Butthisconsequencealsowemustnotforget，thatitfollowsthat
　　therearepriorandposterior2andsimilarlywiththeother
　　numbers。Forletthe2’sinthe4besimultaneous；yettheseareprior
　　tothoseinthe8andasthe2generatedthem，theygeneratedthe
　　4’sinthe8-itself。Thereforeifthefirst2isanIdea，these2’s
　　alsowillbeIdeasofsomekind。Andthesameaccountappliestothe
　　units；fortheunitsinthefirst2generatethefourin4，sothat
　　alltheunitscometobeIdeasandanIdeawillbecomposedof
　　Ideas。Clearlythereforethosethingsalsoofwhichthesehappentobe
　　theIdeaswillbecomposite，e。g。onemightsaythatanimalsare
　　composedofanimals，ifthereareIdeasofthem。
　　Ingeneral，todifferentiatetheunitsinanywayisan
　　absurdityandafiction；andbyafictionImeanaforcedstatement
　　madetosuitahypothesis。Forneitherinquantitynorinqualitydo
　　weseeunitdifferingfromunit，andnumbermustbeeitherequalor
　　unequal-allnumberbutespeciallythatwhichconsistsofabstract
　　units-sothatifonenumberisneithergreaternorlessthan
　　another，itisequaltoit；butthingsthatareequalandinnowise
　　differentiatedwetaketobethesamewhenwearespeakingofnumbers。
　　Ifnot，noteventhe2inthe10-itselfwillbeundifferentiated，
　　thoughtheyareequal；forwhatreasonwillthemanwhoallegesthat
　　theyarenotdifferentiatedbeabletogive？
　　Again，ifeveryunitanotherunitmakestwo，aunitfromthe
　　2-itselfandonefromthe3-itselfwillmakea2。Nowathiswill
　　consistofdifferentiatedunits；andwillitbepriortothe3or
　　posterior？Itratherseemsthatitmustbeprior；foroneoftheunits
　　issimultaneouswiththe3andtheotherissimultaneouswiththe2。
　　Andwe，forourpart，supposethatingeneral1and1，whetherthe
　　thingsareequalorunequal，is2，e。g。thegoodandthebad，oraman
　　andahorse；butthosewhoholdtheseviewssaythatnoteventwo
　　unitsare2。
　　Ifthenumberofthe3-itselfisnotgreaterthanthatofthe2，
　　thisissurprising；andifitisgreater，clearlythereisalsoa
　　numberinitequaltothe2，sothatthisisnotdifferentfromthe
　　2-itself。Butthisisnotpossible，ifthereisafirstandasecond
　　number。
　　NorwilltheIdeasbenumbers。Forinthisparticularpointthey
　　arerightwhoclaimthattheunitsmustbedifferent，ifthereare
　　tobeIdeas；ashasbeensaidbefore。FortheFormisunique；butif
　　theunitsarenotdifferent，the2’sandthe3’salsowillnotbe
　　different。Thisisalsothereasonwhytheymustsaythatwhenwe
　　countthus-’1，2’-wedonotproceedbyaddingtothegivennumber；
　　forifwedo，neitherwillthenumbersbegeneratedfromthe
　　indefinitedyad，norcananumberbeanIdea；forthenoneIdeawill
　　beinanother，andallFormswillbepartsofoneForm。Andsowith
　　aviewtotheirhypothesistheirstatementsareright，butasa
　　wholetheyarewrong；fortheirviewisverydestructive，sincethey
　　willadmitthatthisquestionitselfaffordssome
　　difficulty-whether，whenwecountandsay-1，2，3-wecountby
　　additionorbyseparateportions。Butwedoboth；andsoitis
　　absurdtoreasonbackfromthisproblemtosogreatadifferenceof
　　essence。
　　Firstofallitiswelltodeterminewhatisthedifferentiaof
　　anumber-andofaunit，ifithasadifferentia。Unitsmustdiffer
　　eitherinquantityorinquality；andneitheroftheseseemstobe
　　possible。Butnumberquanumberdiffersinquantity。Andifthe
　　unitsalsodiddifferinquantity，numberwoulddifferfromnumber，
　　thoughequalinnumberofunits。Again，arethefirstunitsgreateror
　　smaller，anddothelateronesincreaseordiminish？Alltheseare
　　irrationalsuppositions。Butneithercantheydifferinquality。For
　　noattributecanattachtothem；foreventonumbersqualityissaid
　　tobelongafterquantity。Again，qualitycouldnotcometothemeither
　　fromthe1orthedyad；fortheformerhasnoquality，andthe
　　lattergivesquantity；forthisentityiswhatmakesthingstobe
　　many。Ifthefactsarereallyotherwise，theyshouldstatethis
　　quiteatthebeginninganddetermineifpossible，regardingthe
　　differentiaoftheunit，whyitmustexist，and，failingthis，what
　　differentiatheymean。
　　Evidentlythen，iftheIdeasarenumbers，theunitscannotall
　　beassociable，norcantheybeinassociableineitherofthetwoways。
　　Butneitheristhewayinwhichsomeothersspeakaboutnumbers
　　correct。ThesearethosewhodonotthinkthereareIdeas，either
　　withoutqualificationorasidentifiedwithcertainnumbers，butthink
　　theobjectsofmathematicsexistandthenumbersarethefirstof
　　existingthings，andthe1-itselfisthestarting-pointofthem。Itis
　　paradoxicalthatthereshouldbea1whichisfirstof1’s，asthey
　　say，butnota2whichisfirstof2’s，nora3of3’s；forthesame
　　reasoningappliestoall。If，then，thefactswithregardtonumber
　　areso，andonesupposesmathematicalnumberalonetoexist，the1
　　isnotthestarting-pointforthissortof1mustdifferfrom
　　the-otherunits；andifthisisso，theremustalsobea2whichis
　　firstof2’s，andsimilarlywiththeothersuccessivenumbers。Butif
　　the1isthestarting-point，thetruthaboutthenumbersmustrather
　　bewhatPlatousedtosay，andtheremustbeafirst2and3and
　　numbersmustnotbeassociablewithoneanother。Butifontheother
　　handonesupposesthis，manyimpossibleresults，aswehavesaid，
　　follow。Buteitherthisortheothermustbethecase，sothatif
　　neitheris，numbercannotexistseparately。
　　Itisevident，also，fromthisthatthethirdversionisthe
　　worst，-theviewidealandmathematicalnumberisthesame。Fortwo
　　mistakesmustthenmeetintheoneopinion。1Mathematicalnumber
　　cannotbeofthissort，buttheholderofthisviewhastospinitout
　　bymakingsuppositionspeculiartohimself。And2hemustalsoadmit
　　alltheconsequencesthatconfrontthosewhospeakofnumberinthe
　　senseof’Forms’。
　　ThePythagoreanversioninonewayaffordsfewerdifficultiesthan
　　thosebeforenamed，butinanotherwayhasotherspeculiarto
　　itself。Fornotthinkingofnumberascapableofexistingseparately
　　removesmanyoftheimpossibleconsequences；butthatbodiesshouldbe
　　composedofnumbers，andthatthisshouldbemathematicalnumber，is
　　impossible。Foritisnottruetospeakofindivisiblespatial
　　magnitudes；andhowevermuchtheremightbemagnitudesofthissort，
　　unitsatleasthavenotmagnitude；andhowcanamagnitudebecomposed
　　ofindivisibles？Butarithmeticalnumber，atleast，consistsofunits，
　　whilethesethinkersidentifynumberwithrealthings；atanyrate
　　theyapplytheirpropositionstobodiesasiftheyconsistedof
　　thosenumbers。
　　If，then，itisnecessary，ifnumberisaself-subsistentreal
　　thing，thatitshouldexistinoneofthesewayswhichhavebeen
　　mentioned，andifitcannotexistinanyofthese，evidentlynumber
　　hasnosuchnatureasthosewhomakeitseparablesetupforit。
　　Again，doeseachunitcomefromthegreatandthesmall，
　　equalized，oronefromthesmall，anotherfromthegreat？aIfthe
　　latter，neitherdoeseachthingcontainalltheelements，norare
　　theunitswithoutdifference；forinonethereisthegreatandin
　　anotherthesmall，whichiscontraryinitsnaturetothegreat。
　　Again，howisitwiththeunitsinthe3-itself？Oneofthemisanodd
　　unit。Butperhapsitisforthisreasonthattheygive1-itselfthe
　　middleplaceinoddnumbers。bButifeachofthetwounitsconsists
　　ofboththegreatandthesmall，equalized，howwillthe2whichis
　　asinglething，consistofthegreatandthesmall？Orhowwillit
　　differfromtheunit？Again，theunitispriortothe2；forwhenit
　　isdestroyedthe2isdestroyed。Itmust，then，betheIdeaofanIdea
　　sinceitispriortoanIdea，anditmusthavecomeintobeing
　　beforeit。Fromwhat，then？Notfromtheindefinitedyad，forits
　　functionwastodouble。
　　Again，numbermustbeeitherinfiniteorfinite；forthese
　　thinkersthinkofnumberascapableofexistingseparately，sothatit
　　isnotpossiblethatneitherofthosealternativesshouldbetrue。
　　Clearlyitcannotbeinfinite；forinfinitenumberisneitherodd
　　noreven，butthegenerationofnumbersisalwaysthegeneration
　　eitherofanoddorofanevennumber；inoneway，when1operates
　　onanevennumber，anoddnumberisproduced；inanotherway，when2
　　operates，thenumbersgotfrom1bydoublingareproduced；in
　　anotherway，whentheoddnumbersoperate，theotherevennumbers
　　areproduced。Again，ifeveryIdeaisanIdeaofsomething，andthe
　　numbersareIdeas，infinitenumberitselfwillbeanIdeaof
　　something，eitherofsomesensiblethingorofsomethingelse。Yet
　　thisisnotpossibleinviewoftheirthesisanymorethanitis
　　reasonableinitself，atleastiftheyarrangetheIdeasastheydo。
　　Butifnumberisfinite，howfardoesitgo？Withregardtothis
　　notonlythefactbutthereasonshouldbestated。Butifnumber
　　goesonlyupto10assomesay，firstlytheFormswillsoonrunshort；
　　e。g。if3isman-himself，whatnumberwillbethehorse-itself？The
　　seriesofthenumberswhicharetheseveralthings-themselvesgoes
　　upto10。Itmust，then，beoneofthenumberswithintheselimits；
　　foritisthesethataresubstancesandIdeas。Yettheywillrun
　　short；forthevariousformsofanimalwilloutnumberthem。Atthe
　　sametimeitisclearthatifinthiswaythe3isman-himself，the
　　other3’saresoalsoforthoseinidenticalnumbersaresimilar，so
　　thattherewillbeaninfinitenumberofmen；ifeach3isanIdea，
　　eachofthenumberswillbeman-himself，andifnot，theywillat
　　leastbemen。Andifthesmallernumberispartofthegreater
　　beingnumberofsuchasortthattheunitsinthesamenumberare
　　associable，thenifthe4-itselfisanIdeaofsomething，e。g。of
　　’horse’orof’white’，manwillbeapartofhorse，ifmanisItis
　　paradoxicalalsothatthereshouldbeanIdeaof10butnotof11，nor
　　ofthesucceedingnumbers。Again，therebothareandcometobe
　　certainthingsofwhichtherearenoForms；why，then，aretherenot
　　Formsofthemalso？WeinferthattheFormsarenotcauses。Again，
　　itisparadoxical-ifthenumberseriesupto10ismoreofareal
　　thingandaFormthan10itself。Thereisnogenerationofthe
　　formerasonething，andthereisofthelatter。Buttheytryto
　　workontheassumptionthattheseriesofnumbersupto10isa
　　completeseries。Atleasttheygeneratethederivatives-e。g。thevoid，
　　proportion，theodd，andtheothersofthiskind-withinthedecade。
　　Forsomethings，e。g。movementandrest，goodandbad，theyassign
　　totheoriginativeprinciples，andtheotherstothenumbers。This
　　iswhytheyidentifytheoddwith1；foriftheoddimplied3how
　　would5beodd？Again，spatialmagnitudesandallsuchthingsare
　　explainedwithoutgoingbeyondadefinitenumber；e。g。thefirst，
　　theindivisible，line，thenthe2&c。；theseentitiesalsoextendonly
　　upto10。
　　Again，ifnumbercanexistseparately，onemightaskwhichis
　　prior-1，or3or2？Inasmuchasthenumberiscomposite，1isprior，
　　butinasmuchastheuniversalandtheformisprior，thenumberis
　　prior；foreachoftheunitsispartofthenumberasitsmatter，
　　andthenumberactsasform。Andinasensetherightangleisprior
　　totheacute，becauseitisdeterminateandinvirtueofits
　　definition；butinasensetheacuteisprior，becauseitisapart
　　andtherightangleisdividedintoacuteangles。Asmatter，then，the
　　acuteangleandtheelementandtheunitareprior，butinrespect
　　oftheformandofthesubstanceasexpressedinthedefinition，the
　　rightangle，andthewholeconsistingofthematterandtheform，
　　areprior；fortheconcretethingisnearertotheformandtowhatis
　　expressedinthedefinition，thoughingenerationitislater。How
　　thenis1thestarting-point？Becauseitisnotdivisiable，they
　　say；butboththeuniversal，andtheparticularortheelement，are
　　indivisible。Buttheyarestarting-pointsindifferentways，onein
　　definitionandtheotherintime。Inwhichway，then，is1the