But，further，allotherthingscannotcomefromtheFormsinany
　　oftheusualsensesof’from’。Andtosaythattheyarepatternsand
　　theotherthingsshareinthemistouseemptywordsandpoetical
　　metaphors。Forwhatisitthatworks，lookingtotheIdeas？Andany
　　thingcanbothbeandcomeintobeingwithoutbeingcopiedfrom
　　somethingelse，sothat，whetherSocratesexistsornot，amanlike
　　Socratesmightcometobe。Andevidentlythismightbesoevenif
　　Socrateswereeternal。Andtherewillbeseveralpatternsofthe
　　samething，andthereforeseveralForms；e。g。’animal’and
　　’two-footed’，andalso’man-himself’，willbeFormsofman。Again，the
　　Formsarepatternsnotonlyofsensiblethings，butofForms
　　themselvesalso；i。e。thegenusisthepatternofthevarious
　　forms-of-a-genus；thereforethesamethingwillbepatternandcopy。
　　Again，itwouldseemimpossiblethatsubstanceandthatwhose
　　substanceitisshouldexistapart；how，therefore，couldtheIdeas，
　　beingthesubstancesofthings，existapart？
　　InthePhaedothecaseisstatedinthisway-thattheFormsare
　　causesbothofbeingandofbecoming。YetthoughtheFormsexist，
　　stillthingsdonotcomeintobeing，unlessthereissomethingto
　　originatemovement；andmanyotherthingscomeintobeinge。g。a
　　houseoraringofwhichtheysaytherearenoForms。Clearly
　　thereforeeventhethingsofwhichtheysaythereareIdeascanboth
　　beandcomeintobeingowingtosuchcausesasproducethethingsjust
　　mentioned，andnotowingtotheForms。ButregardingtheIdeasitis
　　possible，bothinthiswayandbymoreabstractandaccurate
　　arguments，tocollectmanyobjectionslikethosewehaveconsidered。
　　Sincewehavediscussedthesepoints，itiswelltoconsideragain
　　theresultsregardingnumberswhichconfrontthosewhosaythat
　　numbersareseparablesubstancesandfirstcausesofthings。Ifnumber
　　isanentityanditssubstanceisnothingotherthanjustnumber，as
　　somesay，itfollowsthateither1thereisafirstinitanda
　　second，eachbeingdifferentinspecies，-andeitherathisistrue
　　oftheunitswithoutexception，andanyunitisinassociablewith
　　anyunit，orbtheyareallwithoutexceptionsuccessive，andanyof
　　themareassociablewithany，astheysayisthecasewith
　　mathematicalnumber；forinmathematicalnumbernooneunitisin
　　anywaydifferentfromanother。Orcsomeunitsmustbeassociable
　　andsomenot；e。g。supposethat2isfirstafter1，andthencomes3
　　andthentherestofthenumberseries，andtheunitsineachnumber
　　areassociable，e。g。thoseinthefirst2areassociablewithone
　　another，andthoseinthefirst3withoneanother，andsowiththe
　　othernumbers；buttheunitsinthe’2-itself’areinassociablewith
　　thoseinthe’3-itself’；andsimilarlyinthecaseoftheother
　　successivenumbers。Andsowhilemathematicalnumberiscounted
　　thus-after1，2whichconsistsofanother1besidestheformer1，
　　and3whichconsistsofanother1besidesthesetwo，andtheother
　　numberssimilarly，idealnumberiscountedthus-after1，adistinct
　　2whichdoesnotincludethefirst1，anda3whichdoesnotinclude
　　the2andtherestofthenumberseriessimilarly。Or2onekind
　　ofnumbermustbelikethefirstthatwasnamed，onelikethatwhich
　　themathematiciansspeakof，andthatwhichwehavenamedlastmustbe
　　athirdkind。
　　Again，thesekindsofnumbersmusteitherbeseparablefrom
　　things，ornotseparablebutinobjectsofperceptionnothowever
　　inthewaywhichwefirstconsidered，inthesensethatobjectsof
　　perceptionconsistsofnumberswhicharepresentinthem-eitherone
　　kindandnotanother，orallofthem。
　　Theseareofnecessitytheonlywaysinwhichthenumberscan
　　exist。Andofthosewhosaythatthe1isthebeginningand
　　substanceandelementofallthings，andthatnumberisformedfrom
　　the1andsomethingelse，almosteveryonehasdescribednumberinone
　　oftheseways；onlynoonehassaidalltheunitsareinassociable。
　　Andthishashappenedreasonablyenough；fortherecanbenoway
　　besidesthosementioned。Somesaybothkindsofnumberexist，that
　　whichhasabeforeandafterbeingidenticalwiththeIdeas，and
　　mathematicalnumberbeingdifferentfromtheIdeasandfromsensible
　　things，andbothbeingseparablefromsensiblethings；andothers
　　saymathematicalnumberaloneexists，asthefirstofrealities，
　　separatefromsensiblethings。AndthePythagoreans，also，believe
　　inonekindofnumber-themathematical；onlytheysayitisnot
　　separatebutsensiblesubstancesareformedoutofit。Forthey
　　constructthewholeuniverseoutofnumbers-onlynotnumbers
　　consistingofabstractunits；theysupposetheunitstohavespatial
　　magnitude。Buthowthefirst1wasconstructedsoastohave
　　magnitude，theyseemunabletosay。
　　Anotherthinkersaysthefirstkindofnumber，thatofthe
　　Forms，aloneexists，andsomesaymathematicalnumberisidentical
　　withthis。
　　Thecaseoflines，planes，andsolidsissimilar。Forsomethink
　　thatthosewhicharetheobjectsofmathematicsaredifferentfrom
　　thosewhichcomeaftertheIdeas；andofthosewhoexpress
　　themselvesotherwisesomespeakoftheobjectsofmathematicsandina
　　mathematicalway-viz。thosewhodonotmaketheIdeasnumbersnor
　　saythatIdeasexist；andothersspeakoftheobjectsof
　　mathematics，butnotmathematically；fortheysaythatneitheris
　　everyspatialmagnitudedivisibleintomagnitudes，nordoanytwo
　　unitstakenatrandommake2。Allwhosaythe1isanelementand
　　principleofthingssupposenumberstoconsistofabstractunits，
　　exceptthePythagoreans；buttheysupposethenumberstohave
　　magnitude，ashasbeensaidbefore。Itisclearfromthisstatement，
　　then，inhowmanywaysnumbersmaybedescribed，andthatalltheways
　　havebeenmentioned；andalltheseviewsareimpossible，butsome
　　perhapsmorethanothers。
　　First，then，letusinquireiftheunitsareassociableor
　　inassociable，andifinassociable，inwhichofthetwowayswe
　　distinguished。Foritispossiblethatanyunityisinassociable
　　withany，anditispossiblethatthoseinthe’itself’are
　　inassociablewiththoseinthe’itself’，and，generally，thatthosein
　　eachidealnumberareinassociablewiththoseinotherideal
　　numbers。Now1allunitsareassociableandwithoutdifference，we
　　getmathematicalnumber-onlyonekindofnumber，andtheIdeas
　　cannotbethenumbers。Forwhatsortofnumberwillman-himselfor
　　animal-itselforanyotherFormbe？ThereisoneIdeaofeachthing
　　e。g。oneofman-himselfandanotheroneofanimal-itself；butthe
　　similarandundifferentiatednumbersareinfinitelymany，sothat
　　anyparticular3isnomoreman-himselfthananyother3。Butifthe
　　Ideasarenotnumbers，neithercantheyexistatall。Forfromwhat
　　principleswilltheIdeascome？Itisnumberthatcomesfromthe1and
　　theindefinitedyad，andtheprinciplesorelementsaresaidtobe
　　principlesandelementsofnumber，andtheIdeascannotberankedas
　　eitherpriororposteriortothenumbers。
　　But2iftheunitsareinassociable，andinassociableinthe
　　sensethatanyisinassociablewithanyother，numberofthissort
　　cannotbemathematicalnumber；formathematicalnumberconsistsof
　　undifferentiatedunits，andthetruthsprovedofitsuitthis
　　character。Norcanitbeidealnumber。For2willnotproceed
　　immediatelyfrom1andtheindefinitedyad，andbefollowedbythe
　　successivenumbers，astheysay’2，3，4’fortheunitsintheidealare
　　generatedatthesametime，whether，asthefirstholderofthetheory
　　said，fromunequalscomingintobeingwhenthesewereequalizedor
　　insomeotherway-since，ifoneunitistobepriortotheother，it
　　willbeprioralsoto2thecomposedofthese；forwhenthereisone
　　thingpriorandanotherposterior，theresultantofthesewillbe
　　priortooneandposteriortotheother。
　　Again，sincethe1-itselfis
　　first，andthenthereisaparticular1whichisfirstamongthe
　　othersandnextafterthe1-itself，andagainathirdwhichisnext
　　afterthesecondandnextbutoneafterthefirst1，-sotheunitsmust
　　bepriortothenumbersafterwhichtheyarenamedwhenwecountthem；
　　e。g。therewillbeathirdunitin2before3exists，andafourthand
　　afifthin3beforethenumbers4and5exist-Nownoneofthese
　　thinkershassaidtheunitsareinassociableinthisway，but
　　accordingtotheirprinciplesitisreasonablethattheyshouldbe
　　soeveninthisway，thoughintruthitisimpossible。Foritis
　　reasonableboththattheunitsshouldhavepriorityandposteriority
　　ifthereisafirstunitorfirst1，andalsothatthe2’sshouldif
　　thereisafirst2；forafterthefirstitisreasonableandnecessary
　　thatthereshouldbeasecond，andifasecond，athird，andsowith
　　theotherssuccessively。Andtosayboththingsatthesametime，
　　thataunitisfirstandanotherunitissecondaftertheideal1，and
　　thata2isfirstafterit，isimpossible。Buttheymakeafirstunit
　　or1，butnotalsoasecondandathird，andafirst2，butnotalsoa
　　secondandathird。Clearly，also，itisnotpossible，ifallthe
　　unitsareinassociable，thatthereshouldbea2-itselfanda
　　3-itself；andsowiththeothernumbers。Forwhethertheunitsare
　　undifferentiatedordifferenteachfromeach，numbermustbecounted
　　byaddition，e。g。2byaddinganother1totheone，3byadding
　　another1tothetwo，andsimilarly。Thisbeingso，numberscannot
　　begeneratedastheygeneratethem，fromthe2andthe1；for2
　　becomespartof3and3of4andthesamehappensinthecaseofthe
　　succeedingnumbers，buttheysay4camefromthefirst2andthe
　　indefinitewhichmakesittwo2’sotherthanthe2-itself；ifnot，the
　　2-itselfwillbeapartof4andoneother2willbeadded。And
　　similarly2willconsistofthe1-itselfandanother1；butifthisis
　　so，theotherelementcannotbeanindefinite2；foritgenerates
　　oneunit，not，astheindefinite2does，adefinite2。
　　Again，besidesthe3-itselfandthe2-itselfhowcantherebe
　　other3’sand2’s？Andhowdotheyconsistofpriorandposterior
　　units？Allthisisabsurdandfictitious，andtherecannotbea
　　first2andthena3-itself。Yettheremust，ifthe1andthe
　　indefinitedyadaretobetheelements。Butiftheresultsare
　　impossible，itisalsoimpossiblethatthesearethegenerating
　　principles。
　　Iftheunits，then，aredifferentiated，eachfromeach，these
　　resultsandotherssimilartothesefollowofnecessity。But3if
　　thoseindifferentnumbersaredifferentiated，butthoseinthesame
　　numberarealoneundifferentiatedfromoneanother，evensothe
　　difficultiesthatfollowarenoless。E。g。inthe10-itselftheir
　　aretenunits，andthe10iscomposedbothofthemandoftwo5’s。But
　　sincethe10-itselfisnotanychancenumbernorcomposedofany
　　chance5’s——or，forthatmatter，units——theunitsinthis10must
　　differ。Foriftheydonotdiffer，neitherwillthe5’sofwhichthe
　　10consistsdiffer；butsincethesediffer，theunitsalsowill
　　differ。Butiftheydiffer，willtherebenoother5’sinthe10but
　　onlythesetwo，orwilltherebeothers？Iftherearenot，thisis
　　paradoxical；andifthereare，whatsortof10willconsistofthem？
　　Forthereisnootherinthe10butthe10itself。Butitis
　　actuallynecessaryontheirviewthatthe4shouldnotconsistof
　　anychance2’s；fortheindefiniteastheysay，receivedthe
　　definite2andmadetwo2’s；foritsnaturewastodoublewhatit
　　received。
　　Again，astothe2beinganentityapartfromitstwounits，and
　　the3anentityapartfromitsthreeunits，howisthispossible？
　　Eitherbyone’ssharingintheother，as’paleman’isdifferent
　　from’pale’and’man’foritsharesinthese，orwhenoneisa
　　differentiaoftheother，as’man’isdifferentfrom’animal’and
　　’two-footed’。
　　Again，somethingsareonebycontact，somebyintermixture，
　　somebyposition；noneofwhichcanbelongtotheunitsofwhichthe2
　　orthe3consists；butastwomenarenotaunityapartfromboth，
　　somustitbewiththeunits。Andtheirbeingindivisiblewillmakeno
　　differencetothem；forpointstooareindivisible，butyetapair
　　ofthemisnothingapartfromthetwo。