The　little　Hexagon　meditated　on　this　a　while　and　then　said　to　me；　〃But　you　have　been　teaching　me　to　raise　numbers　to　the　third　power：　I　suppose　3^3　must　mean　something　in　Geometry；　what　does　it　mean？〃　〃Nothing　at　all；〃　replied　I；　〃not　at　least　in　Geometry；　for　Geometry　has　only　Two　Dimensions。〃　　And　then　I　began　to　shew　the　boy　how　a　Point　by　moving　through　a　length　of　three　inches　makes　a　Line　of　three　inches；　which　may　be　represented　by　3；　and　how　a　Line　of　three　inches；　moving　parallel　to　itself　through　a　length　of　three　inches；　makes　a　Square　of　three　inches　every　way；　which　may　be　represented　by　3^2。
  Upon　this；　my　Grandson；　again　returning　to　his　former　suggestion；　took　me　up　rather　suddenly　and　exclaimed；　〃Well；　then；　if　a　Point　by　moving　three　inches；　makes　a　Line　of　three　inches　represented　by　3；　and　if　a　straight　Line　of　three　inches；　moving　parallel　to　itself；　makes　a　Square　of　three　inches　every　way；　represented　by　3^2；　it　must　be　that　a　Square　of　three　inches　every　way；　moving　somehow　parallel　to　itself　（but　I　don't　see　how）　must　make　Something　else　（but　I　don't　see　what）　of　three　inches　every　way　　and　this　must　be　represented　by　3^3。〃
  〃Go　to　bed；〃　said　I；　a　little　ruffled　by　this　interruption：　〃if　you　would　talk　less　nonsense；　you　would　remember　more　sense。〃
  So　my　Grandson　had　disappeared　in　disgrace；　and　there　I　sat　by　my　Wife's　side；　endeavouring　to　form　a　retrospect　of　the　year　1999　and　of　the　possibilities　of　the　year　2000；　but　not　quite　able　to　shake　off　the　thoughts　suggested　by　the　prattle　of　my　bright　little　Hexagon。　　Only　a　few　sands　now　remained　in　the　half…hour　glass。　Rousing　myself　from　my　reverie　I　turned　the　glass　Northward　for　the　last　time　in　the　old　Millennium；　and　in　the　act；　I　exclaimed　aloud；　〃The　boy　is　a　fool。〃
  Straightway　I　became　conscious　of　a　Presence　in　the　room；　and　a　chilling　breath　thrilled　through　my　very　being。　〃He　is　no　such　thing；〃　cried　my　Wife；　〃and　you　are　breaking　the　Commandments　in　thus　dishonouring　your　own　Grandson。〃　But　I　took　no　notice　of　her。　　Looking　round　in　every　direction　I　could　see　nothing；　yet　still　I　FELT　a　Presence；　and　shivered　as　the　cold　whisper　came　again。　　I　started　up。　　〃What　is　the　matter？〃　said　my　Wife；　〃there　is　no　draught；　what　are　you　looking　for？　There　is　nothing。〃　　There　was　nothing；　and　I　resumed　my　seat；　again　exclaiming；　〃The　boy　is　a　fool；　I　say；　3^3　can　have　no　meaning　in　Geometry。〃　　At　once　there　came　a　distinctly　audible　reply；　〃The　boy　is　not　a　fool；　and　3^3　has　an　obvious　Geometrical　meaning。〃
  My　Wife　as　well　as　myself　heard　the　words；　although　she　did　not　understand　their　meaning；　and　both　of　us　sprang　forward　in　the　direction　of　the　sound。　　What　was　our　horror　when　we　saw　before　us　a　Figure！　　At　the　first　glance　it　appeared　to　be　a　Woman；　seen　sideways；　but　a　moment's　observation　shewed　me　that　the　extremities　passed　into　dimness　too　rapidly　to　represent　one　of　the　Female　Sex；　and　I　should　have　thought　it　a　Circle；　only　that　it　seemed　to　change　its　size　in　a　manner　impossible　for　a　Circle　or　for　any　regular　Figure　of　which　I　had　had　experience。
  But　my　Wife　had　not　my　experience；　nor　the　coolness　necessary　to　note　these　characteristics。　　With　the　usual　hastiness　and　unreasoning　jealousy　of　her　Sex；　she　flew　at　once　to　the　conclusion　that　a　Woman　had　entered　the　house　through　some　small　aperture。　〃How　comes　this　person　here？〃　she　exclaimed；　〃you　promised　me；　my　dear；　that　there　should　be　no　ventilators　in　our　new　house。〃　〃Nor　are　there　any；〃　said　I；　〃but　what　makes　you　think　that　the　stranger　is　a　Woman？　　I　see　by　my　power　of　Sight　Recognition　〃　〃Oh；　I　have　no　patience　with　your　Sight　Recognition；〃　replied　she；　〃'Feeling　is　believing'　and　'A　Straight　Line　to　the　touch　is　worth　a　Circle　to　the　sight'〃　　two　Proverbs；　very　common　with　the　Frailer　Sex　in　Flatland。
  〃Well；〃　said　I；　for　I　was　afraid　of　irritating　her；　〃if　it　must　be　so；　demand　an　introduction。〃　　Assuming　her　most　gracious　manner；　my　Wife　advanced　towards　the　Stranger；　〃Permit　me；　Madam；　to　feel　and　be　felt　by　〃　then；　suddenly　recoiling；　〃Oh！　it　is　not　a　Woman；　and　there　are　no　angles　either；　not　a　trace　of　one。　Can　it　be　that　I　have　so　misbehaved　to　a　perfect　Circle？〃
  〃I　am　indeed；　in　a　certain　sense　a　Circle；〃　replied　the　Voice；　〃and　a　more　perfect　Circle　than　any　in　Flatland；　but　to　speak　more　accurately；　I　am　many　Circles　in　one。〃　　Then　he　added　more　mildly；　〃I　have　a　message；　dear　Madam；　to　your　husband；　which　I　must　not　deliver　in　your　presence；　and；　if　you　would　suffer　us　to　retire　for　a　few　minutes　〃　　But　my　Wife　would　not　listen　to　the　proposal　that　our　august　Visitor　should　so　incommode　himself；　and　assuring　the　Circle　that　the　hour　of　her　own　retirement　had　long　passed；　with　many　reiterated　apologies　for　her　recent　indiscretion；　she　at　last　retreated　to　her　apartment。
  I　glanced　at　the　half…hour　glass。　　The　last　sands　had　fallen。　The　third　Millennium　had　begun。
  Section　16。　　How　the　Stranger　vainly　endeavoured　to　reveal　to　me　　　　　　　　　　　　　　　　in　words　the　mysteries　of　Spaceland
  As　soon　as　the　sound　of　the　Peace…cry　of　my　departing　Wife　had　died　away；　I　began　to　approach　the　Stranger　with　the　intention　of　taking　a　nearer　view　and　of　bidding　him　be　seated：　but　his　appearance　struck　me　dumb　and　motionless　with　astonishment。　Without　the　slightest　symptoms　of　angularity　he　nevertheless　varied　every　instant　with　gradations　of　size　and　brightness　scarcely　possible　for　any　Figure　within　the　scope　of　my　experience。　　The　thought　flashed　across　me　that　I　might　have　before　me　a　burglar　or　cut…throat；　some　monstrous　Irregular　Isosceles；　who；　by　feigning　the　voice　of　a　Circle；　had　obtained　admission　somehow　into　the　house；　and　was　now　preparing　to　stab　me　with　his　acute　angle。
  In　a　sitting…room；　the　absence　of　Fog　（and　the　season　happened　to　be　remarkably　dry）；　made　it　difficult　for　me　to　trust　to　Sight　Recognition；　especially　at　the　short　distance　at　which　I　was　standing。　　Desperate　with　fear；　I　rushed　forward　with　an　unceremonious；　〃You　must　permit　me；　Sir　〃　and　felt　him。　My　Wife　was　right。　　There　was　not　the　trace　of　an　angle；　not　the　slightest　roughness　or　inequality：　　never　in　my　life　had　I　met　with　a　more　perfect　Circle。　　He　remained　motionless　while　I　walked　round　him；　beginning　from　his　eye　and　returning　to　it　again。　Circular　he　was　throughout；　a　perfectly　satisfactory　Circle；　there　could　not　be　a　doubt　of　it。　　Then　followed　a　dialogue；　which　I　will　endeavour　to　set　down　as　near　as　I　can　recollect　it；　omitting　only　some　of　my　profuse　apologies　　for　I　was　covered　with　shame　and　humiliation　that　I；　a　Square；　should　have　been　guilty　of　the　impertinence　of　feeling　a　Circle。　　It　was　commenced　by　the　Stranger　with　some　impatience　at　the　lengthiness　of　my　introductory　process。
  STRANGER。　　Have　you　felt　me　enough　by　this　time？　　Are　you　not　introduced　to　me　yet？
  I。　　Most　illustrious　Sir；　excuse　my　awkwardness；　which　arises　not　from　ignorance　of　the　usages　of　polite　society；　but　from　a　little　surprise　and　nervousness；　consequent　on　this　somewhat　unexpected　visit。　　And　I　beseech　you　to　reveal　my　indiscretion　to　no　one；　and　especially　not　to　my　Wife。　　But　before　your　Lordship　enters　into　further　communications；　would　he　deign　to　satisfy　the　curiosity　of　one　who　would　gladly　know　whence　his　Visitor　came？
  STRANGER。　　From　Space；　from　Space；　Sir：　　whence　else？
  I。　　Pardon　me；　my　Lord；　but　is　not　your　Lordship　already　in　Space；　your　Lordship　and　his　humble　servant；　even　at　this　moment？
  STRANGER。　　Pooh！　what　do　you　know　of　Space？　　Define　Space。
  I。　　Space；　my　Lord；　is　height　and　breadth　indefinitely　prolonged。
  STRANGER。　　Exactly：　　you　see　you　do　not　even　know　what　Space　is。　You　think　it　is　of　Two　Dimensions　only；　but　I　have　come　to　announce　to　you　a　Third　　height；　breadth；　and　length。
  I。　　Your　Lordship　is　pleased　to　be　merry。　　We　also　speak　of　length　and　height；　or　breadth　and　thickness；　thus　denoting　Two　Dimensions　by　four　names。
  STRANGER。　　But　I　mean　not　only　three　names；　but　Three　Dimensions。
  I。　　Would　your　Lordship　indicate　or　explain　to　me　in　what　direction　is　the　Third　Dimension；　unknown　to　me？
  STRANGER。　　I　came　from　it。　　It　is　up　above　and　down　below。
  I。　　My　Lord　means　seemingly　that　it　is　Northward　and　Southward。
  STRANGER。　　I　mean　nothing　of　the　kind。　　I　mean　a　direction　in　which　you　cannot　look；　because　you　have　no　eye　in　your　side。
  I。　　Pardon　me；　my　Lord；　a　moment's　inspection　will　convince　your　Lordship　that　I　have　a　perfect　luminary　at　the　juncture　of　two　of　my　sides。
  STRANGER。　　Yes：　　but　in　order　to　see　into　Space　you　ought　to　have　an　eye；　not　on　your　Perimeter；　but　on　your　side；　that　is；　on　what　you　would　probably　call　your　inside；　but　we　in　Spaceland　should　call　it　your　side。
  I。　　An　eye　in　my　inside！　　An　eye　in　my　stomach！　　Your　Lordship　jests。
  STRANGER。　　I　am　in　no　jesting　humour。　　I　tell　you　that　I　come　from　Space；　or；　since　you　will　not　understand　what　Space　means；　from　the　Land　of　Three　Dimensions　whence　I　but　lately　looked　down　upon　your　Plane　which　you　call　Space　forsooth。　　From　that　position　of　advantage　I　discerned　all　that　you　speak　of　as　SOLID　（by　which　you　mean　〃enclosed　on　four　sides〃）；　your　houses；　your　churches；　your　very　chests　and　safes；　yes　even　your　insides　and　stomachs；　all　lying　o