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。