circles,andothergeometricfigureswithoutwhichitishumanlyimpossibletounderstandasinglewordofit;withoutthese,onewandersaboutinadarklabyrinth.”Keplerexpressedevengreaterreverenceforgeometry.Hedescribeditas“coeternalwiththedivinemind”andbelievedthatit“suppliedGodwithpatternsforthecreationoftheworld.”ThechallengeforGalileo,Kepler,andotherlike-mindedmathematiciansof

theearlyseventeenthcenturywastotaketheirbelovedgeometry,sowellsuitedtoaworldatrest,andextendittoaworldinflux.Theproblemstheyfacedweremorethanmathematical;theyhadtoovercomephilosophical,scientific,andtheologicalresistanceaswell.

TheWorldAccordingtoAristotle

Beforetheseventeenthcentury,motionandchangewerepoorlyunderstood.Notonlyweretheydifficulttostudy;theywereconsidereddownrightdistasteful.Platohadtaughtthattheobjectofgeometrywastogain“knowledgeofwhateternallyexists,andnotofwhatcomesforamomentintoexistence,andthenperishes.”Hisphilosophicalcontemptforthetransitoryreturnedonagranderscaleinthecosmologyofhismostillustriousstudent,Aristotle.AccordingtoAristotelianteaching,whichdominatedWesternthoughtfor

almosttwomillennia(andwhichCatholicismembracedafterThomasAquinasexpungeditspaganparts),theheavenswereeternal,unchanging,andperfect.EarthsatmotionlessatthecenterofGod’screationwhilethesun,moon,stars,andplanetsrevolvedarounditinperfectcircles,carriedalongbytherotationoftheheavenlyspheres.Accordingtothiscosmology,everythingintheterrestrialrealmbelowthemoonwascorruptedandplaguedbyrot,death,anddecay.Thevagariesoflife,muchlikethefallingofleaves,werebytheirverynaturefleeting,erratic,anddisorderly.

AlthoughanEarth-centeredcosmologyseemedreassuringandcommonsensical,themotionoftheplanetspresentedanawkwardproblem.Thewordplanetmeans“wanderer.”Inantiquitytheplanetswereknownasthewanderingstars;insteadofmaintainingtheirplacesinthesky,likethefixedstarsinOrion’sBeltandtheladleoftheBigDipper,whichnevermovedrelativetooneanother,theplanetsappearedtodriftacrosstheheavens.Theyprogressedfromoneconstellationtoanotherastheweeksandmonthswentby.Mostofthetimetheymovedeastwardrelativetothestars,butoccasionallytheyappearedtoslowdown,stop,andgobackward,westward,inwhatastronomerscalledretrogrademotion.Mars,forexample,wasseentomoveinretrogradeforaboutelevenweeks

overthecourseofitsnearlytwo-yearcircuitaroundthesky.Nowadayswecancapturethisreversalphotographically.In2005theastrophotographerTunçTezeltookaseriesofthirty-fivesnapshotsofMars,eachaboutaweekapart,andalignedtheimagestothestarsinthebackground.Intheresultingcomposite,theelevendotsinthemiddleshowMarsmovinginretrograde.

Todayweunderstandthatretrogrademotionisanillusion.It’scausedbyour

vantagepointonEarthaswepasstheslower-movingMars.

It’slikewhathappenswhenyoupassacaronthehighway.Imaginedriving

onalonghighwayoutinthedesert,withmountainsoffinthedistance.Asyouapproachaslowercarfrombehind,itlookslikeit’smovingforwardwhenviewedagainstthebackdropofthemountains.Butwhenyoupullalongsideandpassit,theslowercarmomentarilyseemstomovebackwardrelativetothemountains.Then,onceyougetfarenoughaheadofit,thecarappearstomoveforwardagain.ThiskindofobservationledtheancientGreekastronomerAristarchusto

proposeasun-centereduniversealmosttwomillenniabeforeCopernicusdid.Itneatlysolvedtheriddleofretrogrademotion.However,asun-centereduniverseraisedquestionsofitsown.IftheEarthmoves,whydon’twefalloff?Andwhydothestarsappearfixed?Theyshouldn’t.AstheEarthmovesaroundthesun,thedistantstarsshouldappeartoshifttheirpositionsslightly.Experienceshowsthatifyoulookatsomethingfarawayandthenmoveandlookagain,thepositionofthatfarawayobjectappearstoshiftwhenviewedagainstamoredistantbackdrop.Thiseffectiscalledparallax.Toexperienceit,holdyourfinger

faroutinfrontofyourface.Closeoneeye,thentheother.Yourfingerseemstoshiftsidewaysagainstthebackdropwhenyouswitcheyes.Likewise,astheEarthmovesaroundthesuninitsorbit,thestarsshouldshifttheirapparentpositionsagainstthebackgroundofevenmoredistantstars.Theonlywayoutofthisparadox(asArchimedeshimselfrealizedwhenreactingtoAristarchus’ssun-centeredcosmology)wouldbeifallthestarswereimmenselydistant,effectivelyinfinitelyfarawayfromtheEarth.Thentheplanet’smotionwouldproducenodetectableshift,becausetheparallaxwouldbetoosmalltobemeasured.Thisconclusionwashardtoacceptatthetime.Noonecouldimagineauniversesoimmensewithstarssoremote,muchfartherawaythantheplanets.Todayweknowthatisexactlythecase,butbackthenitwasinconceivable.SotheEarth-centeredcosmology,forallitsfaults,seemedlikethemore

plausiblepicture.SuitablymodifiedbytheancientGreekastronomerPtolemywithepicycles,equants,andotherfudgefactors,thetheorycouldbemadetoaccountreasonablywellforplanetarymotionanditkeptthecalendarinlinewithseasonalcycles.ThePtolemaicsystemwasclunkyandcomplicated,butitworkedwellenoughtolastintothelateMiddleAges.Twobookspublishedin1543markedaturningpoint,thebeginningofthe

scientificrevolution.Inthatyear,theFlemishdoctorAndreasVesaliusreportedtheresultsofhisdissectionsofhumancadavers,apracticethathadbeenforbiddeninearliercenturies.Hisfindingscontradictedfourteencenturiesofreceivedwisdomabouthumananatomy.Inthatsameyear,thePolishastronomerNicolausCopernicusfinallyallowedpublicationofhisradicaltheorythattheEarthmovedaroundthesun.He’dwaiteduntilhewasneardeath(anddiedjustasthebookwasbeingpublished)becausehe’dfearedthattheCatholicChurchwouldbeinfuriatedbyhisdemotionoftheworldfromthecenterofGod’screation.Hewasrighttobescared.AfterGiordanoBrunoproposed,amongotherheresies,thattheuniversewasinfinitelylargewithinfinitelymanyworlds,hewastriedbytheInquisitionandburnedatthestakeinRomein1600.

EnterGalileo

Intothisclimate,asauthorityanddogmawerebeingchallengedbydangerousideas,GalileoGalileiwasbornonFebruary15,1564,inPisa,Italy.Theeldestsonofaonce-noblefamilynowdownonitsluck,Galileowaspushedbyhisfathertowardacareerinmedicine,amuchmorelucrativeprofessionthanhisfather’sownfieldofmusictheory.ButGalileosoonfoundhispassionwas

mathematics.HestudiedEuclidandArchimedesandmasteredboth.Thoughheneverfinishedhisdegree(hisfamilycouldn’taffordthetuition),hecontinuedtoteachhimselfmathandscience,gotaluckybreakasatemporaryinstructoratPisa,andgraduallyrosethroughtheacademicranksasaprofessorofmathematicsattheUniversityofPadua.Hewasabrilliantlecturer,clearandirreverentwithacausticwit.Studentsflockedtohisclassestohearhim.HemetavivaciousandmuchyoungerwomannamedMarinaGambawith

whomhehadalongandlovingbutillicitrelationship.Theyhadtwodaughtersandasontogetherbutdidnotmarry;itwouldhavebeenconsidereddishonorableforhim,givenMarina’syouthandlowersocialstanding.Withthestrainofhismeagersalaryasamathteacher,thecostofraisingtheirthreechildren,andtheadditionalresponsibilitytoprovideforhisunwedsister,Galileofeltforcedtoplacehisdaughtersinaconvent,whichbrokehisheart.Hiselderdaughter,Virginia,washisfavorite,thejoyofhislife.Helaterdescribedheras“awomanofexquisitemind,singulargoodness,andmosttenderlyattachedtome.”Whenshetookhervowsasanun,shechoseSisterMariaCelesteasherreligiousnameinhonoroftheVirginMaryandherfather’sfascinationwithastronomy.Galileoisperhapsmostoftenrememberedtodayforhisworkwiththe

telescopeandasachampionoftheCopernicantheorythattheEarthmovesaroundthesun,acontradictionoftheviewsofAristotleandtheCatholicChurch.AlthoughGalileodidnotinventthetelescope,heimproveditandwasthefirsttomakegreatscientificdiscoverieswithit.In1610and1611,heobservedthatthemoonhadmountains,thesunhadspots,andJupiterhadfourmoons(othershavebeendiscoveredsincethen).Alltheseobservationsflewinthefaceoftheprevailingdogma.Mountainson

themoonmeantitwasnotaglistening,perfectorb,contrarytoAristotelianteaching.Likewise,spotsonthesunmeantitwasnotaperfectcelestialbody;itwasmarredbyblemishes.AndsinceJupiteranditsmoonslookedlikealittleplanetarysystemofitsown,withfoursmallmoonsorbitingaroundabiggercentralplanet,thenclearlynotallheavenlybodiesrevolvedsolelyaroundtheEarth.Furthermore,thosemoonsmanagedtostaywithJupiterastheyallmovedacrossthesky.Atthetime,oneofthestandardargumentsagainstheliocentricitywasthat,iftheEarthwasorbitingthesun,itwouldleavethemoonbehind,butnowJupiteranditsmoonsshowedthatthisreasoningmustbefalse.ThisisnottosaythatGalileowasanatheistorirreligious.Hewasagood

CatholicandbelievedthathewasrevealingthegloryofGod’sworkbydocumentingitasittrulywasratherthanbyrelyingonthereceivedwisdomofAristotleandhislaterscholasticinterpreters.TheCatholicChurch,however,did

notseeitthisway.Galileo’swritingswerecondemnedasheresy.HewasbroughtbeforetheInquisitionin1633andorderedtorecant,whichhedid.Hewassentencedtolifeinprison,apunishmentimmediatelycommutedtopermanenthousearrestinhisvillainArcetriinthehillsofFlorence.HelookedforwardtoseeinghisbeloveddaughterMariaCeleste,butsoonafterhisreturn,shefellillanddied,atonlythirty-threeyearsofage.Galileowasbereftandforawhilelostallinterestinworkandlife.Hespenthisremainingyearsunderhousearrest,anoldmanlosinghisvision

andracingagainsttime.Somehow,withintwoyearsofhisdaughter’sdeath,hefoundthestrengthwithinhimselftosummarizehisunpublishedinvestigationsofmotionfromdecadesearlier.Theresultingbook,DiscoursesandMathematicalDemonstrationsConcerningTwoNewSciences,wastheculminationofhislife’sworkandthefirstgreatmasterpieceofmodernphysics.HewroteitinItalianratherthanLatinsothatitcouldbeunderstoodbyanyoneandarrangedforittobesmuggledouttoHolland,whereitwaspublishedin1638.Itsradicalinsightshelpedlaunchthescientificrevolutionandbroughthumanitytothecuspofdiscoveringthesecretoftheuniverse:thatthegreatbookofnatureiswrittenincalculus.

Falling,Rolling,andtheLawofOddNumbers

Galileowasthefirstpractitionerofthescientificmethod.Ratherthanquotingauthoritiesorphilosophizingfromanarmchair,heinterrogatednaturethroughmeticulousobservations,ingeniousexperiments,andelegantmathematicalmodels.Hisapproachledhimtomanyremarkablediscoveries.Oneofthesimplestandmostsurprisingisthis:Theoddnumbers1,3,5,7,andsofortharehidinginhowthingsfall.BeforeGalileo,Aristotlehadproposedthatheavyobjectsfallbecausetheyare

seekingtheirnaturalplaceatthecenterofthecosmos.Galileothoughtthesewereemptywords.Insteadofspeculatingaboutwhythingsfell,hewantedtoquantifyhowtheyfell.Todoso,heneededtofindawaytomeasurefallingbodiesthroughouttheirdescentandkeeptrackofwheretheyweremomentbymoment.Itwasn’teasy.Anyonewhohasdroppedarockoffabridgeknowsthatrocks

fallfast.Itwouldtakeaveryaccurateclock,ofakindnotavailableinGalileo’sday,andseveralverygoodvideocameras,alsonotavailableintheearly1600s,totrackafallingrockateachmomentofitsrapiddescent.

Galileocameupwithabrilliantsolution:Heslowedthemotion.Insteadofdroppingarockoffabridge,heallowedaballtorollslowlydownaramp.Inthejargonofphysics,thissortoframpisknownasaninclinedplane,althoughinGalileo’soriginalexperiments,itwasmorelikealong,thinpieceofwoodenmoldingwithagroovecutalongitslengthtoactasachannelfortheball.Byreducingtheslopeoftherampuntilitwasnearlyhorizontal,hecouldmaketheball’sdescentasslowashewished,thusallowinghimtomeasurewheretheballwasateachmoment,evenwiththeinstrumentsavailableinhisday.Totimetheball’sdescentheusedawaterclock.Itworkedlikeastopwatch.

Tostarttheclockhewouldopenavalve.Waterwouldthenflowsteadily,ataconstantrate,straightdownthroughathinpipeandintoacontainer.Tostoptheclock,hewouldclosethevalve.Byweighinghowmuchwaterhadaccumulatedduringtheball’sdescent,Galileocouldquantifyhowmuchtimehadelapsedtowithin“one-tenthofapulse-beat.”Herepeatedtheexperimentmanytimes,sometimesvaryingthetiltofthe

ramp,othertimeschangingthedistancesrolledbytheball.Whathefound,inhisownwords,wasthis:“Thedistancestraversed,duringequalintervalsoftime,byabodyfallingfromrest,standtooneanotherinthesameratioastheoddnumbersbeginningwithunity.”Tospelloutthislawofoddnumbersmoreexplicitly,let’ssupposetheball

rollsacertaindistanceinthefirstunitoftime.Then,inthenextunitoftime,itwillrollthreetimesasfar.Andinthenextunitoftimeafterthat,itwillrollfivetimesasfarasitdidoriginally.It’samazing;theoddnumbers1,3,5,andsoonaresomehowinherentinthewaythingsrolldownhill.Andiffallingisjustthelimitofrollingasthetiltapproachesvertical,thesamerulemustholdforfalling.WecanonlyimaginehowpleasedGalileomusthavebeenwhenhe

discoveredthisrule.Butnoticehowhephrasedit—withwordsandnumbersandproportions,notlettersandformulasandequations.Ourcurrentpreferenceforalgebraoverspokenlanguagewouldhaveseemedcutting-edgebackthen,anavant-garde,newfangledwayofthinkingandspeaking.It’snothowGalileowouldhavethoughtorexpressedhimself,norwouldhisreadershaveunderstoodhimifhehad.ToseethemostimportantimplicationofGalileo’srule,let’slookatwhat

happensifweaddconsecutiveoddnumbers.Afteroneunitoftime,theballhastraveledoneunitofdistance.Afterthenextunitoftimetheballhastraveledanotherthreeunitsofdistance,foratotalof1+3=4unitstraveledsincethemotionstarted.Afterthethirdunitoftime,thetotalbecomes1+3+5=9unitsofdistance.Noticethepattern:thenumbers1,4,and9arethesquaresofconsecutiveintegers—12=1,22=4,32=9.SoGalileo’sodd-numberrule

seemstobeimplyingthatthetotaldistancefallenisproportionaltothesquareofthetimeelapsed.Thischarmingrelationshipbetweenoddnumbersandsquarescanbeproved

visually.ThinkoftheoddnumbersasL-shapedarraysofdots:

Thennestlethemtogethertoformasquare.Forexample,1+3+5+7=16=

4×4,becausewecanpackthefirstfouroddnumberstogethertomakea4-by-4square.

Alongwithhislawaboutthedistancetraversedbyafallingbody,Galileoalso

discoveredalawforitsspeed.Asheputit,thespeedincreasesinproportiontothetimeoffalling.What’sinterestingaboutthisisthathewasreferringtothespeedofthebodyataninstant,aseeminglyparadoxicalconcept.HetookpainsinTwoNewSciencestoexplainthatwhenabodyfallsfromrest,itdoesn’tjumpsuddenlyfromzerospeedtosomehigherspeed,ashiscontemporariesthought.Rather,itpassessmoothlythrougheveryintermediatespeed—infinitelymanyof

them—inafiniteamountoftime,startingfromzeroandcontinuouslygainingspeedasitfalls.Sointhislawoffallingbodies,Galileowasinstinctivelythinkingabout

instantaneousspeed,adifferentialcalculusconceptthatwe’llexamineinchapter6.Atthetimehecouldn’tmakeitprecise,butheknewwhathemeantintuitively.

TheArtofScientificMinimalism

BeforeweleaveGalileo’sinclined-planeexperiment,let’sbesuretonoticetheartistrybehindit.Hecoaxedabeautifulansweroutofnaturebyaskingabeautifulquestion.Likeanabstractexpressionistpainter,hehighlightedwhathewasinterestedinandcasttherestaside.Forexample,indescribinghisapparatus,hesayshemadethe“groovevery

straight,smooth,andpolished”and“rolledalongitahard,smooth,andveryroundbronzeball.”Whywashesoconcernedwithsmoothness,straightness,hardness,androundness?Becausehewantedtheballtorolldownhillunderthesimplest,mostidealconditionshecouldcontrive.Hedideverythinghecouldtoreducethepotentialcomplicationscomingfromfrictionorfromtheball’scollisionswiththesidewallsofthegroove(whichcouldoccurifthechannelwasnotstraight)orfromtheball’ssoftness(whichcouldcausetheballtoloseenergyifitdeformedtoomuch)orfromanythingelsethatcouldcausedeviationsfromtheidealcase.Thoseweretherightaestheticchoices.Simple.Elegant.Andminimal.CompareAristotle,whogotthelawoffallingbodieswrongbecausehewas

ledastraybycomplications.Heclaimedthatheavybodiesfellfasterthanlightoneswithspeedsproportionaltotheirweight.That’strueoftinyparticlessinkinginaverythick,viscousmediumlikemolassesorhoney,butnotofcannonballsormusketballsdroppedthroughtheair.Aristotleseemstohavebeensoconcernedwiththedragforcesproducedbyairresistance(admittedlyanimportanteffectforfallingfeathers,leaves,snowflakes,andotherlightobjectsthatalsoofferanunusualamountofsurfaceareafortheairtopushupagainst)thatheforgottotesthistheoryonmoretypicalobjectslikerocksandbricksandshoes,thingsthatarecompactandheavy.Inotherwords,hefocusedtoomuchonthenoise(airresistance)andnotenoughonthesignal(inertiaandgravity).Galileodidn’tlethimselfbedistracted.Heknewthatairresistanceand

frictionwereinescapableintherealworld,asinhisexperiment,buttheywere

notoftheessence.Anticipatingthecriticismthatheoverlookedtheminhisanalysis,heconcededthatapelletofbirdshotdoesnotfallquiteasfastasacannonballbutnotedthattheerrorincurredismuch,muchlessthanthatproducedbyAristotle’stheory.InthedialogueofTwoNewSciences,Galileo’ssurrogateurgeshissimple-mindedAristotelianquestionernotto“divertthediscussionfromitsmainintentandfastenuponsomestatementofminewhichlacksahair’s-breadthofthetruthand,underthishair,hidethefaultofanotherwhichisasbigasaship’scable.”That’sthepoint.Inscience,beingoffbyahairsbreadthisacceptable.Being

offbyaship’scableisnot.Galileowentontostudyprojectilemotion,liketheflightofamusketballora

cannonball.Whatsortofarcdotheyfollow?Galileohadtheideathataprojectile’smotionwascompoundedoftwodifferenteffectsthatcouldbetreatedseparately:amotionsideways,paralleltotheground,forwhichgravityplayednopart,andaverticalmotionupwardordownward,onwhichgravityactedandhislawoffallingbodiesapplied.Puttingthosetwokindsofmotiontogether,hediscoveredthatprojectilesfollowparabolicpaths.Youseethemwheneveryouplayagameofcatchortakeadrinkfromawaterfountain.Thiswasanotherstunningconnectionbetweennatureandmathandafurther

cluethatthebookofnatureiswritteninthelanguageofmathematics.Galileowaselatedtodiscoverthataparabola,anabstractcurvestudiedbyhisheroArchimedes,wasoutthereintherealworld.Naturewasusinggeometry.Toarriveatthisinsight,however,Galileoagainhadtoknowwhattoneglect.

Asbefore,hehadtoignoreairresistance—theeffectofdragontheprojectileasitmovesthroughtheair.Thatfrictionaleffectwouldslowtheprojectiledown.Forsomekindsofprojectiles(athrownrock),frictionisnegligiblecomparedtogravity;forothers(abeachballoraPing-Pongball),itisnot.Allformsoffriction,includingdragcausedbyairresistance,aresubtleanddifficulttostudy.Tothisday,frictionremainsmysteriousandisatopicofactiveresearch.Togetthesimpleparabola,Galileoneededtoassumethesidewaysmotion

wouldcontinueforeverandneverslowdown.Thiswasaninstanceofhislawofinertia,whichstatesthatabodyinmotionstaysinmotionatthesamespeedandinthesamedirectionunlessactedonbyanoutsideforce.Forarealprojectile,airresistancewouldbethatoutsideforce.ButinGalileo’smind,itwasbettertostartbyignoringit,tocapturethelion’sshareofthetruth—andthebeauty—ofhowthingsmove.

FromaSwingingChandeliertotheGlobalPositioningSystem

LegendhasitthatGalileomadehisfirstscientificdiscoverywhenhewasateenagemedicalstudent.Oneday,whileattendingaMassattheCathedralofPisa,henoticedachandelierswayingoverhead,movingtoandfrolikeapendulum.Aircurrentskeptjostlingit,andGalileoobservedthatitalwaystookthesametimetocompleteitsswingwhetherittraversedawidearcorasmallone.Thatsurprisedhim.Howcouldabigswingandalittleswingtakethesameamountoftime?Butthemorehethoughtaboutit,themoreitmadesense.Whenthechandeliermadeabigswing,ittraveledfartherbutitalsomovedfaster.Maybethetwoeffectsbalancedout.Totestthisidea,Galileotimedtheswingingchandelierwithhispulse.Sureenough,everyswinglastedthesamenumberofheartbeats.Thislegendiswonderful,andIwanttobelieveit,butmanyhistoriansdoubtit

happened.ItcomesdowntousfromGalileo’sfirstandmostdevotedbiographer,VincenzoViviani.Asayoungman,hehadbeenGalileo’sassistantanddiscipleneartheendoftheolderman’slife,whenGalileowascompletelyblindandunderhousearrest.Inhisunderstandablereverenceforhisoldmaster,VivianiwasknowntohaveembellishedataleortwowhenhewroteGalileo’sbiographyyearsafterhisdeath.Butevenifthestoryisapocryphal(anditmaynotbe!),wedoknowforsure

thatGalileoperformedcarefulexperimentswithpendulumsasearlyas1602andthathewroteaboutthemin1638inTwoNewSciences.Inthatbook,whichisstructuredasaSocraticdialogue,oneofthecharacterssoundslikehewasrightthereinthecathedralwiththedreamyyoungstudent:“ThousandsoftimesIhaveobservedvibrationsespeciallyinchurcheswherelamps,suspendedbylongcords,hadbeeninadvertentlysetintomotion.”Therestofthedialogueexpoundsontheclaimthatapendulumtakesthesameamountoftimetotraverseanarcofanysize.SoweknowthatGalileowasthoroughlyfamiliarwiththephenomenondescribedinViviani’sstory;whetherheactuallydiscovereditasateenagerisanybody’sguess.Inanycase,Galileo’sassertionthatapendulum’sswingalwaystakesthe

sameamountoftimeisnotexactlytrue;biggerswingstakealittlelonger.Butifthearcissmallenough—lessthan20degrees,say—it’sverynearlytrue.Thisinvarianceoftempoforsmallswingsisknowntodayasthependulum’sisochronism,fromtheGreekwordsfor“equaltime.”Itformsthetheoreticalbasisformetronomesandpendulumclocks,fromordinarygrandfatherclocksto

thetoweringclockusedinLondon’sBigBen.Galileohimselfdesignedtheworld’sfirstpendulumclockinthelastyearofhislife,buthediedbeforeitcouldbebuilt.Thefirstworkingpendulumclockappearedfifteenyearslater,inventedbytheDutchmathematicianandphysicistChristiaanHuygens.Galileowasparticularlyintrigued—andfrustrated—byacuriousfacthe

discoveredaboutpendulums,theelegantrelationshipbetweenitslengthanditsperiod(thetimeittakesthependulumtoswingoncebackandforth).Asheexplained,“Ifonewishestomakethevibration-timeofonependulumtwicethatofanother,hemustmakeitssuspensionfourtimesaslong.”Usingthelanguageofproportions,hestatedthegeneralrule.“Forbodiessuspendedbythreadsofdifferentlengths,”hewrote,“thelengthsaretoeachotherasthesquaresofthetimes.”Unfortunately,Galileonevermanagedtoderivethisrulemathematically.Itwasanempiricalpatterncryingoutforatheoreticalexplanation.Heworkedatitforyearsbutfailedtosolveit.Inretrospect,hecouldn’thave.Itsexplanationrequiredanewkindofmathematicsbeyondanythatheorhiscontemporariesknew.ThederivationwouldhavetowaitforIsaacNewtonandhisdiscoveryofthelanguageGodtalks,thelanguageofdifferentialequations.Galileoconcededthatthestudyofpendulums“mayappeartomany

exceedinglyarid,”althoughitwasanythingbutthat,aslaterworkshowed.Inmathematics,pendulumsstimulatedthedevelopmentofcalculusthroughtheriddlestheyposed.Inphysicsandengineering,pendulumsbecameparadigmsofoscillation.LikethelineinWilliamBlake’spoemaboutseeingtheworldinagrainofsand,physicistsandengineerslearnedtoseetheworldinapendulum’sswing.Thesamemathematicsappliedwhereveroscillationsoccurred.Theworrisomemovementsofafootbridge,thebouncingofacarwithmushyshockabsorbers,thethumpingofawashingmachinewithanunbalancedload,theflutteringofvenetianblindsinagentlebreeze,therumblingoftheearthintheaftershockofanearthquake,thesixty-cyclehumoffluorescentlights—everyfieldofscienceandtechnologytodayhasitsownversionofto-and-fromotion,ofrhythmicreturn.Thependulumisthegranddaddyofthemall.Itspatternsareuniversal.Aridisnottherightwordforthem.Insomecases,theconnectionsbetweenpendulumsandotherphenomenaare

soexactthatthesameequationscanberecycledwithoutchange.Onlythesymbolsneedtobereinterpreted;thesyntaxstaysthesame.It’sasifnaturekeepsreturningtothesamemotifagainandagain,apendularrepetitionofapendulartheme.Forexample,theequationsfortheswingingofapendulumcarryoverwithoutchangetothoseforthespinningofgeneratorsthatproducealternatingcurrentandsendittoourhomesandoffices.Inhonorofthat

pedigree,electricalengineersrefertotheirgeneratorequationsasswingequations.Thesameequationspopupyetagain,Zelig-like,inthequantumoscillations

ofahigh-techdevicethat’sbillionsoftimesfasterandmillionsoftimessmallerthananygeneratororgrandfatherclock.In1962BrianJosephson,thenatwenty-two-year-oldgraduatestudentattheUniversityofCambridge,predictedthatattemperaturesclosetoabsolutezero,pairsofsuperconductingelectronscouldtunnelbackandforththroughanimpenetrableinsulatingbarrier,anonsensicalstatementaccordingtoclassicalphysics.Yetcalculusandquantummechanicssummonedthesependulum-likeoscillationsintoexistence—or,toputitlessmystically,theyrevealedthepossibilityoftheiroccurrence.TwoyearsafterJosephsonpredictedtheseghostlyoscillations,theconditionsneededtoconjurethemweresetupinthelaboratoryand,indeed,theretheywere.TheresultingdeviceisnowcalledaJosephsonjunction.Itspracticalusesarelegion.Itcandetectultra-faintmagneticfieldsahundredbilliontimesweakerthanthatoftheEarth,whichhelpsgeophysicistshuntforoildeepunderground.NeurosurgeonsusearraysofhundredsofJosephsonjunctionstopinpointthesitesofbraintumorsandlocatetheseizure-causinglesionsinpatientswithepilepsy.Theproceduresareentirelynoninvasive,unlikeexploratorysurgery.Theyworkbymappingthesubtlevariationsinmagneticfieldproducedbyabnormalelectricalpathwaysinthebrain.Josephsonjunctionscouldalsoprovidethebasisforextremelyfastchipsinthenextgenerationofcomputersandmightevenplayaroleinquantumcomputation,whichwillrevolutionizecomputerscienceifitevercomestopass.Pendulumsalsogavehumanitythefirstwaytokeeptimeaccurately.Until

pendulumclockscamealong,thebestclockswerepitiful.Theywouldloseorgainfifteenminutesaday,evenunderidealconditions.Pendulumclockscouldbemadeahundredtimesmoreaccuratethanthat.TheyofferedthefirstrealhopeofsolvingthegreatesttechnologicalchallengeofGalileo’sera:findingawaytodeterminelongitudeatsea.Unlikelatitude,whichcanbeascertainedbylookingatthesunorthestars,longitudehasnocounterpartinthephysicalenvironment.Itisanartificial,arbitraryconstruct.Buttheproblemofmeasuringitwasreal.Intheageofexploration,sailorstooktotheoceanstowagewarorconducttrade,buttheyoftenlosttheirwayorranagroundbecauseofconfusionaboutwheretheywere.ThegovernmentsofPortugal,Spain,England,andHollandofferedvastrewardstoanyonewhocouldsolvethelongitudeproblem.Itwasachallengeofthegravestconcern.WhenGalileowastryingtodeviseapendulumclockinhislastyearoflife,he

hadthelongitudeproblemfirmlyinmind.Heknew,asscientistshadknown

sincethe1500s,thatthelongitudeproblemcouldbesolvedifonehadaveryaccurateclock.Anavigatorcouldsettheclockathisportofdepartureandcarryhishometimeouttosea.Todeterminetheship’slongitudeasittraveledeastorwest,thenavigatorcouldconsulttheclockattheexactmomentoflocalnoon,whenthesunwashighestinthesky.SincetheEarthspinsthrough360degreesoflongitudeinatwenty-four-hourday,eachhourofdiscrepancybetweenlocaltimeandhometimecorrespondsto15degreesoflongitude.Intermsofdistance,15degreestranslatestoawhoppingonethousandmilesattheequator.Soforthisschemetohaveanyhopeofguidingashiptoitsdesireddestination,giveortakeafewmilesoftolerableerror,aclockhadtoruntruetowithinafewsecondsaday.Andithadtomaintainthisunwaveringaccuracyinthefaceofheavingseasandviolentfluctuationsinairpressure,temperature,salinity,andhumidity,factorsthatcouldrustaclock’sgears,stretchitssprings,orthickenitslubricants,causingittospeedup,slowdown,orstop.Galileodiedbeforehecouldbuildhisclockanduseittotacklethelongitude

problem.ChristiaanHuygenspresentedhispendulumclockstotheRoyalSocietyofLondonasapossiblesolution,buttheywerejudgedunsatisfactorybecausetheyweretoosensitivetodisturbancesintheirenvironment.Huygenslaterinventedamarinechronometerwhoseticktockoscillationswereregulatedbyabalancewheelandaspiralspringinsteadofapendulum,aninnovativedesignthatpavedthewayforpocketwatchesandmodernwristwatches.Intheend,however,thelongitudeproblemwassolvedbyanewkindofclock,developedinthemid-1700sbyJohnHarrison,anEnglishmanwithnoformaleducation.Whentestedatseainthe1760s,hisH4chronometertrackedlongitudetoanaccuracyoftenmiles,sufficienttowintheBritishParliament’sprizeoftwentythousandpounds(equivalenttoafewmilliondollarstoday).Inourownera,thechallengeofnavigatingonEarthstillreliesontheprecise

measurementoftime.Considertheglobalpositioningsystem.Justasmechanicalclockswerethekeytothelongitudeproblem,atomicclocksarethekeytopinpointingthelocationofanythingonEarthtowithinafewmeters.Anatomicclockisamodern-dayversionofGalileo’spendulumclock.Likeitsforebear,itkeepstimebycountingoscillations,butinsteadoftrackingthemovementsofapendulumbobswingingbackandforth,anatomicclockcountstheoscillationsofcesiumatomsastheyswitchbackandforthbetweentwooftheirenergystates,somethingtheydo9,192,631,770timespersecond.Thoughthemechanismisdifferent,theprincipleisthesame.Repetitivemotion,backandforth,canbeusedtokeeptime.Andtime,inturn,candetermineyourlocation.WhenyouusetheGPSinyour

phoneorcar,yourdevicereceiveswirelesssignalsfromatleastfourofthe

twenty-foursatellitesintheglobalpositioningsystemthatareorbitingabouttwelvethousandmilesoverhead.Eachsatellitecarriesfouratomicclocksthataresynchronizedtowithinabillionthofasecondofoneanother.Thevarioussatellitesvisibletoyourreceiversenditacontinuousstreamofsignals,eachofwhichistime-stampedtothenanosecond.That’swheretheatomicclockscomein.Theirtremendoustemporalprecisiongetsconvertedintothetremendousspatialprecisionwe’vecometoexpectfromGPS.Thecalculationreliesontriangulation,anancientgeolocationtechniquebased

ongeometry.ForGPS,itworkslikethis:Whenthesignalsfromthefoursatellitesarriveatthereceiver,yourGPSgadgetcomparesthetimetheywerereceivedtothetimetheyweretransmitted.Thosefourtimesareallslightlydifferent,becausethesatellitesareatfourdifferentdistancesawayfromyou.YourGPSdevicemultipliesthosefourtinytimedifferencesbythespeedoflighttocalculatehowfarawayyouarefromthefoursatellitesoverhead.Becausethepositionsofthesatellitesareknownandcontrolledextremelyaccurately,yourGPSreceivercanthentriangulatethosefourdistancestodeterminewhereitisonthesurfaceoftheEarth.Itcanalsofigureoutitselevationandspeed.Inessence,GPSconvertsveryprecisemeasurementsoftimeintoveryprecisemeasurementsofdistanceandtherebyintoveryprecisemeasurementsoflocationandmotion.TheglobalpositioningsystemwasdevelopedbytheUSmilitaryduringthe

ColdWar.TheoriginalintentwastokeeptrackofUSsubmarinescarryingnuclearmissilesandgivethempreciseestimatesoftheircurrentlocationssothatiftheyneededtolaunchanuclearstrike,theycouldtargettheirintercontinentalballisticmissilesveryaccurately.PeacetimeapplicationsofGPSnowadaysincludeprecisionfarming,blindlandingsofairplanesinheavyfog,andenhanced911systemsthatautomaticallycalculatethefastestroutesforambulancesandfiretrucks.ButGPSismorethanalocationandguidancesystem.Itallowstime

synchronizationtowithinahundrednanoseconds,whichisusefulforcoordinatingbanktransfersandotherfinancialtransactions.Italsokeepswirelessphoneanddatanetworksinsync,allowingthemtosharethefrequenciesintheelectromagneticspectrummoreefficiently.I’vegoneintoallthisdetailbecauseGPSisaprimeexampleofthehidden

usefulnessofcalculus.Asissooftenthecase,calculusoperatesquietlybehindthescenesofourdailylives.InthecaseofGPS,almosteveryaspectofthefunctioningofthesystemdependsoncalculus.Thinkaboutthewirelesscommunicationbetweensatellitesandreceivers;calculuspredictedtheelectromagneticwavesthatmakewirelesspossiblethroughtheworkofMaxwell

thatwediscussedearlier.Withoutcalculus,there’dbenowirelessandnoGPS.Likewise,theatomicclocksontheGPSsatellitesusethequantummechanicalvibrationsofcesiumatoms;calculusunderpinstheequationsofquantummechanicsandthemethodsforsolvingthem.Withoutcalculus,there’dbenoatomicclocks.Icouldgoon—calculusunderliesthemathematicalmethodsforcalculatingthetrajectoriesofthesatellitesandcontrollingtheirlocationsandforincorporatingEinstein’srelativisticcorrectionstothetimemeasuredbyatomicclocksastheymoveathighspeedsandinweakgravitationalfields—butIhopethemainpointisclear.Calculusenabledthecreationofmuchofwhatmadetheglobalpositioningsystempossible.Calculusdidn’tdoitonitsown,ofcourse.Itwasasupportingplayer,butanimportantone.Alongwithelectricalengineering,quantumphysics,aerospaceengineering,andalltherest,calculuswasanindispensablepartoftheteam.Solet’sreturntoyoungGalileosittingintheCathedralofPisaponderingthat

chandelierswingingbackandforth.Wecanseenowthathisidlethoughtsaboutpendulumsandtheequaltimesoftheirswingshadanoutsizeimpactonthecourseofcivilization,notjustinhisownerabutinourown.

KeplerandtheMysteryofPlanetaryMotion

WhatGalileodidforthemotionofobjectsonEarth,JohannesKeplerdidforthemotionoftheplanetsintheheavens.HesolvedtheancientriddleofplanetarymotionandfulfilledthePythagoreandreambyshowingthatthesolarsystemwasruledbyakindofcelestialharmony.LikePythagoraswithhispluckedstringsandGalileowithhispendulums,projectiles,andfallingbodies,Keplerdiscoveredthatplanetarymotionsfollowmathematicalpatterns.AndlikeGalileo,hewasenthralledbythepatternsheglimpsedandyetfrustratedthathecouldn’texplainthem.AlsolikeGalileo,Keplerwasbornintoafamilyonthewaydown.Buthis

circumstanceswerefarworse.Hisfatherwasadrunkenmercenarysoldier,“criminallyinclined,”asKeplerrecalled,andhismotherwas(perhapsunderstandably)“bad-tempered.”Ontopofthat,Keplercontractedsmallpoxasachildandnearlydiedfromit.Hishandsandvisionwerepermanentlydamaged,whichmeanthecouldneverhaveaphysicallystrenuousjobasanadult.Fortunately,hewasbright.Asateenagerhelearnedmathematicsand

CopernicanastronomyatTübingen,wherehewasrecognizedashaving“suchasuperiorandmagnificentmindthatsomethingspecialmaybeexpectedofhim.”

Afterreceivinghismaster’sdegreein1591,KeplerstudiedtheologyatTübingenandplannedtobecomeaLutheranminister.ButwhenamathteacherattheLutheranschoolinGrazdiedandthechurchauthoritiescalledforasubstitute,Keplerwaschosen,andhereluctantlygaveuptheideaofalifeintheclergy.Nowadays,allstudentsofphysicsandastronomylearnaboutKepler’sthree

lawsofplanetarymotion.Whatisoftenleftoutisthestoryofhisagonizing,almostfanaticalstruggletouncoverthoselaws.Hespentdecadestoiling,searchingforregularities,propelledbymysticismandhisfaiththattherehadtobesomedivineorderinthenightlypositionsofMercury,Venus,Mars,Jupiter,andSaturn.AyearafterhisarrivalinGraz,asecretofthecosmoswasrevealedtohim,he

believed.Onedaywhileteachinghisclass,hesuddenlyhadavisionofhowtheplanetsmustarrangethemselvesaroundthesun.Theideawasthattheplanetswerecarriedbycelestialspheresnestedinsideoneanother,likeRussiandolls,withthedistancesbetweenthemdictatedbythefivePlatonicsolids:thecube,tetrahedron,octahedron,icosahedron,anddodecahedron.PlatohadknownandEuclidhadprovedthatnootherthree-dimensionalshapescouldbebuiltfromidenticalregularpolygons.ToKepler,theiruniquenessandsymmetryseemedfitforeternity.Heperformedhiscalculationsintensely,feverishly.“DayandnightIwas

consumedbythecomputing,toseewhetherthisideawouldagreewiththeCopernicanorbits,orifmyjoywouldbecarriedawaybythewind.Withinafewdayseverythingworked,andIwatchedasonebodyafteranotherfitpreciselyintoitsplaceamongtheplanets.”HecircumscribedanoctahedronaboutthecelestialsphereofMercuryand

placedthesphereofVenusthroughitscorners.ThenhecircumscribedanicosahedronaboutthesphereofVenusandplacedthesphereofEarththroughitscorners,andsoonwiththeotherplanets,interlockingthecelestialspheresandPlatonicsolidslikeathree-dimensionalpuzzle.HedepictedtheresultingsysteminacutawaydrawinginhisCosmicMysteryof1596.

Hisepiphanyexplainedsomuch.JustastherewereonlyfivePlatonicsolids,

therewereonlysixplanets(includingtheEarth)andhencefivegapsbetweenthem.Everythingmadesense.Geometryruledthecosmos.Hehadwantedtobecomeatheologian,andnowhecouldwritewithsatisfactiontooneofhismentors,“BeholdhowthroughmyeffortGodisbeingcelebratedinastronomy.”Actually,thetheorydidn’tquitematchthedata,particularlyasregardsthe

positionsofMercuryandJupiter.Thatmismatchmeantsomethingwaswrong,butwhatwasit—histheory,thedata,orboth?Keplersuspectedthedatamightbewrong,buthedidn’tinsistonthecorrectnessofhistheory(whichwaswise,inretrospect,sincethetheoryhadnochanceofsuccess;aswenowknow,therearemorethansixplanets).Nevertheless,hedidn’tgiveup.Hecontinuedtopondertheplanetsandsoon

gotabreakwhenTychoBraheaskedhimtobehisassistant.Tycho(ashistoriansalwayscallhim)wastheworld’sbestobservationalastronomer.His

dataweretentimesmoreaccuratethananyobtainedpreviously.Inthedaysbeforetheinventionofthetelescope,he’ddevisedspecialinstrumentsthatallowedhim,withthenakedeye,toresolvetheangularpositionsoftheplanetstowithintwoarcminutes.That’sone-thirtiethofadegree.Togetasenseofwhatatinyanglethisis,imaginelookingupatthefullmoon

onaclearnightwhileholdingyourlittlefingerallthewayoutinfrontofyourface.Yourlittlefingerturnsouttobeaboutsixtyarcminuteswide,andthemoonisabouthalfthat.SowhenwesayTychocouldresolvetwominutesofarc,thatmeansifyoudrewthirtyevenlyspaceddotsacrossthewidthofyourlittlefinger(orfifteenacrossthemoon),Tychocouldseethedifferencebetweenonedotandthenext.AfterTychodied,in1601,KeplerinheritedhistroveofdataonMarsandthe

otherplanets.Toexplaintheirmotion,hetriedonetheoryafteranother,allowingtheplanetstomoveinepicycles,invariousegg-shapedorbits,andineccentriccircleswiththesunslightlyoffcenter.ButallproduceddiscrepancieswithTycho’sdatathatcouldn’tbeignored.“Dearreader,”helamentedafteronesuchcalculation,“ifyouaretiredbythistediousprocedure,takepityonme,forIcarrieditoutatleast70times.”

Kepler’sFirstLaw:EllipticalOrbits

Inhissearchtoexplainthemotionsoftheplanets,Keplereventuallytriedawell-knowncurvecalledanellipse.LikeGalileo’sparabola,ellipseshadbeenstudiedinantiquity.Aswesawinchapter2,theancientGreekshaddefinedellipsesastheoval-shapedcurvesformedbycuttingthroughaconewithaplaneatashallowangle,lesssteepthantheslopeoftheconicalsurfaceitself.Ifthetiltofthecuttingplaneisshallow,theresultingellipseisalmostcircular.Attheotherextreme,ifthetiltoftheplaneisonlyslightlylessthanthetiltoftheconicalsurface,theellipseisverylongandthin,liketheshapeofacigar.Ifyouadjustthetiltoftheplane,anellipsecanbemorphedfromveryroundtoverysquashedoranywhereinbetween.Anotherwaytodefineanellipseisindown-to-earthtermsandwiththehelp

ofafewhouseholditems.

Getapencil,acorkboard,asheetofpaper,twopushpins,andapieceofstring.Placethepaperonthecorkboard.Pintheendsofthestringdownthroughthepaper,makingsuretoleavesomeslackinthestring.Thenpullthestringtautwiththepencilandbegindrawingacurve,keepingthestringtautasyoumovethepencil.Afterthepencilhasgonearoundbothpinsandreturnedtoitsstartingpoint,theresultingclosedcurveisanellipse.Thelocationsofthepinsplayaspecialrolehere.Keplernamedthemthefoci,

orfocalpoints,oftheellipse.Theyareasmeaningfultoanellipseasthecenteristoacircle.Acircleisdefinedasasetofpointswhosedistancefromagivenpoint(itscenter)isconstant.Likewise,anellipseisasetofpointswhosecombineddistancefromtwogivenpoints(itsfoci)isconstant.Inthestring-and-pushpinconstruction,thatconstantcombineddistanceispreciselythelengthoftheloosestringbetweenthepins.Kepler’sfirstgreatdiscovery—andthistimehereallydidgetitrightand

didn’tneedtorevisehisideas—isthatalltheplanetsmoveinellipticalorbits.Notcirclesorcirclescompoundedwithcircularepicycles,asAristotle,Ptolemy,Copernicus,andevenGalileohadthought.No.Ellipses.Moreover,hefoundthatforeveryplanet,thesunwaslocatedatoneofthefocioftheplanet’sellipticalorbit.Itwasastonishing,justthesortofholyclueKeplerhadbeenhopingfor.The

planetsweremovinginaccordancewithgeometry.Ithadn’tturnedouttobethegeometryofthefivePlatonicsolidsashe’doriginallyguessed,buthisinstinctshadbeenrightnonetheless.Geometrydidruletheheavens.

Kepler’sSecondLaw:EqualAreasinEqualTimes

Keplerfoundanotherregularityinthedata.Whereasthefirstonewasaboutthepathsoftheplanets,thisonewasabouttheirspeeds.KnowntodayasKepler’ssecondlaw,itsaysthatanimaginarylinedrawnfromaplanettothesunsweepsoutequalareasinequalintervalsoftimeastheplanetgoesaroundinitsorbit.Toclarifywhatthislawmeans,supposewelookatwhereMarsistonightin

itsellipticalorbit.Connectthatpointtothesunwithastraightline.

Nowthinkofthislineasbeingsomethinglikethebladeofawindshieldwiper

withthesunatthepivotpointandMarsatthetipofthewiper(exceptthewiperdoesn’toscillatebackandforthlikearealwindshieldwiper;italwaysadvances,anditdoessovery,veryslowly).AsMarstravelsforwardinitsorbitonsubsequentnights,thewipermovesalongwithitandtherebysweepsoutanareainsidetheellipse.IfwelookatMarsagainsometimelater,sayafterthreeweeks,theslow-movingwiperwillhavesweptoutashapecalledasector.

WhatKeplerdiscoveredisthattheareaofathree-weeksectoralwaysstays

thesamenomatterwhereMarshappenstobeinitsorbitaroundthesun.Andthere’snothingspecialaboutthreeweeks.IfwelookatMarsatanytwopointsinitsorbitseparatedbyequalamountsoftime,theresultingsectorswillalwayshaveequalareas,nomatterwheretheyareintheorbit.Inanutshell,thesecondlawsaysthattheplanetsdonotmoveataconstant

speed.Instead,theclosertheygettothesun,thefastertheymove.Thestatementaboutequalareasinequaltimesisawayofmakingthisprecise.

Iftime(P1→P2)=time(P3→P4),theirsectorshaveequalareas.

HowdidKeplermeasuretheareaofanellipticalsector,giventhatithada

curvedside?HedidwhatArchimedeswouldhavedone.Heslicedthesectorintolotsofthinsliversandapproximatedthemwithtriangles.Nexthecomputedtheareasofthetriangles(easy,becausealltheirsidesarestraight)andaddedthemtogether,integratingthemtoestimatetheareaoftheoriginalsector.Ineffect,heusedanArchimedeanversionofintegralcalculusandappliedittorealdata.

Kepler’sThirdLawandtheSacredFrenzy

Thelawsthatwe’vediscussedsofar—eachplanetmovesinanellipsewiththesunatafocus,andeachplanetsweepsoutequalareasinequaltimes—areabouttheplanetsindividually.Keplerdiscoveredboththeselawsin1609.Incontrast,ittookhimanothertenyearstodiscoverhisthirdlaw,whichisaboutalltheplanetscollectively.Itbindsthewholesolarsystemintoasinglenumerologicalpattern.Itcametohimaftermonthsoffuriouslyrenewedcalculationsandmorethan

twentyyearsafterhisagonizingnearmisswiththePlatonicsolids.IntheprefacetoHarmoniesoftheWorld(1619),hewroteinecstasyaboutfinallyseeingthepatterninGod’splan:“Now,sincethedawneightmonthsago,sincethebroaddaylightthreemonthsago,andsinceafewdaysago,whenthefullsunilluminatedmywonderfulspeculations,nothingholdsmeback.Iyieldfreelytothesacredfrenzy.”ThenumerologicalpatternthatenrapturedKeplerwashisdiscoverythatthe

squareoftheperiodofrevolutionofaplanetisproportionaltothecubeofitsaveragedistancefromthesun.Equivalently,thenumberT2/a3isthesameforalltheplanets.Here,Tmeasureshowlongittakesaplanettogoaroundthesunonce(ayearfortheEarth,1.9yearsforMars,11.9yearsforJupiter,andsoon),whileameasureshowfarawaytheplanetisfromthesun.That’sabittrickytodefine,becausetheactualdistancechangesfromweektoweekasaplanetmovesinitsellipticalorbit;sometimesit’sclosertothesunandsometimesit’sfartheraway.Toaccountforthiseffect,Keplerdefinedaastheaverageoftheplanet’snearestandfarthestdistancestothesun.Thegistofthethirdlawissimple:Thefartheraplanetisfromthesun,the

sloweritmovesandthelongerittakestocompleteitsorbit.Butwhat’sinterestingandsubtleaboutthislawisthattheorbitalperiodisnotsimply

proportionaltotheorbitaldistance.Forexample,ournearestneighbor,Venus,hasaperiodthat’s61.5percentaslongasouryear,yetitsaveragedistancefromthesunis72.3percentofours(not61.5percent,asonemightnaivelyexpect).That’sbecauseperiodsquaredisproportionaltodistancecubed(notsquared),andsotherelationshipbetweenperiodanddistanceismorecomplicatedthanadirectproportion.WhenTandaareexpressedaspercentagesofEarth-yearsandEarth-

distances,asabove,Kepler’sthirdlawsimplifiestoT2=a3.Itbecomesanequationinsteadofamereproportionality.Toseehowwellitworks,pluginthenumbersforVenus:T2=(0.615)2≈0.378,whereasa3=(0.723)3≈0.378.Sothelawholdstothreesignificantfigures.That’swhatgotKeplersoexcited.It’sequallyimpressivewhenappliedtotheotherplanets.

KeplerandGalileo,theSameandNottheSame

KeplerandGalileonevermet,buttheycorrespondedabouttheirCopernicanviewsandthediscoveriestheyweremakinginastronomy.WhensomepeoplerefusedtolookthroughGalileo’stelescope,fearingtheinstrumentwastheworkofthedevil,GalileowrotetoKeplerinatoneofamusedresignation:“MydearKepler,Iwishwecouldlaughattheextraordinarystupidityofthemob.WhatsayyouabouttheforemostphilosophersofthisUniversity,whowiththeobstinacyofastuffedsnake,anddespitemyattemptsandinvitationsathousandtimestheyhaverefusedtolookattheplanets,orthemoon,ormytelescope?”Insomeways,KeplerandGalileowerealike.Bothwerefascinatedby

motion.Bothworkedonintegralcalculus,Kepleronthevolumesofcurvedshapes,likewinebarrels,Galileooncentersofgravityofparaboloids.InthistheychanneledthespiritofArchimedes,carvingsolidobjectsintheirmindsintoimaginarythinwafers,likesomanyslicesofsalami.Yetinotherways,theywerecomplementarytoeachother.Mostobviously,

theywerecomplementaryintheirgreatestscientificcontributions,GalileoforthelawsofmotiononEarth,Keplerforthelawsofmotioninthesolarsystem.Butthecomplementaritygoesdeeper,downtoscientificstyleanddisposition.WhereGalileowasrational,Keplerwasmystical.GalileowastheintellectualdescendantofArchimedes,entrancedby

mechanics.Inhisfirstpublication,hegavethefirstplausibleaccountofthe“Eureka!”legendbyshowinghowArchimedescouldhaveusedabalanceanda

bathtubtodeterminethatKingHiero’scrownwasnotmadeofpuregoldandtocalculatethepreciseamountofsilverthatthethievinggoldsmithhadmixedin.GalileocontinuedtoelaborateonArchimedes’sworkthroughouthiscareer,oftenbyextendinghismechanicsfromequilibriumtomotion.Kepler,however,wasmoretheheirtoPythagoras.Fiercelyimaginativeand

withanumerologicalcastofmind,hesawpatternseverywhere.Hegaveusthefirstexplanationforwhysnowflakesformsix-corneredshapes.Heponderedthemostefficientwaytopackcannonballs,andguessed(correctly)thattheoptimalpackingarrangementisthesameonethatnatureusestopackpomegranateseedsandthatgrocersusetostackoranges.Kepler’sobsessionwithgeometry,bothsacredandprofane,vergedontheirrational.Buthisfervormadehimwhohewas.AsthewriterArthurKoestlerastutelyobserved,“JohannesKeplerbecameenamoredwiththePythagoreandream,andonthisfoundationoffantasy,bymethodsofreasoningequallyunsound,builtthesolidedificeofmodernastronomy.Itisoneofthemostastonishingepisodesinthehistoryofthought,andanantidotetothepiousbeliefthattheProgressofScienceisgovernedbylogic.”

StormCloudsGathering

Likeallgreatdiscoveries,Kepler’slawsofplanetarymotionintheheavensandGalileo’slawsoffallingbodiesonEarthraisedmanymorequestionsthantheyanswered.Onthescientificside,itwasnaturaltoaskaboutultimatecauses.Wheredidthelawscomefrom?Didadeepertruthunderliethem?Forexample,itseemedtoocoincidentalthatthesunoccupiedsuchaspecialpositioninalltheplanetaryellipses,alwaysresidingatafocus.Didthatmeanthesunwasaffectingtheplanetssomehow?Influencingthemthroughsomekindofoccultforce?Keplerthoughtso.Hewonderedifmagneticemanations,recentlystudiedbyWilliamGilbertinEngland,mightbepullingontheplanets.Whateveritwas,anunknown,invisibleforceseemedtobeactingatgreatdistancesacrosstheemptinessofspace.TheworkofGalileoandKepleralsoraisedquestionsformathematics.In

particular,curveswerebackinthelimelight.Galileohadshownthatthearcofaprojectilewasaparabola,andAristotle’scircleshadnowgivenwaytoKepler’sellipses.Otherscientificandtechnologicaladvancesoftheearly1600sonlyheightenedtheinterestincurves.Inoptics,theshapeofacurvedlensdeterminedhowmuchanimagewasmagnified,ordistorted,orblurred.Those

werevitalconsiderationsforthedesignoftelescopesandmicroscopes,thehotnewinstrumentsthatwererevolutionizingastronomyandbiology,respectively.TheFrenchpolymathRenéDescartesasked:Couldalensbedesignedtobefreeofallblurring?Itamountedtoaquestionaboutcurves:Whatcurvedshapewouldalensneedtohavesothatalltheraysoflightemanatingfromasinglepointortravelingparalleltooneanotherwouldbeguaranteedtoconvergeatanotheruniquepointafterpassingthroughthelens?Curves,inturn,raisedquestionsaboutmotion.Kepler’ssecondlawimplied

thattheplanetsmovednonuniformlyaroundtheirellipses,sometimeshesitating,sometimesaccelerating.Likewise,Galileo’sprojectilesmovedatever-changingspeedsontheirparabolicarcs.Theysloweddownastheyclimbed,pausedatthetop,thenspedupastheyfellbacktoearth.Thesamewastrueforpendulums.Theysloweddownastheyclimbedtotheendsoftheirarcs,reversedandspedupastheyswungthroughthebottom,thensloweddownonceagainattheotherextreme.Howcouldonequantifymotionsinwhichspeedchangedfrommomenttomoment?Amidthisswirlofquestions,aninfluxofideasfromIslamicandIndian

mathematicsofferedEuropeanmathematiciansanewwayforward,achancetogobeyondArchimedesandbreaknewground.TheideasfromtheEastwouldleadtofreshwaysofthinkingaboutmotionandcurvesandthen,withathunderclap,todifferentialcalculus.

4

TheDawnofDifferentialCalculus

FROMAMODERNperspective,therearetwosidestocalculus.Differentialcalculuscutscomplicatedproblemsintoinfinitelymanysimplerpieces.Integralcalculusputsthepiecesbacktogetheragaintosolvetheoriginalproblem.Giventhatcuttingcomesnaturallybeforerebuilding,itseemssensiblefora

novicetolearndifferentialcalculusfirst.Andindeed,that’showallcalculuscoursesbegintoday.Theystartwithderivatives—therelativelyeasytechniquesforslicinganddicing—andthenworktheirwayuptointegrals,themuchhardertechniquesforreassemblingthepiecesintoanintegratedwhole.Studentsfinditmorecomfortabletolearncalculusinthisorderbecausetheeasiermaterialcomesfirst.Theirteacherslikeitbecausethesubjectseemsmorelogicalthisway.Yet,strangelyenough,historyunfoldedintheoppositeorder.Integralswere

alreadyinfullswinginancientGreeceinArchimedes’sworkaround250BCE,whereasderivativesweren’tevenagleaminanybody’seyeuntilthe1600s.Whydiddifferentialcalculus—theeasiersideofthesubject—developsomuchlaterthanintegralcalculus?It’sbecausedifferentialcalculusgrewoutofalgebra,andalgebratookcenturiestomature,migrate,andmutate.InitsoriginalforminChina,India,andtheIslamicworld,algebrawasentirelyverbal.Unknownswerewords,nottoday’sxandy.Equationsweresentences,andproblemswere

paragraphs.ButsoonafteralgebraarrivedinEurope,around1200,itevolvedintoanartofsymbols.Thatmadealgebramoreabstract...andmorepowerful.Thisnewbreed,symbolicalgebra,thencoupledwithgeometryandspawnedanevenstrongerhybrid,analyticgeometry,whichinturnbegatazooofnewcurves,thestudyofwhichledthewaytodifferentialcalculus.Thischapterexploreshowthathappened.

TheRiseofAlgebraintheEast

ThementionofChina,India,andtheIslamicworldshouldcorrecttheimpressionImayhavegivensofarthatthecreationofcalculuswasaEurocentricaffair.AlthoughcalculusculminatedinEurope,itsrootslieelsewhere.Inparticular,algebracamefromAsiaandtheMiddleEast.ItsnamederivesfromtheArabicwordal-jabr,meaning“restoration”or“thereunionofbrokenparts.”Thesearethekindsofoperationsneededtobalanceequationsandsolvethem;forinstance,bysubtractinganumberfromonesideofanequationandaddingittotheother,ineffectrestoringwhatwasbroken.Likewise,geometry,aswe’veseen,wasborninancientEgypt;thefoundingfatherofGreekgeometry,Thales,issaidtohavelearnedthesubjectthere.Andthegreatesttheoremofgeometry,thePythagoreantheorem,didnotoriginatewithPythagoras;itwasknowntotheBabyloniansforatleastathousandyearsbeforehim,asevidencedbyexamplesofitonMesopotamianclaytabletsfromaround1800BCE.WeshouldalsokeepinmindthatwhenwespeakofancientGreece,weare

referringtoahugeswathofterritorythatreachedfarbeyondAthensandSparta.Atitslargest,itstretchedtoEgyptinthesouth,toItalyandSicilyinthewest,andeastacrosstheshoresoftheMediterraneantoTurkey,theMiddleEast,CentralAsia,andpartsofPakistanandIndia.PythagorashimselfwasfromSamos,anislandoffthewestcoastofAsiaMinor(nowTurkey).ArchimedeslivedinSyracuse,onthesoutheasterncoastofSicily.EuclidworkedinAlexandria,thegreatportandscholarlyhubatthemouthoftheNileinEgypt.AftertheRomansconqueredtheGreeks,andespeciallyafterthelibraryin

AlexandriawasburnedandthewesternRomanEmpirefell,thecenterofmathematicsswungbacktotheEast.ThewritingsofArchimedesandEuclidweretranslatedintoArabic,aswerethoseofPtolemy,Aristotle,andPlato.ScholarsandscribesinConstantinopleandBaghdadkepttheoldlearningaliveandaddedideasoftheirown.

HowAlgebraWaxedWhileGeometryWaned

Duringthosecenturiesbeforealgebraarrived,geometryslowedtoacrawl.AfterArchimedesdied,in212BCE,itseemedthatnobodycouldbeathimathisowngame.Well,almostnobody.Around250CE,theChinesegeometerLiuHuiimprovedonArchimedes’smethodforcalculatingpi.Twocenturieslater,ZuChongzhiappliedLiuHui’smethodtoapolygonwith24,576sides.Throughwhatmusthavebeenheroicfeatsofarithmetic,hetightenedtheviseonpitoeightdigits:

3.1415926<π<3.1415927.ThenextstepforwardtookanotherfivecenturiesandcamefromthesageAl-

HasanIbnal-Haytham,knowntoEuropeansasAlhazen.BorninBasra,Iraq,around965CE,heworkedinCairoduringtheIslamicgoldenageoneverythingfromtheologyandphilosophytoastronomyandmedicine.Inhisworkongeometry,Ibnal-HaythamcalculatedvolumesofsolidsthatArchimedesneverconsidered.Still,impressiveastheseadvanceswere,theywereraresignsoflifeforgeometry,andtheytooktwelvecenturiestooccur.Duringthatsamelongspanoftime,rapidandsubstantialadvanceswerebeing

madeinalgebraandarithmetic.Hindumathematiciansinventedtheconceptsofzeroandthedecimalplace-valuesystemfornumbers.AlgebraictechniquesforsolvingequationssprangupinEgypt,Iraq,Persia,andChina.Muchofthiswasdrivenbypracticalproblemsinvolvinginheritancelaw,taxassessment,commerce,bookkeeping,interestcalculations,andothertopicswellsuitedtonumbersandequations.Inthosedays,whenalgebrawasstillallaboutwordproblems,solutionsweregivenasrecipes,step-by-steproutestoanswers,aselucidatedinthefamoustextbookbyMuhammadIbnMusaal-Khwarizmi(c.780–850CE),whoselastnamelivesoninthestep-by-stepprocedurescalledalgorithms.Eventuallytraders,merchants,andexplorersbroughtthisverbalformofalgebraandHindu-ArabicdecimalswestwardtoEurope.Meanwhile,peoplestartedtranslatingArabictextsintoLatin.Thestudyofalgebrainitsownright,asasymbolicsystemapartfromits

applications,begantoflourishinRenaissanceEurope.Itreacheditspinnacleinthe1500s,whenitstartedtolooklikewhatweknowtoday,withlettersusedtorepresentnumbers.InFrancein1591,FrançoisViètedesignatedunknownquantitieswithvowels,likeAandE,andusedconsonants,likeBandG,forconstants.(Today’suseofx,y,zforunknownsanda,b,cforconstantscame

fromtheworkofRenéDescartesaboutfiftyyearslater.)Replacingwordswithlettersandsymbolsmadeitmucheasiertomanipulateequationsandfindsolutions.AnequallybigadvanceintherealmofarithmeticcamewhenSimonStevinin

HollandshowedhowtogeneralizeHindu-Arabicdecimalnumberstodecimalfractions.Insodoing,hedestroyedtheoldAristoteliandistinctionbetweennumbers(meaningwholenumbersofindivisibleunits)andmagnitudes(continuousquantitiesthatcouldbedividedinfinitelyintoarbitrarilysmallparts).BeforeStevin,decimalshadbeenappliedonlytothewhole-numberpartofaquantity,andanypartlessthanaunitwasexpressedasafraction.InStevin’snewapproach,evenaunitcouldbechoppedintopiecesandwrittenindecimalnotationbyplacingthecorrectdigitsafterthedecimalpoint.Itsoundssimpletousnow,butitwasarevolutionaryideathathelpedmakecalculuspossible.Oncetheunitwasnolongersacrosanctandindivisible,allquantities—whole,fractional,orirrational—coalescedintoonebigfamilyofnumbers,allonequalfooting.Thatgavecalculustheinfinitelypreciserealnumbersitneededtodescribethecontinuityofspace,time,motion,andchange.Justbeforegeometrypartneredwithalgebra,therewasonelasthurrahforthe

old-schoolgeometricmethodsofArchimedes.Atthebeginningoftheseventeenthcentury,Keplerfoundthevolumesofcurvedshapeslikewinebarrelsanddoughnut-shapedsolidsbyslicingtheminhismindintoaninfinitenumberofinfinitesimallythindisks,whileGalileoandhisstudentsEvangelistaTorricelliandBonaventuraCavalierisimilarlycomputedareas,volumes,andcentersofgravityofvariousshapesbytreatingthemasinfinitestacksoflinesandsurfaces.Becausethesemenhadadevil-may-careapproachtoinfinityandinfinitesimals,theirtechniqueswerenotrigorous,buttheywerepotentandintuitive.Theyproducedanswersmuchmoreeasilyandquicklythanthemethodofexhaustion,sothisseemedlikeanexcitingadvance(thoughwenowknowthatArchimedeshadbeatenthemtoit;thesameidealayhiddeninhistreatiseontheMethod,atthattimestilllanguishingundetectedinaprayerbookinamonastery,whereitwouldremainuntil1899).Atanyrate,althoughtheprogressmadebytheneo-Archimedeansseemed

promisingatthetime,thiscontinuationoftheoldapproachwasnotdestinedtocarrytheday.Symbolicalgebrawasnowwheretheactionwas.Andwithit,theseedsforitsmostvigorousoffshoots—analyticgeometryanddifferentialcalculus—werefinallyabouttobesown.

AlgebraMeetsGeometry

Thefirstbreakthroughcamearound1630whentwoFrenchmathematicians(andsoon-to-berivals),PierredeFermatandRenéDescartes,independentlylinkedalgebratogeometry.Theirworkcreatedanewkindofmathematics,analyticgeometry,whosecentraltheaterwasthexyplane,anarenawhereequationscamealiveandtookform.Weusethexyplanetodaytographrelationshipsbetweenvariables.For

example,considerthecaloricimplicationsofmyoccasionallydisgracefuleatinghabits.SometimesItreatmyselftoacoupleofslicesofcinnamon-raisinbreadforbreakfast.Onthepackageittellsmethateachslicepacksawhopping200calories.(IfIwantedtoeathealthier,Icouldalwayssettlefortheseven-grainbreadmywifebuys,withits130caloriesperslice,butforthisexample,Ipreferthecinnamon-raisinbreadbecause200isamorecongenialnumber,mathematicallyifnotnutritionally,than130.)Here’sagraphofhowmanycaloriesIconsumewhenIeatone,two,orthree

slicesofbread.

Sinceeachsliceamountsto200calories,twoslicesamountto400calories,andthreeto600calories.Whenplottedasdatapointsonthegraph,allthreepointsfallonthesamestraightline.Inthatsense,there’salinearrelationshipbetweencaloriesconsumedandnumberofsliceseaten.Ifweusetheletterxtorepresentthenumberofsliceseatenandyforthenumberofsinfulcaloriesingested,the

linearrelationshipcanbesummarizedasy=200x.Thisrelationshipalsoappliesbetweenthedatapoints.Forexample,oneandahalfslicesamountto300calories,andthecorrespondingdatapointfallsrightontheline.Soitmakessensetoconnectthedotsingraphslikethese.Irealizeallofthismightseemobvious,butthat’smypoint.Itwasn’talways

obvious.Itwasn’tobviousinthepast—someonehadtocomeupwiththeideatodepictrelationshipsonanabstractvisualchart—anditstillisn’tobvioustoday,atleastnottokidswhentheyfirstlearnaboutgraphslikethis.Thereareseveralimaginativeleapshere.Oneistouseapicturetorepresent

foodintake.Thatrequiresmentalflexibility.Thereisnothinginherentlypictorialaboutcalories.Thegraphwe’relookingatisnotaphotorealisticpaintingshowingraisinsandbrownswirlsofcinnamonembeddedinbread.Thegraphisanabstraction.Itallowsdifferentmathematicaldomainstointeractandcooperate:thedomainofnumbers,likenumbersofcaloriesorslicesofbread;thedomainofsymbolicrelationships,likey=200x;andthedomainofshapes,likedotslyingonastraightlineonagraphwithtwoperpendicularaxes.Throughthisconfluenceofideas,thehumblechartblendsnumbers,relationships,andshapesandhenceletsarithmeticandalgebramergeintogeometry.That’sthebigdealhere.Differentstreamsofmathematicshavebeenbroughttogetheraftercenturiesofrunningontheirseparatecourses.(RecallthattheancientGreekselevatedgeometryoverarithmeticandalgebraanddidn’tletthemmingle,atleastnotveryoften.)Anotherconfluencehereinvolvesthehorizontalandverticalaxes.Theyare

oftencalledthexandyaxes,namedforthevariablesweusetolabelthem.Theseaxesarenumberlines.Thinkaboutthatterm:numberlines.Numbersarebeingrepresentedaspointsonaline.Arithmeticisconsortingwithgeometry.Andthey’reminglingbeforeweevenplotanydata!TheancientGreekswouldhavescreamedbloodymurderatthatbreachof

protocol.Tothem,numbersmeantexclusivelydiscretequantities,likewholenumbersandfractions.Bycontrast,continuousquantitiesofthesortmeasuredbythelengthofalinewereregardedasmagnitudes,aconceptuallydistinctcategoryfromnumbers.SoforthenearlytwothousandyearsfromArchimedestothebeginningoftheseventeenthcentury,numberswereabsolutelynotseenasequivalenttothecontinuumofpointsonaline.Inthissense,theideaofanumberlinewasradicallytransgressive.Nowadayswedon’tgiveitasecondthought.Weexpectelementary-schoolchildrentounderstandthatnumberscanberepresentedvisuallyinthisway.Furtherblasphemyhere,fromthestandpointoftheancientGreeks,isthe

graph’sutterdisregardforcomparinglikewithlike,appleswithapplesor

calorieswithcalories.Instead,thegraphshowscaloriesononeaxisandslicesontheother.Theyarenotdirectlycomparable.Andyetwedon’tblinkaneyeatmakingsuchcomparisonstodaywhenwedrawgraphslikethis.Wesimplyconvertcaloriesandslicestonumbers,meaningrealnumbers,infinitedecimals,theuniversalcurrencyofcontinuousmathematics.TheGreeksdrewsharpdistinctionsbetweenlengths,areas,andvolumes,butthey’realljustrealnumberstous.

EquationsasCurves

Tobesure,FermatandDescartesneverusedthexyplanetostudyanythingastangibleascinnamon-raisinbread.Forthem,thexyplanewasatooltostudypuregeometry.Workingseparately,theyeachnoticedthatanylinearequation(meaningan

equationinwhichxandyappeartothefirstpoweronly)producedastraightlineonthexyplane.Thisconnectionbetweenlinearequationsandlinessuggestedthepossibilityofadeeperconnection,onebetweennonlinearequationsandcurves.Inalinearequationlikey=200x,thevariablesxandyappearontheirown,unadulterated,anddonotgetsquaredorcubedorraisedtoanyhigherpower.FermatandDescartesrealizedtheycouldplaythesamegamewithotherpowersandotherequations.Theycouldcookupanyequationtheydesiredanddowhatevertheywantedtoxandy—squareoneofthem,cubetheother,multiplythemtogether,addthem,whatever—andtheninterprettheresultasacurve.Withanyluck,itwouldbeaninterestingcurve,maybeonethatnobodyhadeverimagined,maybeonethatArchimedeshadneverstudied.Anyequationwithxandyinitwasanewadventure.Itwasalsoagestaltswitch.Insteadofstartingwithacurve,youstartwithanequationandseewhatkindofcurveitmakes.Letalgebradrive,andputgeometryinthebackseat.FermatandDescartesbeganbylookingatquadraticequations.Theseare

equationsinwhich,alongwiththeusualconstants(like200)andlinearterms(likexandy),thevariablescanalsogetsquaredormultipliedtogether,creatingquadratictermslikex2,y2,andxy.(InLatin,quadratusmeans“square.”)Squaredquantitieshadtraditionallybeeninterpretedastheareasofsquareregions.Thus,x2meanttheareaofanx-by-xsquare.Intheolddays,anareawasseenasafundamentallydifferentkindofquantityfromalengthoravolume.ButtoFermatandDescartes,x2wasjustanotherrealnumber,whichmeantit

couldbegraphedonanumberline,justasxorx3oranyotherpowerofxcouldbe.Today,studentsinhigh-schoolalgebraareroutinelyexpectedtobeableto

graphequationslikey=x2,whoseassociatedcurveturnsouttobeaparabola.Remarkably,allotherequationsinvolvingquadratictermsinxandybutnohigherpowersgivecurvesofjustfourpossibletypes:parabolas,ellipses,hyperbolas,orcircles.Andthat’sit.(Exceptforsomedegeneratecasesthatyieldlines,points,ornographatall,butthesearerareodditiesthatwecansafelyignore.)Forexample,thequadraticequationxy=1givesahyperbola,whilex2+y2=4isacircleandx2+2y2=4isanellipse.Evenaquadraticasbeastlyasx2+2xy+y2+x+3y=2hastobeoneofthefourpossibilitiesabove.Itturnsouttobeaparabola.

FermatandDescarteswerethefirsttodiscoverthiswonderfulcoincidence:

Thequadraticequationsinxandyarethealgebraiccounterpartsoftheconic

sectionsoftheGreeks,thefourkindsofcurvesobtainedbyslicingthroughaconeatdifferentangles.Here,inFermat’sandDescartes’snewarena,classicalcurveswerereappearinglikeghostsfromthemist.

BetterTogether

Thenewfoundtiesbetweenalgebraandgeometrywereaboontobothsubjects.Eachcouldhelptheothercompensateforitsdeficits.Geometryappealedtotherightsideofthebrain.Itwasintuitiveandvisual,andthetruthsofitspropositionswereoftenclearataglance.Butitcalledforacertainkindofingenuity.Withgeometry,therewasoftennoclueaboutwheretostartaproof.Beginninganargumentrequiredstrokesofgenius.Algebra,however,wassystematic.Equationscouldbemassagedalmost

mindlessly,peacefully;youcouldaddthesametermtobothsidesofanequation,cancelcommonterms,solveforanunknownquantity,orperformadozenotherproceduresandalgorithmsaccordingtostandardrecipes.Theprocessesofalgebracouldbesoothinglyrepetitive,likethepleasuresofknitting.Butalgebrasufferedfromitsemptiness.Itssymbolswerevacuous.Theymeantnothinguntiltheyweregivenmeaning.Therewasnothingtovisualize.Algebrawasleft-brainedandmechanical.Together,though,algebraandgeometrywereunstoppable.Algebragave

geometryasystem.Insteadofneedingingenuity,itnowdemandedtenacity.Ittransformeddifficultquestionsrequiringinsightintostraightforward,iflaborious,calculations.Theuseofsymbolsfreedthemindandsavedtimeandenergy.Foritspart,geometrygavealgebrameaning.Equationswerenolongersterile;

theywerenowembodimentsofsinuousgeometricforms.Awholenewcontinentofcurvesandsurfacesopenedupassoonasequationswereviewedgeometrically.Lushjunglesofgeometricfloraandfaunawaitedtobediscovered,cataloged,classified,anddissected.

FermatVersusDescartes

AnyonewhohasstudiedalotofmathandphysicswillhaverunintothenamesofFermatandDescartes.ButnoneofmyteachersortextbooksevertoldmeabouttheirrivalryorhowviciousDescartescouldbe.Tounderstandwhatwasatstakeintheirfights,youneedtoknowmoreabouttheirlives,theirpersonalities,andwhattheyhopedtoachieve.RenéDescartes(1596–1650)wasoneofthemostambitiousthinkersofall

time.Daring,intellectuallyfearless,andcontemptuousofauthority,hehadanegoasbigashisgenius.Forexample,oftheGreekapproachtogeometry,whichallothermathematicianshadreveredfortwothousandyears,hewrotedismissively:“WhattheancientshavetaughtusissoscantyandforthemostpartsolackingincredibilitythatImaynothopeforanykindofapproachtowardtruthexceptbyrejectingallthepathswhichtheyhavefollowed.”Atapersonallevel,hecouldbeparanoidandthin-skinned.Themostfamousportraitofhimshowsamanwithagauntface,haughtyeyes,andasnidelittlemustache.Helookslikeacartoonvillain.Descartessetouttorebuildhumanknowledgeonafoundationofreason,

science,andskepticism.Heisbestknownforhisworkinphilosophy,immortalizedbyhisfamouslineCogito,ergosum(“Ithink,thereforeIam”).Inotherwords,whenallisindoubt,atleastonethingiscertain:thedoubtingmindexists.Hisanalyticapproach,whichappearstohavebeeninspiredbytherigorouslogicofmathematics,iswidelyseentodayasthebeginningofmodernphilosophy.Inhismostfamousbook,hisDiscourseonMethod,Descartesintroducedabracingnewstyleofthinkingaboutphilosophicalproblems,buthealsoincludedthreeappendicesofinterestintheirownright—oneongeometry,inwhichhepresentedhisapproachtoanalyticgeometry;anotheronoptics,ofgreatimportatatimewhentelescopes,microscopes,andlenseswerethelatesttechnology;andathirdonweather,whichhasmostlybeenforgottenexceptforhiscorrectexplanationofrainbows.Hiscapaciousintellectroamedfarandwide.Heviewedthelivingbodyasasystemofmechanicaldevicesandlocatedtheseatofthesoulinthebrain’spinealgland.Heproposedagrand(butwrong)systemoftheuniverseaccordingtowhichinvisiblevorticespervadedallspace,withtheplanetscarriedalonglikeleavesinawhirlpool.Bornintoawealthyfamily,Descarteswassicklyasalittleboyandwas

allowedtostayinbedandreadandthinkaslongasheliked,ahabithekepthiswholelife,neverrisingbeforenoon.Hismotherdiedwhenhewasjustayearold,butfortunatelyshelefthimasizableinheritancethatlaterallowedhimtolivealifeofleisureandadventureasawanderinggentleman.HevolunteeredfortheDutcharmybutneversawcombat,andhehadplentyoftimeforphilosophy.HespentmuchofhisadultlifeinHolland,workingonhisideasand

correspondingandbickeringwithothergreatthinkers.In1650,hereluctantlytookapositioninSweden(whichhescornedas“thecountryofbears,amidrocksandice”)asQueenChristina’spersonalphilosophytutor.UnfortunatelyforDescartes,theenergeticyoungqueenwasanearlyriser.Sheinsistedonlessonsatfiveinthemorning,anungodlyhourforanyonebutespeciallyforDescartes,accustomedtogettingupatnoonhiswholelife.ThatwinterinStockholmwasthecoldestindecades.Afterafewweeks,Descartescaughtpneumoniaanddied.PierredeFermat(1601–1665),whowasfiveyearsyoungerthanDescartes,

livedapeaceful,upper-middle-class,comparativelyuneventfullife.BydayhewasalawyerandprovincialjudgeinToulouse,farfromthehubbubinParis.Bynighthewasahusbandandfather.Hecamehomefromwork,atedinnerwithhiswifeandfivekids,andthenspentafewhourswithhisonetruepassion:doingmath.WhereasDescarteswasabigthinkerofcolossalambitions,Fermatwasashyman,quiet,even-tempered,andnaive.HehadmoremodestgoalsthanDescartesdid.Hedidn’tseehimselfasaphilosopherorascientist.Mathwasenoughforhim.Hepursueditasanamateur,lovingly.Hesawnoneedtopublish,andhedidn’t.Hewrotelittlenotestohimselfinthebookshewasreading,classicGreektomesbyDiophantusandArchimedes,andoccasionallymailedhisideastoscholarshethoughtmightappreciatethem.HenevertraveledfarfromToulouseormetanyofthemajormathematiciansofhisday,althoughhecorrespondedwiththemthroughMarinMersenne,aFranciscanfriar,mathematician,andsocialconnector.ItwasthroughMersennethatFermatandDescarteslockedhorns.Among

mathematicians,Mersennewasthego-toguyinParis.InatimebeforeFacebook,hekepteveryoneintouchwitheveryoneelse,arealbusybodywithacertainlackoftactanddiscretion.Hehadawayofstirringuptrouble;forexample,heshowedpeoplepersonallettershereceivedandreleasedconfidentialmanuscriptsbeforetheywerepublished.Therewasacirclearoundhimoftopmathematicians,notquiteinthesameleagueasFermatandDescartes,butstrongnonetheless,andtheyapparentlyhaditinforDescartes.TheywerealwayssnipingathimandhisgrandioseDiscourseonMethod.SowhenDescartesheardviaMersennethatsomenobodyinToulouse,some

amateurnamedFermat,claimedtohavedevelopedanalyticgeometryadecadeearlierthanhehadandthatthissameamateur(whowasthisguy?)hadraiseddoubtsabouthistheoryofoptics,Descartesconsidereditanothercaseofsomeoneouttogethim.Intheyearstocome,hefoughtvehementlyagainstFermatandtriedtoruinhisreputation.Afterall,Descarteshadalottolose.IntheDiscourse,he’dclaimedthathisanalyticalmethodwastheonesurerouteto

knowledge.IfFermatcouldoutdohimwithoutevenusinghismethod,hiswholeprojectwasatrisk.DescartesbadmouthedFermatmercilesslyandtosomeextentsucceededin

diminishinghim.Fermat’sworkwasneverproperlypublisheduntil1679.Hisresultstrickledoutthroughwordofmouthorincopiesofhisletters,buthewasnottrulyappreciateduntillongafterhisdeath.Descartes,however,hititbig.HisDiscoursebecamefamous.Thenextgenerationlearnedanalyticgeometryfromit.Eventoday,ourstudentslearnaboutCartesiancoordinates,eventhoughFermatcameupwiththemfirst.

TheSearchforAnalysis,theLong-LostMethodofDiscovery

ThesquabblesbetweenDescartesandFermattookplaceagainstthebackdropoftheearlyseventeenthcentury,atimewhenmathematiciansdreamedoffindingamethodofanalysisforgeometry.Hereanalysis,asinanalyticgeometry,istobeunderstoodinthearchaicsenseoftheword—asameansofdiscoveringresultsratherthanprovingthem.Therewaswidespreadsuspicionatthetimethattheancientshadpossessedsuchamethodofdiscoverybuthaddeliberatelyconcealedit.Descartes,forexample,allegedthattheancientGreeks“hadknowledgeofaspeciesofmathematicsverydifferentfromthatwhichpassescurrentinourtime...butmyopinionisthatthesewritersthen,withasortoflowcunning,deplorableindeed,suppressedthisknowledge.”Symbolicalgebraseemedlikeitmightbethislong-lostmethodofdiscovery.

Butinmoreconservativequarters,algebrametwithreactionaryskepticism.Agenerationlater,whenIsaacNewtonsaid,“Algebraistheanalysisofthebunglersinmathematics,”hewasthrowingathinlyveiledinsultatDescartes,theprimeexampleofa“bungler”whohadreliedonalgebraasacrutchtosolveproblemsbyworkingbackward.Inlaunchinghisattack,Newtonwasadheringtoatraditionaldistinction

betweenanalysisandsynthesis.Inanalysis,onesolvesaproblembystartingattheend,asiftheanswerhadalreadybeenobtained,andthenworksbackwishfullytowardthebeginning,hopingtofindapathtothegivenassumptions.It’swhatkidsinschoolthinkofasworkingbackwardfromtheanswertofigureouthowtogetthere.

Synthesisgoesintheotherdirection.Itstartswiththegivens,andthen,bystabbinginthedark,tryingthings,youaresomehowsupposedtomoveforwardtoasolution,stepbylogicalstep,andeventuallyarriveatthedesiredresult.Synthesistendstobemuchharderthananalysisbecauseyoudon’teverknowhowyou’regoingtogettothesolutionuntilyoudo.TheancientGreeksregardedsynthesisascarryingmorelogicalforce,more

persuasivepower,thananalysis.Synthesiswasconsideredtheonlyvalidwaytoprovearesult;analysiswasapracticalwaytofindtheresult.Ifyouwantedarigorousdemonstration,youhadtodosynthesis.That’swhy,forexample,Archimedesusedhisanalyticalmethodofbalancingshapesonseesawstofindhistheoremsbutthenswitchedtothesyntheticmethodofexhaustiontoprovethem.Still,althoughNewtonlookeddownhisnoseatalgebraicanalysis,wewillsee

inchapter7thatheusedithimself,andtotremendouseffect.Buthewasn’titsfirstmaster.Fermatwas.Fermat’sstyleofthinkingisfuntoexamine,becauseit’selegantandaccessibleandyetforeignandsurprising.Hismethodsforstudyingcurvesarenolongerinuse,havingbeensupersededbythemoresophisticatedtechniquesinthetextbookstoday.

OptimizingfortheOverheadBin

Fermat’sembryonicversionofdifferentialcalculusgrewoutofhisapplicationofalgebratooptimizationproblems.Optimizationisthestudyofhowtodothingsinthebestpossibleway.Dependingoncontext,bestmightmeanfastest,cheapest,biggest,mostprofitable,mostefficient,orsomeothernotionofoptimality.Toillustratehisideasinthesimplestfashion,Fermatcontrivedafewproblemsthatsoundalotliketheexerciseswemathteachersarestillassigningtoourstudentstoday.Theycanblameitallonhim.Oneofthoseproblems,updatedforourtime,goessomethinglikethis.

Imagineyouwanttodesignarectangularboxtoholdasmuchstuffaspossible,subjecttotwoconstraints.First,theboxhastohaveasquarecrosssection,xincheswidebyxinchesdeep.Second,ithastofitintheoverheadbinofacertainairline.Accordingtotheirrulesforcarry-onbaggage,thewidthplusdepthplusheightoftheboxcannotexceed45inches.Whatchoiceofxproducesaboxofmaximumvolume?Onewaytosolvethisiswithcommonsense.Tryafewpossibilities.Saythe

widthanddepthare10incheseach.Thatwouldallowforaheightof25inches,

since10+10+25=45.Aboxwiththosedimensionswouldhaveavolumeof10×10×25,whichequals2,500cubicinches.Wouldacube-shapedboxbebetter?Sinceacubemusthaveequalheight,width,anddepth,itwouldhavetohavedimensions15×15×15,whichmultipliesouttoaroomy3,375cubicinches.Fiddlingaroundwithafewotherpossibilitiesmakesitseemlikelythatacubeistheoptimalchoicefortheshapeofthebox.Andindeeditis.Sothisisnotaparticularlyhardprobleminitself.Thepointofitistoshow

howFermatreasonedaboutsuchproblems,becausehisapproachledtomuchgreaterthings.Asinmostalgebraproblems,thefirststepistotranslateallthegiven

informationintosymbols.Sincethewidthanddepthoftheboxarebothx,theyaddupto2x.Andsincetheheightpluswidthplusdepthcannotexceed45inches,thatleaves45−2xfortheheight.Thusthevolumewillbextimesxtimes(45−2x).Multiplyingthatoutgives45x2−2x3.That’sthevolumeofourbox.CallitV(x).Thus

V(x)=45x2−2x3.Ifwecheatmomentarilyanduseacomputeroragraphingcalculatortoplotx

horizontallyandVvertically,weseethatthecurverisesupandreachesitsmaximumwhenx=15inches,asexpected,andthendescendsbacktozero.

Alternatively,tofindthatmaximumwithdifferentialcalculusaspracticed

today,ourstudentswouldreflexivelytakethederivativeofVandsetitequaltozero.Thethinkingisthatatthetopofthecurve,theslopeiszero.Thecurveisneitherrisingnorfallingthere.So,sincetheslopeismeasuredbythederivative(aswe’llseeinchapter6),thederivativemustbezeroatthemaximum.Aftera

bitofalgebraandtheincantationofvariousmemorizedrulesforderivatives,thislineofreasoningwouldalsoyieldx=15atthemaximum.ButFermatdidn’thavegraphicalcalculatorsorcomputers,andhecertainly

didn’thavetheconceptofderivatives;onthecontrary,heinventedtheideasthatledtoderivatives!Sohowdidhesolvetheproblem?Heusedaspecialpropertyofthemaximum:horizontallinesbelowthemaximumintersectthecurveattwopoints,asshownhere,

whereashorizontallinesabovethemaximumdon’tintersectthecurveatall.

Thatsuggestedanintuitivestrategytosolvetheproblem.Imagineslowly

liftingahorizontallinethatstartsbelowthemaximum.Asthelinegraduallymovesup,itstwointersectionpointsslidetowardeachotheralongthecurvelikebeadsonanecklace.

Atthemaximum,thosetwopointscollide.Lookingforthatcollisionwashow

Fermatdeterminedthemaximum.Hederivedaconditionfortwopointstomergeintoone,formingwhat’sknownasadoubleintersection.Withthatinsightinplace,therestisalgebra,themeremanipulationofsymbols.Itgoesasfollows.Saythetwointersectionsoccuratx=aandx=b.Thensince(by

construction)theylieonthesamehorizontalline,wemusthaveV(a)=V(b).Hence

45a2−2a3=45b2−2b3.Tomakeheadway,ithelpstorearrangethisequation.Ifweputthesquaresononesideandthecubesontheother,weget

45a2−45b2=2a3−2b3.Withsomeskillinhigh-schoolalgebra,wecanthenfactorbothsidestoobtain

45(a−b)(a+b)=2(a−b)(a2+ab+b2).Next,dividebothsidesbythecommonfactorofa−b.That’slegal,sinceaandbareassumedtobedifferent.(Iftheywereequal,dividingbothsidesbya−bwouldamounttodividingbyzero,whichisprohibited,asdiscussedinchapter1.)Aftercancellation,theresultingequationis

45(a+b)=2(a2+ab+b2).

Buckleupnowforaconfusingpointoflogic.Fermathasjustassumedthataandbarenotequal.Yethegoesontoimaginethattheequationhehasjustderivedwillcontinuetoholdwhenaandbdobecomeequalastheymergeatthemaximum.Hetriestojustifythisbyinvokingamurkyconcepthecallsadequality.Itexpressestheideathataandbbecomesortofequalbutnotreallyequalatthemaximum(todaywewouldphraseitusingtheconceptofalimitoradoubleintersection).Anyway,hesetsa≈b,wherethesquigglyequalssignmeansadequal,andthencavalierlysubstitutesaforbintheequationabovetoget

45(2a)=2(a2+a2+a2).Thissimplifiesto90a=6a2,whosesolutionsarea=0anda=15.Thefirstofthesesolutions,a=0,givesaboxofminimumvolume;ithaszerowidthanddepthandhencehaszerovolume.That’sofnointerest.Thesecondsolution,a=15,givestheboxofmaximumvolume.There’stheanswerwe’vebeenexpecting:15inchesistheoptimalwidthanddepth.Fromtoday’sperspective,Fermat’sreasoningseemsstrange.Hefindsa

maximumwithoutusingderivatives.Todayweteachderivativesbeforeoptimization;Fermatdidittheotherwayaround.Butitdoesn’tmatter.Hisideasareequivalenttoours.

HowFermatHelpedtheFBI

ThelegacyofFermat’searlyworkonoptimizationisallaroundus.Ourlivestodaydependonalgorithmsthatsolveoptimizationproblemsusingthenotionofdoubleintersectionsandequivalentconditionsexpressedwithderivatives.Today’sproblemstendtobemuchmorecomplicatedthanFermat’s,butthespiritisthesame.Oneimportantapplicationinvolvesbigdatasets,whereit’softenhelpfulto

codethedataascompactlyaspossible.Forexample,theFederalBureauofInvestigationhasmillionsofrecordsoffingerprints.Tostorethem,searchthem,andretrievethemefficientlyforbackgroundchecks,theyusecalculus-basedmethodsofdatacompression.Cleveralgorithmsreducethesizeofthedigitizedfingerprintfileswithoutsacrificinganydetailsthatmatter.Thesameistruewhenyoustoremusicandpicturesonyourphone.Ratherthankeepeverynote

andpixel,compressionalgorithmsnamedMP3andJPEGsavespacebydistillingtheinformationdowntoamuchmoreefficientform.Theyalsoletusdownloadsongsandphotosquicklyandsendthemtoourlovedoneswithoutclogginguptheirinboxestoomuch.Toseewhatcalculusandoptimizationhavetodowithdatacompression,let’s

takealookattherelatedstatisticalproblemoffittingacurvetodata,anissuethatcomesupeverywherefromclimatesciencetobusinessforecasting.Thedatasetwe’llexamineshowshowdaylengthvarieswiththeseasons.Asweallknow,thedaysarelongerinthesummerandshorterinthewinter,butwhatdoestheoverallpatternlooklike?Inthegraphbelow,I’veplottedthedataforNewYorkCityfortheyear2018,withtimerunninghorizontallyfromJanuary1onthefarlefttoDecember31onthefarright.Theverticalaxisshowsthenumberofminutesbetweensunriseandsunsetatdifferenttimesoftheyear.Toavoidclutteringupthepicture,I’veshownthedataforonlytwenty-sevendays,sampledeverytwoweeksstartingonJanuary1.

Thegraphshowsthatdaylengthrisesandfallsthroughouttheyear,as

expected.Thedaysarelongestaroundthesummersolstice(June21,correspondingtothepeakatday172nearthemiddleofthegraph)andshortestaroundthewintersolstice,halfayearlater.Overallthedataappeartolieonasmoothlyundulatingwave.Inhigh-schooltrigonometryclasses,teacherstalkaboutacertainkindof

wave,asinewave.LaterinthisbookIwillhavemoretosayaboutwhatsinewavesareandwhytheyarespecialfromthestandpointofcalculus.Fornow,the

mainthingweneedtoknowisthatsinewavesareconnectedtocircularmotion.Toseetheconnection,imagineapointmovingaroundacircleataconstantspeed.Ifwetrackitsup-and-downpositionasafunctionoftime,thepointtracesoutasinewave.

Andbecausecirclesareintimatelyconnectedtocycles,sinewavescomeup

wherevercyclicphenomenaoccur,fromthecycleoftheseasonstothevibrationsofatuningforktothesixty-cyclehumoffluorescentlightsandpowerlines.Thatannoyinghumisthesoundofsinewavesbobbingupanddownsixtytimesasecond.It’sthetelltalesignofalternatingcurrentproducedbygeneratorsinthepowergridwhosemachineryisspinningatthatsamefrequency.Wherethereiscircularmotion,therearesinewaves.Anysinewaveiscompletelydefinedbyfourvitalstatistics:itsperiod,

average,amplitude,andphase.

Thesefourparametershavesimpleinterpretations.TheperiodTindicateshowlongittakesthewavetocompleteafullcycle.Fortheday-lengthdatawe’reconsideringhere,Tisaboutayearor,tobemoreprecise,365.25days.(Thatextraquarterofadayiswhyweneedleapyearseveryfourthyear,tokeepthe

calendarinsyncwithnaturalcycles.)Theaverageofthesinewaveisitsbaselinevalue,b.Forourdata,it’sthetypicalnumberofminutesofdaylightinNewYorkCityaveragedacrossallthedaysoftheyear2018.Thewave’samplitudeatellsushowmanyadditionalminutesoflightthereareonthelongestdayoftheyearascomparedtotheaverageday.Thewave’sphasectellsusthedayonwhichthewavecrossesupwardthroughitsaveragevalue,sometimearoundthespringequinox.It’shelpfultothinkofthesefourparameters—a,b,c,T—asfourknobswe

canturntoadjustvariousfeaturesofthesinewave’sshapeandlocation.Theb-knobmovesthesinewaveupordown.Thec-knobmovesitleftorright.TheT-knobcontrolshowrapidlyitoscillates.Andthea-knobdetermineshowpronouncedthoseoscillationsare.Ifwecouldsomehowsettheknobstomakethesinewavegothroughallthe

datapointsweplottedearlier,thatwouldamounttoasignificantcompressionofinformation.Itwouldmeanwewerecapturingthetwenty-sevendatapointswithjustthefourparametersinthesinewave,therebycompressingthedatabyafactorof27/4,or6.75.Actually,sinceweknowoneoftheparametersisayear,wereallyhaveonlythreeparameterstofiddlewith,givingusacompressionfactorof27/3,or9.Areductionofthissizeisconceivablebecausethedataarenotrandom.Theyfollowapattern.Thesinewaveembodiesthatpatternanddoestheworkforus.Theonlycatchisthatthereisnosinewavethatgoesthroughthedata

perfectly.That’stobeexpectedwhenfittinganidealizedmodeltoreal-worlddata;thereareboundtobesomediscrepancies.Thehopeisthatthediscrepanciesarenegligible.Tominimizethem,weneedtofindthesinewavethathugsthedatapointsascloselyaspossible.That’swherecalculuscomesin.Thefigurebelowshowsthebest-fittingsinewave,asdeterminedbyan

optimizationalgorithmI’llexplaininaminute.

Butfirst,noticetheresultingfitisnotperfect.Forinstance,thewavedoesn’tquitedipdownlowenoughinDecember,whenthedaysareveryshortandthedatafallbelowthecurve.Nevertheless,asimplesinewavecertainlycapturestheessenceofwhat’sgoingon.Dependingonourgoals,afitofthisqualitymaybeadequate.Sohowdoescalculuscomein?Ithelpsuschoosethefourparameters

optimally.Imagineturningthefourknobstogetthebestpossiblefit,somewhatliketuningthedialonaradiotogetthestrongestpossiblesignal.ThisisessentiallywhatFermatdidintheoverhead-binproblemwhenhefoundtheroomiestdimensionsforthebox.Inthatcase,hewastuningasingleparameter,x,thesidelengthofthebox,andlookingforadoubleintersectionasasignalthatthevolumeoftheboxwasamaximum.Inourcase,wehavefourparameterstotune.Butthebasicideaisthesame.We’lllookforadoubleintersection,andthat’llgiveusouroptimalchoiceofthefourparameters.Inalittlemoredetail,here’showitworks.Foranygivenchoiceofthefour

parameters,wecalculatethediscrepancy(inotherwords,theerror)betweenthesine-wavefitandtheactualdataateveryoneofthetwenty-sevenpointsrecordedthroughouttheyear.Anaturalcriterionforchoosingthebestfitisthatthetotalerror,summedoveralltwenty-sevenpoints,shouldbeassmallaswecanmakeit.Buttotalerrorisnotquitetherightconcept,becausewedon’twantthenegativeerrorstocancelthepositiveonesandgivethefalseimpressionthatthefithaslesserrorthanitdoes.Undershootsarejustasbadasovershoots,andbothshouldbepenalized;theyshouldn’tbeallowedtocancelout.Forthisreason,mathematicianssquaretheerrorsateachpointtomakethenegativeones

becomepositive.Thatway,theycan’tpossiblyproduceanyspuriouscancellations.(Here’soneplacewherethefactthatanegativetimesanegativeisapositiveisusefulinapracticalsetting.Itmakesthesquareofanegativeerrorcountasapositivediscrepancy,asitshould.)Sothebasicideaistochoosethefourparametersinthesinewaveinsuchawaythattheyminimizethetotalsquarederrorofthefittothedata.Accordingly,thisapproachiscalledthemethodofleastsquares.Itworksbestwhenthedatafollowapattern,astheydohere.Allofwhichraisesanextremelyimportantgeneralpoint:Patternsarewhat

makecompressionpossibleinthefirstplace.Onlypatterneddatacanbecompressed.Randomdatacannot.Happily,manyofthethingspeoplecareabout,likesongsandfacesandfingerprints,arehighlystructuredandpatterned.Justasdaylengthfollowsasimplewavepattern,aphotographofafacecontainseyebrows,blemishes,cheekbones,andothercharacteristicpatterns.Songshavemelodiesandharmonies,rhythmsanddynamics.Fingerprintscontainridgesandloopsandwhorls.Ashumanbeings,werecognizethesepatternsinstantly.Computerscanbetaughttorecognizethemtoo.Thetrickistofindtherightkindsofmathematicalobjectstoencodeparticularpatterns.Sinewavesareidealforrepresentingperiodicpatterns,buttheyarelesswellsuitedtorepresentingsharplylocalizedfeatures,liketheedgeofanostrilorabeautymark.Forthispurpose,researchersinseveraldifferentfieldscameupwitha

generalizationofsinewavescalledwavelets.Theselittlewavesaremorelocalizedthansinewaves.Insteadofextendingperiodicallyouttoinfinityinbothdirections,theyaresharplyconcentratedintimeorspace.

Waveletssuddenlyturnon,oscillateforawhile,andthenturnoff.Theylook

almostlikethesignalsonheartmonitorsortheburstsofactivityrecordedon

seismographsduringanearthquake.Theyareidealforrepresentingasuddenspikeinabrain-waverecording,aboldstrokeonaVanGoghpainting,orawrinkleonaface.TheFederalBureauofInvestigationusedwaveletstomodernizetheir

fingerprintfiles.Fromthetimefingerprintswereintroduced,atthebeginningofthetwentiethcentury,fingerprintrecordshadbeenstoredasinkedimpressionsonpapercards.Thefilesweredifficulttosearchquickly.Bythemid-1990s,thecollectionhadgrowntoroughlytwohundredmillionfingerprintcardsonfileandoccupiedanacreofofficespace.WhentheFBIdecidedtodigitizethefiles,theyturnedthemintograyscaleimageswith256differentlevelsofgrayataresolutionof500dotsperinch,enoughtocaptureallthefinewhorls,loops,ridgeendings,bifurcations,andotheridentifyingminutiaeoffingerprints.Theproblem,however,wasthatatthetime,asingledigitizedcardcontained

about10megabytesofdata.ThatmadeitprohibitivefortheFBItosenddigitalfilesquicklytolocalpolice.Remember,thiswasinthemid-1990s,whenphonemodemsandfaxmachineswerestateoftheartandtransmittinga10-megabytefiletookhours.Plusitwastoughtoexchangefilesthatbigwhen1.5-megabytefloppydiskswerethemediumofchoice.Giventhegrowingdemandforfasterturnaroundtimesonthethirtythousandnewfingerprintcardsthatfloodedineverydaywithurgentrequestsforbackgroundchecks,therewasadesperateneedtomodernizethesystem.TheFBIhadtofindawaytocompressthefileswithoutdistortingthem.Waveletswereidealforthejob.Byrepresentingfingerprintsascombinations

ofmanywaveletsandbyturningtheknobsonthemoptimallyusingcalculus,mathematiciansfromtheLosAlamosNationalLabteamedupwiththeFBItoshrinktheirfilesbyafactorofmorethantwenty.Itwasarevolutionforforensics.ThankstoFermat’sideasintheirmodernform(alongwithanevengreaterroleforwaveletanalysis,computerscience,andsignalprocessing),a10-megabytefilecouldbecompressedtoonly500kilobytes,amanageablesizetosendoverthephonelines.Anditcouldbedonewithoutsacrificingfidelity.Human-fingerprintexpertsnoddedtheirapproval.Sodidcomputers;thecompressedfilespassedtheFBI’sautomaticidentificationsystemwithflyingcolors.Itwasgoodnewsforcalculusandbadnewsforcriminals.

ThePrincipleofLeastTime

IwonderwhatFermatwouldhavethoughtofthisuseofhisideas.Hehimselfwasneverespeciallyinterestedinapplyingmathematicstotherealworld.Hewascontenttodomathforitsownsake.Buthedidmakeonecontributiontoappliedmathematicsoflastingimportance:hewasthefirstpersontodeducealawofnaturefromadeeperlawbyusingcalculusasalogicalengine.JustasMaxwellwoulddowithelectricityandmagnetismtwocenturieslater,Fermattranslatedahypotheticallawofnatureintothelanguageofcalculus,startedtheengine,andfedthelawin,andoutpoppedanotherlaw,aconsequenceofthefirstone.Insodoing,Fermat,theaccidentalscientist,initiatedastyleofreasoningthathasdominatedtheoreticalscienceeversince.Thestorybeganin1637whenagroupofParisianmathematiciansasked

FermatforhisopiniononDescartes’srecenttreatiseonoptics.Descarteshadatheoryabouthowlightbentwhenitpassedfromairintowaterorfromairintoglass,aneffectknownasrefraction.Anyonewhohaseverplayedwithamagnifyingglassknowsthatlightcanbe

bentandfocused.Inmyyouth,Ilikedtosetleavesonfirebyholdingamagnifyingglassoverthemonthedrivewayandliftingtheglassupanddownuntilthesun’sraysfocusedintoatightwhitespotofblazingintensity,causingtheleaftosmolderandeventuallyignite.Refractionisusedinlessspectacularwaysinourspectacles.Thelensesinoureyeglassesbendandfocusthelightrayswheretheybelong,attherightplaceontheretinatocorrectfaultyvision.Thebendingoflightalsoexplainsanillusionyoumayhavenoticedwhile

loungingbyaswimmingpoolonasunnyday.Supposethatatthebottomofthepooltherehappenstobeashiny,tragicallymisplacedobjectlikeapieceofjewelry.

Youlookdownthroughthewaterattheshinyobject,butit’snotquitewhereitappearstobebecausethelightraysbouncingoffitgetbentastheypassfromwaterintoairontheirwayoutofthepool.Forthesamereason,aspearfishermanneedstoaimbelowtheapparentpositionofafishtohaveachanceofhittingit.Refractionphenomenaliketheseobeyasimplerule.Whenarayoflight

passesfromathinnermediumlikeairintoadensermediumlikewaterorglass,theraybendstowardtheperpendiculartotheinterfacebetweenthetwomedia.Whenitpassesfromadensermediumintoathinnerone,itbendsawayfromtheperpendicular,asillustratedhere.

In1621,theDutchscientistWillebrordSnellsharpenedthisruleandmadeit

quantitativebydoingacleverexperiment.Bysystematicallychangingtheangleaoftheincomingrayandobservinghowtheangleboftheoutgoingraychangedinresponse,hediscoveredthattheratiosina/sinbalwaysstayedthesameforagivenpairofmedia.(Heresinreferstothesinefunctionoftrigonometry,thesamesinefunctionwhosewavygraphappearedinouranalysisofdaylength.)However,Snellfoundthatthevalueofsina/sinbdiddependonwhatthetwo

mediaweremadeof.Airandwaterproducedoneconstantratio,whereasairandglassproducedanother.Hehadnoideawhythesinelawworked.Itjustdid.Itwasabrutefactaboutlight.DescartesrediscoveredSnell’ssinelawandpublisheditinhis1637essay

Dioptrics,unawarethatithadbeenfoundbyatleastthreeothersbeforehim:

Snellin1621,theEnglishastronomerThomasHarriotin1602,andthePersianmathematicianAbuSa’dal-A’laIbnSahlwaybackin984.Descarteshadgivenamechanicalexplanationforthesinelawinwhichhe

(incorrectly)assumedthatlightmovedfasterinadensermedium.ToFermat,thatsoundedupsidedownandcontrarytocommonsense.Tryingtobehelpful,andbeinganaiveandinnocentfellow,FermatofferedwhathethoughtwereafewgentlecriticismsofDescartes’stheoryandmailedthembacktotheParisianmathematicianswho’daskedforhisopinion.WhatFermatdidnotknowwasthatthosemathematicianswereDescartes’s

bitterenemies.TheywereusingFermatfortheirownsinisterpurposes.Andasanyteenagercouldhaveanticipated,whenDescartesheardthroughthegrapevineaboutFermat’scomments,hefelthewasbeingattacked.He’dneverheardofthelawyerfromToulouse.Tohim,Fermatwasanobscureamateurworkingoutinthecountryside,someoneeasilydismissedasanothergnatbuzzingaroundhishead.Overthenextfewyears,DescartestreatedFermatcondescendinglyandclaimedthathe’dblunderedintohisresultsbyaccident.Fast-forwardtwentyyears.In1657,afterDescarteshaddied,Fermatwas

askedbyacolleaguenamedMarinCureaudelaChambretorevisittheoldcontroversyaboutrefraction.Cureau’sinquirypromptedFermattotakealookattheproblemhimself,usingwhatheknewaboutoptimization.Fermathadahunchthatlightoptimized.Moreprecisely,heguessedthatlight

alwaysfollowedthepathofleastresistancebetweenanytwopoints,whichhetooktomeanthatittraveledalongthefastestpossibleroute.Hecouldseethatthisprincipleofleasttimewouldexplainwhylightmovedinastraightlineinauniformmediumandwhy,whenitreflectedoffamirror,itsangleofincidenceequaleditsangleofreflection.Butcouldtheprincipleofleasttimealsocorrectlypredicthowlightbentwhenitpassedfromonemediumintoanother?Woulditexplainthesinelawofrefraction?Fermatwasn’tsure.Thecalculationwouldn’tbeeasy.Aninfinitenumberof

straight-linepaths,eachbentlikeanelbowattheinterface,couldtakethelightfromasourcepointinonemediumtoatargetpointintheother.

Computingtheminimumofallthosetraveltimeswasgoingtobedifficult,

especiallyatthatembryonicstageinthedevelopmentofdifferentialcalculus.Therewerenotoolsavailableotherthanhisolddouble-intersectionmethod.Plushewasafraidofgettingthewronganswer.AshewrotetoCureau,“Thefearoffinding,afteralonganddifficultcalculation,someirregularandfantasticproportion,andmynaturalinclinationtolaziness,leftthematterinthatstate.”FiveyearspassedwhileFermatworkedonotherproblems.Buteventuallyhis

curiositygotthebetterofhim.In1662heforcedhimselftodothecalculation.Itwasarduousandunpleasant.Butashehackedawayatthethicketofsymbols,hestartedtoseesomething.Thetermsbegantocancel.Thealgebrawasworking.Andthereitwas:thesinelaw.InalettertoCureau,Fermatcalledthiscalculation“themostextraordinary,themostunforeseenandthehappiest”onehe’deverdone.“Iwassosurprisedatsuchanunexpectedevent,thatIcanscarcelyrecoverfrommyastonishment.”Fermathadappliedhisembryonicversionofdifferentialcalculustophysics.

Noonehadeverdonethatbefore.Andinsodoing,heshowedthatlighttravelsinthemostefficientway—notthemostdirectway,butthefastest.Ofallthepossiblepathslightcantake,itsomehowknows,orbehavesasifitknows,howtogetfromheretothereasquicklyaspossible.Thiswasanimportantearlycluethatcalculuswassomehowbuiltintotheoperatingsystemoftheuniverse.Theprincipleofleasttimewaslatergeneralizedtotheprincipleofleast

action,whereactionhasatechnicalmeaningthatweneedn’tgointohere.Thisoptimizationprinciple—thatnaturebehavesinthemosteconomicalway,inacertainprecisesense—wasfoundtocorrectlypredictthelawsofmechanics.Inthetwentiethcentury,theprincipleofleastactionwasextendedtogeneralrelativityandquantummechanicsandotherpartsofmodernphysics.Iteven

madeanimpressiononphilosophyintheseventeenthcentury,whenGottfriedWilhelmLeibnizarguedthatallisforthebestinthebestofallpossibleworlds,anoptimisticpointofviewlaterparodiedbyVoltaireinCandide.TheideaofusinganoptimalityprincipletoexplainphysicalphenomenaandtodeduceitsconsequenceswithcalculusbeganwiththisverycalculationbyFermat.

TheTussleoverTangents

Fermat’soptimizationtechniquesalsoallowedhimtofigureouttangentlinestocurves.ThiswastheproblemthatreallymadeDescartes’sbloodboil.ThewordtangentcomesfromaLatinrootfor“touching.”Theterminologyis

apt,sinceinsteadofcuttingacrossacurveintwoplaces,atangentlinetouchesthecurveatonepoint,barelygrazingit.

Theconditionfortangencyissimilartothatforamaximumoraminimum.If

weintersectacurvewithalineandthenslidethelineupordowncontinuously,tangencyoccurswhentwointersectionscoalesceintoone.Bysometimeinthelate1620s,Fermatwasabletofindthetangentlineto

essentiallyanyalgebraiccurve(meaningacurveexpressiblesolelyintermsofwholenumberpowersofxandy,withoutanylogarithms,sinefunctions,orotherso-calledtranscendentalfunctionsinit).Usinghisbigideaofthedoubleintersection,hecouldcalculateeverythingwithhismethodsthatwecantodaywithderivatives.

Descarteshadhisownmethodoffindingtangentlines.InhisGeometryof1637,heproudlyannouncedhismethodtotheworld.UnawarethatFermathadalreadysolvedtheproblem,Descartesindependentlyhitonthedouble-intersectionidea,butheusedcirclesinsteadoflinestocutthroughthecurvesofinterest.Nearthepointoftangency,atypicalcirclewouldcutthroughthecurveattwopoints,oratnone.

Byadjustingthelocationandradiusofthecircle,Descartescouldforcethe

twointersectionpointstomergeintoone.Atthatdoubleintersection—bingo!—thecircleintersectedthecurvetangentially.

ThatgaveDescarteseverythingheneededtofindthetangenttothecurve.Italsogavehimthenormaltothecurve,whichlayatrightanglestothetangent,alongthecircle’sradius.Hismethodwascorrectbutclumsy.Itgeneratedbushelsofalgebra,much

morethanFermat’s.ButDescarteshadn’tevenheardofFermatyet,soinhisusualcocksurefashion,hepresumedhehadoutdoneeveryone.AshecrowedinGeometry,“Ihavegivenageneralmethodofdrawingastraightlinemakingrightangleswiththecurveatanarbitrarilychosenpointuponit.AndIdaresaythatthisisnotonlythemostusefulandmostgeneralproblemingeometrythatIknow,buteventhatIeverdesiredtoknow.”Latein1637,whenDescarteslearnedfromhiscorrespondentsinParisthat

Fermathadbeatenhimtothesolutionofthetangentproblembyabouttenyearsbuthadnevergottenaroundtopublishingit,hewasdismayed.In1638hestudiedFermat’smethod,lookingforholes.Oh,thereweresomany!Writingthroughanintermediary,hesaid,“Idonotevenwanttonamehim,sothathewillfeellessshameattheerrorsthatIhavefound.”HechallengedFermat’slogic,which,tobefair,wassketchyandpoorlyexplained.Buteventually,afterseverallettersbackandforth,withFermatcalmlytryingtoclarifyhisideas,Descarteshadtoconcedethathisreasoningwasvalid.Butbeforeadmittingdefeat,hetriedtostumpFermatbychallenginghimto

findthetangentlinetoacurvedefinedbythecubicequationx3+y3=3axy,whereawasaconstant.Descartesknewthathehimselfcouldn’tfindthetangentwithhisownclunkycirclemethod—thealgebrabecameunmanageable—sohewasconfidentthatFermatwouldn’tbeabletofinditwithhislinemethod.ButFermatwasastrongermathematician,andhehadabettermethod.HedispatchedDescartes’scurvewithoutbreakingasweat,muchtoDescartes’schagrin.

WithinSightofthePromisedLand

Fermatpavedthewayforcalculusinitsmodernform.Hisprincipleofleasttimerevealedthatoptimizationiswovendeeplyintothefabricofnature.Hisworkonanalyticgeometryandtangentlinesblazedatrailtodifferentialcalculusthatotherssoonfollowed.Andhisvirtuositywithalgebraenabledhimtofindtheareasundercertaincurvesthathadeludedevenhismostillustriouspredecessors.Inparticular,hefoundtheareaunderthecurvey=xnforanypositiveintegern,usinglittlemorethanhisbarehands.(Othershadsolvedthefirstninecases,n=

1,2,...,9,butcouldn’tfindastrategythatworkedforalln.)Fermat’sadvancewasagiantstepforwardforintegralcalculus,onethatwouldsetthestageforbreakthroughstocome.Yetforallthat,hisstudiesstillfellshortofthesecretthatNewtonandLeibniz

wouldsoondiscover,thesecretthatrevolutionizedandunifiedthetwosidesofcalculus.It’sapitythatFermatmissedit,forhecamesoclose.Themissinglinkwasrelatedtosomethinghecreatedbutneverrecognizedascrucial,somethingimplicitinhismethodofmaximaandtangents.Itwouldlaterbecalledthederivative.Itsapplicationswouldgofarbeyondcurvesandtheirtangentstoincludeanysortofchangeatall.

5

TheCrossroads

WEHAVECOMEtoacrossroadsinourstory.Thisiswherecalculusbecomesmodernandprogressesfromthemysteryofcurvestothemysteriesofmotionandchange.It’swherecalculusstartstowonderabouttherhythmsoftheuniverse,itsupsanddowns,itsineffablepatternsintime.Nolongercontentinthestaticworldofgeometry,calculusbecomesfascinatedwithdynamics.Itasks:Whataretherulesofmotionandchange?Whatcanwepredictaboutthefuturewithcertainty?Inthefourcenturiessincecalculusreachedthiscrossroads,ithasbranched

outfromalgebraandgeometrytophysicsandastronomy,biologyandmedicine,engineeringandtechnology,andeveryotherfieldwhereallisinfluxandchangeneverstops.Calculushasmathematizedtime.Andithasofferedushopethattheworldwelivein,forallitsunfairnessandmiseryandchaos,maybereasonabledeepdown,deepinitsheart,whereitfollowsmathematicallaws.Sometimeswecanfindthoselawsthroughscience.Sometimeswecanunderstandthemthroughcalculus.Andsometimeswecanusethemtoimproveourlives,helpoursocieties,andchangethecourseofhistoryforthebetter.Thepivotalmomentinthestoryofcalculusoccurredinthemiddleofthe

seventeenthcenturywhenthemysteriesofcurves,motion,andchangecollidedonatwo-dimensionalgrid,thexyplaneofFermatandDescartes.Backthen,FermatandDescarteshadnoideawhataversatiletoolthey’dcreated.Theyintendedthexyplaneasatoolforpuremathematics.Yetfromthestart,ittoowasacrossroadsofsorts,aplacewhereequationsmetcurves,algebramet

geometry,andthemathematicsoftheEastmetthatoftheWest.Then,inthenextgeneration,IsaacNewtonbuiltontheirworkaswellasontheworkofGalileoandKeplerandbroughtgeometryandphysicstogetherinagreatsynthesis.Newton’ssparksetoffthefirethatlittheEnlightenmentandcausedarevolutioninWesternscienceandmathematics.Buttotellthatstory,weneedtobeginwiththearenawhereitalltookplace,

thexyplane.Whenstudentstodaytaketheirfirstcourseincalculus,theyspendtheentireyearinthatplane.Thetermofartforthissubjectiscalculusoffunctionsofonevariable.Ourdiscussionofitwilloccupyusforthenextseveralchapters.Webeginherewithfunctions.Inthecenturiessincecurvescollidedwithmotionandchange,thexyplane

hasbecomeevermorevitalasahub.It’susedtodayineveryquantitativefieldtographdataanduncoverhiddenrelationships.Wecanuseittovisualizehowonevariabledependsonanother,howxrelatestoywheneverythingelseisheldconstant.Suchrelationshipsaremodeledbyfunctionsofonevariable.Theyarewrittensymbolicallyasy=f(x),whichispronounced“yequalsfofx.”Herefdenotesafunctionthatdescribeshowthevariabley(calledthedependentvariable)dependsonthevariablex(theindependentvariable),assumingeverythingelseisnaileddownandheldconstant.Suchfunctionsmodelhowtheworldbehavesatitstidiest.Acauseproducesapredictableeffect.Adosestimulatesapredictableresponse.Moreformally,afunctionfisarulethatassignsauniqueytoeachx.It’slikeaninput-outputmachine:feeditxanditspitsouty,anditdoessoreliablyandpredictably.AfewdecadesbeforeFermatandDescartes,Galileounderstoodthepowerof

thisdeliberatesimplificationofreality.Hemeticulouslychangedjustonethingatatimeinhisexperimentswhileholdingeverythingelseconstant.Heletaballrolldownarampandmeasuredhowfaritwentinacertainamountoftime.Niceandsimple—distanceasafunctionoftime.Likewise,Keplerstudiedhowlongittookaplanettoorbitthesunandrelatedthatperiodtotheplanet’saveragedistancefromthesun.Onevariableversusanother,periodversusdistance.Thiswasthewaytomakeprogress.Thiswasthewaytoreadthegreatbookofnature.We’veencounteredexamplesoffunctionsinpreviouschapters.Inthe

cinnamon-raisinbreadexample,xwasthenumberofsliceseatenandywasthenumberofcaloriesconsumed.Therelationshipinthatcasewasy=200x,whichproducedastraight-linegraphinthexyplane.AnotherexamplecameupwhenwestudiedhowthelengthofthedaychangedwiththeseasonsinNewYorkCityin2018.Inthatsetting,thevariablexrepresentedthedayoftheyearandywasthenumberofminutesofdaylightonthatday,definedasthetimefrom

sunrisetosunset.Wefoundthatthegraphinthatcaseoscillatedlikeasinewave,withthelongestdaysinthesummerandtheshortestinthewinter.

TheFunctionofFunctions

Somefunctionsaresoimportantthatthey’vebeengiventheirownbuttonsonascientificcalculator.Thesearemathematicalcelebritieslikex2andlogxand10x.Admittedly,mostpeopledon’thavemuchuseforthem.Theyaren’tneededformakingchangeordecidinghowmuchtotip.Ineverydaylife,numbersareusuallyenough.That’swhywhenyoupressthecalculatorapponyourphone,bydefaultitoffersyouabasiccalculatorwiththenumbersfrom0to9onit,aswellasthefourbasicoperationsofarithmetic—addition,subtraction,multiplicationanddivision—andabuttonforpercentages.Thoseareallmostofusneedaswegoaboutourbusiness.Butforpeopleintechnicalprofessions,numbersarejustthebeginning.

Scientists,engineers,financialquants,andmedicalresearchersneedtoworkwithrelationshipsbetweennumbers,whichshowhowonethingaffectsanother.Todescriberelationshipslikethat,functionsareindispensable.Theyprovidethetoolsneededtomodelmotionandchange.Generallyspeaking,thingscanchangeinoneofthreeways:theycangoup,

theycangodown,ortheycangoupanddown.Inotherwords,theycangrow,decay,orfluctuate.Differentfunctionsaresuitablefordifferentoccasions.Sincewe’regoingtobemeetingvariousfunctionsinthepagesahead,it’shelpfultorecallsomeofthemostusefulones.

PowerFunctions

Toquantifygrowthinitsmostgradualforms,weoftenusepowerfunctionslikex2orx3,inwhichavariablexisraisedtosomepower.Thesimplestoftheseisalinearfunction,inwhichthedependentvariabley

growsindirectproportiontox.Forexample,ifyisthenumberofcaloriesconsumedbyeating1,2,or3slicesofcinnamon-raisinbread,thenygrowsaccordingtotheequationy=200x,wherexisthenumberofslicesand200isthenumberofcaloriesperslice.However,thereisnoneedforaseparatex

buttononthecalculatorbecausemultiplicationservesthesamepurpose;here,200caloriestimesthenumberofslicesofbreadequalsthenumberofcaloriesconsumed.Butforthenextkindofgrowthinthehierarchy,thetypeknownasquadratic

growth,it’sveryhelpfultohaveanx2buttononthecalculator.Quadraticgrowthislessintuitivethanlineargrowth.It’snotjustamatterofmultiplication.Forexample,ifwechangexfrom1to2to3againandaskhowthecorrespondingvaluesofy=x2change,weseetheygofrom12=1to22=4to32=9.Thusthey-valuesgrowinincreasingsteps,firstbyΔy=4−1=3,thenbyΔy=9−4=5.Ifwekeepgoing,they’dincreaseby7,9,11,andsoon,followingthepatternofoddnumbers.Thus,forquadraticgrowth,theamountofchangeitselfgoesupasweincreasex.Thegrowthgrowsfasterasitproceeds.Wealreadysawthiscuriousodd-numberpatterninGalileo’sinclined-plane

experimentsinwhichhetimedballsastheyrolledslowlydownaramp.Heobservedthatwhenaballwasreleasedfromrest,itrolledfasterastimepassed,suchthatineachsuccessiveincrementoftime,ittraveledfartherandfarther,withthesuccessivedistancesgrowinginproportiontothesuccessiveoddnumbers1,3,5,andsoon.Galileorealizedwhatthiscrypticruleimplied.Itmeantthatthetotaldistancetheballrolledwasn’tproportionaltotime;itwasproportionaltotimesquared.Sointhestudyofmotion,thesquaringfunctionx2aroseverynaturally.

ExponentialFunctions

Incontrasttoamildpowerfunctionlikexorx2,anexponentialfunctionlike2xor10xdescribesamuchmoreexplosivekindofgrowth,agrowththatsnowballsandfeedsonitself.Insteadofaddingaconstantincrementateachstepasinlineargrowth,exponentialgrowthinvolvesmultiplyingbyaconstantfactor.Forexample,abacterialpopulationgrowingonapetridishdoublesevery20

minutes.Ifthereare1000bacterialcellsinitially,after20minutes,therewillbe2000cells.Afteranother20minutes,4000cells,and20minutesafterthat,8000cells,then16,000,32,000,andsoon.Inthisexample,theexponentialfunction2xcomesintoplay.Specifically,ifwemeasuretimeinunitsof20minutes,thenumberofbacteriaafterxunitsoftimewouldbe1000×2xcells.Similarexponentialgrowthisrelevanttoallsortsofsnowballingprocesses,fromthe

multiplicationofrealvirusestotheviralspreadofinformationinasocialnetwork.Exponentialgrowthisalsorelevanttothegrowthofmoney.Imaginealump

sumof$100sittinginabankaccountthatearnsaconstantannualinterestrateof1percent.Afteroneyear,thatsumwouldgrowto$101.Aftertwoyears,itwouldbecome$101times1.01,whichequals$102.01.Afterxyears,theamountofmoneyinthebankaccountwouldbe100×(1.01)x.Inexponentialfunctionslike2xand(1.01)x,thenumbers2and1.01arecalled

thefunction’sbase.Themostcommonlyusedbaseinprecalculusmathematicsis10.There’snomathematicalreasonforpreferring10overanyotherbase.It’satraditionalfavoritebecauseofanaccidentofbiologicalevolution:wehappentohavetenfingers.Accordingly,wehavebasedoursystemofarithmetic,thedecimalsystem,onpowersoften.Forthesamereason,theexponentialfunctionthatallbuddingscientists

encounterfirst,usuallyinhighschool,is10x.Herethenumberxiscalledtheexponent.Whenxis1,2,3,oranyotherpositivewholenumber,thatvalueofxindicateshowmanyfactorsoftenarebeingmultipliedtogetherin10x.Butwhenxiszero,negative,orinbetweentwowholenumbers,themeaningof10xisabitsubtler,aswe’reabouttosee.

PowersofTen

Therearemanysituationsinsciencewhereweusepowersoftentoeasecalculations.Inparticular,whennumbersareeitherverybigorverysmall,rewritingtheminscientificnotationisagoodidea.Scientificnotationusespowersoftentoexpressnumbersascompactlyaspossible.Takethenumbertwenty-onetrillion,muchtalkedaboutthesedaysin

connectionwiththeUSnationaldebt.Twenty-onetrillioncanbewritteneitherindecimalnotationas21,000,000,000,000,ormorecompactlyinscientificnotationas21×1012=2.1×1013.Ifforsomereasonweneedtomultiplythatbignumberby,say,onebillion,it’seasiertowrite(2.1×1013)×109=2.1×1022thanitistokeeptrackofallthosezerosindecimalnotation.Thefirstthreepowersoftenarenumberswerunintoeveryday:

1101=102102=100

3103=1000

Noticethetrend:Theleftcolumn(x)growsadditively,whereastherightcolumn(10x)growsmultiplicatively,asweexpectforexponentialgrowth.Thus,intheleftcolumn,eachupwardstepadds1totheprecedingnumber,whileintherightcolumnitmultipliestheprecedingnumberby10.Thisintriguingcorrespondencebetweenadditionandmultiplicationisahallmarkofexponentialfunctionsingeneralandpowersofteninparticular.Becauseofthiscorrespondencebetweenthetwocolumns,ifweaddtwo

numbersintheleftcolumn,thatoperationcorrespondstomultiplyingtheirpartnersintherightcolumn.Forexample,1+2=3onthelefttranslatesinto10×100=1000ontheright.Thetranslationfromadditiontomultiplicationmakessensebecause

101+2=103=101×102.Thus,whenwemultiplypowersoften,theirexponentsadd,as1and2do

here.Thegeneralruleis

10a×10b=10a+b.Arelatedtrendisthatsubtractionintheleftcolumncorrespondstodivisionin

therightcolumn:

3-2=1correspondsto1000⁄100=10

Theseniftypatternssuggesthowtocontinuethetwocolumnsdownward

towardsmallerandsmallernumbers.Theprincipleis,wheneverwesubtractby1intheleftcolumn,weshoulddivideby10intherightcolumn.Nowlookatthetoprowagain:

1101=102102=1003103=1000

Sincesubtracting1ontheleftamountstodividingby10ontheright,thecorrespondencecontinueswithanewtoprowthathas1−1=0ontheleftand10/10=1ontheright:

0100=11101=102102=1003103=1000

Thisreasoningexplainswhy100isdefinedas1(andhastobedefinedthatway),adefinitionthatmanypeoplefindpuzzling.Anyotherchoicewouldbreakthepattern.It’stheonlydefinitionthatcontinuesthetrendsestablishedfartherdowninthetwocolumns.Goingoninthisway,wecanextrapolatethecorrespondenceevenfurther,

nowtonegativenumbersintheleftcolumn.Thecorrespondingnumbersontherightthenbecomefractions,equivalenttopowersof1/10:

-21⁄100-11⁄1001110210031000

Noticethatthenumbersintherightcolumnalwaysremainpositive,evenwhenthenumbersintheleftcolumnbecomezeroornegative.Apotentialcognitivepitfallwhenusingpowersoftenisthattheycanmake

vastlydifferentnumbersseemmoresimilarthantheyreallyare.Toavoidthistrap,it’sgoodtopretendthatdifferentpowersoftenformconceptuallydistinctcategories.Sometimeshumanlanguagesdothisontheirownbyassigningdistinctnamestodifferentpowersoften,asiftheywereunrelatedspecies.InEnglishwereferto10,100,and1000withthreeunrelatedwords—ten,ahundred,andathousand.That’sgood.Itconveystherightideathatthesenumbersarequalitativelydifferent,eventhoughtheyareneighboringpowersoften.Anyonewhoappreciatesthedifferencebetweenafive-figureandasix-figuresalaryknowsthatoneextrazerocanmatteragreatdeal.

Whenthewordsforpowersoftensoundtoomuchalike,weareledastray.Duringthe2016USpresidentialcampaign,SenatorBernieSandersfrequentlyrailedagainsttheexorbitanttaxbreaksgoingto“millionairesandbillionaires.”Whetheryouagreedwithhimornotaboutthepolitics,heunfortunatelymadeitsoundlike,intermsofwealth,millionairesandbillionaireswerecomparable.Infact,billionairesaremuch,muchricherthanmillionaires.Tograsphowdifferentamillionisfromabillion,thinkaboutitlikethis:Amillionsecondsisalittleundertwoweeks;abillionsecondsisaboutthirty-twoyears.Thefirstisthelengthofavacation;thesecondisasignificantfractionofalifetime.Thelessonhereisthatweneedtousepowersoftenwithcare.Theyare

dangerouslystrongcompressors,capableofshrinkingenormousnumbersdowntosizeswecanfathommoreeasily.That’salsowhythey’resopopularwithscientists.Incontextsinwhichsomequantityvariesovermanyordersofmagnitude,powersoftenareoftenusedtodefineanappropriatemeasurementscale.ExamplesincludethepHscaleofacidityandbasicity,theRichterscaleofearthquakemagnitudes,andthedecibelmeasureofloudness.Forinstance,ifthepHofasolutionchangesfrom7(neutral,likepurewater)to2(acidic,likelemonjuice),theconcentrationofhydrogenionsincreasesbyfiveordersofmagnitude,meaningafactorof105,orahundredthousand.ThedropinpHfrom7to2makesitseemlikejustfiveitty-bittysteps,notmuchofachangeatall,eventhoughit’sreallyahundred-thousand-foldchangeinhydrogen-ionconcentration.

Logarithms

Intheexampleswehaveconsideredsofar,thenumbersintherightcolumn,like100and1000,havealwaysbeenroundnumbers.Sincepowersoftenaresoconvenient,itwouldbewonderfulifwecouldexpressnon-roundnumbersinthesamemanner.Take90,forinstance.Giventhat90isalittlelessthan100,and100equals102,itseemslike90shouldequal10raisedtosomenumberslightlylessthan2.Butraisedtowhatnumber,exactly?Logarithmswereinventedtoanswersuchquestions.Onacalculator,ifyou

typein90andthenpressthelogbutton,youget

log90=1.9542....

That’stheanswer:101.9542...=90.Inthisway,logarithmsenableustowriteanypositivenumberasapowerof

ten.Doingthatmakesmanycalculationseasierandalsorevealssurprisingconnectionsbetweennumbers.Lookwhathappensifwemultiply90byafactorof10or100andthentakeitslogagain:

log900≈2.9542...and

log9000=3.9542....Observetwostrikingthingshere:

1. Allthelogsherehavethesamedecimalpart,.9542....2. Multiplyingtheoriginalnumber,90,by10increaseditslogby1.

Multiplyingitby100increaseditslogby2,etc.

Wecanexplainbothofthesefactsbyappealingtoaruleoflogs:Thelogofaproductisthesumofthelogs.Here,

log90=log(9×10)=log9+log10=.9542...+1

and

log900=log(9×100)=log9+log100=.9542...+2

andsoon.Thisexplainswhythelogsof90and900and9000allhavethesamedecimalpart,.9542....Thatdecimalpartisthelogof9,and9isafactorthatappearsinallofthenumberswe’vebeendiscussing.Thedifferentpowersoftenshowupasthedifferentwhole-numberpartsinthelogs(inthiscase,1,2,or3infrontofthedecimalpart).Becauseofthis,ifweareinterestedinthelogsofothernumbers,weneedonlyworkoutthelogsofnumbersfrom1to10.Thattakescareofthedecimalparts.Thelogofeveryotherpositivenumbercanthen

beexpressedintermsofthoselogsalone.Powersoftenhavetheirownjob;theyaccountforthewhole-numberpart.Thegeneralruleinsymbolicformis

log(a×b)=loga+logb.

Inotherwords,whenwemultiplytwonumberstogetherandthentaketheirlog,theresultisthesum(nottheproduct!)oftheirindividuallogs.Inthatsense,logarithmsreplacemultiplicationproblemswithadditionproblems,whicharemucheasier.Thisiswhylogarithmswereinvented.Theyspedupcalculationstremendously.InsteadofhavingtodealwithHerculeanmultiplicationproblems,squareroots,cuberoots,andthelike,suchcalculationscouldbeturnedintoadditionproblemsandthensolvedwiththehelpofalookuptableknownasatableoflogarithms.Theideaoflogarithmswasintheairintheearlyseventeenthcentury,butmuchofthecreditforpopularizingthemgoestotheScottishmathematicianJohnNapier,whopublishedhisDescriptionoftheWonderfulRuleofLogarithmsin1614.Adecadelater,JohannesKeplerenthusiasticallyusedthenewcalculationaltoolwhenhewascompilingastronomicaltablesaboutthepositionsoftheplanetsandotherheavenlybodies.Logarithmswerethesupercomputersoftheirera.Manypeoplefindlogarithmsconfusing,buttheymakealotofsenseifyou

thinkaboutthembyanalogywithcarpentry.Logarithmsandotherfunctionsareliketools.Differenttoolshavedifferentpurposes.Hammersareforpoundingnailsintowood;drillsareforboringholes;sawsareforcutting.Likewise,exponentialfunctionsareformodelinggrowththatfeedsonitself,andpowerfunctionsareformodelinglessviolentformsofgrowth.Logarithmsareusefulforthesamereasonthatstapleremoversareuseful:theyundotheactionofanothertool.Specifically,logarithmsundotheactionsofexponentialfunctions,andviceversa.Considertheexponentialfunction10xandapplyittoanumber,say3.The

resultis1000.Toundothataction,pressthelogxbutton.Applyingitto1000returnsthenumberwestartedwith:3.Thebase-10logarithmfunctionlogxundoestheactionofthe10xfunction.Theyareinversefunctionsinthissense.Asidefromtheirroleasinversefunctions,logarithmsalsodescribemany

naturalphenomena.Forexample,ourperceptionofpitchisapproximatelylogarithmic.Whenamusicalpitchgoesupbysuccessiveoctaves,fromonedotothenext,thatincreasecorrespondstosuccessivedoublingsofthefrequencyoftheassociatedsoundwaves.Yetalthoughthewavesoscillatetwiceasfastfor

everyoctaveincrease,wehearthedoublings—whicharemultiplicativechangesinfrequency—asequalupwardstepsinpitch,meaningequaladditivesteps.It’sfreaky.Ourmindsfoolusintobelievingthat1isasfarfrom2as2isfrom4,andas4isfrom8,andsoon.Wesomehowsensefrequencylogarithmically.

TheNaturalLogarithmandItsExponentialFunction

Asusefulasbase10wasinitsheyday,itisrarelydeployedinmoderncalculus.Ithasbeensupersededbyanotherbasethatlooksabstrusebutturnsouttobefarmorenaturalthan10.Thisnaturalbaseiscallede.It’sanumbercloseto2.718(I’llexplainwhereitcomesfrominaminute),butitsnumericalvalueisbesidethepoint.Theimportantpointabouteisthatanexponentialfunctionwiththisbasegrowsataratepreciselyequaltothefunctionitself.Letmesaythatagain.Therateofgrowthofexisexitself.Thismarvelouspropertysimplifiesallcalculationsaboutexponential

functionswhentheyareexpressedinbasee.Nootherbaseenjoysthissimplicity.Whetherweareworkingwithderivatives,integrals,differentialequations,oranyoftheothertoolsofcalculus,exponentialfunctionsexpressedinbaseearealwaysthecleanest,mostelegant,andmostbeautiful.Asidefromitssimplifyingroleincalculus,baseearisesnaturallyinfinance

andbanking.Thefollowingexamplewillrevealwherethenumberecomesfromandhowitisdefined.Imagineyoudeposit$100inabankthatpaysinterestatanimplausiblebut

irresistibleannualrateof100percent.Thatmeansthatafteroneyear,your$100wouldbecome$200.Nowstartoverandconsideranevenmorefavorablescenario.Imaginethatyoucouldpersuadethebanktocompoundyourmoneytwiceayearsothatyoucouldgaininterestontheinterestasyourmoneygrows.Howmuchmorewouldyouearninthatcase?Sinceyou’reaskingthebanktocompoundthemoneytwiceasoften,it’sonlyfairthattheinterestrateforeachsix-monthperiodshouldbehalfaslarge,namely50percent.Thus,aftersixmonths,you’dhave$100×1.50,whichequals$150.Sixmonthslater,atyear’send,theamountwouldbeanother50percentmore:$150×1.50,whichequals$225.That’smorethanthe$200yougotundertheoriginalarrangementbecauseyougainedinterestontheinterestduringtheyear.

Thenextquestionis,whathappensifyoucouldgetthebanktocompoundyourmoneymoreandmorefrequently,atcorrespondinglysmallerinterestratesduringeachcompoundingperiod?Wouldyouachievefabulouswealth?Unfortunately,no.Compoundingquarterlywouldyield$100×(1.25)4≈$244.14,notmuchofanimprovementover$225.Compoundingstillmorerapidly,onceadayforthe365daysintheyear,wouldleaveyouwithonly

$100×(1+1⁄365)365≈$271.46

attheendoftheyear.Herethe365inboththedenominatorandtheexponentreferstothenumberofcompoundingperiodsintheyear,andthe1inthenumeratorof1⁄365isthe100percentinterestrateexpressedasadecimal.Finally,supposewetakethiscompoundingmadnesstothelimit.Ifthebank

compoundsyourmoneyntimesayearwherenisamonstrouslyhugenumber,withcorrespondinglytinyinterestratesduringeachsub-nanosecondperiod,thenbyanalogywiththeresultfor365dailyperiods,you’dhave

$100×(1+1⁄n)n

inyouraccountatyear’send.Asnapproachesinfinity,thisamountapproaches100timesthelimitof

(1+1⁄n)n

asnapproachesinfinity.Thatlimitisdefinedasthenumbere.It’snotatallobviouswhatthelimitingnumberis,butitturnsouttobeapproximately2.71828....Inthebankingworld,thefinancialarrangementaboveiscalledcontinuous

compounding.Ourresultsshowthatitisnothingtogetexcitedabout.Intheproblemabove,itwouldyieldayear-endbalanceof

$100×e≈$271.83.That’sthebestdealyet,butit’sonly37centsmorethantheresultofdailycompounding.Wejustjumpedthroughalotofhoopstodefinee.Intheend,eturnedoutto

beacomplicatedlimit.Ithasinfinitybuiltintoitinmuchthesamewaythatthe

numberπdoesforcircles.Recallthatπinvolvedacalculationoftheperimeterofamany-sidedpolygoninscribedinacircle.Thatpolygonapproachedthecircleasthenumberofsides,n,approachedinfinityandthelengthsofthosesidesapproachedzero.Thenumbereisdefinedinasomewhatsimilarwayasalimit,exceptthatitarisesinthedifferentcontextofcontinuouslycompoundedgrowth.Theexponentialfunctionassociatedwitheiswrittenasex,justasthe

exponentialfunctionforbase10iswrittenas10x.Itlooksweirdatfirst,butatastructurallevelit’sjustlikebase10.Alltheprinciplesandpatternsarethesame.Forexample,ifwewanttofindanxsuchthatexisagivennumber,say90,wecanagainuselogarithmsaswedidbefore,exceptnowwewheeloutthebase-elogarithm,betterknownasthenaturallogarithmanddenotedlnx.Tofindtheunknownxsuchthatex=90,turnonascientificcalculator,enter90,pressthelnxbutton,andthere’syouranswer:

ln90≈4.4498.Tocheckit,keepthatnumberonthescreenandhittheexbutton.Youshouldget90.Asbefore,logsandexponentialsundoeachother’sactionslikeastaplerandastapleremover.Reconditeasallthismaysound,thenaturallogarithmisextremelypractical,

thoughofteninconspicuously.Foronething,itunderliesaruleofthumbknowntoinvestorsandbankersastheruleof72.Toestimatehowlongitwilltaketodoubleyourmoneyatagivenannualrateofreturn,divide72bytherateofreturn.Thus,moneygrowingata6percentannualratedoublesafterabout72/6=12years.Thisruleofthumbfollowsfromthepropertiesofthenaturallogarithmandexponentialgrowthandworkswelliftheinterestrateislowenough.Naturallogarithmsalsooperatebehindthescenesinthecarbondatingofancienttreesandbonesandinart-authenticationdisputes.AfamouscaseinvolvedpaintingsallegedlybyVermeerthatturnedouttobeforgeries;thiswasrevealedbyanalysisoftheradioactivelydecayingisotopesofleadandradiuminthepaint.Astheseexamplessuggest,thenaturallogarithmnowpervadesallfieldswhereexponentialgrowthanddecayarise.

TheMechanismBehindExponentialGrowthandDecay

Toreiteratethemainpoint,thethingthatmakesespecialisthattherateofchangeofexisex.Hence,asthegraphofthisexponentialfunctionsoarshigherandhigher,itsslopealwaystiltstomatchitscurrentheight.Thehigheritgets,thesteeperitclimbs.Inthejargonofcalculus,exisitsownderivative.Nootherfunctioncansaythat.It’sthefairestofthemall—atleastasfarascalculusisconcerned.Althoughbaseeisuniquelydistinguished,otherexponentialfunctionsobeya

similarprincipleofgrowth.Theonlydifferenceisthattherateofexponentialgrowthisproportionaltothefunction’scurrentlevel,notstrictlyequaltoit.Still,thatproportionalityissufficienttogeneratetheexplosivenessweassociatewithexponentialgrowth.Theexplanationfortheproportionalityshouldbeintuitivelyclear.Inbacterial

growth,forexample,biggerpopulationsgrowfasterbecausethemorecellsthereare,themoreofthemareavailabletodivideandmakeoffspring.Thesameistruewiththegrowthofmoneyinanaccountbeingcompoundedataconstantinterestrate.Moremoneymeansmoreinterestonthatmoneyandhenceafasterrateofgrowthoftheoverallaccount.Thiseffectalsoaccountsforthehowlofamicrophonewhenitpicksupthe

soundofitsownloudspeaker.Theloudspeakercontainsanamplifierthatmakesasoundlouder.Ineffect,itmultipliesthevolumeofthesoundbyaconstantfactor.Ifthatloudersoundgetspickedupbythemicrophoneandthensentbackthroughtheamplifieragain,itsvolumewillbeamplifiedrepeatedlyinapositivefeedbackloop.Thiscausesasuddenexponentialrunawayofvolume,growingatarateproportionaltothecurrentvolumeandleadingtotheawfulscreechingsound.Nuclearchainreactionsaregovernedbyexponentialgrowthforthesame

reason.Whenauraniumatomsplits,itfiresoutneutronsthatcanpotentiallysmashintootheratomsandcausethemtosplit,sendingoutstillmoreneutrons,andsoon.Theexponentialgrowthofthenumberofneutrons,ifleftunchecked,cansetoffanuclearexplosion.Alongwithgrowth,decaycanbedescribedbyexponentialfunctions.

Exponentialdecayoccurswhensomethingisbeingdepletedorconsumedatarateproportionaltoitscurrentlevel.Forexample,halftheatomsinanisolatedlumpofuraniumalwaystakethesameamountoftimetodecayradioactively,nomatterhowmanyatomswerepresentinthelumpinitially.Thatdecaytimeisknownasthehalf-life.Theconceptappliestootherfieldsaswell.Inchapter8we’lldiscusswhatdoctorslearnedaboutAIDSaftertheydiscoveredthatthenumberofvirusparticlesinthebloodstreamsofHIV-infectedpatientsdropped

exponentially,withahalf-lifeofonlytwodays,afteramiracledrugcalledaproteaseinhibitorwasadministered.Thesediverseexamples,fromthedynamicsofchainreactionsand

microphonefeedbackhowltotheaccumulationofmoneyinabankaccount,makeitseemlikeexponentialfunctionsandtheirlogarithmsarefirmlyplantedinthepartofcalculusthatdealswithchangesintime.Andit’struethatexponentialgrowthanddecayareprominenttopicsonthemodernsideofthecrossroadsofcalculus.Butlogarithmswerefirstsightedontheotherside,backwhencalculuswasstillfocusedonthegeometryofcurves.Indeed,thenaturallogarithmaroseearlyoninstudiesoftheareaunderthehyperbolay=1/x.Theplotthickenedinthe1640swhenitwasdiscoveredthattheareaunderthehyperboladefinedafunctionthatbehaveduncannilylikealogarithm.Infact,itwasalogarithm.Itobeyedthesamestructuralrulesandturnedproblemsofmultiplicationintoproblemsofaddition,justlikeanyotherlogarithm,butitsbasewasunknown.Therewasstillmuchtobelearnedabouttheareasundercurves.Thatwasto

beoneofthetwogreatchallengesaheadforcalculus.Theotherwastodeviseamoresystematicmethodtofindthetangentlinesandslopesofcurves.Thesolutionofthesetwoproblemsandthediscoveryofthesurprisingconnectionbetweenthemwouldsoontakecalculus,andtheworld,decisivelyintomodernity.

6

TheVocabularyofChange

FROMATWENTY-FIRST-CENTURYvantagepoint,calculusisoftenseenasthemathematicsofchange.Itquantifieschangeusingtwobigconcepts:derivativesandintegrals.Derivativesmodelratesofchangeandarethemaintopicofthischapter.Integralsmodeltheaccumulationofchangeandwillbediscussedinchapters7and8.Derivativesanswerquestionslike“Howfast?”“Howsteep?”and“How

sensitive?”Theseareallquestionsaboutratesofchangeinoneformoranother.Arateofchangemeansachangeinadependentvariabledividedbyachangeinanindependentvariable.Insymbols,arateofchangealwaystakestheformΔy/Δx,achangeinydividedbyachangeinx.Sometimesotherlettersareused,butthestructureisthesame.Forexample,whentimeistheindependentvariable,it’scustomaryandclearertowritetherateofchangeasΔyΔt,wheretdenotestime.Themostfamiliarexampleofarateisaspeed.Whenwesayacarisgoing

100kilometersanhour,thatnumberqualifiesasarateofchangebecauseitdefinesspeedasaΔyΔtwhenitstateshowfarthecargoes(Δy=100kilometers)inagivenamountoftime(Δt=1hour).Likewise,accelerationisarate.It’sdefinedastherateofchangeofspeed,

usuallywrittenΔvΔt,wherevstandsforvelocity.WhentheAmericancarmanufacturerChevroletclaimsthatoneofitsmusclecars,theV-8CamaroSS,

cangofrom0to60milesperhourin4secondsflat,they’requotingaccelerationasarate:achangeinspeed(from0to60milesperhour)dividedbyachangeintime(4seconds).Theslopeofarampisathirdexampleofarateofchange.It’sdefinedasthe

ramp’sverticalriseΔydividedbyitshorizontalrunΔx.Asteepramphasalargeslope.Awheelchair-accessiblerampisrequiredbyUSlawtohaveaslopelessthan1/12.Flatgroundhaszeroslope.Ofallthevariousratesofchangethatexist,theslopeofacurveinthexy

planeisthemostimportantanduseful,becauseitcanstandinforalltherest.Dependingonwhatxandyrepresent,theslopeofacurvecanindicateaspeed,anacceleration,arateofpay,anexchangerate,themarginalreturnonaninvestment,oranyotherkindofrate.Forexample,whenweplottedthenumberofcalories,y,containedinxslicesofcinnamon-raisinbread,thegraphwasalinewithaslopeof200caloriesperslice.Thatslope,ageometricalfeature,toldustherateatwhichthebreaddeliverscalories,anutritionalfeature.Similarly,onagraphofdistanceversustimeforamovingcar,theslopeindicatesthecar’sspeed.Thus,slopeisasortofuniversalrate.Sinceanyfunctionofonevariablecanalwaysbegraphedasacurveonthexyplane,wecanfinditsrateofchangebyreadingofftheslopeofitsgraph.Thecatchisthatratesofchangearerarelyconstantintherealworldorin

mathematics.Inthatcase,definingaratebecomesproblematic.Thefirstbigissueindifferentialcalculusistodefinewhatwemeanbytheratewhentherateofchangekeepschanging.SpeedometersandGPSdeviceshavesolvedthisproblem.Theyalwaysknowwhatspeedtoreportevenifacarspeedsupandslowsdown.Howdothesegadgetsdoit?Whatcalculationaretheymaking?Withcalculus,we’llsee.Justasspeedsdon’tneedtobeconstant,slopesdon’tneedtobeconstant

either.Onacurvelikeacircleoraparabolaoranyothersmoothpath(aslongasit’snotaperfectlystraightline),theslopeisboundtobesteeperinsomeplacesandshallowerinothers.That’strueintherealworldtoo.Mountaintrailshavetreacheroussteepsectionsandrestfulflatsections.Sothequestionremains:Howdowedefinetheslopewhentheslopekeepschanging?Thefirstthingtorealizeisthatweneedtoexpandourconceptofwhatarate

ofchangeis.Inalgebraproblemsthatinvolvedistanceequalsratetimestime,therateisalwaysaconstant.Thatisnotthecaseincalculus.Becausespeeds,slopes,andotherratesvaryastheindependentvariablexortchanges,theyhavetoberegardedasfunctionsthemselves.Ratesofchangecannolongerbemerenumbers.Theyneedtobecomefunctions.

Thisiswhattheconceptofaderivativedoesforus.Itdefinesarateofchangeasafunction.Itspecifiesarateatagivenpointoratagiventime,evenifthatrateisvariable.Inthischapter,we’llseehowderivativesaredefined,whattheymean,andwhytheymatter.Toletthecatoutofthebag,derivativesmatterbecausethey’reubiquitous.At

theirdeepestlevel,thelawsofnatureareexpressedintermsofderivatives.It’sasiftheuniverseknewaboutratesofchangebeforewedid.Atamoremundanelevel,derivativescomeupwheneverwewanttoquantifyhowachangeinsomethingisrelatedtoachangeinsomethingelse.Howmuchdoesraisingthepriceofanappaffecttheconsumerdemandforit?Howmuchdoesincreasingthedoseofastatindrugenhanceitsabilitytolowerapatient’scholesterolorincreaseitsriskoftriggeringsideeffectslikeliverdamage?Wheneverwestudyarelationshipofanykind,wewanttoknow:Ifonevariablechanges,howmuchdoesarelatedvariablechange?Andinwhatdirection,upordown?Thesearequestionsaboutderivatives.Theaccelerationofarocketship,thegrowthrateofapopulation,themarginalreturnonaninvestment,thetemperaturegradientinabowlofsoup—derivatives,oneandall.Incalculus,thesymbolforthederivativeisdy/dx.It’ssupposedtoremindyou

ofanordinaryrateofchangeΔy/Δx,exceptthatthetwochangesdyanddxarenowimaginedtobeinfinitesimallytiny.That’sawildnewideathatwe’llworkourwayupto,slowlyandgently,thoughitshouldn’tcomeasasurprise.WeknowfromtheInfinityPrinciplethatthewaytomakeprogressoncomplicatedproblemsistochopthemintoinfinitesimalbits,analyzethebits,andthenputthebitsbacktogethertofindtheanswer.Thelittlechangesdxanddyarethoseinfinitesimalbitsinthecontextofdifferentialcalculus.Puttingthembacktogetheristhejobofintegralcalculus.

TheThreeCentralProblemsofCalculus

Toprepareourselvesforwhatliesahead,weneedtohavethebigpictureinmindfromthestart.Therearethreecentralproblemsincalculus.Theyareshownschematicallyonthediagrambelow.

1. Theforwardproblem:Givenacurve,finditsslopeeverywhere.2. Thebackwardproblem:Givenacurve’sslopeeverywhere,findthecurve.3. Theareaproblem:Givenacurve,findtheareaunderit.

Thediagramshowsthegraphofagenericfunctiony(x).Ihaven’tsaidwhatxandyrepresentbecauseitdoesn’tmatter.Thepictureiscompletelygeneral.Itshowsacurveintheplane.Thatcurvecouldrepresentanyfunctionofonevariableandsocouldapplytoanybranchofmathematicsorsciencewheresuchfunctionsarise,whichisessentiallyeverywhere.Thesignificanceofitsslopeandareawillbeexplainedlater.Fornow,justthinkofthemaswhattheyare:aslopeandanarea.Thekindofthingthatgeometerswouldworryabout.Wecanviewthiscurveintwoways,oneoldandonenew.Intheearly

seventeenthcentury,beforecalculusarrived,suchcurveswereviewedasgeometricalobjects.Theywereconsideredfascinatingintheirownright.Mathematicianswantedtoquantifytheirgeometricalproperties.Givenacurve,theywantedtobeabletofigureouttheslopeofitstangentlineateachpoint,thearclengthofthecurve,theareabeneaththecurve,andsoon.Inthetwenty-firstcentury,wearemoreinterestedinthefunctionthatproducedthecurve,whichmodelssomenaturalphenomenonortechnologicalprocessthatmanifesteditselfinthecurve.Thecurveisdata,butsomethingdeeperunderliesit.Todaywethinkofthecurveasfootprintsinthesand,acluetotheprocessthatmadeit.Thatprocess—modeledbyafunction—iswhatweareinterestedin,notthetracesitleftbehind.Thiscollisionbetweenthesetwopointsofviewishowthemysteryofcurves

collidedwiththemysteriesofmotionandchange.It’showancientgeometrycollidedwithmodernscience.Eventhoughweareinmoderntimesnow,I’vechosentodrawthepicturefromtheolderperspectivebecausethexyplaneissofamiliar.Itofferstheclearestwaytograspthethreecentralproblemsof

calculus,becauseallthreecanbereadilyvisualizedwhenweposethemingeometricterms.(Thesameideascanalsobereformulatedintermsofmotionandchangeusingdynamicalideaslikespeedanddistanceinsteadofcurvesandslope,butwewilldothatlater,oncewehaveabettergrasponthegeometry.)Thequestionsshouldbeinterpretedinthesenseoffunctions.Inotherwords,

whenIspeakabouttheslopeofthecurve,Idon’tjustmeanatonespecificpoint.Imeanatanarbitrarypointx.Theslopechangesaswemovealongthecurve.Ourgoalistounderstandhowitchangesasafunctionofx.Similarly,theareaunderthecurvedependsonx.I’veshownitshadedingray,andlabeleditwiththesymbolA(x).Thatareashouldalsoberegardedasafunctionofx.Asweincreasex,theverticaldashedlineslidestotheright,andtheareaexpands.Sotheareadependsonwhichxwechoose.These,then,arethethreecentralproblems.Howcanwefigureoutthe

changingslopeofacurve?Howcanwereconstructacurvefromitsslope?Andhowcanwefigureoutthechangingareabeneaththecurve?Asphrasedinthecontextofgeometry,thosequestionsmightsoundpretty

dry.Butoncewereinterpretthemintherealworld,fromthetwenty-first-centurypointofviewasproblemsaboutmotionandchange,theybecomephenomenallywide-reaching.Slopesmeasureratesofchange;areasmeasuretheaccumulationofchange.Assuch,slopesandareasariseineveryfield—physics,engineering,finance,medicine,younameit,anywherethatchangeisanabidingconcern.Understandingtheproblemsandtheirsolutionsopensuptheuniverseofmodernquantitativethinking,atleastaboutfunctionsofonevariable.Forthesakeoffulldisclosure,Ishouldmentionthere’smuchmoretocalculusthanthat;therearefunctionsofmanyvariables,differentialequations,andthelike.Allingoodtime.We’llgettothoselater.Thischapterisconcernedwithfunctionsofonevariableandtheirderivatives

(theirratesofchange),startingwithfunctionsthatchangeataconstantrateandthenmovingontotheknottierissueoffunctionsthatchangeatachangingrate.That’swheredifferentialcalculusreallyshines—inmakingsenseofever-changingchange.Oncewe’vegottencomfortablewithratesofchange,we’llbereadytotackle

theaccumulationofchange,themorechallengingtopicofthenextchapter.Thereitwillberevealedthattheforwardproblemandthebackwardproblem,asdifferentastheyseem,aretwinsseparatedatbirth,ashockercalledthefundamentaltheoremofcalculus.Itrevealedthatratesofchangeandtheaccumulationofchangearemuchmorecloselyrelatedthananyonehadsuspected,adiscoverythatunifiedthetwohalvesofcalculus.Butfirst,let’sbeginatthebeginningwithrates.

LinearFunctionsandTheirConstantRates

Manysituationsineverydaylifearedescribedbylinearrelationshipsinwhichonevariableisproportionaltoanother.Forexample:

1. Lastsummermyolderdaughter,Leah,gotherfirstjob,ataclothingstoreinthemall.Sheearned$10anhour,sowhensheworkedfortwohours,shemade$20.Moregenerally,whensheworkedforthours,shemadeydollars,wherey=10t.

2. Acardrivesdownthehighwayat60milesperhour.Thus,afteronehouritgoes60miles.Aftertwohours,itgoes120miles.Afterthours,itgoes60tmiles.Therelationshiphereisy=60t,whereyisthenumberofmilesdriveninthours.

3. AccordingtotheAmericanswithDisabilitiesAct,awheelchair-accessiblerampmustnotrisebymorethan1inchforevery12inchesofhorizontalrun.Forarampwiththismaximumpermissiblegradient,therelationshipbetweenriseandrunisy=x/12,whereyistheriseandxistherun.

slope=rise/run

Ineachoftheselinearrelationships,thedependentvariablechangesataconstantratewithrespecttotheindependentvariable.Mydaughter’srateofpaywasaconstant$10perhour.Thecar’sspeedisaconstant60milesperhour.Andthewheelchair-accessibleramphasaconstantslope,definedasitsriseoverrun,equalto1/12.Thesameistrueofthatcinnamon-raisinbreadIliketoeat;itdeliverscaloriesataconstantrateof200caloriesperslice.

Inthetechnicaljargonofcalculus,aratealwaysmeansaquotientoftwochanges:achangeinydividedbyachangeinx,writteninsymbolsasΔy/x.Forexample,ifIeattwomoreslicesofbread,Ipackonanother400calories.Thusthecorrespondingrateis

Δy⁄Δx=400calories⁄2slices

whichsimplifiesto200caloriesperslice.Nosurprisethere.Butwhat’sinterestingtoobserveisthatthisrateisconstant.It’sthesamenomatterhowmanyslicesI’vealreadyeaten.

Whenarateisconstant,it’stemptingtothinkofitassimplybeinganumber,

like200caloriespersliceor$10anhouroraslopeof1/12.Thatcausesnoharmhere,butitwouldgetusintotroublelater.Inmorecomplicatedsituations,rateswillnotbeconstant.Forexample,considerawalkthrougharollinglandscape,wheresomepartsofthehikearesteepandothersareflat.Onarollinglandscape,slopeisafunctionofposition.Itwouldbeamistaketothinkofitasamerenumber.Likewise,whenacaracceleratesorwhenaplanetorbitsthesun,itsspeedchangesincessantly.Thenit’svitaltoregardspeedasafunctionoftime.Soweshouldgetinthathabitnow.Weshouldstopthinkingofratesofchangeasnumbers.Ratesarefunctions.Thepotentialconfusionarisesbecausetheratefunctionsareconstantforthe

linearrelationshipswe’vebeenconsidering.That’swhyitdoesnoharmtotreatthemasnumbersinalinearcontext.Theydon’tchangeaswechangetheindependentvariable.Mydaughter’srateofpayis$10anhour,nomatterhowmuchsheworks,andtheslopeoftherampis1/12everywherealongitslength.Butdon’tletthatfoolyou.Thoseratesarestillfunctions.Theyjusthappentobeconstantfunctions.Thegraphofaconstantfunctionisaflatline,asshownhereforthecinnamon-raisinbreadwithitsconstantpayloadof200caloriesperslice.

Whenwedealinthenextsectionwitharelationshipthatisnotlinear,wewill

seethatitgeneratesacurve,notaline,whengraphedinthexyplane.Eitherway,alineoracurvealwaysrevealsalotabouttherelationshipthatproducedit.It’slikearelationship’smugshotorsignature.It’sacluethatrevealswhatmadeit.Noticethedistinctionbetweenafunctionandthegraphofthefunction.A

functionisadisembodiedrulethateatsxsandspitsoutysanddoessouniquely,oneyforeachx.Inthatsense,afunctionisincorporeal.There’snothingtolookatwhenyoulookatafunction.It’saghostlyentity,anabstractrule.Forexample,therulemightbe“FeedmeanumberandIwillreturn10timesthenumber.”Bycontrast,thegraphofafunctionisavisible,almosttangiblething,ashapeyoucansee.Specifically,thegraphofthefunctionIjustdescribed

wouldbealinethroughtheoriginwithaslopeof10,definedbytheequationy=10x.Butthefunctionitselfisnottheline.Thefunctionistherulethatproducestheline.Tomakeafunctionmanifestitself,youneedtofeeditanx,letitspitoutay,andrepeatthatforallxsandplottheresults.Whenyoudothat,thefunctionitselfstaysinvisible.Whatyou’reseeingisitsgraph.

ANonlinearFunctionandItsChangingRate

Whenafunctionisnotlinear,itsrateofchangeΔy/Δxisnotconstant.Ingeometricalterms,thatmeansthegraphofthefunctionisacurvewithaslopethatchangesfrompointtopoint.Asanexample,considertheparabolashownbelow.

It’sthecurvey=x2,whichcorrespondstothesimplestnonlinearbuttononthecalculator,thesquaringfunctionx2.Thisexamplewillgiveussomepracticewiththedefinitionofaderivativeastheslopeofthetangentlineandalsoclarifywhylimitsenterthatdefinition.Inspectingtheparabola,weseethatsomepartsofitaresteepandsomeparts

arerelativelyflat.Theflattestpartofalloccursatthebottomoftheparabolaatthepointwherex=0.Therewecansee,withoutdoinganywork,thatthederivativemustbezero.Ithastobezerobecausethetangentlineatthebottom

isevidentlythex-axis.Viewedasaramp,thatlineisnoriseandallrunandhencehasaslopeofzero.Butatotherpointsontheparabola,it’snotimmediatelyobviouswhatthe

slopeofthetangentlineshouldbe.Infact,it’snotobviousatall.Tofigureitout,let’sdoanEinstein-stylethoughtexperiment.We’llimaginewhatwewouldseeifwecouldzoominonanarbitrarypoint(x,y)ontheparabolaasifweweremakingphotographicenlargementsofthatpoint,alwayskeepingitinthecenterofourfieldofview.It’slikewe’relookingatapieceofthecurveunderamicroscopeandincreasingthemagnificationprogressively.Aswezoomincloserandcloser,thatpieceoftheparabolashouldbegintolookstraighterandstraighter.Inthelimitofinfinitemagnification(whichamountstozoominginonaninfinitesimalpieceofthecurvearoundthepointofinterest),thatmagnifiedpieceshouldapproachastraightline.Ifitdoes,thatlimitingstraightlineisdefinedasthetangentlineatthatpointonthecurve,anditsslopeisdefinedasthederivativethere.NoticethatweareusingtheInfinityPrinciplehere—wearetryingtomakea

complicatedcurvesimplerbychoppingitintoinfinitesimalstraightpieces.Thisiswhatwealwaysdoincalculus.Curvedshapesarehard;straightshapesareeasy,evenifthereareinfinitelymanyofthemandeveniftheyareinfinitesimallysmall.CalculatingaderivativeinthiswayisaquintessentialcalculusmoveandoneofthemostfundamentalapplicationsoftheInfinityPrinciple.Toconductthethoughtexperiment,weneedtoselectapointonthecurveto

zoominon.Anypointwilldo,butanumericallyconvenientchoiceisthepointthatliesontheparabolaabovex=½.Inthediagramabove,I’vemarkedthatpointwithadot.Inthexyplaneitliesat

(x,y)=(½,¼)

or,indecimalnotation,(x,y)=(0.5,0.25).Thereasonthatyequals¼atthispointisthat,inordertoqualifyasapointontheparabola,thepointmustobeyy=x2,asallpointsontheparabolado;afterall,thisiswhatdefinesapointasamemberoftheparaboliccurve.Thus,atx=½,thepointmusthaveay-valueof

y=x2=(½)2=¼

Nowwearereadytozoominonthepointofinterest.Placethepoint(x,y)=(0.5,0.25)atthecenterofthemicroscope.Withthehelpofcomputergraphics,

zoominonalittlepieceofthecurvesurroundingthatpoint.Thefirstmagnificationisshownhere.

Theoverallshapeoftheparabolaislostinthismagnifiedview.Instead,wejustseeaslightlycurvedarc.Thissmallpieceoftheparabola,whichliesbetweenx=0.3and0.7,appearsalotlesscurvedthantheparabolaasawhole.Zoominfurtherbyblowingupthepiecebetweenx=0.49and0.51.Thisnew

enlargementlooksevenstraighterthanthelastonedid,thoughit’snottrulystraight,sinceit’sstillaportionoftheparabola.

Thetrendisclear.Aswekeepzoomingin,thepieceslookstraighter.By

measuringtheriseoverrun,Δy/x,forthisalmost-straightpieceandzoomingincloserandcloser,weareeffectivelytakingthelimitofthepiece’sslope,Δy/Δx,asΔxgoestozero.Thecomputergraphicsstronglysuggestthattheslopeofthealmost-straightlineisgettingcloserandcloserto1,correspondingtoalineata45-degreeangle.Withabitofalgebra,wecanprovethatthelimitingslopeisexactly1.(In

chapter8we’llseehowsuchcalculationsaredone.)Furthermore,performingthesamecalculationatanyx,notjustatx=½,revealsthatthelimitingslope—andhencetheslopeofthetangentline—equals2xatanypoint(x,y)ontheparabola.Orinthelingoofcalculus:

Thederivativeofx2is2x.Temptingasitistoprovethisderivativerulebeforemovingon,fornowlet’s

acceptitandseewhatitmeans.Foronething,itsaysthatatthedotwherex=½,theslopeshouldequal2x=2×(½)=1,whichisjustwhatwesawinthecomputergraphics.Italsopredictsthatatthebottomoftheparabolaatx=0,theslopeshouldbe2×0,whichiszero,andwe’vealreadyseenthat’scorrecttoo.

Finally,the2xformulapredictsthattheslopeshouldincreaseasweascendtheparabolatotheright;whenxgetsbigger,theslope(=2x)shouldalsogetbigger,whichmeanstheparabolashouldgetsteeper,anditdoes.Ourexperimentwiththeparabolahelpsusunderstandacoupleofcaveats

aboutderivatives.Aderivativeisdefinedonlyifacurveapproachesalimitingstraightlineaswezoominonit.Thatwon’tbethecaseforcertainpathologicalcurves.Forexample,ifacurvehasaVshapewithasharpcorneratonepoint,thenwhenwezoominonthatpoint,itwillcontinuetolooklikeacorner.Thecornernevergoesaway,nomatterhowmuchwemagnifythecurve.Itwillneverlookstraightthere.Becauseofthis,aV-shapedcurvedoesnothaveawell-definedtangentlineoraslopeatthecorner,andhenceitdoesnothaveaderivativethere.However,whenacurvedoeslookincreasinglystraightwhenwezoominonit

sufficientlyatanypoint,thatcurveissaidtobesmooth.Throughoutthisbook,Ihavebeenassumingthatthecurvesandprocessesofcalculusaresmooth,justastheearlypioneersdid.Inmoderncalculus,however,wehavelearnedhowtocopewithcurvesthatarenotsmooth.Theinconveniencesandpathologiesofnon-smoothcurvessometimesariseinapplicationsduetosuddenjumpsorotherdiscontinuitiesinthebehaviorofaphysicalsystem.Forexample,whenweflipaswitchinanelectricalcircuit,thecurrentgoesfromnotflowingatalltosuddenlyflowingsignificantly.Agraphofcurrentversustimewouldshowanabrupt,almost-verticalriseapproximatedbyadiscontinuousjumpasthecurrentturnson.Sometimesit’smoreconvenienttomodelthatabrupttransitionasatrulydiscontinuousjump,inwhichcasethecurrentasafunctionoftimewillnothaveaderivativeatthemomenttheswitchflipped.Muchofthefirstcourseincalculusinhighschoolorcollegeisdevotedto

calculatingderivativerulesliketheoneaboveforx2butfortheotherbuttonsonthecalculator,like“thederivativeofsinxequalscosx”or“thederivativeoflnxequals1/x.”Forourpurposes,however,it’smoreimportanttounderstandtheideaofthederivativeandtoseehowitsabstractdefinitionappliesinpractice.Forthat,let’sturntotherealworld.

DerivativesasRatesofChangeofDayLength

Inchapter4,welookedatdataonseasonalchangesofdaylength.Althoughourpurposeatthetimewastoillustrateideasaboutsinewaves,curvefitting,and

datacompression,wecannowrepurposethosedatatoilluminatevariableratesofchangeandbringderivativesdowntoearthinanothersetting.Theearlierdataconcernedthenumberofminutesofdaylight—thetime

betweensunriseandsunset—inNewYorkCityoneachdayoftheyearin2018.Therelevantderivativeinthiscontextistherateatwhichthedayslengthenedorshortenedfromonedaytothenext.OnJanuary1,forinstance,thetimefromsunrisetosunsetwas9hours,19minutes,and23seconds.OnJanuary2itgotalittlelonger:9hours,20minutes,and5seconds.Thatextra42secondsofdaylight(equivalentto0.7minutes)wasameasureofhowrapidlythedayswerelengtheningonthatparticulardayoftheyear.Theyweregettinglongeratarateofabout0.7minutesperday.Forcomparison,considertherateofchangetwoweekslater,onJanuary15.

Betweenthatdayandthenext,theamountofdaylightincreasedby90seconds,correspondingtoarateoflengtheningof1.5minutesperday,morethantwicetherateof0.7measuredtwoweeksearlier.Thus,thedayswerenotonlylengtheninginJanuary;theywerelengtheningfasterwitheachpassingday.Thiswelcometrendcontinuedforthenextseveralweeks.Daytimekept

gettinglonger—anddidsomorerapidly—withthecomingofspring.Onthespringequinox,March20,therateofincreasetoppedoutataglorious2.72minutesofextrasunlighteachday.Youcanspotthatdayontheearliergraphinchapter4.It’sday79,aboutaquarterofthewayinfromtheleft,wherethewaveofdaylengthrisesmoststeeply.Thatmakessense—wherethegraphissteepest,it’sclimbingmostrapidly,whichmeansthederivativeislargestandthedaysarelengtheningasquicklyaspossible.Allofthishappensonthefirstdayofspring.Foramelancholycontrast,considertheshortestdaysoftheyear.Theypacka

doublewhammy.Inthosedarkdaysofwinter,thedaysarenotonlydepressinglyshort;theyalsodonotchangemuchfromonedaytothenext,whichonlyaddstothetorpor.Butthisalsomakessense.Theshortestdaysoccuratthebottomofthewaveofdaylength,andatthebottom,thewaveisflat(otherwiseitwouldn’tbeabottom;itwouldbeimprovingorworsening).Butbecauseitisflatatthebottom,itsderivativeiszerothere,whichmeansitsrateofchangegrindstoahalt,atleastmomentarily.Ondayslikethat,itcanfeellikespringwillnevercome.I’vehighlightedtwotimesofyearthathaveemotionalmeaningformanyof

us,aroundthespringequinoxandthewintersolstice,butit’sevenmoreinstructivetoconsidertheyearasawhole.Totracktheseasonalvariationsintherateofchangeofdaylength,I’vecomputeditatperiodicintervalsthroughouttheyear,startingonJanuary1andcontinuingeverytwoweeksafterthat.Theresultsareplottedinthegraphbelow.

Theverticalaxisshowsthedailyrateofchange,thatis,theadditionalminutes

ofdaylightfromonedaytothenext.Thehorizontalaxisshowswhatdayitis,withdaysnumberedfrom1(January1)to365(December31).Therateofchangebobsupanddownlikeawave.Itstartsoutpositiveinthe

latewinterandearlyspring,whenthedaysaregettinglonger,andpeaksaroundday79(thespringequinox,March20).Aswealreadyknow,that’swhenthedaysarelengtheningmostrapidly,around2.72minutesperday.Butafterthat,it’salldownhill.Theratestartstodropandgoesnegativeafterthesummersolsticeonday172(June21).Itbecomesnegativebecausethedaysstartshorteningthen;thenextdayhasfewerhoursofdaylightthanthecurrentone.TheratebottomsoutaroundSeptember22whenthelightisfadingfastest,anditstaysnegative(butnotasnegative)untilthewintersolsticeonday355(December21)whenthedaysstartgettinglongeragain,evenifimperceptibly.It’sfascinatingtocomparethiswavetothewavewemetearlierinchapter4.

Whenthey’replottedtogetherandrescaledtohavecomparableamplitudes,here’swhattheylooklike.

(I’mshowingtwoyears’worthofdataheretoemphasizetherepetitivenessofthewaves.Andtoheightenthecomparisonbetweenthem,I’vealsoconnectedthedotsandremovedthenumbersfromtheverticalaxistofocusmoreattentiononthewaves’shapeandtiming.)Thefirstthingtonoticeisthatthewavesareoutofsync.Theydon’tpeak

simultaneously.Thewaveofdaylengthpeaksaroundhalfwaythroughtheyear,whereasitsrateofchangepeaksaboutthreemonthsearlier.Thatamountstoaquarterofacycleearlier,giventhateachwavetakestwelvemonthstocompleteitsup-and-downmovement.Theotherthingtonoticeisthatthewavesresembleeachother,withslight

differences.Althoughtheyshowclearfamilyties,thedashedwaveislesssymmetricalthanthesolidoneanditspeaksandtroughsareflatter.I’mgoingintoallthisbecausethesereal-worldwavesofferaglimpse,as

throughaglassdarkly,ofaremarkablepropertyofsinewaves,namely,whenavariablefollowsaperfectsine-wavepattern,itsrateofchangeisalsoaperfectsinewavetimedaquarterofacycleahead.Thisself-regenerationpropertyisuniquetosinewaves.Nootherkindsofwaveshaveit.Itcouldevenbetakenasadefinitionofsinewaves.Inthatsense,ourdatahintatamarvelousphenomenonofrebirthinherentinperfectsinewaves.(WewillhavemoretosayaboutthiswhenitcomesupagaininconnectionwithFourieranalysis,apowerfuloffshootofcalculusthathasledtosomeofitsmostexcitingapplicationstoday.)

Letmetrytogiveyousomeinsightintowherethequarter-cycleshiftcomesfrom.Thesameconceptexplainswhysinewavesbegetsinewaveswhenwecomputetheirratesofchange.Thekeyisthatsinewavesareconnectedtouniformcircularmotion.Recallthatwhenapointmovesaroundacircleataconstantspeed,itsup-and-downmotiontracesasinewaveintime.(Forthatmatter,sodoesitsleftandrightmotion.)Withthatinmind,considerthediagrambelow.

Itshowsapointmovingclockwisearoundacircle.Thepointisnotsupposed

torepresentanythingphysicalorastronomical.It’snottheEarthorbitingthesun,anditdoesn’thaveanythingtodowiththeseasons.It’sjustanabstractpointmovingaroundacircle.Itseastwarddisplacement(or“eastiness,”forshort)increasesanddecreaseslikeasinewave.Whenthepointreachesitsmaximumeastiness,asshowninthediagram,that’sanalogoustothemaximumofasinewave,orthelongestdayoftheyear.Thequestionis:Whenthepointismaximallyeastandthesinewaveisatitspeakeastiness,whathappensnext?Asthediagramshows,atitseasternmostpoint,thepointheadssouth,asindicatedbythedownwardarrow.Butsouthis90degreesawayfromeastonacompass,and90degreesisaquarterofacycle.Aha!That’swherethequarter-cycleoffsetcomesfrom.Becauseofthegeometryofacircle,there’salwaysaquarter-cycle

offsetbetweenanysinewaveandthewavederivedfromitasitsderivative,itsrateofchange.Inthisanalogy,thepoint’sdirectionoftravelislikeitsrateofchange.Itdetermineswherethepointwillgonextandhencehowitchangesitslocation.Moreover,thiscompassheadingofthearrowitselfrotatesinacircularfashionataconstantspeedasthepointgoesaroundthecircle,sothecompassheadingofthearrowfollowsasine-wavepatternintime.Andsincethecompassheadingisliketherateofchange,voilà!Therateofchangefollowsasine-wavepatterntoo.That’stheself-regenerationpropertyweweretryingtounderstand.Sinewavesbegetsinewaveswitha90-degreeshift.(ExpertswillrealizethatI’mtryingtoexplainwithoutformulaswhythederivativeofthesinefunctionisthecosinefunction,whichisitselfjustasinefunctionshiftedbyaquartercycle.)Asimilar90-degreephaselagoccursinotheroscillatingsystems.Whena

pendulumswingsbackandforth,itsspeedisatitsmaximumwhenitgoesthroughitsbottom,whereasitsangleisatitsmaximumaquartercyclelaterwhenthependulumisfarthesttotheright.Agraphoftheangleversustimeandthespeedversustimeshowstwoapproximatesinewaves,oscillatingoutofphaseby90degrees.Anotherexamplecomesfromasimplifiedmodelofpredator-preyinteractions

inbiology.Imagineapopulationofsharkspreyingonapopulationoffish.Whenthefishareattheirmaximumpopulationlevel,thesharkpopulationgrowsatitsmaximumratebecausetherearesomanyfishtoeat.Thesharkpopulationcontinuestoclimbandreachesitsownmaximumlevelaquartercyclelater,bywhichtimethefishpopulationhasdropped,havingbeenpreyedonsoseverelyaquartercycleearlier.Ananalysisofthismodelshowsthatthetwopopulationsoscillateoutofphaseby90degrees.Similarpredator-preyoscillationsareseenelsewhereinnature,forexample,inannualfluctuationsofCanadianhareandlynxpopulationsasrecordedbyfur-trappingcompaniesinthe1800s(thoughtherealexplanationforthoseoscillationsisundoubtedlymorecomplicated,asisoftenthecaseinbiology).Returningtotheday-lengthdata,weseethat,alas,theyarenotperfectsine

waves.They’realsoaninherentlydiscretesetofpoints,justoneperday,withnodataexistinginbetween.Assuch,theydonotprovidethesortofcontinuumofpointsthatcalculusinsistson.Soforourfinalexampleofaderivative,let’sturntoacasewherewecancollectdatawithasmuchresolutionaswelike,rightdowntothemillisecond.

DerivativesasInstantaneousSpeeds

TheeveningofAugust16,2008,waswindlessinBeijing.Attenthirty,theeightfastestmenintheworldlinedupfortheOlympicfinalsofthe100-meterdash.Oneofthem,atwenty-one-year-oldJamaicansprinternamedUsainBolt,wasarelativenewcomertothisevent.Knownmoreasa200-meterman,he’dbeggedhiscoachforyearstolethimtryrunningtheshorterrace,andoverthepastyearhe’dbecomeverygoodatit.Hedidn’tlookliketheothersprinters.Hewasgangly,6feet,5inches(1.96

meters)tall,withalong,lopingstride.Asaboyhehadfocusedonsoccerandcricketuntilhiscricketcoachnoticedhisspeedandsuggestedthathetryoutfortrack.Asateenagerhekeptimprovingasarunner,buthenevertookthesportorhimselftooseriously.Hewasgoofyandmischievousandhadafondnessforpracticaljokes.OnthatnightinBeijing,afteralltheathleteshadbeenintroducedandfinished

muggingfortheTVcameras,thestadiumgotquiet.Thesprintersplacedtheirfeetintheblocksandcrouchedintoposition.Anofficialcalledout,“Onyourmarks.Set,”andthenfiredthestartingpistol.Boltshotoutoftheblocks,butnotquiteasexplosivelyastheother

Olympians.Hisslowerreactiontimelefthimseventhoutofeightnearthestart.Gainingspeed,bythirtymetershemoveduptothemiddleofthepack.Then,stillacceleratinglikeabullettrain,heputdaylightbetweenhimselfandtherestofthefield.

Ateightymeters,heglancedtohisrighttoseewherehismaincompetitorswere.Whenherealizedhowfaraheadhewas,hesloweddownvisibly,droppedhisarmstohissides,andslappedhischestashecruisedacrossthefinishline.Somecommentatorssawthisasbragging,othersasgleefulcelebration,butinanycase,Boltclearlydidn’tfeeltheneedtorunhardattheend,whichledtospeculationaboutjusthowfasthecouldhaverun.Asitwas,evenwithhiscelebration(andanuntiedshoelace)hesetanewworldrecordof9.69seconds.Oneofficialcriticizedhimforbeingunsportsmanlike,butBoltdidn’tmeananydisrespect.Ashelatertoldreporters,“That’sjustme.Iliketohavefun,juststayrelaxed.”Howfastdidherun?Well,100metersin9.69secondstranslatesto100/9.69

=10.32meterspersecond.Inmorefamiliarunits,that’sabout37kilometersperhour,or23milesperhour.Butthatwashisaveragespeedoverthewholerace.Hewentslowerthanthatatthebeginningandendandfasterthanthatinthemiddle.Moredetailedinformationisavailablefromhissplittimesrecordedevery10

metersdownthetrack.Hecoveredthefirst10metersin1.83seconds,correspondingtoanaveragespeedof5.46meterspersecondthere.Hisfastestsplitsoccurredat50to60meters,60to70meters,and70to80meters.Heblazedthroughthose10-metersectionsin0.82secondeach,foranaveragespeedof12.2meterspersecond.Inthefinal10meters,whenheeasedupandbrokeform,hedeceleratedtoanaveragespeedof11.1meterspersecond.Humanbeingshaveevolvedtospotpatterns,soinsteadofporingover

numberslikewe’vejustbeendoing,it’susuallymoreinformativetovisualizethem.ThefollowinggraphshowstheelapsedtimesatwhichBoltcrossed10meters,20meters,30meters,andsoon,uptothe9.69secondsittookhimtocrossthefinishlineatthe100-metermark.

I’veconnectedthedotswithstraightlinesasaguidetotheeye,butkeepin

mindthatonlythedotsarerealdata.Togetherthedotsandthelinesegmentsbetweenthemformapolygonalcurve.Theslopesofthesegmentsareshallowestontheleft,correspondingtoBolt’slowerspeedatthestartoftherace.Theybendupwardastheymovetotheright;thatmeanshe’saccelerating.Thentheyjointoformanearlystraightline,indicatingthehighandsteadyspeedthathemaintainedformostoftherace.It’snaturaltowonderatwhattimehewasrunninghisabsolutefastestand

whereonthetrackthatoccurred.Weknowthathisfastestaveragespeed,overa10-metersection,occurredsomewherebetween50and80meters,butanaveragespeedover10metersisnotquitewhatwewant;weareinterestedinhispeakspeed.ImaginethatUsainBoltwaswearingaspeedometer.Atwhatprecisemomentwasherunningthefastest?Andexactlyhowfastwasthat?Whatwe’relookingforhereisawayofmeasuringhisinstantaneousspeed.

Theconceptseemsalmostparadoxical.Atanyinstant,UsainBoltwasatpreciselyoneplace.Hewasfrozen,asinasnapshot.Sowhatwoulditmeantospeakofhisspeedatthatinstant?Speedcanonlyoccuroveratimeinterval,notinasingleinstant.Theenigmaofinstantaneousspeedgoesfarbackinthehistoryof

mathematicsandphilosophy,toaround450BCEwithZenoandhisredoubtableparadoxes.RecallthatinhisparadoxofAchillesandthetortoise,Zenoclaimedthatafasterrunnercouldneverovertakeaslowerrunner,despitewhatUsainBoltprovedthatnightinBeijing.AndintheParadoxoftheArrow,Zenoargued

thatanarrowinflightcouldnevermove.Mathematiciansarestillunsurewhatpointhewastryingtomakewithhisparadoxes,butmyguessisthatthesubtletiesinherentinthenotionofspeedataninstanttroubledZeno,Aristotle,andotherGreekphilosophers.TheiruneasinessmayexplainwhyGreekmathematicsalwayshadsolittletosayaboutmotionandchange.Likeinfinity,thoseunsavorytopicsseemtohavebeenbanishedfrompoliteconversation.TwothousandyearsafterZeno,thefoundersofdifferentialcalculussolved

theriddleofinstantaneousspeed.Theirintuitivesolutionwastodefineinstantaneousspeedasalimit—specifically,thelimitofaveragespeedstakenovershorterandshortertimeintervals.It’slikewhatwedidwhenwezoomedinontheparabola.There,we

approximatedasmallerandsmallerpieceofasmoothcurvewithastraightline.Thenweaskedwhathappensinthelimitofinfinitemagnification.Bystudyingthelimitingvalueoftheline’sslope,wewereabletodefinethederivativeataparticularpointonthesmoothlycurvingparabola.Here,byanalogy,wewouldliketoapproximatesomethingchanging

smoothlyintime:UsainBolt’sdistancedownthetrack.Theideaistoreplacethegraphofhisdistanceversustimewithapolygonalcurvechangingataconstantaveragespeedovershorttimeintervals.Iftheaveragespeedoneachintervalapproachesalimitasthosetimeintervalsgetshorterandshorter,thatlimitingvalueiswhatwemeanbytheinstantaneousspeedatagiventime.Likeslopeatapoint,speedataninstantisaderivative.Forallthistosucceed,wehavetoassumehisdistancedownthetrackvaried

smoothly.Otherwisethelimitwe’reinvestigatingwon’texist,andneitherwillthederivative.Theresultswon’tapproachanythingsensibleastheintervalsgetshorter.Butdidhisdistanceactuallyvarysmoothlyasafunctionoftime?Wedon’tknowforsure.TheonlydatawehavearediscretesamplesofBolt’selapsedtimesateachoftheten-metermarkersonthetrack.Toestimatehisinstantaneousspeed,weneedtogobeyondthedataandmakeaneducatedguessaboutwherehewasattimesinbetweenthosepoints.Asystematicwaytomakesuchaguessisknownasinterpolation.Theideais

todrawasmoothcurvebetweenthedataavailable.Inotherwords,wewanttoconnectthedots,notbystraight-linesegmentsaswe’vealreadydone,butbythemostplausiblesmoothcurvethatgoesthroughthedots,oratleastthatgoesveryclosetothem.Theconstraintsweimposeonthiscurvearethatitshouldbetautandnotundulatetoomuch;itshouldpassasclosetoallthedotsaspossible;anditshouldshowthatBolt’sinitialspeedwaszero,sinceweknowhewasmotionlesswhenhewasinthecrouchposition.Therearemanydifferentcurvesthatmeetthesecriteria.Statisticianshavedevisedahostoftechniquesforfitting

smoothcurvestodata.Allofthemgivesimilarresults,andsincetheyallinvolveabitofguessworkanyway,let’snotbothertoomuchaboutwhichonetouse.Here’soneexampleofasmoothcurvethatmeetsalltherequirements.

Sincethecurveissmoothbydesign,wecancalculateitsderivativeateverypoint.TheresultinggraphgivesusanestimateofUsainBolt’svelocityateachinstantofhisrecord-settingracethatnightinBeijing.

ItindicatesthatBoltreachedatopspeedofaround12.3meterspersecondat

aboutthethree-quarterpointintherace.Untilthen,he’dbeenaccelerating,gainingspeedateachmoment.Afterthathedecelerated,somuchsothathisspeeddroppedto10.1meterspersecondashecrossedthefinishline.Thegraphconfirmswhateveryonesaw;Boltsloweddowndramaticallyneartheend,especiallyinthelasttwentymeters,whenherelaxedandcelebrated.Thenextyear,atthe2009WorldChampionshipsinBerlin,Boltputanendto

thespeculationabouthowfasthecouldgo.Nochestthumpsthistime.HeranhardtothefinishandshatteredhisBeijingworldrecordof9.69secondswithanevenmoreastonishingtimeof9.58seconds.Becauseofthegreatanticipationsurroundingthisevent,biomechanicalresearcherswereonhandwithlaserguns,similartotheradargunsusedbypolicetocatchspeeders.Thesehigh-techinstrumentsallowedtheresearcherstomeasurethesprinters’positionsahundredtimesasecond.WhentheycomputedBolt’sinstantaneousspeed,thisiswhattheyfound:

Thelittlewigglesontheoveralltrendrepresenttheupsanddownsinspeed

thatinevitablyoccurduringstrides.Running,afterall,isaseriesofleapingsandlandings.Bolt’sspeedchangedalittlewheneverhelandedafootonthegroundandmomentarilybraked,thenpropelledhimselfforwardandlaunchedhimselfairborneagain.Intriguingastheyare,theselittlewigglesareannoyingandbothersometoa

dataanalyst.Whatwereallywantedtoseewasthetrend,notthewiggles,andforthatpurpose,theearlierapproachoffittingasmoothcurvetothedatawas

justasgoodandarguablybetter.Aftercollectingallthathigh-resolutiondataandobservingthewiggles,theresearchershadtocleanthemoffanyway.Theyfilteredthemouttounmaskthemoremeaningfultrend.Tome,thesewigglesholdalargerlesson.Iseethemasametaphor,akindof

instructionalfableaboutthenatureofmodelingrealphenomenawithcalculus.Ifwetrytopushtheresolutionofourmeasurementstoofar,ifwelookatanyphenomenoninexcruciatinglyfinedetailintimeorspace,wewillstarttoseeabreakdownofsmoothness.InUsainBolt’sspeeddata,thewigglestookasmoothtrendandmadeitlookasbushyasapipecleaner.Thesamethingwouldhappenwithanyformofmotionifwecouldmeasureitatthemolecularscale.Downatthatlevel,motionbecomesjitteryandfarfromsmooth.Calculuswouldnolongerhavemuchtotellus,atleastnotdirectly.Yetifwhatwecareaboutaretheoveralltrends,smoothingoutthejittersmaybegoodenough.Theenormousinsightthatcalculushasgivenusintothenatureofmotionandchangeinthisuniverseisatestamenttothepowerofsmoothness,approximatethoughitmaybe.There’sonelastlessonhere.Inmathematicalmodeling,asinallofscience,

wealwayshavetomakechoicesaboutwhattostressandwhattoignore.Theartofabstractionliesinknowingwhatisessentialandwhatisminutia,whatissignalandwhatisnoise,whatistrendandwhatiswiggle.It’sanartbecausesuchchoicesalwaysinvolveanelementofdanger;theycomeclosetowishfulthinkingandintellectualdishonesty.Thegreatestscientists,likeGalileoandKepler,somehowmanagetowalkalongthatprecipice.“Art,”saidPicasso,“isaliethatmakesusrealizetruth.”Thesamecouldbe

saidforcalculusasamodelofnature.Inthefirsthalfoftheseventeenthcentury,calculusbegantobeusedasapowerfulabstractionofmotionandchange.Inthesecondhalfofthatcentury,thesamekindsofartisticchoices—theliesthatrevealedthetruth—preparedthewayforarevolution.

7

TheSecretFountain

INTHESECONDhalfoftheseventeenthcentury,IsaacNewtoninEnglandandGottfriedWilhelmLeibnizinGermanychangedthecourseofmathematicsforever.Theytookaloosepatchworkofideasaboutmotionandcurvesandturneditintoacalculus.Noticetheindefinitearticle.WhenLeibnizintroducedthewordcalculusin

thiscontextin1673,hespokeof“acalculus”andsometimes,moreaffectionately,“mycalculus.”Hewasusingthewordinitsgenericsense,asystemofrulesandalgorithmsforperformingcomputations.Later,afterhissystemwasbroughttoahighpolish,itsaccompanyingarticlewasupgradedtothedefinite,andthefieldbecameknownasthecalculus.Butnow,sadtosay,itsarticlesandpossessiveshaveallgoneaway.Whatremainsiscalculus,humdrumandgray.Articlesaside,thewordcalculusitselfhasstoriestotell.Itcomesfromthe

Latinrootcalx,meaningasmallstone,areminderofatimelongagowhenpeopleusedpebblesforcountingandthusforcalculations.Thesamerootgivesuswordslikecalcium,chalk,andcaulk.Yourdentistmightusethewordcalculustorefertothatgunkonyourteeth,thetinypebblesofsolidifiedplaquethehygienistscrapesoffwhenyougoforacleaning.Doctorsusethesamewordforgallstones,kidneystones,andbladderstones.Inacruelirony,bothNewton

andLeibniz,thepioneersofcalculus,diedinexcruciatingpainwhilesufferingfromcalculi—abladderstoneforNewton,akidneystoneforLeibniz.

Areas,Integrals,andtheFundamentalTheorem

Althoughcalculushadoncebeenaboutcountingwithstones,bythetimeofNewtonandLeibnizitwasdevotedtocurvesandtheirnewfangledanalysisthroughalgebra.Thirtyyearsearlier,FermatandDescarteshaddiscoveredhowtousealgebratofindthemaxima,minima,andtangentsofcurves.Whatremainedelusiveweretheareasofcurvesor,moreprecisely,theareasofregionsboundedbycurves.Thisareaproblem,classicallyknownasthequadrature,orsquaring,of

curves,hadconsumedandfrustratedmathematiciansfortwothousandyears.Manyingenioustrickshadbeendevisedtosolveparticularcases,fromArchimedes’sworkontheareaofthecircleandthequadratureoftheparabolatoFermat’ssolutionfortheareaunderthecurvey=xn.Butwhatwaslackingwasasystem.Areaproblemsweretackledonanadhocbasis,casebycase,asifthemathematicianwerestartingovereachtime.Thesamedifficultybesetproblemsaboutthevolumesofcurvedsolidsandthe

lengthsofcurvedarcs.Indeed,Descartesthoughtarclengthswerebeyondhumancomprehension.Inhisbookongeometry,hewrote,“Theratiowhichexistsbetweenstraightandcurvedlinesisnotknown,andevencannot,inmyjudgment,beknownbyman.”Alltheseproblems—areas,arclengths,andvolumes—requiredinfinitesumsofinfinitesimallysmallpieces.Inmodernparlance,theyallinvolvedintegrals.Nobodyhadasurefiresystemforanyofthem.ThisiswhatchangedafterNewtonandLeibniz.Theyindependently

discoveredandprovedafundamentaltheoremthatmadesuchproblemsroutine.Thetheoremconnectedareastoslopesandtherebylinkedintegralstoderivatives.Itwasastonishing.LikeatwistoutofaDickensnovel,twoseeminglydistantcharactersweretheclosestofkin.Integralsandderivativeswererelatedbyblood.Theimpactofthisfundamentaltheoremwasbreathtaking.Almostovernight,

areasbecametractable.Questionsthatearliersavantshadstrainedtosolvecouldnowbedispatchedinamatterofminutes.AsNewtonwrotetoafriendofhis,“Thereisnocurvedlineexpressedbyanyequation...butIcaninlessthanhalf

aquarterofanhourtellwhetheritmaybesquared.”Realizinghowincrediblethisclaimwouldsoundtohiscontemporaries,hecontinued,“Thismayseemaboldassertion...butit’splaintomebythefountainIdrawitfrom,thoughIwillnotundertaketoproveittoothers.”Newton’ssecretfountainwasthefundamentaltheoremofcalculus.Although

heandLeibnizweren’tthefirsttonoticethistheorem,theygetthecreditforitbecausetheywerethefirsttoproveitingeneral,recognizeitsoverwhelmingutilityandimportance,andbuildanalgorithmicsystemaroundit.Themethodstheydevelopedarenowcommonplace.Integralshavebeendefangedandturnedintohomeworkexercisesforteenagers.Rightnow,millionsofstudentsinhighschoolandcollegeallaroundthe

worldaregrindingawayontheircalculusproblemsets,solvingintegralafterintegralwiththehelpofthefundamentaltheorem.Yetmanyofthemareoblivioustothegiftthey’vebeengiven.Perhapsunderstandablyso—it’sliketheoldjokeaboutthefishwhoaskshisfriend,“Aren’tyougratefulforwater?”towhichtheotherfishsays,“What’swater?”Studentsincalculusareswimminginthefundamentaltheoremallthetime,sonaturallytheytakeitforgranted.

VisualizingtheFundamentalTheoremwithMotion

Thefundamentaltheoremcanbeunderstoodintuitivelybythinkingaboutthedistancetraveledbyamovingbodylikearunneroracar.Byacquaintingourselveswiththiswayofthinking,we’lllearnwhatthefundamentaltheoremsays,whyit’strue,andwhyitmatters.It’snotjustatrickforfindingareas.It’sthekeytopredictingthefutureofanythingwecareabout(inthecaseswherewecan)andforunlockingthesecretsofmotionandchangeintheuniverse.ThefundamentaltheoremoccurredtoNewtonwhenhelookedatthearea

problemdynamically.Hisbrainstormwastoinvitetimeandmotionintothepicture.Lettheareaflow,saidhe.Letitexpandcontinuously.Thesimplestillustrationofhisideatakesusbacktothefamiliarproblemofa

carmovingataconstantspeedforwhichdistanceequalsratetimestime.Aselementaryasthisexamplemaybe,itstillcapturestheessenceofthefundamentaltheorem,soit’sagoodplacetostart.Imagineacarcruisingdownthehighwayat60milesperhour.Ifweplotits

distanceversustimeand,beneaththat,itsspeedversustime,theresulting

distanceandspeedgraphslooklikethis:

Lookatdistanceversustimefirst.Afteronehourthecarhastraveled60

miles,andaftertwohours,120miles,andsoon.Ingeneral,distanceandtimearerelatedbyy=60t,whereydenotesthedistancethecarhastraveleduptotimet.I’llrefertoy(t)=60tasthedistancefunction.Asshowninthetopdiagram,thegraphofthedistancefunctionisastraightlinewithaslopeof60milesperhour.Thatslopetellsusthecar’sspeedateveryinstantifwedidn’talreadyknowit.Inaharderproblem,thespeedmightfluctuate,buthereit’sasimpleconstantfunction,v(t)=60atallt,graphedastheflatlineinthebottompanelofthediagram.(Herevstandsforvelocity,asynonymforspeed.)

Havingseenhowspeedmanifestsitselfonthedistancegraph(astheslopeoftheline),wenowturnthequestionaroundandask:Howdoesdistancerevealitselfonthespeedgraph?Inotherwords,istheresomevisualorgeometricfeatureofthespeedgraphthatwouldallowustoinferhowfarthecarhastraveleduptoanygiventimet?Yes.Thedistancetraveledistheareaaccumulatedunderthespeedcurve(theflatline)uptotimet.Toseewhy,supposethecardrivesforsomeparticularamountoftime,saya

halfanhour.Inthatcasethedistancetraveledwouldbe30miles,sincedistanceequalsratetimestimeand60×½=30.Thecoolthing,andthepointofallthis,isthatwecanreadoffthatdistanceastheareaofthegrayrectangleundertheflatlinebetweentimest=0andt=½hour.

Therectangle’sheightof60milesperhourtimesitsbaseof½hourgivesthe

rectangle’sarea,30miles,whichrecoversthedistancetraveled,asclaimed.Thesamereasoningworksforanytimet.Thebaseoftherectanglethen

becomestanditsheightisstill60soitsareais60t,and,indeed,that’sthedistancewewereexpectingtofind,y=60t.So,atleastinthisexample,wherespeedwasperfectlyconstantandthespeed

curvewassimplyaflatline,thekeytorecoveringdistancefromspeedwastocomputetheareaunderthespeedcurve.Newton’sinsightwasthisequalitybetweenareaanddistancealwaysholds,evenifthespeedisnotconstant.Nomatterhowerraticallysomethingmoves,theareaaccumulatedunderitsspeedcurveuptotimetalwaysequalsthetotaldistanceithastraveleduptothattime.

That’soneversionofthefundamentaltheorem.Itseemstooeasytobetrue,butitistrue.Newtonwasledtoitbythinkingofareaasaflowing,movingquantity,notas

afrozenmeasureofashape,aswasthencustomaryingeometry.Hebroughttimeintogeometryandvieweditlikephysics.Ifhewerealivetoday,perhapshewouldhavevisualizedthepictureaboveasananimation,morelikeaflipbookthanasnapshot.Todothis,lookatthepictureaboveonelasttime,butnowimagineitasasingleframeinamovieorasinglepageinaflipbook.Astheanimationplaysinourminds,whatwouldweseethegrayrectangledo?Wewouldseeitexpandingsideways.Why?Becauseitsbasehaslengtht,whichgrowsastimepasses.Ifwecouldmakeaframeforeachtimeandreplaytheminsequence,likeflippingthepagesofaflipbook,theanimatedversionofthegrayrectanglewouldlooklikeitwasstretchingtotheright.Itwouldresembleapistonexpandingorasyringelyingonitsside,pullinggrayfluidintoitself.Thatgrayfluidrepresentstheexpandingareaoftherectangle.Wethinkofthe

areaas“accumulating”underthespeedcurvev(t).Inthiscase,theareaaccumulateduptotimetisA(t)=60t,andthatcoincideswiththedistancethecarhastraveled,y(t)=60t.Thus,theaccumulatedareaunderthespeedcurvegivesthedistanceasafunctionoftime.That’sthemotionversionofthefundamentaltheorem.

ConstantAcceleration

We’reworkingourwayuptoNewton’sgeneralgeometricversionofthefundamentaltheorem,whichisphrasedintermsofanabstractcurvey(x)andtheareaA(x)accumulatedbeneathit.Theideaofareaaccumulationisthekeytoexplainingthetheorem,butIrealizethisideatakessomegettingusedto,solet’sapplyittoonemoreconcreteproblemaboutmotionbeforetacklingtheabstractgeometriccase.Consideranobjectthatmoveswithaconstantacceleration.Thatmeansit

keepsgoingfasterandfasterwithaspeedthatrampsupataconstantrate.It’sroughlylikewhatwouldhappenifyouweretofloorthegaspedalinyourcar,startingfromrest.Afteronesecond,thecarmightbegoing,perhaps,10milesperhour;aftertwoseconds,20milesperhour;afterthreeseconds,30milesperhour,andsoon.Inthishypotheticalexample,thecaralwaysgains10milesperhourwitheachpassingsecond.Thisrateofchangeofspeed,10milesperhourpersecond,isdefinedasthecar’sacceleration.(Forsimplicity,weareignoring

thefactthatarealcarhasatopspeeditcan’texceedandthatitsaccelerationmightnotbestrictlyconstantwhenyoufloorthegaspedal.)Inouridealizedexample,thecar’sspeedateachmomentisgivenbythe

linearfunctionv(t)=10t.Herethenumber10signifiesthecar’sacceleration.Iftheaccelerationweresomeotherconstant,saya,theformulawouldgeneralizeto

v(t)=at.Whatwewanttoknowis,foracarpeelingoutlikethis,howfardoesitgo

betweentime0andtimet?Inotherwords,howdoesitsdistancefromthestartingpointincreaseasafunctionoftime?Itwouldbeahorribleblundertoinvokethemiddle-schoolformulaofdistanceequalsratetimestimebecausethatformulaisvalidonlywhentherate—thecar’sspeed—isconstant,whichitcertainlyisn’there.Onthecontrary,inthisproblemthecar’sspeedisrampingupateveryinstant.Wearenolongerinthesleepyworldofconstantspeed.Thisisthethrillingworldofconstantacceleration.ScholarsintheMiddleAgesalreadyknewtheanswer.WilliamHeytesbury,a

philosopherandlogicianatMertonCollege,Oxford,solvedtheproblemaround1335,andNicoleOresme,aFrenchclericandmathematician,elucidateditfurtherandanalyzeditpictoriallyaround1350.Unfortunatelytheirworkswerenotwidelystudiedandweresoonforgotten.Abouttwohundredandfiftyyearslater,Galileodemonstratedexperimentallythatconstantaccelerationisnotapurelyacademicassumption.ItisactuallyhowheavyobjectslikeironballsmovewhentheyfallfreelynearthesurfaceoftheEarthorwhentheyrolldownagentlyslopingramp.Inbothcases,aball’sspeedvreallydoesgrowinproportiontotime,v=at,asexpectedformotionwithaconstantacceleration.Next,knowingthatthespeedgrowslinearlyaccordingtov=at,howdoesthe

distancegrow?Thefundamentaltheoremsaysthedistancetraveledequalstheareaaccumulatedunderthespeedcurveuptotimet.Andsinceherethespeedcurveistheslopinglinev=at,therelevantareaiseasytocompute.It’sgivenbytheareaofthetrianglebelow.

Likethegrayrectangleinthepreviousproblem,thegraytrianglehereis

expandingastimepasses.Thedifferenceisthattherectangleexpandedonlyhorizontallywhereasthistriangleisexpandinginbothdirections.Tocomputehowfastitsareaisexpanding,observethatatanytimetthetriangle’sbaseistanditsheightisthebody’scurrentspeed,v=at.Sincetheareaofatriangleishalfitsbasetimesitsheight,theaccumulatedareaequals½×t×at=(½)at2.Bythefundamentaltheorem,thatareaunderthespeedcurvetellsushowfarthebodyhastraveled:

y(t)=½at2.

Hence,forabodythatstartsfromrestandacceleratesuniformly,thedistancetraveledincreasesinproportiontothesquareofthetimeelapsed.ThisisexactlywhatGalileodiscoveredexperimentallyandexpressedinsuchacharmingfashionwithhislawofoddnumbers,aswesawinchapter3.ThescholarsintheMiddleAgesknewittoo.ButwhatwasnotknownintheMiddleAges,oreveninthetimeofGalileo,

washowthevelocitywouldbehavewhentheaccelerationwasnotsimplyconstant.Inotherwords,givenabodymovingwithanarbitraryaccelerationa(t),whatcouldonesayaboutitsspeedv(t)?ThisislikethebackwardproblemImentionedinthelastchapter.It’satricky

question.Tounderstanditproperly,it’scrucialtoappreciatewhatweknowanddon’tknow.Theaccelerationisdefinedastherateofchangeofspeed.Soifweweregiven

thespeedv(t),findingthecorrespondingaccelerationa(t)wouldbeeasy.That’scalledsolvingtheforwardproblem.Wecouldsolveitbycomputingtherateofchangeofthegivenspeedfunctioninmuchthesamewaythatwecalculatedtheslopeoftheparabolainthelastchapterbyplacingitunderthemicroscope.Findingarateofchangeofaknownfunctionrequiresnothingmorethan

invokingthedefinitionofthederivativeandapplyingthemanyrulesforcalculatingderivativesofvariousfunctions.Butwhatmakesthebackwardproblemsotrickyisthatwearenotgiventhe

speedfunction.Onthecontrary,wearebeingaskedtofindthespeedfunction.Weareassumingthatwehavebeengivenitsrateofchange—itsacceleration—asafunctionoftime,andwearetryingtofigureoutwhatspeedfunctionhasthataccelerationfunctionasitsgivenrateofchange.Howcanwegobackwardtoinferanunknownspeedfromitsknownrateofchange?It’slikeachildren’sgame:“I’mthinkingofaspeedfunctionwhoserateofchangeissuchandsuch.WhatspeedfunctionamIthinkingof?”Thesamepuzzleofhavingtoreasonbackwardariseswhenwetrytoinfer

distancefromspeed.Justasaccelerationistherateofchangeofspeed,speedistherateofchangeofdistance.Reasoningforwardiseasy;ifweknowamovingbody’sdistanceasafunctionoftime,aswedidinthecaseofUsainBoltrunningdownthetrackinBeijing,it’seasytocalculatethebody’sspeedateveryinstant.Wedidthatcalculationinthelastchapter.Butreasoningbackwardisdifficult.IfItoldyouhowfastUsainBoltwasrunningateveryinstantintherace,couldyouinferwherehewasonthetrackateachmoment?Moregenerally,givenanarbitraryspeedfunctionv(t),couldyouinferthecorrespondingdistancefunctiony(t)?Newton’sfundamentaltheoremshedlightonthisverydifficultbackward

problemofinferringanunknownfunctionfromitsgivenrateofchangeandinmanycasessolveditcompletely.Thekeywastoreframeitasaquestionaboutareasthatflowandexpand.

APaint-RollerProofoftheFundamentalTheorem

Thefundamentaltheoremofcalculuswastheculminationofeighteencenturiesofmathematicalthought.Bydynamicmeans,itansweredastaticgeometricquestionthatArchimedescouldhaveaskedinancientGreecein250BCEorthatcouldhaveoccurredtoLiuHuiinChinain250CEortoIbnal-HaythaminCairoin1000ortoKeplerinPraguein1600.Considerashapelikethegrayregionshownhere.

Isthereawaytocomputetheexactareaofanarbitraryshapelikethis,giventhatthecurveontopcouldbealmostanything?Inparticular,itneedn’tbeaclassiccurve.Itcouldbeanexoticnewcurvedefinedbyanequationinthexyplane,thejungleopenedupbyFermatandDescartes.Orwhatifthecurvewasdefinedbysomethingofphysicalinterest,likeatrajectoryofamovingparticleorthepathofalightray—wasthereanywaytofindtheareaundersuchanarbitrarycurveanddoitsystematically?Thiswastheareaproblem,thethirdcentralproblemofcalculusImentionedearlierandthemostpressingmathematicalchallengeofthemid-1600s.Itwasthelastremainingpuzzleinthemysteryofcurves.IsaacNewtonapproacheditfromanewdirection,usingideasinspiredbythemysteriesofmotionandchange.Historically,theonlywaytosolveproblemslikethishadbeentobeclever.

Youhadtofindsomecunningwaytosliceacurvedregionintostripsorsmashitintoshardsandthenreassemblethepiecesinyourmindorweighthemonanimaginaryseesaw,asArchimedeshaddone.Butaround1665Newtongavetheareaproblemitsfirstmajoradvanceinnearlytwomillennia.HeincorporatedtheinsightsofIslamicalgebraandFrenchanalyticgeometrybutwentfarbeyondthem.Thefirststep,accordingtohisnewsystem,wastolaytheareadowninthexy

planeanddetermineanequationforitscurvedtop.Thisrequiredcomputinghowfarthecurvewasabovethex-axis,oneverticalsliceatatime(asindicatedbythedottedverticallineinthediagram)toobtainthecorrespondingy.That

computationconvertedthecurveintoanequationrelatingytox,whichmadeitsusceptibletotheinstrumentsofalgebra.Thirtyyearsearlier,FermatandDescarteshadalreadyunderstoodthismuchandhadusedthesetechniquestofindtangentlinestocurves,ahugebreakthroughinitself.Butwhattheymissedwasthattangentlinespersewerenotthatimportant.

Moreimportantthansuchlinesweretheirslopes,foritwasslopesthatledtotheconceptofthederivative.Aswesawinthelastchapter,thederivativearoseverynaturallyingeometryastheslopeofacurve.Andderivativesalsoaroseinphysicsasotherratesofchange,suchasspeeds.Thus,derivativessuggestedalinkbetweenslopesandspeedsand,morebroadly,betweengeometryandmotion.OncetheideaofthederivativewasfirmlyinNewton’smind,itspowertobridgegeometryandmotionmadethefinalbreakthroughpossible.Itwasthederivativethatfinallyunlockedtheareaproblem.Thedeeplyhiddenconnectionsamongalltheseideas—slopesandareas,

curvesandfunctions,ratesandderivatives—emergedfromtheshadowswhenNewtonlookedattheareaproblemdynamically.Inthespiritofourearlierworkinthelasttwosections,ponderthediagramaboveandimagineslidingxtotherightataconstantspeed.Youcouldeventhinkofxastime;Newtonoftendid.Thentheareaofthegrayregionchangescontinuouslyasxmoves.Becausethatareadependsonx,itshouldberegardedasafunctionofx,sowewriteitasA(x).Whenwewanttostressthatthisareaisafunctionofx(asopposedtoafrozennumber),werefertoitastheareaaccumulationfunction,orsometimesjusttheareafunction.Mycalculusteacherinhighschool,Mr.Joffray,hadamemorablemetaphor

forthisfluidscenario,withitsslidingxanditschangingarea.Heaskedustoimagineamagicalpaintrollermovingsideways.Asitrollssteadilytotheright,itpaintstheregionunderthecurvegray.

Thedottedlineatxmarksthecurrentpositionofthisimaginaryrollerasit

rollstotheright.Meanwhile,toensurethattheregionispaintedneatly,therollerinstantlyandmagicallyshrinksorstretchesintheverticaldirection,exactlyasneededtoreachthecurveontopandthex-axisonthebottomwithoutevercrossingthoseboundaries.Themagicalaspectisthatitalwaysadjustsitslengthtoy(x)asitrolls,soitpaintstheareaimmaculately.Havingsetupthisfar-fetchedscenario,weask:Atwhatratedoesthegray

areaexpandasxmovestotheright?Or,equivalently,what’stherateatwhichpaintisbeinglaiddownwhentherollerisatx?Toanswerthat,thinkaboutwhathappensinthenextinfinitesimalintervaloftime.Therollerrollstotherightthroughsomeinfinitesimaldistancedx.Meanwhile,asittraversesthattinydistance,itkeepsitslengthyintheverticaldirectionalmostperfectlyconstant,sincethere’salmostnotimeforittochangeitslengthduringtheinfinitesimallybriefroll(afinepointthatwe’lldiscussinthenextchapter).Duringthatbriefinterval,itpaintswhatisessentiallyatall,thinrectangleofheighty,infinitesimalwidthdx,andinfinitesimalareadA=ydx.Dividingthisequationbydxthenrevealstherateatwhichareaaccumulates.Itisgivenby

dA⁄dx=y.

Thistidyformulasaysthatthetotalpaintedareaunderthecurveincreasesatarategivenbythecurrentlengthyofthepaintroller.Itmakessense;thelongertherollercurrentlyis,themorepaintitlaysdowninthenextinstant,andsothefastertheareaaccumulates.Withalittlemoreeffortwecouldshowthatthisgeometricversionofthe

theoremisequivalenttothemotionversionweusedearlier,whichstatedthattheareaaccumulatedunderaspeedcurveequalsthedistancetraveledbyamovingbody.Butwehavemoreurgenttasksahead.Weneedtounderstandwhatthetheoremmeans,whyitmatters,andhowitultimatelychangedtheworld.

TheMeaningoftheFundamentalTheorem

Thediagrambelowsummarizeswhatwe’vejustlearned.

A(x) derivative→ y(x) derivative→ dy/dx

areaundercurve

curve slopeofcurve

Itshowsthethreefunctionswe’reinterestedinandtherelationshipsbetweenthem.Thegivencurveisinthemiddle,itsunknownslopeisontheright,anditsunknownareaisontheleft.Aswesawinchapter6,thesearethefunctionsthatoccurinthethreecentralproblemsofcalculus.Giventhecurvey,wearetryingtofigureoutitsslopeanditsarea.IhopethediagramnowmakesclearwhyIreferredtofindingtheslopeas“the

forwardproblem.”Tofindtheslopefromthecurve,wesimplyfollowthearrowontherightbymovingforwardalongit.Wecomputethederivativeofytofinditsslope.That’sthestraightforwardproblem(1)wediscussedinthelastchapter.Whatwedidnotknowbeforeandwhatwehavejustlearnedfromthe

fundamentaltheoremisthattheareaAandthecurveyarealsorelatedbyaderivative—thefundamentaltheoremhasrevealedthatthederivativeofAisy.Thisisastupendousfact.Itgivesusanavenueforfiguringouttheareaunderneathanarbitrarycurve,theage-oldmysterythatstumpedthegreatestmindsforalmosttwothousandyears.Thepicturenowsuggestsapathtotheanswer.Butbeforeweuncorkthechampagne,weshouldrealizethatthe

fundamentaltheoremdoesnotquitegiveuswhatwewant.Itdoesnotgiveustheareadirectly.Butittellsushowtoobtainit.

TheHolyGrailofIntegralCalculus

AsI’vetriedtomakeclear,thefundamentaltheoremdoesn’tfullysolvetheareaproblem.Itprovidesinformationabouttherateatwhichtheareachanges,butwestillneedtoinfertheareaitself.Intermsofsymbols,thefundamentaltheoremtellsusthatdA/dx=y,where

y(x)isourgivenfunction.We’restillleftwiththechoreoffindinganA(x)thatsatisfiesthisequation.Waitaminute—thismeanswe’resuddenlyfacedwiththebackwardproblemagain!It’saremarkableturnofevents.Weweretryingtosolvetheareaproblem,centralproblemnumber3onourlistinchapter6,andsuddenlywe’rebeingconfrontedbythebackwardproblem,centralproblemnumber2onthelist.I’mcallingitthebackwardproblembecause,asthediagramaboveshows,findingAfromymeansgoingupstreamagainstthearrow,goingbackwardagainstthederivative.Inthissettingthechildren’sgamemightgosomethinglikethis:“I’mthinkingofanareafunctionA(x)whosederivativeis12x+x10−sinx.WhatfunctionamIthinkingof?”Developingmethodstosolvethebackwardproblem,notjustfor12x+x10−

sinxbutforanycurvey(x),becametheholygrailofcalculus.Moreprecisely,itbecametheholygrailofintegralcalculus.Solvingthebackwardproblemwouldallowtheareaproblemtobesolvedonceandforall.Givenanycurvey(x),we’dknowtheareaA(x)underneathit.Bysolvingthebackwardproblem,we’dalsosolvetheareaproblem.ThisiswhatImeantaboutthosetwoproblemsbeingseparated-at-birthtwinsandtwosidesofthesamecoin.Asolutiontothebackwardproblemwouldalsohavemuchlarger

implications,forthefollowingreason:Anareais,fromanArchimedeanstandpoint,aninfinitesumofinfinitesimalrectangularstrips.Assuch,anareaisanintegral.It’stheintegratedcollectionofallthepiecesputbacktogether,anaccumulationofinfinitesimalchange.Andjustasderivativesaremoreimportantthanslopes,integralsaremoreimportantthanareas.Areasarecrucialtogeometry;integralsarecrucialtoeverything,aswe’llseeinthechaptersahead.Onewaytoapproachthedifficultbackwardproblemistoignoreit.Shuntit

aside.Replaceitwiththeeasierforwardproblem(givenA,computeitsrateofchangedA/dx;bythefundamentaltheorem,weknowthatthismustequalthey

we’reseeking).Thisforwardproblemismucheasierbecauseweknowwheretostart.WecanstartwithaknownareafunctionA(x)andthencrankoutitsrateofchangebyapplyingstandardformulasforderivatives.TheresultingrateofchangedA/dxthenmustplaytheroleofthepartnerfunctiony;that’swhatthefundamentaltheoremassuresus:dA/dx=y.Havingdoneallthat,wenowhaveapairofpartnerfunctions,A(x)andy(x),whichrepresentanareafunctionanditsassociatedcurve.Thehopeisthatifweareluckyenoughtostumbleacrossaproblemwhereweneedtofindtheareaunderthisparticularcurvey(x),itscorrespondingareafunctionwillbeitspartnerA(x).It’snotasystematicapproachanditworksonlyifwehappentogetlucky,butatleastit’sastartandit’seasy.Toincreaseouroddsofsuccess,wecanmakeabiglookuptablethatlistshundredsofareafunctionsandtheirassociatedcurvesas(A(x),y(x))pairs.Thenthesheersizeanddiversityofthattablewillimproveourchancesofstumblingacrossthepairweneedtosolveagenuineareaproblemofinterest.Havingfoundthenecessarypair,wewouldn’tneedtodoanyfurtherwork.Theanswerwouldberightthereinthetable.Forexample,inthenextchapterwe’llseethatthederivativeofx3is3x2.

We’llobtainthatresultbysolvingaforwardproblem,simplytakingaderivative.What’swonderfulaboutit,however,isthatittellsusthatx3couldplaytheroleofA(x),and3x2couldplaytheroleofy(x).Withoutbreakingasweat,we’vesolvedtheareaproblemfor3x2(shouldweeverhappentobeinterestedinit).Continuinginthisfashion,wecanfillinthetablewithotherpowerfunctionsofx.Similarcalculationswouldshowthatthederivativeofx4is4x3,thederivativeofx5is5x4,andingeneralthederivativeofxnisnxn–1.Thesearealleasysolutionsoftheforwardproblemforpowerfunctions.Thusthecolumnsofthetablewouldlooklikethis:

Curvey(x) ItsareafunctionA(x)

3x2 x3

4x3 x4

5x4 x5

6x5 x6

7x6 x7

Inhiscollegenotebook,atwenty-two-year-oldIsaacNewtonwroteout

similartablesforhimself.

Noticethathislanguageisabitdifferentfromours.Thecurvesintheleft

columnare“Theequationsexpressingthenatureofyelines.”Theirareafunctionsare“Theiresquare”(becauseheviewstheareaproblemasthe“squaringofcurves”).Healsofeelstheneedtoinsertvariouspowersofa,anarbitraryunitoflength,toensurethatallquantitieshavethepropernumberofdimensions.Forexample,hisbottomrightA(x),fivelinesdownfromthetopofthelist,isx7/a5(insteadofoursimplerx7)becauseinhismind,itrepresentsanareaandhenceneedstohaveunitsoflengthsquared.Allofthiscomesafewpagesafter“Amethodwherebytosquarethosecrookedlineswhichmaybesquared”—thebirthannouncementofthefundamentaltheoremofcalculus.Armedwiththattheorem,Newtonfilledmanymorepageswithlistsof“crookedlines”andtheir“squares.”InNewton’shands,themachineryofcalculuswasbeginningtowhir.Thenexttask,afantasy,really,wastofindamethodtosquareanycurve,not

justpowerfunctions.Perhapsitdoesn’tsoundlikeaparticularlyscintillatingfantasy.Butthat’sbecauseit’ssogeneral.Letmeputitthisway:Thisproblemcontainsthedistilledessenceofwhatmakesintegralcalculussochallenging.Ifthisproblemcouldbesolved,itwouldbelikesettingoffachainreaction.Itwouldbeliketopplingdominoes;oneproblemafteranotherwouldfall.Ifthisproblemcouldbesolved,itcouldbeusedtoanswerthequestionthatDescartesthoughtwasbeyondhumancomprehension,findingthearclengthofanarbitrary

curve.Itwouldbepossibletofindtheareaofanyamoeba-shapedregionintheplane.Itwouldbepossibletocalculatethesurfaceareas,volumes,andcentersofgravityofspheres,paraboloids,urns,barrels,andallothersurfacesmadebyspinningacurvearoundanaxis,likeavaseonapotter’swheel.TheclassicproblemsaboutcurvedshapesthatArchimedesponderedandthatanothereighteencenturiesofmathematicaltalentponderedafterhimwouldallbecometractableinstantly,inasinglestroke.Notonlythat,butcertainproblemsofpredictionwouldbeovercomeaswell.

Predictingthepositionofamovingobjectfarintothefuture—forinstance,whereaplanetwillbeatacertainpointinitsorbit,evenaplanetthatobeysadifferentforceofattractionthantheoneoperatinginourownuniverse—wouldbecomepossibleifjustthisoneproblemcouldbesolved.That’swhatImeanbycallingittheholygrailofintegralcalculus.Many,manyotherproblemsboildowntosolvingthisone.Ifitgoes,theyallgo.Thisiswhyitwassoimportanttobeabletofindtheareaunderanarbitrary

curve.Becauseofitsintimateconnectiontothebackwardproblem,theareaproblemisnotjustaboutarea.It’snotjustaboutshapeortherelationshipbetweendistanceandspeedoranythingthatnarrow.It’scompletelygeneral.Fromamodernperspective,theareaproblemisaboutpredictingtherelationshipbetweenanythingthatchangesatachangingrateandhowmuchthatthingbuildsupovertime.It’saboutthefluctuatinginflowtoabankaccountandtheaccumulatedbalanceofmoneyinit.It’saboutthegrowthrateoftheworld’spopulationandthenetnumberofpeopleonEarth.It’saboutthechangingconcentrationofachemotherapydruginapatient’sbloodandtheaccumulatedexposuretothatdrugovertime.Thattotalexposureaffectshowpotentthechemowillbe,aswellashowtoxic.Areamattersbecausethefuturematters.Newton’snewmathematicswasexquisitelysuitedtoaworldinflux.

Accordingly,hechristeneditfluxions.Hespokeoffluentquantities(whichwenowthinkofasfunctionsoftime)andtheirfluxions(theirderivatives,theirratesofchangeintime).Heidentifiedtwocentralproblems:

1. Giventhefluents,howcanonefindtheirfluxions?(Thisisequivalenttotheforwardproblemwementionedearlier,theeasyproblemoffindingtheslopeofagivencurveor,moregenerally,findingtherateofchangeorderivativeofaknownfunction,theprocessknowntodayasdifferentiation.)

2. Giventhefluxions,howcanonefindtheirfluents?(Thisisequivalenttothebackwardproblemandthekeytotheareaproblem;itisthedifficultproblemofinferringacurvefromitsslopeor,moregenerally,inferringan

unknownfunctionfromitsrateofchange,theprocessknowntodayasintegration.)

Problem2ismuchharderthanproblem1.It’salsomuchmoreimportantforpredictionandfortappingintothecodeoftheuniverse.BeforewelookathowfarNewtongotonit,letmetrytoclarifywhyit’ssohard.

LocalVersusGlobal

Thereasonwhyintegrationissomuchharderthandifferentiationhastodowiththedistinctionbetweenlocalandglobal.Localproblemsareeasy.Globalproblemsarehard.Differentiationisalocaloperation.Aswe’veseen,whenwearecalculatinga

derivative,it’slikewe’relookingunderamicroscope.Wezoominonacurveorafunction,repeatedlymagnifyingthefieldofview.Aswezoominonthatlittlelocalpatch,thecurveappearstobecomelessandlesscurved.Weseeablown-upversionofthecurve,atinyramp,almostperfectlystraight,withariseΔyandarunΔx.Inthelimitofinfinitemagnification,itapproachesacertainstraightline,thetangentlineatthepointinthecenterofthemicroscope.Theslopeofthatlimitinglinegivesusthederivativethere.Theroleofthemicroscopeistoletusfocusonthepartofthecurvewecareabout.Everythingelsegetsignored.That’sthesenseinwhichfindingthederivativeisalocaloperation.Itdiscardsalldetailsoutsidetheinfinitesimalneighborhoodofapoint,theonlypointofinterest.Integrationisaglobaloperation.Insteadofamicroscope,wearenowusinga

telescope.Wearetryingtopeerfaroffintothedistance—orfaraheadintothefuture,althoughinthatcaseweneedacrystalball.Naturally,theseproblemsarealotharder.Alltheinterveningeventsmatterandcannotbediscarded.Orsoitwouldseem.Letmeofferananalogytobringoutthesedistinctionsbetweenlocaland

global,betweendifferentiationandintegration,andtoclarifywhyintegrationissohardandsoscientificallyimportant.TheanalogytakesusbacktoBeijingandUsainBolt’srecord-breakingrace.Recallthattofindhisspeedateachinstant,wefitasmoothcurvetothedatashowinghispositiononthetrackasafunctionoftime.Then,tofindhisspeedatacertainpoint,say7.2secondsintotherace,weusedthefittedcurvetoestimatehispositionashorttimelater,sayat7.25seconds,andthenlookedatthechangeindistancedividedbythechangeintime

toestimatehisspeedatthatmoment.Thesewerealllocalcalculations.Theonlyinformationtheyusedwashowhewasrunninginthefewhundredthsofasecondaroundthatgiventime.Everythinghedidintherestoftherace,beforeandafter,wasirrelevant.That’swhatImeanbylocal.Bycontrast,thinkaboutwhatwouldbeinvolvedifwewerehandedan

infinitelylongspreadsheetshowinghisspeedateverymomentintheraceandaskedtoreconstructwherehewas7.2secondsafterthestart.Ashecomesoutoftheblocks,wecouldusehisinitialspeedtoestimatewherehewasat,let’ssay,ahundredthofasecondlaterbyusingdistanceequalsratetimestimetoadvancehimdownthetrack.Fromthatnewpositionandthatnewelapsedtime,wecouldagainadvancehimdownthetrackoverthenexthundredthofasecondwiththecorrespondingspeedandthecorrespondingdistancehewouldcover.Onandon,inchingdownthetrack,accumulatinginformationonehundredthofasecondatatime,wecouldupdatehispositionthroughouttherace.Itwouldbeanarduousgrind.Computationally,Imean.Thisiswhatmakesaglobalcalculationsodifficult.Weneedtocomputeeverysteptogettoadesiredanswerfarintothefuture,inthiscase7.2secondsafterthestartinggunwentoff.Butimagineifwecouldsomehowfast-forwardandzapstraighttotheinstant

wecaredabout—now,thatwouldbeuseful.Andthatisexactlywhatasolutiontothebackwardproblemofintegrationwouldachieve.Itwouldgiveusashortcut,awormholethroughtime.Itwouldconvertaglobalproblemintoalocalone.That’swhysolvingthebackwardproblemwouldbelikefindingtheholygrailofcalculus.Itwasfirstsolved,assomanythingsare,byastudent.

ALonesomeBoy

IsaacNewtonwasborninastonefarmhouseonChristmasDay1642.Apartfromthedate,therewasnothingauspiciousabouthisarrival.Hewasbornprematureandwassotiny,itwassaid,hecouldfitinsideaquartmug.Hewasalsofatherless.TheelderIsaacNewton,ayeomanfarmer,haddiedthreemonthsearlier,leavingbehindbarley,furniture,andsomesheep.WhenlittleIsaacwasthree,hismother,Hannah,remarriedandlefthiminthe

careofhismaternalgrandparents.(Hismother’snewhusband,ReverendBarnabasSmith,insistedonthisarrangement;hewasawealthymantwiceherageandwantedayoungwifebutnotayoungson.)Understandably,Isaacresentedhisstepfatherandfeltabandonedbyhismother.Laterinlife,onalist

ofsinshe’dcommittedbeforetheageofnineteen,heincludedthisentry:“13.ThreatningmyfatherandmotherSmithtoburnethemandthehouseoverthem.”Thenextentrywasdarker:“14.Wishingdeathandhopingittosome.”Andthenthis:“15.Strikingmany.16.Havinguncleanethoughtswordsandactionsanddreamese.”Hewasatroubled,lonelylittleboywithnocompanionsandtoomuchtimeon

hishands.Hepursuedscholarlyinvestigationsonhisown,buildingsundialsinthefarmhouse,measuringtheplayoflightandshadowsonthewall.Whenhewasten,hismotherreturned,widowedagain,withthreenewchildrenintow,twodaughtersandason.ShesentIsaacawaytoaschoolinGrantham,eightmilesuptheroad,toofarforhimtowalkeachday.HeboardedwithMr.WilliamClark,anapothecaryandchemist,fromwhomhelearnedcuresandremedies,boilingandmixing,andhowtogrindwithamortarandpestle.Theschoolmaster,Mr.HenryStokes,taughthimLatin,abitoftheology,Greek,Hebrew,andsomepracticalmathforfarmersaboutsurveyingandmeasuringacreage,aswellassomedeeperthings,likehowArchimedeshadestimatedpi.Althoughhisschoolreportsdescribedhimasanidleandinattentivestudent,whenIsaacwasaloneinhisroomatnight,hedrewshapesonthewall,Archimedeandiagramsofcirclesandpolygons.Whenhewassixteenhismotherpulledhimoutofschoolandforcedhimto

runthefamilyfarm.Hehatedfarming.Heallowedhisswinetotrespassonhisneighbors’fieldsandlethisfencesfallapart,andhewasdulyfinedbythemanorcourt.Hegotinfightswithhismotherandhalfsisters.Hewouldoftenlieinthefieldsandreadbyhimself.Hebuiltwaterwheelsinthestreamandstudiedthewhorlstheymadeintheflow.Finally,hismotherdidtherightthing.Attheurgingofherbrotherand

schoolmasterStokes,sheallowedIsaactogobacktoschool.Heperformedwellenoughacademicallythatin1661,hewasabletoenterTrinityCollege,Cambridge,asasizar.Beingasizarmeanthehadtoearnhiskeepbywaitingontablesandservingthericherstudents.Sometimesheatetheirleftovers.(Hismothercouldhaveaffordedtosupporthim,butshedidn’t.)Hemadefewfriendsincollege,apatternthatwouldcontinuefortherestofhislife.Henevermarriedand,asfarasweknow,neverhadaromanticrelationship.Herarelylaughed.HisfirsttwoyearsofcollegeweretakenupwithAristotelianscholasticism,

stillstandardatthattime.Butthenhismindbegantostir.Hebecamecuriousaboutmathematicsafterreadingabookonastrology.Hefoundhecouldn’tunderstanditwithoutknowingsometrigonometryandthathecouldn’tunderstandtrigonometrywithoutknowingsomegeometry,sohetookalookat

Euclid’sElements.Atfirstalltheresultsseemedobvioustohim,buthechangedhismindwhenhecametothePythagoreantheorem.In1664hewasawardedascholarship,andhedelvedintomathematicsin

earnest.Teachinghimselffromsixstandardtextsoftheera,hegotuptospeedonthebasicsofdecimalarithmetic,symbolicalgebra,Pythagoreantriples,permutations,cubicequations,conicsections,andinfinitesimals.Twoauthorsespeciallyenthralledhim:Descartes,onanalyticgeometryandtangents,andJohnWallis,oninfinityandquadrature.

AtPlaywithPowerSeries

WhileporingoverWallis’sArithmeticaInfinitoruminthewinterof1664–65,Newtonchanceduponsomethingmagical.Itwasanewwaytofindtheareasundercurves,awaythatwasbotheasyandsystematic.Inessence,heturnedtheInfinityPrincipleintoanalgorithm.Thetraditional

InfinityPrinciplesaysthattocomputeacomplicatedarea,reimagineitasaninfiniteseriesofsimplerareas.Newtonfollowedthatstrategy,butheupdateditbyusingsymbols,notshapes,ashisbuildingblocks.Insteadoftheusualshardsorstripsorpolygons,heusedpowersofasymbolx,likex2andx3.Todaywecallhisstrategythemethodofpowerseries.Newtonviewedpowerseriesasanaturalgeneralizationofinfinitedecimals.

Aninfinitedecimal,afterall,isnothingbutaninfiniteseriesofpowersof10and1/10.Thedigitsinthenumbertellushowmuchofeachpowerof10or1/10tomixin.Forexample,thenumberpi=3.14...correspondstothisparticularmix:

3.14...=3×100+1×(1⁄10)1+4×(1⁄10)2+···.

Ofcourse,towriteanynumberinthismanner,weneedtoallowourselvestouseinfinitelymanydigits,whichiswhatinfinitedecimalsdemandandrequire.Byanalogy,Newtonsuspectedhecouldconcoctanycurveorfunctionoutofinfinitelymanypowersofx.Thetrickwastofigureouthowmuchofeachpowertomixin.Inthecourseofhisstudieshedevelopedseveralmethodsforfindingtherightmix.Hehitonhismethodwhilethinkingabouttheareaofacircle.Bymakingthis

ancientproblemmoregeneral,heuncoveredastructurewithinitthatnobodyhadevernoticedbefore.Ratherthanrestrictinghisattentiontoastandardshape,

likeawholecircleoraquartercircle,helookedattheareaofanoddlyshaped“circularsegment”ofwidthx,wherexcouldbeanynumberfrom0to1andwhere1wastheradiusofthecircle.

Thiswashisfirstcreativemove.Theadvantageofusingthevariablexwas

thatitletNewtonadjusttheshapeoftheregioncontinuously,asifbyturningaknob.Asmallvalueofxnear0wouldproduceathin,uprightsegmentofthecircle,likeathinstripstandingonitsedge.Increasingxwouldfattenthesegmentintoablockyregion.Goingallthewayuptoanx-valueof1wouldgivehimthefamiliarshapeofaquartercircle.Bydialingxupordown,hecouldgoanywherehelikedinbetween.Throughafreewheelingprocessofexperimentation,patternrecognition,and

inspiredguesswork(astyleofthinkinghelearnedfromWallis’sbook),Newtondiscoveredthattheareaofthecircularsegmentcouldbeexpressedbythefollowingpowerseries:

A(x)=x−1⁄6x3−1⁄40x5−1⁄112x7−5⁄1152x9−···.

Asforwherethosepeculiarfractionscamefromorwhythepowersofxwerealloddnumbers,well,thatwasNewton’ssecretsauce.Hecookeditupbyanargumentthatcanbesummarizedasfollows.(Feelfreetoskiptherestofthisparagraphifyouarenotespeciallyinterestedinhisargument.However,ifyouwouldliketoseethedetails,checkoutthenotesforreferences.)Newtonbeganhisworkonthecircularsegmentbyusinganalyticgeometry.Heexpressedthecircleasx2+y2=1andthensolvedforytogety=√1−x2.Nexthearguedthatthesquarerootwasequivalenttoahalfpowerandhencethaty=(1–x2)½;notethe½powertotherightoftheparenthesis.Then,sinceneitherhenoranyoneelseknewhowtofindtheareasofsegmentsforhalfpowers,hesidesteppedtheproblem—hissecondcreativemove—andsolveditforwholepowersinstead.Findingtheareasforwholepowerswaseasy;heknewhowfromhisreadingofWallis.SoNewtoncrankedouttheareasofsegmentsfory=(1−x2)1and(1−x2)2and(1−x2)3andsoon,allofwhichhavewhole-numberpowerslike1,2,and3outsidetheirparentheses.Heexpandedtheexpressionswiththebinomialtheoremandsawthattheybecamesumsofsimplepowerfunctions,theindividualareafunctionsofwhichhehadalreadytabulated,aswesawonthepagefromhishandwrittennotebook.Thenhelookedforpatternsintheareasofthesegmentsasfunctionsofx.Basedonwhathesawforwholepowers,heguessedtheanswer—histhirdcreativemove—forhalfpowersandthencheckeditinvariousways.Theanswerforthe½powerledhimtohisformulaforA(x),theamazingpowerserieswiththepeculiarfractionsdisplayedabove.Thederivativeofthepowerseriesforthecircularsegmentthenledhimtoan

equallyamazingseriesforthecircleitself:

y=√1−x2=1−1/2x2−1/8x4−1/16x6−5/128x8−···.

Therewasmuchmoretocome,butalreadythiswasremarkable.He’dconcoctedacircleoutofinfinitelymanysimplerpieces—simpler,thatis,fromthestandpointofintegrationanddifferentiation.Allitsingredientswerepowerfunctionsoftheformxn,wherethepowernwasawholenumber.Alltheindividualpowerfunctionshadeasyderivativesandintegrals(areafunctions).Likewise,thenumericalvaluesofxncouldbecalculatedwithsimplearithmeticusingnothingmorethanrepeatedmultiplicationandcouldthenbecombinedintoaseries,againusingnothingmorethanaddition,subtraction,multiplication,anddivision.Therewerenosquarerootstotakeoranyothermessyfunctionstoworryabout.Ifhecouldfindpowerserieslikethisforothercurvesbesidescircles,integratingthemwouldbecomeeffortlesstoo.

Atbarelytwenty-two,IsaacNewtonhadfoundapathtotheholygrail.Byconvertingcurvestopowerseries,hecouldfindtheirareassystematically.Thebackwardproblemwasacinchforpowerfunctions,giventhepairsoffunctionshehadtabulated.Soanycurvethathecouldexpressasaseriesofpowerfunctionswaseverybitaseasytosolve.Thiswashisalgorithm.Itwastremendouslypowerful.Thenhetriedadifferentcurve,thehyperbolay=1/(1+x),andfoundhe

couldwriteit,too,asapowerseries:

1/1+x=1−x+x2−x3+x4−x5+···.

Thisseriesinturnledhimtoapowerseriesfortheareaofasegmentunderthehyperbolafrom0tox,thehyperboliccounterpartofthecircularsegmenthe’dstudiedearlier.Itdefinedafunctionthathecalledthehyperboliclogarithmandthattodaywecallthenaturallogarithm:

ln(1+x)=x−½x2+⅓x3−¼x4+1⁄5x5−1⁄6x6+···.

LogarithmsexcitedNewtonfortworeasons.First,theycouldbeusedtospeedupcalculationsenormously,andsecond,theywererelevanttoacontroversialprobleminmusictheoryhewasworkingon:howtodivideanoctaveintoperfectlyequalmusicalstepswithoutsacrificingthemostpleasingharmoniesofthetraditionalscale.(Inthejargonofmusictheory,Newtonwasusinglogarithmstoassesshowfaithfullyanequal-tempereddivisionoftheoctavecouldapproximatethetraditionaltuningofjustintonation.)ThankstothemarvelsoftheinternetandthehistoriansattheNewtonProject,

youcantravelbackto1665rightnowandwatchyoungNewtonatplay.(Hishandwrittencollegenotebookisfreelyavailableathttp://cudl.lib.cam.ac.uk/view/MS-ADD-04000/.)Lookoverhisshoulderatpage223(105vintheoriginal)andyou’llseehimcomparingmusicalandgeometricalprogressions.Zoominonthebottomofthatpagetoseehowheconnectshiscalculationstologarithms.Thengotopage43(20rintheoriginal)towatchhim“squarethehyperbola”andusehispowerseriestocalculatethenaturallogarithmof1.1tofiftydigits.Whatkindofpersoncalculateslogarithmsbyhandtofiftydigits?Heseemed

toberevelinginthenewfoundstrengthhispowerseriesgavehim.Whenhelaterreflectedontheextravaganceofthiscalculation,hesoundedabitsheepish:“IamashamedtotelltohowmanyplacesIcarriedthesecomputations,havingno

otherbusinessatthattime:forthenItookreallytoomuchdelightintheseinventions.”Ifit’sanyconsolation,nobody’sperfect.Whenhefirstdidthese

computations,Newtonmadeasmallarithmeticerror.Hiscalculationwascorrecttoonlytwenty-eightdigits.Helatercaughttheerrorandfixedit.Afterhisforaywiththenaturallogarithm,Newtonextendedhispowerseries

tothetrigonometricfunctions,whicharisewhenevercirclesorcyclesortrianglesappear,asinastronomy,surveying,andnavigation.Here,however,Newtonwasnotthefirst.Morethantwocenturiesearlier,mathematiciansinKerala,India,haddiscoveredpowerseriesforthesine,cosine,andarctangentfunctions.Writingintheearly1500s,JyesthadevaandNilakanthaSomayajiattributedtheseformulastoMadhavaofSangamagrama(c.1350–c.1425),thefounderoftheKeralaschoolofmathematicsandastronomy,whoderivedthemandexpressedtheminverseabouttwohundredandfiftyyearsbeforeNewton.InawayitmakessensethatpowerseriesshouldhavebeenanticipatedinIndia.DecimalswerealsodevelopedinIndia,andaswe’veseen,Newtonregardedwhathewasdoingforcurvesasananalogofwhatinfinitedecimalshaddoneforarithmetic.ThepointofallthisisthatNewton’spowerseriesgavehimaSwissarmy

knifeforcalculus.Withthem,hecoulddointegrals,findrootsofalgebraicequations,andcalculatethevaluesofnon-algebraicfunctionslikesines,cosines,andlogarithms.Asheputit,“Bytheirhelpanalysisreaches,Imightalmostsay,toallproblems.”

NewtonasMash-UpArtist

Idon’tbelieveNewtonwasconsciouslyawareofit,butinhisworkonpowerserieshebehavedlikeamathematicalmash-upartist.HeapproachedareaproblemsingeometryviatheInfinityPrincipleoftheancientGreeksandinfuseditwithIndiandecimals,Islamicalgebra,andFrenchanalyticgeometry.Someofhismathematicaldebtsarevisibleinthearchitectureofhisequations.

Forexample,comparetheinfiniteseriesofnumbersthatArchimedesusedinhisquadratureoftheparabola,

4⁄3=1+1⁄4+1⁄16+1⁄64+···.

withtheinfiniteseriesofsymbolsthatNewtonusedinhisquadratureofthehyperbola:

1⁄1+x=1−x+x2−x3+x4−x5+···.

Ifyouplugx=−¼intoNewton’sseries,itbecomesArchimedes’sseries.Inthatsense,Newton’sseriessubsumesArchimedes’sasaspecialcase.What’smore,thesimilarityintheirworkextendstothegeometricproblems

theyconsidered.Bothofthemwerefondofsegments;Archimedesusedhisnumberseriestosquare(orfindtheareaof)aparabolicsegment,whereasNewtonusedhisjacked-uppowerseries,

Acircular(x)=x−1⁄6x3−1⁄40x5−1⁄112x7−5⁄1152x9−···.

tosquareacircularsegment,andheusedadifferentpowerseries,

Ahyperbolic(x)=x−1⁄2x2+1⁄3x3−1⁄4x4+1⁄5x5−1⁄6x6+···.

tosquareahyperbolicsegment.Actually,Newton’sserieswereinfinitelymorepowerfulthanArchimedes’sin

thattheyenabledhimtofindtheareasofnotjustonebutawholecontinuousinfinityofcircularandhyperbolicsegments.That’swhathisabstractsymbolxdidforhim.Itlethimchangehisproblemscontinuouslyandeffortlessly.Itenabledhimtotunetheshapeofsegmentsbyslidingxtotheleftorrightsothatwhatappearedtobeasingleinfiniteserieswasinfactaninfinitefamilyofinfiniteseries,oneforeachchoiceofx.Thatwasthepowerofpowerseries.TheyletNewtonsolveinfinitelymanyproblemsinasinglestroke.Butagain,hecouldn’thavedoneanyofthiswithoutstandingontheshoulders

ofgiants.Heunified,synthesized,andgeneralizedideasfromhisgreatpredecessors:HeinheritedtheInfinityPrinciplefromArchimedes.HelearnedhistangentlinesfromFermat.HisdecimalscamefromIndia.HisvariablescamefromArabicalgebra.HisrepresentationofcurvesasequationsinthexyplanecamefromhisreadingofDescartes.Hisfreewheelingshenaniganswithinfinity,hisspiritofexperimentation,andhisopennesstoguessworkandinductioncamefromWallis.Hemashedallofthistogethertocreatesomethingfresh,somethingwe’restillusingtodaytosolvecalculusproblems:theversatilemethodofpowerseries.

APrivateCalculus

WhileNewtonwasworkingonpowerseriesduringthewinterof1664–65,aterriblepestilencewassweepingnorthacrossEurope,movinglikeawave,propagatingupfromtheMediterraneanandintoHolland.WhenthebubonicplaguereachedLondon,itkilledhundredsinaweek,andthenthousands.Inthesummerof1665,CambridgeUniversitytemporarilyshutdownindefense.NewtonwenthometothefamilyfarmhouseinLincolnshire.Overthenexttwoyearshebecamethebestmathematicianintheworld.But

inventingmoderncalculuswasn’tenoughtokeephismindoccupied.Healsodiscoveredtheinverse-squarelawofgravityandappliedittothemoon,inventedthereflectingtelescope,andshowedexperimentallythatwhitelightiscomposedofallthecolorsoftherainbow.Hewasnotyettwenty-five.Ashelaterrecalled,“InthosedaysIwasintheprimeofmyageforinventionandmindedmathematicsandphilosophymorethanatanytimesince.”In1667,aftertheplagueabated,NewtonreturnedtoCambridgeand

continuedhissolitarystudies.By1671,hehadunifiedthedisparatepartsofcalculusintoaseamlesswhole.He’ddevelopedthemethodofpowerseries,vastlyimprovedonexistingtheoriesoftangentlinesbyexploitingideasaboutmotion,foundandprovedthefundamentaltheorem,whichcrackedtheareaproblem,compiledtablesofcurvesandtheirareafunctions,andweldedalloftheseintoafinelytuned,systematic,computationalmachine.ButbeyondthecloistersofTrinityCollege,hewasinvisible.Thatwashowhe

wantedit.Hekepthissecretfountaintohimself.Reclusiveandsuspicious,hewaspainfullysensitivetocriticismandhatedgettingintoargumentswithanyone,especiallythosewhodidn’tunderstandhim.Ashelaterputit,hedidn’tenjoybeing“baitedbylittlesmatterersinmathematics.”Hehadanotherreasontobewary:Heknewthathisworkcouldbeattacked

onlogicalgrounds.He’dusedalgebra,notgeometry,andhe’dplayednonchalantlywithinfinity,theoriginalsinofcalculus.JohnWallis,whosebookhadsoinfluencedNewtoninhisstudentdays,hadbeenbrutallycriticizedforthosesametransgressions.ThomasHobbes,apoliticalphilosopherandsecond-ratemathematician,hadblastedWallis’sArithmeticaInfinitorumasa“scabofsymbols”foritsrelianceonalgebraanda“scurvybook”foritsuseofinfinity.AndNewtonhadtoadmitthathisownworkwasmerelyanalysis,notsynthesis.Itwasgoodonlyformakingdiscoveries,notprovingthem.Hedownplayedhisinfinitemethodsasnot“worthyofpublicutterance”andsaid,manyyearslater,

“Ourspeciousalgebraisfitenoughtofindout,butentirelyunfittoconsigntowritingandcommittoposterity.”Fortheseandotherreasons,Newtonkepthisworkhidden.Yetpartofhim

wantedcreditforit.HefelttornanddistressedwhenNicholasMercatorpublishedalittlebookaboutlogarithmsin1668thatcontainedthesameinfiniteseriesforthenaturallogarithmthatNewtonhaddiscoveredthreeyearsearlier.TheshockanddisappointmentofbeingscoopedpromptedNewtontowriteashortmanuscriptin1669aboutpowerseriesandcirculateitprivatelyamongafewtrustedacolytes.Itwentfarbeyondlogarithms.KnownasDeAnalysi,itsfulltitleinEnglishisOnAnalysisbyEquationsUnlimitedinTheirNumberofTerms.In1671,heenlargeditintohismaintractoncalculus,ATreatiseoftheMethodsofSeriesandFluxions,knownasDeMethodis,butthemanuscriptdidn’tseethelightofdayduringhislifetime;heguardeditcloselyandkeptitforhisprivateuse.DeAnalysiwasnotpublisheduntil1711;DeMethodisappearedposthumously,in1736.Newton’sestateincludedfivethousandpagesofunpublishedmathematicalmanuscript.SoittooktheworldawhiletodiscoverIsaacNewton.Withinthewallsof

Cambridge,however,hewasknownasagenius.In1669,IsaacBarrow,thefirstLucasianProfessorandtheclosestthingtoamentorthatNewtoneverhad,steppeddownandrecommendedthatNewtonbeappointedtotheLucasianChairofMathematics.ItwasanidealpostforNewton.Forthefirsttimeinhislife,hewas

financiallysecure.Thepositionrequiredlittleteaching.Hehadnograduatestudents,andhislecturestoundergraduateswerepoorlyattended,whichwasjustaswell.Thestudentsdidn’tunderstandhimanyway.Theydidn’tknowwhattomakeofthestrange,gaunt,monkishfigureinhisscarletrobes,withhisgrimfaceandsilveryshoulder-lengthhair.AfterNewtoncompletedhisworkonDeMethodis,hismindwasasfebrileas

ever,butcalculuswasnolongerhismaininterest.Hewasnowdeepintobiblicalprophecyandchronology,opticsandalchemy,splittinglightintocolorswithprisms,experimentingwithmercury,sniffinghischemicalsandsometimestastingthem,stokinghistinfurnacedayandnightashetriedtoturnleadintogold.LikeArchimedes,heneglectedhisfoodandhissleep.Hewaslookingforthesecretsoftheuniverse,andhehadnopatiencefordistractions.Adistractioncameonedayin1676intheformofaletterfromParis.Itwas

fromsomeonenamedLeibniz.Hehadafewquestionsaboutpowerseries.

8

FictionsoftheMind

HOWHADLEIBNIZheardaboutNewton’sunpublishedwork?Itwasn’tdifficult.WordofNewton’sdiscoverieshadbeenleakingoutforyears.In1669,IsaacBarrow,hopingtopromotehisyoungprotégé,hadsentananonymouscopyofDeAnalysitoamannamedJohnCollins,amathematicalwannabeandimpresario.CollinshadputhimselfatthehubofacorrespondencenetworkinvolvingBritishandContinentalmathematicians.HewasflooredbytheresultsinDeAnalysiandaskedBarrowwhoitsauthorwas.WithNewton’spermission,Barrowunmaskedhim:“Iamgladmyfriendspapergivethyousomuchsatisfaction.HisnameisMr.Newton;afellowofourCollege,&veryyoung...butofanextraordinarygeniusandproficiencyinthesethings.”Collinswasneversomeonetokeepasecretinconfidence.Heteasedhis

correspondentswithsnippetsofDeAnalysiandwowedthemwithNewton’sresultswithoutexplainingwheretheycamefrom.In1675heshowedNewton’spowerseriesfortheinversesineandsinefunctionstoaDanishmathematiciannamedGeorgBohr,andheinturntoldLeibnizaboutthem.LeibnizsentarequesttothesecretaryoftheRoyalSocietyofLondon,aGerman-bornschmoozerandpromoterofsciencenamedHenryOldenburg:“SinceIsay,he[Bohr]hasbroughtusthesestudieswhichseemtometobeveryingenious,thelatterseriesinparticularhavingacertainrareelegance,soIshallbegrateful,IllustriousSir,ifyouwillsendmetheproof.”

OldenburgpassedtherequestalongtoNewton,andNewtonwasnotpleased.Sendtheproof?Ha.Instead,herepliedtoLeibniz,throughOldenburg,withpageafterpageofcryptic,intimidatingformulas,thefullarmamentofDeAnalysi.OutsideofNewton’sinnercircle,noonehadeverseenmathlikethis.Andforgoodmeasure,Newtonstressedthatthematerialwasoldhat:“Iwriterathershortlybecausethesetheorieslongagobegantobedistastefultome,tosuchanextentthatIhavenowrefrainedfromthemfornearlyfiveyears.”Undeterred,LeibnizwrotebackandpokedNewton,hopingtoextractabit

more.Hewasanewcomertoallthis.Adiplomat,logician,linguist,andphilosopher,he’donlyrecentlybecomeinterestedinadvancedmathematics.He’dspenttimewithChristiaanHuygens,theleadingmathematicalmindinEurope,togetuptospeedonthelatestdevelopments.Afterjustthreeyearsofstudy,LeibnizhadalreadyoutpacedeveryoneontheContinent.AllheneedednowwastofigureoutwhatNewtonknew...andwhathewaswithholding.ToprytheinformationoutofNewton,Leibniztriedadifferenttack.Hemade

themistakeoftryingtoimpresshim.Heproducedsomeofhisownwares—inparticular,aninfiniteserieshewasproudof—andofferedittoNewton,ostensiblyasagiftbutactuallyasasignalthathewasworthytoreceivethesecret.NewtonrepliedthroughOldenburgtwomonthslater,onOctober24,1676.He

openedwithflattery,callingLeibniz“verydistinguished”andpraisinghisinfiniteseries,sayingthatit“leadsusalsotohopeforverygreatthingsfromhim.”Werethesecomplimentsmeanttobetakenseriously?Apparentlynot,forthenextlineburnedwithacidsarcasm:“Thevarietyofwaysbywhichthesamegoalisapproachedhasgivenmethegreaterpleasure,becausethreemethodsofarrivingatseriesofthatkindhadalreadybecomeknowntome,sothatIcouldscarcelyexpectanewonetobecommunicatedtous.”Inotherwords,ThanksforshowingmesomethingIalreadyknowhowtodothreeotherways.Intherestofhisletter,NewtontoyedwithLeibniz.Herevealedsomeofhis

ownmethodsforinfiniteseries,explainingtheminthepedagogicalmanneronewouldusetolectureaschoolchild.Fortunatelyforposterity,thesepartsoftheletteraresoclearthatwecanunderstandexactlywhatNewtonhadinmind.Butwhenhegottohismostprizedpossessions(therevolutionarytechniques

ofhissecondtractoncalculus,DeMethodis,includingthefundamentaltheorem,whichhadn’tleakedoutyet),Newton’sgentleexpositioncametoahalt:“Thefoundationoftheseoperationsisevidentenough,infact;butbecauseIcannotproceedwiththeexplanationofitnow,Ihavepreferredtoconcealitthus:6accdae13eff7i3l9n4o4qrr4s8t12vx.OnthisfoundationIhavealsotriedto

simplifythetheorieswhichconcernthesquaringofcurves,andIhavearrivedatcertaingeneraltheorems.”Andwiththatencryptedcode,Newtondangledhismostcherishedsecretin

frontofLeibniz,essentiallytellinghim,Iknowsomethingyoudon’t,andevenifyoudiscoveritlater,thiscryptogramwillproveIknewitfirst.WhatNewtondidnotrealizewasthatLeibnizhadalreadydiscoveredthe

secretonhisown.

IntheTwinklingofanEyelid

Between1672and1676,Leibnizhadcreatedhisownversionofcalculus.LikeNewton,hespottedandprovedthefundamentaltheorem,recognizeditssignificance,andbuiltanalgorithmicsystemaroundit.Withitshelp,hewrote,he’dbeenabletoderive“inthetwinklingofaneyelid”nearlyallthetheoremsaboutquadraturesandtangentsknownatthattime—exceptfortheonesNewtonwasstillhidingfromtheworld.WhenLeibnizwrotehistwoletterstoNewtonin1676,nosingaroundand

askingforproofs,heknewhewasbeingpushybuthecouldn’thelpit.Asheoncetoldafriend,“Ifeelmyselfburdenedwithadeficiencythatcountsforagreatdealinthisworld,namely,thatIlackpolishedmannersandtherebyoftenspoilthefirstimpressionofmyperson.”Skinny,stooped,andpale,Leibnizmightnothavebeenmuchtolookat,but

hismindwasbeautiful.HewasthemostversatilegeniusinacenturyofgeniusesthatincludedDescartes,Galileo,Newton,andBach.AlthoughLeibnizfoundhiscalculusadecadeafterNewtondid,heis

generallyconsidereditsco-inventorforseveralreasons.Hepublisheditfirst,inagracefulanddigestibleform,andhecoucheditinacarefullydesigned,elegantnotationthat’sstillusedtoday.Moreover,heattracteddiscipleswhospreadthewordwithevangelicalzeal.Theywroteinfluentialtextbooksanddevelopedthesubjectinluxuriantdetail.Muchlater,whenLeibnizwasaccusedofstealingcalculusfromNewton,hisdisciplesdefendedhimvigorouslyandcounterattackedNewtonwithequalfervor.Leibniz’sapproachtocalculusismoreelementary—and,insomeways,more

intuitive—thanNewton’s.Italsoexplainswhythestudyofderivativeshaslongbeencalleddifferentialcalculusandwhytheoperationoftakingaderivativeiscalleddifferentiation—it’sbecause,inLeibniz’sapproach,conceptscalled

differentialsarethetrueheartofcalculus;derivativesaresecondary,anafterthought,alaterrefinement.Nowadays,wetendtoforgethowimportantdifferentialswere.Modern

textbooksdownplaythem,redefinethem,orwhitewashthemawaybecausetheyare(gasp!)infinitesimals.Assuch,theyareseenasparadoxical,transgressive,andscary,sojusttobeonthesafeside,manybookskeepinfinitesimalslockedintheattic,likeNormanBates’smotherinPsycho.Butthey’rereallynothingtobeafraidof.Really.Let’sgomeetMother.

Infinitesimals

Aninfinitesimalisahazything.Itissupposedtobethetiniestnumberyoucanpossiblyimaginethatisn’tactuallyzero.Moresuccinctly,aninfinitesimalissmallerthaneverythingbutgreaterthannothing.Evenmoreparadoxically,infinitesimalscomeindifferentsizes.An

infinitesimalpartofaninfinitesimalisincomparablysmallerstill.Wecouldcallitasecond-orderinfinitesimal.Justasthereareinfinitesimalnumbers,thereareinfinitesimallengthsand

infinitesimaltimes.Aninfinitesimallengthisnotapoint—it’sbiggerthanthat—butitissmallerthananylengthyoucanenvision.Likewise,aninfinitesimaltimeintervalisnotaninstant,notasinglepointintime,butitisshorterthananyconceivableduration.Theconceptofinfinitesimalsaroseasawayofspeakingaboutlimits.Recall

theexampleinchapter1wherewelookedatasequenceofregularpolygonsstartingwithanequilateraltriangleandasquareandproceedingupwardthroughpentagons,hexagons,andotherregularpolygonshavingmoreandmoresides.Wenoticedthatthemoresidesweconsideredandtheshorterwemadethem,themorethepolygonbegantolooklikeacircle.Weweretemptedtosaythatacircleisaninfinitepolygonhavinginfinitesimalsidesbutbitourtonguesbecausethenotionseemedtoleadtononsense.Wealsofoundthatifwechoseanypointonthecircumferenceofthecircle

andlookedatitunderamicroscope,anytinyarccontainingthatpointlookedstraighterandstraighterasthemagnificationincreased.Inthelimitofinfinitemagnification,thattinyarclooksperfectlystraight.Inthatsense,itreallydoesseemhelpfultothinkofthecircleasaninfinitecollectionofstraightpiecesandthereforeasaninfinitepolygonwithinfinitesimalsides.

BothNewtonandLeibnizusedinfinitesimals,butwhileNewtonlaterdisavowedtheminfavoroffluxions(whichareratiosoffirst-orderinfinitesimalsandhencefiniteandpresentable,justlikederivatives),Leibniztookamorepragmaticview.Hedidn’tfretaboutwhethertheyactuallyexisted.Hesawthemasusefulshorthand,anefficientwaytorecastargumentsaboutlimits.Healsoregardedthemashelpfulbookkeepingdevicesthatfreedtheimaginationformoreproductivework.Asheexplainedtoacolleague,“Philosophicallyspeaking,Inomorebelieveininfinitelysmallquantitiesthanininfinitelygreatones,thatis,ininfinitesimalsratherthaninfinituples.Iconsiderbothasfictionsofthemindforsuccinctwaysofspeaking,appropriatetothecalculus.”Andwhatdomathematiciansthinktoday?Doinfinitesimalsreallyexist?It

dependsonwhatyoumeanbyreally.Physiciststellusinfinitesimalsdon’texistintherealworld(butthenagain,neitherdoestherestofmathematics).Withintheidealworldofmathematics,infinitesimalsdon’texistintherealnumbersystem,buttheydoexistincertainnonstandardnumbersystemsthatgeneralizetherealnumbers.ForLeibnizandhisfollowers,theyexistedasfictionsofthemindthatcameinhandy.That’sthewaywewillbethinkingaboutthem.

TheCubeofNumbersnear2

Toseehowilluminatinginfinitesimalscanbe,let’sstartveryconcretely.Considerthisarithmeticproblem:What’s2cubed(meaning2×2×2)?It’s8,ofcourse.Whatabout2.001×2.001×2.001?Slightlymorethan8,sure,buthowmuchmore?Whatweareafterhereisawayofthinking,notanumericalanswer.The

generalquestionis,whenwechangetheinputtoaproblem(here,bychanging2to2.001),howmuchdoestheoutputchange?(Here,itchangesfrom8to8plussomethingwhosestructurewewanttounderstand.)Sinceit’shardtoresistpeeking,let’sgoaheadandseewhatacalculatorhas

tosay.Punchingin2.001andhittingthex3buttongives

(2.001)3=8.012006001.Thestructuretonoticeisthattheextrabitafterthedecimalpointisreallythreeextrabitsofverydifferentsizes:

.012006001=.012+.000006+.000000001.

Thinkofthisassmallplussuper-smallplussuper-super-small.Wecanunderstandthestructurewe’reseeingbyworkingwithalgebra.

Supposeaquantityx(playedherebythenumber2)changesslightlytox+Δx(inthiscase,becoming2.001).ThesymbolΔxdenotesthedifferenceinx,meaningatinychangeinx(here,Δx=0.001).Then,whenweaskwhat(2.001)3is,wearereallyaskingwhat(x+Δx)3is.Multiplyingitout(orusingPascal’striangleorthebinomialtheorem),wefindthat

(x+Δx)3=x3+3x2Δx+3x(Δx)2+(Δx)3.Forourproblemwherex=2,thisequationbecomes

(2+Δx)3=23+3(2)2(Δx)+3(2)(Δx)2+(Δx)3=8+12Δx+6(Δx)2+(Δx)3.

Nowweseewhytheextrabitbeyond8consistsofthreebitsofdifferentsizes.

Thesmallbutdominantbitis12Δx=12(.001)=.012.Theremainingbits6(Δx)2and(Δx)3accountforthesuper-small.000006andthesuper-super-small.000000001.ThemorefactorsofΔxthereareinabit,thesmalleritis.That’swhythebitsaregradedinsize.EveryadditionalmultiplicationbythetinyfactorΔxmakesasmallbitevensmaller.Thekeyinsightbehinddifferentialcalculusisdisplayedrighthereinthis

humbleexample.Inmanyproblemsofcauseandeffect,doseandresponse,inputandoutput,oranyothersortofrelationshipbetweenavariablexandanothervariableythatdependsonit,asmallchangeintheinput,Δx,producesasmallchangeintheoutput,Δy.Thatsmallchangeistypicallyorganizedinastructuredwaywecanexploit—namely,thechangeintheoutputconsistsofahierarchyofbits.Theyaregradedinsizefromsmalltosuper-smalltoevensmallercontributions.Thatgradationallowsustofocusonthesmallbutdominantchangeandneglectalltherest,thesuper-smallandevensmallerones.That’sthekeyinsight.Althoughthesmallchangeissmall,itisgiganticcomparedtotheothers(muchlike.012wasgiganticcomparedto.000006and.000000001).

Differentials

Thiswayofthinking,inwhichweneglectallcontributionstotherightanswerexceptforthebiggestone,thelion’sshare,mightseemonlyapproximate.Anditis—ifthechangesintheinput,likethe.001wetackedontothe2above,arefinitechanges.Butifweconsiderinfinitesimalchangesintheinput,thenourthinkingbecomesexact.Wemakenoerrorwhatsoever.Thelion’ssharebecomeseverything.And,aswe’veseenthroughoutthisbook,infinitesimalchangesarepreciselywhatweneedtomakesenseofslopes,instantaneousvelocities,andtheareasofcurvedregions.Toseehowthisworksinpractice,let’sgobacktotheexampleabove,where

weweretryingtocalculatethecubeofanumberslightlygreaterthan2.Exceptnow,let’schange2to2+dx,wheredxissupposedtorepresentaninfinitesimallysmalldifferenceΔx.Thisnotionisinherentlynonsensicalsodon’tthinkaboutittoohard.Thepointisthatlearninghowtoworkwithitmakescalculusabreeze.Inparticular,theearliercalculationof(2+Δx)3as8+12Δx+6(Δx)2+(Δx)3

nowshrinkstosomethingmuchsimpler:

(2+dx)3=8+12dx.Whathappenedtotheothertermslike6(dx)2+(dx)3?Wediscardedthem.

Theyarenegligible.Theyaresuper-smallandsuper-super-smallinfinitesimalsandareutterlyinconsequentialcomparedto12dx.Butthenwhydowekeep12dx?Isn’titequallynegligiblecomparedto8?Itis,butifweweretodiscardittoo,wewouldn’tbeconsideringanychangeatall.Ouranswerwouldbefrozenat8.Sotherecipeisthis:tostudyinfinitesimalchange,keeptermsthatinvolvedxtothefirstpowerandignoretherest.Thiswayofthinking,usinginfinitesimalslikedx,canberephrasedintermsof

limitsandmadeperfectlykosherandrigorous.That’showmoderntextbooksdealwiththem.Butit’seasierandfastertouseinfinitesimals.Thetermofartfortheminthiscontextisdifferentials.TheirnamecomesfromthinkingofthemasbeinglikethedifferencesΔxandΔy,inthelimitasthosedifferencestendtozero.Theyarelikewhatwesawwhenwelookedataparabolaunderamicroscopeandwatchedthecurvegetstraighterandstraighteraswezoomedinonit.

DerivativesviaDifferentials

Letmeshowyouhoweasycertainideasbecomewhencouchedindifferentials.Forexample,what’stheslopeofacurvewhenit’sviewedasagraphinthexyplane?Aswelearnedfromourworkwiththeparabolainchapter6,theslopeisthederivativeofy,definedasthelimitofΔy/ΔxasΔxapproacheszero.Butwhatisitintermsofdifferentials?It’ssimplydy/dx.It’sasifthecurveismadeupoflittlestraightpieces:

Ifwethinkofdyasaninfinitesimalriseanddxasaninfinitesimalrun,theslopeissimplytheriseovertherun,justasitalwaysis,andhenceisdy/dx.Toapplythisapproachtoaspecificcurve(sayy=x3,thecaseweconsidered

whilecubingnumbersslightlygreaterthan2),wecalculatedyasfollows.Write

y+dy=(x+dx)3.Asbefore,theright-handsideexpandsto

(x+dx)3=x3+3x2dx+3x(dx)2+(dx)3.Butnow,followingtherecipe,wediscardtheterms(dx)2and(dx)3,sincethey’renotpartofthelion’sshare.Thus

y+dy=(x+dx)3=x3+3x2dx.

Andsincey=x3,wecansimplifytheequationabovetoobtain

dy=3x2dx.Dividingbothsidesbydxyieldsthecorrespondingslope,

dy⁄dx=3x2.

Atx=2,thisgivesaslopeof3(2)2=12.That’sthesame12wesawearlier.It’swhychanging2to2.001gaveus(2.001)3≈8.012.Itmeansthataninfinitesimalchangeinxnear2(callitdx)getsconvertedtoaninfinitesimalchangeinynear8(callitdy)that’s12timesbigger(dy=12dx).Incidentally,similarreasoningshowsthatforanypositiveintegern,the

derivativeofy=xnis=dy/dx=nxn-1,aresultwe’vementionedearlier.Withalittlemorework,wecouldextendthisresulttonegative,fractional,andirrationaln.Thegreatadvantageofinfinitesimalsingeneralanddifferentialsinparticular

isthattheymakecalculationseasier.Theyprovideshortcuts.Theyfreethemindformoreimaginativethought,justasalgebradidforgeometryinanearlierera.ThisiswhatLeibnizadoredabouthisdifferentials.AshewrotetohismentorHuygens,“Mycalculusgaveme,almostwithoutmeditation,thegreatpartofthediscoverieswhichhavebeenmadeconcerningthissubject.ForwhatIlovemostaboutmycalculusisthatitgivesusthesameadvantagesovertheAncientsinthegeometryofArchimedes,thatVièteandDescarteshavegivenusinthegeometryofEuclidorApollonius,infreeingusfromhavingtoworkwiththeimagination.”Theonlythingwrongwithinfinitesimalsisthattheydon’texist,atleastnot

withinthesystemofrealnumbers.Oh,andoneotherthing—theyareparadoxical.Theywouldn’tmakesenseeveniftheydidexist.OneofLeibniz’sdisciples,JohannBernoulli,realizedthey’dhavetosatisfynonsensicalequationslikex+dx=x,eventhoughdxisn’tzero.Hmmm.Well,youcan’thaveeverything.Infinitesimalsdogivetherightanswersoncewelearnhowtoworkwiththem,andthebenefitstheyprovidemorethanmakeupforanypsychicdistresstheymaycause.TheyarelikePicasso’sliethathelpsusrealizethetruth.

Asafurtherdemonstrationofinfinitesimals’power,LeibnizusedthemtoderiveSnell’ssinelawfortherefractionoflight.Recallfromchapter4thatwhenlightpassesfromonemediumintoanother—let’ssayfromairintowater—itbendsinaccordancewithamathematicallawthatwasdiscoveredandrediscoveredseveraltimesoverthecenturies.Fermathadexplaineditwithhisprincipleofleasttime,buthestruggledmightilytosolvetheoptimizationproblemthathisprincipleimplied.Withhisnewcalculusofdifferentials,Leibnizdeducedthesinelawwitheaseandnotedwithevidentpridethat“otherverylearnedmenhavesoughtinmanydeviouswayswhatsomeoneversedinthiscalculuscanaccomplishintheselinesasbymagic.”

TheFundamentalTheoremviaDifferentials

AnothertriumphofLeibniz’sdifferentialsisthattheymadethefundamentaltheoremtransparent.RecallthatthefundamentaltheoremconcernstheareaaccumulationfunctionA(x),whichgivestheareaunderthecurvey=f(x)overtheintervalfrom0tox.Thetheoremsaysthatasweslidextotheright,theareaunderthecurveaccumulatesatarategivenbyf(x)itself.Thusf(x)isthederivativeofA(x).

Toseewherethisresultcomesfrom,supposewechangexbyaninfinitesimalamounttox+dx.HowmuchdoestheareaA(x)change?Bydefinition,itchangesbyanamountdA.HencethenewareaequalstheoldareaplusthechangeinareaandisthereforeA+dA.ThefundamentaltheoremdropsoutimmediatelyoncewevisualizewhatdA

mustbe.Assuggestedbythepicturebelow,theareachangesbytheinfinitesimalamountdAgivenbytheareaoftheinfinitesimallythinverticalstripbetweenxandx+dx:

Thatstripisarectangleofheightyandbasedx.Soitsareaisitsheighttimesitsbase,whichisydxor,ifyouprefer,f(x)dx.Actually,thestripisarectangleonlywhenviewedinfinitesimally.Inreality,

forastripofanyfinitewidthΔx,thechangeinareaΔAhastwocontributions.ThedominantoneisarectangleofareayΔx.Amuchsmalleroneistheareaofthetiny,curved,triangular-lookingcapontopoftherectangle.

Here’sanothercasewheretheinfinitesimalworldisnicerthantherealworld.

Intherealworld,wewouldhavetoaccountfortheareaofthecap,whichwouldn’tbeeasytoestimatebecauseitwoulddependonthedetailsofthecurveontop.Butasthewidthoftherectangleapproacheszeroand“becomes”dx,theareaofthecapbecomesnegligiblecomparedtotheareaoftherectangle.It’ssuper-smallcomparedtosmall.TheupshotisthatdA=ydx=f(x)dx.Boom—that’sthefundamentaltheorem

ofcalculus.Or,asitismorepolitelyphrasednowadays(inourmisguidederawhendifferentialshavebeenforsakenforderivatives),

dA⁄dx=y=f(x).

Thisisexactlywhatwefoundinchapter7withthepaint-rollerargument.Onelastthing:Whenweregardtheareaunderacurveasasumofinfinitely

manyinfinitesimalrectangularstrips,wewriteitas

A(x)=∫x₀f(x)dx.

Thatlong-necked,swan-likesymbolisactuallyastretched-outS.TheSremindsusthatasummationistakingplace.It’sasummationofapeculiarkind,distinctivetointegralcalculus,involvingasumofinfinitelymanyinfinitesimalstrips,allbeingintegratedintoasingle,coherentarea.Asasymbolofintegration,it’scalledanintegralsign.Leibnizintroduceditina1677manuscriptandpublisheditin1686.It’scalculus’smostrecognizableicon.Thezeroatthebottomofitandthexatthetopofitindicatetheendpointsoftheintervalofthex-axisoverwhichtherectanglesstand.Thoseendpointsarecalledthelimitsofintegration.

WhatLedLeibniztoDifferentialsandtheFundamentalTheorem?

NewtonandLeibnizarrivedatthefundamentaltheoremofcalculusbytwoseparateroutes.Newtoncameatitbythinkingaboutmotionandflow,thecontinuoussideofmath.Leibnizcameatitfromtheotherside.Althoughhewasnotamathematicianbytraining,earlierinhislifehe’dspentsometimethinkingaboutdiscretemath—wholenumbersandcounting,combinationsandpermutations,andfractionsandsumsofaparticularsort.HebeganwadingintodeeperwatersafterhemetChristiaanHuygens.Atthe

time,LeibnizwasservingonadiplomaticmissioninParis,buthefoundhimselfentrancedbywhatHuygenswastellinghimaboutthelatestdevelopmentsinmathematicsandhewantedtolearnmore.Withamazingpedagogicalprescience(orwasitluck?),Huygenschallengedhisstudentwithaproblemthatledhimtothefundamentaltheorem.Theproblemhegavehimwastocalculatethisinfinitesum:

1⁄1·2+1⁄2·3+1⁄3·4+···+1⁄n·(n+1)+···=?

(Thedotsinthedenominatorsmeanmultiplication.)Tobringtheproblemdowntoearth,let’sbeginwithawarm-upversion.Supposethesumhas,say,99termsinitinsteadofinfinitelymany.Thenwewouldhavetocalculate

S=1⁄1·2+1⁄2·3+1⁄3·4+···+1⁄n·(n+1)+···+1⁄99·100.

Ifyoudon’tseethetrick,thisisatediousbutstraightforwardcalculation.Withsufficientpatience(oracomputer),wecouldploddinglyaddupthe99fractions.Butthatwouldbemissingthepoint.Thepointistofindanelegantsolution.Elegantsolutionsarevaluedinmathinpartbecausethey’reprettybutalsobecausethey’repowerful.Thelighttheyshedcanoftenbeusedtoilluminateotherproblems.Inthiscase,theelegantlightthatLeibnizdiscoveredquicklypointedhimtothefundamentaltheorem.HesolvedHuygens’sproblemwithabrillianttrick.ThefirsttimeIsawit,I

feltlikeIwaswatchingamagicianpullarabbitoutofahat.Ifyouwanttoexperiencethatsamefeeling,skiptheanalogyI’mabouttopresent.Butifyouprefertounderstandwhat’sbehindthemagic,here’swhatmakesitwork.Imaginesomeoneclimbingaverylongandirregularstaircase.

Supposeourclimberwantedtomeasurethetotalverticalrisefromthebottomofthestaircasetothetop.Howcouldhedothat?Well,hecouldalwaysaddupalltherisesoftheindividualstepsinbetween.Thatuninspiredstrategywouldbelikeaddingupthe99termsinthesumSabove.Itcouldbedone,butitwouldbeunpleasantbecausethestaircaseissoirregular.Andifthestaircasehasmillionsofsteps,addingupalltheirriseswouldbeahopelesstask.Therehastobeabetterway.Thebetterwayistouseanaltimeter.Analtimeterisadevicethatmeasures

altitude.IfZenointhepicturehadanaltimeter,hecouldsolvehisproblembysubtractingthealtitudeatthebottomofthestaircasefromthealtitudeatthetop.That’sallthereistoit:thetotalverticalriseequalsthedifferenceinthosetwoaltitudes.Thedifferencebetweenthemhastoequalthesumofalltherisesinbetween.Nomatterhowirregularthestaircaseis,thistrickwillalwayswork.Thesuccessofthetrickhingesonthefactthatthealtimeterreadingsare

intimatelyrelatedtotherisesofthesteps—theriseofanygivenstepisthedifferenceofconsecutivealtimeterreadings.Inotherwords,theheightofastepequalsthealtitudeatitstopminusthealtitudeatitsbottom.Bynowyou’reprobablythinking,Whatdoesanaltimeterhavetodowiththe

originalmathproblemofaddingupalonglistofcomplicated,irregularnumbers?Well,ifwecouldsomehowfindtheanalogofanaltimeterforacomplicated,irregularsum,thatsumwouldbecomeeasy.Itwouldjustadduptothedifferencebetweenthehighestandlowestaltimeterreadings.ThisisessentiallywhatLeibnizdid.HefoundanaltimeterforthesumS.Itenabledhimtowriteeachterminthesumasadifferenceofconsecutivealtimeterreadings,

whichinturnallowedhimtocomputethedesiredsumusingtheideamentionedabove.Thenhegeneralizedhisaltimetertootherproblems.Ultimatelyitledhimtothefundamentaltheoremofcalculus.Withthisanalogyinmind,let’sexamineSagain:

S=1⁄1·2+1⁄2·3+1⁄3·4+···+1⁄n·(n+1)+···+1⁄99·100.

We’regoingtorewriteeachtermasadifferenceoftwoothernumbers.Thisislikesayingthattheriseofeachstepisthedifferenceofthealtimeterreadingsatitstopandbottom.Forthefirststep,therewritinggoeslikethis:

1⁄1·2=2−1⁄1·2=1⁄1−1⁄2.

Admittedly,it’snotobviousyetwherethisisgoing,butstaytuned.Inamomentwe’llseehowhelpfulitistorewritethefraction1/(1⋅2)asadifferenceoftwoconsecutiveunitfractions,1/1and½.(Aunitfractionmeansafractionwitha1inthenumerator.Theseconsecutiveunitfractionsaregoingtoplaytheroleofconsecutivealtimeterreadings.)Also,ifthearithmeticaboveseemsunclear,trysimplifyingtheequationsbyworkingthemfromrighttoleft.Onthefarrightwearesubtractingaunitfraction(½)fromanotherunitfraction(1/1);inthemiddleweareputtingthemoveracommondenominator;andonthefarleftwearesimplifyingthenumerator.Similarly,wecanwriteeveryotherterminSasadifferenceofconsecutive

unitfractions:

1⁄2·3=3−2⁄2·3=1⁄2−1⁄3.1⁄3·4=4−3⁄3·4=1⁄3−1⁄4.

andsoon.Whenweaddupallthesedifferencesofunitfractions,Sbecomes

S=(1⁄1−1⁄2)+(1⁄2−1⁄3)+(1⁄3−1⁄4)+···+(1⁄98−1⁄99)+(1⁄99−1⁄100).

Nowweseethemethodinthemadness.Lookcarefullyatthestructureofthissum.Nearlyalltheunitfractionsappeartwice,oncewithanegativesignandoncewithapositivesign.Forexample,½issubtractedandthenaddedbackin;theneteffectisthatthe½termscanceleachotherout.Thesameistruefor⅓.It

occurstwiceandcancelsitself.Nearlyalltheotherunitfractions,uptoandincluding1/99,dothesame.Theonlyexceptionsarethefirstandlastunitfractions,1/1and1/100.BeingattheendsofthelineinS,theyhavenopartnerstocancelwith.Afterthesmokeclears,theyaretheonlyunitfractionsleftstanding.Sotheresultis

S=1⁄1−1⁄100.

Thismakesperfectsenseintermsofthestaircaseanalogy.Itsaysthetotalriseofallthestepsisthealtitudeatthetopofthestaircaseminusthealtitudeatthebottom.Incidentally,Ssimplifiesto99/100.That’stheanswertothepuzzlewith99

terms.Leibnizrealizedthathecouldaddanynumberoftermsusingthesametrick.IfthesumhadNtermsinsteadof99,theresultwouldbe

S=1⁄1−1⁄N+1.

ThustheanswertoHuygens’soriginalquestionabouttheinfinitesumbecomesclear:AsNapproachesinfinity,theterm1/(N+1)approacheszero,andsoSapproaches1.Thatlimitingvalueof1istheanswertoHuygens’spuzzle.ThekeythatallowedLeibniztofindthesumwasthatithadaveryparticular

structure:itcouldberewrittenasasumofconsecutivedifferences(inthiscase,differencesofconsecutiveunitfractions).Thatdifferencestructurecausedthemassivecancellationswesawabove.Sumswiththispropertyarenowtermedtelescopingsumsbecausetheycalltomindoneofthoseoldcollapsibletelescopesyouseeinpiratemovies,thekindofspyglassthatcanbestretchedoutorcontractedatwill.Theanalogyisthattheoriginalsumappearsinitsstretched-outform,but,becauseofitsdifferencestructure,itcanbetelescopeddowntoamuchmorecompactresult.Theonlytermsthatsurvivethecollapsearethetermswithoutpartnerstocancelthem,theonesattheveryendsofthetelescope.NaturallyLeibnizwonderedifhecouldusethetelescopingtrickonother

problems.Itwasanideaworthpursuing,givenhowpowerfulitcouldbe.Confrontedwithalonglistofnumberstosum,ifhecouldwriteeachnumberasadifferenceofconsecutivenumbers(tobedetermined),thetelescopingtrickwouldworkagain.AndthatgotLeibnizthinkingaboutareas.Approximatingtheareaundera

curveinthexyplane,afterall,amountedtosummingalonglistofnumbers,the

areasoflotsofthin,verticalrectangularstrips.

Theideabehindwhathehadinmindisdemonstratedinthefigureabove.It

showsonlyeightrectangularareas,butyoushouldtrytoimagineasimilarimagewithmillionsorbillionsofmuchthinnerrectanglesor,betteryet,infinitelymanyinfinitesimallythinrectangles.That,unfortunately,ishardtodraworvisualize.That’swhyI’musingeightblockyrectanglesfornow.Forsimplicity,supposealltherectangleshavethesamewidth.CallitΔx.The

heightsoftherectanglesarey1,y2,...y8.Thenthetotalareaoftheapproximatingrectanglesis

y1Δx+y2Δx+...+y8Δx.ThissumofeightnumberswouldconvenientlytelescopeifwecouldsomehowfindmagicnumbersA0,A1,A2,...,A8whosedifferencesgivetherectangularareas

y1Δx=A1−A0y2Δx=A2−A1y3Δx=A3−A2

andsoon,downtoy8Δx=A8−A7.Thenthetotalareaoftherectangleswouldtelescopetothis:

y1Δx+y2Δx+···+y8Δx=(A1−A0)+(A2−A1)+···+(A8−A7)=A8−A0

Nowthinkaboutthelimitofinfinitesimallythinstrips.TheirwidthΔxturnsintothedifferentialdx.Theirvaryingheightsy1,y2,...,y8becomey(x),afunctionthatgivestheheightoftherectanglestandingoverthepointlabeledbythevariablex.Thesumoftheinfinitelymanyrectangularareasbecomestheintegral∫y(x)dx.Andasfortheearliertelescoping,thesumthatwaspreviouslyA8−A0nowbecomesA(b)−A(a),whereaandbarethevaluesofxontheleftandrightendsoftheareabeingcalculated.Theinfinitesimalversionoftelescopingthenyieldstheexactareaunderthecurve:

y(x)dx=A(b)−A(a).

AndhowdowefindthemagicfunctionA(x)thatmakesallthispossible?Well,lookattheearlierequationslikey1Δx=A1−A0.Theymorphintoy(x)dx=dA

astherectanglesbecomeinfinitesimallythin.Toputthesameresultintermsofderivativesinsteadofdifferentials,dividebothsidesoftheequationabovebydxtoget

dA⁄dx=y(x).

ThisishowwefindtheanalogsofthemagicnumbersA0,A1,A2,...,A8thatcausetelescopingtooccur.Inthelimitofinfinitesimallythinstrips,theyaregivenbytheunknownfunctionA(x)whosederivativeisthegivencurvey(x).

AllofthisisLeibniz’sversionofthebackwardproblemandthefundamentaltheoremofcalculus.Asheputit,“Findingtheareasoffiguresisreducedtothis:givenaseries,tofindthesums,or(toexplainthisbetter)givenaseries,tofindanotheronewhosedifferencescoincidewiththetermsofthegivenseries.”Inthisway,differencesandtelescopingsumsguidedLeibniztodifferentialsandintegralsandfromtheretothefundamentaltheorem,justasfluxionsandexpandingareashadledNewtontothatsamesecretfountain.

FightingHIVwithanAssistfromCalculus

Althoughdifferentialsarefictionsofthemind,theyhaveaffectedourworld,oursocieties,andourlivesinprofoundlynonfictionalwayseversinceLeibnizinventedthem.Foranexampleinourowntime,considerthesupportingrolethatdifferentialsplayedintheunderstandingandtreatmentofHIV,thehumanimmunodeficiencyvirus.Inthe1980s,amysteriousdiseasebegankillingtensofthousandsofpeoplea

yearintheUnitedStatesandhundredsofthousandsworldwide.Nooneknewwhatitwas,whereitcamefrom,orwhatwascausingit,butitseffectswereclear—itweakenedpatients’immunesystemssoseverelythattheybecamevulnerabletorarekindsofcancer,pneumonia,andopportunisticinfections.Deathfromthediseasewasslow,painful,anddisfiguring.Doctorsnameditacquiredimmunedeficiencysyndrome,orAIDS.Patientsanddoctorsweredesperate.Nocurewasinsight.Basicresearchdemonstratedthataretroviruswastheculprit.Itsmechanism

wasinsidious:ThevirusattackedandinfectedwhitebloodcellscalledhelperTcells,akeycomponentoftheimmunesystem.Onceinside,thevirushijackedthecell’sgeneticmachineryandco-opteditintomakingmoreviruses.Thosenewvirusparticlesthenescapedfromthecell,hitchedarideinthebloodstreamandotherbodilyfluids,andlookedformoreTcellstoinfect.Thebody’simmunesystemrespondedtothisinvasionbytryingtoflushoutthevirusparticlesfromthebloodandkillasmanyinfectedTcellsasitcouldfind.Insodoing,theimmunesystemwaskillinganimportantpartofitself.ThefirstantiretroviraldrugapprovedtotreatHIVappearedin1987.Although

itslowedHIVdownbyinterferingwiththehijackingprocess,itwasn’taseffectiveashoped,andthevirusoftenbecameresistanttoit.Adifferentclassofdrugscalledproteaseinhibitorsappearedin1994.TheythwartedHIVbyinterferingwiththenewlyproducedvirusparticles,keepingthemfrommaturing

andrenderingthemnoninfectious.Althoughalsonotacure,proteaseinhibitorswereagodsend.Soonafterproteaseinhibitorsbecameavailable,ateamofresearchersledby

Dr.DavidHo(aformerphysicsmajoratCaltechandso,presumably,someonecomfortablewithcalculus)andamathematicalimmunologistnamedAlanPerelsoncollaboratedonastudythatchangedhowdoctorsthoughtaboutHIVandrevolutionizedhowtheytreatedit.BeforetheworkofHoandPerelson,itwasknownthatuntreatedHIVinfectiontypicallyprogressedthroughthreestages:anacuteprimarystageofafewweeks,achronicandparadoxicallyasymptomaticstageofuptotenyears,andaterminalstageofAIDS.Inthefirststage,soonafterapersonbecomesinfectedwithHIV,heorshe

displaysflu-likesymptomsoffever,rash,andheadaches,andthenumberofhelperTcells(alsoknownasCD4cells)inthebloodstreamplummets.AnormalT-cellcountisabout1000cellspercubicmillimeterofblood;afterprimaryHIVinfection,T-cellcountdropstothelowhundreds.SinceTcellshelpthebodyfightinfections,theirdepletionseverelyweakenstheimmunesystem.Meanwhile,thenumberofvirusparticlesintheblood,knownastheviralload,spikesandthendropsastheimmunesystembeginstocombattheHIVinfection.Theflu-likesymptomsdisappearandthepatientfeelsbetter.Attheendofthisfirststage,theviralloadstabilizesatalevelthatcan,

puzzlingly,lastformanyyears.Doctorsrefertothislevelasthesetpoint.ApatientwhoisuntreatedmaysurviveforadecadewithnoHIV-relatedsymptomsandnolabfindingsotherthanapersistentviralloadandalowandslowlydecliningT-cellcount.Eventually,however,theasymptomaticstageendsandAIDSsetsin,markedbyafurtherdecreaseintheT-cellcountandasharpriseintheviralload.Onceanuntreatedpatienthasfull-blownAIDS,opportunisticinfections,cancers,andothercomplicationsusuallycausethepatient’sdeathwithintwotothreeyears.Thekeytothemysterywasinthedecade-longasymptomaticstage.Whatwas

goingonthen?WasHIVlyingdormantinthebody?Otherviruseswereknowntohibernatelikethat.Thegenital-herpesvirus,forexample,hunkersdowninnervegangliatoevadetheimmunesystem.Thechickenpoxvirusalsodoesthis,hidingoutinnervecellsforyearsandsometimesawakeningtocauseshingles.ForHIV,thereasonforthelatencywasunknown,butitbecameclearafterHoandPerelson’swork.Ina1995study,theygavepatientsaproteaseinhibitor,notasatreatmentbut

asaprobe.Thisnudgedapatient’sbodyoffitssetpointandallowedHoandPerelson—forthefirsttimeever—totrackthedynamicsoftheimmunesystemasitbattledHIV.Theyfoundthataftereachpatienttooktheproteaseinhibitor,

thenumberofvirusparticlesinhisbloodstreamdroppedexponentiallyfast.Therateofdecaywasincredible;halfofallthevirusparticlesinthebloodstreamwereclearedbytheimmunesystemeverytwodays.DifferentialcalculusenabledPerelsonandHotomodelthisexponentialdecay

andextractitssurprisingimplications.Firsttheyrepresentedthechangingconcentrationofvirusinthebloodasanunknownfunction,V(t),wheretdenotestheelapsedtimesincetheproteaseinhibitorwasadministered.Thentheyhypothesizedhowmuchtheconcentrationofviruswouldchange,dV,inaninfinitesimallyshorttimeinterval,dt.Theirdataindicatedthataconstantfractionofthevirusinthebloodwasclearedeachday,soperhapsthesameconstancywouldholdwhenextrapolateddowntoaninfinitesimaltimeintervaldt.SincedV/Visthefractionalchangeinthevirusconcentration,theirmodelcouldbetranslatedintosymbolsasthefollowingequation:

dV⁄V=−cdt.

Heretheconstantofproportionality,c,istheclearancerate,ameasureofhowfastthebodyflushedoutthevirus.Theequationaboveisanexampleofadifferentialequation.Itrelatesthe

differentialdVtoVitselfandtothedifferentialdtoftheelapsedtime.Byusingthefundamentaltheoremtointegratebothsidesoftheequation,PerelsonandHosolvedforV(t)andfounditsatisfied

ln[V(t)/V0]=–ctwhereV0istheinitialviralloadandlndenotesthenaturallogarithm(thesamelogarithmicfunctionthatNewtonandMercatorstudiedinthe1660s).Invertingthisfunctionthenimplied

V(t)=V0e–ct,whereeisthebaseofthenaturallogarithm,thusconfirmingthattheviralloaddidindeeddecayexponentiallyfastinthemodel.Finally,byfittinganexponential-decaycurvetotheirexperimentaldata,HoandPerelsonestimatedthepreviouslyunknownvalueoftheclearanceratec.Forthosewhopreferderivativestodifferentials,themodelequationcanbe

rewrittenas

dV⁄dt=−cV.

Here,dV/dtisthederivativeofV.Itmeasureshowfastthevirusconcentrationgrowsordeclines.Positivevaluesofthederivativesignifygrowth;negativevaluesindicatedecline.SincetheconcentrationVispositive,then–cVmustbenegative,sothederivativemustalsobenegative,whichmeansthevirusconcentrationmustdecline,asweknowitdoesintheexperiment.Furthermore,theproportionalitybetweendV/dtandVmeansthatthecloserVgetstozero,themoreslowlyitdeclines.Intuitively,thisslowingdeclineofVislikewhathappensifyoufillasinkwithwaterandthenallowittodrain.Thelesswaterthereisinthesink,themoreslowlyitflowsoutbecausethere’slesswaterpressurepushingitdown.Inthisanalogy,theamountofvirusislikethewater,andthedrainingisliketheoutflowofthevirusduetoitsclearancebytheimmunesystem.Havingmodeledtheeffectoftheproteaseinhibitor,PerelsonandHomodified

theirequationtodescribetheconditionsbeforethedrugwasadministered.Theyassumedtheequationwouldbecome

dV⁄dt=P−cV.

Inthisequation,Preferstotheuninhibitedrateofproductionofnewvirusparticles,anothercrucialunknownatthattime.PerelsonandHoimaginedthatbeforeadministrationoftheproteaseinhibitor,ateverymomentinfectedcellswerereleasingnewinfectiousvirusparticles,whichtheninfectedothercells,andsoon.ThispotentialforaragingfireiswhatmakesHIVsodevastating.Intheasymptomaticphase,however,thereisevidentlyabalancebetweenthe

productionofthevirusanditsclearancebytheimmunesystem.Atthissetpoint,thevirusisproducedasfastasit’scleared.Thatgavenewinsightintowhytheviralloadcouldstaythesameforyears.Inthewater-in-the-sinkanalogy,it’slikewhathappensifyouturnonthefaucetandopenthedrainatthesametime.Thewaterwillreachasteady-statelevelatwhichoutflowequalsinflow.Atthesetpoint,theconcentrationofvirusdoesn’tchange,soitsderivative

mustbezero:dV/dt=0.Hence,thesteady-stateviralload,V0,satisfies

P=cV0.

PerelsonandHousedthissimpleequation,P=cV0,toestimateavitallyimportantnumberthatnoonehadfoundawaytomeasurebefore:thenumberofvirusparticlesbeingclearedeachdaybytheimmunesystem.Itturnedouttobeabillionvirusparticlesaday.Thatnumberwasunexpectedandtrulystunning.Itindicatedthatatitanic

strugglewastakingplaceduringtheseeminglycalmtenyearsoftheasymptomaticphaseinapatient’sbody.Everyday,theimmunesystemclearedabillionvirusparticlesandtheinfectedcellsreleasedabillionnewones.Theimmunesystemwasinafurious,all-outwarwiththevirusandfightingittoanearstandstill.Ho,Perelson,andtheircolleaguesconductedafollow-upstudyin1996toget

abetterhandleonsomethingthey’dseenin1995butcouldn’tresolvebackthen.Thistimetheycollectedviral-loaddataatshortertimeintervalsaftertheproteaseinhibitorwasadministeredbecausetheywantedtoobtainmoreinformationaboutaninitiallagthey’dobservedinthemedicine’sabsorption,distribution,andpenetrationintothetargetcells.Afterthedrugwasgiven,theteammeasuredthepatients’viralloadeverytwohoursuntilthesixthhour,theneverysixhoursuntildaytwo,andthenonceadaythereafteruntildayseven.Onthemathematicalside,Perelsonrefinedthedifferential-equationmodeltoaccountforthelagandtotrackthedynamicsofanotherimportantvariable,thechangingnumberofinfectedTcells.Whentheresearchersredidtheexperiment,fitthedatatothemodel’s

predictions,andestimateditsparametersagain,theyobtainedresultsevenmorestaggeringthanbefore:tenbillionvirusparticleswerebeingproducedandthenclearedfromthebloodstreameachday.Moreover,theyfoundthatinfectedTcellshadalifespanofonlyabouttwodays.Thesurprisinglyshortlifespanaddedanotherpiecetothepuzzle,giventhatT-celldepletionisthehallmarkofHIVinfectionandAIDS.ThediscoverythatHIVreplicationwassoastonishinglyrapidchangedthe

waythatdoctorstreatedtheirHIV-positivepatients.UntiltheworkofHoandPerelson,physicianswaiteduntilHIVemergedfromitssupposedhibernationbeforetheyprescribedantiviraldrugs.Theideawastoconserveforcesuntilthepatient’simmunesystemreallyneededhelp,becausethevirusoftenbecameresistanttothedrugs,andthenthere’dbenothingelsetotry.Soitwasgenerallythoughtwisertowaituntilpatientswerefaralongintheirillness.HoandPerelson’sworkturnedthispictureupsidedown.Therewasno

hibernation.HIVandthebodywerelockedinapitchedstruggleeverysecondofeveryday,andtheimmunesystemneededallthehelpitcouldgetandassoonas

possibleafterthecriticalearlydaysofinfection.Andnowitwasobviouswhynosinglemedicationworkedforverylong.Thevirusreplicatedsorapidlyandmutatedsoquickly,itcouldfindawaytoescapealmostanytherapeuticdrug.Perelson’smathematicsgaveaquantitativeestimateofhowmanydrugshad

tobeusedincombinationtobeatHIVdownandkeepitdown.BytakingintoaccountthemeasuredmutationrateofHIV,thesizeofitsgenome,andthenewlyestimatednumberofvirusparticlesthatwereproduceddaily,hedemonstratedmathematicallythatHIVwasgeneratingeverypossiblemutationateverybaseinitsgenomemanytimesaday.Sinceevenasinglemutationcouldconferdrugresistance,therewaslittlehopeofsuccesswithsingle-drugtherapy.Twodrugsgivenatthesametimewouldstandabetterchanceofworking,butPerelson’scalculationsshowedthatasizablefractionofallpossibledoublemutationsalsooccurredeachday.Threedrugsincombination,however,wouldbehardfortheHIVvirustoovercome.ThemathsuggestedthattheoddsweresomethingliketenmilliontooneagainstHIVbeingabletoundergothenecessarythreesimultaneousmutationstoescapetriple-combinationtherapy.WhenHoandhiscolleaguestestedathree-drugcocktailonHIV-infected

patientsinclinicalstudies,theresultswereremarkable.Thelevelofvirusintheblooddroppedaboutahundredfoldintwoweeks.Overthenextmonth,itbecameundetectable.ThisisnottosaythatHIVwaseradicated.Studiessoonafterwardshowedthe

viruscouldreboundaggressivelyifpatientstookabreakfromtherapy.TheproblemisthatHIVcanhideoutinvariousplacesinthebody.Itcanlielowinsanctuarysitesthatthedrugscannotreadilypenetrateorlurkinlatentlyinfectedcellsandrestwithoutreplicating,asneakywayofevadingtreatment.Atanytime,thesedormantcellscanwakeupandstartmakingnewviruses.That’swhyit’ssoimportantforHIV-positivepeopletokeeptakingtheirmeds,evenwhentheirviralloadsareloworundetectable.Still,eventhoughitdidnotcureHIV,triple-combinationtherapychangedit

toachronicconditionthatcouldbemanaged,atleastforthosewhohadaccesstotreatment.Itgavehopewherealmostnonehadexistedbefore.In1996,Dr.DavidHowasnamedTimemagazine’sManoftheYear.In2017,

AlanPerelsonreceivedamajorprizeforhis“profoundcontributionstotheoreticalimmunology,whichbringinsightandsavelives.”Heisstillusingcalculusanddifferentialequationstoanalyzeviraldynamics.HislatestworkconcernshepatitisC,avirusthataffectsabout170millionpeopleworldwideandkillsabout350,000peopleeachyear.Itistheleadingcauseofcirrhosisandlivercancer.In2014,withthehelpofPerelson’smath,newtreatmentsforhepatitisC

weredevelopedthataresafeandeasytotakeasaonce-a-daypill.Incredibly,thetreatmentcurestheinfectioninnearlyeverypatient.

9

TheLogicalUniverse

CALCULUSUNDERWENTametamorphosisinthesecondhalfoftheseventeenthcentury.Itbecamesosystematic,sopenetrating,andsopowerfulthatmanyhistorianssaycalculuswas“invented”then.Accordingtothisview,beforeNewtonandLeibniz,therewasproto-calculus;afterward,calculus.Iwouldn’tputitthatwaymyself.Tome,it’sbeencalculusallalong,eversinceArchimedesharnessedinfinity.Whateverit’scalled,calculustransformeddramaticallybetween1664and

1676,anditchangedtheworldalongwithit.Inscience,itallowedhumanitytostartreadingthebookofnaturethatGalileohaddreamedof.Intechnology,itlaunchedtheindustrialrevolutionandtheinformationage.Inphilosophyandpolitics,itleftitsmarkonmodernconceptionsofhumanrights,society,andlaws.Iwouldn’tsaycalculuswasinventedinthelateseventeenthcentury;rather,I

woulddescribewhathappenedasanevolutionarybreakthrough,analogoustoapivotaleventinbiologicalevolution.Intheearlydaysoflife,organismswererelativelysimple.Theyweresingle-celledcreatures,somethinglikethebacteriaoftoday.Thateraofunicellularlifecontinuedforaboutthreeandahalfbillionyears,dominatingmostoftheEarth’shistory.Butaroundhalfabillionyearsago,anastonishingdiversityofmulticellularlifeburstforthinwhatbiologistscalltheCambrianexplosion.Injustafewtensofmillionsofyears—an

evolutionarysplitsecond—manyofthemajoranimalphylasuddenlyemerged.Similarly,calculuswastheCambrianexplosionformathematics.Onceitarrived,anamazingdiversityofmathematicalfieldsbegantoevolve.Theirlineageisvisibleintheircalculus-basednames,inadjectiveslikedifferentialandintegralandanalytic,asindifferentialgeometry,integralequations,andanalyticnumbertheory.Theseadvancedbranchesofmathematicsarelikethemanybranchesandspeciesofmulticellularlife.Inthisanalogy,themicrobesofmathematicsaretheearliesttopics:numbers,shapes,andwordproblems.Likeunicellularorganisms,theydominatedthemathematicalsceneformostofitshistory.ButaftertheCambrianexplosionofcalculusthreehundredandfiftyyearsago,newmathematicallifeformsbegantoproliferateandflourish,andtheyalteredthelandscapearoundthem.Muchofthestoryoflifeisataleofprogresstowardgreatersophisticationand

complexitybuildingonearlierprecursors.That’strueofcalculusaswell.Butwhatisthestorybuildingtoward?Isthereanydirectiontotheevolutionofcalculus?Orisit,assomewouldsayofbiologicalevolution,undirectedandrandom?Withinpuremathematics,theevolutionofcalculushasbeenastoryof

crossbreedinganditsbenefits.Olderpartsofmathwereinvigoratedaftertheywerecrossedwithcalculus.Forexample,theancientstudyofnumbersandtheirpatternswasrevitalizedbyaninfusionofcalculus-basedtoolslikeintegrals,infinitesums,andpowerseries.Theresultinghybridfieldiscalledanalyticnumbertheory.Likewise,differentialgeometryusedcalculustoshedlightonthestructureofsmoothsurfacesandrevealedcousinstheyneverknewtheyhad,unimaginablecurvedshapesinfourdimensionsandbeyond.Inthisway,theCambrianexplosionofcalculusmademathematicsmoreabstractandmorepowerful.Italsomadeitmorelikeafamily.Calculusexposedawebofhiddenrelationshipstyingallpartsofmathematicstogether.Inappliedmathematics,theevolutionofcalculushasbeenastoryofour

expandingunderstandingofchange.Aswe’veseen,calculusbeganwiththestudyofcurves,wherethechangeswerechangesindirection,anditcontinuedwiththestudyofmotion,wherethechangesbecamechangesinposition.IntheaftermathofitsCambrianexplosion,andespeciallywiththeriseofdifferentialequations,calculusmovedontothestudyofchangemuchmoregenerally.Today,differentialequationshelpuspredicthowepidemicswillspread,whereahurricanewillhitland,andhowmuchtopayforanoptiontobuyastockinthefuture.Ineveryfieldofhumanendeavor,differentialequationshaveemergedasacommonframeworkfordescribinghowthingschangearoundusandinsideus,fromthesubatomicdomaintothefarthestreachesofthecosmos.

TheLogicofNature

TheearliesttriumphofdifferentialequationsalteredthecourseofWesternculture.In1687,IsaacNewtonproposedasystemoftheworldthatdemonstratedthepowerofreasonandusheredintheEnlightenment.Hediscoveredasmallsetofequations—hislawsofmotionandgravity—thatcouldexplainthemysteriouspatternsGalileoandKeplerhadfoundinfallingbodiesonEarthandplanetaryorbitsinthesolarsystem.Insodoing,heerasedthedistinctionbetweentheearthlyandcelestialrealms.AfterNewton,therewasjustoneuniverse,withthesamelawsapplyingeverywhereandalways.Inhismagisterialthree-volumemasterpiece,MathematicalPrinciplesof

NaturalPhilosophy(oftenknownasthePrincipia),Newtonappliedhistheoriestomuchmore:theshapeoftheEarth,withitsslightlybulgingwaistlinecausedbythecentrifugalforceofitsspin;therhythmofthetides;theeccentricorbitsofcomets;andthemotionofthemoon,aproblemsodifficultthatNewtoncomplainedtohisfriendEdmondHalleythatithad“madehisheadache,andkepthimawakesooften,thathewouldthinkofitnomore.”Today,whencollegestudentsstudyphysics,theyaretaughtclassical

mechanicsfirst—themechanicsofNewtonandhissuccessors—afterwhichtheyaretoldthatithasbeensupersededbyEinstein’srelativitytheoryandthequantumtheoryofPlanck,Einstein,Bohr,Schrödinger,Heisenberg,andDirac.There’scertainlyalotoftruthtothat.ThenewtheoriesoverturnedNewtonianconceptionsofspaceandtime,massandenergy,anddeterminismitself,replacingitinthecaseofquantumtheorywithamoreprobabilistic,statisticaldescriptionofnature.Butwhathasnotchangedistheroleofcalculus.Inrelativity,asinquantum

mechanics,thelawsofnaturearestillwritteninthelanguageofcalculus,withsentencesintheformofdifferentialequations.That,tome,isNewton’sgreatestlegacy.Heshowedthatnatureislogical.Causeandeffectinthenaturalworldbehavemuchlikeaproofingeometry,withonetruthfollowingfromanotherbylogic,exceptthatwhatisfollowingisoneeventfromanotherintheworld,notoneideafromanotherinourminds.Thisuncannyconnectionbetweennatureandmathematicsharksbacktothe

Pythagoreandream.ThelinkbetweenmusicalharmonyandnumbersdiscoveredbythePythagoreansledthemtoproclaimthatallisnumber.Theywereontosomething.Numbersareimportanttotheworkingsoftheuniverse.Shapesareimportanttoo;inthebookofnaturethatGalileodreamedof,thewordsweregeometricalfigures.Butasimportantasnumbersandshapesmightbe,they’re

notthetruedriversoftheplay.Inthedramaoftheuniverse,shapesandnumbersarelikeactors;theyarequietlydirectedbyanunseenpresence,thelogicofdifferentialequations.Newtonwasthefirsttotapintothislogicoftheuniverseandbuildasystem

aroundit.Itwasn’tpossiblebeforehim,becausethenecessaryconceptshadn’tbeenbornyet.Archimedesdidn’tknowaboutdifferentialequations.NeitherdidGalileo,Kepler,Descartes,orFermat.Leibnizdid,buthewasn’tasinclinedtowardscienceasNewtonornearlyasvirtuosicmathematically.ThesecretlogicoftheuniversewasvouchsafedtoNewtonalone.Thecenterpieceofhistheoryishisdifferentialequationofmotion:

F=ma.

Itranksasoneofthemostconsequentialequationsinhistory.Itsaysthattheforce,F,onamovingbodyisequaltothebody’smass,m,timesitsacceleration,a.It’sadifferentialequationbecauseaccelerationisaderivative(therateofchangeofthebody’svelocity)or,inLeibnizianterms,theratiooftwodifferentials:

a=dv⁄dt.

Heredvistheinfinitesimalchangeinthebody’svelocityvduringaninfinitesimaltimeintervaldt.SoifweknowtheforceFonthebody,andifweknowitsmassm,wecanuseF=matofinditsaccelerationviaa=F/m.Thataccelerationinturndetermineshowthebodywillmove.Ittellsushowthebody’svelocitywillchangeinthatnextinstant,anditsvelocitytellsushowitspositionwillchange.Inthisway,F=maisanoracle.Itpredictsthebody’sfuturebehavior,onetinystepatatime.Considerthesimplest,bleakestsituationimaginable:anisolatedbodyalonein

anemptyuniverse.Howwoulditmove?Well,sincethere’snothingaroundtopushitorpullit,theforceonthebodyiszero:F=0.Then,sincemisnotzero(assumingthebodyhassomemass),Newton’slawyieldsF/m=a=0,whichimpliesthatdv/dt=0aswell.Butdv/dt=0meansthelonesomebody’svelocitydoesn’tchangeduringtheinfinitesimaltimeintervaldt.Nordoesitchangeduringthenextinterval,ortheoneafterthat.TheupshotisthatwhenF=0,abodymaintainsitsvelocityforever.ThisisGalileo’sprincipleofinertia:Intheabsenceofanoutsideforce,abodyatreststaysatrest,andabodyinmotionstaysinmotionandmovesataconstantvelocity.Itsspeedanddirectionnever

change.WehavejustdeducedthelawofinertiaasalogicalconsequenceofNewton’sdeeperlawofmotion,F=ma.Newtonseemedtohaveunderstoodearlyon,backinhiscollegedays,that

accelerationwasproportionaltoforce.HeknewfromstudyingGalileothatiftherewasnoforceonabody,itwouldeitherstayatrestorcontinuemovinginastraightlineataconstantspeed.Force,herealized,wasnotneededtoproducemotion;itwasneededtoproducechangesinmotion.Itwasforcethatwasresponsibleformakingbodiesspeedup,slowdown,ordepartfromastraightpath.ThisinsightwasabigadvanceoverearlierAristotelianthinking.Aristotle

didn’tappreciateinertia.Heimaginedthataforcewasneededjusttokeepabodymoving.Andtobefair,that’strueinsituationsdominatedbyfriction.Ifyou’retryingtoslideadeskacrossthefloor,youhavetokeeppushingit;onceyoustoppushing,thedeskstopsmoving.Butfrictionismuchlessrelevantforplanetsglidingthroughspaceorapplesdroppingtotheground.Inthosecases,theforceoffrictionisnegligible.Itcanbeignoredwithoutlosingtheessenceofthephenomenon.InNewton’spictureoftheuniverse,thedominantforceisgravity,notfriction.

Whichisasitshouldbe,giventhatNewtonandgravityaresocloselyassociatedinthepopularmind.WhenmostpeoplethinkofNewton,theyimmediatelyrecallwhattheylearnedaschildren,thatNewtondiscoveredgravitywhenanapplefellonhishead.Spoileralert:That’snotwhathappened.Newtondidn’tdiscovergravity;peoplealreadyknewthatheavythingsfell.Butnobodyknewhowfargravitywent.Diditendatthesky?Newtonhadahunchthatgravitymightextendtothemoonandpossibly

beyond.Hisideawasthatthemoon’sorbitwasakindofnever-endingfalltotheEarth.Butunlikeafallingapple,thefallingmoondoesn’tcrashtothegroundbecauseit’salsosimultaneouslycruisingsidewaysduetoinertia.It’slikeoneofGalileo’scannonballs,glidingsidewaysandfallingatthesametime,tracingacurvedpath,exceptthatit’sglidingsofastthatitneverreachesthesurfaceofthesphericalEarthcurvingawaybeneathit.Asitsorbitdeviatesfromastraightline,themoonaccelerates—notinthesensethatitsspeedchanges,butitsdirectionofmotionchanges.Whatpullsitoffastraight-linepathistheincessanttugoftheEarth’sgravity.Theresultingtypeofaccelerationiscalledcentripetalacceleration,atendencytobepulledtowardacenter—inthiscase,thecenteroftheEarth.NewtoninferredfromKepler’sthirdlawthattheforceofgravityweakened

withdistance,whichexplainedwhythemoredistantplanetstooklongertogoaroundthesun.Hiscalculationssuggestedthatifthesunwaspullingonthe

planetswiththesamekindofforcethatdrewanappletotheEarthandthatkeptthemooninitsorbit,thatforcehadtoweakeninverselywiththesquareofthedistance.SoiftheseparationbetweentheEarthandthemooncouldsomehowbedoubled,thegravitationalforcebetweenthemwouldweakenbyafactoroffour(twosquared,nottwo).Iftheseparationwastripled,theforcewoulddecreasebyninefold,notthreefold.Admittedly,thereweresomedubiousassumptionsbuiltintoNewton’scalculations,particularlytheassumptionthatgravityactedinstantaneouslyatadistance,asifthevastnessofspacewereirrelevant.Hehadnoideahowthiscouldbepossible,butstill,theinverse-squarelawintriguedhim.Totestitquantitatively,heestimatedthecentripetalaccelerationofthemoon

asitcircledtheEarthatitsknowndistance(about60timestheradiusoftheEarth)anditsknownperiodofrevolution(about27days).Thenhecomparedthemoon’saccelerationtotheaccelerationoffallingbodiesonEarth,whichGalileohadmeasuredinhisinclined-planeexperiments.Newtonfoundthatthetwoaccelerationsdifferedbyafactorencouraginglycloseto3,600,whichequals60squared.Thatwasjustwhathisinverse-squarelawpredicted.Afterall,themoonwasabout60timesfartherfromthecenteroftheEarththananapplefallingfromatreeontheEarth’ssurface,soitsaccelerationshouldbeabout60squaredtimesless.Inlateryears,Newtonrecalledthathe’d“comparedtheforcerequisitetokeeptheMooninherOrbwiththeforceofgravityatthesurfaceoftheearth,&foundthemanswerprettynearly.”Thenotionthatthetugofgravitymightextendtothemoonwasawildideaat

thetime.RememberthatinAristoteliandoctrine,everythingbelowthemoonwasheldtobecorruptibleandimperfect,andeverythingbeyondthemoonwasperfect,eternal,andunchanging.Newtonshatteredthisparadigm.Heunifiedheavenandearthandshowedthatthesamelawsofphysicsdescribedboth.Abouttwentyyearsafterhisinsightwiththeinverse-squarelaw,Newtontook

abreakfromhisinterestsinalchemyandbiblicalchronologyandrevisitedthequestionofmotionduetogravity.He’dbeenprovokedbyhiscolleaguesandrivalsattheRoyalSocietyofLondon.They’dchallengedhimtosolveamuchharderproblemthananyhe’dpreviouslyconsideredandthatnoneofthemknewhowtosolve:Iftherewasaforceofattractionemanatingfromthesunthatweakenedaccordingtoaninverse-squarelaw,howwouldtheplanetsmove?“Inellipses,”NewtonissaidtohaverepliedatoncewhenhisfriendEdmondHalleyposedthequestion.“But,”askedaflabbergastedHalley,“howdoyouknow?”“Why,Ihavecalculatedit,”saidNewton.WhenHalleyurgedhimtoexplainhisreasoning,Newtonsetaboutreconstructinghisoldwork.Inafurioustorrentof

activity,acreativeoutpouringalmostasfrenziedaswhathehaddoneasastudentduringtheplagueyears,NewtonwrotethePrincipia.Byassuminghislawsofmotionandgravityasaxiomsandusinghiscalculus

asadeductiveinstrument,NewtonprovedthatallthreeofKepler’slawsfollowedaslogicalnecessities.ThesamewastrueforGalileo’slawofinertia,theisochronismofpendulums,theodd-numberruleforballsrollingdownramps,andtheparabolicarcsofprojectiles.Eachofthemwasacorollaryoftheinverse-squarelawandF=ma.ThisappealtodeductivereasoningshockedNewton’scolleaguesanddisturbedthemonphilosophicalgrounds.Manyofthemwereempiricists.Theythoughtthatlogicappliedonlywithinmathematicsitself.Naturehadtobestudiedbyexperimentandobservation.Theyweredumbfoundedbythethoughtthatnaturehadaninnermathematicalcoreandthatphenomenainnaturecouldbededucedbylogicfromempiricalaxiomslikethelawsofgravityandmotion.

TheTwo-BodyProblem

ThequestionthatHalleyposedtoNewtonwasmonstrouslydifficult.Itrequiredtheconversionoflocalinformationintoglobalinformation,thecentraldifficultyofintegralcalculusandpredictionthatwediscussedinchapter7.Thinkaboutwhatwouldbeinvolvedinpredictingthegravitationalinterplay

oftwobodies.Tosimplifytheproblem,pretendthatoneofthem,thesun,isinfinitelymassiveandhencemotionless,whiletheother,theplanetinorbit,movesaroundit.Initially,theplanetisatsomedistancefromthesun,atagivenlocation,andmovingwithagivenspeedinagivendirection.Inthenextinstant,theplanet’svelocitycarriesittoitsnextlocation,aninfinitesimaldistancefromwhereitwasamomentago.Sinceit’snowataslightlydifferentplace,itfeelsaslightlydifferentgravitationalpullfromthesun,differentinbothdirectionandmagnitude.Thatnewforce(computablefromtheinverse-squarelaw)tugstheplanetagainandchangesitsspeedanditsdirectionoftravelbyanotherinfinitesimalamount(computablefromF=ma)duringthenextinfinitesimalincrementoftime.Theprocesscontinuesadinfinitum.Alltheseinfinitesimallocalstepshavetobeintegratedsomehow,addedtogethertoproducethewholeorbitofthemovingplanet.IntegratingF=maforthetwo-bodyproblemisthusanexerciseintheuseof

theInfinityPrinciple.ArchimedesandothershadappliedtheInfinityPrincipletothemysteryofcurves,butNewtonwasthefirsttoapplyittothemysteryof

motion.Ashopelessasthetwo-bodyproblemseemed,Newtonmanagedtosolveitwiththehelpofthefundamentaltheoremofcalculus.Insteadofinchingtheplanetforwardinstantbyinstantinhismind,heusedcalculustothrustitforwardbyleapsandbounds,asifbymagic.Hisformulascouldpredictwheretheplanetwouldbe—aswellashowfastitwouldbemoving—asfarintothefutureashedesired.TheInfinityPrincipleandthefundamentaltheoremofcalculusentered

Newton’sworkinanothernovelrespect.Inhisfirstattackonthetwo-bodyproblem,hehadidealizedtheplanetandthesunaspoint-likeparticles.Couldhemodelthemmorerealisticallyasthecolossalsphericalballsthattheyactuallywereandstillsolvetheproblem?Andifhecould,wouldhisresultschange?Thiswasanotherextraordinarilydifficultcalculationatthattimeinthe

developmentofcalculus.ConsiderwhatwouldbeneededtotallyupthenettugofthegiantsphereofthesunonthesmallerbutstillgiantsphereoftheEarth.EveryatominthesunpullsoneveryatomintheEarth.Thedifficultyisthatallthoseatomsareatdifferentdistancesfromoneanother.Theatomsatthebackofthesunarefartheraway,andhenceexertaweakergravitationalpullontheatomsoftheEarth,thantheatomsinthefrontofthesun.Moreover,theatomsontheleftandrightsidesofthesunpulltheEarthinconflictingdirectionsandwithvaryingstrengthsdependingontheirowndistancesfromtheEarth.Alloftheseeffectshavetobeaddedup.Puttingthepiecesbacktogetheragainforthisproblemwasharderthananythinganyonehadeverdoneinintegralcalculus.Whenwesolveittoday,weuseamethodcalledtripleintegration.It’sabear.Newtonmanagedtosolvethistripleintegralandfoundsomethingsobeautiful

andsosimple,itisalmostunbelievable,eventoday.Hefoundthathecouldgetawaywithpretendingthatallthemassofthesphericalsunwasconcentratedatitscenter;likewisefortheEarth.HiscalculationsshowedthattheorbitoftheEarthwouldbethesameeitherway.Inotherwords,hecouldreplacethegiantsphereswithinfinitesimalpointswithoutincurringanyerror.How’sthatforaliethatrevealsthetruth!ThereweremanyotherapproximationsinNewton’scalculations,however,

whoseeffectsweremoreseriousandproblematic.Forthesakeofsimplicity,he’dcompletelyignoredthegravitationalpullsexertedbyalltheotherplanets.Plushe’dcontinuedtoassumethatgravityactedinstantaneously.Heknewthatbothoftheseapproximationscouldn’tpossiblybecorrect,buthedidn’tseeanywaytomakeprogresswithoutthem.Healsoconfessedthathehadnoexplanationforwhatgravityactuallywasorwhyitobeyedthemathematicaldescriptionhe’dgivenit.Heknewthathiscriticswouldbesuspiciousofhiswholeprogram.Tomakehisworkasconvincingandpersuasiveaspossible,he

coucheditinthereassuringlanguageofgeometry,thegoldstandardofrigorandcertaintyasunderstoodatthattime.Butitwasn’ttraditionalEuclideangeometry;itwasapeculiar,idiosyncraticadmixtureofclassicalgeometryandcalculus.Itwascalculusingeometricclothing.Nonetheless,hedidhisbesttogiveitaclassicalveneer.Thestyleofthe

Principiaisold-schoolEuclidean.Followingtheformatofclassicalgeometry,Newtonstartedfromaxiomsandpostulates—hislawsofmotionandgravity—andtreatedthemasunquestionedfoundationstones.Onthemhebuiltanedificeoflemmas,propositions,theorems,andproofs,alldeducedbylogic,onefromtheotherinanunbrokenchainreachingallthewaybacktotheaxioms.JustasEuclidgavetheworldhisimmortalthirteenbooksoftheElements,Newtongavetheworldthreebooksofhisown.Withoutfalsemodesty,hecalledthethirdoneTheSystemoftheWorld.Hissystemdepictednatureasamechanism.Intheyearstocome,itwould

oftenbecomparedtoaclockwork,itsgearsspinning,itsspringsstretching,allitspartsmovinginsequence,awonderofcauseandeffect.Applyingthefundamentaltheoremofcalculusandarmedwithpowerseries,ingenuity,andluck,Newtoncouldoftensolvehisdifferentialequationsexactly.Insteadofcrabbingforwardinstantbyinstant,hecouldleapaheadandforecastthestateofhisclockworkindefinitelyfarintothefuture,justashe’ddoneforthetwo-bodyproblemofaplanetorbitingthesun.InthecenturiesafterNewton,hissystemwasrefinedbymanyother

mathematicians,physicists,andastronomers.Itwassotrustedthatwhenthemotionofaplanetdisagreedwithitspredictions,astronomersassumedtheyweremissingsomethingimportant.ThiswashowtheplanetNeptunewasdiscoveredin1846.IrregularitiesintheorbitofUranussuggestedthepresenceofanunknownplanetbeyondit,anunseenneighborthatwasperturbingUranusgravitationally.Calculuspredictedwherethemissingplanetshouldbe,andwhenastronomerslooked,thereitwas.

NewtonMeetsHiddenFigures

Bythemid-twentiethcentury,itseemedthatphysicshadfinallymovedonfromNewtonianmechanics.Quantumtheoryandrelativityhadputtheoldworkhorseouttopasture.Yeteventhenitenjoyedonelasthurrah,thankstothespaceracebetweentheUnitedStatesandtheSovietUnion.

Intheearly1960s,KatherineJohnson,theAfrican-AmericanmathematicianandheroineofHiddenFigures,usedthetwo-bodyproblemtobringastronautJohnGlenn,thefirstAmericantoorbittheEarth,safelybackhome.Johnsonbrokenewgroundinsomanyways.Inheranalysis,thetwogravitatingbodieswereaspacecraftandtheEarth,notaplanetandthesunastheyhadbeenforNewton.SheusedcalculustopredictthepositionofthemovingspacecraftasitorbitedtheEarthrotatingunderneathitandtocalculateitstrajectoryforsuccessfulreentryintotheatmosphere.Todothat,sheneededtoincludecomplicationsthatNewtonhadleftout,themostvitalofwhichwasthattheEarthisnotperfectlyspherical;itbulgesslightlyattheequatorandflattensatthepoles.Gettingthedetailsrightwasamatteroflifeanddeath.Thespacecapsulehadtoreentertheatmosphereattherightangleoritwouldburnup.Andithadtolandattherightspotintheocean.Ifitsplasheddowntoofarawayfromtherendezvoussite,Glennmightdrowninhisspacecapsulebeforeanyonecouldreachhim.OnFebruary20,1962,ColonelJohnGlenncompletedthreeorbitsofour

planet,andthen,guidedbyJohnson’scalculations,hereenteredtheatmosphereandlandedsafelyintheNorthAtlanticOcean.Hewasanationalhero.YearslaterhewouldbeelectedaUSsenator.Fewpeoplewereawarethatonthedayhemadehistory,hehadrefusedtoflyhismissionuntilKatherineJohnsonherselfhadcheckedallthelast-minutecalculationsforit.Hetrustedherwithhislife.KatherineJohnsonwasacomputerfortheNationalAeronauticsandSpace

Administrationatatimewhencomputerswerewomen,notmachines.Shewastherenearthestart,whenshehelpedAlanShepardbecomethefirstAmericaninspace,andshewasthereneartheend,whensheworkedonthetrajectoryforthefirstmoonlanding.Fordecades,herworkwasunknowntothepublic.Thankfully,herpioneeringcontributions(andherinspiringlifestory)havenowbeenrecognized.In2015,atageninety-seven,shereceivedthePresidentialMedalofFreedomfromPresidentBarackObama.Ayearlater,NASAnamedabuildingafterher.Atthededicationceremony,theNASAofficialremindedtheaudiencethat“millionsofpeoplearoundtheworldwatched[Alan]Shepard’sflight,butwhattheydidn’tknowatthetimewasthatthecalculationsthatgothimintospaceandsafelyhomeweredonebytoday’sguestofhonor,KatherineJohnson.”

CalculusandtheEnlightenment

Newton’spictureofaworldruledbymathematicsreverberatedfarbeyondscience.Inthehumanities,itservedasafoilforRomanticpoetslikeWilliamBlake,JohnKeats,andWilliamWordsworth.Ataraucousdinnerpartyin1817,WordsworthandKeats,amongothers,agreedthatNewtonhaddestroyedthepoetryoftherainbowbyreducingittoitsprismaticcolors.Theyraisedtheirglassesinaboisteroustoast:“Newton’shealth,andconfusiontomathematics.”Newtongotawarmerreceptioninphilosophy,wherehisideasinfluenced

Voltaire,DavidHume,JohnLocke,andotherEnlightenmentthinkers.Theyweretakenwiththepowerofreasonandtheexplanatorysuccessesofhissystem,withitsclockworkuniversedrivenbycausality.Hisempirical-deductiveapproach,anchoredinfactsandfueledbycalculus,sweptawaytheapriorimetaphysicsofearlierphilosophers(I’mlookingatyou,Aristotle).Beyondscience,itleftitsmarkonEnlightenmentconceptionsofeverythingfromdeterminismandlibertytonaturallawandhumanrights.Consider,forexample,Newton’sswayoverThomasJefferson—architect,

inventor,farmer,thirdpresidentoftheUnitedStates,andauthoroftheDeclarationofIndependence.ThereareechoesofNewtonthroughouttheDeclaration.Rightfromthestart,thephrase“Weholdthesetruthstobeself-evident”announcestherhetoricalstructure.AsEucliddidintheElementsandasNewtondidinthePrincipia,Jeffersonbeganwiththeaxioms,theself-evidenttruthsofhissubject.Then,byforceoflogic,hededucedaseriesofinescapablepropositionsfromthoseaxioms,themostimportantofwhichwasthatthecolonieshadtherighttoseverthemselvesfromBritishrule.TheDeclarationjustifiesthatseparationbyappealingto“theLawsofNatureandofNature’sGod.”(Incidentally,noticethepost-NewtoniandeismimplicitinJefferson’sordering:Godcomesafterthelawsofnatureandonlyinasubordinaterole,as“Nature’sGod.”)Theargumentisclinchedbythe“causeswhichimpel[thecolonists]totheseparation”fromtheBritishCrown.ThosecausesplaytheroleofNewtonianforces,impellingtheclockwork’smotion,determiningtheeffectsthatmustfollow—inthiscase,theAmericanRevolution.Ifallofthisseemsfar-fetched,keepinmindthatJeffersonreveredNewton.In

amacabreactofdevotion,heacquiredacopyofNewton’sdeathmask.Andafterhewasnolongerpresident,JeffersonwrotetohisoldfriendJohnAdamsonJanuary21,1812,aboutthepleasuresofleavingpoliticsbehind:“IhavegivenupnewspapersinexchangeforTacitusandThucydides,forNewtonandEuclid;andIfindmyselfmuchthehappier.”Jefferson’sfascinationwithNewtonianprinciplescarriedovertohisinterest

inagriculture.Hewonderedaboutthebestshapeforthemoldboardofaplow.(Amoldboardisthecurvedpartofaplowthatliftsandturnsthesoilcutbythe

plowshare.)Jeffersonframedthequestionasoneofefficiency:Howshouldthemoldboardbecurvedsothatitwouldencountertheleastresistancetotherisingsod?Thesurfaceofthemoldboardneededtobehorizontalinfrontsothatitcouldgetunderthecutsoiltoliftit,anditshouldthengraduallycurvetobecomeperpendiculartothegroundtowardthebacksothatitcouldturnthesoilandpushitaside.Jeffersonaskedamathematicalfriendofhistoaddressthisoptimization

problem.Inmanyways,thequestionwasreminiscentofonethatNewtonhimselfhadposedinthePrincipiaontheshapeofasolidbodyofleastresistancetomotionthroughwater.Guidedbythattheory,Jeffersonhadaplowfittedwithawoodenmoldboardofhisowndesign.

Hereportedin1798that“anexperienceoffiveyearshasenabledmetosay,itanswersinpracticetowhatitpromisesintheory.”ItwasNewtoniancalculusintheserviceoffarming.

FromDiscretetoContinuousSystems

Forthemostpart,Newtonhadappliedcalculustooneortwobodiesatmost—aswingingpendulum,aflyingcannonball,aplanetcirclingthesun.Solving

differentialequationsforthreeormorebodieswasanightmare,ashe’dlearnedthehardway.Theproblemofamutuallygravitatingsun,Earth,andmoonhadalreadygivenhimamigraine.Soanalyzingthewholesolarsystemwasoutofthequestion,farbeyondwhatevenNewtoncoulddowithcalculus.Asheputitinoneofhisunpublishedpapers,“UnlessIammuchmistaken,itwouldexceedtheforceofhumanwittoconsidersomanycausesofmotionatthesametime.”Butsurprisingly,goingevenhigher,allthewayuptoinfinitelymany

particles,madedifferentialequationstractableagain...aslongasthoseparticlesformedacontinuousmedium,notadiscreteset.Recallthedifference:Adiscretesetofparticlesislikeacollectionofmarblesspreadoutonthefloor.It’sdiscreteinthesensethatyoucouldtouchonemarble,moveyourfingerthroughemptyspace,touchanotherone,andsoon.Therearegapsbetweenthemarbles.Incontrast,withacontinuousmediumlike,forinstance,aguitarstring,youwouldneverhavetoliftyourfingerfromthestringasyoutracedalongitslength.Alltheparticlesintheguitarstringhangtogether.Notreally,ofcourse,becauseaguitarstring,likeallothermaterialobjects,isdiscreteandgranularattheatomicscale.Butinourminds,aguitarstringismoreaptlyregardedasacontinuum.Thisusefulfictionfreesusfromthechoreofhavingtocontemplatetrillionsandtrillionsofparticles.Itwasbyaddressingthemysteriesofhowcontinuousmediamoveandchange

—howguitarstringsvibratetomakesuchwarmmusicorhowheatflowsfromwarmspotstocoldspots—thatcalculusmadeitsnextgreatstridestowardchangingtheworld.Butfirstcalculushadtochangeitself.Itneededtoenlargeitsconceptofwhatdifferentialequationswereandwhattheycoulddescribe.

OrdinaryVersusPartialDifferentialEquations

WhenIsaacNewtonexplainedtheellipticalorbitsoftheplanetsandwhenKatherineJohnsoncalculatedthetrajectoryofJohnGlenn’sspacecapsule,theywerebothsolvingaclassofdifferentialequationsknownasordinarydifferentialequations.Thewordordinaryisnotmeanttobepejorative.It’sthetermofartfordifferentialequationsthatdependonjustoneindependentvariable.Forexample,inNewton’sequationsforthetwo-bodyproblem,thepositionof

aplanetwasafunctionoftime.ItkeptchangingitslocationfrommomenttomomentaccordingtothedictatesofF=ma.Thatordinarydifferentialequationdeterminedhowmuchtheplanet’spositionwouldchangeduringthenextinfinitesimalincrementoftime.Inthisexample,theplanet’spositionisthe

dependentvariable,sinceitdependsontime(theindependentvariable).Likewise,timewastheindependentvariableinAlanPerelson’smodelofHIVdynamics.Hewasmodelinghowtheconcentrationofvirusparticlesintheblooddecreasedafteradministrationofanantiretroviraldrug.Theissueagainwaschangesintime—howtheviralconcentrationchangedfrommomenttomoment.Here,concentrationplayedtheroleofthedependentvariable;theindependentvariablewasstilltime.Moregenerally,anordinarydifferentialequationdescribeshowsomething

(thepositionofaplanet,theconcentrationofavirus)changesinfinitesimallyastheresultofaninfinitesimalchangeinsomethingelse(suchasaninfinitesimalincrementoftime).Whatmakessuchanequation“ordinary”isthatthereisexactlyonesomethingelse,oneindependentvariable.Curiously,itdoesn’tmatterhowmanydependentvariablesthereare.Aslong

asthereisonlyoneindependentvariable,thedifferentialequationisconsideredordinary.Forexample,ittakesthreenumberstopinpointthepositionofaspacecraftmovinginthree-dimensionalspace.Callthosenumbersx,y,andz.Theyindicatewherethespacecraftisatagiventimebylocatingitleftorright,upordown,frontorback,andthustellingushowfarawayitisfromsomearbitraryreferencepointcalledtheorigin.Asthespacecraftmoves,itsx,y,andzcoordinateschangefrommomenttomoment.Thus,they’refunctionsoftime.Toemphasizetheirtimedependence,wecouldwritethemasx(t),y(t),andz(t).Ordinarydifferentialequationsareperfectlytailoredtodiscretesystems

consistingofoneormorebodies.Theycandescribethemotionofasinglespaceshipreenteringtheatmosphere,asinglependulumswingingbackandforth,orasingleplanetasitorbitsthesun.Thecatchisthatweneedtoidealizeeachoftheindividualbodiesasapoint-likeobject,aninfinitesimalspeckwithnospatialextent.Doingthatallowsustothinkofitasexistingatapointwithcoordinatesx,y,z.Thesameapproachworksiftherearemanypoint-likeparticles—aswarmoftinyspaceships,achainofpendulumsconnectedbysprings,asolarsystemofeightornineplanetsandcountlessasteroids.Allthesesystemsaredescribedbyordinarydifferentialequations.InthecenturiesafterNewton,mathematiciansandphysicistsdevelopedmany

ingenioustechniquesforsolvingordinarydifferentialequationsandthusforecastingthefutureofthereal-worldsystemstheydescribe.ThemathematicaltechniquesinvolvedextensionsofNewton’sideasaboutpowerseries,Leibniz’sideasaboutdifferentials,clevertransformationsthatcouldallowthefundamentaltheoremofcalculustobeinvoked,andsoon.Thiswasanenormousindustry,anditcontinuestothisday.

Butnotallsystemsarediscrete—oratleast,notallofthemarebestviewedthatway,aswesawwiththeexampleofaguitarstring.Consequently,notallsystemscanbedescribedbyordinarydifferentialequations.Tounderstandwhynot,let’shaveanotherlookatourimaginarybowlofsoupcoolingoffonthekitchentable.Abowlofsoupis,atonelevel,adiscretecollectionofmolecules,all

bouncingarounderratically.Yetthere’snohopeofseeingthem,measuringthem,orquantifyingtheirmotion,sonobodywouldeverthinkofusingordinarydifferentialequationstomodelthecoolingofabowlofsoup.Therearesimplytoomanyparticlestodealwith,andtheirmotionistooirregular,haphazard,andunknowable.Amuchmorepracticalwaytodescribewhat’shappeningistothinkofthe

soupasacontinuum.Thisisnotreallytrue,butit’suseful.Inacontinuumapproximation,wepretendthatthesoupexistsateverypointinsidethethree-dimensionalvolumeofthesoupbowl.Thetemperature,T,atagivenpoint(x,y,z)dependsontime,t.AllofthisinformationiscapturedbyafunctionT(x,y,z,t).Aswewillseeshortly,therearedifferentialequationsfordescribinghowthisfunctionchangesinspaceandtime.Suchadifferentialequationisnotanordinarydifferentialequation.Itcan’tbe,becauseitdoesn’tdependonjustoneindependentvariable.Infact,itdependsonfourofthem:x,y,z,andt.It’sanewkindofbeast—apartialdifferentialequation,socalledbecauseeachofitsindependentvariablesplaysitsown“part”incausingchangetooccur.Partialdifferentialequationsaremuchricherthanordinarydifferential

equations.Theydescribecontinuoussystemsmovingandchanginginspaceandtimesimultaneouslyorintwoormoredimensionsofspace.Alongwithacoolingbowlofsoup,thesaggyshapeofahammockisdescribedbysuchanequation.Soisthespreadingofapollutantinalakeortheflowofairoverthewingofafighterplane.Partialdifferentialequationsareextremelydifficulttohandle.Theymake

ordinarydifferentialequations,whicharealreadydifficult,seemlikechild’splay.Yettheyarealsoextremelyimportant.Ourlivesdependonthemwheneverwetaketotheskies.

PartialDifferentialEquationsandtheBoeing787

Modernairplaneflightisawonderofcalculus.Butitwasn’talwaysso;inasimplertime,atthedawnofaviation,thefirstflyingmachineswereinventedby

analogywithbirdsandkites,byengineeringsavvy,andbypersistenttrialanderror.TheWrightbrothers,forexample,usedtheirknowledgeofbicyclestodevisetheirthree-axissystemforcontrollingairplanesinflightandovercomingtheirinherentinstabilities.Asaircraftbecameincreasinglysophisticated,however,itbecamenecessary

tousemoresophisticatedmeanstodesignthem.Windtunnelsallowedengineerstotesttheaerodynamicpropertiesoftheirflyingmachineswithoutthecraftleavingtheground.Scalemodels,inwhichthedesignerbuilttinymockupsoftherealplanes,allowedairworthinesstobetestedwithoutbuildingcostlyfull-sizemodels.AfterWorldWarII,aeronauticalengineersaddedcomputerstotheirdesign

arsenal.Thevacuum-tubebehemothsthathadbeenusedforcode-breaking,artillerycalculations,andweatherforecastingweredeployedtohelpcreatemodernjetaircraft.Computerscouldbeusedtosolvethecomplexpartialdifferentialequationsthatinevitablyaroseinthedesignprocess.Themathinvolvedcouldbehorrendouslydifficultforseveralreasons.For

onething,thegeometryofanairplaneiscomplicated.It’snotlikeasphereorakiteorabalsa-woodglider.It’samuchmorecomplexshape,withwings,fuselage,engines,tail,flaps,andlandinggear.Eachofthesedeflectstheairrushingpasttheplaneathighspeed.Andwheneveronrushingairisdeflected,itexertsaforceonwhateverdeflectedit(asanyonewhohaseverstuckhisorherhandoutthewindowofacarspeedingdownthehighwayknows).Ifanairplanewingisshapedproperly,theonrushingairtendstoliftit.Iftheplaneismovingfastenoughdowntherunway,thisupwardforceliftstheplaneoffthegroundandkeepsitaloft.Butwhereasliftisaforceperpendiculartothedirectionofoncomingairflow,anotherkindofforce—drag—actsinadirectionparalleltotheflow.Dragislikefriction.Itresiststheplane’smotionandslowsitdown,causingitsenginestoworkharderandburnmorefuel.Calculatingthesizeoftheseliftanddragforcesisabrutallydifficultcalculusproblem,farbeyondtheabilityofanyhumanbeingtosolveforarealisticallyshapedairplane.Yetsuchproblemsmustbesolved.Theyarecrucialtoairplanedesign.ConsidertheBoeing787Dreamliner.In2011,Boeing—theworld’slargest

aerospacecompany—rolledoutitsnext-generationmidsizejetfortransportingtwohundredtothreehundredpeopleonlong-haulflights.Theplanewastoutedas60percentquieterand20percentmorefuelefficientthantheBoeing767,whichitwasdesignedtoreplace.Oneofitsmostinnovativefeatureswasitsuseofcarbon-fiber-reinforcedpolymersinthefuselageandwings.Thesespace-agecompositematerialsarelighterandstrongerthanaluminum,steel,andtitanium,

theconventionalmaterialsofchoiceforjetaircraft.Becausethey’relighterthanmetals,theysavefuelandalsomakeiteasierfortheplanetoflyfaster.ButperhapsthemostinnovativethingabouttheBoeing787wasthe

mathematicalandcomputationalforesightthatwentintoit,whichfarexceededthatinthedesignofanyotherpreviousplane.CalculusandcomputerssavedBoeinganenormousamountoftime—simulatinganewprototypeisalotfasterthanbuildingit.TheyalsosavedBoeingmoney—computersimulationsaremuchcheapertorunthanwind-tunneltests,thepriceofwhichhasskyrocketedinthepastfewdecades.DouglasBall,thechiefengineerofEnablingTechnologyandResearchatBoeing,pointedoutinaninterviewthatduringthedesignprocessfortheBoeing767inthe1980s,thecompanybuiltandtestedseventy-sevenprototypewings.Twenty-fiveyearslater,byusingsupercomputerstosimulatetheBoeing787’swings,theyhadtobuildandtestonlysevenofthem.Partialdifferentialequationsenteredintotheprocessinmyriadways.For

example,alongwiththeircalculationsofliftanddrag,Boeing’sappliedmathematiciansusedcalculustoanticipatehowtheairplane’swingswouldflexwhenmovingatsixhundredmilesperhour.Whenawingissubjectedtolift,theliftforcecausesthewingtoflexupwardandtwist.Onephenomenonengineerswanttoavoidisadangerouseffectcalledaeroelasticflutter,anastierversionoftheflutteringofvenetianblindswhenabreezeblowspastthem.Inthebestcase,suchunwantedvibrationsofthewingsproduceabumpy,unpleasantride.Intheworstcase,theoscillationscreateapositive-feedbackloop:asthewingsflutter,theyaltertheairflowovertheminawaythatmakesthemflutterevenmore.Aeroelasticflutterhasbeenknowntodamagethewingsoftestaircraftandtocausestructuralfailuresandcrashes(asoccurredoncewithaLockheedF-117Nighthawkstealthfighterduringanairshow).Ifasevereflutteroccurredonacommercialflight,itcouldputhundredsofpassengersatrisk.Theequationsthatgovernaeroelasticflutterarecloselyrelatedtothosewe

mentionedearlierinourdiscussionoffacialsurgery.There,themodelerschanneledthespiritofArchimedeswhentheyapproximatedapatient’ssofttissueandskullusinghundredsofthousandsofgem-shapedpolyhedronsandpolygons.Inthesamespirit,Boeing’smathematiciansapproximatedawingwithhundredsofthousandsoftinycubes,prisms,andtetrahedrons.Thesesimplershapesplayedtheroleofelementalbuildingblocks.Stiffnessandelasticpropertieswereassignedtoeachofthem,justasinthefacial-surgerymodeling,andthenthebuildingblocksweresubjectedtotherelevantpushesandpullsimpartedbytheirneighbors.Thepartialdifferentialequationsofelasticitytheorypredictedhoweachsimpleelementwouldrespondtothoseforces.Finally,with

thehelpofasupercomputer,allthoseresponseswerecombinedandusedtopredicttheoverallvibrationofthewing.Similarly,partialdifferentialequationswereusedtooptimizethecombustion

processintheaircraftengines.Thisisanespeciallycomplicatedproblemtomodel.Itinvolvestheinterplayofthreedifferentbranchesofscience:chemistry(thefuelundergoeshundredsofchemicalreactionsathightemperature);heatflow(theheatredistributesitselfwithintheengineaschemicalenergyisconvertedintothemechanicalenergyspinningtheturbineblades);andfluidflow(hotgasesswirlinthecombustionchamber,andpredictingtheirbehaviorisanexceedinglydifficultprobleminlightoftheturbulenceofsuchgases).Asbefore,theBoeingteamusedanArchimedeanapproach—theycuttheproblemintopieces,solvedtheproblemforeachpiece,andputthepiecestogetheragain.It’stheInfinityPrincipleinaction,thedivide-and-conquerstrategyonwhichallofcalculusrests.Hereitwasaidedbysupercomputersandanumericalmethodknownasfiniteelementanalysis.Butattheheartofitallisstillcalculus,embodiedindifferentialequations.

TheUbiquityofPartialDifferentialEquations

Theapplicationofcalculustomodernscienceislargelyanexerciseintheformulation,solution,andinterpretationofpartialdifferentialequations.Maxwell’sequationsforelectricityandmagnetismarepartialdifferentialequations.Soarethelawsofelasticity,acoustics,heatflow,fluidflow,andgasdynamics.Thelistgoeson:theBlack-Scholesmodelforpricingfinancialoptions,theHodgkin-Huxleymodelforthespreadofelectricalimpulsesalongnervefibers—partialdifferentialequationsall.Evenatthecuttingedgeofmodernphysics,partialdifferentialequationsstill

providethemathematicalinfrastructure.ConsiderEinstein’sgeneraltheoryofrelativity.Itreimaginesgravityasamanifestationofcurvatureinthefour-dimensionalfabricofspace-time.Thestandardmetaphorinvitesustopicturespace-timeasastretchy,deformablefabric,likethesurfaceofatrampoline.Normallythefabricispulledtaut,butitcancurveundertheweightofsomethingheavyplacedonit,sayamassivebowlingballsittingatitscenter.Inmuchthesameway,amassivecelestialbodylikethesuncancurvethefabricofspace-timearoundit.Nowimaginesomethingmuchsmaller,sayatinymarble(whichrepresentsaplanet),rollingonthetrampoline’scurvedsurface.Becausethesurfacesagsunderthebowlingball’sweight,itdeflectsthemarble’strajectory.

Insteadoftravelinginastraightline,themarblefollowsthecontoursofthecurvedsurfaceandorbitsaroundthebowlingballrepeatedly.That,saysEinstein,iswhytheplanetsgoaroundthesun.They’renotfeelingaforce;they’rejustfollowingthepathsofleastresistanceinthecurvedfabricofspace-time.Asmind-bogglingasthistheoryis,atitsmathematicalcorearepartial

differentialequations.Thesameistrueofquantummechanics,thetheoryofthemicroscopicrealm.Itsgoverningequation,theSchrödingerequation,isapartialdifferentialequationtoo.Thenextchaptertakesacloserlookatsuchequationstogiveyouafeelforwhattheyare,wheretheycamefrom,andwhytheymatterinoureverydaylives.Aswe’llsee,partialdifferentialequationsdomorethandescribethatbowlofsoupcoolingoffonthekitchentable.Theyalsoexplainhowthemicrowavenukedit.

10

MakingWaves

BEFORETHEEARLY1800s,heatwasariddle.Whatwasit,exactly?Wasitaliquidlikewater?Itdidseemtoflow.Butyoucouldn’tholditinyourhandsorseeit.Youcouldmeasureitindirectlybytrackingthetemperatureofsomethinghotasitcooleddown,butnooneknewwhatwasgoingoninsidethecoolingobject.Thesecretsofheatwereunraveledbyamanwhooftenfeltcold.Orphanedat

theageoften,JeanBaptisteJosephFourierwasasickly,dyspepticasthmaticasateenager.Asanadult,hebelievedheatwasessentialtohealth.Hekepthisroomoverheatedandswathedhimselfinaheavyovercoat,eveninthesummer.Inallaspectsofhisscientificlife,Fourierwasobsessedwithheat.HeoriginatedtheconceptofglobalwarmingandwasthefirsttoexplainhowthegreenhouseeffectregulatestheEarth’saveragetemperature.In1807,Fourierusedcalculustosolvetheriddleofheatflow.Hecameup

withapartialdifferentialequationthatallowedhimtopredicthowthetemperatureofanobject,suchasared-hotironrod,wouldchangeasitcooled.Amazingly,hefoundhecouldsolveproblemslikethisnomatterhowerraticallytherod’stemperaturevariedalongitslengthatthebeginningofthecooling-offprocess.Therodcouldstartwithhotspotshereandcoldspotsthere.Nosweat—Fourier’sanalyticalmethodcouldhandleit.Imaginealong,thin,cylindricalironrod,heatedunevenlyinablacksmith’s

forgesothatithaspatchesofhotandcoldscatteredalongitslength.Forsimplicity,assumeaperfectlyinsulatingsleevesurroundstherodsothatheatcan’tescape.Theonlywayheatcanflowistodiffusealongtherod’slength

fromhotspotstocoldspots.Fourierpostulated(andexperimentsconfirmed)thattherateofchangeoftemperatureatagivenpointontherodwasproportionaltothemismatchbetweenthetemperatureatthatpointandtheaverageofthetemperaturesofitsneighborsoneithersideofit.AndwhenIsayneighbors,Ireallymeanneighbors—picturetwopointsflankingthepointwe’refocusingon,eachinfinitesimallyclosetothatpoint.Undertheseidealizedconditions,thephysicsofheatflowissimple.Ifapoint

iscoolerthanitsneighbors,itheatsup.Ifit’shotter,itcoolsdown.Thegreaterthemismatch,thefasterthetemperatureevensout.Ifapointhappenstobeatpreciselytheaverageofitsneighbors’temperatures,everythingbalances,heatdoesn’tflow,andthetemperatureofthatpointstaysthesameinthenextinstant.Thisprocessofcomparingapoint’sinstantaneoustemperaturewiththatofits

neighborsledFouriertoapartialdifferentialequationthat’snowknownastheheatequation.Itinvolvesderivativeswithrespecttotwoindependentvariables,oneforinfinitesimalchangesintime(t)andoneforinfinitesimalchangesinposition(x)alongtherod.ThehardpartabouttheproblemFouriersetforhimselfisthatthehotspots

andcoldspotscouldbeinitiallyarrangedhiggledy-piggledy.Tosolvesuchageneralproblem,Fourierproposedaschemethatseemedwildlyoptimistic,almostfoolhardy.Heclaimedhecouldreplaceanyinitialtemperaturepatternwithanequivalentsumofsimplesinewaves.Sinewaveswerehisbuildingblocks.Hechosethembecausetheymadethe

problemeasier.Heknewthatifthetemperaturestartedinasine-wavepattern,itwouldstayinthatpatternastherodcooledoff.

Thatwasthekey:Sinewavesdidn’tmovearound.Theyjuststoodthere.

True,theydampeddownastheirhotspotscooledoffandtheircoldspotswarmedup,butthatdecaywaseasytohandle.Itmerelymeantthatthe

temperaturevariationsflattenedoutastimepassed.Assketchedinthediagrambelow,atemperaturepatternthatstartedoutlookinglikethedashedsinewavewouldgraduallydampdowntolooklikethesolidsinewave.

Theimportantthingwasthatthesinewavesstoodstillastheydamped.They

werestandingwaves.Soifhecouldfigureouthowtotakeaninitialtemperaturepatternapartand

breakitintosinewaves,hecouldsolvetheheat-flowproblemforeachsinewaveseparately.Healreadyknewtheanswertothatproblem:Eachsinewavedecayedexponentiallyfastataratethatdependedonhowmanycrestsandtroughsithad.Sinewaveswithmorecrestsdecayedfasterbecausetheirhotspotsandcoldspotswerepackedclosertogether,whichmadeformorerapidexchangeofheatbetweenthemandhencefasterequilibration.Then,knowinghoweachsinusoidalbuildingblockdecayed,allFourierhadtodowasputthembacktogethertosolvetheoriginalproblem.TherubinallthiswasthatFourierhadcasuallyinvokedaninfiniteseriesof

sinewaves.Hehadsummonedthegolemofinfinityintocalculusyetagain,andhe’ddoneitevenmorerecklesslythanhispredecessorshad.Insteadofusinganinfinitesumoftriangularshardsornumbers,hehadcavalierlyusedaninfinitesumofwaves.ItwasreminiscentofwhatNewtonhaddonewithhisinfinitesumsofpowerfunctionsxn,exceptthatNewtonhadneverclaimedhecouldrepresentarbitrarilycomplicatedcurvesthatincludedsuchhorrorsasdiscontinuousjumpsorsharpcornersinthem.Fourierwasnowclaimingexactlythat—curveswithcornersandjumpsdidn’tscarehim.Also,Fourier’swavesarosenaturallyfromthedifferentialequationitself,inthesensethattheywereitsnaturalmodesofvibration,itsnaturalstanding-wavepatterns.Theyweretailoredtoheatflow.Newton’spowerfunctionshadhadnospecialclaimas

buildingblocks;Fourier’ssinewavesdid.Theywereorganicallysuitedtotheproblemathand.Althoughhisdaringuseofsinewavesasbuildingblockssparkedcontroversy

andraisedknottyproblemsofrigorthattookmathematiciansacenturytoresolve,inourowntime,Fourier’sbigideahasplayedastarringroleinsuchtechnologiesassynthesizersforcomputerizedvoicesandMRIscansformedicalimaging.

StringTheory

Sinewavesalsoariseinmusic.They’rethenaturalmodesofvibrationforthestringsofguitars,violins,andpianos.ApartialdifferentialequationforsuchvibrationscanbederivedbyapplyingNewtonianmechanicsandLeibniziandifferentialstoanidealizedmodelofatautstring.Inthismodel,thestringisregardedasacontinuousarrayofinfinitesimalparticlesstackedsidebysideandbondedtotheirneighborsbyelasticforces.Atanyinstantoftimet,eachparticleinthestringmovesinaccordancewiththeforcesimpingingonit.Thoseforcesareproducedbythetensioninthestringasneighboringparticlesyankononeanother.Giventhoseforces,eachparticlemovesaccordingtoNewton’slawF=ma.Thishappensateverypointxalongthestring.Thus,theresultingdifferentialequationdependsonbothxandtandisanotherexampleofapartialdifferentialequation.It’scalledthewaveequationbecause,asexpected,itpredictsthatthetypicalmotionofavibratingstringisawave.Asintheheat-flowproblem,certainsinewavesproveusefulbecausethey

regeneratethemselvesastheyvibrate.Iftheendsofthestringarepinneddown,thesesinewavesdon’tpropagate;theysimplystandstillandvibrateinplace.Ifairresistanceandinternalfrictioninthestringarenegligible,anidealstringwillvibrateforeverinsuchasine-wavepatternifitstartsinasine-wavepattern.Anditsfrequencyofvibrationwillneverchange.Forallthesereasons,sinewavescontinuetoserveasidealbuildingblocksforthisproblemtoo.

Othervibrationshapescanbebuiltoutofinfinitesumsofsinewaves.For

example,intheharpsichordsusedinthe1700s,astringwasoftenpulledbyaquillanddrawnintoatriangularshapebeforeitwasreleased.

Eventhoughatrianglewavehasasharpcorner,itcanberepresentedbyan

infinitesumofperfectlysmoothsinewaves.Inotherwords,itdoesn’ttakesharpcornerstomakesharpcorners.Inthediagrambelow,I’veapproximatedatrianglewave,showndashedonthebottom,withthreeprogressivelymorefaithfulapproximationsbysinewaves.

Thefirstapproximationshowsasinglesinewavewiththebestpossible

amplitude(bestinthesensethatitminimizesthetotalsquarederrorfromthetrianglewave,thesameoptimalitycriterionwemetinchapter4).Thesecondapproximationistheoptimalsumoftwosinewaves.Andthethirdisthebestsumofthreesinewaves.TheamplitudesoftheoptimalsinewavesfollowaprescriptionthatFourierdiscovered:

Trianglewave=sinx−1⁄9sin3x+1⁄25sin5x−1⁄49sin7x+···.

ThisinfinitesumiscalledtheFourierseriesforthetrianglewave.Noticethecoolnumericalpatternsinit.Onlyoddfrequencies1,3,5,7,...appearinthesinewaves,andtheircorrespondingamplitudesaretheinversesquaresoftheoddnumberswithalternatingplusandminussigns.Unfortunately,Ican’teasilyexplainwhythisprescriptionworks;wewouldhavetoplowthroughtoomuchnitty-grittycalculustoseewherethosemagicamplitudescomefrom.ButthepointisthatFourierknewhowtocomputethem.Bydoingsohewasabletosynthesizeatrianglewaveoranyotherarbitrarilycomplicatedcurveoutofmuchsimplersinewaves.Fourier’sbigideaisthebasisformusicsynthesizers.Toseewhy,considerthe

soundofanote,suchastheAabovemiddleC.Togeneratethatprecisepitch,wecouldstrikeatuningforksettooscillateatthecorrespondingfrequencyof440cyclespersecond.Atuningforkconsistsofahandleandtwometaltines.Whentheforkishitwitharubberhammer,thetinesvibratebackandforth440timeseverysecond.Theirvibrationsexcitetheairnearby.Whenatinevibratesoutward,itcompressestheair;whenitvibratesback,itrarefiesthesurroundingair.Astheairmoleculesjigglebackandforth,theyproduceasinusoidalpressuredisturbancethatourearsperceiveasapuretone,aboringandcolorlessA.Itlackswhatmusicianscalltimbre.WecouldplaythesameAwithaviolinorapiano,andbothwouldsoundcolorfulandwarm.Eventhoughtheytooemit

vibrationsatafundamentalfrequencyof440cyclespersecond,theysounddifferentfromatuningfork(andfromeachother)becauseoftheirdistinctsetofovertones.That’sthemusicaltermforthewaveslikesin3xandsin5xintheearlierformulaforthetrianglewave.Overtonesaddcolortoanotebyaddinginmultiplesofthefundamentalfrequency.Inadditiontothesinewaveat440cyclesasecond,asynthesizedtrianglewaveincludesasine-waveovertoneatthreetimesthatfrequency(3×440=1320cyclespersecond).Thatovertoneisnotasstrongasthefundamentalsinxmode.Itsrelativeamplitudeisonly1/9aslargeasthefundamental,andtheotherodd-numberedmodesareevenweaker.Inmusicalterms,theseamplitudesdeterminetheloudnessoftheovertones.Therichnessofthesoundofaviolinhastodowithitsparticularcombinationofsofterandlouderovertones.TheunifyingpowerofFourier’sideaisthatthesoundofanymusical

instrumentcanbesynthesizedbyanarrayofinfinitelymanytuningforks.Allweneedtodoisstrikethetuningforkswiththerightstrengthsandattherighttimesand,incredibly,outpopsthesoundofaviolinorapianoorevenatrumpetoranoboe,althoughwe’reusingnothingmorethancolorlesssinewaves.Thisisessentiallyhowthefirstelectronicsynthesizersworked:theyreproducedthesoundofanyinstrumentbycombiningalargenumberofsinewaves.BackinhighschoolItookaclassinelectronicmusicthatgavemeafeeling

forwhatsinewavescoulddo.Thiswasinthedarkagesofthe1970s,whenelectronicmusicwasproducedbyabigboxthatlookedlikeanold-fashionedswitchboard.MyclassmatesandIwouldplugcablesintovariousjacksandturnknobsupanddown,andoutwouldcomethesoundofsinewaves,squarewaves,andtrianglewaves.Myrecollectionisthatsinewaveshadaclear,opensound,likeflutes.Squarewavessoundedpiercing,likefirealarms.Trianglewaveswerebrassy.Withoneknob,wecouldchangeawave’sfrequencytoraiseorloweritspitch.Withanother,wecouldchangeitsamplitudetomakeitsoundlouderorsofter.Byplugginginseveralcablesatonce,wecouldaddwavesandtheirovertonestogetherindifferentcombinations,justasFourierhaddoneabstractly,butforustheexperiencewassensory.Wecouldseethewaves’shapesonanoscilloscopeatthesametimeaswelistenedtothem.Youcouldtryallthisforyourselfnowontheinternet.Searchforsomethinglikethesoundoftrianglewavesandyou’llfindinteractivedemosthatwillletyoufeellikeyou’resittingrightthereinmyclassroomin1974,playingwithwavesforthefunofit.ThelargersignificanceofFourier’sworkisthathetookthefirststeptoward

usingcalculusasasoothsayertopredicthowacontinuumofparticlescouldmoveandchange.ThiswasanenormousadvancebeyondNewton’sworkonthemotionofdiscretesetsofparticles.Inthecenturiestocome,scientistswould

extendFourier’smethodstoforecastthebehaviorofothercontinuousmedia,liketheflutterofaBoeing787wing,theappearanceofapatientafterfacialsurgery,theflowofbloodthroughanartery,ortherumblingofthegroundafteranearthquake.Todaythesetechniquesareubiquitousinscienceandengineering.Theyareusedtoanalyzeshockwavesfromathermonuclearblast;radiowavesforcommunications;thewavesofdigestionintheintestinethatallownutrientstobeabsorbedandsendwasteproductsmovingintherightdirection;thepathologicalelectricalwavesinthebrainassociatedwithepilepsyandParkinson’stremors;andthecongestionwavesoftrafficonahighway,asseenintheexasperatingphenomenonofphantomjams,wheretrafficslowsdownfornoapparentreason.Fourier’sideasandtheiroffshootshaveenabledallofthesewavephenomenatobeunderstoodmathematically,sometimeswiththehelpofformulas,othertimesthroughmassivecomputersimulations,sowecanexplain,predict,and,insomecases,controlorabolishthem.

WhySineWaves?

Beforeweleavesinewavesandmoveontotheirtwo-andthree-dimensionalcounterparts,it’sworthclarifyingwhatmakesthemsospecial.Afterall,othertypesofcurvescanserveasbuildingblocks,andsometimestheyworkbetterthansinewaves.Forinstance,tocapturelocalizedfeatureslikefingerprintridges,waveletsgotthenodfromtheFBI.Waveletsareoftensuperiortosinewavesformanyimage-andsignal-processingtasksinfieldslikeearthquakeanalysis,artrestorationandauthentication,andfacialrecognition.Sowhyaresinewavessowellsuitedtothesolutionofthewaveequationand

theheatequationandotherpartialdifferentialequations?Theirvirtueisthattheyplayverynicelywithderivatives.Specifically,thederivativeofasinewaveisanothersinewave,shiftedbyaquartercycle.That’saremarkableproperty.It’snottrueofotherkindsofwaves.Typically,whenwetakethederivativeofacurveofanykind,thatcurvewillbecomedistortedbybeingdifferentiated.Itwon’thavethesameshapebeforeandafter.Beingdifferentiatedisatraumaticexperienceformostcurves.Butnotforasinewave.Afteritsderivativeistaken,itdustsitselfoffandappearsunfazed,assinusoidalasever.Theonlyinjuryitsuffers—anditisn’tevenaninjury,really—isthatthesinewaveshiftsintime.Itpeaksaquarterofacycleearlierthanitusedto.Wesawanimperfectversionofthisinchapter4whenwelookedattheday-

to-dayvariationsindaylengthinNewYorkCityintheyear2018andcompared

themtothedailychangesindaylength,thenumberofminutesofsunlightfromonedaytothenext.Wesawthatbothcurveslookedapproximatelysinusoidal,exceptthatthedifferenceindaylightfromonedaytothenextformedawavethatwasshiftedthreemonthsearlierthanthedatafromwhichitcame.Putsimply,thelongestdayin2018wasJune21,whilethefastest-lengtheningdaywasthreemonthsearlier,March20.Thisiswhatweexpectfromsinusoidaldata.Ifday-lengthdatawereaperfectsinewave,andifwelookedatitsdifferencenotfromonedaytothenextbutfromoneinstanttothenext,thenitsinstantaneousrateofchange(the“derivative”wavederivedfromit)woulditselfbeaperfectsinewave,shiftedexactlyaquarterofacycleearlier.Backinchapter4wealsosawwhythequarter-cycleshiftoccurs.Itfollowsfromthedeepconnectionbetweensinewavesanduniformcircularmotion.(Youmaywanttolookbackatthatargumentifitseemshazynow.)Thatquarter-cycleshifthasafascinatingconsequence.Itimpliesthatifwe

taketwoderivativesofthesinewave,itshiftsaquarterplusanotherquartercycleearlier.Sointotalitgetsshiftedhalfacycleearlier.Thatmeansthatitsformerpeakisnowavalley,andviceversa.Thesinewavehasturnedupsidedown.Inmathematicalterms,thisisexpressedbytheformula

d⁄dx(d⁄dxsinx)=−sinx

wheretheLeibniziandifferentiationsymbold/dxmeans“takethederivativeofwhateverexpressionappearstotheright.”Theformulashowsthattakingtwoderivativesofsinxamountstonothingmorethanmultiplyingitby–1.Thisreplacementoftwoderivativesbyasimplemultiplicationisafantasticsimplification.Takingtwoderivativesisafull-borecalculusoperation,whereasmultiplyingby–1ismiddle-schoolarithmetic.Butwhy,youmaybeaskingyourself,wouldanyoneeverwanttotaketwo

derivativesofsomething?Becausenaturedoes—anditdoesitallthetime.Or,rather,ourmodelsofnaturedoitallthetime.Forexample,inNewton’slawofmotion,F=ma,theaccelerationainvolvestwoderivatives.Toseewhy,rememberthattheaccelerationisthederivativeofspeedandspeedisthederivativeofdistance.Thatmakesaccelerationthederivativeofthederivativeofdistance,ortoputitmoreconcisely,thesecondderivativeofdistance.Secondderivativescomeupeverywhereinphysicsandengineering.AlongwithNewton’sequation,theyalsostarintheheatequationandthewaveequation.Sothat’swhysinewavesaresowellsuitedtothoseequations.Forsine

waves,twoderivativesboildowntomeremultiplicationby–1.Ineffect,the

inherentcalculusthatmadetheheatandwaveequationshardtoanalyzeisnolongeranissuewhenwerestrictourattentiontosinewaves.Thecalculusgetsstrippedoutandreplacedbymultiplication.Thisiswhatmadethevibrating-stringproblemandtheheat-flowproblemsomucheasiertosolveforsinewaves.Ifanarbitrarycurvecouldbebuiltfromthem,thatcurvewouldinheritthevirtuesofsinewaves.Theonlyhitchwasthatinfinitelymanysinewavesneededtobeaddedtogethertobuildupanarbitrarycurve,butthatwasasmallpricetopay.Thisisthecalculusperspectiveonwhysinewavesarespecial.Physicistshave

theirownperspective,onethatisalsoworthunderstanding.Toaphysicist,what’sremarkableaboutsinewaves(inthecontextofthevibrationandheatflowproblems)isthattheyformstandingwaves.Theydon’ttravelalongthestringortherod.Theyremaininplace.Theyoscillateupanddownbutneverpropagate.Evenmoreremarkably,standingwavesvibrateatauniquefrequency.That’sararityintheworldofwaves.Mostwavesareacombinationofmanyfrequencies,justaswhitelightisacombinationofallthecolorsoftherainbow.Inthatrespect,astandingwaveispure,notamixture.

VisualizingModesofVibration:ChladniPatterns

Thewarmsoundofaguitarandtheplaintivesoundofaviolinarerelatedtothevibrationssetupinthebellyandbodyoftheinstrument,inthewoodandthecavitiesinside,wheresoundwavesvibrateandresonate.Thosevibrationpatternsdeterminethequalityandvoiceoftheinstrument.That’spartofwhatmakesaStradivariussospecial,itsuniquelyevocativevibrationpatternsofwoodandair.Westilldon’tunderstandexactlywhatmakescertainviolinssoundbetterthanothers,butthekeymustbesomethingabouttheirmodesofvibration.In1787,aGermanphysicistandmusical-instrumentmakernamedErnst

Chladnipublishedanarticleshowingacleverwaytovisualizethesevibrationalpatterns.Insteadofusingashapeascomplicatedasaguitaroraviolin,though,heplayedamuchsimplerinstrument—athinmetalplate—bydrawingaviolinbowacrossitsedge.Inthisway,hewasabletogettheplatetovibrateandsing(abitlikethewayyoucangetahalf-filledwineglasstosingbyrubbingyourfingerarounditsrim).Tovisualizethevibrations,Chladnisprinkledafinedustofsandontotheplatebeforehebowedit.Whenhestrokedtheplate,thesand

bouncedoffthepartsthatwerevibratingthemostandsettledinthepartsthatweren’tvibratingatall.TheresultingcurvesarenowcalledChladnipatterns.

YoumayhaveseenademonstrationofChladnipatternsatsciencemuseums.

Ametalplateisplacedoveraloudspeakerandcoveredwithsand,thenit’sdriventovibratebyanelectronicsignalgenerator.Asthefrequencyofthesoundcomingoutoftheloudspeakerisadjusted,theplatecanbeexcitedintodifferentresonantpatterns.Whenevertheloudspeakertunesintoanewresonantfrequency,thesandrearrangesitselfintoadifferentstanding-wavepattern.Theplatedividesitselfintoneighboringregionsthatvibrateinoppositedirections,boundedbynodalcurveswheretheplateremainsmotionless.Perhapsitseemsoddthatsomepartsoftheplatedon’tmove.Butthat

shouldn’tbesurprising.Wesawthesamethingwithsinewavesonastring.Thepointswherethestringdoesn’tmovearethenodesofvibration.Foraplate,therearesimilarnodes,excepttheyarenotisolatedpoints.Rather,theylinktogethertoformnodallinesandcurves.ThesearethecurvesthatChladnimademanifestinhisexperiments.TheywereconsideredsoastonishingatthetimethatChladniwasinvitedtoshowthemtoEmperorNapoleonhimself.Napoleon,whohadsometraininginmathandengineering,wassointriguedthathe

establishedacontestandchallengedthegreatestmathematiciansofEuropetoexplainChladni’spatterns.Thenecessarymathematicsdidnotexistatthattime.Thepreeminent

mathematicianinEurope,JosephLouisLagrange,feltthattheproblemwasbeyondreachandthatnoonewouldsolveit.Indeed,onlyonepersontried.HernamewasSophieGermain.

TheNoblestCourage

SophieGermainhadtaughtherselfcalculusatayoungage.Thedaughterofawealthyfamily,shehadbecomeentrancedbymathematicsafterreadingabookaboutArchimedesinherfather’slibrary.Whenherparentsfoundoutthatshelovedmathematicsandwasstayinguplateatnighttoworkonit,theytookawayhercandles,leftherfireunlit,andconfiscatedhernightgowns.Sophiepersisted.Shewrappedherselfinquiltsandworkedbythelightofstolencandles.Eventuallyherfamilyrelentedandgavehertheirblessing.Germain,likeallwomenofherera,wasnotpermittedtoattenduniversity,so

shecontinuedtoteachherself,insomecasesbyobtaininglecturenotesfromthecoursesatthenearbyÉcolePolytechniqueusingthenameMonsieurAntoine-AugustLeBlanc,astudentwhohadlefttheschool.Unawareofhisdeparture,academyadministratorscontinuedtoprintlecturenotesandproblemsetsforhim.Shesubmittedworkunderhisnameuntiloneoftheschool’steachers,thegreatLagrange,noticedtheremarkableimprovementinMonsieurLeBlanc’spreviouslyabysmalperformance.LagrangerequestedameetingwithLeBlancandwasdelightedandastonishedtodiscoverhertrueidentity.HetookGermainunderhiswing.Herearliesttriumphswereinnumbertheory,whereshemadeimportant

contributionstooneofthemostdifficultunsolvedproblemsinthatfield,knownasFermat’slasttheorem.Whenshefeltshe’dmadeabreakthrough,shewrotetotheworld’sgreatestnumbertheorist(andoneofthegreatestmathematiciansofalltime),CarlFriedrichGauss,oncemoreusingthepseudonymofAntoineLeBlanc.Gaussadmiredthebrillianceofhismysteriouscorrespondentandtheyconductedalivelyexchangeoflettersforthreeyears.Mattersdarkenedonedayin1806wheneventsthreatenedGauss’slife.Napoleon’sarmyhadbegunstormingthroughPrussia,andGauss’shomecityofBrunswickwastaken.Usingfamilyconnections,GermainwrotetoafriendwhowasageneralintheFrencharmyandaskedhimtoguaranteeGauss’ssafety.WhenwordgotbacktoGauss

thathislifehadbeenprotectedbytheinterventionofaMademoiselleSophieGermain,hewasgratefulbutpuzzled,sinceheknewnoonebythatname.Inhernextletter,Germainunmaskedherself.Gausswasflabbergastedtolearnthathehadbeencorrespondingwithawoman.Giventhedepthofherinsightsandrecognizingalltheprejudicesandobstaclesshemusthaveendured,hetoldherthat“withoutdoubtshemusthavethenoblestcourage,quiteextraordinarytalentsandsuperiorgenius.”SowhensheheardofthecompetitiontosolvethemysteryofChladni

patterns,Germainrosetothechallenge.Shewastheonlypersonbraveenoughtotakeastabatdevelopingthenecessarytheoryfromscratch.Hersolutioninvolvedcreatinganewsubfieldofmechanics,thetheoryofelasticityforflat,thin,two-dimensionalplates,goingbeyondtheearlierandmuchsimplertheoriesforone-dimensionalstringsandbeams.Shebuiltitonprinciplesofforcesanddisplacementsandcurvatures,andsheusedtechniquesofcalculustoformulateandsolvetherelevantpartialdifferentialequationsforChladni’svibratingplatesandthemarvelouspatternstheyproduced.ButgiventhegapsinGermain’seducationandherlackofformaltraining,herattemptedsolutioncontainedflawsthatthejudgesnoticed.Theyfeltthattheproblemhadnotbeenfullysolved,andtheyrenewedthecontestforanothertwoyears,andthenanothertwoafterthat.Onherthirdtry,Germainwasawardedtheprize,thefirstwomanevertobesohonoredbytheParisAcademyofSciences.

MicrowaveOvens

Chladnipatternsallowustovisualizestandingwavesintwodimensions.Inourdailylives,werelyonthethree-dimensionalcounterpartofChladnipatternswheneverweuseamicrowaveoven.Theinsideofamicrowaveovenisathree-dimensionalspace.Whenyoupressthestartbutton,theovenfillswithastanding-wavepatternofmicrowaves.Thoughyoucan’tseetheseelectromagneticvibrationswithyoureyes,youcanvisualizethemindirectlybymimickingwhatChladnididwithhissand.Takeamicrowave-safeplateandcoveritcompletelywithathinlayerof

processedshreddedcheese(oranythingelsethatwilllieflatandmelteasily,likeathinslabofchocolateorasprinklingofmini-marshmallows).Beforeyouputtheplateintheoven,besuretotakeouttherotatingturntable.That’simportantbecauseyouwanttheplateofcheese(orwhateveryou’reusing)tostandstilltoallowyoutodetectthehotspots.Oncetheturntableisoutandtheplateisinside,

closethedoorandturnonthemicrowave.Letitgoforaboutthirtyseconds,nomore.Thentakeouttheplate.You’llseeplaceswherethecheesehasmeltedcompletely.Thosearethehotspots.Theycorrespondtoanti-nodesofthemicrowavepattern,theplaceswherethevibrationsaremostvigorous.They’relikethepeaksandtroughsofasinewaveorliketheplacesintheChladnipatternwherethesandisnot(becausethevigorousoscillationshaveshakenitoff).Forastandardmicrowaveoventhatrunsat2.45GHz(meaningthewaves

vibratebackandforth2.45billiontimesasecond),youshouldfindthatthedistancebetweenneighboringmeltedspotsisabouttwoandahalfinches,orsixcentimeters.Keepinmind,that’sonlythedistancefromapeaktoatroughandhenceishalfawavelength.Togetthefullwavelength,wedoublethatdistance.Thusthewavelengthofthestanding-wavepatternintheovenisaboutfiveinches,ortwelvecentimeters.Incidentally,youcanuseyouroventocalculatethespeedoflight.Multiply

thefrequencyofvibration(listedontheoven’sdoorframe)bythewavelengthyoumeasuredinyourexperiment,andyoushouldgetthespeedoflightorsomethingprettyclosetoit.Here’showitwouldgowiththenumbersIjustgave:Thefrequencyis2.45billioncyclespersecond.Thewavelengthis12centimeters(percycle).Multiplyingthemtogethergives29.4billioncentimeterspersecond.That’sprettyclosetotheacceptedvalueforthespeedoflight,30billioncentimeterspersecond.Notbadforsuchacrudemeasurement.

WhyMicrowaveOvensUsedtoBeCalledRadarRanges

AttheendofWorldWarII,theRaytheonCompanywaslookingfornewapplicationsforitsmagnetrons,thehigh-poweredvacuumtubesusedinradar.Amagnetronistheelectronicanalogofawhistle.Justasawhistlesendsoutsoundwaves,amagnetronsendsoutelectromagneticwaves.Thesewavescanbebouncedoffanairplaneoverheadtodetecthowfarawayitisandhowfastit’smoving.Nowadays,radarisusedtotrackthemovementofeverythingfromshipsandspeedingcarstofastballs,tennisserves,andweatherpatterns.Afterthewar,in1946,Raytheonhadnoideawhatitwasgoingtodowithall

themagnetronsithadbeenmanufacturing.AnengineernamedPercySpencernoticedonedaythatapeanut-clusterbarinhispockethadturnedintoagooey,stickymesswhilehewasworkingwithamagnetron.Herealizedthatthe

microwavesitemittedcouldwarmfoodveryeffectively.Toexploretheideafurther,hetriedpointingamagnetronatanegg,anditgotsohotitexplodedinsomeone’sface.Spenceralsodemonstratedthathecouldmakepopcornwithit.Thisconnectionbetweenradarandmicrowavesiswhythefirstmicrowaveovenswerecalledradarranges.Theywerenotacommercialhituntilthelate1960s.Thefirstmicrowaveovensweretoobig,almostsixfeettall,andextremelyexpensive,costingtheequivalentoftensofthousandsofdollarsintoday’smoney.Buteventuallymicrowavesbecamesufficientlyminiaturizedandcheapenoughthatordinaryfamiliescouldaffordthem.Today,atleast90percentofhouseholdsinindustrializedcountrieshavethem.Thestoryofradarandmicrowaveovensisatestamenttothe

interconnectednessofscience.Thinkofwhatwentintothem:physics,electricalengineering,materialsscience,chemistry,andgoodoldserendipitousinvention.Calculusplayedanimportantparttoo.Itprovidedthelanguagefordescribingwavesandthetoolsforanalyzingthem.Thediscoveryofthewaveequation,whichstartedasanoutgrowthofmusicinconnectionwithvibratingstrings,wasultimatelyusedbyMaxwelltopredicttheexistenceofelectromagneticwaves.Fromthereitwasashorthoptovacuumtubes,transistors,computers,radar,andmicrowaveovens.Alongtheway,Fourier’smethodswereindispensable.Andasweareabouttosee,histechniquesplayedaroleinthediscoveryofanewuseforhigher-energyelectromagneticwaves.Thesemuchmoreenergeticwaveswerediscoveredbyaccidentattheturnofthetwentiethcentury.Noonewassurewhattheywere,soinhonoroftheunknowninmathematics,theywerenamedx-rays.

ComputerizedTomographyandBrainImaging

Microwavesaregoodforcooking,butx-raysaregoodforseeingintoourbodies.Theyallownoninvasivediagnosisofbrokenbones,skullfractures,andcurvedspines.Unfortunately,traditionalx-rayscapturedonblack-and-whitefilmareinsensitivetosubtlevariationsintissuedensity.Thislimitstheirusefulnessforexaminingsofttissuesandorgans.Amoremodernformofmedicalimaging,calledCTscanning,ishundredsoftimesmoresensitivethanconventionalx-rayfilms.Theirprecisionhasrevolutionizedmedicine.TheCstandsforcomputerizedandtheTstandsfortomography,meaningthe

processofvisualizingsomethingbycuttingitintoslices.ACTscanusesx-raystoimageanorganoratissueonesliceatatime.Whenapatientisplacedina

CTscanner,x-raysaresentthroughtheperson’sbodyatmanydifferentanglesandrecordedbyadetectorontheotherside.Fromallthatinformation—fromallthoseviewsatdifferentangles—it’spossibletoreconstructmuchmoreclearlywhatthex-rayspassedthrough.Inotherwords,CTisnotjustamatterofseeing;it’samatterofinferring,deducing,andcalculating.Indeed,themostbrilliantandrevolutionarypartofCTisitsuseofsophisticatedmathematics.Withthehelpofcalculus,Fourieranalysis,signalprocessing,andcomputers,theCTsoftwareinfersthepropertiesofthetissue,organ,orbonethroughwhichthex-rayspassedandthengeneratesadetailedpictureofthatpartofthebody.Toseehowcalculusplaysaroleinallthis,firstweneedtounderstandwhat

problemCTsolvesandhowitsolvesit.Imaginefiringabeamofx-raysthroughasliceofbraintissue.Asthex-rays

travel,theyencountergraymatter,whitematter,possiblybraintumors,bloodclots,andsoon.Thesetissuesabsorbthex-rays’energytoagreaterorlesserdegree,dependingonthetypeoftissueitis.ThegoalofCTistomaptheabsorptionpatterninthewholeslice.Fromthatinformation,CTcanrevealwheretumorsorclotsmaybe.CTdoesn’tseethebraindirectly;itseesthex-ray–absorptionpatterninthebrain.Themathworkslikethis.Asanx-raytravelsthroughagivenpointinthe

brainslice,itlosessomeofitsintensity.Thislossofintensityislikewhathappenswhenordinarylightpassesthroughsunglassesandbecomeslessbright.Thecomplicationhereisthatthereisasequenceofdifferentbraintissuesalongthex-ray’spath,sothetissuesactmorelikeasequenceofsunglasses,oneinfrontofanother,allofdifferentopacities.Andwedon’tknowtheopacityofanyofthesunglasses;that’swhatwe’retryingtofigureout!Becauseofthisvariabilityintheabsorptionpropertiesofthedifferenttissues,

whenthex-raysemergefromthebrainandstrikethex-raydetectorontheotherside,theirintensityhasbeenreducedbydisparateamountsalongtheway.Tocomputetheneteffectofallofthesereductions,wehavetofigureouthowmuchtheintensitywasreduced,stepbyinfinitesimalstepasthex-raystraveledthroughthetissue,andthencombinealltheresultsappropriately.Thiscomputationamountstoanintegral.Theappearanceofintegralcalculushereshouldn’tcomeasasurprise.It’sthe

mostnaturalwaytomakethisverycomplicatedproblemmoretractable.Asalways,weappealtotheInfinityPrinciple.First,weimaginechoppingthex-rays’pathintoinfinitelymanyinfinitesimalsteps,thenwefigureouthowmuchtheirintensityattenuateswitheachstep,andfinallyweputalltheanswersbacktogethertocomputethenetattenuationalongthegivenlineoftravel.

Sadly,havingdonethis,we’veobtainedonlyasinglepieceofinformation.Weknowthetotalattenuationofthex-raysonlyalongtheparticularpaththatthex-raysfollowed.Thatdoesn’ttellusmuchaboutthebrainsliceasawhole.Itdoesn’teventellusmuchabouttheparticularlinethex-raystraveledon.Itjusttellsusthenetattenuationalongtheline,notthepoint-to-pointpatternofattenuationalongit.Letmetrytoillustratethedifficultybyanalogy:Thinkaboutallthedifferent

wayswecouldaddupnumberstomake6.Justasthenumber6canresultfrom1+5or2+4or3+3,thesamenetattenuationofthex-rayscouldresultfrommanydifferentsequencesoflocalattenuations.Forexample,therecouldbehighattenuationatthebeginningofthelineandlowattenuationattheend.Oritcouldbetheotherwayaround.Ortherecouldbeaconstant,mediumlevelofattenuationthewholewaythrough.Wehavenowayofdistinguishingamongthesepossibilitiesfromjustonemeasurement.However,oncewerecognizethedifficulty,wecanimmediatelyseehowto

solveit.Weneedtofirex-raysalongmanydifferentdirections.That’stheheartofcomputerizedtomography.Byfiringx-raysfrommultipledirectionsthroughthesamepointoftissueandthenrepeatingthemeasurementformanydifferentpoints,weshould,inprinciple,beabletomapouttheattenuationfactorseverywhereinthebrain.Thisisnotquitethesamethingaslookingatthebrain,butit’salmostasgood,becauseitprovidesinformationaboutwhichtypesoftissuesoccurinwhichbrainregions.Themathematicalchallenge,then,istoreassembletheinformationobtained

fromallthemeasurementsalonglinesintoacoherenttwo-dimensionalpictureofthewholebrainslice.ThisiswhereFourieranalysiscamein.ItallowedaSouthAfricanphysicistnamedAllanCormacktosolvethereassemblyproblem.Fourieranalysisenteredbecausetherewasacirclelurkingintheproblem.Thatcirclewasthecircleofallthelines—allthedirectionsalongwhichthex-rayscouldbefired,edgeon,intoatwo-dimensionalslice.Rememberthatcirclesarealwaysassociatedwithsinewaves,andsinewaves

arethebuildingblocksofFourierseries.BywritingthereassemblyproblemintermsofFourierseries,Cormackwasabletoboilatwo-dimensionalreassemblyproblemdowntoaneasierone-dimensionalproblem.Ineffect,hegotridofthe360degreesofpossibleangles.Then,withgreatprowessinintegralcalculus,hemanagedtosolvetheone-dimensionalreassemblyproblem.Theupshotwasthat,giventhemeasurementsalongafullcircleoflines,hecoulddeducethepropertiesofthetissueinside.Hecouldinfertheabsorptionmap.Itwasalmostlikeseeingthebrainitself.

In1979,CormacksharedtheNobelPrizeinPhysiologyorMedicinewithGodfreyHounsfieldfortheirdevelopmentofcomputer-assistedtomography.Neitherofthemwasamedicaldoctor.CormackdevelopedtheFourier-basedmathematicaltheoryofCTscanninginthelate1950s.Hounsfield,aBritishelectricalengineer,inventedthescannerincollaborationwithradiologistsintheearly1970s.Theinventionofthescannerprovidesanotherdemonstrationofthe

unreasonableeffectivenessofmathematics.Inthiscase,theideasthatmadeCTscanningpossiblehadexistedformorethanhalfacenturyandhadnoconnectionwhatsoevertomedicine.Thenextpartofthestorybeganinthelate1960s.Hounsfieldhadalready

testedaprototypeofhisinventiononpigs’brains.Hewasdesperatetofindaclinicalradiologisttohelphimextendhisworktohumanpatients,butonedoctorafteranotherrefusedtomeetwithhim.Theyallthoughthewasacrackpot.Theyknewsofttissuescouldn’tbevisualizedwithx-rays.Atraditionalx-rayofahead,forexample,showedtheskullbonesclearly,butthebrainlookedlikeafeaturelesscloud.Tumors,hemorrhages,andbloodclotsweren’tvisible,despitewhatHounsfieldclaimed.Finally,oneradiologistagreedtohearhimout.Theconversationdidn’tgo

well.Attheendofthemeeting,theskepticalradiologisthandedHounsfieldajarcontainingahumanbrainwithatumorinitandchallengedhimtoimageitwithhisscanner.Hounsfieldsoonbroughtbackimagesofthebrainthatpinpointednotonlythetumorbutalsoareasofbleedingwithinit.Theradiologistwasstunned.Wordspread,andsoonotherradiologistscame

onboard.WhenHounsfieldpublishedthefirstcomputerizedtomographsin1972,theyshockedthemedicalworld.Radiologistscouldsuddenlyusex-raystoseetumors,cysts,graymatter,whitematter,andthefluid-filledcavitiesofthebrain.Ironically,giventhatwavetheoryandFourieranalysisbeganwiththestudy

ofmusic,atakeymomentinthedevelopmentofcomputerizedtomography,musicprovedindispensableagain.Hounsfieldhadhisbreakthroughideasinthemid-1960swhenhewasworkingforacompanycalledElectricandMusicalIndustries.HehadfirstworkedonEMI’sradarandguidedweaponry,andthenheturnedhisattentiontodevelopingBritain’sfirstall-transistorcomputer.Afterthatsmashingsuccess,EMIdecidedtosupportHounsfieldandlethimdowhateverhewantedforhisnextproject.Atthattime,EMIwasflushwithmoneyandcouldaffordtotakerisks.Theirprofitshaddoubledafterthey’dsignedabandfromLiverpoolcalledtheBeatles.

Hounsfieldapproachedmanagementwithhisideaofimagingorganswithx-rays,andEMI’sdeeppocketshelpedhimtakethefirststep.Hecameupwithhisownapproachtosolvingthereassemblyprobleminthemathematics,unawarethatCormackhadsolveditadecadeearlier.AndCormack,inturn,didn’tknowthatapuremathematiciannamedJohannRadonhadsolveditfortyyearsbeforehim,withnoapplicationinmind.ThequestforpuremathematicalunderstandinghadgivenCTscanningthetoolsitneeded,halfacenturyaheadoftime.InCormack’sNobelPrizeaddress,hementionedthatheandhiscolleague

ToddQuintohadlookedintoRadon’sresultsandweretryingtogeneralizethemtothree-andevenfour-dimensionalregions.Thatmusthavebeenhardforhisaudiencetofathom.Weliveinathree-dimensionalworld.Whywouldanyonewanttostudyafour-dimensionalbrain?Cormackexplained:

Whatistheuseoftheseresults?TheansweristhatIdon’tknow.Theywillalmostcertainlyproducesometheoremsinthetheoryofpartialdifferentialequations,andsomeofthemmayfindapplicationinimagingwithMRIorultrasound,butthatisbynomeanscertain.Itisalsobesidethepoint.QuintoandIarestudyingthesetopicsbecausetheyareinterestingintheirownrightasmathematicalproblems,andthatiswhatscienceisallabout.

11

TheFutureofCalculus

THETITLEOFthischaptermightraiseafeweyebrowsamongthosewhobelievethatcalculusisfinished.Howcouldithaveafuture?It’sovernow,isn’tit?Thisissomethingyouhearsurprisinglyofteninmathematicalcircles.Accordingtothisnarrative,calculusbeganwithabang,thankstothebreakthroughsofNewtonandLeibniz.Theirdiscoveriessparkedagold-rushmentalityinthe1700s,aperiodmarkedbyplayful,almostgiddyexplorationduringwhichthegolemofinfinitywasallowedtorunwild.Bygivingitfreerein,mathematiciansproducedaraftofspectacularresultsbutalsogeneratedalotofnonsenseandconfusion.Sointhe1800s,thenextfewgenerationsofmathematicians,amorerigorouslot,proddedthegolembackintoitscage.Theyexpungedinfinityandinfinitesimalsfromcalculus,shoredupthefoundationsofthesubject,andfinallyclarifiedwhatlimits,derivatives,integrals,andrealnumbersactuallymeant.Byaround1900,theirmopping-upoperationwascomplete.Tomymind,thatvisionofcalculusisfartooblinkered.Calculusisnotjust

theworkofNewtonandLeibnizandtheirsuccessors.Itstartedmuchearlierthanthatandit’sstillgoingstrongtoday.Calculus,tome,isdefinedbyitscredo:tosolveahardproblemaboutanythingcontinuous,sliceitintoinfinitelymanypartsandsolvethem.Byputtingtheanswersbacktogether,youcanmakesenseoftheoriginalwhole.I’vecalledthiscredotheInfinityPrinciple.

TheInfinityPrinciplewastherefromthebeginning,inArchimedes’sworkoncurvedshapes,anditwasthereinthescientificrevolution,inNewton’ssystemoftheworld,andit’swithustodayinourhomes,atourjobs,andinourcars.IthelpedgiveusGPS,cellphones,lasers,andmicrowaveovens.TheFBIusedittocompressmillionsoffingerprintfiles.AllanCormackusedittocreatethetheoryforCTscanning.BoththeFBIandCormacksolvedahardproblembyreassemblingitfromsimplerparts:waveletsforfingerprints,sinewavesforCT.Fromthispointofview,calculusisthesprawlingcollectionofideasandmethodsusedtostudyanything—anypattern,anycurve,anymotion,anynaturalprocess,system,orphenomenon—thatchangessmoothlyandcontinuouslyandhenceisgristfortheInfinityPrinciple.ThisbroaddefinitiongoesfarbeyondthecalculusofNewtonandLeibniztoincludeitsdescendants:multivariablecalculus,ordinarydifferentialequations,partialdifferentialequations,Fourieranalysis,complexanalysis,andanyotherpartofhighermathematicswherelimits,derivatives,andintegralsappear.Viewedthisway,calculusisnotover.It’sashungryasever.ButI’mintheminorityhere.Actually,aminorityofone.Noneofmy

colleaguesinthemathdepartmentwouldagreethatallofthisiscalculus,andforgoodreason:Itwouldbeabsurd.Halfthecoursesinthecurriculumwouldhavetoberenamed.AlongwithCalculus1,2,and3,we’dnowhaveCalculus4through38.Notverydescriptive.Soinstead,wegivedifferentnamestoeachoffshootofcalculusandobscurethecontinuityamongthem.Weslicethewholeofcalculusintoitssmallestconsumableparts.That’sironic,orperhapsfitting,giventhatcalculusitselfisaboutslicingcontinuousthingsintopartstomakethemeasiertounderstand.Letmebeclear:Ihavenoobjectiontoallthedifferentcoursenames.AllI’msayingisthatslicingcanbemisleadingwhenitmakesusforgetthatthepartsbelongtogether,thatthey’reallpartofsomethingbigger.Mygoalinthisbookhasbeentoshowcalculusasawhole,togiveafeelingforitsbeauty,unity,andgrandeur.What,then,mightthefutureholdforcalculus?Astheysay,predictionis

alwaysdifficult,especiallyaboutthefuture,butIthinkit’ssafetoassumethatseveraltrendsarelikelytobeimportantintheyearsahead.Theseinclude

Newapplicationsofcalculustothesocialsciences,music,thearts,andthehumanitiesOngoingapplicationsofcalculustomedicineandbiologyCopingwiththerandomnessinherentinfinance,economics,andtheweatherCalculusintheserviceofbigdata,andviceversa

Thecontinuingchallengeofnonlinearity,chaos,andcomplexsystemsTheevolvingpartnershipbetweencalculusandcomputers,includingartificialintelligencePushingtheboundariesofcalculusinthequantumrealm.

Thisisalotofgroundtocover.Ratherthansayingalittleabouteachofthetopicsmentionedhere,I’llfocusonafewofthem.AfterabriefforayintothedifferentialgeometryofDNA,wherethemysteryofcurvesmeetsthesecretoflife,we’llconsidersomecasestudiesthatIhopeyou’llfindphilosophicallyprovocative.Theseincludethechallengestoinsightandpredictioncausedbytheriseofchaos,complexitytheory,computers,andartificialintelligence.Forallofthattomakesense,however,wewillneedtoreviewthefundamentalsofnonlineardynamics.Examiningthatcontextwillallowustobetterappreciatethechallengesahead.

TheWrithingNumberofDNA

Calculushastraditionallybeenappliedinthe“hard”scienceslikephysics,astronomy,andchemistry.Butinrecentdecades,ithasmadeinroadsintobiologyandmedicine,infieldslikeepidemiology,populationbiology,neuroscience,andmedicalimaging.We’veseenexamplesofmathematicalbiologythroughoutourstory,rangingfromtheuseofcalculusinpredictingtheoutcomeoffacialsurgerytothemodelingofHIVasitbattlestheimmunesystem.Butallthoseexampleswereconcernedwithsomeaspectofthemysteryofchange,themostmodernobsessionofcalculus.Incontrast,thefollowingexampleisdrawnfromtheancientmysteryofcurves,whichwasgivennewlifebyapuzzleaboutthethree-dimensionalpathofDNA.ThepuzzlehadtodowithhowDNA,anenormouslylongmoleculethat

containsallthegeneticinformationneededtomakeaperson,ispackagedincells.EveryoneofyourtentrillionorsocellscontainsabouttwometersofDNA.Iflaidendtoend,thatDNAwouldreachtothesunandbackdozensoftimes.Still,askepticmightarguethatthiscomparisonisnotasimpressiveasitsounds;itmerelyreflectshowmanycellseachofushas.Amoreinformativecomparisoniswiththesizeofthecell’snucleus,thecontainerthatholdstheDNA.Thediameterofatypicalnucleusisaboutfive-millionthsofameter,anditisthereforefourhundredthousandtimessmallerthantheDNAthathastofit

insideit.Thatcompressionfactorisequivalenttostuffingtwentymilesofstringintoatennisball.Ontopofthat,theDNAcan’tbestuffedintothenucleushaphazardly.It

mustn’tgettangled.ThepackaginghastobedoneinanorderlyfashionsotheDNAcanbereadbyenzymesandtranslatedintotheproteinsneededforthemaintenanceofthecell.OrderlypackagingisalsoimportantsothattheDNAcanbecopiedneatlywhenthecellisabouttodivide.Evolutionsolvedthepackagingproblemwithspools,thesamesolutionwe

usewhenweneedtostorealongpieceofthread.TheDNAincellsiswoundaroundmolecularspoolsmadeofspecializedproteinscalledhistones.Toachievefurthercompaction,thespoolsarelinkedendtoend,likebeadsonanecklace,andthenthenecklaceiscoiledintoropelikefibersthatarethemselvescoiledintochromosomes.ThesecoilsofcoilsofcoilscompacttheDNAenoughtofititintothecrampedquartersofthenucleus.Butspoolswerenotnature’soriginalsolutiontothepackagingproblem.The

earliestcreaturesonEarthweresingle-celledorganismsthatlackednucleiandchromosomes.Theyhadnospools,justastoday’sbacteriaandvirusesdon’t.Insuchcases,thegeneticmaterialiscompactedbyamechanismbasedongeometryandelasticity.Imaginepullingarubberbandtightandthentwistingitfromoneendwhileholdingitbetweenyourfingers.Atfirst,eachsuccessiveturnoftherubberbandintroducesatwist.Thetwistsaccumulate,andtherubberbandremainsstraightuntiltheaccumulatedtorsioncrossesathreshold.Thentherubberbandsuddenlybucklesintothethirddimension.Itbeginstocoilonitself,asifwrithinginpain.Thesecontortionscausetherubberbandtobunchupandcompactitself.DNAdoesthesamething.Thisphenomenonisknownassupercoiling.Itisprevalentincircularloopsof

DNA.AlthoughwetendtopictureDNAasastraighthelixwithfreeends,inmanycircumstancesitclosesonitselftoformacircle.Whenthishappens,it’sliketakingoffyourbelt,puttingafewtwistsinit,andthenbucklingitclosedagain.Afterthatthenumberoftwistsinthebeltcannotchange.Itislockedin.Ifyoutrytotwistthebeltsomewherealongitslengthwithouttakingitoff,countertwistswillformelsewheretocompensate.Thereisaconservationlawatworkhere.Thesamethinghappenswhenyoustoreagardenhosebypilingitonthefloorwithmanycoilsstackedontopofeachother.Whenyoutrytopullthehoseoutstraight,ittwistsinyourhands.Coilsconverttotwists.Theconversioncanalsogointheotherdirection,fromtwiststocoils,aswhenarubberbandwritheswhentwisted.TheDNAofprimitiveorganismsmakesuseofthiswrithing.CertainenzymescancutDNA,twistit,andthencloseitbackup.WhentheDNArelaxesitstwiststoloweritsenergy,theconservationlawforces

ittobecomemoresupercoiledandthereforemorecompact.TheresultingpathoftheDNAmoleculenolongerliesinaplane.Itwrithesaboutinthreedimensions.Intheearly1970sanAmericanmathematiciannamedBrockFullergavethe

firstmathematicaldescriptionofthisthree-dimensionalcontortionofDNA.HeinventedaquantitythathedubbedthewrithingnumberofDNA.Hederivedformulasforitusingintegralsandderivativesandprovedcertaintheoremsaboutthewrithingnumberthatformalizedtheconservationlawfortwistsandcoils.ThestudyofthegeometryandtopologyofDNAhasbeenathrivingindustryeversince.MathematicianshaveusedknottheoryandtanglecalculustoelucidatethemechanismsofcertainenzymesthatcantwistDNAorcutitorintroduceknotsandlinksintoit.TheseenzymesalterthetopologyofDNAandhenceareknownastopoisomerases.TheycanbreakstrandsofDNAandresealthem,andtheyareessentialforcellstodivideandgrow.Theyhaveprovedtobeeffectivetargetsforcancer-chemotherapydrugs.Themechanismofactionisnotcompletelyclear,butitisthoughtthatbyblockingtheactionoftopoisomerases,thedrugs(knownastopoisomeraseinhibitors)canselectivelydamagetheDNAofcancercells,whichcausesthemtocommitcellularsuicide.Goodnewsforthepatient,badnewsforthetumor.IntheapplicationofcalculustosupercoiledDNA,thedoublehelixismodeled

asacontinuouscurve.Asusual,calculuslikestoworkwithcontinuousobjects.Inreality,DNAisadiscretecollectionofatoms.There’snothingtrulycontinuousaboutit.Buttoagoodapproximation,itcanbetreatedasifitwereacontinuouscurve,likeanidealrubberband.Theadvantageofdoingthatisthattheapparatusofelasticitytheoryanddifferentialgeometry,twospinoffsofcalculus,canthenbeappliedtocalculatehowDNAdeformswhensubjectedtoforcesfromproteins,fromtheenvironment,andfrominteractionswithitself.Thelargerpointisthatcalculusistakingitsusualcreativelicense,treating

discreteobjectsasiftheywerecontinuoustoshedlightonhowtheybehave.Themodelingisapproximatebutuseful.Anyway,it’stheonlygameintown.Withouttheassumptionofcontinuity,theInfinityPrinciplecannotbedeployed.AndwithouttheInfinityPrinciple,wehavenocalculus,nodifferentialgeometry,andnoelasticitytheory.Iexpectinthefuturewewillseemanymoreexamplesofcalculusand

continuousmathematicsbeingbroughttobearontheinherentlydiscreteplayersofbiology:genes,cells,proteins,andtheotheractorsinthebiologicaldrama.Thereissimplytoomuchinsighttobegainedfromthecontinuumapproximationnottouseit.Untilwedevelopanewformofcalculusthatworksaswellfordiscretesystemsastraditionalcalculusdoesforcontinuumones,the

InfinityPrinciplewillcontinuetoguideusinthemathematicalmodelingoflivingthings.

DeterminismandItsLimits

Ournexttwotopicsaretheriseofnonlineardynamicsandtheimpactofcomputersoncalculus.I’vechosenthembecausethey’resophilosophicallyintriguingintheirimplications.Theycouldalterthenatureofpredictionforeverandleadtoaneweraincalculus—andinsciencemoregenerally—wherehumaninsightmaybegintofade,althoughscienceitselfwillstillgoon.ToclarifywhatImeanbythissomewhatapocalypticwarning,weneedtounderstandhowpredictionispossibleatall,whatitmeantclassically,andhowourclassicalnotionsarebeingrevisedbydiscoveriesmadeinthepastseveraldecadesinstudiesofnonlinearity,chaos,andcomplexsystems.Earlyinthe1800s,theFrenchmathematicianandastronomerPierreSimon

LaplacetookthedeterminismofNewton’sclockworkuniversetoitslogicalextreme.Heimaginedagodlikeintellect(nowknownasLaplace’sdemon)thatcouldkeeptrackofallthepositionsofalltheatomsintheuniverseaswellasalltheforcesactingonthem.“Ifthisintellectwerealsovastenoughtosubmitthesedatatoanalysis,”hewrote,“nothingwouldbeuncertainandthefuturejustlikethepastwouldbepresentbeforeitseyes.”Astheturnofthetwentiethcenturyapproached,thisextremeformulationof

theclockworkuniversebegantoseemscientificallyandphilosophicallyuntenable,forseveraldifferentreasons.Thefirstcamefromcalculus,andwehaveSofiaKovalevskayatothankforit.Kovalevskayawasbornin1850andgrewupinanaristocraticfamilyinMoscow.Whenshewaselevenshefoundherselfsurroundedbycalculus,literally—onewallofherbedroomwaspaperedwithnotesfromacalculuscourseherfatherhadattendedinhisyouth.Shelaterwrotethatshe“spentwholehoursofmychildhoodinfrontofthatmysteriouswall,tryingtomakeoutevenasinglesentenceandfindtheorderinwhichthepagesoughttohavefollowedoneanother.”ShewentontobecomethefirstwomaninhistorytoearnaPhDinmathematics.AlthoughKovalevskayashowedaflairformathematicsearlyon,Russianlaw

preventedherfromenrollingincollege.Sheenteredamarriageofconvenience,whichcausedhermuchheartacheintheyearstocomebutthatatleastallowedhertotraveltoGermany,wheresheimpressedseveralprofessorsasanextraordinarytalent.Yeteventhere,shewasnotofficiallyallowedtoattendtheir

classes.ShearrangedtostudyprivatelywiththeanalystKarlWeierstrassand,athisrecommendation,wasawardedadoctorateforsolvingseveraloutstandingproblemsinanalysis,dynamics,andpartialdifferentialequations.SheeventuallybecameafullprofessorattheUniversityofStockholmandtaughtthereforeightyearsbeforedyingfrominfluenzaattheageofforty-one.In2009,theNobelPrize–winningauthorAliceMunropublishedashortstoryabouthercalled“TooMuchHappiness.”Kovalevskaya’sinsightsonthelimitsofdeterminismcamefromherworkon

thedynamicsofrigidbodies.Arigidbodyisamathematicalabstractionofanobjectthatcan’tbebentordeformed;allofitspointsarerigidlyattachedtooneanother.Anexampleisaspinningtop.It’scompletelysolidandcomposedofinfinitelymanypointsandisthereforeamorecomplicatedmechanicalobjectthanthesinglepoint-likeparticlesthatNewtonhadconsidered.ThemotionofrigidbodiesisimportantinastronomyandspacesciencefordescribingphenomenarangingfromthechaotictumblingofHyperion,alittlepotato-shapedmoonofSaturn,totheregularrotationofaspacecapsuleorsatellite.Whilestudyingrigid-bodydynamics,Kovalevskayaproducedtwomajor

results.Thefirstwasanexampleofaspinningtopwhosemotioncouldbecompletelyanalyzedandsolved,inthesamesensethatNewtonhadsolvedthetwo-bodyproblem.Twoothersuch“integrabletops”werealreadyknown,butherswasmoresubtleandsurprising.Moreimportant,sheprovedthatnoothersolvabletopscouldexist.Shehad

foundthelastone.Allothersfromthenonwouldbenon-integrable,meaningthattheirdynamicswouldbeimpossibletosolvewithNewtonian-styleformulas.Itwasn’tamatterofinsufficientcleverness;sheprovedthattheresimplycouldn’tbeanyformulasofacertaintype(inthejargon,ameromorphicfunctionoftime)thatcoulddescribethemotionofthetopforever.Inthisway,sheputlimitsonwhatcalculuscoulddo.IfevenaspinningtopcoulddefyLaplace’sdemon,therewasnohope—eveninprinciple—offindingaformulaforthefateoftheuniverse.

Nonlinearity

TheunsolvabilitythatSofiaKovalevskayadiscoveredisrelatedtoastructuralaspectoftheequationsforatop:theequationsarenonlinear.Thetechnicalmeaningofnonlinearneednotconcernushere.Forourpurposes,allweneedis

afeelforthedistinctionbetweenlinearandnonlinearsystems,whichwecangetbyconsideringsomehomeyexamplesfromeverydaylife.Toillustratewhatlinearsystemsarelike,supposetwopeopletrytoweigh

themselvesbysteppingonascaleatthesametime,justforthefunofit.Theircombinedweightwillbethesumoftheirindividualweights.That’sbecauseascaleisalineardevice.Thepeople’sweightsdon’tinteractwitheachotherordoanythingtrickythatweneedtobeawareof.Forexample,theirbodiesdon’tsomehowconspirewitheachothertoseemlighterorsabotageeachothertoseemheavier.Theysimplyaddup.Onalinearsystemlikeascale,thewholeisequaltothesumoftheparts.That’sthefirstkeypropertyoflinearity.Thesecondisthatcausesareproportionaltoeffects.Imaginepullingonthestringofanarcher’sbow.Ifittakesacertainamountofforcetopullthestringbackacertaindistance,ittakestwiceasmuchforcetopullitbackbytwicethatdistance.Causeandeffectareproportional.Thesetwoproperties—theproportionalitybetweencauseandeffect,andtheequalityofthewholetothesumoftheparts—aretheessenceofwhatitmeanstobelinear.Yetmanythingsinnaturearemorecomplicatedthanthis.Wheneverpartsof

asysteminterfereorcooperateorcompetewitheachother,therearenonlinearinteractionstakingplace.Mostofeverydaylifeisspectacularlynonlinear;ifyoulistentoyourtwofavoritesongsatthesametime,youwon’tgetdoublethepleasure.Thesamegoesforconsumingalcoholanddrugs,wheretheinteractioneffectscanbedeadly.Bycontrast,peanutbutterandjellyarebettertogether.Theydon’tjustaddup—theysynergize.Nonlinearityisresponsiblefortherichnessintheworld,foritsbeautyand

complexityand,often,itsinscrutability.Forexample,allofbiologyisnonlinear;soissociology.That’swhythesoftsciencesarehard—andthelasttobemathematized.Becauseofnonlinearity,there’snothingsoftaboutthem.Thesamedistinctionbetweenlinearandnonlinearappliestodifferential

equations,thoughinalessintuitivefashion.Theonlythingweneedtosayisthatwhendifferentialequationsarenonlinear,astheywereforKovalevskaya’stops,theyareextremelydifficulttoanalyze.EversinceNewton,mathematicianshaveavoidednonlineardifferentialequationswhereverpossible.They’reseenasnastyandrecalcitrant.Incontrast,lineardifferentialequationsaresweetanddocile.Mathematicians

lovethembecausethey’reeasy.There’sanenormousbodyoftheoryforsolvingthem.Indeed,untilaboutthe1980s,thetraditionaleducationofanappliedmathematicianwasalmostentirelydevotedtolearningmethodstoexploitlinearity.YearswerespentmasteringFourierseriesandothertechniquestailoredtolinearequations.

Thegreatadvantageoflinearityisthatitallowsforreductionistthinking.Tosolvealinearproblem,wecanbreakitdowntoitssimplestparts,solveeachpartseparately,andputthepartsbacktogethertogettheanswer.Fouriersolvedhisheatequation—whichwaslinear—withthisreductioniststrategy.Hebrokeacomplicatedtemperaturedistributionintosinewaves,figuredouthoweachsinewavewouldchangeonitsown,thenrecombinedthosesinewavestopredicthowtheoveralltemperaturewouldchangealongthelengthofaheatedmetalrod.Thestrategyworkedbecausetheheatequationislinear.Itcanbechoppedintobitswithoutlosingitsessence.SofiaKovalevskayahelpedusunderstandhowdifferenttheworldappears

whenwefinallyfaceuptononlinearity.Sherealizedthatnonlinearityplaceslimitsonhumanhubris.Whenasystemisnonlinear,itsbehaviorcanbeimpossibletoforecastwithformulas,eventhoughthatbehavioriscompletelydetermined.Inotherwords,determinismdoesnotimplypredictability.Ittookthemotionofatop—achild’splaything—tomakeusmorehumbleaboutwhatwecaneverhopetoknow.

Chaos

Inretrospect,wecanseemoreclearlywhyNewton’sheadachedwhenhetriedtosolvethethree-bodyproblem.Thatproblemisinescapablynonlinear,unlikethetwo-bodyproblem,whichcanbemassagedtobecomelinear.Thenonlinearitywasn’tcausedbytheleapfromtotwotothreebodies.Itwascausedbythestructureoftheequationsthemselves.Fortwogravitatingbodies,butnotforthreeormore,thenonlinearitycouldbeeliminatedbyafelicitouschoiceofnewvariablesinthedifferentialequations.Ittookalongtimeforthehumblingimplicationsofnonlinearitytobefully

appreciated.Mathematiciansthrashedaroundforcenturiestryingtosolvethethree-bodyproblem,andalthoughprogresswasmade,noonemanagedtocrackitcompletely.Inthelate1800s,theFrenchmathematicianHenriPoincaréthoughthe’dsolvedit,buthe’dmadeamistake.Whenherectifiedhiserror,hestillcouldn’tsolvethethree-bodyproblem,buthediscoveredsomethingfarmoreimportant:thephenomenonthatwenowcallchaos.Chaoticsystemsarefinicky.Alittlechangeinhowthey’restartedcanmakea

bigdifferenceinwheretheyendup.That’sbecausesmallchangesintheirinitialconditionsgetmagnifiedexponentiallyfast.Anytinyerrorordisturbancesnowballssorapidlythatinthelongterm,thesystembecomesunpredictable.

Chaoticsystemsarenotrandom—they’redeterministicandhencepredictableintheshortrun—butinthelongrun,they’resosensitivetotinydisturbancesthattheylookeffectivelyrandominmanyrespects.Chaoticsystemscanbepredictedperfectlywelluptoatimeknownasthe

predictabilityhorizon.Beforethat,thedeterminismofthesystemmakesitpredictable.Forexample,thehorizonofpredictabilityfortheentiresolarsystemhasbeencalculatedtobeaboutfourmillionyears.Fortimesmuchshorterthanthat,likethesingleyearittakesourEarthtogoaroundthesun,everythingbehaveslikeclockwork.Butoncewemovepastafewmillionyears,allbetsareoff.Thesubtlegravitationalperturbationsamongallthebodiesinthesolarsystemaccumulateuntilwecannolongerforecastthesystemaccurately.TheexistenceofthepredictabilityhorizonemergedfromPoincaré’swork.

Beforehim,itwasthoughtthaterrorswouldgrowonlylinearlyintime,notexponentially;ifyoudoubledthetime,there’dbedoubletheerror.Withalineargrowthoferrors,improvingthemeasurementscouldalwayskeeppacewiththedesireforlongerprediction.Butwhenerrorsgrowexponentiallyfast,asystemissaidtohavesensitivedependenceonitsinitialconditions.Thenlong-termpredictionbecomesimpossible.Thisisthephilosophicallydisturbingmessageofchaos.It’simportanttounderstandwhat’snewaboutthis.Peoplealwaysknewthat

bigcomplexsystemsliketheweatherwerehardtopredict.Thesurprisewasthatsomethingassimpleasaspinningtoporthreegravitatingbodieswassimilarlyunpredictable.ThatwasashockerandanotherblowtoLaplace’snaiveconflationofdeterminismwithpredictability.Onthepositiveside,vestigesoforderexistwithinchaoticsystemsbecauseof

theirdeterministiccharacter.Poincarédevelopednewmethodsforanalyzingnonlinearsystems,includingchaoticones,andfoundwaystoextractsomeoftheorderhiddenwithinthem.Insteadofformulasandalgebra,heusedpicturesandgeometry.Hisqualitativeapproachhelpedsowtheseedsforthemodernmathematicalfieldsoftopologyanddynamicalsystems.Wenowhaveamuchbetterunderstandingoforderandchaosbecauseofhisseminalwork.

Poincaré’sVisualApproach

TogiveanexampleofhowPoincaré’sapproachworks,considertheoscillationsofasimplependulumofthesortthatGalileostudied.UsingNewton’slawofmotionandtakingnoteoftheforcesthatapendulumexperiencesasitswings,

wecandrawanabstractpictureshowinghowthependulumchangesitsangleandvelocityfrommomenttomoment.ThatpictureisessentiallyavisualtranslationofwhatNewton’slawsays.Thereisnonewcontentinthepicturebeyondwhat’salreadyinthedifferentialequation.It’sjustanotherwayoflookingatthesameinformation.Thepicturelookslikeamapofaweatherpatterntravelingacrossthe

countryside.Onsuchmaps,weseearrowsshowingthelocaldirectionofpropagation,whichwaytheweatherfrontwillmoveinstantbyinstant.Thisisthesamekindofinformationthatadifferentialequationprovides.It’salsothesamekindofinformationgivenindanceinstructions:putyourleftfoothere,putyourrightfootthere.Suchamapiscalledagraphofavectorfield.Thelittlearrowsonitarevectorsshowingthatiftheangleandvelocityofthependulumarecurrentlyhere,thisiswheretheyshouldgoamomentlater.Thevector-fieldpictureforthependulumlookslikethis:

Beforeweinterpretthepicture,pleaseunderstandthatitisabstractinthe

sensethatit’snotshowingarealisticportraitofapendulum.Thepatternofswirlingarrowsdoesnotresembleaweighthangingfromastring.It’snotwhat

aphotographofapendulumwouldlooklike.(Cartoonsofsuchsnapshotsareshownbelowthevector-fieldpicturetogiveyouafeelingforwhatitmeans.)Insteadofarealisticdepictionofthependulum,thevector-fieldpictureshowsanabstractmapofhowthestateofthependulumchangesfromonemomenttothenext.Eachpointonthemaprepresentsapossiblecombinationofthependulum’sangleandvelocityataninstant.Thehorizontalaxisrepresentsthependulum’sangle.Theverticalaxisrepresentsitsvelocity.Atanymoment,aknowledgeofthosetwonumbers,angleandvelocity,definethedynamicalstateofthependulum.Theyprovidetheinformationweneedtopredictwhattheangleandvelocityofthependulumwillbeamomentlater,andthenamomentafterthat,andsoon.Allweneedtodoisfollowthearrows.Theswirlingarrangementofthearrowsnearthecentercorrespondstoa

simpleback-and-forthmotionofthependulumwhenitishangingnearlystraightdown.Thewavystructureofthearrowsonthetopandbottomcorrespondtoapendulumrotatingvigorouslyoverthetop,whirlinglikeapropeller.Newtonneverconsideredsuchwhirlingmotions;neitherdidGalileo.Theywereoutsidetherealmofwhatcouldbecalculatedwithclassicalmethods.YetwhirlingmotionsareplaintoseeonPoincaré’spicture.Thisqualitativewayoflookingatdifferentialequationsisnowastapleineveryfieldwherenonlineardynamicsarise,fromlaserphysicstoneuroscience.

NonlinearityGoestoWar

Nonlineardynamicscanbeintenselypractical.InthehandsoftheBritishmathematiciansMaryCartwrightandJohnLittlewood,Poincaré’stechniquescontributedtothewartimedefenseofBritainagainstNaziairraids.In1938,theBritishgovernment’sDepartmentofScientificandIndustrialResearchaskedtheLondonMathematicalSocietyforhelpwithaproblemrelatedtotop-secretdevelopmentsinradiodetectionandranging,thetechnologyknowntodayasradar.Britishgovernmentengineersworkingontheprojecthadbeenperplexedbynoisy,erraticoscillationstheywereobservingintheiramplifiers,especiallywhenthedevicesweredrivenbyhigh-power,high-frequencyradiowaves.Theyfearedthatsomethingmightbewrongwiththeirequipment.Thegovernment’scallforhelpcaughtCartwright’sattention.Shehadalready

beenstudyingmodelsofoscillatingsystemsgovernedbysimilar“veryobjectionable-lookingdifferentialequations,”asshelaterdescribedthem.SheandLittlewoodwentontodiscoverthesourceoftheerraticoscillationsinthe

radarelectronics.Theamplifierswerenonlinear,andtheycouldrespondchaoticallyiftheyweredriventoofastandtoohard.Decadeslater,thephysicistFreemanDysonrecalledhearingCartwright

lectureonherworkin1942.Hewrote:

ThewholedevelopmentofradarinWorldWarIIdependedonhighpoweramplifiers,anditwasamatteroflifeanddeathtohaveamplifiersthatdidwhattheyweresupposedtodo.Thesoldierswereplaguedwithamplifiersthatmisbehaved,andblamedthemanufacturersfortheirerraticbehaviour.CartwrightandLittlewooddiscoveredthatthemanufacturerswerenottoblame.Theequationitselfwastoblame.

TheinsightsofCartwrightandLittlewoodenabledthegovernment’s

engineerstoworkaroundtheproblembyoperatingtheamplifiersinregimeswheretheybehavedmorepredictably.Cartwrightwascharacteristicallymodestabouthercontribution.WhenshereadwhatDysonhadwrittenaboutherwork,shescoldedhimformakingtoomuchofit.DameMaryCartwrightpassedawayin1998attheageofninety-seven.She

wasthefirstfemalemathematicianelectedtotheRoyalSociety.Sheleftstrictinstructionsthatnoeulogiesweretobegivenathermemorialservice.

TheAllianceBetweenCalculusandComputers

Theneedtosolvedifferentialequationsinwartimespurredthedevelopmentofcomputers.Mechanicalandelectronicbrains,astheyweresometimescalledinthosedays,couldbeusedtocalculatethetrajectoriesofrocketsandcannonshellsunderrealisticconditionsbyaccountingforcomplicationslikeairresistanceandwinddirection.Suchinformationwasneededbyartilleryofficersinthefieldtohelpthemhittheirtargets.Allthenecessaryballisticdatawerecomputedaheadoftimeandcompiledinstandardtablesandcharts.High-speedcomputerswereessentialforthistask.Inamathematicalsimulation,thecomputerscouldinchanidealizedcannonshellforwardonitsflightpath,onesmallstepatatime,usingtheappropriatedifferentialequationtoupdatetheshell’spositionandvelocitybyonesmallincrementafteranother,proceedingtothesolutionbybruteforcethroughanenormousnumberofadditions.Onlyamachinecouldchugforwardrelentlesslyandperformallthenecessaryadditionsandmultiplicationsquickly,correctly,andtirelessly.

Thelegacyofcalculusinthisendeavorisevidentinthenamesofsomeoftheearliestcomputers.OnewasamechanicaldevicecalledtheDifferentialAnalyzer.Itsjobwastosolvethedifferentialequationsneededtocomputeartillery-firingtables.AnotherwascalledENIAC,for“ElectronicNumericalIntegratorandComputer.”Herethewordintegratorwasusedinthecalculussense,asindoingintegralsorintegratingadifferentialequation.Completedin1945,ENIACwasoneofthefirstreprogrammable,general-purposecomputers.Alongwithcomputingfiringtables,italsoassessedthetechnicalfeasibilityofahydrogenbomb.Althoughmilitaryapplicationsofcalculusandnonlineardynamicsstimulated

thedevelopmentofcomputers,manypeacetimeuseswerefoundforboththemathandthemachines.Inthe1950sscientistsbegantousethemtosolveproblemsarisingintheirowndisciplines,outsideofphysics.Forexample,theBritishbiologistsAlanHodgkinandAndrewHuxleyneededcomputerstohelpthemunderstandhownervecellstalkedtooneanotherand,morespecifically,howelectricalsignalstraveledalongnervefibers.Theyperformedpainstakingexperimentstocalculatetheflowofsodiumandpotassiumionsacrossthemembraneofaverybigandexperimentallyconvenientkindofnervefiber—thegiantaxonofasquid—andworkedoutempiricallyhowthoseflowsdependedonthevoltageacrossthemembraneandhowthevoltagewasalteredbytheflowingions.Butwhattheywerenotabletodowithoutacomputerwascalculatethespeedandshapeofaneuralimpulseasittraveleddownanaxon.Calculatingitsmotionrequiredsolvinganonlinearpartialdifferentialequationforthevoltageasafunctionoftimeandspace.AndrewHuxleysolveditoverthecourseofthreeweeksonahand-crankedmechanicalcalculator.In1963,HodgkinandHuxleysharedaNobelPrizefortheirdiscoveriesabout

theionicbasisofhownervecellswork.Theirapproachhasbeenabiginspirationtoallthoseinterestedinapplyingmathematicstobiology.Thisissuretobeagrowthareafortheapplicationsofcalculus.Mathematicalbiologyisano-holds-barredexerciseinnonlineardifferentialequations.WiththehelpofNewton-styleanalyticalmethods,Poincaré-stylegeometricmethods,andanunabashedrelianceoncomputers,mathematicalbiologistsarelookingforandstartingtomakeheadwayonthedifferentialequationsthatgovernheartrhythms,thespreadofepidemics,thefunctioningoftheimmunesystem,theorchestrationofgenes,thedevelopmentofcancer,andmanyothermysteriesoflife.Wecouldn’tdoanyofitwithoutcalculus.

ComplexSystemsandtheCurseofHighDimensions

ThemostseriouslimitationofPoincaré’sapproachhastodowiththehumanbrain,whichcan’timaginespaceshavingmorethanthreedimensions.Naturalselectionhastunedournervoussystemstoperceiveupanddown,frontandback,andleftandright,thethreedirectionsofordinaryspace.Tryaswemight,wecan’tpictureafourthdimension,notinthesenseofseeingitinthemind’seye.Withabstractsymbols,however,wecantrytodealwithanynumberofdimensions.FermatandDescartesshowedushow.Theirxyplanetaughtusthatnumberscouldbeattachedtodimensions.Leftandrightcorrespondedtothenumberx.Upanddowncorrespondedtothenumbery.Byincludingmorenumbers,wecouldincludemoredimensions.Forthreedimensions,x,y,andzsufficed.Whynothavefourdimensions,orfive?Therewerestillplentyoflettersleft.Youmayhaveheardthattimeisthefourthdimension.Indeed,inEinstein’s

specialandgeneraltheoriesofrelativity,spaceandtimearefusedintoasingleentity,space-time,andrepresentedinafour-dimensionalmathematicalarena.Roughlyspeaking,ordinaryspacegetsplottedonthefirstthreeaxesandtimegetsplottedonthefourth.Thisconstructioncanbeviewedasageneralizationofthetwo-dimensionalxyplaneofFermatandDescartes.Butwearenottalkingaboutspace-timehere.Thelimitationinherentin

Poincaré’sapproachinvolvesamuchmoreabstractarena.It’sageneralizationoftheabstractstatespacewemetwhenwelookedatthevectorfieldforapendulum.Inthatexample,weconstructedanabstractspacewithoneaxisforthependulum’sangleandanotherforitsvelocity.Ateachinstant,theangleandvelocityoftheswingingpendulumhadcertainvalues;hence,atthatinstant,theycorrespondedtoasinglepointintheangle-velocityplane.Thearrowsonthatplane(theonesthatlookedlikedanceinstructions)dictatedhowthestatechangedfrominstanttoinstant,asdeterminedbyNewton’sdifferentialequationforthependulum.Byfollowingthearrows,wecouldforecasthowthependulumwouldmove.Dependingonwhereitstarted,itcouldoscillatebackandforthoritcouldwhirloverthetop.Allofthatwascontainedinthepicture.Thekeythingtorealizeisthatthependulum’sstatespacehadtwodimensions

becausetwovariables—thependulum’sangleanditsvelocity—werenecessaryandsufficienttopredictitsfuture.Theygaveusexactlytheinformationweneededtopredictitsangleandvelocityaninstantlater,andaninstantafterthat,

onandonintothefuture.Inthatsense,thependulumisaninherentlytwo-dimensionalsystem.Ithasatwo-dimensionalstatespace.Thecurseofhighdimensionsariseswhenweconsidersystemsmore

complicatedthanapendulum.Forexample,let’staketheproblemthatgaveNewtonaheadache,theproblemofthreemutuallygravitatingbodies.Itsstatespacehaseighteendimensions.Toseewhy,concentrateononeofthebodies.Atanyinstant,itislocatedsomewhereinordinarythree-dimensionalphysicalspace.Itslocationcanthereforebespecifiedbythreenumbers:x,y,z.Itcanalsomoveineachofthosethreedirections,correspondingtothreevelocities.Soasinglebodyrequiressixpiecesofinformation:threecoordinatesforitslocationplusthreeforitsvelocityinthedifferentdirections.Thosesixnumbersspecifywhereitisandhowit’smoving.Multiplythatsixbyeachofthethreebodiesintheproblemandnowyouhave6×3=18dimensionsinstatespace.Thus,inPoincaré’sapproach,thechangingstateofasystemofthreemutuallygravitatingbodiesisrepresentedbyasingleabstractpointmovingaroundinaneighteen-dimensionalspace.Astimepasses,theabstractpointtracesoutatrajectory,analogoustothetrajectoryofarealcometoracannonball,exceptthisabstracttrajectorylivesinPoincaré’sfantasticarena,theeighteen-dimensionalstatespaceofthethree-bodyproblem.Whenweapplynonlineardynamicstobiology,weoftenfinditnecessaryto

imagineevenhigher-dimensionalspaces.Forexample,inneuroscienceweneedtokeeptrackofallthechangingconcentrationsofsodium,potassium,calcium,chloride,andotherionsinvolvedinthenerve-membraneequationsofHodgkinandHuxley.Modernversionsoftheirequationscaninvolvehundredsofvariables.Thosevariablesrepresentthechangingconcentrationsofionsinthenervecell,thechangingvoltageacrossthecellmembrane,andthemembrane’schangingabilitytoconductthevariousionsandallowthemtopassintothecelloroutofit.Theabstractstatespaceinthiscasehashundredsofdimensions,oneforeachvariable—oneforpotassiumconcentration,anotherforsodiumconcentration,athirdforvoltage,afourthforsodiumconductance,afifthforpotassiumconductance,andsoon.Atanygiveninstant,allthosevariablestakecertainvalues.TheHodgkin-Huxleyequations(ortheirgeneralizations)givethevariablestheirdanceinstructionsandtellthemhowtomoveontheirtrajectories.Inthisway,thedynamicsofnervecells,braincells,andheartcellscanbepredicted,sometimeswithsurprisingaccuracy,withthehelpofcomputerstostepthetrajectoriesforwardthoughstatespace.Thefruitsofthisapproacharebeingusedtostudyneuralpathologiesandcardiacarrhythmiasandtodesignbetterdefibrillators.

Today,mathematiciansregularlythinkaboutabstractspaceshavingarbitrarynumbersofdimensions.Wespeakaboutn-dimensionalspace,andwehavedevelopedgeometryandcalculusinanynumberofdimensions.Aswesawinchapter10,AllanCormack,theinventorofthetheorybehindCTscanning,wonderedhowCTwouldworkinfourdimensions,purelyoutofintellectualcuriosity.Greatthingshavecomefromthisspiritofpureadventure.WhenEinsteinneededfour-dimensionalgeometryforcurvedspaceandtimeingeneralrelativity,hewaspleasedtolearnitalreadyexisted,thankstoBernhardRiemann,whohadcreateditdecadesearlierforthepurestofmathematicalreasons.Sothereisalottobesaidforfollowingone’scuriosityinmathematics.It

oftenhasscientificandpracticalpayoffsthatcan’tbeforeseen.Italsogivesmathematiciansgreatpleasureforitsownsakeandrevealshiddenconnectionsbetweendifferentpartsofmathematics.Forallthesereasons,thepursuitofhigher-dimensionalspaceshasbeenavigorouspartofmathematicsforthepasttwohundredyears.However,althoughwehaveanabstractsystemfordoingmathinhigh-

dimensionalspaces,mathematiciansstillhavetroublevisualizingthem.Actually,letmebemorefrank—wecan’tvisualizethem.Ourbrainsjustaren’tuptoit.Wearen’twiredthatway.ThatcognitivelimitationdealsaseriousblowtoPoincaré’sprogram,atleast

indimensionshigherthanthree.Hisapproachtononlineardynamicsdependsonvisualintuition.Ifwecan’tpicturewhat’sgoingtohappeninfouroreighteenorahundreddimensions,hisapproachcan’thelpusallthatmuch.Thishasbecomeabigobstacletoprogressinthefieldofcomplexsystems,wherehigh-dimensionalspacesareexactlywhatweneedtounderstandifwewanttomakesenseofthethousandsofbiochemicalreactionstakingplaceinahealthylivingcellorexplainhowtheygoawryincancer.Ifwearetohaveanyhopeofmakingsenseofcellbiologyusingdifferentialequations,weneedtobeabletosolvethoseequationswithformulas(whichSofiaKovalevskayashowedwecannot)orpicturethem(whichourlimitedbrainswon’tallow).Sothemathematicsofcomplexnonlinearsystemsisdiscouraging.Itseems

likeitwillalwaysbehard,ifnotimpossible,foranyonetomakeheadwayonthemostdifficultproblemsofourtime,fromthebehaviorofeconomies,societies,andcellstotheworkingsoftheimmunesystem,genes,brains,andconsciousness.Afurtherdifficultyisthatwedon’tevenknowifsomeofthosesystems

harborpatternsakintothoseuncoveredbyKeplerandGalileo.Nervecellsapparentlydo,butwhatabouteconomiesorsocieties?Inmanyfields,human

understandingisstillinthepre-Galilean,pre-Keplerianphase.Wehaven’tfoundthepatterns.Sohowcanwefinddeepertheoriesthatwouldgiveinsightintothosepatterns?BiologyandpsychologyandeconomicsarenotNewtonianyet,becausetheyaren’tevenGalileanandKeplerian.Wehavealongwaytogo.

Computers,ArtificialIntelligence,andtheMysteryofInsight

Atthispoint,thecomputertriumphalistsdemandtobeheard.Withcomputers,theysay,withartificialintelligence,alloftheseproblemswillfall.Andthatmaywellbetrue.Computershavelonghelpedusinthestudyofdifferentialequations,nonlineardynamics,andcomplexsystems.WhenHodgkinandHuxleyopenedthedoorinthe1950stounderstandinghownervecellswork,theysolvedtheirpartialdifferentialequationsonahand-crankedmachine.WhenengineersatBoeingdesignedthe787Dreamlinerin2011,theyusedsupercomputerstocalculatetheliftanddragontheplaneandfigureouthowtopreventunwantedvibrationsofitswings.Computersbeganascalculatingmachines—literally,compute-ers—butthey

arenowmuchmorethanthat.Theyhaveachievedartificialintelligenceofasort.Forexample,GoogleTranslatenowdoesasurprisinglygoodjobofprovidingidiomatictranslations.AndtherearemedicalAIsystemsthatdiagnosediseasesmoreaccuratelythanthebesthumanexperts.Still,Idon’tbelieveanyonewouldsaythatGoogleTranslatehasinsightinto

languagesorthatmedicalAIsystemsunderstanddiseases.Couldcomputerseverbeinsightful?Ifso,couldtheysharetheirinsightswithusaboutthingswereallycareabout,likecomplexsystems,whicharecentraltomostofthegreatestunsolvedproblemsofscience?Toexplorethecaseforandagainstthepossibilityofcomputerinsight,

considerhowcomputerchesshasevolved.In1997,IBM’schess-playingprogramDeepBluemanagedtobeatthereigninghumanworldchesschampion,GarryKasparov,inasix-gamematch.Althoughunexpectedatthetime,therewasnogreatmysteryinthisachievement.Themachinecouldevaluatetwohundredmillionpositionspersecond.Itdidn’thaveinsight,butithadrawspeed,itnevergottired,itneverblunderedinacalculation,anditneverforgotwhatitwasthinkingaminuteago.Still,itplayedlikeacomputer,mechanicallyandmaterialistically.ItcouldoutcomputeKasparovbutitcouldn’toutthinkhim.The

currentgenerationoftheworld’sstrongestchessprograms,withintimidatingnameslikeStockfishandKomodo,stillplayinthisinhumanstyle.Theyliketocapturematerial.Theydefendlikeiron.Butalthoughtheyarefarstrongerthananyhumanplayer,theyarenotcreativeorinsightful.Allthatchangedwiththeriseofmachinelearning.OnDecember5,2017,the

DeepMindteamatGooglestunnedthechessworldwithitsannouncementofadeep-learningprogramcalledAlphaZero.Theprogramtaughtitselfchessbyplayingmillionsofgamesagainstitselfandlearningfromitsmistakes.Inamatterofhours,itbecamethebestchessplayerinhistory.Notonlycoulditeasilydefeatallthebesthumanmasters(itdidn’tevenbothertotry),itcrushedthereigningcomputerworldchampionofchess.Inahundred-gamematchagainstStockfish,atrulyformidableprogram,AlphaZeroscoredtwenty-eightwinsandseventy-twodraws.Itdidn’tloseasinglegame.ThescariestpointisthatAlphaZeroshowedinsight.Itplayedlikeno

computereverhas,intuitivelyandbeautifully,witharomantic,attackingstyle.Itplayedgambitsandtookrisks.InsomegamesitparalyzedStockfishandtoyedwithit.Itseemedmalevolentandsadistic.Anditwascreativebeyondwords,playingmovesnograndmasterorcomputerwouldeverdreamofmaking.Ithadthespiritofahumanandthepowerofamachine.Itwashumankind’sfirstglimpseofaterrifyingnewkindofintelligence.SupposewecouldunleashAlphaZeroorsomethinglikeit—let’scallit

AlphaInfinity—onthegreatestunsolvedproblemsintheoreticalscience,problemsofimmunologyandcancerbiologyandconsciousness.Tocontinuethefantasy,supposethatGalileanandKeplerianpatternsexistinthesephenomenaandareripeforthepicking,butonlybyanintelligencefarsuperiortoours.Assumingthatsuchlawsexist,wouldthissuperhumanintelligencebeabletoworkthemout?Idon’tknow.Nooneknows.Anditallmaybemoot,becausesuchlawsmaynotevenexist.Butiftheydo,andifAlphaInfinitycouldfindthem,itwouldseemlikean

oracletous.We’dsitatitsfeetandlistentoit.Wewouldn’tunderstandwhyitwasalwaysrightorevenwhatitwassaying,butwecouldcheckitscalculationsagainstexperimentsorobservations,anditwouldseemtoknoweverything.Wewouldbereducedtospectators,gapinginwonderandconfusion.Evenifitcouldexplainitself,wewouldn’tbeabletofollowitsreasoning.Atthatmoment,theageofinsightthatbeganwithNewtonwouldcometoaclose,atleastforhumanity,andanewageofinsightwouldbegin.Sciencefiction?Perhaps.ButIthinkascenariolikethisisnotoutofthe

question.Inpartsofmathematicsandscience,wearealreadyexperiencingtheduskofinsight.Therearetheoremsthathavebeenprovedbycomputers,yetno

humanbeingcanunderstandtheproof.Thetheoremsarecorrectbutwehavenoinsightintowhy.Andatthispoint,themachinescannotexplainthemselves.Considerthefamouslong-standingmathproblemcalledthefour-colormap

theorem.Itsaysthatundercertainreasonableconstraints,anymapofcontiguouscountriescanalwaysbecoloredwithjustfourcolorssuchthatnotwoneighboringcountriesarecoloredthesame.(LookatatypicalmapofEuropeorAfricaoranyothercontinentbesidesAustraliaandyou’llseewhatImean.)Thefour-colortheoremwasprovedin1977withthehelpofacomputer,butnohumanbeingcouldcheckallthestepsintheargument.Althoughtheproofhasbeenvalidatedandsimplifiedsincethen,therearepartsofitthatunavoidablyentailbrute-forcecomputation,likethewaycomputersusedtoplaychessbeforeAlphaZero.Whenthisproofcameout,manyworkingmathematicianswerecrankyaboutit.Theyalreadybelievedthefour-colortheorem.Theydidnotneedanyassurancethatitwastrue.Theywantedtounderstandwhyitwastrue,andthisproofdidn’thelp.Likewise,considerafour-hundred-year-oldgeometryproblemposedby

JohannesKepler.Itasksforthedensestwaytopackequal-sizespheresinthreedimensions,akintotheproblemfacedbygrocerswhentheypackorangesinacrate.Woulditbemostefficienttostackthespheresinidenticallayers,onedirectlyontopofanother?Orwoulditbebettertostaggerthelayerssothateachspherenestlesinthehollowformedbyfourothersbeneathit,thesamewaygrocersstackoranges?Ifso,isthatpackingthebestpossibleone,orcouldsomeother,possiblyirregular,packingarrangementbedenser?Kepler’sconjecturewasthatthegrocers’packingisthebest.Butthiswasn’tproveduntil1998.ThomasHales,withthehelpofhisstudentSamuelFergusonand180,000linesofcomputercode,reducedthecalculationtoalargebutfinitenumberofcases.Then,withthehelpofbrute-forcecomputationandingeniousalgorithms,hisprogramverifiedtheconjecture.Themathematicalcommunityshrugged.WenowknowthattheKeplerconjectureistruebutwestilldon’tunderstandwhy.Wedon’thaveinsight.NorcouldHales’scomputerexplainittous.ButwhataboutwhenweunleashAlphaInfinityonsuchproblems?Amachine

likethatwouldcomeupwithbeautifulproofs,asbeautifulasthechessgamesthatAlphaZeroplayedagainstStockfish.Itsproofswouldbeintuitiveandelegant.Theywouldbe,inthewordsoftheHungarianmathematicianPaulErdős,proofsstraightfromtheBook.ErdősimaginedthatGodkeptabookwithallthebestproofsinit.SayingthataproofwasstraightfromtheBookwasthehighestpossiblepraise.Itmeantthattheproofrevealedwhyatheoremwastrueanddidn’tmerelybludgeonthereaderintoacceptingitwithsomeugly,difficultargument.Icanimagineaday,nottoofarinthefuture,whenartificial

intelligencewillgiveusproofsfromtheBook.Whatwillcalculusbelikethen,andwhatwillmedicinebelike,andsociology,andpolitics?

Conclusion

Bywieldinginfinityinjusttherightway,calculuscanunlockthesecretsoftheuniverse.We’veseenthathappenagainandagain,butitstillseemsalmostmiraculous.Asystemofreasoninghumansinventedissomehowintunewiththeharmonyofnature.It’sreliablenotjustatthescaleswhereitwasinvented—attheeverydayscalesofordinarylife,withitsspinningtopsanditsbowlsofsoup—butalsoatthesmallestscalesofatomsandatthegrandestscalesofthecosmos.Soitcan’tjustbeatrickofcircularreasoning.It’snotthatwe’restuffingthingsintocalculusthatwealreadyknow,andcalculusishandingthembacktous;calculustellsusaboutthingswe’veneverseen,nevercouldsee,andneverwillsee.Insomecases,ittellsusaboutthingsthatneverexistedbutcould—ifonlywehadthewittoconjurethem.This,tome,isthegreatestmysteryofall:Whyistheuniverse

comprehensible,andwhyiscalculusinsyncwithit?Ihavenoanswer,butIhopeyou’llagreeit’sworthcontemplating.Inthatspirit,letmetakeyoutotheTwilightZoneforthreefinalexamplesoftheeerieeffectivenessofcalculus.

EightDecimalPlaces

Thefirstexampletakesusbacktowherewestarted,withRichardFeynman’squipthatcalculusisthelanguageGodtalks.TheexampleisrelatedtoFeynman’sownworkonanextensionofquantummechanicscalledquantumelectrodynamics,orQEDforshort.QEDisthequantumtheoryofhowlightandmatterinteract.ItmergesMaxwell’stheoryofelectricityandmagnetismwithHeisenberg’sandSchrödinger’squantumtheoryandEinstein’sspecialtheoryofrelativity.FeynmanwasoneoftheprincipalarchitectsofQED,andafterlookingatthestructureofhistheory,Icanseewhyhehadsuchadmirationforcalculus.Histheoryischock-fullofit,bothintacticsandinstyle.It’steemingwithpowerseries,integrals,anddifferentialequationsandincludesplentyofhijinkswithinfinity.Moreimportant,it’sthemostaccuratetheoryanyonehaseverdevised...

aboutanything.Withthehelpofcomputers,physicistsarestillbusysummingtheseriesthatariseinQED,usingwhatareknownasFeynmandiagrams,tomakepredictionsaboutthepropertiesofelectronsandotherparticles.Bycomparingthosepredictionstoextremelypreciseexperimentalmeasurements,they’veshownthatthetheoryagreeswithrealitytoeightdecimalplaces,betterthanonepartinahundredmillion.Thisisafancywayofsayingthatthetheoryisessentiallyright.It’salways

hardtofindhelpfulanalogiestomakesenseofsuchbignumbers,butletmetryputtingitlikethis:ahundredmillionsecondsequals3.17years,sogettingsomethingrighttowithinonepartinahundredmillionislikeplanningtosnapyourfingersexactly3.17yearsfromnowandtimingitrighttothenearestsecond—withoutthehelpofaclockoranalarm.There’ssomethingastonishingaboutthis,philosophicallyspeaking.The

differentialequationsandintegralsofquantumelectrodynamicsarecreationsofthehumanmind.Theyarebasedonexperimentsandobservations,certainly,sotheyhaverealitybuiltintothemtothatextent.Yettheyareproductsoftheimaginationnonetheless.Theyarenotslavishimitationsofreality.Theyareinventions.AndwhatissoastonishingisthatbymakingcertainscribblesonpaperanddoingcertaincalculationswithmethodsanalogoustothosedevelopedbyNewtonandLeibnizbutsoupedupforthetwenty-firstcentury,wecanpredictnature’sinnermostpropertiesandgetthemrighttoeightdecimalplaces.Nothingthathumanityhaseverpredictedisasaccurateasthepredictionsofquantumelectrodynamics.Ithinkthisisworthmentioningbecauseitputsthelietothelineyou

sometimeshear,thatscienceislikefaithandotherbeliefsystems,thatithasnospecialclaimontruth.Comeon.Anytheorythatagreestoonepartinahundredmillionisnotjustamatteroffaithorsomebody’sopinion.Itdidn’thaveto

matchtoeightdecimalplaces.Plentyoftheoriesinphysicshaveturnedouttobewrong.Notthisone.Notyet,atleast.Nodoubtit’salittlebitoff,aseverytheoryalwaysis,butitsurecomesclosetothetruth.

SummoningthePositron

Thesecondexampleoftheeerieeffectivenessofcalculushastodowithanearlierextensionofquantummechanics.In1928,theBritishphysicistPaulDiractriedtofindawaytoreconcileEinstein’sspecialtheoryofrelativitywiththegoverningprinciplesofquantummechanicsasappliedtoanelectronmovingnearthespeedoflight.Hecameupwithatheorythatstruckhimasbeautiful.Hechoseitlargelyonaestheticgrounds.Hehadnoparticularempiricalevidenceforthetheory,justanartisticsensethatitsbeautywasasignofitscorrectness.Thoseconstraintsalone—compatibilitywithrelativityandquantummechanicsalongwithmathematicalelegance—tiedhishandstoalargeextent.Afterstrugglingwithvarioustheories,hefoundonethatmatchedallhisaestheticdesiderata.Thetheory,inotherwords,wasguidedbyaquestforharmony.Andlikeanygoodscientist,Diracsoughttotesthistheorybyextractingpredictionsfromit.Forhim,asatheoreticalphysicist,thatmeantusingcalculus.Whenhesolvedhisdifferentialequation,nowknownastheDiracequation,

andkeptanalyzingitoverthenextfewyears,hefoundthatitmadeseveralstartlingpredictions.Onewasthatantimattershouldexist.Thereshouldbe,inotherwords,aparticleequivalenttoanelectronbutwithapositivecharge.Atfirsthethoughtthatparticlemightbeaproton,butaprotonhadtoomuchmass;theparticlehepredictedwasabouttwothousandtimessmallerthanaproton.Nosuchpositivelychargedparticlethatwispyhadeverbeenseen.Yethisequationwaspredictingit.Diraccalleditananti-electron.In1931hepublishedapaperinwhichhepredictedthatwhenthisstill-unobservedparticlecollidedwithanelectron,thetwowouldannihilateeachother.“Thisnewdevelopmentrequiresnochangewhateverintheformalismwhenexpressedintermsofabstractsymbols,”hewrote,andheaddeddryly,“UnderthesecircumstancesonewouldbesurprisedifNaturehadmadenouseofit.”Thenextyear,anexperimentalphysicistnamedCarlAndersonsawanodd

trackinhiscloudchamberwhenhewasstudyingcosmicrays.Somesortofparticlewascoilinglikeanelectronbutcurvingintheoppositedirection,asifithadapositivecharge.HewasunawareofDirac’sprediction,buthegotthegistofwhathewasseeing.WhenAndersonpublishedapaperaboutitin1932,his

editorsuggestedhecallitapositron.Thenamestuck.DiracwonaNobelPrizeforhisequationthenextyear;Andersonwonforthepositronin1936.Intheyearssincethen,positronshavebeenputtoworksavinglives.They

underliePETscans(PETstandsforpositronemissiontomography),aformofmedicalimagingthatallowsdoctorstoseeregionsofabnormalmetabolicactivityinsofttissuesinthebrainorotherorgans.Inanoninvasivefashionthatrequiresnosurgeryorotherdangerousintrusionsintotheskull,PETscanscanhelplocatebraintumorsanddetecttheamyloidplaquesassociatedwithAlzheimer’sdisease.Sohereisanothersterlingexampleofcalculusasthehandmaidento

somethingmarvelouslypracticalandimportant.Becausecalculusisthelanguageoftheuniverseaswellasthelogicalengineforextractingitssecrets,Diracwasabletowritedownadifferentialequationfortheelectronthattoldhimsomethingnewandtrueandbeautifulaboutnature.Itledhimtoconjureupanewparticleandrealizethatitoughttoexist.Logicandbeautydemandedit.Butnotontheirown—theyhadtoalignwithknownfactsandmeshwithknowntheories.Whenallofthatwasstirredintothepot,itwasalmostasifthesymbolsthemselvesbroughtthepositronintoexistence.

TheMysteryofaComprehensibleUniverse

Forourthirdexampleoftheeerieeffectivenessofcalculus,itseemsappropriatetoendourjourneyinthecompanyofAlbertEinstein.Heembodiedsomanyofthethemeswe’vetouchedon:areverencefortheharmonyofnature,aconvictionthatmathematicsisatriumphoftheimagination,asenseofwonderatthecomprehensibilityoftheuniverse.Nowherearethesethemesmoreclearlyvisiblethaninhisgeneraltheoryof

relativity.Inthistheory,hismagnumopus,EinsteinoverturnedNewton’sconceptionsofspaceandtimeandredefinedtherelationshipbetweenmatterandgravity.ToEinstein,gravitywasnolongeraforceactinginstantaneouslyatadistance.Instead,itwasanalmostpalpablething,awarpinthefabricoftheuniverse,amanifestationofthecurvatureofspaceandtime.Curvature—anideathatgoesbacktothebirthofcalculus,totheancientfascinationwithcurvedlinesandcurvedsurfaces—inEinstein’shandsbecameapropertynotjustofshapesbutofspaceitself.It’sasifthexyplaneofFermatandDescartestookonalifeofitsown.Insteadofbeinganarenaforthedrama,spacebecameanactorinitsownright.InEinstein’stheory,mattertellsspace-timehowtocurve,while

curvaturetellsmatterhowtomove.Thedancebetweenthemmakesthetheorynonlinear.Andweknowwhatthatmeans:Understandingwhattheequationsimplyis

boundtobedifficult.Tothisday,thenonlinearequationsofgeneralrelativityconcealmanysecrets.Einsteinwasabletoexcavatesomeofthemthroughhismathematicalskillanddoggedness.Hepredicted,forexample,thatstarlightwouldbendasitpassedaroundthesunonitswaytoourplanet,apredictionthatwasconfirmedduringasolareclipsein1919andthatmadeEinsteinaninternationalsensation,front-pagenewsintheNewYorkTimes.Thetheoryalsopredictedthatgravitycouldhaveastrangeeffectontime:The

passageoftimecouldspeeduporslowdownasanobjectmovesthroughagravitationalfield.Bizarreasthissounds,itreallydoesoccur.ItneedstobetakenintoaccountinthesatellitesoftheglobalpositioningsystemastheymovehighabovetheEarth.Thegravitationalfieldisweakerupthere,whichreducesthecurvatureofspace-timeandcausesclockstorunfasterthantheydoontheground.Withoutcorrectingforthiseffect,theclocksaboardtheGPSsatelliteswouldn’tkeepaccuratetime.They’dgetaheadofground-basedclocksbyabout45microsecondsperday.Thatmaynotsoundlikemuch,butkeepinmindthatthewholeglobalpositioningsystemrequiresnanosecondaccuracytoworkproperly,and45microsecondsis45,000nanoseconds.Withoutthecorrectionforgeneralrelativity,errorsinglobalpositionswouldaccumulateatabouttenkilometerseachday,andthewholesystemwouldbecomeworthlessfornavigationinamatterofminutes.Thedifferentialequationsofgeneralrelativitymakeseveralotherpredictions,

suchastheexpansionoftheuniverseandtheexistenceofblackholes.Allseemedoutlandishwhentheywerepredicted,yetallhaveturnedouttobetrue.The2017NobelPrizeinPhysicswasawardedforthedetectionofanother

outrageouseffectpredictedbygeneralrelativity:gravitationalwaves.Thetheoryshowedthatapairofblackholesrotatingaroundeachotherwouldswirlthespace-timearoundthem,stretchingitandsqueezingitrhythmically.Theresultingdisturbanceinthefabricofspace-timewaspredictedtospreadoutwardlikearipplemovingatthespeedoflight.Einsteindoubtedthatitwouldeverbepossibletomeasuresuchawave;heworrieditmightbeamathematicalillusion.TheachievementoftheteamthatwontheNobelPrizewastodesignandbuildthemostsensitivedetectorevermade.OnSeptember14,2015,theirapparatusdetectedaspace-timetremorathousandtimessmallerthanthediameterofaproton.Forcomparison,that’sliketweakingthedistancetotheneareststarbythewidthofahumanhair.

It’saclearwinternightasIwritetheselastwords.I’vesteppedouttolookatthesky.Withthestarsupaboveandtheblacknessofspace,Ican’tavoidfeelingawe.Howcouldwe,Homosapiens,aninsignificantspeciesonaninsignificant

planetadriftinamiddleweightgalaxy,havemanagedtopredicthowspaceandtimewouldtrembleaftertwoblackholescollidedinthevastnessoftheuniverseabillionlight-yearsaway?Weknewwhatthatwaveshouldsoundlikebeforeitgothere.And,courtesyofcalculus,computers,andEinstein,wewereright.Thatgravitationalwavewasthefaintestwhispereverheard.Thatsoftlittle

wavehadbeenheadedourwayfrombeforewewereprimates,beforeweweremammals,fromatimeinourmicrobialpast.Whenitarrivedthatdayin2015,becausewewerelistening—andbecauseweknewcalculus—weunderstoodwhatthesoftwhispermeant.

Acknowledgments

Writingaboutcalculusforthegeneralpublichasbeenawonderfulchallengeandalotoffun.I’vebeeninlovewithcalculuseversinceIfirstlearneditinhighschoolandhavelongdreamedofsharingthatlovewithawidereadership,butIsomehownevermanagedtogetaroundtoit.Somethingwouldalwayscomeup.Therewereresearchpaperstowrite,gradstudentstomentor,classestoprepare,kidstoraise,andadogtowalk.Then,abouttwoyearsago,itdawnedonmethatmyage(likeyours,Ibet)wasincreasingatarateofoneyearperyear,andsoitseemedlikethatwasasgoodatimeasanytotrytosharethejoyofcalculuswitheveryone.Myfirstacknowledgmentisthereforetoyou,dearreader.I’vebeenimaginingyoufordecades.Thanksforbeingherenow.Asitturnedout,writingthebookIalwaysmeanttowritehasbeenharderthan

expected.Thisshouldn’thavebeenasurprise,butitwas.I’vebeenimmersedincalculusforsolongthatit’sbeenhardtoseeitthroughtheeyesofanewcomer.Fortunately,I’vebeenhelpedbysomeverysmart,generous,andpatientpeoplewhodidn’thavethefoggiestideawhatcalculuswasorwhyitmatteredandwhocertainlydidn’tspendeverywakingminutethinkingaboutmaththewaymycolleaguesandIdo.Thankyoutomyliteraryagent,KatinkaMatson.Alongtimeago,whenI

offhandedlymentionedthatcalculuswasoneofthegreatestideasanyoneeverhad,yousaidthatwasabookyou’dliketoread.Well,hereitis.Thankyousomuchforbelievinginmeandinthisproject.I’vebeenblessedtoworkwithtwobrillianteditors,EamonDolanandAlex

Littlefield.Eamon,Ican’tbegintothankyouenough.Workingwithyouwasfantastic,frombeginningtoend.YouwerethereaderIalwayshadinmind:whip-smart,alittlebitskeptical,curious,andeagertobethrilled.Bestofall,youfoundthestructureinthestorybeforeIdidandguidedmewithafirmbutgentlehand.Iforgiveyouforaskingmetododraftafterdraft,becauseyoumadethebookbettereverytime.Truly,Icouldn’thavedoneitwithoutyou.Alex,thankyouforshepherdingthismanuscripttothefinishlineandforbeingsuchapleasuretoworkwithineveryway.Speakingofpleasure,whatatreatitistobecopyeditedbyTracyRoe.Tracy,

italmostmakesmewanttowriteanotherbook,justforthegood-naturededucationyougivemeeverytimeweworktogether.EditorialassistantRosemaryMcGuinness,thankyouforyourcheerfulness,

efficiency,andattentiontodetail.AndthankstoeveryoneatHoughtonMifflinHarcourtforallyourhardworkandforbeingsuchgreatteamplayers.Ifeelluckytoworkwithyouall.MargyNelsondidtheillustrationsforthisbook,justasshehasformyothers.

Thanksasalwaysforyoursenseofwhimsyandcollaborativespirit.I’mgratefultomycolleaguesMichaelBarany,BillDunham,PaulGinsparg,

andManilSuri,whokindlyreadsectionsofthebookorentiredrafts,improvedmyphrasing,correctedmyerrors(whoknewthatthereweretwoMercators?),andofferedhelpfulsuggestionsinthejoviallynitpickingfashionthateveryacademichopesfor.Michael,IlearnedsomuchfromyourcommentsandwishI’dshownyouthebookearlier.Bill,youareahero.Paul,youarewhatyoualwaysare(andthebestatit).Manil,thankyouforreadingmyfirstdraftsocarefullyandbestofluckwithyournewbook,whichIcan’twaittoreadinprint.TomGilovich,HerbertHui,andLindaWoodard:Thankyouforbeingsuch

goodfriends.Youletmeblabberonaboutthebookforclosetotwoyearsasitwashatchingandneverwaveredinyourencouragementor,asfarasIcouldtell,yourattention.AlanPerelsonandJohnStillwell:Iadmireyourworkenormouslyandfeelhonoredthatyousharedyourthoughtswithmeaboutthisbook.Thankyou,too,toRodrigoTetsuoArgenton,TonyDeRose,PeterSchröder,TunçTezel,andStefanZachow,whoallowedmetodiscusstheirresearchandreproducetheirpublishedillustrations.ToMurray:You’veheardmesayitamilliontimes,andeventhoughyou

don’tunderstandwhatitmeans,Iknowyougetthedrift.Who’sagoodboy?Youare.Finally,thankyoutomywife,Carole,anddaughters,JoandLeah,forallyour

loveandsupportandforputtingupwithmydistractedair,whichmusthavebeenevenmoreannoyingthanusual.Zeno’sparadoxaboutwalkinghalfwaytothewalltookonnewmeaninginourhouseholdwhenitseemedlikethisprojectwasapproachingcompletionbutneverquitegettingthere.Iamsogratefultoyouallforyourpatienceandloveyouverymuch.

StevenStrogatzIthaca,NewYork

IllustrationCredits

page51 PeterSchröderpage52 EntertainmentPictures/Alamypage53 EntertainmentPictures/Alamypage54 StefanZachow,ZuseInstituteBerlin(ZIB)page56 StefanZachow,ZuseInstituteBerlin(ZIB)page61 TunçTezelpage160 WENNLtd/Alamypage182 ReproducedbykindpermissionoftheSyndicsofCambridgeUniversityLibrary.

MS-ADD-04000–000–00259.tif(MSAdd.4000,page124r).page240 ©ThomasJeffersonFoundationatMonticellopage260 ImagereproducedbykindpermissionofRodrigoTetsuoArgenton

Notes

Introduction

vii“It’sthelanguageGodtalks”:Wouk,TheLanguageGodTalks,5.viiuniverseisdeeplymathematical:Forphysicsperspectives,seeBarrowandTipler,AnthropicCosmologicalPrinciple;Rees,JustSixNumbers;Davies,TheGoldilocksEnigma;Livio,IsGodaMathematician?;Tegmark,OurMathematicalUniverse;andCarroll,TheBigPicture.Foraphilosophyperspective,seeSimonFriederich,“Fine-Tuning,”StanfordEncyclopediaofPhilosophy,https://plato.stanford.edu/archives/spr2018/entries/fine-tuning/.

viiianswertotheultimatequestionoflife,theuniverse,andeverything:Adams,Hitchhiker’sGuide,andGill,DouglasAdams’AmazinglyAccurateAnswer.

ix“amathematicalignoramuslikeme”:Wouk,TheLanguageGodTalks,6.xtellitdifferently:Forhistoricaltreatments,seeBoyer,TheHistoryoftheCalculus,andGrattan-Guinness,FromtheCalculus.Dunham,TheCalculusGallery;Edwards,TheHistoricalDevelopment;andSimmons,CalculusGems,tellthestoryofcalculusbywalkingusthroughsomeofitsmostbeautifulproblemsandsolutions.

xTobeanappliedmathematician:Stewart,InPursuitoftheUnknown;Highametal.,ThePrincetonCompanion;andGoriely,AppliedMathematics,conveythespirit,breadth,andvitalityofappliedmathematics.

xpristine,hermeticallysealedworld:Kline,MathematicsinWesternCulture,andNewman,TheWorldofMathematics,connectmathtothewiderculture.Ispentmanyhoursinhighschoolreadingthesetwomasterpieces.

xielectricityandmagnetism:Forthemathematicsandphysics,seeMaxwell,“OnPhysicalLinesofForce,”andPurcell,ElectricityandMagnetism.Forconceptsandhistory,seeKline,MathematicsinWesternCulture,304–21;Schaffer,“TheLairdofPhysics”;andStewart,InPursuitoftheUnknown,chapter11.ForabiographyofMaxwellandFaraday,seeForbesandMahon,Faraday,Maxwell.

xiwaveequation:Stewart,InPursuitoftheUnknown,chapter8.xiii“Theeternalmysteryoftheworld”:Einstein,PhysicsandReality,51.Thisaphorismisoftenrephrasedas“Themostincomprehensiblethingaboutthe

universeisthatitiscomprehensible.”ForfurtherexamplesofEinsteinquotesbothrealandimaginary,seeCalaprice,TheUltimateQuotableEinstein,andRobinson,“EinsteinSaidThat.”

xiii“UnreasonableEffectivenessofMathematics”:Wigner,“TheUnreasonableEffectiveness”;Hamming,“TheUnreasonableEffectiveness”;andLivio,IsGodaMathematician?

xiiiPythagoras:Asimov,Asimov’sBiographicalEncyclopedia,4–5;Burkert,LoreandScience;Guthrie,PythagoreanSourcebook;andC.Huffman,“Pythagoras,”https://plato.stanford.edu/archives/sum2014/entries/pythagoras/.Martínez,inCultofPythagorasandScienceSecrets,debunksmanyofthemythsaboutPythagoraswithalighttouchanddevastatinghumor.

xivthePythagoreans:Katz,HistoryofMathematics,48–51,andBurton,HistoryofMathematics,section3.2,discussPythagoreanmathematicsandphilosophy.

xxiistimulatedemission:Ball,“ACenturyAgoEinsteinSparked,”andPais,SubtleIstheLord.TheoriginalpaperisEinstein,“ZurQuantentheoriederStrahlung.”

1.Infinity

1beginningsofmathematics:Burton,HistoryofMathematics,andKatz,HistoryofMathematics,providegentleyetauthoritativeintroductionstothehistoryofmathematicsfromancienttimestothetwentiethcentury.Atamoreadvancedmathematicallevel,Stillwell,MathematicsandItsHistory,isexcellent.Forawide-ranginghumanistictreatmentwithahealthydoseofcrotchetyopinionthrownin,Kline,MathematicsinWesternCulture,isdelightful.

3anoutgrowthofgeometry:Seesection4.5ofBurton,HistoryofMathematics;chapters2and3inKatz,HistoryofMathematics;andchapter4inStillwell,MathematicsandItsHistory.

4areaofacircle:Katz,HistoryofMathematics,section1.5,discussesancientestimatesoftheareaofacirclemadebyvariousculturesaroundtheworld.ThefirstproofoftheformulawasgivenbyArchimedesusingthemethodofexhaustion;seeDunham,JourneyThroughGenius,chapter4,andHeath,TheWorksofArchimedes,91–93.

15Aristotle:HenryMendell,“AristotleandMathematics,”StanfordEncyclopediaofPhilosophy,

https://plato.stanford.edu/archives/spr2017/entries/aristotle-mathematics/.15completedinfinity:Katz,HistoryofMathematics,56,andStillwell,MathematicsandItsHistory,54,discussAristotle’sdistinctionbetweencompleted(oractual)infinityandpotentialinfinity.

16GiordanoBruno:Drawingonnewevidence,Martínez,BurnedAlive,arguesthatBrunowasexecutedforhiscosmology,nothistheology.AlsoseeA.A.Martínez,“WasGiordanoBrunoBurnedattheStakeforBelievinginExoplanets?,”ScientificAmerican(2018),https://blogs.scientificamerican.com/observations/was-giordano-bruno-burned-at-the-stake-for-believing-in-exoplanets/.SeealsoD.Knox,“GiordanoBruno,”StanfordEncyclopediaofPhilosophy,https://plato.stanford.edu/entries/bruno/.

16immeasurablysubtleandprofound:Russell’sessayonZenoandinfinityis“MathematicsandtheMetaphysicians,”reprintedinNewman,TheWorldofMathematics,vol.3,1576–90.

17Zeno’sparadoxes:Mazur,Zeno’sParadox.SeealsoBurton,HistoryofMathematics,101–2;Katz,HistoryofMathematics,section2.3.3;Stillwell,MathematicsandItsHistory,54;JohnPalmer,“ZenoofElea,”StanfordEncyclopediaofPhilosophy,https://plato.stanford.edu/archives/spr2017/entries/zeno-elea/;andNickHuggett,“Zeno’sParadoxes,”StanfordEncyclopediaofPhilosophy,https://plato.stanford.edu/entries/paradox-zeno/.

21Quantummechanics:Greene,TheElegantUniverse,chapters4and5.22Schrödinger’sequation:Stewart,InPursuitoftheUnknown,chapter14.23Plancklength:Greene,TheElegantUniverse,127–31,explainswhyphysicistsbelievethatspacedissolvesintoquantumfoamattheultramicroscopicscaleofthePlancklength.Forphilosophy,seeS.WeinsteinandD.Rickles,“QuantumGravity,”StanfordEncyclopediaofPhilosophy,https://plato.stanford.edu/entries/quantum-gravity/.

2.TheManWhoHarnessedInfinity

27Archimedes:Forhislife,seeNetzandNoel,TheArchimedesCodex,andC.Rorres,“Archimedes,”https://www.math.nyu.edu/~crorres/Archimedes/contents.html.Forascholarlybiography,seeM.Clagett,“Archimedes,”inGillispie,CompleteDictionary,vol.1,withamendmentsbyF.Acerbiinvol.19.For

Archimedes’smathematics,Stein,Archimedes,andEdwards,TheHistoricalDevelopment,chapter2,arebothoutstanding,butseealsoKatz,HistoryofMathematics,sections3.1–3.3,andBurton,HistoryofMathematics,section4.5.AscholarlycollectionofArchimedes’sworkisHeath,TheWorksofArchimedes.

27storiesabouthim:Martínez,CultofPythagoras,chapter4,tracestheevolutionofthemanylegendsaboutArchimedes,includingthecomicalEurekataleandthetragicstoryofArchimedes’sdeathatthehandsofaRomansoldierduringthesiegeofSyracusein212BCE.WhileitseemslikelythatArchimedeswaskilledduringthesiege,there’snoreasontobelievehisfinalwordswere“Don’tdisturbmycircles!”

27Plutarch:ThePlutarchquotesarefromJohnDryden’stranslationofPlutarch’sMarcellus,availableonlineathttp://classics.mit.edu/Plutarch/marcellu.html.ThespecificpassagesaboutArchimedesandthesiegeofSyracusearealsoavailableathttps://www.math.nyu.edu/~crorres/Archimedes/Siege/Plutarch.html.

27“madehimforgethisfood”:http://classics.mit.edu/Plutarch/marcellu.html.27“carriedbyabsoluteviolencetobathe”:Ibid.27Vitruvius:TheEurekastory,asfirsttoldbyVitruvius,isavailableinLatinandEnglishathttps://www.math.nyu.edu/~crorres/Archimedes/Crown/Vitruvius.html.Thatsitealsoincludesachildren’sversionofthestorybytheacclaimedwriterJamesBaldwin,takenfromThirtyMoreFamousStoriesRetold(NewYork:AmericanBookCompany,1905).Unfortunately,BaldwinandVitruviusoversimplifyArchimedes’ssolutiontotheproblemoftheking’sgoldencrown.Rorresoffersamoreplausibleaccountathttps://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html,alongwithGalileo’sguessregardinghowArchimedesmighthavesolvedit(https://www.math.nyu.edu/~crorres/Archimedes/Crown/bilancetta.html).

28“Ashipwasfrequentlyliftedup”:http://classics.mit.edu/Plutarch/marcellu.html.

29estimatepi:Stein,Archimedes,chapter11,showsindetailhowArchimedesdidit.Bepreparedforsomehairyarithmetic.

33existenceofirrationalnumbers:Noonereallyknowswhofirstprovedthatthesquarerootof2isirrationalor,equivalently,thatthediagonalofasquareisincommensurablewithitsside.There’sanirresistibleoldyarnthataPythagoreannamedHippasuswasdrownedatseaforit.Martínez,CultofPythagoras,chapter2,tracksdowntheoriginofthismythanddebunksit.SodoestheAmericanfilmmakerErrolMorrisinalongandwonderfullyquirky

essayintheNewYorkTimes;seeErrolMorris,“TheAshtray:HippasusofMetapontum(Part3),”NewYorkTimes,March8,2001,https://opinionator.blogs.nytimes.com/2011/03/08/the-ashtray-hippasus-of-metapontum-part-3/.

35QuadratureoftheParabola:AtranslationofArchimedes’soriginaltextisinHeath,TheWorksofArchimedes,233–52.ForthedetailsIglossedoverinthetriangular-shardargument,seeEdwards,TheHistoricalDevelopment,35–39;Stein,Archimedes,chapter7;LaubenbacherandPengelley,MathematicalExpeditions,section3.2;andStillwell,MathematicsandItsHistory,section4.4.Therearealsomanytreatmentsavailableontheinternet.OneoftheclearestisbyMarkReederathttps://www2.bc.edu/mark-reeder/1103quadparab.pdf.AnotherisbyR.A.G.Seelyathttp://www.math.mcgill.ca/rags/JAC/NYB/exhaustion2.pdf.Asanalternative,Simmons,CalculusGems,sectionB.3,usesananalyticgeometryapproachthatyoumayfindeasiertofollow.

40“Whenyouhaveeliminatedtheimpossible”:ArthurConanDoyle,TheSignoftheFour(London:SpencerBlackett,1890),https://www.gutenberg.org/files/2097/2097-h/2097-h.htm.

42TheMethod:Fortheoriginaltext,seeHeath,TheWorksofArchimedes,326andfollowing.FortheapplicationoftheMethodtothequadratureoftheparabola,seeLaubenbacherandPengelley,MathematicalExpeditions,section3.3,andNetzandNoel,TheArchimedesCodex,150–57.FortheapplicationoftheMethodtoseveralotherproblemsaboutareas,volumes,andcentersofgravity,seeStein,Archimedes,chapter5,andEdwards,TheHistoricalDevelopment,68–74.

42“doesnotfurnishanactualdemonstration”:QuotedinStein,Archimedes,33.

42“theoremswhichhavenotyetfallentoourshare”:QuotedinNetzandNoel,TheArchimedesCodex,66–67.

47“madeupofalltheparallellines”:Heath,TheWorksofArchimedes,17.47“drawninsidethecurve”:Dijksterhuis,Archimedes,317.Dijksterhuisargues,asIhavehere,thattheMethodairedsomedirtylaundry.Itrevealedthattheuseofcompletedinfinity“hadonlybeenbanishedfromthepublishedtreatises,”butthatdidn’tstopArchimedesfromusingitinprivate.AsDijksterhuisputit,“Intheworkshopoftheproducingmathematician,”argumentsbasedoncompletedinfinity“heldundiminishedsway.”

47“asortofindication”:Heath,TheWorksofArchimedes,17.48volumeofasphere:Stein,Archimedes,39–41.49“inherentinthefigures”:Heath,TheWorksofArchimedes,1.

50ArchimedesPalimpsest:SeeNetzandNoel,TheArchimedesCodex;theauthorstellthestoryofthelostmanuscriptanditsrediscoverywithgreatpanache.TherewasalsoaterrificNovaepisodeaboutit,andtheaccompanyingwebsiteofferstimelines,interviews,andinteractivetools;seehttp://www.pbs.org/wgbh/nova/archimedes/.SeealsoStein,Archimedes,chapter4.

50Archimedes’slegacy:Rorres,ArchimedesintheTwenty-FirstCentury.50computer-animatedmovies:Forthemathbehindcomputer-generatedmoviesandvideo,seeMcAdamsetal.,“CrashingWaves.”

50triangulationsofamannequin’shead:ZorinandSchröder,“SubdivisionforModeling,”18.

51Shrek:DreamWorks,“WhyComputerAnimationLooksSoDarnReal,”July9,2012,https://mashable.com/2012/07/09/animation-history-tech/#uYHyf6hO.Zq3.

51forty-fivemillionpolygons:Shrek,productioninformation,http://cinema.com/articles/463/shrek-production-information.phtml.

51Avatar:“NVIDIACollaborateswithWetatoAccelerateVisualEffectsforAvatar,”http://www.nvidia.com/object/wetadigital_avatar.html,andBarbaraRobertson,“HowWetaDigitalHandledAvatar,”StudioDaily,January5,2010,http://www.studiodaily.com/2010/01/how-weta-digital-handled-avatar/.

51firstmovietousepolygonsbythebillions:“NVIDIACollaborateswithWeta.”

51ToyStory:BurrSnider,“TheToyStoryStory,”Wired,December1,1995,https://www.wired.com/1995/12/toy-story/.

51“morePhDsworkingonthisfilm”:Ibid.51Geri’sGame:IanFailes,“‘Geri’sGame’Turns20:DirectorJanPinkavaReflectsontheGame-ChangingPixarShort,”November25,2017,https://www.cartoonbrew.com/cgi/geris-game-turns-20-director-jan-pinkava-reflects-game-changing-pixar-short-154646.html.ThemovieisonYouTubeathttps://www.youtube.com/watch?v=gLQG3sORAJQ(originalsoundtrack)andhttps://www.youtube.com/watch?v=9IYRC7g2ICg(modifiedsoundtrack).

52subdivisionprocess:DeRoseetal.,“SubdivisionSurfaces.”ExploresubdivisionsurfacesforcomputeranimationinteractivelyatKhanAcademyincollaborationwithPixarathttps://www.khanacademy.org/partner-content/pixar/modeling-character.Studentsandtheirteachersmightalsoenjoytryingtheotherlessonsofferedin“PixarinaBox,”a“behind-the-sceneslookathowPixarartistsdotheirjobs,”athttps://www.khanacademy.org/partner-

content/pixar.It’sagreatwaytoseehowmathisbeingusedtomakemoviesthesedays.

53doublechin:DreamWorks,“WhyComputerAnimationLooksSoDarnReal.”

53facialsurgery:Deuflhardetal.,“MathematicsinFacialSurgery”;Zachowetal.,“Computer-AssistedPlanning”;andZachow,“ComputationalPlanning.”

56Archimedeanscrew:Rorres,ArchimedesintheTwenty-FirstCentury,chapter6,andhttps://www.math.nyu.edu/~crorres/Archimedes/Screw/Applications.html.

57Archimedeswassilent:Infairness,Archimedesdiddoonestudyrelatedtomotion,thoughitwasanartificialformofmotionmotivatedbymathematicsratherthanphysics.Seehisessay“OnSpirals,”reproducedinHeath,TheWorksofArchimedes,151–88.HereArchimedesanticipatedthemodernideasofpolarcoordinatesandparametricequationsforapointmovinginaplane.Specifically,heconsideredapointmovinguniformlyintheradialdirectionawayfromtheoriginatthesametimeastheradialrayrotateduniformly,andheshowedthatthetrajectoryofthemovingpointisthecurvenowknownasanArchimedeanspiral.Then,bysumming12+22+...+n2andapplyingthemethodofexhaustion,hefoundtheareaboundedbyoneloopofthespiralandtheradialray.SeeStein,Archimedes,chapter9;Edwards,TheHistoricalDevelopment,54–62;andKatz,HistoryofMathematics,114–15.

3.DiscoveringtheLawsofMotion

60“thisgrandbook”:Galileo,TheAssayer(1623).SelectionstranslatedbyStillmanDrake,DiscoveriesandOpinionsofGalileo(NewYork:Doubleday,1957),237–38,https://www.princeton.edu/~hos/h291/assayer.htm.

60“coeternalwiththedivinemind”:JohannesKepler,TheHarmonyoftheWorld,translatedbyE.J.Aiton,A.M.Duncan,andJ.V.Field,MemoirsoftheAmericanPhilosophicalSociety209(1997):304.

60“suppliedGodwithpatterns”:Ibid.60Platohadtaught:Plato,Republic(Hertfordshire:Wordsworth,1997),240.60Aristotelianteaching:Asimov,Asimov’sBiographicalEncyclopedia,17–20.61retrogrademotion:Katz,HistoryofMathematics,406.62Aristarchus:Asimov,Asimov’sBiographicalEncyclopedia,24–25,andJamesEvans,“AristarchusofSamos,”EncyclopediaBritannica,

https://www.britannica.com/biography/Aristarchus-of-Samos.63Archimedeshimselfrealized:Evans,“AristarchusofSamos.”63Ptolemaicsystem:Katz,HistoryofMathematics,145–57.64GiordanoBruno:Martínez,BurnedAlive.64GalileoGalilei:TheGalileoProject,http://galileo.rice.edu/galileo.html,isanexcellentonlineresourceforGalileo’slifeandwork.FermiandBernardini,GalileoandtheScientificRevolution,originallypublishedin1961,isadelightfulbiographyofGalileoforgeneralreaders.Asimov’sBiographicalEncyclopedia,91–96,isagoodquickintroductiontoGalileo,andsoisKline,MathematicsinWesternCulture,182–95.Forascholarlytreatment,seeDrake,GalileoatWork,andMicheleCamerota,“Galilei,Galileo,”inGillispie,CompleteDictionary,96–103.

64MarinaGamba:http://galileo.rice.edu/fam/marina.html.64washisfavorite:Sobel,Galileo’sDaughter.SisterMariaCeleste’sletterstoherfatherareathttp://galileo.rice.edu/fam/daughter.html#letters.

65TwoNewSciences:Thebookisavailablefreeonlineathttp://oll.libertyfund.org/titles/galilei-dialogues-concerning-two-new-sciences.

66proposedthatheavyobjectsfall:Kline,MathematicsinWesternCulture,188–90.

67“one-tenthofapulse-beat”:Galileo,Discourses,179,http://oll.libertyfund.org/titles/753#Galileo_0416_607.

67“sameratioastheoddnumbersbeginningwithunity”:Ibid.,190,http://oll.libertyfund.org/titles/753#Galileo_0416_516.

69“verystraight,smooth,andpolished”:Ibid.,178,http://oll.libertyfund.org/titles/753#Galileo_0416_607.

70“asbigasaship’scable”:Ibid.,109,http://oll.libertyfund.org/titles/753#Galileo_0416_242.

71chandelierswayingoverhead:FermiandBernardini,GalileoandtheScientificRevolution,17–20,andKline,MathematicsinWesternCulture,182.

72“ThousandsoftimesIhaveobserved”:Galileo,Discourses,140,http://oll.libertyfund.org/titles/753#Galileo_0416_338.

72“thelengthsaretoeachotherasthesquares”:Ibid.,139,http://oll.libertyfund.org/titles/753#Galileo_0416_335.

73“mayappeartomanyexceedinglyarid”:Ibid.,138,http://oll.libertyfund.org/titles/753#Galileo_0416_329.

74Josephsonjunction:Strogatz,Sync,chapter5,andRichardNewrock,“WhatAreJosephsonJunctions?HowDoTheyWork?,”ScientificAmerican,https://www.scientificamerican.com/article/what-are-josephson-juncti/.

74longitudeproblem:Sobel,Longitude.

75globalpositioningsystem:Thompson,“GlobalPositioningSystem,”andhttps://www.gps.gov.

78JohannesKepler:ForKepler’slifeandwork,seeOwenGingerich,“JohannesKepler,”inGillispie,CompleteDictionary,vol.7,onlineathttps://www.encyclopedia.com/people/science-and-technology/astronomy-biographies/johannes-kepler#kjen14,withamendmentsbyJ.R.Voelkelinvol.22.SeealsoKline,MathematicsinWesternCulture,110–25;Edwards,TheHistoricalDevelopment,99–103;Asimov,Asimov’sBiographicalEncyclopedia,96–99;Simmons,CalculusGems,69–83;andBurton,HistoryofMathematics,355–60.

78“criminallyinclined”:QuotedinGingerich,“JohannesKepler,”https://www.encyclopedia.com/people/science-and-technology/astronomy-biographies/johannes-kepler#kjen14.

78“bad-tempered”:Ibid.78“suchasuperiorandmagnificentmind”:Ibid.79“DayandnightIwasconsumedbythecomputing”:Ibid.80“Godisbeingcelebratedinastronomy”:Ibid.81“thistediousprocedure”:KeplerinAstronomiaNova,quotedbyOwenGingerich,TheBookNobodyRead:ChasingtheRevolutionsofNicolausCopernicus(NewYork:Penguin,2005),48.

84“sacredfrenzy”:QuotedinGingerich,“JohannesKepler,”https://www.encyclopedia.com/people/science-and-technology/astronomy-biographies/johannes-kepler#kjen14.

85“MydearKepler,Iwishwecouldlaugh”:QuotedinMartínez,ScienceSecrets,34.

86“JohannesKeplerbecameenamored”:Koestler,TheSleepwalkers,33.

4.TheDawnofDifferentialCalculus

90China,India,andtheIslamicworld:Katz,“IdeasofCalculus”;Katz,HistoryofMathematics,chapters6and7;andBurton,HistoryofMathematics,238–85.

91Al-HasanIbnal-Haytham:Katz,“IdeasofCalculus,”andJ.J.O’ConnorandE.F.Robertson,“AbuAlial-Hasanibnal-Haytham,”http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Haytham.html.

92FrançoisViète:Katz,HistoryofMathematics,369–75.92decimalfractions:Ibid.,375–78.

93EvangelistaTorricelliandBonaventuraCavalieri:Alexander,Infinitesimal,discussestheirbattleswiththeJesuitsoverinfinitesimals,whichwereseenasdangerousreligiously,notjustmathematically.

99RenéDescartes:Forhislife,seeClarke,Descartes;Simmons,CalculusGems,84–92;andAsimov,Asimov’sBiographicalEncyclopedia,106–8.Forsummariesofhismathandphysicsintendedforgeneralreaders,seeKline,MathematicsinWesternCulture,159–81;Edwards,TheHistoricalDevelopment;Katz,HistoryofMathematics,sections11.1and12.1;andBurton,HistoryofMathematics,section8.2.Forascholarlyhistoricaltreatmentofhisworkinmathematicsandphysics,seeMichaelS.Mahoney,“Descartes:MathematicsandPhysics,”inGillispie,CompleteDictionary,alsoonlineatEncyclopediaBritannica,https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/descartes-mathematics-and-physics.

99“Whattheancientshavetaughtusissoscanty”:RenéDescartes,LesPassionsdel’Ame(1649),quotedinGuicciardini,IsaacNewton,31.

100“thecountryofbears,amidrocksandice”:HenryWoodhead,MemoirsofChristina,QueenofSweden(London:HurstandBlackett,1863),285.

100PierredeFermat:Mahoney,MathematicalCareer,isthedefinitivetreatment.Simmons,CalculusGems,96–105,isbriskandentertainingaboutFermat(justastheauthorwaswitheverythinghewrote;ifyouhaven’treadSimmons,youmust).

100FermatandDescarteslockedhorns:Mahoney,MathematicalCareer,chapter4.

101triedtoruinhisreputation:Ibid.,171.101Fermatcameupwiththemfirst:IagreewiththeassessmentinSimmons,CalculusGems,98,abouthowthecreditforanalyticgeometryshouldbeapportioned:“SuperficiallyDescartes’sessaylooksasifitmightbeanalyticgeometry,butisn’t;whileFermat’sdoesn’tlookit,butis.”Formoreeven-handedviews,seeKatz,HistoryofMathematics,432–42,andEdwards,TheHistoricalDevelopment,95–97.

101findingamethodofanalysis:Guicciardini,IsaacNewton,andKatz,HistoryofMathematics,368–69.

102“lowcunning,deplorableindeed”:Descartes,rule4inRulesfortheDirectionoftheMind(1629),asquotedinKatz,HistoryofMathematics,368–69.

102“analysisofthebunglersinmathematics”:QuotedinGuicciardini,IsaacNewton,77.

103optimizationproblems:Mahoney,MathematicalCareer,199–201,discussesFermat’sworkonthemaximizationproblemconsideredinthemaintext.

106adequality:Ibid.,162–65,andKatz,HistoryofMathematics,470–72.107JPEG:Austin,“WhatIs...JPEG?,”andHighametal.,ThePrincetonCompanion,813–16.

108howdaylengthvaries:Timeanddate.comwillgiveyoutheinformationforanylocationofinterest.

112sinewavescalledwavelets:Foraclearintroductiontowaveletsandtheirmanyapplications,seeDanaMackenzie,“Wavelets:SeeingtheForestandtheTrees,”inBeyondDiscovery:ThePathfromResearchtoHumanBenefit,aprojectoftheNationalAcademyofSciences;gotohttp://www.nasonline.org/publications/beyond-discovery/wavelets.pdf.ThentryKaiser,FriendlyGuide,Cipra,“Parlez-VousWavelets?,”orGoriely,AppliedMathematics,chapter6.Daubechies,TenLectures,wasalandmarkseriesoflecturesonwaveletmathematicsbyapioneerinthefield.

112FederalBureauofInvestigationusedwavelets:Bradleyetal.,“FBIWavelet/ScalarQuantization.”

113mathematiciansfromtheLosAlamosNationalLabteamedupwiththeFBI:BradleyandBrislawn,“TheWavelet/ScalarQuantization”;Brislawn,“FingerprintsGoDigital”;andhttps://www.nist.gov/itl/iad/image-group/wsq-bibliography.

115Snell’ssinelaw:Kwanetal.,“WhoReallyDiscoveredSnell’sLaw?,”andSabra,TheoriesofLight,99–105.

116principleofleasttime:Mahoney,MathematicalCareer,387–402.117“mynaturalinclinationtolaziness”:Ibid.,398.117“Icanscarcelyrecoverfrommyastonishment”:Ibid.,400(mytranslationofFermat’sFrench).

118principleofleastaction:Fermat’sprincipleofleasttimeanticipatedthemoregeneralprincipleofleastaction.Forentertaininganddeeplyenlighteningdiscussionsofthisprinciple,includingitsbasisinquantummechanics,seeR.P.Feynman,R.B.Leighton,andM.Sands,“ThePrincipleofLeastAction,”FeynmanLecturesonPhysics,vol.2,chapter19(Reading,MA:Addison-Wesley,1964),andFeynman,QED.

119Descarteshadhisownmethod:Katz,HistoryofMathematics,472–73.120“Ihavegivenageneralmethod”:QuotedinGrattan-Guinness,FromtheCalculus,16.

120“Idonotevenwanttonamehim”:QuotedinMahoney,MathematicalCareer,177.

121foundtheareaunderthecurve:Simmons,CalculusGems,240–41;andKatz,HistoryofMathematics,481–84.

121hisstudiesstillfellshort:Katz,HistoryofMathematics,485,explainswhyhefeelsFermatdoesnotdeservetobeconsideredaninventorofcalculus,andhemakesagoodcase.

5.TheCrossroads

131Logarithmswereinvented:Stewart,InPursuitoftheUnknown,chapter2,andKatz,HistoryofMathematics,section10.4.

137paintingsallegedlybyVermeer:Braun,DifferentialEquations,section1.3.

6.TheVocabularyofChange

159UsainBolt:Bolt,FasterthanLightning.160OnthatnightinBeijing:JonathanSnowden,“RememberingUsainBolt’s100mGoldin2008,”Bleacherreport.com(August19,2016),https://bleacherreport.com/articles/2657464-remembering-usain-bolts-100m-gold-in-2008-the-day-he-became-a-legend,andEriksenetal.,“HowFast.”Forlivevideoofhisastonishingperformance,seehttps://www.youtube.com/watch?v=qslbf8L9nl0andhttp://www.nbcolympics.com/video/gold-medal-rewind-usain-bolt-wins-100m-beijing.

160“That’sjustme”:Snowden,“RememberingUsainBolt’s.”163wewanttoconnectthedots:MyanalysisisbasedonthatinA.Oldknow,“AnalysingMen’s100mSprintTimeswithTI-Nspire,”https://rcuksportscience.wikispaces.com/file/view/Analysing+men+100m+Nspire.pdfThedetailsmaydifferslightlybetweenthetwostudiesbecauseweuseddifferentcurve-fittingprocedures,butourqualitativeconclusionsarethesame.

165researcherswereonhandwithlaserguns:GraubnerandNixdorf,“BiomechanicalAnalysis.”

166“Art,”saidPicasso:Thequoteisfrom“PicassoSpeaks,”TheArts(May1923),excerptedinhttp://www.gallerywalk.org/PM_Picasso.htmlfromAlfred

H.BarrJr.,Picasso:FiftyYearsofHisArt(NewYork:ArnoPress,1980).

7.TheSecretFountain

167IsaacNewton:Forbiographicalinformation,seeGleick,IsaacNewton.SeealsoWestfall,NeveratRest,andI.B.Cohen,“IsaacNewton,”invol.10ofGillispie,CompleteDictionary,withamendmentsbyG.E.SmithandW.Newmaninvol.23.ForNewton’smathematics,seeWhiteside,TheMathematicalPapers,vols.1and2;Edwards,TheHistoricalDevelopment;Grattan-Guinness,FromtheCalculus;Rickey,“IsaacNewton”;Dunham,JourneyThroughGenius;Katz,HistoryofMathematics;Guicciardini,ReadingthePrincipia;Dunham,TheCalculusGallery;Simmons,CalculusGems;Guicciardini,IsaacNewton;Stillwell,MathematicsandItsHistory;andBurton,HistoryofMathematics.

168“betweenstraightandcurvedlines”:RenéDescartes,TheGeometryofRenéDescartes:WithaFacsimileoftheFirstEdition,translatedbyDavidE.SmithandMarciaL.Latham(Mineola,NY:Dover,1954),91.Withintwentyyears,Descarteswasprovedwrongabouttheimpossibilityoffindingarclengthsexactlyforcurves;seeKatz,HistoryofMathematics,496–98.

169“Thereisnocurvedline”:I’veupdatedNewton’sspellinghereforeasierreading.Theoriginalwas“Thereisnocurvelineexprestbyanyæquation...butIcaninlessthenhalfaquarterofanhowertellwhetheritmaybesquared.”Letter193fromNewtontoCollins,November8,1676,inTurnbull,CorrespondenceofIsaacNewton,179.Theomittedmaterialinvolvestechnicalcaveatsabouttheclassoftrinomialequationstowhichhisclaimapplied.See“AManuscriptbyNewtononQuadratures,”manuscript192,inibid.,178.

169“thefountainIdrawitfrom”:Letter193fromNewtontoCollins,November8,1676,inibid.,180.Again,I’veupdatedthespelling;Newtonwrote“yefountain.”

169weren’tthefirsttonoticethistheorem:Katz,HistoryofMathematics,498–503,showsthatJamesGregoryandIsaacBarrowhadbothrelatedtheareaproblemtothetangentproblemandsohadanticipatedthefundamentaltheorembutconcludesthat“neitherofthesemenin1670couldmoldthesemethodsintoatruecomputationalandproblem-solvingtool.”Fiveyearsbeforethat,however,Newtonalreadyhad.Inasidebaronpage521,Katz

makesaconvincingcasethatNewtonandLeibniz(asopposedto“FermatorBarroworsomeoneelse”)deservecreditfortheinventionofcalculus.

173ScholarsintheMiddleAges:Katz,HistoryofMathematics,section8.4.182collegenotebook:YoucanexploreNewton’shandwrittencollegenotebookonline.Thepageshowninthemaintextishttp://cudl.lib.cam.ac.uk/view/MS-ADD-04000/260.

186IsaacNewtonwasborn:MyaccountofNewton’searlylifeisbasedonGleick,IsaacNewton.

188Newtonchanceduponsomethingmagical:Whiteside,TheMathematicalPapers,vol.1,96–142,andKatz,HistoryofMathematics,section12.5.EdwardsgivesafascinatingtreatmentofWallis’sworkoninterpolationandinfiniteproductsandshowshowNewton’sworkonpowerseriesarosefromhisattempttogeneralizeit;seeEdwards,TheHistoricalDevelopment,chapter7.WeknowwhenNewtonmadethesediscoveriesbecausehedatedtheminanentryonpage14vofhiscollegenotebook(onlineathttps://cudl.lib.cam.ac.uk/view/MS-ADD-04000/32).Newtonwrote,“Ifindthatinyeyear1664alittlebeforeChristmasI...borrowedWallis’works&byconsequencemadetheseAnnotations...inwinterbetweentheyears1664&1665.AtwchtimeIfoundthemethodofInfiniteseries.Andinsummer1665beingforcedfromCambridgebythePlagueIcomputedyeareaofyeHyperbola...totwo&fiftyfiguresbythesamemethod.”

190Hecookeditupbyanargument:Edwards,TheHistoricalDevelopment,178–87,andKatz,HistoryofMathematics,506–59,showthestepsinNewton’sthinkingashederivedhisresultsforpowerseries.

192“reallytoomuchdelightintheseinventions”:Letter188fromNewtontoOldenburg,October24,1676,inTurnbull,CorrespondenceofIsaacNewton,133.

193mathematiciansinKerala,India:Katz,“IdeasofCalculus”;Katz,HistoryofMathematics,494–96.

193“Bytheirhelpanalysisreaches”:Thislineappearsinthefamousepistolaprior,Newton’sreplytoLeibniz’sfirstinquiry,sentviaHenryOldenburgasintermediary;seeletter165fromNewtontoOldenburg,June13,1676,inTurnbull,CorrespondenceofIsaacNewton,39.

195“primeofmyageforinvention”:DraftletterfromNewtontoPierredesMaizeaux,writtenin1718,whenNewtonwasseekingtoestablishhispriorityoverLeibnizintheinventionofcalculus;availableonlineathttps://cudl.lib.cam.ac.uk/view/MS-ADD-03968/1349inthecollectionofCambridgeUniversityLibrary.Thefullquoteisbreathtaking:“Inthebeginningoftheyear1665IfoundtheMethodofapproximatingseries&the

RuleforreducinganydignityofanyBinomialintosuchaseries.ThesameyearinMayIfoundthemethodofTangentsofGregory&Slusius,&inNovemberhadthedirectmethodoffluxions&thenextyearinJanuaryhadtheTheoryofColours&inMayfollowingIhadentranceintoyeinversemethodoffluxions.AndthesameyearIbegantothinkofgravityextendingtoyeorboftheMoon&(havingfoundouthowtoestimatetheforcewithwhichagloberevolvingwithinaspherepressesthesurfaceofthesphere)fromKepler’sruleoftheperiodicaltimesofthePlanetsbeinginsesquialterate[three-halfpower]proportionoftheirdistancesfromthecentersoftheirOrbs,IdeducedthattheforceswhichkeepthePlanetsintheirOrbsmustbereciprocallyasthesquaresoftheirdistancesfromthecentersaboutwhichtheyrevolve:&therebycomparedtheforcerequisitetokeeptheMooninherOrbwiththeforceofgravityatthesurfaceoftheearth,&foundthemanswerprettynearly.Allthiswasinthetwoplagueyearsof1665and1666.ForinthosedaysIwasintheprimeofmyageforinvention&mindedMathematicks&Philosophymorethanatanytimesince.”

195“baitedbylittlesmatterersinmathematics”:QuotedinWhiteside,“TheMathematicalPrinciples,”referenceinhisref.2.

196ThomasHobbes:Alexander,Infinitesimal,tellsthestoryofHobbes’sfuriousbattleswithWallis,whichwereaspoliticalastheyweremathematical.Chapter7focusesonHobbesaswould-begeometer.

196a“scabofsymbols”:QuotedinStillwell,MathematicsandItsHistory,164.196“scurvybook”:Ibid.196not“worthyofpublicutterance”:QuotedinGuicciardini,IsaacNewton,343.

196“Ourspeciousalgebra”:Ibid.

8.FictionsoftheMind

199“HisnameisMr.Newton”:LetterfromIsaacBarrowtoJohnCollins,August20,1669,quotedinGleick,IsaacNewton,68.

199“sendmetheproof”:Letter158,fromLeibniztoOldenburg,May2,1676,inTurnbull,CorrespondenceofIsaacNewton,4.FormoreontheNewton-Leibnizcorrespondence,seeMackinnon,“Newton’sTeaser.”Guicciardini,IsaacNewton,354–61,offersaparticularlyclearandhelpfulanalysisofthemathematicalcat-and-mousegametakingplacebetweenNewtonandLeibniz

intheletters.TheoriginallettersappearinTurnbull,CorrespondenceofIsaacNewton;seeespeciallyletters158(Leibniz’sinitialinquirytoNewtonviaOldenburg),165(Newton’sepistolaprior,terseandintimidating),172(Leibniz’srequestforclarification),188(Newton’sepistolaposterior,gentlerandclearerbutstillintendedtoshowLeibnizwhowasboss),and209(Leibnizfightingback,thoughgraciously,andmakingitclearthatheknewcalculustoo).

200“distastefultome”:Oneofthebestzingersintheepistolaprior,letter165fromNewtontoOldenburg,June13,1676.SeeTurnbull,CorrespondenceofIsaacNewton,39.

200“verydistinguished”:Fromtheepistolaposterior,letter188fromNewtontoOldenburg,October24,1676,inibid.,130.

200“hopeforverygreatthingsfromhim”:Ibid.200“thesamegoalisapproached”:Ibid.201“Ihavepreferredtoconcealitthus”:Ibid.,134.TheencryptionencodesNewton’sunderstandingofthefundamentaltheoremandthecentralproblemsofcalculus:“givenanyequationinvolvinganynumberoffluentquantities,tofindthefluxions,andconversely.”Seealsopage153,note25.

201“inthetwinklingofaneyelid”:LetterfromLeibniztoMarquisdeL’Hospital,1694,excerptedinChild,EarlyMathematicalManuscripts,221.AlsoquotedinEdwards,TheHistoricalDevelopment,244.

201“burdenedwithadeficiency”:Mates,PhilosophyofLeibniz,32.201Skinny,stooped,andpale:Ibid.201themostversatilegenius:ForLeibniz’slife,seeHofmann,LeibnizinParis;Asimov,Asimov’sBiographicalEncyclopedia;andMates,PhilosophyofLeibniz.ForLeibniz’sphilosophy,seeMates,PhilosophyofLeibniz.ForLeibniz’smathematics,seeChild,EarlyMathematicalManuscripts;Edwards,TheHistoricalDevelopment;Grattan-Guinness,FromtheCalculus;Dunham,JourneyThroughGenius;Katz,HistoryofMathematics;Guicciardini,ReadingthePrincipia;Dunham,TheCalculusGallery;Simmons,CalculusGems;Guicciardini,IsaacNewton;Stillwell,MathematicsandItsHistory;andBurton,HistoryofMathematics.

202Leibniz’sapproachtocalculus:Edwards,TheHistoricalDevelopment,chapter9,isespeciallygood.SeealsoKatz,HistoryofMathematics,section12.6,andGrattan-Guinness,FromtheCalculus,chapter2.

203morepragmaticview:Forexample,Leibnizwrote:“Wehavetomakeaneffortinordertokeeppuremathematicschastefrommetaphysicalcontroversies.Thiswewillachieveif,withoutworryingwhethertheinfinitesandinfinitelysmallsinquantities,numbers,andlinesarereal,weuseinfinites

andinfinitelysmallsasanappropriateexpressionforabbreviatingreasonings.”QuotedinGuicciardini,ReadingthePrincipia,160.

203“fictionsofthemind”:LeibnizinalettertoDesBossesin1706,quotedinGuicciardini,ReadingthePrincipia,159.

208“Mycalculus”:Quotedinibid.,166.209Leibnizdeducedthesinelawwithease:Edwards,TheHistoricalDevelopment,259.

209“otherverylearnedmen”:Quotedinibid.212problemthatledhimtothefundamentaltheorem:Ibid.,236–38.Actually,thesumthatconcernedLeibnizwasthesumofthereciprocalsofthetriangularnumbers,whichistwiceaslargeasthesumconsideredinthemaintext.SeealsoGrattan-Guinness,FromtheCalculus,60–62.

218“Findingtheareasoffigures”:FromalettertoEhrenfriedWaltervonTschirnhausin1679,quotedinGuicciardini,ReadingthePrincipia,145.

218thehumanimmunodeficiencyvirus:ForHIVandAIDSstatistics,seehttps://ourworldindata.org/hiv-aids/.Forthehistoryofthevirusandattemptstocombatit,seehttps://www.avert.org/professionals/history-hiv-aids/overview.

219HIVinfectiontypicallyprogressedthroughthreestages:“TheStagesofHIVInfection,”AIDSinfo,https://aidsinfo.nih.gov/understanding-hiv-aids/fact-sheets/19/46/the-stages-of-hiv-infection.

220HoandPerelson’swork:Hoetal.,“RapidTurnover”;Perelsonetal.,“HIV-1Dynamics”;Perelson,“ModellingViralandImmuneSystem”;andMurray,MathematicalBiology1.

224triple-combinationtherapy:TheresultsoftheprobabilitycalculationfirstappearedinPerelsonetal.,“DynamicsofHIV-1.”

225ManoftheYear:Gorman,“Dr.DavidHo.”225Perelsonreceivedamajorprize:AmericanPhysicalSociety,2017MaxDelbruckPrizeinBiologicalPhysicsRecipient,https://www.aps.org/programs/honors/prizes/prizerecipient.cfm?first_nm=Alan&last_nm=Perelson&year=2017.

225hepatitisC:“MultidisciplinaryTeamAidsUnderstandingofHepatitisCVirusandPossibleCure,”LosAlamosNationalLaboratory,March2013,http://www.lanl.gov/discover/publications/connections/2013–03/understanding-hep-c.php.ForanintroductiontothemathematicalmodelingofhepatitisC,seePerelsonandGuedj,“ModellingHepatitisC.”

9.TheLogicalUniverse

228Cambrianexplosionformathematics:Forthemanyoffshootsofcalculusintheyearsfrom1700tothepresent,seeKline,MathematicsinWesternCulture;Boyer,TheHistoryoftheCalculus;Edwards,TheHistoricalDevelopment;Grattan-Guinness,FromtheCalculus;Katz,HistoryofMathematics;Dunham,TheCalculusGallery;Stewart,InPursuitoftheUnknown;Highametal.,ThePrincetonCompanion;andGoriely,AppliedMathematics.

229systemoftheworld:Peterson,Newton’sClock;Guicciardini,ReadingthePrincipia;Stewart,InPursuitoftheUnknown;andStewart,CalculatingtheCosmos.

229usheredintheEnlightenment:Kline,MathematicsinWesternCulture,234–86,chroniclestheprofoundimpactthatNewton’sworkhadonthecourseofWesternphilosophy,religion,aesthetics,andliteratureaswellasonscienceandmathematics.SeealsoW.Bristow,“Enlightenment,”https://plato.stanford.edu/entries/enlightenment/.

229“madehisheadache”:D.Brewster,MemoirsoftheLife,Writings,andDiscoveriesofSirIsaacNewton,vol.2(Edinburgh:ThomasConstable,1855),158.

232whenanapplefell:Forthesurprisinghistoryoftheapplestory,seeGleick,IsaacNewton,55–57,andnote18on207.SeealsoMartínez,ScienceSecrets,chapter3.

233“forcerequisitetokeeptheMooninherOrb”:DraftletterfromNewtontoPierredesMaizeaux,writtenin1718,availableonlineathttps://cudl.lib.cam.ac.uk/view/MS-ADD-03968/1349inthecollectionofCambridgeUniversityLibrary.

234“Inellipses”:Asimov,Asimov’sBiographicalEncyclopedia,138,givesoneversionofthisoft-toldstory.

234followedaslogicalnecessities:Katz,HistoryofMathematics,516–19,outlinesNewton’sgeometricarguments.Guicciardini,ReadingthePrincipia,discusseshowNewton’scontemporariesreactedtothePrincipiaandwhattheircriticismsofitwere(someoftheirobjectionswerecogent).AmodernderivationofKepler’slawsfromtheinverse-squarelawisgivenbySimmons,CalculusGems,326–35.

237Neptune:Jones,JohnCouchAdams,andSheehanandThurber,“JohnCouchAdams’sAspergerSyndrome.”

237KatherineJohnson:Shetterly,HiddenFigures,gaveKatherineJohnsontherecognitionshesolongdeserved.Formoreaboutherlife,seehttps://www.nasa.gov/content/katherine-johnson-biography.Forhermathematics,seeSkopinskiandJohnson,“DeterminationofAzimuthAngle.”Seealsohttp://www-groups.dcs.st-and.ac.uk/history/Biographies/Johnson_Katherine.htmlandhttps://ima.org.uk/5580/hidden-figures-impact-mathematics/.

238NASAofficialremindedtheaudience:SarahLewin,“NASAFacilityDedicatedtoMathematicianKatherineJohnson,”Space.com,May5,2016,https://www.space.com/32805-katherine-johnson-langley-building-dedication.html.

239boisteroustoast:QuotedinKline,MathematicsinWesternCulture,282.Theaccountofthedinnerpartycomesfromthediaryoftheparty’shost,thepainterBenjaminHaydon,excerptedinAinger,CharlesLamb,84–86.

239ThomasJefferson:Cohen,ScienceandtheFoundingFathers,makesapersuasivecaseforNewton’sinfluenceonJeffersonandthe“Newtonianechoes”intheDeclarationofIndependence;alsosee“TheDeclarationofIndependence,”http://math.virginia.edu/history/Jefferson/jeff_r(4).htm.FormoreonJeffersonandmathematics,seethelecturebyJohnFauvel,“‘WhenIWasYoung,MathematicsWasthePassionofMyLife’:MathematicsandPassionintheLifeofThomasJefferson,”onlineathttp://math.virginia.edu/history/Jefferson/jeff_r.htm.

240“Ihavegivenupnewspapers”:LetterfromThomasJeffersontoJohnAdams,January21,1812,onlineathttps://founders.archives.gov/documents/Jefferson/03-04-02-0334.

240moldboardofaplow:Cohen,ScienceandtheFoundingFathers,101.Seealso“MoldboardPlow,”ThomasJeffersonEncyclopedia,https://www.monticello.org/site/plantation-and-slavery/moldboard-plow,and“DigDeeper—AgriculturalInnovations,”https://www.monticello.org/site/jefferson/dig-deeper-agricultural-innovations.

240“whatitpromisesintheory”:LetterfromThomasJeffersontoSirJohnSinclair,March23,1798,https://founders.archives.gov/documents/Jefferson/01-30-02-0135.

241“UnlessIammuchmistaken”:HallandHall,UnpublishedScientificPapers,281.

242ordinarydifferentialequations:Forordinarydifferentialequationsandtheirapplications,seeSimmons,DifferentialEquations.SeealsoBraun,DifferentialEquations;Strogatz,NonlinearDynamics;Highametal.,ThePrincetonCompanion;andGoriely,AppliedMathematics.

244partialdifferentialequation:Forpartialdifferentialequationsandtheirapplications,seeFarlow,PartialDifferentialEquations,andHaberman,AppliedPartialDifferentialEquations.SeealsoHighametal.,ThePrincetonCompanion,andGoriely,AppliedMathematics.

245Boeing787Dreamliner:NorrisandWagner,Boeing787,andhttp://www.boeing.com/commercial/787/by-design/#/featured.

246aeroelasticflutter:JasonPaur,“Why‘Flutter’Isa4-LetterWordforPilots,”Wired(March25,2010),https://www.wired.com/2010/03/flutter-testing-aircraft/.

247Black-Scholesmodelforpricingfinancialoptions:Szpiro,PricingtheFuture,andStewart,InPursuitoftheUnknown,chapter17.

248Hodgkin-Huxleymodel:ErmentroutandTerman,MathematicalFoundations,andRinzel,“Discussion.”

248Einstein’sgeneraltheoryofrelativity:Stewart,InPursuitoftheUnknown,chapter13,andFerreira,PerfectTheory.SeealsoGreene,TheElegantUniverse,andIsaacson,Einstein.

248Schrödingerequation:Stewart,InPursuitoftheUnknown,chapter14.

10.MakingWaves

249Fourier:Körner,FourierAnalysis,andKline,MathematicsinWesternCulture,chapter19.Forhislifeandwork,seeDirkJ.Struik,“JosephFourier,”EncyclopediaBritannica,https://www.britannica.com/biography/Joseph-Baron-Fourier.SeealsoGrattan-Guinness,FromtheCalculus;Stewart,InPursuitoftheUnknown;Highametal.,ThePrincetonCompanion;andGoriely,AppliedMathematics.

249heatflow:ThemathematicsofFourier’sheatequationisdiscussedinFarlow,PartialDifferentialEquations,Katz,HistoryofMathematics,andHaberman,AppliedPartialDifferentialEquations.

252waveequation:Forthemathematicsofvibratingstrings,Fourierseries,andthewaveequation,seeFarlow,PartialDifferentialEquations;Katz,HistoryofMathematics;Haberman,AppliedPartialDifferentialEquations;Stillwell,MathematicsandItsHistory;Burton,HistoryofMathematics;Stewart,InPursuitoftheUnknown;andHighametal.,ThePrincetonCompanion.

259Chladnipatterns:Theoriginalimagesarereproducedathttps://publicdomainreview.org/collections/chladni-figures-1787/andhttp://www.sites.hps.cam.ac.uk/whipple/explore/acoustics/ernstchladni/chladniplates/

Foramoderndemo,seethevideobySteveMouldcalled“RandomCouscousSnapsintoBeautifulPatterns,”https://www.youtube.com/watch?v=CR_XL192wXw&feature=youtu.beandthevideobyPhysicsGirlcalled“SingingPlates—StandingWavesonChladniPlates,”https://www.youtube.com/watch?v=wYoxOJDrZzw.

261SophieGermain:HertheoryofChladnipatternsisdiscussedinBucciarelliandDworsky,SophieGermain.Forbiographies,see:https://www.agnesscott.edu/lriddle/women/germain.htmandhttp://www.pbs.org/wgbh/nova/physics/sophie-germain.htmlandhttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Germain.html.

262“thenoblestcourage”:QuotedinNewman,TheWorldofMathematics,vol.1,333.

262microwaveoven:ForaveryclearexplanationofhowamicrowaveovenworksaswellasademonstrationoftheexperimentIsuggested,see“HowaMicrowaveOvenWorks,”https://www.youtube.com/watch?v=kp33ZprO0Ck.Tomeasurethespeedoflightwithamicrowaveoven,youcanalsousechocolate,asshownhere:https://www.youtube.com/watch?v=GH5W6xEeY5U.Forthebackstoryofmicrowaveovensandthegooey,stickymessthatPercySpencerfeltinhispocket,seeMattBlitz,“TheAmazingTrueStoryofHowtheMicrowaveWasInventedbyAccident,”PopularMechanics(February23,2016),https://www.popularmechanics.com/technology/gadgets/a19567/how-the-microwave-was-invented-by-accident/.

265CTscanning:Kevles,NakedtotheBone,145–72;Goriely,AppliedMathematics,85–89;andhttps://www.nobelprize.org/nobel_prizes/medicine/laureates/1979/.TheoriginalpaperthatsolvesthereconstructionproblemwithcalculusandFourierseriesisCormack,“RepresentationofaFunction.”

267AllanCormack:Theoriginalpaperthatsolvesthereconstructionproblemforcomputerizedtomographybyusingcalculus,Fourierseries,andintegralequationsisCormack,“RepresentationofaFunction.”HisNobelPrizelectureisavailableonlineathttps://www.nobelprize.org/nobel_prizes/medicine/laureates/1979/cormack-lecture.pdf.

268theBeatles:ForthestoryofGodfreyHounsfield,theBeatles,andtheinventionoftheCTscanner,seeGoodman,“TheBeatles,”andhttps://www.nobelprize.org/nobel_prizes/medicine/laureates/1979/perspectives.html

269Cormackexplained:Thequoteappearsonpage563ofhisNobellecture:https://www.nobelprize.org/nobel_prizes/medicine/laureates/1979/cormack-

lecture.pdf.

11.TheFutureofCalculus

275writhingnumber:Fuller,“TheWrithingNumber.”SeealsoPohl,“DNAandDifferentialGeometry.”

275geometryandtopologyofDNA:BatesandMaxwell,DNATopology,andWassermanandCozzarelli,“BiochemicalTopology.”

275knottheoryandtanglecalculus:ErnstandSumners,“CalculusforRationalTangles.”

276targetsforcancer-chemotherapydrugs:Liu,“DNATopoisomerasePoisons.”

277PierreSimonLaplace:Kline,MathematicsinWesternCulture;C.Hoefer,“CausalDeterminism,”https://plato.stanford.edu/entries/determinism-causal/.

277“nothingwouldbeuncertain”:Laplace,PhilosophicalEssayonProbabilities,4.

277SofiaKovalevskaya:Cooke,MathematicsofSonyaKovalevskaya,andGoriely,AppliedMathematics,54–57.Sheisoftenreferredtobyothernames;SoniaKovalevskyisacommonvariant.Foronlinebiographies,seeBeckyWilson,“SofiaKovalevskaya,”BiographiesofWomenMathematicians,https://www.agnesscott.edu/lriddle/women/kova.htm,andJ.J.O’ConnorandE.F.Robertson,“SofiaVasilyevnaKovalevskaya,”http://www-groups.dcs.st-and.ac.uk/history/Biographies/Kovalevskaya.html.

278chaotictumblingofHyperion:Wisdometal.,“ChaoticRotation.”281Poincaréthoughthe’dsolvedit:DiacuandHolmes,CelestialEncounters.281Chaoticsystems:Gleick,Chaos;Stewart,DoesGodPlayDice?;andStrogatz,NonlinearDynamics.

281predictabilityhorizon:Lighthill,“TheRecentlyRecognizedFailure.”281horizonofpredictabilityfortheentiresolarsystem:SussmanandWisdom,“ChaoticEvolution.”

282Poincaré’sVisualApproach:Gleick,Chaos;Stewart,DoesGodPlayDice?;Strogatz,NonlinearDynamics;andDiacuandHolmes,CelestialEncounters.

284MaryCartwright:McMurranandTattersall,“MathematicalCollaboration,”andL.Jardine,“Mary,QueenofMaths,”BBCNewsMagazine,https://www.bbc.com/news/magazine-21713163.Forbiographies,see

http://www.ams.org/notices/199902/mem-cartwright.pdfandhttp://www-history.mcs.st-and.ac.uk/Biographies/Cartwright.html.

284“veryobjectionable-lookingdifferentialequations”:QuotedinL.Jardine,“Mary,QueenofMaths.”

285“equationitselfwastoblame”:Dyson,“ReviewofNature’sNumbers.”287HodgkinandHuxley:ErmentroutandTerman,MathematicalFoundations;Rinzel,“Discussion”;andEdelstein-Keshet,MathematicalModels.

287Mathematicalbiology:Forintroductionstothemathematicalmodelingofepidemics,heartrhythms,cancer,andbraintumors,seeEdelstein-Keshet,MathematicalModels;Murray,MathematicalBiology1;andMurray,MathematicalBiology2.

290complexsystems:Mitchell,Complexity.291computerchess:ForbackgroundonAlphaZeroandcomputerchess,seehttps://www.technologyreview.com/s/609736/alpha-zeros-alien-chess-shows-the-power-and-the-peculiarity-of-ai/.TheoriginalpreprintdescribingAlphaZeroisathttps://arxiv.org/abs/1712.01815.ForvideoanalysesofthegamesbetweenAlphaZeroandStockfish,startwithhttps://www.youtube.com/watch?v=Ud8F-cNsa-kandhttps://www.youtube.com/watch?v=6z1o48Sgrck.

293theduskofinsight:Davies,“WhitherMathematics?,”https://www.ams.org/notices/200511/comm-davies.pdf.

294PaulErdős:Hoffman,TheManWhoLovedOnlyNumbers.

Conclusion

296quantumelectrodynamics:Feynman,QED,andFarmelo,TheStrangestMan.

296themostaccuratetheory:PeskinandSchroeder,IntroductiontoQuantumFieldTheory,196–98.Forbackground,seehttp://scienceblogs.com/principles/2011/05/05/the-most-precisely-tested-theo/.

297PaulDirac:ForDirac’slifeandwork,seeFarmelo,TheStrangestMan.The1928paperthatintroducedtheDiracequationisDirac,“TheQuantumTheory.”

298In1931hepublishedapaper:Dirac,“QuantisedSingularities.”298“onewouldbesurprised”:Ibid.,71.298PETscans:Kevles,NakedtotheBone,201–27,andHighametal.,ThePrincetonCompanion,816–23.ForpositronsinPETscanning,seeFarmelo,

TheStrangestMan,andRich,“BriefHistory.”299AlbertEinstein:Isaacson,Einstein,andPais,SubtleIstheLord.299generalrelativity:Ferreira,PerfectTheory,andGreene,TheElegantUniverse.

299strangeeffectontime:FormoreonGPSandrelativisticeffectsontimekeeping,seeStewart,InPursuitoftheUnknown,andhttp://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html.

300gravitationalwaves:Levin,BlackHoleBlues,isalyricalbookaboutthesearchforgravitationalwaves.Formorebackground,seehttps://brilliant.org/wiki/gravitational-waves/andhttps://www.nobelprize.org/nobel_prizes/physics/laureates/2017/press.html.Fortheroleofcalculus,computers,andnumericalmethodsinthediscovery,seeR.A.Eisenstein,“NumericalRelativityandtheDiscoveryofGravitationalWaves,”https://arxiv.org/pdf/1804.07415.pdf.

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Index

A|B|C|D|E|F|G|H|I|J|K|L|M|N|O|P|Q|R|S|T|U|V|W|X|Y|ZNote:Illustrationsareindicatedbyitalics

AAbuSa’dal-A’laIbnSahl,116acceleration,141–42,172–75,230–31,258AchillesandtheTortoiseparadox,17–19,17,162acquiredimmunedeficiencysyndrome(AIDS),218–25Adams,Douglas,viiiAdams,John,239adequality,106–7aeroelasticflutter,246–47AIDS,218–25airresistance,69–71aircraftengineering,244–47algebra

AchillesandtheTortoiseparadox,19birthplaceof,90distance,rateandtime,145geometry,mergewith,93–96,98historyof,89–93Newtonon,102optimizationproblems,103–7popularityof,xx–xxisymbolsvsspokenlanguage,67tangentsofcurves,168Wallison,196xyplane,177

al-Khwarizmi,MuhammadIbnMusa,92

AlphaInfinity,292–93,294AlphaZero,292,293altimeter,213–16AmericanRevolution,239Ampère,André-Marie,x–xiAmpère’slaw,xiamplitude,109–10,254analysisvssynthesis,102–3analyticgeometry,101–3analyticnumbertheory,228ancientcivilizations

algebraandgeometry,90–91,95areaofacircle,7Descarteson,102Egypt,1geographyofGreece,90geometryvsarithmetic,33Greeksoncurves,3–4instantaneousspeed,162musicalharmony,xiv,48planetarymotion,62,63synthesisvsanalysis,102viewofnumbersandsymbols,95–96

Anderson,Carl,298animatedmovies,50–53anti-electron,298antimatter,297–98anti-nodes,263antiretroviraldrugs,219Aquinas,Thomas,60Archimedeanscrew,56–57Archimedes,27–57

analysisvssynthesis,102onareaofacircle,7cheeseexample,39–41contributionsof,28–29deathof,312n27Galileoand,86geographyof,90

legacyof,50–56,84,92–93,193–94“MeasurementofaCircle,”7Method,the,42–50,93,313n47motionstudy,315n57“OntheSphereandCylinder,”49onparabolas,35–39onpi,29–32portrayalsof,27–28QuadratureoftheParabola,The,35–39,41,43ratiosvsnumbers,33–35,48writings,91

ArchimedesPalimpsest,50area

ofacircle,4–8,33,188–91underacurve,121,144–46,168–69,176–78withdifferentials,209–11,216–17distanceand,171–72underahyperbola,139integralnatureof,181parabolicsegment,37–39

areaproblem,144–46,176–79Aristarchus,heliocentricmodel,62–63Aristotle

circlesvsellipses,82,87onEarth-centricmodelofuniverse,60–64,65onfallingbodies,66,69inertia,231–32oninfinity,15–16Zeno’sparadoxes,17

ArithmeticaInfinitorum(Wallis),188,196artificialintelligence(AI),291–94astronomy,78,80,85–86,87,91,278atomicclocks,75–76Avatar(movie),51average,109–10

B

backwardproblem,144–46,175,180–85bacterialgrowth,127,138Ball,Douglas,246Barrow,Isaac,196,199,321n169base10,127,134basee,134–37Beatles,The,268Bernoulli,Johann,208Blake,William,73Boeing787,244–47Bohr,Georg,199Bolt,Usain,159–66,160,175,185–86Brahe,Tycho,80Bruno,Giordano,16,64bubonicplague,195buoyancy,28

Ccalculus

breakthroughin,227–29computersand,285–87contributionsof,x–xi,xvii–xxiiicurves,motion,andchangein,xvii–xxdatacompression,107–13descendantsof,271–72discretevscontinuoussystems,241Enlightenmentperiod,238–40asGod’slanguage,vii–viii,ix,xix,295–97hiddenusefulnessof,76–77HIVtherapy,218–25limitson,278–79logicandcreativityin,xii–xiii,xvi–xvii,42,49namingof,167originsof,3–4phasesof,xiv–xvipredictionsforfuture,273

threecentralproblemsof,144–46,144,175,176–85Seealsodifferentialcalculus;integralcalculus

Cameron,James,51car,speedanddistance,169–75carbondating,137Cartwright,Mary,284–85Cavalieri,Bonaventura,93centerofgravity,Archimedeson,48chainreactions,138chaos,281–82chemotherapydrugs,276chessplaying,291–94Chladni,Ernst,259–60Chladnipatterns,259–60Church,the,60–61,63–64,65cinnamon-raisinbreadexample,93–96,147–49,148circles

areaof,4–8,33,188–91centerof,82circumference,4–5,7,29–32diameter,30equationsfor,97importanceof,2–3asinfinitepolygons,11–12,203piand,29–32assliceofcone,35

circularmotion,sinewavesand,108–12,109circumference,4–5,7,29–32Clark,William,187clocks,67,74–75,299–300Collins,John,199completedinfinity,15–16,313n47compoundinterest,135–36computer-animatedmovies,50–53,51computerizedtomography,265–69,289computers,245,277–79,285–87,291–94constantacceleration,172–75constantrates,inlinearfunctions,146–49continuouscompounding,136

continuousvsdiscretesystems,16–21,241Copernicantheory,64Copernicus,Nicolaus,62,63–64,79Cormack,Allan,267,269,272,289CosmicMystery(Kepler),78–81,79cosmology.Seexuniversecreativity.SeeintuitionandcreativityCTscanning,265–69,289cuneiform,1CureaudeLaChambre,Marin,116curvature,299–300curves

Archimedeson,47–48areaof,168–69,176–79,209–11equations,96–97interpolation,163Keplerand,87nonlinearequations,149–54slopeof,142,206–9smoothness,153,163–64strugglewith,xviiithreecentralproblemsof,144–46

Ddatacompression,107–13daylengthexample,108–12,154–59DeAnalysi(Newton),196,199,200DeMethodis(Newton),196,197,201decayandexponentialgrowth,137–39,220–24,251decimals,9–10,91,92,189,193,295–97DeclarationofIndependence,xxi–xxii,239DeepBlue,291–92DeepMind,292dependentvariables,124,141,147,242derivatives

daylengthexample,154–59

vsdifferentials,206–9instantaneousspeed,159–66integralsand,168–69linearrelationships,146–49nonlinearrelationships,149–54purposeandtypesof,141–44sinewaves,256–59slopeand,177–79symbolfor,143

Descartes,Renéanalyticgeometry,101–3background,99–100oncurvedarcs,168Dioptrics,115–16DiscourseonMethod,99,101Fermatrivalry,98–99,116Geometry,119legacyof,93,188lenses,87tangents,119–20unknownsandconstants,92xyplane,96–97

DescriptionoftheWonderfulRuleofLogarithms(Napier),133determinism,277–79,280Deuflhard,Peter,53–55diameter,ofacircle,30DifferentialAnalyzer,286differentialcalculus,89–121

aircraftengineering,244–47algebraandgeometryconvergence,93–96,98analyticgeometry,101–3derivativesvsdifferentials,206–9Descartes-Fermatrivalry,98–101Fermat’scontributionsto,120–21fundamentaltheorem,209–11infinitesimals,205–6vsintegralcalculus,89,185–86Leibnizand,201–208Newtonand,184–85

optimizationproblems,103–7ordinaryvspartialequations,242–44originsof,59,68–69overviewof,vii–viii,xx–xxipartialequations,applicationsof,247–48asphaseofcalculus,xv–xvisinelaw,117–18

dimensions,fourormore,287–91Dioptrics(Descartes),115–16Dirac,Paul,xiv,297–98DiscourseonMethod(Descartes),99,101DiscoursesandMathematicalDemonstrationsConcerningTwoNewSciences

(Galilei),65discretevscontinuoussystems,16–21,241distancefunction,170DNA,273–76doubleintersection,106,111,119DreamWorks,51,52–53Dyson,Freeman,285

EEarth

ascenterofuniverse,60–65freefallingobjects,173,233GPS,76,299–300greenhouseeffect,249Kepleron,79moon’sorbit,232–33navigationandlongitude,75Newtonand,229,235–36periodof,84–85retrogrademotion,62tunnelingphenomenon,22two-bodyproblem,237–38

eightdecimalplaces,295–97Einstein,Albert,xiii,xxii,77,287,289,297,299–301ElectricandMusicalIndustries(EMI),268

electronicsynthesizers,255Elements(Euclid),32,188,236ellipses

equationsfor,97planetarymotion,81–82,83,87,234assliceofcone,35

ENIAC,286Enlightenmentperiod,238–40equations.SeeformulasEratosthenes,42,49Erdős,Paul,294Euclid,32,90–91,188,236Eudoxus,methodofexhaustionusedby,32exponentialfunctions,127–28exponentialgrowthanddecay,137–39,220–24,251

Ffacialsurgery,53–56,56fallingbodies,66–69Faraday,Michael,x–xiFBIfingerprintingtechnology,107–13,257feedbackloop,138Ferguson,Samuel,294Fermat,Pierrede

analyticgeometry,101–3background,100contributionsof,93,120–21,194Descartesrivalry,98–99FBIfingerprintingtechnology,107–13optimization,103–7principleofleasttime,113–18,319n118tangents,118–20xyplane,96–97

Feynman,Richard,vii,viii–ix,295–97FindingNemo(movie),50fingerprintsdatabase,107–13finitedecimals,10

fluxions,184foci(focalpoints),81–82force,230–31,252,258formulas

circle,areaof,7,33forceandmotion,230–31,252,258functionsofonevariable,124fundamentaltheorem,179–80,211HIVdecay,221–22Kepler’sthirdlaw,85parabolicsegment,38,39pi,boundsof,32powerseries(areaofcircularsegment),190,191sinewaves,derivativeof,258velocity,173

forwardproblem,144–46,175,179–80four-colormaptheorem,293Fourier,JeanBaptisteJoseph

applicationsofwork,256Fourieranalysis,267heatflow,249–52stringtheory,252–56

Fourieranalysis,267Fourierseries,254fourthdimension,287–91frequencies,254,256,259–60friction,69–70,232–33,245Fuller,Brock,275functions

applicationsof,125–26exponentialfunctions,127–28exponentialgrowthanddecay,137–39linearfunctions,146–49logarithms,131–34naturalbase(e),134–37nonlinearequations,149–54powerfunctions,126–27,182scientificnotation,128–31threecentralproblemsof,144–46

xyplane,124–25Seealsoderivatives

fundamentaltheorembackwardproblem,180–85constantacceleration,172–75differentials,209–11discoveryof,168–69equationfor,179–80,211Leibniz’sapproachto,211–18,213localvsglobaloperations,185–86meaningof,179–80motionandchange,169–72Newtonon,182,193–94“paint-roller”proof,175–79,178

futuredirections,271–94chaos,281–82computers,285–87determinism,277–79dimensions,fourormore,287–91DNA,273–76nonlinearity,279–80Poincaré’svectorfields,282–84predictions,273radar,284–85

GGalilei,Galileo,64–76

background,64constantacceleration,173contributionsof,59–60,86–88DiscoursesandMathematicalDemonstrationsConcerningTwoNew

Sciences,65fallingbodies,66–69functionsofonevariable,124housearrest,65–66idealconditions,69–71vsKepler,85–86

observationswithtelescope,65pendulums,71–77powerfunctions,126–27principleofinertia,231religiousbeliefs,65TwoNewSciences,68,70,71–72

Galilei,Virginia(MariaCeleste),64,65Gamba,Marina,64Gauss,CarlFriedrich,261geometricseries,39geometry

algebra,mergewith,93–96,98analyticgeometry,101–3areaofacircle,4–8birthplaceof,90harmonyand,49Kepleron,60,79–80,82inNature,70Platoon,60

Geometry(Descartes),119Geri’sGame(movie),51–52,52Germain,Sophie,260,261–62Gilbert,William,87Glenn,John,237–38globaloperations.Seeintegralcalculusglobalpositioningsystem,75–77,299–300globalwarming,249God

Brunoon,16calculusaslanguageof,vii–viii,ix,xix,295–97Erdőson,294Jeffersonon,239Kepleron,60,80Seealsoreligionandspirituality

golemofinfinity,xvi–xvii,11,47,251,271GoogleTranslate,291GPS,75–77,299–300gravitationalwaves,300gravity

Archimedes’suseof,46,48constantof,23Einsteinon,248,299Galileoonprojectilemotion,70inverse-squarelawof,195Newton’slawsof,231–34theoryofrelativity,299–300two-bodyproblem,235–37

Gregory,James,321n169

HHales,Thomas,294Halley,Edmond,229,234HarmoniesoftheWorld(Kepler),84Harriot,Thomas,116Harrison,John,75heatequation,250–52,258heatflow,249–52heliocentricmodel,60–65,82hepatitisC,225Hertz,Heinrich,xihexagon,30–31Heytesbury,William,173HiddenFigures(movie),xxii,237–38histones,274Hitchhiker’sGuidetotheGalaxy,The(Adams),viiiHIV

exponentialdecaymodeling,220–24mutationrateanddrugtherapy,224–25progressin,218–19stagesof,219–20

Ho,David,219–25Hobbes,Thomas,196Hodgkin,Alan,286–87,289Hodgkin-Huxleyequations,289Holmes,Sherlock,39–40Hounsfield,Godfrey,267–69

Huxley,Andrew,286–87,289Huygens,Christiaan,72,75,200,208,212,215hydrogenbomb,286hyperbola,97,139,191–92Hyperion,278

IIbnal-Haytham,Al-Hasan“Alhazen,”91IbnSahl,AbuSa’dal-A’la,116idealconditions,69–71inclinedplane,Galileo’suseof,66–69independentvariables,124,141–43,147,242,250infinitedecimals,10,189infinitesimals

cubesofsmallnumbers,204–5differentials,205–6fundamentaltheorem,withdifferentials,210–11Leibniz’suseof,202–4,324n203two-bodyproblem,236–37

infinity,xxi,1–25atatomicscale,21–25circle,areaof,4–8,11–12copingwith,xvi–xviiindecimals,9–11golemof,xvi–xvii,11,47,251,271historyofmathematics,1–3limits,8–9,13–14Method,the,47originsof,3–4piand,34skepticismof,195–96Zeno’sparadoxes,16–21zeroand,14–16

InfinityPrincipleasalgorithm,188–93Archimedes’suseof,28,35CTscanningand,266–67

curves,150–51defined,xvioverviewof,xiv–xvi,271–72quantummechanics,21–25two-bodyproblem,235–37

instantaneousspeed,68–69,159–66instantaneoustemperature,250integralcalculus

backwardproblem,180–85CTscanning,266vsdifferentialcalculus,89,185–86discoveryof,168–69Newtonand,184–85originsof,28,36–37,47–48,84,89–90asphaseofcalculus,xv–xvi

integralsign,211interpolation,163intuitionandcreativity

ofAlphaZero,292Archimedes’suseof,42incalculus,xii–xiii,xvi–xvii,276inmathematics,49Newton’suseof,190–91

inventions.Seetechnologyinverse-squarelawofgravity,195,232–33irrationalnumbers,discoveryof,33,312n33isochronism,72

JJefferson,Thomas,xxi–xxii,239–40,326n239Jobs,Steve,51Johnson,Katherine,xxii,237–38Josephson,Brian,73–74Jupiter,65,80,84Jyesthadeva,193

KKasparov,Garry,291–92Kepler,Johannes

aweof,xixbackground,78CosmicMystery,78–81,79firstlawofplanetarymotion,81–82functionsofonevariable,124–25vsGalileo,85–86geometryand,59–60HarmoniesoftheWorld,84legacyof,86–88,234logarithms,useof,133packingspheresproblem,293–94onplanetarymotion,79Pythagoreanfever,xivsecondlawofplanetarymotion,82–84thirdlawofplanetarymotion,84–85,232–33volumes,92–93

Khwarizmi,MuhammadIbnMusaal-,92KingHiero’scrown,86Koestler,Arthur,86Komodo,292Kovalevskaya,Sofia,277–79,280,290

LLaChambre,MarinCureaude,116Lagrange,JosephLouis,260Laplace,PierreSimon,277Laplace’sdemon,277,279lasers,xxiilawofinertia,71lawofoddnumbers,66–69lawofthelever,28,46lawsofmotion

Aristotelianunderstandingof,60–64circularmotion,sinewavesand,108–12firstlawofplanetarymotion,81–82force,230–31,252,258planetarymotion,81–85secondlawofplanetarymotion,82–84thirdlawofplanetarymotion,84–85,232–33

LeBlanc,Antoine-August(aspseudonym),261–62Leibniz,GottfriedWilhelm

approachtofundamentaltheorem,211–18,213asco-inventorofcalculus,201–2deathof,167–68differentials,useof,208–9fundamentaltheorem,withdifferentials,209–11infinitesimals,202–5,324n203integralsign,211Newtoncorrespondence,197,199,200–201,323n199

levers,28,44–46light

bendingof,xix,114–18,195,209compositionofwhitelight,195,197,259aselectromagneticwave,xi–xiiiquantumelectrodynamics,296–97speedof,23,263,328n262

limitsconceptof,8–9indecimals,9–11ofdeterminism,278–80infinityand,13–14ParadoxoftheArrow,19–21

linearrelationships,95–96,126,146–49,173Littlewood,John,284–85LiuHui,91localoperations,185–86logarithms,131–34,192,196,221longitude,74–75

M

MadhavaofSangamagrama,193magicnumbers,217–18magnetrons,263–64Marconi,Guglielmo,xiMars

periodof,84retrogrademotion,61–62,61,62sectorareas,82–83TychoandKepleron,78,80

MathematicalPrinciplesofNaturalPhilosophy(Newton),229,234,236,240mathematics

analysisvssynthesis,102–3discoveryof,49Galileoon,60Seealsoalgebra;calculus;geometry

Maxwell,JamesClerk,xi,xii–xiii,77,264“MeasurementofaCircle”(Archimedes),7medicalfield

CTscanning,265–69DNA,273–76facialsurgery,53–56hepatitisC,225HIVprogress,218–25Hodgkin-Huxleyequations,289PETscans,298

Mercator,Nicholas,196Mercury,79,80Mersenne,Marin,100–101Method,the(ofArchimedes),42–50,93,313n47methodofexhaustion,32,47,93,102methodofleastsquares,111methodofpowerseries,188–93microwaveovens,262–64,328n262MiddleAges,50,62,173,174military

aircraft,245–47Archimedesand,27–28ballisticdata,285–87

GPS,76–77nonlineardynamicsandradar,284–85

modesofvibration,259–60moon

Aristotleon,60–61Galileo’sobservations,65gravityand,232–33inverse-squarelawofgravity,195Newtonon,229,232–34ofSaturn,278Tychoon,80

motion,123–39Archimedes’sstudyof,315n57exponentialfunctions,127–28exponentialgrowthanddecay,137–39functions,roleof,125–26fundamentaltheorem,169–72logarithms,131–34atamolecularscale,165–66naturallogarithm(e),134–37ofplanets,78–81powerfunctions,126–27scientificnotation,128–31strugglewith,xxtwo-bodyproblem,234–37xyplaneand,123–24

Munro,Alice,278music

continuousvsdiscrete,20–21CTscanning,268harmonyand,48–49,230logarithmicperceptionofpitch,134Newtonon,192Pythagorason,xiii–xiv,230stringtheory,252–56vibrationmodes,259–60

N

Napier,John,133Napoleon,260,261NASA’stwo-bodyproblem,237–38naturallogarithm(e),134–37nature

calculusaslanguageof,vii–viii,xiii–xiv,xix–xx,166circlesin,2Galileoon,67,69,70logicof,229–34nonlinearity,279–80optimizationprinciple,118,120predator-preyinteractions,159quantummechanics,21ratesofchange,143,258

negativepowers,130Neptune,237nervecellcommunication,286–87,289Newton,Hannah,187,188Newton,Isaac

analysisvssynthesis,102–3areaproblem,176–79background,186–88constantacceleration,172–75correspondence,320n169,322n195,323n199DeAnalysi,196,199,200DeMethodis,196,197,201deathof,167–68onDescartes,102discoveriesof,viii,195–97,199–201,322n195discretevscontinuoussystems,241Einstein’stheoryofrelativity,299fundamentaltheorem,169–72,182–83gravity,forceandnature,229–34legacyandinfluenceof,xxi–xxii,238–40,325n229,326n239asmash-upartist,193–94MathematicalPrinciplesofNaturalPhilosophy,229,234,236,240methodofpowerseries,188–93,321n188notebook,182

pendulums,72Principia,229,234,236,240SystemoftheWorld,The,236–37three-bodyproblem,229,281,288two-bodyproblem,234–38xyplane,124

NewtonProject,192NobelPrizewinners,267,269,278,287,298,300nonlinearequations,96–97,149–54nonlinearity,279–80,299–300nuclearreactions,138

OObama,Barack,238Oldenburg,Henry,199–200OnAnalysisbyEquationsUnlimitedinTheirNumberofTerms(Newton),196,

199,200“OntheSphereandCylinder”(Archimedes),49“OntheUnreasonableEffectivenessofMathematicsintheNaturalSciences”

(Wigner),xiiioptics

curvedlenses,87,99principleofleasttime,114–18reflectingtelescopes,195

optimizationalgorithm,110–11optimizationproblems,103–7,116–18orbits

areaof,82–83gravityand,232–33periodof,84–85shapeof,81–82Seealsoellipses;moon;planetarymotion

ordinarydifferentialequations,242–44Oresme,Nicole,173oscillations,73–74,158–59overtones,254–55“paint-roller”proof,fundamentaltheorem,175–79,178

Pparabolas,35–39

equationsfor,97,150projectilesand,70assliceofcone,36slopeand,207–9thoughtexperimentfor,150–53

parabolicsegment,36–39,43–44ParadoxoftheArrow,19–21parallax,63partialdifferentialequations,242–48,249–50patterns,111–12pendulums,71–77,158,282–84,288Perelson,Alan,219–25,242period,109–10period,ororbit,84–85PETscans,298phase,109–10physics

electricityandmagnetism,xigravitationalwaves,300heatflow,249–52lawsofplanetarymotion,81–85NASA’stwo-bodyproblem,237–38Newton’slegacy,229–34pendulums,71–77,158,282–84,288quantummechanics,21–25,77,295–97SeealsoGalilei,Galileo;Newton,Isaac

piArchimedes’sestimationof,29–32,52Chinesecontributionto,91historicalviewofasratio,33–35infinitedecimals,24powerseriesmethod,189

Picasso,166Pixar,51,52,314n52pizzaproof,4–8

Planck,Max,23Plancklength,23planetarymotion,78–81

Einsteinon,248ellipticalorbits,81–82Kepleron,xixNewtonon,234–37orbitalperiod,84–85sectorareas,82–84

planetsvsstars,61Plato,17,60,91Plutarch,onArchimedes,27Poincaré,Henri,281,282–84,288–89polygons,11–12positron,297–98powerfunctions,126–27,182powerseries,methodof,188–93powersoften,128–31predator-preyinteractions,158–59predictabilityhorizon,281–82predictions

antimatter,297–98forcalculus,273electromagneticwaves,264fundamentaltheorem,183–84Josephsoneffect,73–74ofnewplanets,237oforbits,235–37ofparticlesinacontinuousmedia,256quantumelectrodynamics,296–97relativityand,300

Principia(Newton),229,234,236,240principleofleastaction,118principleofleasttime,113–18,319n118projectilemotion,69–70proportions,asGreekway,33–35,48proteaseinhibitors,219,220–22,223“proto-calculus,”227Ptolemaicsystem,63

Ptolemy,63,91Pythagoras,xiii–xiv,86,90,230Pythagoreandream,78,86,230Pythagoreantheorem,31–32,90Pythagoreantheoryofmusicalharmony,48–49

Qquadraticequations,96–97,126quadrature,36,168–69QuadratureoftheParabola,The(Archimedes),35–39,41,43quantumelectrodynamics(QED),296–97quantummechanics,21–25,77,295–97quartercycle,ofsinewaves,109,109,154,156,157–59,257–59Quinto,Todd,269

Rradar,263–64,284–85“radarranges.”Seemicrowaveovensradiology,265–69radius,5Radon,Johann,269rate,defined,147,148ratesofchange.SeederivativesRaytheonCompany,263–64reductionistthinking,280refraction,oflight,114–16,114religionandspirituality

AlphaInfinity,292–93,294Aristotleand,60–61calculusasGod’slanguage,vii–viii,ix,xix,295–97cosmologyand,63–64differentialequations,72God.seeGod“God’sbook,”294Newton’sinfluenceon,239SeealsoChurch,the

Renaissance,xix,50,59,92retrogrademotion,61–62,61,62RiddleoftheWall,8–9,8,21–25Riemann,Bernhard,290rigidbodies,278ruleof72,137ruleoflogs,132–33Russell,Bertrand,16

SSanders,Bernie,130–31satellites,76,299–300Saturn,278Schrödinger,Erwin,22science

idealconditions,69–71physics.seexPhysicsplanetarymotion,78–81thescientificmethod,66

scientificnotation,128–31scientificrevolution,50,63,66,87,92,124,227,272secondderivatives,258sectors,83–84self-regenerationproperty,157–59Shepard,Alan,238Shrek(movie),50,51,53,53sinelawofrefraction,115–117,209sinewaves

Chladnipatterns,259–60daylengthexample,156–59derivativesand,256–59heatflow,250–52overviewof,108–12,109stringtheory,252–56x-raysand,267

slopechangingrateof,149–54

equationfor,147,147,207,208optimizationproblems,104–5problemsconcerning,144–46ofaramp,142

Smith,Barnabas,187smoothcurves,153Snell,Willebrord,115Snell’slaw,115–117,209Somayaji,Nilakantha,193soupexample,243–44space-time,248,287–88,299–300speed,xx,68,141–42,175Spencer,Percy,264,328n262spheres.Seecircles;xcurvessquarewaves,255squaring,111,127,150,168staircaseanalogy,212–18,213standingwaves,251,259starsvsplanets,61steady-state,222–23Stevin,Simon,92Stockfish,292Stokes,Henry,187,188stringtheory,252–56Strogatz,Steven

asappliedmathematician,xonArchimedes,49cinnamon-raisinbread,93–96,147–49,148onlearningaboutinfinity,9–10“paint-roller”proofand,177–78,178onwritingthisbook,ix–x

sun,ascenterofuniverse,60–65,82supercoiling,275synthesisvsanalysis,102–3SystemoftheWorld,The(Newton),236–37

T

Tcells,219,223tangents,38,118–20,145technology

Boeing787,244–47computer-animatedmovies,50–53GPS,75–77,299–300Jefferson’splow,240lasers,xxiimicrowaveovens,262–64optics,87,114–18,195partialdifferentialequations,applicationsof,247–48wirelesscommunication,x–xi

telescopes,195telescopingsums,216–18temperaturechange,249–52Tesla,Nikola,xiTezel,Tunç,61Thales,90three-bodyproblem,281–82,288–89time,effectofgravityon,299–300toneandovertone,254–55“TooMuchHappiness”(shortstory),278topoisomerases,276Torricelli,Evangelista,93ToyStory(movie),50,51TreatiseoftheMethodsofSeriesandFluxions,A(Newton),196,197,201trianglewaves,253,254,255,256triangles

Archimedes’suseof,37–39,43–44computer-animatedmovies,50–53,51facialsurgery,modeling,54–56

triangulation,76tripleintegration,236tuningfork,254–55tunnelingphenomenon,22TwoNewSciences(Galilei),68,70,71–72two-bodyproblem,234–38

Uuniformcircularmotion,157–59unitfraction,214universe

Earth-centricviewof,60–64Einsteinon,299–301heliocentricmodel,60–65,82lawsofplanetarymotion,81–85space-time,248,287–88,299–300

uranium,138Uranus,237

Vvariables,124,288vectorfield,283–84,288velocity,141–42,170,171,173,175,231Venus,79,85Vesalius,Andreas,63vibration,259–60,263Viète,François,92violins,259visualization

Archimedes’suseof,43–44Chladnipatterns,259–62Kovalevskaya’sspinningtop,277–80limitationsof,290–91vectorfields,283–84,288

Vitruvius,onArchimedes,27Viviani,Vincenzo,71

WWallis,John,188,194,195,196,321n188WaltersArtMuseum,50waterclock,67

waveequation,xii,252–56,258wavephenomena,256wavelets,112–13,257waves

Chladnipatterns,259–62CTscanning,265–69heatflow,249–52microwaveovens,262–64sinewavesandderivatives,256–59stringtheory,252–56tunnelingphenomenon,22

Weierstrass,Karl,278Weiser,Martin,53–55Wigner,Eugene,xiiiWilhelmLeibniz,Gottfried,118wirelesscommunication,x–xiWouk,Herman,vii,ix–xwrithingnumber,275

Xx-rays,264,265–69xyplane,93–96

generalizedtohigherdimensions,287slopeofacurvein,142slopeproblemsof,144–46slopewithdifferentials,206–9astoolforFermatandDescartes,96–97,123–24

ZZachow,Stefan,53–55ZenoofElea,16–21,162zero,13,14–16,91ZuChongzhi,91

THEBESTINTRODUCTIONtonumbersI’veeverseen—theclearestandfunniestexplanationofwhattheyareandwhyweneedthem—appearsinaSesameStreetvideocalled123CountwithMe.Humphrey,anamiablebutdimwittedfellowwithpinkfurandagreennose,isworkingthelunchshiftattheFurryArmsHotelwhenhetakesacallfromaroomfulofpenguins.Humphreylistenscarefullyandthencallsouttheirordertothekitchen:“Fish,fish,fish,fish,fish,fish.”ThispromptsErnietoenlightenhimaboutthevirtuesofthenumbersix.

Childrenlearnfromthisthatnumbersarewonderfulshortcuts.Insteadof

sayingtheword“fish”exactlyasmanytimesastherearepenguins,Humphreycouldusethemorepowerfulconceptofsix.Asadults,however,wemightnoticeapotentialdownsidetonumbers.Sure,

theyaregreattimesavers,butataseriouscostinabstraction.Sixismoreetherealthansixfish,preciselybecauseit’smoregeneral.Itappliestosixofanything:sixplates,sixpenguins,sixutterancesoftheword“fish.”It’stheineffablethingtheyallhaveincommon.Viewedinthislight,numbersstarttoseemabitmysterious.Theyapparently

existinsomesortofPlatonicrealm,alevelabovereality.Inthatrespecttheyaremorelikeotherloftyconcepts(e.g.,truthandjustice),andlessliketheordinary

objectsofdailylife.Theirphilosophicalstatusbecomesevenmurkieruponfurtherreflection.Whereexactlydonumberscomefrom?Didhumanityinventthem?Ordiscoverthem?Anadditionalsubtletyisthatnumbers(andallmathematicalideas,forthat

matter)havelivesoftheirown.Wecan’tcontrolthem.Eventhoughtheyexistinourminds,oncewedecidewhatwemeanbythemwehavenosayinhowtheybehave.Theyobeycertainlawsandhavecertainproperties,personalities,andwaysofcombiningwithoneanother,andthere’snothingwecandoaboutitexceptwatchandtrytounderstand.Inthatsensetheyareeerilyreminiscentofatomsandstars,thethingsofthisworld,whicharelikewisesubjecttolawsbeyondourcontrol...exceptthatthosethingsexistoutsideourheads.Thisdualaspectofnumbers—aspartheaven,partearth—isperhapstheir

mostparadoxicalfeature,andthefeaturethatmakesthemsouseful.ItiswhatthephysicistEugeneWignerhadinmindwhenhewroteof“theunreasonableeffectivenessofmathematicsinthenaturalsciences.”Incaseit’snotclearwhatImeanaboutthelivesofnumbersandtheir

uncontrollablebehavior,let’sgobacktotheFurryArms.SupposethatbeforeHumphreyputsinthepenguins’order,hesuddenlygetsacallonanotherlinefromaroomoccupiedbythesamenumberofpenguins,allofthemalsoclamoringforfish.Aftertakingbothcalls,whatshouldHumphreyyellouttothekitchen?Ifhehasn’tlearnedanything,hecouldshout“fish”onceforeachpenguin.Or,usinghisnumbers,hecouldtellthecookheneedssixordersoffishforthefirstroomandsixmoreforthesecondroom.Butwhathereallyneedsisanewconcept:addition.Oncehe’smasteredit,he’llproudlysayheneedssixplussix(or,ifhe’sashowoff,twelve)fish.Thecreativeprocesshereisthesameastheonethatgaveusnumbersinthe

firstplace.Justasnumbersareashortcutforcountingbyones,additionisashortcutforcountingbyanyamount.Thisishowmathematicsgrows.Therightabstractionleadstonewinsight,andnewpower.Beforelong,evenHumphreymightrealizehecankeepcountingforever.Yetdespitethisinfinitevista,therearealwaysconstraintsonourcreativity.

Wecandecidewhatwemeanbythingslike6and+,butoncewedo,theresultsofexpressionslike6+6arebeyondourcontrol.Logicleavesusnochoice.Inthatsense,mathalwaysinvolvesbothinventionanddiscovery:weinventtheconceptsbutdiscovertheirconsequences.Aswe’llseeinthecomingchapters,inmathematicsourfreedomliesinthequestionsweask—andinhowwepursuethem—butnotintheanswersawaitingus.

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AbouttheAuthor

CourtesyofCornellUniversity

STEVENSTROGATZistheJacobGouldSchurmanProfessorofAppliedMathematicsatCornellUniversity.Arenownedteacherandoneoftheworld’smosthighlycitedmathematicians,hehasbloggedaboutmathfortheNewYorkTimesandTheNewYorkerandhasbeenafrequentguestonRadiolabandScienceFriday.HeistheauthorofSyncandTheJoyofx.HelivesinIthaca,NewYork.

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