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AssoonasthesoundofthePeace-cryofmydepartingWifehaddiedaway,IbegantoapproachtheStrangerwiththeintentionoftakinganearerviewandofbiddinghimbeseated:buthisappearancestruckmedumbandmotionlesswithastonishment.
WithouttheslightestsymptomsofangularityheeneverthelessvariedeveryinstantwithgraduationsofsizeandbrightnessscarcelypossibleforanyFigurewithinthescopeofmyexperience.
ThethoughtflashedacrossmethatImighthavebeforemeaburglarorcut-throat,somemonstrousIrregularIsosceles,who,byfeigningthevoiceofaCircle,hadobtainedadmissionsomehowintothehouse,andwasnowpreparingtostabmewithhisacuteangle.
Inasitting-room,theabsenceofFog(andtheseasonhappenedtoberemarkablydry),madeitdifficultformetotrusttoSightRecognition,especiallyattheshortdistanceatwhichIwasstanding.
Desperatewithfear,Irushedforwardwithanunceremonious,"Youmustpermitme,Sir—"andfelthim.MyWifewasright.
Therewasnotthetraceofanangle,nottheslightestroughnessorinequality:neverinmylifehadImetwithamoreperfectCircle.
HeremainedmotionlesswhileIwalkedaroundhim,beginningfromhiseyeandreturningtoitagain.
Circularhewasthroughout,aperfectlysatisfactoryCircle;therecouldnotbeadoubeofit.
Thenfollowedadialogue,whichIwillendeavourtosetdownasnearasIcanrecollectit,omittingonlysomeofmyprofuseapologies—forIwascoveredwithshameandhumiliationthatI,aSquare,shouldhavebeenguiltyoftheimpertinenceoffeelingaCircle.
ItwascommencedbytheStrangerwithsomeimpatienceatthelengthinessofmyintroductoryprocess.
Stranger.Haveyoufeltmeenoughbythistime?Areyounotintroducedtomeyet?
I.MostillustriousSir,excusemyawkwardness,whicharisesnotfromignoranceoftheusagesofpolitesociety,butfromalittlesurpriseandnervousness,consequentonthissomewhatunexpectedvisit.
AndIbeseechyoutorevealmyindiscretiontonoone,andespeciallynottomyWife.
ButbeforeyourLordshipentersintofurthercommunications,wouldhedeigntosatisfythecuriosityofonewhowouldgladlyknowwhencehisvisitorcame?
Stranger.FromSpace,fromSpace,Sir:whenceelse?
I.Pardonme,myLord,butisnotyourLordshipalreadyinSpace,yourLordshipandhishumbleservant,evenatthismoment?
Stranger.Pooh!whatdoyouknowofSpace?DefineSpace.
I.Space,myLord,isheightandbreadthindefinitelyprolonged.
Stranger.Exactly:youseeyoudonotevenknowwhatSpaceis.YouthinkitisofTwoDimensionsonly;butIhavecometoannouncetoyouaThird—height,breadth,andlength.
I.YourLordshipispleasedtobemerry.Wealsospeakoflengthandheight,orbreadthandthickness,thusdenotingTwoDimensionsbyfournames.
Stranger.ButImeannotonlythreenames,butThreeDimensions.
I.WouldyourLordshipindicateorexplaintomeinwhatdirectionistheThirdDimension,unknowntome?
Stranger.Icamefromit.Itisupaboveanddownbelow.
I.MyLordmeansseeminglythatitisNorthwardandSouthward.
Stranger.Imeannothingofthekind.Imeanadirectioninwhichyoucannotlook,becauseyouhavenoeyeinyourside.
I.Pardonme,myLord,amoment'sinspectionwillconvinceyourLordshipthatIhaveaperfectlyluminaryatthejunctureofmytwosides.
Stranger:Yes:butinordertoseeintoSpaceyououghttohaveaneye,notonyourPerimeter,butonyourside,thatis,onwhatyouwouldprobablycallyourinside;butweinSpacelandshouldcallityourside.
I.Aneyeinmyinside!Aneyeinmystomach!YourLordshipjests.
Stranger.Iaminnojestinghumour.
ItellyouthatIcomefromSpace,or,sinceyouwillnotunderstandwhatSpacemeans,fromtheLandofThreeDimensionswhenceIbutlatelylookeddownuponyourPlanewhichyoucallSpaceforsooth.
FromthatpositionofadvantageIdiscernedallthatyouspeakofassolid(bywhichyoumean"enclosedonfoursides"),yourhouses,yourchurches,yourverychestsandsafes,yesevenyourinsidesandstomachs,alllyingopenandexposedtomyview.
I.Suchassertionsareeasilymade,myLord.
Stranger.Butnoteasilyproved,youmean.ButImeantoprovemine.
WhenIdescendedhere,IsawyourfourSons,thePentagons,eachinhisapartment,andyourtwoGrandsonstheHexagons;IsawyouryoungestHexagonremainawhilewithyouandthenretiretohisroom,leavingyouandyourWifealone.
IsawyourIsoscelesservants,threeinnumber,inthekitchenatsupper,andthelittlePageinthescullery.
ThenIcamehere,andhowdoyouthinkIcame?
I.Throughtheroof,Isuppose.
Strange.Notso.
Yourroof,asyouknowverywell,hasbeenrecentlyrepaired,andhasnoaperturebywhichevenaWomancouldpenetrate.ItellyouIcomefromSpace.
AreyounotconvincedbywhatIhavetoldyouofyourchildrenandhousehold?
I.YourLordshipmustbeawarethatsuchfactstouchingthebelongingsofhishumbleservantmightbeeasilyascertainedbyanyoneoftheneighbourhoodpossessingyourLordship'samplemeansofinformation.
Stranger.(Tohimself.)WhatmustIdo?Stay;onemoreargumentsuggestsitselftome.WhenyouseeaStraightLine—yourwife,forexample—howmanyDimensionsdoyouattributetoher?
I.YourLordshipwouldtreatmeasifIwereoneofthevulgarwho,beingignorantofMathematics,supposethataWomanisreallyaStraightLine,andonlyofOneDimension.
No,no,myLord;weSquaresarebetteradvised,andareaswellawareofyourLordshipthataWoman,thoughpopularlycalledaStraightLine,is,reallyandscientifically,averythinParallelogram,possessingTwoDimensions,liketherestofus,viz.,lengthandbreadth(orthickness).
Stranger.ButtheveryfactthataLineisvisibleimpliesthatitpossessesyetanotherDimension.
I.MyLord,IhavejustacknowledgethataWomanisbroadaswellaslong.Weseeherlength,weinferherbreadth;which,thoughveryslight,iscapableofmeasurement.
Stranger.Youdonotunderstandme.
ImeanthatwhenyouseeaWoman,youought—besidesinferringherbreadth—toseeherlength,andtoseewhatwecallherheight;althoughthelastDimensionisinfinitesimalinyourcountry.
IfaLineweremerelengthwithout"height,"itwouldceasetooccupySpaceandwouldbecomeinvisible.Surelyyoumustrecognizethis?
I.ImustindeedconfessthatIdonotintheleastunderstandyourLordship.
WhenweinFlatlandseeaLine,weseelengthandbrightness.
Ifthebrightnessdisappears,theLineisextinguished,and,asyousay,ceasestooccupySpace.
ButamItosupposethatyourLordshipgivesthebrightnessthetitleofaDimension,andthatwhatwecall"bright"youcall"high"?
Stranger.No,indeed.By"height"ImeanaDimensionlikeyourlength:only,withyou,"height"isnotsoeasilyperceptible,beingextremelysmall.
I.MyLord,yourassertioniseasilyputtothetest.
YousayIhaveaThirdDimension,whichyoucall"height."
Now,Dimensionimpliesdirectionandmeasurement.
Dobutmeasuremy"height,"ormerelyindivatetomethedirectioninwhichmy"height"extends,andIwillbecomeyourconvert.
Otherwise,yourLordship'sownunderstandmustholdmeexcused.
Stranger.(Tohimself.)Icandoneither.HowshallIconvincehim?Surelyaplainstatementoffactsfollowedbyoculardemonstrationoughttosuffice.—Now,Sir;listentome.
YouarelivingonaPlane.WhatyoustyleFlatlandisthevastlevelsurfaceofwhatImaycallafluid,orin,thetopofwhichyouandyourcountrymenmoveabout,withoutrisingaboveorfallingbelowit.IamnotaplaneFigure,butaSolid.
YoucallmeaCircle;butinrealityIamnotaCircle,butaninfinitenumberofCircles,ofsizevaryingfromaPointtoaCircleofthirteeninchesindiameter,oneplacedonthetopoftheother.
WhenIcutthroughyourplaneasIamnowdoing,Imakeinyourplaneasectionwhichyou,veryrightly,callaCircle.
ForevenaSphere—whichismypropernameinmyowncountry—ifhemanifesthimselfatalltoaninhabitantofFlatland—mustneedsmanfesthimselfasaCircle.
Doyounotremember—forI,whoseeallthings,discernedlastnightthephantasmalvisionofLinelandwrittenuponyourbrain—doyounotremember,Isay,howwhenyouenteredtherealmofLineland,youwerecompelledtomanifestyourselftotheKing,notasaSquare,butasaLine,becausethatLinearRealmhadnotDimensionsenoughtorepresentthewholeofyou,butonlyasliceorsectionofyou?
Inpreciselythesameway,yourcountryofTwoDimensionsisnotspaciousenoughtorepresentme,abeingofThree,butcanonlyexhibitasliceorsectionofme,whichiswhatyoucallaCircle.
Thediminishedbrightnessofyoureyeindicatesincredulity.
Butnowpreparetoreceiveproofpositiveofthetruthofmyassertions.
Youcannotindeedseemorethanoneofmysections,orCircles,atatime;foryouhavenopowertoraiseyoureyeoutoftheplaneofFlatland;butyoucanatleastseethat,asIriseinSpace,somysectionsbecomesmaller.
Seenow,Iwillrise;andtheeffectuponyoureyewillbethatmyCirclewillbecomesmallerandsmallertillitdwindlestoapointandfinallyvanishes.
Therewasno"rising"thatIcouldsee;buthediminishedandfinallyvanished.
IwinkedonceortwicetomakesurethatIwasnotdreaming.Butitwasnodream.
Forfromthedepthsofnowherecameforthahollowvoice—closetomyheartitseemed—"AmIquitegone?Areyouconvincednow?
Well,nowIwillgraduallyreturntoFlatlandandyoushallseemysectionbecomelargerandlarger."
EveryreaderinSpacelandwilleasilyunderstandthatmymysteriousGuestwasspeakingthelanguageoftruthandevenofsimplicity.
Buttome,proficientthoughIwasinFlatlandMathematics,itwasbynomeansasimplematter.
TheroughdiagramgivenabovewillmakeitcleartoanySpacelandchildthattheSphere,ascendinginthethreepositionsindicatedthere,mustneedshavemanifestedhimselftome,ortoanyFlatlander,asaCircle,atfirstoffullsize,thensmall,andatlastverysmallindeed,approachingtoaPoint.
Buttome,althoughIsawthefactsbeforeme,thecauseswereasdarkasever.
AllthatIcouldcomprehendwas,thattheCirclehadmadehimselfsmallerandvanished,andthathehadnowre-appearedandwasrapidlymakinghimselflarger.
Whenheregainedhisoriginalsize,heheavedadeepsigh;forheperceivedbymysilencethatIhadaltogetherfailedtocomprehendhim.
AndindeedIwasnowincliningtothebeliefthathemustbenoCircleatall,butsomeextremelycleverjuggler;orelsethattheoldwives'talesweretrue,andthatafterallthereweresuchpeopleasEnchantersandMagicians.
Afteralongpausehemutteredtohimself,"Oneresourcealoneremains,ifIamnottoresorttoaction.ImusttrythemethodofAnalogy."
Thenfollwedastilllongersilence,afterwhichhecontinuedourdialogue.
Sphere.Tellme,Mr.Mathematician;ifaPointmovesNorthward,andleavesaluminouswake,whatnamewouldyougivetothewake?
I.AstraightLine.
Sphere.AndastraightLinehashowmanyextremities?
I.Two.
Sphere.NowconceivetheNorthwardstraightLinemomvingparalleltoitself,EastandWest,sothateverypointinitleavesbehinditthewakeofastraightLine.
WhatnamewillyougivetotheFiguretherebyformed?
Wewillsupposethatitmovesthroughadistanceequaltotheoriginalstraightline.—-Whatname,Isay?
I.Asquare.
Sphere.AndhowmanysideshasaSquare?Howmanyangles?
I.Foursidesandfourangles.
Sphere.Nowstretchyourimaginationalittle,andconceiveaSquareinFlatland,movingparalleltoitselfupward.
I.What?Northward?
Sphere.No,notNorthward;upward;outofFlatlandaltogether.
IfitmovedNorthward,theSouthernpointsintheSquarewouldhavetomovethroughthepositionspreviouslyoccupiedbytheNorthernpoints.Butthatisnotmymeaning.
ImeanthateveryPointinyou—foryouareaSquareandwillservethepurposeofmyillustration—everyPointinyou,thatistosayinwhatyoucallyourinside,istopassupwardsthroughSpaceinsuchawaythatnoPointshallpassthroughthepositionpreviouslyoccupiedbyanyotherPoint;buteachPointshalldescribeastraightLineofitsown.
ThisisallinaccordancewithAnalogy;surelyitmustbecleartoyou.
Restrainingmyimpatience—forIwasnowunderastrongtemptationtorushblindlyatmyVisitorandtoprecipitatehimintoSpace,oroutofFlatland,anywhere,sothatIcouldgetridofhim—Ireplied:—
"AndwhatmaybethenatureoftheFigurewhichIamtoshapeoutbythismotionwhichyouarepleasedtodenotebytheword'upward'?IpresumeitisdescribableinthelanguageofFlatland."
Sphere.Oh,certainly.
Itisallplainandsimple,andinstrictaccordancewithAnalogy—only,bytheway,youmustnotspeakoftheresultasbeingaFigure,butasaSolid.ButIwilldescribeittoyou.OrrathernotI,butAnalogy.
WebeganwithasinglePoint,whichofcourse—beingitselfaPoine—hasonlyoneterminalPoint.
OnePointproducesaLinewithtwoterminalPoints.
OneLineproducesaSquarewithfourterminalPoints.
Nowyoucangiveyourselftheanswertoyourownquestion:1,2,4,areevidentlyinGeometricalProgression.Whatisthenextnumber?
I.Eight.
Sphere.Exactly.TheoneSquareproducesaSomething-which-you-do-not-as-yet-know-a-name-for-but-which-we-call-a-CubewitheightterminalPoints.Nowareyouconvinced?
I.AndhasthisCreaturesides,aswellasAnglesorwhatyoucall"terminalPoints"?
Sphere.Ofcourse;andallaccordingtoAnalogy.But,bytheway,notwhatyoucallsides,butwhatwecallsides.Youwouldcallthemsolids.
I.AndhowmanysolidsorsideswillappertaintothisBeingwhomIamtogeneratebythemotionofmyinsideinan"upward"direction,andwhomyoucallaCube?
Sphere.Howcanyouask?Andyouamathematician!
Thesideofanythingisalways,ifImaysosay,oneDimensionbehindthething.
Consequently,asthereisnoDimensionbehindaPoint,aPointhas0sides;aLine,ifImaysosay,has2sides(forthepointsofaLinemaybecalledbycourtesy,itssides);aSquarehas4sides;0,2,4;whatProgressiondoyoucallthat?
I.Arithmetical.
Sphere.Andwhatisthenextnumber?
I.Six.
Sphere.Exactly.Thenyouseeyouhaveansweredyourownquestion.TheCubewhichyouwillgeneratewillbeboundedbysixsides,thatistosay,sixofyourinsides.Youseeitallnow,eh?
"Monster,"Ishrieked,"bethoujuggler,enchanter,dream,ordevil,nomorewillIendurethymockeries.EitherthouorImustperish."AndsayingthesewordsIprecipitatedmyselfuponhim.
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