ARomanceofManyDimensionsEdwinA。Abbott（1838-1926。Englishscholar，theologian，andwriter。）
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|\"Fie，fie，howfranticlyIsquaremytalk！\"|
WithIllustrationsbytheAuthor，ASQUARE（EdwinA。Abbott）
ToTheInhabitantsofSPACEINGENERAL
AndH。C。INPARTICULAR
ThisWorkisDedicatedByaHumbleNativeofFlatlandIntheHopethatEvenashewasInitiatedintotheMysteriesOfTHREEDimensionsHavingbeenpreviouslyconversantWithONLYTWO
SotheCitizensofthatCelestialRegionMayaspireyethigherandhigherTotheSecretsofFOURFIVEOREVENSIXDimensionsTherebycontributingTotheEnlargementofTHEIMAGINATION
AndthepossibleDevelopmentOfthatmostrareandexcellentGiftofMODESTY
AmongtheSuperiorRacesOfSOLIDHUMANITY
PrefacetotheSecondandRevisedEdition，1884。
BytheEditorIfmypoorFlatlandfriendretainedthevigourofmindwhichheenjoyedwhenhebegantocomposetheseMemoirs，Ishouldnotnowneedtorepresenthiminthispreface，inwhichhedesires，firstly，toreturnhisthankstohisreadersandcriticsinSpaceland，whoseappreciationhas，withunexpectedcelerity，requiredasecondeditionofhiswork;secondly，toapologizeforcertainerrorsandmisprints（forwhich，however，heisnotentirelyresponsible）;
and，thirdly，toexplainoneortwomisconceptions。ButheisnottheSquareheoncewas。Yearsofimprisonment，andthestillheavierburdenofgeneralincredulityandmockery，havecombinedwiththenaturaldecayofoldagetoerasefromhismindmanyofthethoughtsandnotions，andmuchalsooftheterminology，whichheacquiredduringhisshortstayinSpaceland。Hehas，therefore，requestedmetoreplyinhisbehalftotwospecialobjections，oneofanintellectual，theotherofamoralnature。
Thefirstobjectionis，thataFlatlander，seeingaLine，seessomethingthatmustbeTHICKtotheeyeaswellasLONG
totheeye（otherwiseitwouldnotbevisible，ifithadnotsomethickness）;andconsequentlyheought（itisargued）
toacknowledgethathiscountrymenarenotonlylongandbroad，butalso（thoughdoubtlessinaveryslightdegree）THICKorHIGH。
Thisobjectionisplausible，and，toSpacelanders，almostirresistible，sothat，Iconfess，whenIfirstheardit，Iknewnotwhattoreply。Butmypooroldfriend’sanswerappearstomecompletelytomeetit。
\"Iadmit，\"saidhe——whenImentionedtohimthisobjection——
\"Iadmitthetruthofyourcritic’sfacts，butIdenyhisconclusions。
ItistruethatwehavereallyinFlatlandaThirdunrecognizedDimensioncalled’height’，justasitisalsotruethatyouhavereallyinSpacelandaFourthunrecognizedDimension，calledbynonameatpresent，butwhichIwillcall’extra-height’。
Butwecannomoretakecognizanceofour’height’thanyoucanofyour’extra-height’。EvenI——whohavebeeninSpaceland，andhavehadtheprivilegeofunderstandingfortwenty-fourhoursthemeaningof’height’——evenIcannotnowcomprehendit，norrealizeitbythesenseofsightorbyanyprocessofreason;
Icanbutapprehenditbyfaith。
\"Thereasonisobvious。Dimensionimpliesdirection，impliesmeasurement，impliesthemoreandtheless。Now，allourlinesareEQUALLYandINFINITESIMALLYthick（orhigh，whicheveryoulike）;consequently，thereisnothinginthemtoleadourmindstotheconceptionofthatDimension。
No’delicatemicrometer’——ashasbeensuggestedbyonetoohastySpacelandcritic——wouldintheleastavailus;forweshouldnotknowWHATTOMEASURE，NORINWHATDIRECTION。WhenweseeaLine，weseesomethingthatislongandBRIGHT;BRIGHTNESS，aswellaslength，isnecessarytotheexistenceofaLine;
ifthebrightnessvanishes，theLineisextinguished。Hence，allmyFlatlandfriends——whenItalktothemabouttheunrecognizedDimensionwhichissomehowvisibleinaLine——say，’Ah，youmeanBRIGHTNESS’：andwhenIreply，’No，ImeanarealDimension’，theyatonceretort，’Thenmeasureit，ortellusinwhatdirectionitextends’;andthissilencesme，forIcandoneither。Onlyyesterday，whentheChiefCircle（inotherwordsourHighPriest）cametoinspecttheStatePrisonandpaidmehisseventhannualvisit，andwhenfortheseventhtimeheputmethequestion，’WasIanybetter？’Itriedtoprovetohimthathewas’high’，aswellaslongandbroad，althoughhedidnotknowit。Butwhatwashisreply？’YousayIam\"high\";measuremy\"high-ness\"andIwillbelieveyou。’WhatcouldIdo？HowcouldI
meethischallenge？Iwascrushed;andhelefttheroomtriumphant。
\"Doesthisstillseemstrangetoyou？Thenputyourselfinasimilarposition。SupposeapersonoftheFourthDimension，condescendingtovisityou，weretosay，’Wheneveryouopenyoureyes，youseeaPlane（whichisofTwoDimensions）andyouINFER
aSolid（whichisofThree）;butinrealityyoualsosee（thoughyoudonotrecognize）aFourthDimension，whichisnotcolournorbrightnessnoranythingofthekind，butatrueDimension，althoughIcannotpointouttoyouitsdirection，norcanyoupossiblymeasureit。’Whatwouldyousaytosuchavisitor？
Wouldnotyouhavehimlockedup？Well，thatismyfate：
anditisasnaturalforusFlatlanderstolockupaSquareforpreachingtheThirdDimension，asitisforyouSpacelanderstolockupaCubeforpreachingtheFourth。Alas，howstrongafamilylikenessrunsthroughblindandpersecutinghumanityinallDimensions！Points，Lines，Squares，Cubes，Extra-Cubes——
weareallliabletothesameerrors，allaliketheSlavesofourrespectiveDimensionalprejudices，asoneofyourSpacelandpoetshassaid——
’OnetouchofNaturemakesallworldsakin’。\"
[Note：TheAuthordesiresmetoadd，thatthemisconceptionofsomeofhiscriticsonthismatterhasinducedhimtoinsertinhisdialoguewiththeSphere，certainremarkswhichhaveabearingonthepointinquestion，andwhichhehadpreviouslyomittedasbeingtediousandunnecessary。]
OnthispointthedefenceoftheSquareseemstometobeimpregnable。
IwishIcouldsaythathisanswertothesecond（ormoral）objectionwasequallyclearandcogent。Ithasbeenobjectedthatheisawoman-hater;andasthisobjectionhasbeenvehementlyurgedbythosewhomNature’sdecreehasconstitutedthesomewhatlargerhalfoftheSpacelandrace，Ishouldliketoremoveit，sofarasIcanhonestlydoso。ButtheSquareissounaccustomedtotheuseofthemoralterminologyofSpacelandthatIshouldbedoinghimaninjusticeifIwereliterallytotranscribehisdefenceagainstthischarge。Acting，therefore，ashisinterpreterandsummarizer，Igatherthatinthecourseofanimprisonmentofsevenyearshehashimselfmodifiedhisownpersonalviews，bothasregardsWomenandasregardstheIsoscelesorLowerClasses。Personally，henowinclinestotheopinionoftheSpherethattheStraightLinesareinmanyimportantrespectssuperiortotheCircles。
But，writingasaHistorian，hehasidentifiedhimself（perhapstooclosely）withtheviewsgenerallyadoptedbyFlatland，and（ashehasbeeninformed）evenbySpaceland，Historians;
inwhosepages（untilveryrecenttimes）thedestiniesofWomenandofthemassesofmankindhaveseldombeendeemedworthyofmentionandneverofcarefulconsideration。
InastillmoreobscurepassagehenowdesirestodisavowtheCircularoraristocratictendencieswithwhichsomecriticshavenaturallycreditedhim。WhiledoingjusticetotheintellectualpowerwithwhichafewCircleshaveformanygenerationsmaintainedtheirsupremacyoverimmensemultitudesoftheircountrymen，hebelievesthatthefactsofFlatland，speakingforthemselveswithoutcommentonhispart，declarethatRevolutionscannotalwaysbesuppressedbyslaughter，andthatNature，insentencingtheCirclestoinfecundity，hascondemnedthemtoultimatefailure——
\"andherein，\"hesays，\"IseeafulfilmentofthegreatLawofallworlds，thatwhilethewisdomofManthinksitisworkingonething，thewisdomofNatureconstrainsittoworkanother，andquiteadifferentandfarbetterthing。\"Fortherest，hebegshisreadersnottosupposethateveryminutedetailinthedailylifeofFlatlandmustneedscorrespondtosomeotherdetailinSpaceland;andyethehopesthat，takenasawhole，hisworkmayprovesuggestiveaswellasamusing，tothoseSpacelandersofmoderateandmodestmindswho——
speakingofthatwhichisofthehighestimportance，butliesbeyondexperience——declinetosayontheonehand，\"Thiscanneverbe，\"andontheotherhand，\"Itmustneedsbepreciselythus，andweknowallaboutit。\"
CONTENTS：
PARTI：THISWORLD
Section1。OftheNatureofFlatland2。OftheClimateandHousesinFlatland3。ConcerningtheInhabitantsofFlatland4。ConcerningtheWomen5。OfourMethodsofRecognizingoneanother6。OfRecognitionbySight7。ConcerningIrregularFigures8。OftheAncientPracticeofPainting9。OftheUniversalColourBill10。OftheSuppressionoftheChromaticSedition11。ConcerningourPriests12。OftheDoctrineofourPriestsPARTII：OTHERWORLDS
13。HowIhadaVisionofLineland14。HowIvainlytriedtoexplainthenatureofFlatland15。ConcerningaStrangerfromSpaceland16。HowtheStrangervainlyendeavouredtorevealtomeinwordsthemysteriesofSpaceland17。HowtheSphere，havinginvaintriedwords，resortedtodeeds18。HowIcametoSpaceland，andwhatIsawthere19。How，thoughtheSphereshewedmeothermysteriesofSpaceland，Istilldesiredmore;andwhatcameofit20。HowtheSphereencouragedmeinaVision21。HowItriedtoteachtheTheoryofThreeDimensionstomyGrandson，andwithwhatsuccess22。HowIthentriedtodiffusetheTheoryofThreeDimensionsbyothermeans，andoftheresultPARTI：THISWORLD
\"Bepatient，fortheworldisbroadandwide。\"
Section1。OftheNatureofFlatlandIcallourworldFlatland，notbecausewecallitso，buttomakeitsnatureclearertoyou，myhappyreaders，whoareprivilegedtoliveinSpace。
ImagineavastsheetofpaperonwhichstraightLines，Triangles，Squares，Pentagons，Hexagons，andotherfigures，insteadofremainingfixedintheirplaces，movefreelyabout，onorinthesurface，butwithoutthepowerofrisingaboveorsinkingbelowit，verymuchlikeshadows——onlyhardandwithluminousedges——andyouwillthenhaveaprettycorrectnotionofmycountryandcountrymen。Alas，afewyearsago，Ishouldhavesaid\"myuniverse\"：butnowmymindhasbeenopenedtohigherviewsofthings。
Insuchacountry，youwillperceiveatoncethatitisimpossiblethatthereshouldbeanythingofwhatyoucalla\"solid\"kind;
butIdaresayyouwillsupposethatwecouldatleastdistinguishbysighttheTriangles，Squares，andotherfigures，movingaboutasIhavedescribedthem。Onthecontrary，wecouldseenothingofthekind，notatleastsoastodistinguishonefigurefromanother。Nothingwasvisible，norcouldbevisible，tous，exceptStraightLines;andthenecessityofthisIwillspeedilydemonstrate。
PlaceapennyonthemiddleofoneofyourtablesinSpace;
andleaningoverit，lookdownuponit。Itwillappearacircle。
Butnow，drawingbacktotheedgeofthetable，graduallyloweryoureye（thusbringingyourselfmoreandmoreintotheconditionoftheinhabitantsofFlatland），andyouwillfindthepennybecomingmoreandmoreovaltoyourview，andatlastwhenyouhaveplacedyoureyeexactlyontheedgeofthetable（sothatyouare，asitwere，actuallyaFlatlander）thepennywillthenhaveceasedtoappearovalatall，andwillhavebecome，sofarasyoucansee，astraightline。
ThesamethingwouldhappenifyouweretotreatinthesamewayaTriangle，orSquare，oranyotherfigurecutoutofpasteboard。
Assoonasyoulookatitwithyoureyeontheedgeonthetable，youwillfindthatitceasestoappeartoyouafigure，andthatitbecomesinappearanceastraightline。TakeforexampleanequilateralTriangle——whorepresentswithusaTradesmanoftherespectableclass。Fig。1representstheTradesmanasyouwouldseehimwhileyouwerebendingoverhimfromabove;
figs。2and3representtheTradesman，asyouwouldseehimifyoureyewereclosetothelevel，orallbutonthelevelofthetable;andifyoureyewerequiteonthelevelofthetable（andthatishowweseehiminFlatland）youwouldseenothingbutastraightline。
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WhenIwasinSpacelandIheardthatyoursailorshaveverysimilarexperienceswhiletheytraverseyourseasanddiscernsomedistantislandorcoastlyingonthehorizon。Thefar-offlandmayhavebays，forelands，anglesinandouttoanynumberandextent;
yetatadistanceyouseenoneofthese（unlessindeedyoursunshinesbrightuponthemrevealingtheprojectionsandretirementsbymeansoflightandshade），nothingbutagreyunbrokenlineuponthewater。
Well，thatisjustwhatweseewhenoneofourtriangularorotheracquaintancescomestowardusinFlatland。Asthereisneithersunwithus，noranylightofsuchakindastomakeshadows，wehavenoneofthehelpstothesightthatyouhaveinSpaceland。
Ifourfriendcomesclosertousweseehislinebecomeslarger;
ifheleavesusitbecomessmaller：butstillhelookslikeastraightline;beheaTriangle，Square，Pentagon，Hexagon，Circle，whatyouwill——astraightLinehelooksandnothingelse。
Youmayperhapsaskhowunderthesedisadvantageouscircumstancesweareabletodistinguishourfriendsfromoneanother：
buttheanswertothisverynaturalquestionwillbemorefitlyandeasilygivenwhenIcometodescribetheinhabitantsofFlatland。
Forthepresentletmedeferthissubject，andsayawordortwoabouttheclimateandhousesinourcountry。
Section2。OftheClimateandHousesinFlatlandAswithyou，soalsowithus，therearefourpointsofthecompassNorth，South，East，andWest。
Therebeingnosunnorotherheavenlybodies，itisimpossibleforustodeterminetheNorthintheusualway;butwehaveamethodofourown。ByaLawofNaturewithus，thereisaconstantattractiontotheSouth;and，althoughintemperateclimatesthisisveryslight——sothatevenaWomaninreasonablehealthcanjourneyseveralfurlongsnorthwardwithoutmuchdifficulty——
yetthehamperingeffectofthesouthwardattractionisquitesufficienttoserveasacompassinmostpartsofourearth。
Moreover，therain（whichfallsatstatedintervals）comingalwaysfromtheNorth，isanadditionalassistance;andinthetownswehavetheguidanceofthehouses，whichofcoursehavetheirside-wallsrunningforthemostpartNorthandSouth，sothattheroofsmaykeepofftherainfromtheNorth。Inthecountry，wheretherearenohouses，thetrunksofthetreesserveassomesortofguide。
Altogether，wehavenotsomuchdifficultyasmightbeexpectedindeterminingourbearings。
Yetinourmoretemperateregions，inwhichthesouthwardattractionishardlyfelt，walkingsometimesinaperfectlydesolateplainwheretherehavebeennohousesnortreestoguideme，Ihavebeenoccasionallycompelledtoremainstationaryforhourstogether，waitingtilltheraincamebeforecontinuingmyjourney。Ontheweakandaged，andespeciallyondelicateFemales，theforceofattractiontellsmuchmoreheavilythanontherobustoftheMaleSex，sothatitisapointofbreeding，ifyoumeetaLadyinthestreet，alwaystogivehertheNorthsideoftheway——bynomeansaneasythingtodoalwaysatshortnoticewhenyouareinrudehealthandinaclimatewhereitisdifficulttotellyourNorthfromyourSouth。
Windowstherearenoneinourhouses：forthelightcomestousalikeinourhomesandoutofthem，bydayandbynight，equallyatalltimesandinallplaces，whenceweknownot。Itwasinolddays，withourlearnedmen，aninterestingandoft-investigatedquestion，\"Whatistheoriginoflight？\"andthesolutionofithasbeenrepeatedlyattempted，withnootherresultthantocrowdourlunaticasylumswiththewould-besolvers。Hence，afterfruitlessattemptstosuppresssuchinvestigationsindirectlybymakingthemliabletoaheavytax，theLegislature，incomparativelyrecenttimes，absolutelyprohibitedthem。
I——alas，IaloneinFlatland——knownowonlytoowellthetruesolutionofthismysteriousproblem;butmyknowledgecannotbemadeintelligibletoasingleoneofmycountrymen;
andIammockedat——I，thesolepossessorofthetruthsofSpaceandofthetheoryoftheintroductionofLightfromtheworldofthreeDimensions——asifIwerethemaddestofthemad！
Butatrucetothesepainfuldigressions：letmereturntoourhouses。
Themostcommonformfortheconstructionofahouseisfive-sidedorpentagonal，asintheannexedfigure。ThetwoNorthernsidesRO，OF，constitutetheroof，andforthemostparthavenodoors;
ontheEastisasmalldoorfortheWomen;ontheWestamuchlargeronefortheMen;theSouthsideorfloorisusuallydoorless。
Squareandtriangularhousesarenotallowed，andforthisreason。
TheanglesofaSquare（andstillmorethoseofanequilateralTriangle），beingmuchmorepointedthanthoseofaPentagon，andthelinesofinanimateobjects（suchashouses）beingdimmerthanthelinesofMenandWomen，itfollowsthatthereisnolittledangerlestthepointsofasquareortriangularhouseresidencemightdoseriousinjurytoaninconsiderateorperhapsabsent-mindedtravellersuddenlytherefore，runningagainstthem：andasearlyastheeleventhcenturyofourera，triangularhouseswereuniversallyforbiddenbyLaw，theonlyexceptionsbeingfortifications，powder-magazines，barracks，andotherstatebuildings，whichitisnotdesirablethatthegeneralpublicshouldapproachwithoutcircumspection。
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Atthisperiod，squarehouseswerestilleverywherepermitted，thoughdiscouragedbyaspecialtax。But，aboutthreecenturiesafterwards，theLawdecidedthatinalltownscontainingapopulationabovetenthousand，theangleofaPentagonwasthesmallesthouse-anglethatcouldbeallowedconsistentlywiththepublicsafety。
ThegoodsenseofthecommunityhassecondedtheeffortsoftheLegislature;andnow，eveninthecountry，thepentagonalconstructionhassupersededeveryother。
Itisonlynowandtheninsomeveryremoteandbackwardagriculturaldistrictthatanantiquarianmaystilldiscoverasquarehouse。
Section3。ConcerningtheInhabitantsofFlatlandThegreatestlengthorbreadthofafullgrowninhabitantofFlatlandmaybeestimatedataboutelevenofyourinches。Twelveinchesmayberegardedasamaximum。
OurWomenareStraightLines。
OurSoldiersandLowestClassesofWorkmenareTriangleswithtwoequalsides，eachaboutelevenincheslong，andabaseorthirdsidesoshort（oftennotexceedinghalfaninch）thattheyformattheirverticesaverysharpandformidableangle。
Indeedwhentheirbasesareofthemostdegradedtype（notmorethantheeighthpartofaninchinsize），theycanhardlybedistinguishedfromStraightLinesorWomen;soextremelypointedaretheirvertices。
Withus，aswithyou，theseTrianglesaredistinguishedfromothersbybeingcalledIsosceles;andbythisnameIshallrefertotheminthefollowingpages。
OurMiddleClassconsistsofEquilateralorEqual-SidedTriangles。
OurProfessionalMenandGentlemenareSquares（towhichclassImyselfbelong）andFive-SidedFiguresorPentagons。
NextabovethesecometheNobility，ofwhomthereareseveraldegrees，beginningatSix-SidedFigures，orHexagons，andfromthencerisinginthenumberoftheirsidestilltheyreceivethehonourabletitleofPolygonal，ormany-sided。Finallywhenthenumberofthesidesbecomessonumerous，andthesidesthemselvessosmall，thatthefigurecannotbedistinguishedfromacircle，heisincludedintheCircularorPriestlyorder;andthisisthehighestclassofall。
ItisaLawofNaturewithusthatamalechildshallhaveonemoresidethanhisfather，sothateachgenerationshallrise（asarule）onestepinthescaleofdevelopmentandnobility。
ThusthesonofaSquareisaPentagon;thesonofaPentagon，aHexagon;andsoon。
ButthisruleappliesnotalwaystotheTradesmen，andstilllessoftentotheSoldiers，andtotheWorkmen;whoindeedcanhardlybesaidtodeservethenameofhumanFigures，sincetheyhavenotalltheirsidesequal。WiththemthereforetheLawofNaturedoesnothold;andthesonofanIsosceles（i。e。aTrianglewithtwosidesequal）remainsIsoscelesstill。Nevertheless，allhopeisnotshutout，evenfromtheIsosceles，thathisposteritymayultimatelyriseabovehisdegradedcondition。For，afteralongseriesofmilitarysuccesses，ordiligentandskilfullabours，itisgenerallyfoundthatthemoreintelligentamongtheArtisanandSoldierclassesmanifestaslightincreaseoftheirthirdsideorbase，andashrinkageofthetwoothersides。
Intermarriages（arrangedbythePriests）betweenthesonsanddaughtersofthesemoreintellectualmembersofthelowerclassesgenerallyresultinanoffspringapproximatingstillmoretothetypeoftheEqual-SidedTriangle。
Rarely——inproportiontothevastnumbersofIsoscelesbirths——
isagenuineandcertifiableEqual-SidedTriangleproducedfromIsoscelesparents。[Note：\"Whatneedofacertificate？\"
aSpacelandcriticmayask：\"IsnottheprocreationofaSquareSonacertificatefromNatureherself，provingtheEqual-sidednessoftheFather？\"IreplythatnoLadyofanypositionwillmarryanuncertifiedTriangle。SquareoffspringhassometimesresultedfromaslightlyIrregularTriangle;butinalmosteverysuchcasetheIrregularityofthefirstgenerationisvisitedonthethird;
whicheitherfailstoattainthePentagonalrank，orrelapsestotheTriangular。]Suchabirthrequires，asitsantecedents，notonlyaseriesofcarefullyarrangedintermarriages，butalsoalong，continuedexerciseoffrugalityandself-controlonthepartofthewould-beancestorsofthecomingEquilateral，andapatient，systematic，andcontinuousdevelopmentoftheIsoscelesintellectthroughmanygenerations。
ThebirthofaTrueEquilateralTrianglefromIsoscelesparentsisthesubjectofrejoicinginourcountryformanyfurlongsaround。
AfterastrictexaminationconductedbytheSanitaryandSocialBoard，theinfant，ifcertifiedasRegular，iswithsolemnceremonialadmittedintotheclassofEquilaterals。HeisthenimmediatelytakenfromhisproudyetsorrowingparentsandadoptedbysomechildlessEquilateral，whoisboundbyoathnevertopermitthechildhenceforthtoenterhisformerhomeorsomuchastolookuponhisrelationsagain，forfearlestthefreshlydevelopedorganismmay，byforceofunconsciousimitation，fallbackagainintohishereditarylevel。
TheoccasionalemergenceofanEquilateralfromtheranksofhisserf-bornancestorsiswelcomed，notonlybythepoorserfsthemselves，asagleamoflightandhopesheduponthemonotonoussqualoroftheirexistence，butalsobytheAristocracyatlarge;forallthehigherclassesarewellawarethattheserarephenomena，whiletheydolittleornothingtovulgarizetheirownprivileges，serveasamostusefulbarrieragainstrevolutionfrombelow。
Hadtheacute-angledrabblebeenall，withoutexception，absolutelydestituteofhopeandofambition，theymighthavefoundleadersinsomeoftheirmanyseditiousoutbreaks，soableastorendertheirsuperiornumbersandstrengthtoomuchevenforthewisdomoftheCircles。ButawiseordinanceofNaturehasdecreedthat，inproportionastheworking-classesincreaseinintelligence，knowledge，andallvirtue，inthatsameproportiontheiracuteangle（whichmakesthemphysicallyterrible）
shallincreasealsoandapproximatetothecomparativelyharmlessangleoftheEquilateralTriangle。Thus，inthemostbrutalandformidableofthesoldierclass——creaturesalmostonalevelwithwomenintheirlackofintelligence——itisfoundthat，astheywaxinthementalabilitynecessarytoemploytheirtremendouspenetratingpowertoadvantage，sodotheywaneinthepowerofpenetrationitself。
HowadmirableisthisLawofCompensation！Andhowperfectaproofofthenaturalfitnessand，Imayalmostsay，thedivineoriginofthearistocraticconstitutionoftheStatesinFlatland！
ByajudicioususeofthisLawofNature，thePolygonsandCirclesarealmostalwaysabletostifleseditioninitsverycradle，takingadvantageoftheirrepressibleandboundlesshopefulnessofthehumanmind。ArtalsocomestotheaidofLawandOrder。
Itisgenerallyfoundpossible——byalittleartificialcompressionorexpansiononthepartoftheStatephysicians——
tomakesomeofthemoreintelligentleadersofarebellionperfectlyRegular，andtoadmitthematonceintotheprivilegedclasses;amuchlargernumber，whoarestillbelowthestandard，alluredbytheprospectofbeingultimatelyennobled，areinducedtoentertheStateHospitals，wheretheyarekeptinhonourableconfinementforlife;oneortwoaloneofthemoreobstinate，foolish，andhopelesslyirregularareledtoexecution。
ThenthewretchedrabbleoftheIsosceles，planlessandleaderless，areeithertransfixedwithoutresistancebythesmallbodyoftheirbrethrenwhomtheChiefCirclekeepsinpayforemergenciesofthiskind;orelsemoreoften，bymeansofjealousiesandsuspicionsskilfullyfomentedamongthembytheCircularparty，theyarestirredtomutualwarfare，andperishbyoneanother’sangles。Nolessthanonehundredandtwentyrebellionsarerecordedinourannals，besidesminoroutbreaksnumberedattwohundredandthirty-five;
andtheyhaveallendedthus。
Section4。ConcerningtheWomenIfourhighlypointedTrianglesoftheSoldierclassareformidable，itmaybereadilyinferredthatfarmoreformidableareourWomen。
ForifaSoldierisawedge，aWomanisaneedle;being，sotospeak，ALLpoint，atleastatthetwoextremities。Addtothisthepowerofmakingherselfpracticallyinvisibleatwill，andyouwillperceivethataFemale，inFlatland，isacreaturebynomeanstobetrifledwith。
Buthere，perhaps，someofmyyoungerReadersmayaskHOWawomaninFlatlandcanmakeherselfinvisible。Thisought，Ithink，tobeapparentwithoutanyexplanation。However，afewwordswillmakeitcleartothemostunreflecting。
Placeaneedleonatable。Then，withyoureyeonthelevelofthetable，lookatitside-ways，andyouseethewholelengthofit;
butlookatitend-ways，andyouseenothingbutapoint，ithasbecomepracticallyinvisible。JustsoisitwithoneofourWomen。Whenhersideisturnedtowardsus，weseeherasastraightline;whentheendcontaininghereyeormouth——
forwithusthesetwoorgansareidentical——isthepartthatmeetsoureye，thenweseenothingbutahighlylustrouspoint;
butwhenthebackispresentedtoourview，then——beingonlysub-lustrous，and，indeed，almostasdimasaninanimateobject——
herhinderextremityservesherasakindofInvisibleCap。
ThedangerstowhichweareexposedfromourWomenmustnowbemanifesttothemeanestcapacityinSpaceland。IfeventheangleofarespectableTriangleinthemiddleclassisnotwithoutitsdangers;iftorunagainstaWorkingManinvolvesagash;
ifcollisionwithanofficerofthemilitaryclassnecessitatesaseriouswound;ifameretouchfromthevertexofaPrivateSoldierbringswithitdangerofdeath;——whatcanitbetorunagainstaWoman，exceptabsoluteandimmediatedestruction？AndwhenaWomanisinvisible，orvisibleonlyasadimsub-lustrouspoint，howdifficultmustitbe，evenforthemostcautious，alwaystoavoidcollision！
ManyaretheenactmentsmadeatdifferenttimesinthedifferentStatesofFlatland，inordertominimizethisperil;
andintheSouthernandlesstemperateclimateswheretheforceofgravitationisgreater，andhumanbeingsmoreliabletocasualandinvoluntarymotions，theLawsconcerningWomenarenaturallymuchmorestringent。ButageneralviewoftheCodemaybeobtainedfromthefollowingsummary：——
1。EveryhouseshallhaveoneentranceintheEasternside，fortheuseofFemalesonly;bywhichallfemalesshallenter\"inabecomingandrespectfulmanner\"andnotbytheMen’sorWesterndoor。[Note：WhenIwasinSpacelandIunderstoodthatsomeofyourPriestlycircleshaveinthesamewayaseparateentranceforVillagers，FarmersandTeachersofBoardSchools（`Spectator’，Sept。1884，p。1255）thattheymay\"approachinabecomingandrespectfulmanner。\"]
2。NoFemaleshallwalkinanypublicplacewithoutcontinuallykeepingupherPeace-cry，underpenaltyofdeath。
3。AnyFemale，dulycertifiedtobesufferingfromSt。Vitus’sDance，fits，chroniccoldaccompaniedbyviolentsneezing，oranydiseasenecessitatinginvoluntarymotions，shallbeinstantlydestroyed。
InsomeoftheStatesthereisanadditionalLawforbiddingFemales，underpenaltyofdeath，fromwalkingorstandinginanypublicplacewithoutmovingtheirbacksconstantlyfromrighttoleftsoastoindicatetheirpresencetothosebehindthem;
othersobligeaWoman，whentravelling，tobefollowedbyoneofhersons，orservants，orbyherhusband;othersconfineWomenaltogethertotheirhousesexceptduringthereligiousfestivals。
ButithasbeenfoundbythewisestofourCirclesorStatesmenthatthemultiplicationofrestrictionsonFemalestendsnotonlytothedebilitationanddiminutionoftherace，butalsototheincreaseofdomesticmurderstosuchanextentthataStatelosesmorethanitgainsbyatooprohibitiveCode。
ForwheneverthetemperoftheWomenisthusexasperatedbyconfinementathomeorhamperingregulationsabroad，theyareapttoventtheirspleenupontheirhusbandsandchildren;
andinthelesstemperateclimatesthewholemalepopulationofavillagehasbeensometimesdestroyedinoneortwohoursofsimultaneousfemaleoutbreak。HencetheThreeLaws，mentionedabove，sufficeforthebetterregulatedStates，andmaybeacceptedasaroughexemplificationofourFemaleCode。
Afterall，ourprincipalsafeguardisfound，notinLegislature，butintheinterestsoftheWomenthemselves。For，althoughtheycaninflictinstantaneousdeathbyaretrogrademovement，yetunlesstheycanatoncedisengagetheirstingingextremityfromthestrugglingbodyoftheirvictim，theirownfrailbodiesareliabletobeshattered。
ThepowerofFashionisalsoonourside。IpointedoutthatinsomelesscivilizedStatesnofemaleissufferedtostandinanypublicplacewithoutswayingherbackfromrighttoleft。
Thispracticehasbeenuniversalamongladiesofanypretensionstobreedinginallwell-governedStates，asfarbackasthememoryofFigurescanreach。ItisconsideredadisgracetoanyStatethatlegislationshouldhavetoenforcewhatoughttobe，andisineveryrespectablefemale，anaturalinstinct。
Therhythmicaland，ifImaysosay，well-modulatedundulationofthebackinourladiesofCircularrankisenviedandimitatedbythewifeofacommonEquilateral，whocanachievenothingbeyondameremonotonousswing，likethetickingofapendulum;
andtheregulartickoftheEquilateralisnolessadmiredandcopiedbythewifeoftheprogressiveandaspiringIsosceles，inthefemalesofwhosefamilyno\"back-motion\"ofanykindhasbecomeasyetanecessityoflife。Hence，ineveryfamilyofpositionandconsideration，\"backmotion\"isasprevalentastimeitself;andthehusbandsandsonsinthesehouseholdsenjoyimmunityatleastfrominvisibleattacks。
NotthatitmustbeforamomentsupposedthatourWomenaredestituteofaffection。Butunfortunatelythepassionofthemomentpredominates，intheFrailSex，overeveryotherconsideration。
Thisis，ofcourse，anecessityarisingfromtheirunfortunateconformation。Forastheyhavenopretensionstoanangle，beinginferiorinthisrespecttotheverylowestoftheIsosceles，theyareconsequentlywhollydevoidofbrain-power，andhaveneitherreflection，judgmentnorforethought，andhardlyanymemory。Hence，intheirfitsoffury，theyremembernoclaimsandrecognizenodistinctions。IhaveactuallyknownacasewhereaWomanhasexterminatedherwholehousehold，andhalfanhourafterwards，whenherragewasoverandthefragmentssweptaway，hasaskedwhathasbecomeofherhusbandandherchildren。
ObviouslythenaWomanisnottobeirritatedaslongassheisinapositionwhereshecanturnround。Whenyouhavethemintheirapartments——whichareconstructedwithaviewtodenyingthemthatpower——youcansayanddowhatyoulike;
fortheyarethenwhollyimpotentformischief，andwillnotrememberafewminuteshencetheincidentforwhichtheymaybeatthismomentthreateningyouwithdeath，northepromiseswhichyoumayhavefounditnecessarytomakeinordertopacifytheirfury。
Onthewholewegetonprettysmoothlyinourdomesticrelations，exceptinthelowerstrataoftheMilitaryClasses。Therethewantoftactanddiscretiononthepartofthehusbandsproducesattimesindescribabledisasters。Relyingtoomuchontheoffensiveweaponsoftheiracuteanglesinsteadofthedefensiveorgansofgoodsenseandseasonablesimulation，theserecklesscreaturestoooftenneglecttheprescribedconstructionofthewomen’sapartments，orirritatetheirwivesbyill-advisedexpressionsoutofdoors，whichtheyrefuseimmediatelytoretract。MoreoverabluntandstolidregardforliteraltruthindisposesthemtomakethoselavishpromisesbywhichthemorejudiciousCirclecaninamomentpacifyhisconsort。
Theresultismassacre;not，however，withoutitsadvantages，asiteliminatesthemorebrutalandtroublesomeoftheIsosceles;
andbymanyofourCirclesthedestructivenessoftheThinnerSexisregardedasoneamongmanyprovidentialarrangementsforsuppressingredundantpopulation，andnippingRevolutioninthebud。
YeteveninourbestregulatedandmostapproximatelyCircularfamiliesIcannotsaythattheidealoffamilylifeissohighaswithyouinSpaceland。Thereispeace，insofarastheabsenceofslaughtermaybecalledbythatname，butthereisnecessarilylittleharmonyoftastesorpursuits;andthecautiouswisdomoftheCircleshasensuredsafetyatthecostofdomesticcomfort。
IneveryCircularorPolygonalhouseholdithasbeenahabitfromtimeimmemorial——andnowhasbecomeakindofinstinctamongthewomenofourhigherclasses——thatthemothersanddaughtersshouldconstantlykeeptheireyesandmouthstowardstheirhusbandandhismalefriends;andforaladyinafamilyofdistinctiontoturnherbackuponherhusbandwouldberegardedasakindofportent，involvinglossofSTATUS。But，asIshallsoonshew，thiscustom，thoughithastheadvantageofsafety，isnotwithoutitsdisadvantages。
InthehouseoftheWorkingManorrespectableTradesman——
wherethewifeisallowedtoturnherbackuponherhusband，whilepursuingherhouseholdavocations——thereareatleastintervalsofquiet，whenthewifeisneitherseennorheard，exceptforthehummingsoundofthecontinuousPeace-cry;
butinthehomesoftheupperclassesthereistoooftennopeace。
TherethevolublemouthandbrightpenetratingeyeareeverdirectedtowardstheMasterofthehousehold;andlightitselfisnotmorepersistentthanthestreamoffemininediscourse。
ThetactandskillwhichsufficetoavertaWoman’sstingareunequaltothetaskofstoppingaWoman’smouth;andasthewifehasabsolutelynothingtosay，andabsolutelynoconstraintofwit，sense，orconsciencetopreventherfromsayingit，notafewcynicshavebeenfoundtoaverthattheypreferthedangerofthedeath-dealingbutinaudiblestingtothesafesonorousnessofaWoman’sotherend。
TomyreadersinSpacelandtheconditionofourWomenmayseemtrulydeplorable，andsoindeeditis。AMaleofthelowesttypeoftheIsoscelesmaylookforwardtosomeimprovementofhisangle，andtotheultimateelevationofthewholeofhisdegradedcaste;
butnoWomancanentertainsuchhopesforhersex。\"OnceaWoman，alwaysaWoman\"isaDecreeofNature;andtheveryLawsofEvolutionseemsuspendedinherdisfavour。YetatleastwecanadmirethewisePrearrangementwhichhasordainedthat，astheyhavenohopes，sotheyshallhavenomemorytorecall，andnoforethoughttoanticipate，themiseriesandhumiliationswhichareatonceanecessityoftheirexistenceandthebasisoftheconstitutionofFlatland。
Section5。OfourMethodsofRecognizingoneanotherYou，whoareblessedwithshadeaswellaslight，you，whoaregiftedwithtwoeyes，endowedwithaknowledgeofperspective，andcharmedwiththeenjoymentofvariouscolours，you，whocanactuallySEEanangle，andcontemplatethecompletecircumferenceofacircleinthehappyregionoftheThreeDimensions——howshallImakecleartoyoutheextremedifficultywhichweinFlatlandexperienceinrecognizingoneanother’sconfiguration？
RecallwhatItoldyouabove。AllbeingsinFlatland，animateorinanimate，nomatterwhattheirform，presentTOOURVIEW
thesame，ornearlythesame，appearance，viz。thatofastraightLine。Howthencanonebedistinguishedfromanother，whereallappearthesame？
Theansweristhreefold。Thefirstmeansofrecognitionisthesenseofhearing;whichwithusisfarmorehighlydevelopedthanwithyou，andwhichenablesusnotonlytodistinguishbythevoiceourpersonalfriends，buteventodiscriminatebetweendifferentclasses，atleastsofarasconcernsthethreelowestorders，theEquilateral，theSquare，andthePentagon——foroftheIsoscelesItakenoaccount。Butasweascendinthesocialscale，theprocessofdiscriminatingandbeingdiscriminatedbyhearingincreasesindifficulty，partlybecausevoicesareassimilated，partlybecausethefacultyofvoice-discriminationisaplebeianvirtuenotmuchdevelopedamongtheAristocracy。Andwhereverthereisanydangerofimposturewecannottrusttothismethod。Amongstourlowestorders，thevocalorgansaredevelopedtoadegreemorethancorrespondentwiththoseofhearing，sothatanIsoscelescaneasilyfeignthevoiceofaPolygon，and，withsometraining，thatofaCirclehimself。
Asecondmethodisthereforemorecommonlyresortedto。
FEELINGis，amongourWomenandlowerclasses——aboutourupperclassesIshallspeakpresently——theprincipaltestofrecognition，atalleventsbetweenstrangers，andwhenthequestionis，notastotheindividual，butastotheclass。
Whattherefore\"introduction\"isamongthehigherclassesinSpaceland，thattheprocessof\"feeling\"iswithus。
\"PermitmetoaskyoutofeelandbefeltbymyfriendMr。So-and-so\"
——isstill，amongthemoreold-fashionedofourcountrygentlemenindistrictsremotefromtowns，thecustomaryformulaforaFlatlandintroduction。Butinthetowns，andamongmenofbusiness，thewords\"befeltby\"areomittedandthesentenceisabbreviatedto，\"LetmeaskyoutofeelMr。So-and-so\";althoughitisassumed，ofcourse，thatthe\"feeling\"istobereciprocal。
Amongourstillmoremodernanddashingyounggentlemen——whoareextremelyaversetosuperfluouseffortandsupremelyindifferenttothepurityoftheirnativelanguage——theformulaisstillfurthercurtailedbytheuseof\"tofeel\"inatechnicalsense，meaning，\"torecommend-for-the-purposes-of-feeling-and-being-felt\";
andatthismomentthe\"slang\"ofpoliteorfastsocietyintheupperclassessanctionssuchabarbarismas\"Mr。Smith，permitmetofeelMr。Jones。\"
LetnotmyReaderhoweversupposethat\"feeling\"iswithusthetediousprocessthatitwouldbewithyou，orthatwefinditnecessarytofeelrightroundallthesidesofeveryindividualbeforewedeterminetheclasstowhichhebelongs。Longpracticeandtraining，begunintheschoolsandcontinuedintheexperienceofdailylife，enableustodiscriminateatoncebythesenseoftouch，betweentheanglesofanequal-sidedTriangle，Square，andPentagon;andIneednotsaythatthebrainlessvertexofanacute-angledIsoscelesisobvioustothedullesttouch。
Itisthereforenotnecessary，asarule，todomorethanfeelasingleangleofanindividual;andthis，onceascertained，tellsustheclassofthepersonwhomweareaddressing，unlessindeedhebelongstothehighersectionsofthenobility。
Therethedifficultyismuchgreater。EvenaMasterofArtsinourUniversityofWentbridgehasbeenknowntoconfuseaten-sidedwithatwelve-sidedPolygon;andthereishardlyaDoctorofScienceinoroutofthatfamousUniversitywhocouldpretendtodecidepromptlyandunhesitatinglybetweenatwenty-sidedandatwenty-foursidedmemberoftheAristocracy。
ThoseofmyreaderswhorecalltheextractsIgaveabovefromtheLegislativecodeconcerningWomen，willreadilyperceivethattheprocessofintroductionbycontactrequiressomecareanddiscretion。OtherwisetheanglesmightinflictontheunwaryFeelerirreparableinjury。ItisessentialforthesafetyoftheFeelerthattheFeltshouldstandperfectlystill。Astart，afidgetyshiftingoftheposition，yes，evenaviolentsneeze，hasbeenknownbeforenowtoprovefataltotheincautious，andtonipinthebudmanyapromisingfriendship。
EspeciallyisthistrueamongthelowerclassesoftheTriangles。
Withthem，theeyeissituatedsofarfromtheirvertexthattheycanscarcelytakecognizanceofwhatgoesonatthatextremityoftheirframe。Theyare，moreover，ofaroughcoarsenature，notsensitivetothedelicatetouchofthehighlyorganizedPolygon。
WhatwonderthenifaninvoluntarytossoftheheadhaserenowdeprivedtheStateofavaluablelife！
IhaveheardthatmyexcellentGrandfather——oneoftheleastirregularofhisunhappyIsoscelesclass，whoindeedobtained，shortlybeforehisdecease，fouroutofsevenvotesfromtheSanitaryandSocialBoardforpassinghimintotheclassoftheEqual-sided——
oftendeplored，withatearinhisvenerableeye，amiscarriageofthiskind，whichhadoccuredtohisgreat-great-great-Grandfather，arespectableWorkingManwithanangleorbrainof59degrees30minutes。Accordingtohisaccount，myunfortunateAncestor，beingafflictedwithrheumatism，andintheactofbeingfeltbyaPolygon，byonesuddenstartaccidentallytransfixedtheGreatManthroughthediagonal;andthereby，partlyinconsequenceofhislongimprisonmentanddegradation，andpartlybecauseofthemoralshockwhichpervadedthewholeofmyAncestor’srelations，threwbackourfamilyadegreeandahalfintheirascenttowardsbetterthings。Theresultwasthatinthenextgenerationthefamilybrainwasregisteredatonly58degrees，andnottillthelapseoffivegenerationswasthelostgroundrecovered，thefull60degreesattained，andtheAscentfromtheIsoscelesfinallyachieved。AndallthisseriesofcalamitiesfromonelittleaccidentintheprocessofFeeling。
AtthispointIthinkIhearsomeofmybettereducatedreadersexclaim，\"HowcouldyouinFlatlandknowanythingaboutanglesanddegrees，orminutes？WecanSEEanangle，becausewe，intheregionofSpace，canseetwostraightlinesinclinedtooneanother;butyou，whocanseenothingbutonestraightlineatatime，oratalleventsonlyanumberofbitsofstraightlinesallinonestraightline——howcanyoueverdiscernanyangle，andmuchlessregisteranglesofdifferentsizes？\"
IanswerthatthoughwecannotSEEangles，wecanINFERthem，andthiswithgreatprecision。Oursenseoftouch，stimulatedbynecessity，anddevelopedbylongtraining，enablesustodistinguishanglesfarmoreaccuratelythanyoursenseofsight，whenunaidedbyaruleormeasureofangles。
NormustIomittoexplainthatwehavegreatnaturalhelps。
ItiswithusaLawofNaturethatthebrainoftheIsoscelesclassshallbeginathalfadegree，orthirtyminutes，andshallincrease（ifitincreasesatall）byhalfadegreeineverygeneration;
untilthegoalof60degreesisreached，whentheconditionofserfdomisquitted，andthefreemanenterstheclassofRegulars。
Consequently，NatureherselfsuppliesuswithanascendingscaleorAlphabetofanglesforhalfadegreeupto60degrees，SpecimensofwhichareplacedineveryElementarySchoolthroughouttheland。Owingtooccasionalretrogressions，tostillmorefrequentmoralandintellectualstagnation，andtotheextraordinaryfecundityoftheCriminalandVagabondClasses，thereisalwaysavastsuperfluityofindividualsofthehalfdegreeandsingledegreeclass，andafairabundanceofSpecimensupto10degrees。Theseareabsolutelydestituteofcivicrights;
andagreatnumberofthem，nothavingevenintelligenceenoughforthepurposesofwarfare，aredevotedbytheStatestotheserviceofeducation。Fetteredimmovablysoastoremoveallpossibilityofdanger，theyareplacedintheclassroomsofourInfantSchools，andtheretheyareutilizedbytheBoardofEducationforthepurposeofimpartingtotheoffspringoftheMiddleClassesthattactandintelligenceofwhichthesewretchedcreaturesthemselvesareutterlydevoid。
InsomeStatestheSpecimensareoccasionallyfedandsufferedtoexistforseveralyears;butinthemoretemperateandbetterregulatedregions，itisfoundinthelongrunmoreadvantageousfortheeducationalinterestsoftheyoung，todispensewithfood，andtorenewtheSpecimenseverymonth——
whichisabouttheaveragedurationofthefoodlessexistenceoftheCriminalclass。Inthecheaperschools，whatisgainedbythelongerexistenceoftheSpecimenislost，partlyintheexpenditureforfood，andpartlyinthediminishedaccuracyoftheangles，whichareimpairedafterafewweeksofconstant\"feeling\"。Normustweforgettoadd，inenumeratingtheadvantagesofthemoreexpensivesystem，thatittends，thoughslightlyyetperceptibly，tothediminutionoftheredundantIsoscelespopulation——anobjectwhicheverystatesmaninFlatlandconstantlykeepsinview。Onthewholetherefore——
althoughIamnotignorantthat，inmanypopularlyelectedSchoolBoards，thereisareactioninfavourof\"thecheapsystem\"
asitiscalled——Iammyselfdisposedtothinkthatthisisoneofthemanycasesinwhichexpenseisthetruesteconomy。
ButImustnotallowquestionsofSchoolBoardpoliticstodivertmefrommysubject。Enoughhasbeensaid，Itrust，toshewthatRecognitionbyFeelingisnotsotediousorindecisiveaprocessasmighthavebeensupposed;anditisobviouslymoretrustworthythanRecognitionbyhearing。Stillthereremains，ashasbeenpointedoutabove，theobjectionthatthismethodisnotwithoutdanger。ForthisreasonmanyintheMiddleandLowerclasses，andallwithoutexceptioninthePolygonalandCircularorders，preferathirdmethod，thedescriptionofwhichshallbereservedforthenextsection。
Section6。OfRecognitionbySightIamabouttoappearveryinconsistent。InprevioussectionsIhavesaidthatallfiguresinFlatlandpresenttheappearanceofastraightline;anditwasaddedorimplied，thatitisconsequentlyimpossibletodistinguishbythevisualorganbetweenindividualsofdifferentclasses：yetnowIamabouttoexplaintomySpacelandcriticshowweareabletorecognizeoneanotherbythesenseofsight。
IfhowevertheReaderwilltakethetroubletorefertothepassageinwhichRecognitionbyFeelingisstatedtobeuniversal，hewillfindthisqualification——\"amongthelowerclasses\"。
ItisonlyamongthehigherclassesandinourtemperateclimatesthatSightRecognitionispractised。
ThatthispowerexistsinanyregionsandforanyclassesistheresultofFog;whichprevailsduringthegreaterpartoftheyearinallpartssavethetorridzones。ThatwhichiswithyouinSpacelandanunmixedevil，blottingoutthelandscape，depressingthespirits，andenfeeblingthehealth，isbyusrecognizedasablessingscarcelyinferiortoairitself，andastheNurseofartsandParentofsciences。Butletmeexplainmymeaning，withoutfurthereulogiesonthisbeneficentElement。
IfFogwerenon-existent，alllineswouldappearequallyandindistinguishablyclear;andthisisactuallythecaseinthoseunhappycountriesinwhichtheatmosphereisperfectlydryandtransparent。ButwhereverthereisarichsupplyofFogobjectsthatareatadistance，sayofthreefeet，areappreciablydimmerthanthoseatadistanceoftwofeeteleveninches;
andtheresultisthatbycarefulandconstantexperimentalobservationofcomparativedimnessandclearness，weareenabledtoinferwithgreatexactnesstheconfigurationoftheobjectobserved。
Aninstancewilldomorethanavolumeofgeneralitiestomakemymeaningclear。
SupposeIseetwoindividualsapproachingwhoserankIwishtoascertain。Theyare，wewillsuppose，aMerchantandaPhysician，orinotherwords，anEquilateralTriangleandaPentagon：
howamItodistinguishthem？
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Itwillbeobvious，toeverychildinSpacelandwhohastouchedthethresholdofGeometricalStudies，that，ifIcanbringmyeyesothatitsglancemaybisectanangle（A）oftheapproachingstranger，myviewwilllieasitwereevenlybetweenhistwosidesthatarenexttome（viz。CAandAB），sothatIshallcontemplatethetwoimpartially，andbothwillappearofthesamesize。
Nowinthecaseof（1）theMerchant，whatshallIsee？IshallseeastraightlineDAE，inwhichthemiddlepoint（A）willbeverybrightbecauseitisnearesttome;butoneithersidethelinewillshadeawayRAPIDLYINTODIMNESS，becausethesidesACandAB
RECEDERAPIDLYINTOTHEFOGandwhatappeartomeastheMerchant’sextremities，viz。DandE，willbeVERYDIMINDEED。