ApproachingoneofthelargestofwhatIthoughttobeWomen，Iaccostedher，butreceivednoanswer。Asecondandathirdappealonmypartwereequallyineffectual。Losingpatienceatwhatappearedtomeintolerablerudeness，Ibroughtmymouthintoapositionfullinfrontofhermouthsoastointercepthermotion，andloudlyrepeatedmyquestion，\"Woman，whatsignifiesthisconcourse，andthisstrangeandconfusedchirping，andthismonotonousmotiontoandfroinoneandthesameStraightLine？\"
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TheKING’SeyesmuchlargerthantherealityshewingthatHISMAJESTY
couldseenothingbutapoint。
\"IamnoWoman，\"repliedthesmallLine。\"IamtheMonarchoftheworld。Butthou，whenceintrudestthouintomyrealmofLineland？\"Receivingthisabruptreply，IbeggedpardonifIhadinanywaystartledormolestedhisRoyalHighness;
anddescribingmyselfasastrangerIbesoughttheKingtogivemesomeaccountofhisdominions。ButIhadthegreatestpossibledifficultyinobtaininganyinformationonpointsthatreallyinterestedme;fortheMonarchcouldnotrefrainfromconstantlyassumingthatwhateverwasfamiliartohimmustalsobeknowntomeandthatIwassimulatingignoranceinjest。However，byperseveringquestionsIelicitedthefollowingfacts：
ItseemedthatthispoorignorantMonarch——ashecalledhimself——
waspersuadedthattheStraightLinewhichhecalledhisKingdom，andinwhichhepassedhisexistence，constitutedthewholeoftheworld，andindeedthewholeofSpace。Notbeingableeithertomoveortosee，saveinhisStraightLine，hehadnoconceptionofanythingoutofit。ThoughhehadheardmyvoicewhenIfirstaddressedhim，thesoundshadcometohiminamannersocontrarytohisexperiencethathehadmadenoanswer，\"seeingnoman\"，asheexpressedit，\"andhearingavoiceasitwerefrommyownintestines。\"UntilthemomentwhenIplacedmymouthinhisWorld，hehadneitherseenme，norheardanythingexceptconfusedsoundsbeatingagainst——whatIcalledhisside，butwhathecalledhisINSIDEorSTOMACH;norhadheevennowtheleastconceptionoftheregionfromwhichIhadcome。
OutsidehisWorld，orLine，allwasablanktohim;nay，notevenablank，forablankimpliesSpace;say，rather，allwasnon-existent。
Hissubjects——ofwhomthesmallLinesweremenandthePointsWomen——wereallalikeconfinedinmotionandeye-sighttothatsingleStraightLine，whichwastheirWorld。ItneedscarcelybeaddedthatthewholeoftheirhorizonwaslimitedtoaPoint;norcouldanyoneeverseeanythingbutaPoint。Man，woman，child，thing——eachwasaPointtotheeyeofaLinelander。Onlybythesoundofthevoicecouldsexoragebedistinguished。Moreover，aseachindividualoccupiedthewholeofthenarrowpath，sotospeak，whichconstitutedhisUniverse，andnoonecouldmovetotherightorlefttomakewayforpassersby，itfollowedthatnoLinelandercouldeverpassanother。Onceneighbours，alwaysneighbours。
Neighbourhoodwiththemwaslikemarriagewithus。
Neighboursremainedneighbourstilldeathdidthempart。
Suchalife，withallvisionlimitedtoaPoint，andallmotiontoaStraightLine，seemedtomeinexpressiblydreary;andIwassurprisedtonotethevivacityandcheerfulnessoftheKing。
Wonderingwhetheritwaspossible，amidcircumstancessounfavourabletodomesticrelations，toenjoythepleasuresofconjugalunion，IhesitatedforsometimetoquestionhisRoyalHighnessonsodelicateasubject;butatlastIplungedintoitbyabruptlyinquiringastothehealthofhisfamily。
\"Mywivesandchildren，\"hereplied，\"arewellandhappy。\"
Staggeredatthisanswer——forintheimmediateproximityoftheMonarch（asIhadnotedinmydreambeforeIenteredLineland）
therewerenonebutMen——Iventuredtoreply，\"Pardonme，butIcannotimaginehowyourRoyalHighnesscanatanytimeeitherseeorapproachtheirMajesties，whenthereareatleasthalfadozeninterveningindividuals，whomyoucanneitherseethrough，norpassby？IsitpossiblethatinLinelandproximityisnotnecessaryformarriageandforthegenerationofchildren？\"
\"Howcanyouasksoabsurdaquestion？\"repliedtheMonarch。
\"Ifitwereindeedasyousuggest，theUniversewouldsoonbedepopulated。No，no;neighbourhoodisneedlessfortheunionofhearts;andthebirthofchildrenistooimportantamattertohavebeenallowedtodependuponsuchanaccidentasproximity。
Youcannotbeignorantofthis。Yetsinceyouarepleasedtoaffectignorance，IwillinstructyouasifyouweretheveriestbabyinLineland。Know，then，thatmarriagesareconsummatedbymeansofthefacultyofsoundandthesenseofhearing。
\"YouareofcourseawarethateveryManhastwomouthsorvoices——aswellastwoeyes——abassatoneandatenorattheotherofhisextremities。Ishouldnotmentionthis，butthatIhavebeenunabletodistinguishyourtenorinthecourseofourconversation。\"
IrepliedthatIhadbutonevoice，andthatIhadnotbeenawarethathisRoyalHighnesshadtwo。\"Thatconfirmsmyimpression，\"
saidtheKing，\"thatyouarenotaMan，butafeminineMonstrositywithabassvoice，andanutterlyuneducatedear。Buttocontinue。
\"NaturehavingherselfordainedthateveryManshouldwedtwowives——\"
\"Whytwo？\"askedI。\"Youcarryyouraffectedsimplicitytoofar\"，hecried。\"HowcantherebeacompletelyharmoniousunionwithoutthecombinationoftheFourinOne，viz。theBassandTenoroftheManandtheSopranoandContraltoofthetwoWomen？\"
\"Butsupposing，\"saidI，\"thatamanshouldpreferonewifeorthree？\"
\"Itisimpossible，\"hesaid;\"itisasinconceivableasthattwoandoneshouldmakefive，orthatthehumaneyeshouldseeaStraightLine。\"Iwouldhaveinterruptedhim;butheproceededasfollows：
\"OnceinthemiddleofeachweekaLawofNaturecompelsustomovetoandfrowitharhythmicmotionofmorethanusualviolence，whichcontinuesforthetimeyouwouldtaketocountahundredandone。Inthemidstofthischoraldance，atthefifty-firstpulsation，theinhabitantsoftheUniversepauseinfullcareer，andeachindividualsendsforthhisrichest，fullest，sweeteststrain。Itisinthisdecisivemomentthatallourmarriagesaremade。SoexquisiteistheadaptationofBasstoTreble，ofTenortoContralto，thatoftentimestheLovedOnes，thoughtwentythousandleaguesaway，recognizeatoncetheresponsivenoteoftheirdestinedLover;and，penetratingthepaltryobstaclesofdistance，Loveunitesthethree。
ThemarriageinthatinstantconsummatedresultsinathreefoldMaleandFemaleoffspringwhichtakesitsplaceinLineland。\"
\"What！Alwaysthreefold？\"saidI。\"Mustonewifethenalwayshavetwins？\"
\"Bass-voicedMonstrosity！yes，\"repliedtheKing。\"HowelsecouldthebalanceoftheSexesbemaintained，iftwogirlswerenotbornforeveryboy？WouldyouignoretheveryAlphabetofNature？\"
Heceased，speechlessforfury;andsometimeelapsedbeforeIcouldinducehimtoresumehisnarrative。
\"Youwillnot，ofcourse，supposethateverybacheloramongusfindshismatesatthefirstwooinginthisuniversalMarriageChorus。
Onthecontrary，theprocessisbymostofusmanytimesrepeated。
Fewaretheheartswhosehappylotitisatoncetorecognizeineachother’svoicesthepartnerintendedforthembyProvidence，andtoflyintoareciprocalandperfectlyharmoniousembrace。
Withmostofusthecourtshipisoflongduration。TheWooer’svoicesmayperhapsaccordwithoneofthefuturewives，butnotwithboth;
ornot，atfirst，witheither;ortheSopranoandContraltomaynotquiteharmonize。InsuchcasesNaturehasprovidedthateveryweeklyChorusshallbringthethreeLoversintocloserharmony。
Eachtrialofvoice，eachfreshdiscoveryofdiscord，almostimperceptiblyinducesthelessperfecttomodifyhisorhervocalutterancesoastoapproximatetothemoreperfect。
Andaftermanytrialsandmanyapproximations，theresultisatlastachieved。Therecomesadayatlast，when，whilethewontedMarriageChorusgoesforthfromuniversalLineland，thethreefar-offLoverssuddenlyfindthemselvesinexactharmony，and，beforetheyareawake，theweddedTripletisraptvocallyintoaduplicateembrace;andNaturerejoicesoveronemoremarriageandoverthreemorebirths。\"
Section14。HowIvainlytriedtoexplainthenatureofFlatlandThinkingthatitwastimetobringdowntheMonarchfromhisrapturestothelevelofcommonsense，Ideterminedtoendeavourtoopenuptohimsomeglimpsesofthetruth，thatistosayofthenatureofthingsinFlatland。SoIbeganthus：
\"HowdoesyourRoyalHighnessdistinguishtheshapesandpositionsofhissubjects？Iformypartnoticedbythesenseofsight，beforeIenteredyourKingdom，thatsomeofyourpeopleareLinesandothersPoints，andthatsomeoftheLinesarelarger——\"
\"Youspeakofanimpossibility，\"interruptedtheKing;
\"youmusthaveseenavision;fortodetectthedifferencebetweenaLineandaPointbythesenseofsightis，aseveryoneknows，inthenatureofthings，impossible;butitcanbedetectedbythesenseofhearing，andbythesamemeansmyshapecanbeexactlyascertained。Beholdme——IamaLine，thelongestinLineland，oversixinchesofSpace——\"\"OfLength\"，Iventuredtosuggest。\"Fool，\"saidhe，\"SpaceisLength。
Interruptmeagain，andIhavedone。\"
Iapologized;buthecontinuedscornfully，\"Sinceyouareimpervioustoargument，youshallhearwithyourearshowbymeansofmytwovoicesIrevealmyshapetomyWives，whoareatthismomentsixthousandmilesseventyyardstwofeeteightinchesaway，theonetotheNorth，theothertotheSouth。Listen，Icalltothem。\"
Hechirruped，andthencomplacentlycontinued：\"Mywivesatthismomentreceivingthesoundofoneofmyvoices，closelyfollowedbytheother，andperceivingthatthelatterreachesthemafteranintervalinwhichsoundcantraverse6。457inches，inferthatoneofmymouthsis6。457inchesfurtherfromthemthantheother，andaccordinglyknowmyshapetobe6。457inches。Butyouwillofcourseunderstandthatmywivesdonotmakethiscalculationeverytimetheyhearmytwovoices。Theymadeit，onceforall，beforeweweremarried。ButtheyCOULDmakeitatanytime。
AndinthesamewayIcanestimatetheshapeofanyofmyMalesubjectsbythesenseofsound。\"
\"Buthow，\"saidI，\"ifaManfeignsaWoman’svoicewithoneofhistwovoices，orsodisguiseshisSouthernvoicethatitcannotberecognizedastheechooftheNorthern？Maynotsuchdeceptionscausegreatinconvenience？Andhaveyounomeansofcheckingfraudsofthiskindbycommandingyourneighbouringsubjectstofeeloneanother？\"Thisofcoursewasaverystupidquestion，forfeelingcouldnothaveansweredthepurpose;butIaskedwiththeviewofirritatingtheMonarch，andIsucceededperfectly。
\"What！\"criedheinhorror，\"explainyourmeaning。\"\"Feel，touch，comeintocontact，\"Ireplied。\"IfyoumeanbyFEELING，\"
saidtheKing，\"approachingsocloseastoleavenospacebetweentwoindividuals，know，Stranger，thatthisoffenceispunishableinmydominionsbydeath。Andthereasonisobvious。
ThefrailformofaWoman，beingliabletobeshatteredbysuchanapproximation，mustbepreservedbytheState;
butsinceWomencannotbedistinguishedbythesenseofsightfromMen，theLawordainsuniversallythatneitherMannorWomanshallbeapproachedsocloselyastodestroytheintervalbetweentheapproximatorandtheapproximated。
\"AndindeedwhatpossiblepurposewouldbeservedbythisillegalandunnaturalexcessofapproximationwhichyoucallTOUCHING，whenalltheendsofsobrutalandcoarseaprocessareattainedatoncemoreeasilyandmoreexactlybythesenseofhearing？
Astoyoursuggesteddangerofdeception，itisnon-existent：
fortheVoice，beingtheessenceofone’sBeing，cannotbethuschangedatwill。Butcome，supposethatIhadthepowerofpassingthroughsolidthings，sothatIcouldpenetratemysubjects，oneafteranother，eventothenumberofabillion，verifyingthesizeanddistanceofeachbythesenseofFEELING：howmuchtimeandenergywouldbewastedinthisclumsyandinaccuratemethod！
Whereasnow，inonemomentofaudition，Itakeasitwerethecensusandstatistics，local，corporeal，mentalandspiritual，ofeverylivingbeinginLineland。Hark，onlyhark！\"
Sosayinghepausedandlistened，asifinanecstasy，toasoundwhichseemedtomenobetterthanatinychirpingfromaninnumerablemultitudeoflilliputiangrasshoppers。
\"Truly，\"repliedI，\"yoursenseofhearingservesyouingoodstead，andfillsupmanyofyourdeficiencies。ButpermitmetopointoutthatyourlifeinLinelandmustbedeplorablydull。ToseenothingbutaPoint！NoteventobeabletocontemplateaStraightLine！
Nay，noteventoknowwhataStraightLineis！Tosee，yetbecutofffromthoseLinearprospectswhicharevouchsafedtousinFlatland！
Bettersurelytohavenosenseofsightatallthantoseesolittle！
IgrantyouIhavenotyourdiscriminativefacultyofhearing;
fortheconcertofallLinelandwhichgivesyousuchintensepleasure，istomenobetterthanamultitudinoustwitteringorchirping。
ButatleastIcandiscern，bysight，aLinefromaPoint。
Andletmeproveit。JustbeforeIcameintoyourkingdom，Isawyoudancingfromlefttoright，andthenfromrighttoleft，withSevenMenandaWomaninyourimmediateproximityontheleft，andeightMenandtwoWomenonyourright。Isnotthiscorrect？\"
\"Itiscorrect，\"saidtheKing，\"sofarasthenumbersandsexesareconcerned，thoughIknownotwhatyoumeanby’right’and’left’。
ButIdenythatyousawthesethings。ForhowcouldyouseetheLine，thatistosaytheinside，ofanyMan？Butyoumusthaveheardthesethings，andthendreamedthatyousawthem。
Andletmeaskwhatyoumeanbythosewords’left’and’right’。
IsupposeitisyourwayofsayingNorthwardandSouthward。\"
\"Notso，\"repliedI;\"besidesyourmotionofNorthwardandSouthward，thereisanothermotionwhichIcallfromrighttoleft。\"
KING。Exhibittome，ifyouplease，thismotionfromlefttoright。
I。Nay，thatIcannotdo，unlessyoucouldstepoutofyourLinealtogether。
KING。OutofmyLine？Doyoumeanoutoftheworld？OutofSpace？
I。Well，yes。OutofYOURWorld。OutofYOURSpace。
ForyourSpaceisnotthetrueSpace。TrueSpaceisaPlane;
butyourSpaceisonlyaLine。
KING。Ifyoucannotindicatethismotionfromlefttorightbyyourselfmovinginit，thenIbegyoutodescribeittomeinwords。
I。Ifyoucannottellyourrightsidefromyourleft，Ifearthatnowordsofminecanmakemymeaningcleartoyou。
Butsurelyyoucannotbeignorantofsosimpleadistinction。
KING。Idonotintheleastunderstandyou。
I。Alas！HowshallImakeitclear？Whenyoumovestraighton，doesitnotsometimesoccurtoyouthatyouCOULDmoveinsomeotherway，turningyoureyeroundsoastolookinthedirectiontowardswhichyoursideisnowfronting？
Inotherwords，insteadofalwaysmovinginthedirectionofoneofyourextremities，doyouneverfeeladesiretomoveinthedirection，sotospeak，ofyourside？
KING。Never。Andwhatdoyoumean？Howcanaman’sinside\"front\"inanydirection？Orhowcanamanmoveinthedirectionofhisinside？
I。Wellthen，sincewordscannotexplainthematter，Iwilltrydeeds，andwillmovegraduallyoutofLinelandinthedirectionwhichIdesiretoindicatetoyou。
AtthewordIbegantomovemybodyoutofLineland。
Aslongasanypartofmeremainedinhisdominionandinhisview，theKingkeptexclaiming，\"Iseeyou，Iseeyoustill;
youarenotmoving。\"ButwhenIhadatlastmovedmyselfoutofhisLine，hecriedinhisshrillestvoice，\"Sheisvanished;
sheisdead。\"\"Iamnotdead，\"repliedI;\"IamsimplyoutofLineland，thatistosay，outoftheStraightLinewhichyoucallSpace，andinthetrueSpace，whereIcanseethingsastheyare。AndatthismomentIcanseeyourLine，orside——
orinsideasyouarepleasedtocallit;andIcanseealsotheMenandWomenontheNorthandSouthofyou，whomIwillnowenumerate，describingtheirorder，theirsize，andtheintervalbetweeneach。\"
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WhenIhaddonethisatgreatlength，Icriedtriumphantly，\"Doesthatatlastconvinceyou？\"And，withthat，IoncemoreenteredLineland，takingupthesamepositionasbefore。
ButtheMonarchreplied，\"IfyouwereaManofsense——though，asyouappeartohaveonlyonevoiceIhavelittledoubtyouarenotaManbutaWoman——but，ifyouhadaparticleofsense，youwouldlistentoreason。YouaskmetobelievethatthereisanotherLinebesidesthatwhichmysensesindicate，andanothermotionbesidesthatofwhichIamdailyconscious。I，inreturn，askyoutodescribeinwordsorindicatebymotionthatotherLineofwhichyouspeak。Insteadofmoving，youmerelyexercisesomemagicartofvanishingandreturningtosight;andinsteadofanyluciddescriptionofyournewWorld，yousimplytellmethenumbersandsizesofsomefortyofmyretinue，factsknowntoanychildinmycapital。Cananythingbemoreirrationaloraudacious？Acknowledgeyourfollyordepartfrommydominions。\"
Furiousathisperversity，andespeciallyindignantthatheprofessedtobeignorantofmysex，Iretortedinnomeasuredterms，\"BesottedBeing！Youthinkyourselftheperfectionofexistence，whileyouareinrealitythemostimperfectandimbecile。
Youprofesstosee，whereasyoucanseenothingbutaPoint！
YouplumeyourselfoninferringtheexistenceofaStraightLine;
butICANSEEStraightLines，andinfertheexistenceofAngles，Triangles，Squares，Pentagons，Hexagons，andevenCircles。
Whywastemorewords？SufficeitthatIamthecompletionofyourincompleteself。YouareaLine，butIamaLineofLines，calledinmycountryaSquare：andevenI，infinitelysuperiorthoughIamtoyou，amoflittleaccountamongthegreatnoblesofFlatland，whenceIhavecometovisityou，inthehopeofenlighteningyourignorance。\"
HearingthesewordstheKingadvancedtowardsmewithamenacingcryasiftopiercemethroughthediagonal;andinthatsamemomenttherearosefrommyriadsofhissubjectsamultitudinouswar-cry，increasinginvehemencetillatlastmethoughtitrivalledtheroarofanarmyofahundredthousandIsosceles，andtheartilleryofathousandPentagons。Spell-boundandmotionless，Icouldneitherspeaknormovetoaverttheimpendingdestruction;
andstillthenoisegrewlouder，andtheKingcamecloser，whenIawoketofindthebreakfast-bellrecallingmetotherealitiesofFlatland。
Section15。ConcerningaStrangerfromSpacelandFromdreamsIproceedtofacts。
Itwasthelastdayofthe1999thyearofourera。
Thepatteringoftherainhadlongagoannouncednightfall;
andIwassittinginthecompanyofmywife，musingontheeventsofthepastandtheprospectsofthecomingyear，thecomingcentury，thecomingMillennium。
[Note：WhenIsay\"sitting\"，ofcourseIdonotmeananychangeofattitudesuchasyouinSpacelandsignifybythatword;
foraswehavenofeet，wecannomore\"sit\"nor\"stand\"
（inyoursenseoftheword）thanoneofyoursolesorflounders。
Nevertheless，weperfectlywellrecognizethedifferentmentalstatesofvolitionimpliedin\"lying\"，\"sitting\"，and\"standing\"，whicharetosomeextentindicatedtoabeholderbyaslightincreaseoflustrecorrespondingtotheincreaseofvolition。
Butonthis，andathousandotherkindredsubjects，timeforbidsmetodwell。]
MyfourSonsandtwoorphanGrandchildrenhadretiredtotheirseveralapartments;andmywifealoneremainedwithmetoseetheoldMillenniumoutandthenewonein。
Iwasraptinthought，ponderinginmymindsomewordsthathadcasuallyissuedfromthemouthofmyyoungestGrandson，amostpromisingyoungHexagonofunusualbrilliancyandperfectangularity。HisunclesandIhadbeengivinghimhisusualpracticallessoninSightRecognition，turningourselvesuponourcentres，nowrapidly，nowmoreslowly，andquestioninghimastoourpositions;andhisanswershadbeensosatisfactorythatIhadbeeninducedtorewardhimbygivinghimafewhintsonArithmetic，asappliedtoGeometry。
TakingnineSquares，eachanincheveryway，IhadputthemtogethersoastomakeonelargeSquare，withasideofthreeinches，andIhadhenceprovedtomylittleGrandsonthat——thoughitwasimpossibleforustoSEEtheinsideoftheSquare——
yetwemightascertainthenumberofsquareinchesinaSquarebysimplysquaringthenumberofinchesintheside：\"andthus，\"
saidI，\"weknowthat3^2，or9，representsthenumberofsquareinchesinaSquarewhosesideis3incheslong。\"
ThelittleHexagonmeditatedonthisawhileandthensaidtome;
\"Butyouhavebeenteachingmetoraisenumberstothethirdpower：
Isuppose3^3mustmeansomethinginGeometry;whatdoesitmean？\"
\"Nothingatall，\"repliedI，\"notatleastinGeometry;
forGeometryhasonlyTwoDimensions。\"AndthenIbegantoshewtheboyhowaPointbymovingthroughalengthofthreeinchesmakesaLineofthreeinches，whichmayberepresentedby3;
andhowaLineofthreeinches，movingparalleltoitselfthroughalengthofthreeinches，makesaSquareofthreeincheseveryway，whichmayberepresentedby3^2。
Uponthis，myGrandson，againreturningtohisformersuggestion，tookmeuprathersuddenlyandexclaimed，\"Well，then，ifaPointbymovingthreeinches，makesaLineofthreeinchesrepresentedby3;andifastraightLineofthreeinches，movingparalleltoitself，makesaSquareofthreeincheseveryway，representedby3^2;itmustbethataSquareofthreeincheseveryway，movingsomehowparalleltoitself（butIdon’tseehow）
mustmakeSomethingelse（butIdon’tseewhat）ofthreeincheseveryway——andthismustberepresentedby3^3。\"
\"Gotobed，\"saidI，alittleruffledbythisinterruption：
\"ifyouwouldtalklessnonsense，youwouldremembermoresense。\"
SomyGrandsonhaddisappearedindisgrace;andthereIsatbymyWife’sside，endeavouringtoformaretrospectoftheyear1999
andofthepossibilitiesoftheyear2000，butnotquiteabletoshakeoffthethoughtssuggestedbytheprattleofmybrightlittleHexagon。Onlyafewsandsnowremainedinthehalf-hourglass。
RousingmyselffrommyreverieIturnedtheglassNorthwardforthelasttimeintheoldMillennium;andintheact，Iexclaimedaloud，\"Theboyisafool。\"
StraightwayIbecameconsciousofaPresenceintheroom，andachillingbreaththrilledthroughmyverybeing。
\"Heisnosuchthing，\"criedmyWife，\"andyouarebreakingtheCommandmentsinthusdishonouringyourownGrandson。\"
ButItooknonoticeofher。LookingroundineverydirectionIcouldseenothing;yetstillIFELTaPresence，andshiveredasthecoldwhispercameagain。Istartedup。\"Whatisthematter？\"
saidmyWife，\"thereisnodraught;whatareyoulookingfor？
Thereisnothing。\"Therewasnothing;andIresumedmyseat，againexclaiming，\"Theboyisafool，Isay;3^3canhavenomeaninginGeometry。\"Atoncetherecameadistinctlyaudiblereply，\"Theboyisnotafool;and3^3hasanobviousGeometricalmeaning。\"
MyWifeaswellasmyselfheardthewords，althoughshedidnotunderstandtheirmeaning，andbothofussprangforwardinthedirectionofthesound。WhatwasourhorrorwhenwesawbeforeusaFigure！AtthefirstglanceitappearedtobeaWoman，seensideways;butamoment’sobservationshewedmethattheextremitiespassedintodimnesstoorapidlytorepresentoneoftheFemaleSex;andIshouldhavethoughtitaCircle，onlythatitseemedtochangeitssizeinamannerimpossibleforaCircleorforanyregularFigureofwhichIhadhadexperience。
ButmyWifehadnotmyexperience，northecoolnessnecessarytonotethesecharacteristics。WiththeusualhastinessandunreasoningjealousyofherSex，sheflewatoncetotheconclusionthataWomanhadenteredthehousethroughsomesmallaperture。
\"Howcomesthispersonhere？\"sheexclaimed，\"youpromisedme，mydear，thatthereshouldbenoventilatorsinournewhouse。\"
\"Norarethereany，\"saidI;\"butwhatmakesyouthinkthatthestrangerisaWoman？IseebymypowerofSightRecognition————\"
\"Oh，IhavenopatiencewithyourSightRecognition，\"repliedshe，\"’Feelingisbelieving’and’AStraightLinetothetouchisworthaCircletothesight’\"——twoProverbs，verycommonwiththeFrailerSexinFlatland。
\"Well，\"saidI，forIwasafraidofirritatingher，\"ifitmustbeso，demandanintroduction。\"Assuminghermostgraciousmanner，myWifeadvancedtowardstheStranger，\"Permitme，Madam，tofeelandbefeltby————\"then，suddenlyrecoiling，\"Oh！
itisnotaWoman，andtherearenoangleseither，notatraceofone。
CanitbethatIhavesomisbehavedtoaperfectCircle？\"
\"Iamindeed，inacertainsenseaCircle，\"repliedtheVoice，\"andamoreperfectCirclethananyinFlatland;buttospeakmoreaccurately，IammanyCirclesinone。\"Thenheaddedmoremildly，\"Ihaveamessage，dearMadam，toyourhusband，whichImustnotdeliverinyourpresence;and，ifyouwouldsufferustoretireforafewminutes————\"ButmyWifewouldnotlistentotheproposalthatouraugustVisitorshouldsoincommodehimself，andassuringtheCirclethatthehourofherownretirementhadlongpassed，withmanyreiteratedapologiesforherrecentindiscretion，sheatlastretreatedtoherapartment。
Iglancedatthehalf-hourglass。Thelastsandshadfallen。
ThethirdMillenniumhadbegun。
Section16。HowtheStrangervainlyendeavouredtorevealtomeinwordsthemysteriesofSpacelandAssoonasthesoundofthePeace-cryofmydepartingWifehaddiedaway，IbegantoapproachtheStrangerwiththeintentionoftakinganearerviewandofbiddinghimbeseated：
buthisappearancestruckmedumbandmotionlesswithastonishment。
WithouttheslightestsymptomsofangularityheneverthelessvariedeveryinstantwithgradationsofsizeandbrightnessscarcelypossibleforanyFigurewithinthescopeofmyexperience。ThethoughtflashedacrossmethatImighthavebeforemeaburglarorcut-throat，somemonstrousIrregularIsosceles，who，byfeigningthevoiceofaCircle，hadobtainedadmissionsomehowintothehouse，andwasnowpreparingtostabmewithhisacuteangle。
Inasitting-room，theabsenceofFog（andtheseasonhappenedtoberemarkablydry），madeitdifficultformetotrusttoSightRecognition，especiallyattheshortdistanceatwhichIwasstanding。Desperatewithfear，Irushedforwardwithanunceremonious，\"Youmustpermitme，Sir——\"andfelthim。
MyWifewasright。Therewasnotthetraceofanangle，nottheslightestroughnessorinequality：neverinmylifehadImetwithamoreperfectCircle。HeremainedmotionlesswhileIwalkedroundhim，beginningfromhiseyeandreturningtoitagain。
Circularhewasthroughout，aperfectlysatisfactoryCircle;
therecouldnotbeadoubtofit。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’sinspectionwillconvinceyourLordshipthatIhaveaperfectluminaryatthejunctureoftwoofmysides。
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。
STRANGER。Notso。Yourroof，asyouknowverywell，hasbeenrecentlyrepaired，andhasnoaperturebywhichevenaWomancouldpenetrate。ItellyouIcomefromSpace。AreyounotconvincedbywhatIhavetoldyouofyourchildrenandhousehold？
I。YourLordshipmustbeawarethatsuchfactstouchingthebelongingsofhishumbleservantmightbeeasilyascertainedbyanyoneintheneighbourhoodpossessingyourLordship’samplemeansofobtaininginformation。
STRANGER。（TOHIMSELF。）WhatmustIdo？Stay;onemoreargumentsuggestsitselftome。WhenyouseeaStraightLine——yourwife，forexample——howmanyDimensionsdoyouattributetoher？
I。YourLordshipwouldtreatmeasifIwereoneofthevulgarwho，beingignorantofMathematics，supposethataWomanisreallyaStraightLine，andonlyofOneDimension。No，no，myLord;
weSquaresarebetteradvised，andareaswellawareasyourLordshipthataWoman，thoughpopularlycalledaStraightLine，is，reallyandscientifically，averythinParallelogram，possessingTwoDimensions，liketherestofus，viz。，lengthandbreadth（orthickness）。
STRANGER。ButtheveryfactthataLineisvisibleimpliesthatitpossessesyetanotherDimension。
I。MyLord，IhavejustacknowledgedthataWomanisbroadaswellaslong。Weseeherlength，weinferherbreadth;
which，thoughveryslight，iscapableofmeasurement。
STRANGER。Youdonotunderstandme。ImeanthatwhenyouseeaWoman，youought——besidesinferringherbreadth——
toseeherlength，andtoSEEwhatwecallherHEIGHT;
althoughthatlastDimensionisinfinitesimalinyourcountry。
IfaLineweremerelengthwithout\"height\"，itwouldceasetooccupySpaceandwouldbecomeinvisible。Surelyyoumustrecognizethis？
I。ImustindeedconfessthatIdonotintheleastunderstandyourLordship。WhenweinFlatlandseeaLine，weseelengthandBRIGHTNESS。Ifthebrightnessdisappears，theLineisextinguished，and，asyousay，ceasestooccupySpace。
ButamItosupposethatyourLordshipgivestobrightnessthetitleofaDimension，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\"，ormerelyindicatetomethedirectioninwhichmy\"height\"extends，andIwillbecomeyourconvert。Otherwise，yourLordship’sownunderstandingmustholdmeexcused。
STRANGER。（TOHIMSELF。）Icandoneither。HowshallI
convincehim？Surelyaplainstatementoffactsfollowedbyoculardemonstrationoughttosuffice。——Now，Sir;listentome。
YouarelivingonaPlane。WhatyoustyleFlatlandisthevastlevelsurfaceofwhatImaycallafluid，on，orin，thetopofwhichyouandyourcountrymenmoveabout，withoutrisingaboveitorfallingbelowit。
IamnotaplaneFigure，butaSolid。YoucallmeaCircle;
butinrealityIamnotaCircle，butaninfinitenumberofCircles，ofsizevaryingfromaPointtoaCircleofthirteeninchesindiameter，oneplacedonthetopoftheother。WhenIcutthroughyourplaneasIamnowdoing，Imakeinyourplaneasectionwhichyou，veryrightly，callaCircle。ForevenaSphere——
whichismypropernameinmyowncountry——ifhemanifesthimselfatalltoaninhabitantofFlatland——mustneedsmanifesthimselfasaCircle。
Doyounotremember——forI，whoseeallthings，discernedlastnightthephantasmalvisionofLinelandwrittenuponyourbrain——
doyounotremember，Isay，how，whenyouenteredtherealmofLineland，youwerecompelledtomanifestyourselftotheKing，notasaSquare，butasaLine，becausethatLinearRealmhadnotDimensionsenoughtorepresentthewholeofyou，butonlyasliceorsectionofyou？Inpreciselythesameway，yourcountryofTwoDimensionsisnotspaciousenoughtorepresentme，abeingofThree，butcanonlyexhibitasliceorsectionofme，whichiswhatyoucallaCircle。
Thediminishedbrightnessofyoureyeindicatesincredulity。Butnowpreparetoreceiveproofpositiveofthetruthofmyassertions。
Youcannotindeedseemorethanoneofmysections，orCircles，atatime;foryouhavenopowertoraiseyoureyeoutoftheplaneofFlatland;butyoucanatleastseethat，asIriseinSpace，somysectionsbecomesmaller。Seenow，Iwillrise;andtheeffectuponyoureyewillbethatmyCirclewillbecomesmallerandsmallertillitdwindlestoapointandfinallyvanishes。
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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，andthathehadnowreappearedandwasrapidlymakinghimselflarger。
Whenheregainedhisoriginalsize，heheavedadeepsigh;
forheperceivedbymysilencethatIhadaltogetherfailedtocomprehendhim。AndindeedIwasnowincliningtothebeliefthathemustbenoCircleatall，butsomeextremelycleverjuggler;
orelsethattheoldwives’talesweretrue，andthatafterallthereweresuchpeopleasEnchantersandMagicians。
Afteralongpausehemutteredtohimself，\"Oneresourcealoneremains，ifIamnottoresorttoaction。ImusttrythemethodofAnalogy。\"
Thenfollowedastilllongersilence，afterwhichhecontinuedourdialogue。
SPHERE。Tellme，Mr。Mathematician;ifaPointmovesNorthward，andleavesaluminouswake，whatnamewouldyougivetothewake？
I。AstraightLine。
SPHERE。AndastraightLinehashowmanyextremities？
I。Two。
SPHERE。NowconceivetheNorthwardstraightLinemovingparalleltoitself，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——beingitselfaPoint——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-CUBE
withEIGHTterminalPoints。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，ifImaysay，has2sides（forthePointsofaLinemaybecalledbycourtesy，itssides）;
aSquarehas4sides;0，2，4;whatProgressiondoyoucallthat？
I。Arithmetical。