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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，itwillmakeal
