Whenspeakingofuninvitingobjects，Imeanthosewhichdonotpassfromonesensationtotheopposite；invitingobjectsarethosewhichdo；inthislattercasethesensecomingupontheobject，whetheratadistanceornear，givesnomorevividideaofanythinginparticularthanofitsopposite。Anillustrationwillmakemymeaningclearer：——herearethreefingers——alittlefinger，asecondfinger，andamiddlefinger。
　　Verygood。
　　Youmaysupposethattheyareseenquiteclose：Andherecomesthepoint。
　　Whatisit？
　　Eachofthemequallyappearsafinger，whetherseeninthemiddleorattheextremity，whetherwhiteorblack，orthickorthin——itmakesnodifference；afingerisafingerallthesame。Inthesecasesamanisnotcompelledtoaskofthoughtthequestion，whatisafinger？forthesightneverintimatestothemindthatafingerisotherthanafinger。
　　True。
　　Andtherefore，Isaid，aswemightexpect，thereisnothingherewhichinvitesorexcitesintelligence。
　　Thereisnot，hesaid。
　　Butisthisequallytrueofthegreatnessandsmallnessofthefingers？Cansightadequatelyperceivethem？andisnodifferencemadebythecircumstancethatoneofthefingersisinthemiddleandanotherattheextremity？Andinlikemannerdoesthetouchadequatelyperceivethequalitiesofthicknessorthinness，orsoftnessorhardness？Andsooftheothersenses；dotheygiveperfectintimationsofsuchmatters？Isnottheirmodeofoperationonthiswise——thesensewhichisconcernedwiththequalityofhardnessisnecessarilyconcernedalsowiththequalityofsoftness，andonlyintimatestothesoulthatthesamethingisfelttobebothhardandsoft？
　　Youarequiteright，hesaid。
　　Andmustnotthesoulbeperplexedatthisintimationwhichthesensegivesofahardwhichisalsosoft？What，again，isthemeaningoflightandheavy，ifthatwhichislightisalsoheavy，andthatwhichisheavy，light？
　　Yes，hesaid，theseintimationswhichthesoulreceivesareverycuriousandrequiretobeexplained。
　　Yes，Isaid，andintheseperplexitiesthesoulnaturallysummonstoheraidcalculationandintelligence，thatshemayseewhethertheseveralobjectsannouncedtoherareoneortwo。
　　True。
　　Andiftheyturnouttobetwo，isnoteachofthemoneanddifferent？
　　Certainly。
　　Andifeachisone，andbotharetwo，shewillconceivethetwoasinastateofdivision，foriftherewereundividedtheycouldonlybeconceivedofasone？
　　True。
　　Theeyecertainlydidseebothsmallandgreat，butonlyinaconfusedmanner；theywerenotdistinguished。
　　Yes。
　　Whereasthethinkingmind，intendingtolightupthechaos，wascompelledtoreversetheprocess，andlookatsmallandgreatasseparateandnotconfused。
　　Verytrue。
　　Wasnotthisthebeginningoftheenquiry’Whatisgreat？’and’Whatissmall？’
　　Exactlyso。
　　Andthusarosethedistinctionofthevisibleandtheintelligible。
　　Mosttrue。
　　ThiswaswhatImeantwhenIspokeofimpressionswhichinvitedtheintellect，orthereverse——thosewhicharesimultaneouswithoppositeimpressions，invitethought；thosewhicharenotsimultaneousdonot。
　　Iunderstand，hesaid，andagreewithyou。
　　Andtowhichclassdounityandnumberbelong？
　　Idonotknow，hereplied。
　　Thinkalittleandyouwillseethatwhathasprecededwillsupplytheanswer；forifsimpleunitycouldbeadequatelyperceivedbythesightorbyanyothersense，then，asweweresayinginthecaseofthefinger，therewouldbenothingtoattracttowardsbeing；butwhenthereissomecontradictionalwayspresent，andoneisthereverseofoneandinvolvestheconceptionofplurality，thenthoughtbeginstobearousedwithinus，andthesoulperplexedandwantingtoarriveatadecisionasks’Whatisabsoluteunity？’Thisisthewayinwhichthestudyoftheonehasapowerofdrawingandconvertingthemindtothecontemplationoftruebeing。
　　Andsurely，hesaid，thisoccursnotablyinthecaseofone；forweseethesamethingtobebothoneandinfiniteinmultitude？
　　Yes，Isaid；andthisbeingtrueofonemustbeequallytrueofallnumber？
　　Certainly。
　　Andallarithmeticandcalculationhavetodowithnumber？
　　Yes。
　　Andtheyappeartoleadthemindtowardstruth？
　　Yes，inaveryremarkablemanner。
　　Thenthisisknowledgeofthekindforwhichweareseeking，havingadoubleuse，militaryandphilosophical；forthemanofwarmustlearntheartofnumberorhewillnotknowhowtoarrayhistroops，andthephilosopheralso，becausehehastoriseoutoftheseaofchangeandlayholdoftruebeing，andthereforehemustbeanarithmetician。
　　Thatistrue。
　　Andourguardianisbothwarriorandphilosopher？
　　Certainly。
　　Thenthisisakindofknowledgewhichlegislationmayfitlyprescribe；andwemustendeavourtopersuadethosewhoareprescribetobetheprincipalmenofourStatetogoandlearnarithmetic，notasamateurs，buttheymustcarryonthestudyuntiltheyseethenatureofnumberswiththemindonly；noragain，likemerchantsorretail—traders，withaviewtobuyingorselling，butforthesakeoftheirmilitaryuse，andofthesoulherself；andbecausethiswillbetheeasiestwayforhertopassfrombecomingtotruthandbeing。
　　Thatisexcellent，hesaid。
　　Yes，Isaid，andnowhavingspokenofit，Imustaddhowcharmingthescienceis！andinhowmanywaysitconducestoourdesiredend，ifpursuedinthespiritofaphilosopher，andnotofashopkeeper！
　　Howdoyoumean？
　　Imean，asIwassaying，thatarithmetichasaverygreatandelevatingeffect，compellingthesoultoreasonaboutabstractnumber，andrebellingagainsttheintroductionofvisibleortangibleobjectsintotheargument。Youknowhowsteadilythemastersoftheartrepelandridiculeanyonewhoattemptstodivideabsoluteunitywhenheiscalculating，andifyoudivide，theymultiply，takingcarethatoneshallcontinueoneandnotbecomelostinfractions。
　　Thatisverytrue。
　　Now，supposeapersonweretosaytothem：Omyfriends，whatarethesewonderfulnumbersaboutwhichyouarereasoning，inwhich，asyousay，thereisaunitysuchasyoudemand，andeachunitisequal，invariable，indivisible，——whatwouldtheyanswer？
　　Theywouldanswer，asIshouldconceive，thattheywerespeakingofthosenumberswhichcanonlyberealisedinthought。
　　Thenyouseethatthisknowledgemaybetrulycallednecessary，necessitatingasitclearlydoestheuseofthepureintelligenceintheattainmentofpuretruth？
　　Yes；thatisamarkedcharacteristicofit。
　　Andhaveyoufurtherobserved，thatthosewhohaveanaturaltalentforcalculationaregenerallyquickateveryotherkindofknowledge；andeventhedulliftheyhavehadanarithmeticaltraining，althoughtheymayderivenootheradvantagefromit，alwaysbecomemuchquickerthantheywouldotherwisehavebeen。
　　Verytrue，hesaid。
　　Andindeed，youwillnoteasilyfindamoredifficultstudy，andnotmanyasdifficult。
　　Youwillnot。
　　And，forallthesereasons，arithmeticisakindofknowledgeinwhichthebestnaturesshouldbetrained，andwhichmustnotbegivenup。
　　Iagree。
　　Letthisthenbemadeoneofoursubjectsofeducation。Andnext，shallweenquirewhetherthekindredsciencealsoconcernsus？
　　Youmeangeometry？
　　Exactlyso。
　　Clearly，hesaid，weareconcernedwiththatpartofgeometrywhichrelatestowar；forinpitchingacamp，ortakingupaposition，orclosingorextendingthelinesofanarmy，oranyothermilitarymanoeuvre，whetherinactualbattleoronamarch，itwillmakeallthedifferencewhetherageneralisorisnotageometrician。
　　Yes，Isaid，butforthatpurposeaverylittleofeithergeometryorcalculationwillbeenough；thequestionrelatesrathertothegreaterandmoreadvancedpartofgeometry——whetherthattendsinanydegreetomakemoreeasythevisionoftheideaofgood；andthither，asIwassaying，allthingstendwhichcompelthesoultoturnhergazetowardsthatplace，whereisthefullperfectionofbeing，whichsheought，byallmeans，tobehold。
　　True，hesaid。
　　Thenifgeometrycompelsustoviewbeing，itconcernsus；ifbecomingonly，itdoesnotconcernus？
　　Yes，thatiswhatweassert。
　　Yetanybodywhohastheleastacquaintancewithgeometrywillnotdenythatsuchaconceptionofthescienceisinflatcontradictiontotheordinarylanguageofgeometricians。
　　Howso？
　　Theyhaveinviewpracticeonly，andarealwaysspeaking？inanarrowandridiculousmanner，ofsquaringandextendingandapplyingandthelike——theyconfusethenecessitiesofgeometrywiththoseofdailylife；whereasknowledgeistherealobjectofthewholescience。
　　Certainly，hesaid。
　　Thenmustnotafurtheradmissionbemade？
　　Whatadmission？
　　Thattheknowledgeatwhichgeometryaimsisknowledgeoftheeternal，andnotofaughtperishingandtransient。
　　That，hereplied，maybereadilyallowed，andistrue。
　　Then，mynoblefriend，geometrywilldrawthesoultowardstruth，andcreatethespiritofphilosophy，andraiseupthatwhichisnowunhappilyallowedtofalldown。
　　Nothingwillbemorelikelytohavesuchaneffect。
　　Thennothingshouldbemoresternlylaiddownthanthattheinhabitantsofyourfaircityshouldbyallmeanslearngeometry。
　　Moreoverthesciencehasindirecteffects，whicharenotsmall。
　　Ofwhatkind？hesaid。
　　Therearethemilitaryadvantagesofwhichyouspoke，Isaid；andinalldepartmentsofknowledge，asexperienceproves，anyonewhohasstudiedgeometryisinfinitelyquickerofapprehensionthanonewhohasnot。
　　Yesindeed，hesaid，thereisaninfinitedifferencebetweenthem。
　　Thenshallweproposethisasasecondbranchofknowledgewhichouryouthwillstudy？
　　Letusdoso，hereplied。
　　Andsupposewemakeastronomythethird——whatdoyousay？
　　Iamstronglyinclinedtoit，hesaid；theobservationoftheseasonsandofmonthsandyearsisasessentialtothegeneralasitistothefarmerorsailor。
　　Iamamused，Isaid，atyourfearoftheworld，whichmakesyouguardagainsttheappearanceofinsistinguponuselessstudies；andIquiteadmitthedifficultyofbelievingthatineverymanthereisaneyeofthesoulwhich，whenbyotherpursuitslostanddimmed，isbythesepurifiedandre—illumined；andismorepreciousfarthantenthousandbodilyeyes，forbyitaloneistruthseen。Nowtherearetwoclassesofpersons：oneclassofthosewhowillagreewithyouandwilltakeyourwordsasarevelation；anotherclasstowhomtheywillbeutterlyunmeaning，andwhowillnaturallydeemthemtobeidletales，fortheyseenosortofprofitwhichistobeobtainedfromthem。Andthereforeyouhadbetterdecideatoncewithwhichofthetwoyouareproposingtoargue。Youwillverylikelysaywithneither，andthatyourchiefaimincarryingontheargumentisyourownimprovement；atthesametimeyoudonotgrudgetoothersanybenefitwhichtheymayreceive。
　　IthinkthatIshouldprefertocarryontheargumentmainlyonmyownbehalf。
　　Thentakeastepbackward，forwehavegonewrongintheorderofthesciences。
　　Whatwasthemistake？hesaid。
　　Afterplanegeometry，Isaid，weproceededatoncetosolidsinrevolution，insteadoftakingsolidsinthemselves；whereasaftertheseconddimensionthethird，whichisconcernedwithcubesanddimensionsofdepth，oughttohavefollowed。
　　Thatistrue，Socrates；butsolittleseemstobeknownasyetaboutthesesubjects。
　　Why，yes，Isaid，andfortworeasons：——inthefirstplace，nogovernmentpatronisesthem；thisleadstoawantofenergyinthepursuitofthem，andtheyaredifficult；inthesecondplace，studentscannotlearnthemunlesstheyhaveadirector。Butthenadirectorcanhardlybefound，andevenifhecould，asmattersnowstand，thestudents，whoareveryconceited，wouldnotattendtohim。That，however，wouldbeotherwiseifthewholeStatebecamethedirectorofthesestudiesandgavehonourtothem；thendiscipleswouldwanttocome，andtherewouldbecontinuousandearnestsearch，anddiscoverieswouldbemade；sinceevennow，disregardedastheyarebytheworld，andmaimedoftheirfairproportions，andalthoughnoneoftheirvotariescantelltheuseofthem，stillthesestudiesforcetheirwaybytheirnaturalcharm，andverylikely，iftheyhadthehelpoftheState，theywouldsomedayemergeintolight。
　　Yes，hesaid，thereisaremarkablecharminthem。ButIdonotclearlyunderstandthechangeintheorder。Firstyoubeganwithageometryofplanesurfaces？
　　Yes，Isaid。
　　Andyouplacedastronomynext，andthenyoumadeastepbackward？
　　Yes，andIhavedelayedyoubymyhurry；theludicrousstateofsolidgeometry，which，innaturalorder，shouldhavefollowed，mademepassoverthisbranchandgoontoastronomy，ormotionofsolids。
　　True，hesaid。
　　ThenassumingthatthesciencenowomittedwouldcomeintoexistenceifencouragedbytheState，letusgoontoastronomy，whichwillbefourth。
　　Therightorder，hereplied。Andnow，Socrates，asyourebukedthevulgarmannerinwhichIpraisedastronomybefore，mypraiseshallbegiveninyourownspirit。Foreveryone，asIthink，mustseethatastronomycompelsthesoultolookupwardsandleadsusfromthisworldtoanother。
　　Everyonebutmyself，Isaid；toeveryoneelsethismaybeclear，butnottome。
　　Andwhatthenwouldyousay？
　　Ishouldrathersaythatthosewhoelevateastronomyintophilosophyappeartometomakeuslookdownwardsandnotupwards。
　　Whatdoyoumean？heasked。
　　You，Ireplied，haveinyourmindatrulysublimeconceptionofourknowledgeofthethingsabove。AndIdaresaythatifapersonweretothrowhisheadbackandstudythefrettedceiling，youwouldstillthinkthathismindwasthepercipient，andnothiseyes。Andyouareverylikelyright，andImaybeasimpleton：but，inmyopinion，thatknowledgeonlywhichisofbeingandoftheunseencanmakethesoullookupwards，andwhetheramangapesattheheavensorblinksontheground，seekingtolearnsomeparticularofsense，Iwoulddenythathecanlearn，fornothingofthatsortismatterofscience；hissoulislookingdownwards，notupwards，whetherhiswaytoknowledgeisbywaterorbyland，whetherhefloats，oronlyliesonhisback。
　　Iacknowledge，hesaid，thejusticeofyourrebuke。Still，I
　　shouldliketoascertainhowastronomycanbelearnedinanymannermoreconducivetothatknowledgeofwhichwearespeaking？
　　Iwilltellyou，Isaid：Thestarryheavenwhichwebeholdiswroughtuponavisibleground，andtherefore，althoughthefairestandmostperfectofvisiblethings，mustnecessarilybedeemedinferiorfartothetruemotionsofabsoluteswiftnessandabsoluteslowness，whicharerelativetoeachother，andcarrywiththemthatwhichiscontainedinthem，inthetruenumberandineverytruefigure。Now，thesearetobeapprehendedbyreasonandintelligence，butnotbysight。
　　True，hereplied。
　　Thespangledheavensshouldbeusedasapatternandwithaviewtothathigherknowledge；theirbeautyislikethebeautyoffiguresorpicturesexcellentlywroughtbythehandofDaedalus，orsomeothergreatartist，whichwemaychancetobehold；anygeometricianwhosawthemwouldappreciatetheexquisitenessoftheirworkmanship，buthewouldneverdreamofthinkingthatinthemhecouldfindthetrueequalorthetruedouble，orthetruthofanyotherproportion。
　　No，hereplied，suchanideawouldberidiculous。
　　Andwillnotatrueastronomerhavethesamefeelingwhenhelooksatthemovementsofthestars？WillhenotthinkthatheavenandthethingsinheavenareframedbytheCreatoroftheminthemostperfectmanner？Buthewillneverimaginethattheproportionsofnightandday，orofbothtothemonth，orofthemonthtotheyear，orofthestarstotheseandtooneanother，andanyotherthingsthatarematerialandvisiblecanalsobeeternalandsubjecttonodeviation——thatwouldbeabsurd；anditisequallyabsurdtotakesomuchpainsininvestigatingtheirexacttruth。
　　Iquiteagree，thoughIneverthoughtofthisbefore。
　　Then，Isaid，inastronomy，asingeometry，weshouldemployproblems，andlettheheavensaloneifwewouldapproachthesubjectintherightwayandsomakethenaturalgiftofreasontobeofanyrealuse。
　　That，hesaid，isaworkinfinitelybeyondourpresentastronomers。
　　Yes，Isaid；andtherearemanyotherthingswhichmustalsohaveasimilarextensiongiventothem，ifourlegislationistobeofanyvalue。Butcanyoutellmeofanyothersuitablestudy？
　　No，hesaid，notwithoutthinking。
　　Motion，Isaid，hasmanyforms，andnotoneonly；twoofthemareobviousenougheventowitsnobetterthanours；andthereareothers，asIimagine，whichmaybelefttowiserpersons。
　　Butwherearethetwo？
　　Thereisasecond，Isaid，whichisthecounterpartoftheonealreadynamed。
　　Andwhatmaythatbe？
　　Thesecond，Isaid，wouldseemrelativelytotheearstobewhatthefirstistotheeyes；forIconceivethatastheeyesaredesignedtolookupatthestars，soaretheearstohearharmoniousmotions；
　　andthesearesistersciences——asthePythagoreanssay，andwe，Glaucon，agreewiththem？
　　Yes，hereplied。
　　Butthis，Isaid，isalaboriousstudy，andthereforewehadbettergoandlearnofthem；andtheywilltelluswhetherthereareanyotherapplicationsofthesesciences。Atthesametime，wemustnotlosesightofourownhigherobject。
　　Whatisthat？
　　Thereisaperfectionwhichallknowledgeoughttoreach，andwhichourpupilsoughtalsotoattain，andnottofallshortof，asIwassayingthattheydidinastronomy。Forinthescienceofharmony，asyouprobablyknow，thesamethinghappens。Theteachersofharmonycomparethesoundsandconsonanceswhichareheardonly，andtheirlabour，likethatoftheastronomers，isinvain。
　　Yes，byheaven！hesaid；and’tisasgoodasaplaytohearthemtalkingabouttheircondensednotes，astheycallthem；theyputtheirearsclosealongsideofthestringslikepersonscatchingasoundfromtheirneighbour’swall——onesetofthemdeclaringthattheydistinguishanintermediatenoteandhavefoundtheleastintervalwhichshouldbetheunitofmeasurement；theothersinsistingthatthetwosoundshavepassedintothesame——eitherpartysettingtheirearsbeforetheirunderstanding。
　　Youmean，Isaid，thosegentlemenwhoteaseandtorturethestringsandrackthemonthepegsoftheinstrument：mightcarryonthemetaphorandspeakaftertheirmanneroftheblowswhichtheplectrumgives，andmakeaccusationsagainstthestrings，bothofbackwardnessandforwardnesstosound；butthiswouldbetedious，andthereforeIwillonlysaythatthesearenotthemen，andthatI
　　amreferringtothePythagoreans，ofwhomIwasjustnowproposingtoenquireaboutharmony。Fortheytooareinerror，liketheastronomers；theyinvestigatethenumbersoftheharmonieswhichareheard，buttheyneverattaintoproblems—thatistosay，theyneverreachthenaturalharmoniesofnumber，orreflectwhysomenumbersareharmoniousandothersnot。
　　That，hesaid，isathingofmorethanmortalknowledge。
　　Athing，Ireplied，whichIwouldrathercalluseful；thatis，ifsoughtafterwithaviewtothebeautifulandgood；butifpursuedinanyotherspirit，useless。Verytrue，hesaid。
　　Now，whenallthesestudiesreachthepointofinter—communionandconnectionwithoneanother，andcometobeconsideredintheirmutualaffinities，then，Ithink，butnottillthen，willthepursuitofthemhaveavalueforourobjects；otherwisethereisnoprofitinthem。
　　Isuspectso；butyouarespeaking，Socrates，ofavastwork。
　　Whatdoyoumean？Isaid；thepreludeorwhat？Doyounotknowthatallthisisbutthepreludetotheactualstrainwhichwehavetolearn？Foryousurelywouldnotregardtheskilledmathematicianasadialectician？
　　Assuredlynot，hesaid；Ihavehardlyeverknownamathematicianwhowascapableofreasoning。
　　Butdoyouimaginethatmenwhoareunabletogiveandtakeareasonwillhavetheknowledgewhichwerequireofthem？
　　Neithercanthisbesupposed。
　　Andso，Glaucon，Isaid，wehaveatlastarrivedatthehymnofdialectic。Thisisthatstrainwhichisoftheintellectonly，butwhichthefacultyofsightwillneverthelessbefoundtoimitate；
　　forsight，asyoumayremember，wasimaginedbyusafterawhiletobeholdtherealanimalsandstars，andlastofallthesunhimself。
　　Andsowithdialectic；whenapersonstartsonthediscoveryoftheabsolutebythelightofreasononly，andwithoutanyassistanceofsense，andperseveresuntilbypureintelligencehearrivesattheperceptionoftheabsolutegood，heatlastfindshimselfattheendoftheintellectualworld，asinthecaseofsightattheendofthevisible。
　　Exactly，hesaid。
　　Thenthisistheprogresswhichyoucalldialectic？
　　True。
　　Butthereleaseoftheprisonersfromchains，andtheirtranslationfromtheshadowstotheimagesandtothelight，andtheascentfromtheundergrounddentothesun，whileinhispresencetheyarevainlytryingtolookonanimalsandplantsandthelightofthesun，butareabletoperceiveevenwiththeirweakeyestheimagesinthewater（whicharedivine），andaretheshadowsoftrueexistence（notshadowsofimagescastbyalightoffire，whichcomparedwiththesunisonlyanimage）——thispowerofelevatingthehighestprincipleinthesoultothecontemplationofthatwhichisbestinexistence，withwhichwemaycomparetheraisingofthatfacultywhichistheverylightofthebodytothesightofthatwhichisbrightestinthematerialandvisibleworld——thispowerisgiven，asIwassaying，byallthatstudyandpursuitoftheartswhichhasbeendescribed。
　　Iagreeinwhatyouaresaying，hereplied，whichmaybehardtobelieve，yet，fromanotherpointofview，isharderstilltodeny。
　　This，however，isnotathemetobetreatedofinpassingonly，butwillhavetobediscussedagainandagain。Andso，whetherourconclusionbetrueorfalse，letusassumeallthis，andproceedatoncefromthepreludeorpreambletothechiefstrain，anddescribethatinlikemanner。Say，then，whatisthenatureandwhatarethedivisionsofdialectic，andwhatarethepathswhichleadthither；forthesepathswillalsoleadtoourfinalrest？
　　DearGlaucon，Isaid，youwillnotbeabletofollowmehere，thoughIwoulddomybest，andyoushouldbeholdnotanimageonlybuttheabsolutetruth，accordingtomynotion。WhetherwhatItoldyouwouldorwouldnothavebeenarealityIcannotventuretosay；butyouwouldhaveseensomethinglikereality；ofthatIamconfident。
　　Doubtless，hereplied。