Forjustastheuniversalpropositionsofmathematicsdealnot
　　withobjectswhichexistseparately，apartfromextendedmagnitudes
　　andfromnumbers，butwithmagnitudesandnumbers，nothoweverqua
　　suchastohavemagnitudeortobedivisible，clearlyitispossible
　　thatthereshouldalsobebothpropositionsanddemonstrationsabout
　　sensiblemagnitudes，nothoweverquasensiblebutquapossessedof
　　certaindefinitequalities。Forastherearemanypropositionsabout
　　thingsmerelyconsideredasinmotion，apartfromwhateachsuchthing
　　isandfromtheiraccidents，andasitisnotthereforenecessarythat
　　thereshouldbeeitheramobileseparatefromsensibles，oradistinct
　　mobileentityinthesensibles，sotoointhecaseofmobilesthere
　　willbepropositionsandsciences，whichtreatthemhowevernotqua
　　mobilebutonlyquabodies，oragainonlyquaplanes，oronlyqua
　　lines，orquadivisibles，orquaindivisibleshavingposition，oronly
　　quaindivisibles。Thussinceitistruetosaywithoutqualification
　　thatnotonlythingswhichareseparablebutalsothingswhichare
　　inseparableexistforinstance，thatmobilesexist，itistrue
　　alsotosaywithoutqualificationthattheobjectsofmathematics
　　exist，andwiththecharacterascribedtothembymathematicians。
　　Andasitistruetosayoftheothersciencestoo，without
　　qualification，thattheydealwithsuchandsuchasubject-notwith
　　whatisaccidentaltoite。g。notwiththepale，ifthehealthything
　　ispale，andthesciencehasthehealthyasitssubject，butwith
　　thatwhichisthesubjectofeachscience-withthehealthyifit
　　treatsitsobjectquahealthy，withmanifquaman:-sotooisit
　　withgeometry；ifitssubjectshappentobesensible，thoughitdoes
　　nottreatthemquasensible，themathematicalscienceswillnotfor
　　thatreasonbesciencesofsensibles-nor，ontheotherhand，of
　　otherthingsseparatefromsensibles。Manypropertiesattachtothings
　　invirtueoftheirownnatureaspossessedofeachsuchcharacter；
　　e。g。thereareattributespeculiartotheanimalquafemaleorqua
　　maleyetthereisno’female’nor’male’separatefromanimals；so
　　thattherearealsoattributeswhichbelongtothingsmerelyas
　　lengthsorasplanes。Andinproportionaswearedealingwith
　　thingswhicharepriorindefinitionandsimpler，ourknowledgehas
　　moreaccuracy，i。e。simplicity。Thereforeasciencewhichabstracts
　　fromspatialmagnitudeismoreprecisethanonewhichtakesitinto
　　account；andascienceismostpreciseifitabstractsfrom
　　movement，butifittakesaccountofmovement，itismostpreciseif
　　itdealswiththeprimarymovement，forthisisthesimplest；andof
　　thisagainuniformmovementisthesimplestform。
　　Thesameaccountmaybegivenofharmonicsandoptics；forneither
　　considersitsobjectsquasightorquavoice，butqualinesand
　　numbers；butthelatterareattributespropertotheformer。And
　　mechanicstooproceedsinthesameway。Thereforeifwesuppose
　　attributesseparatedfromtheirfellowattributesandmakeanyinquiry
　　concerningthemassuch，weshallnotforthisreasonbeinerror，any
　　morethanwhenonedrawsalineonthegroundandcallsitafootlong
　　whenitisnot；fortheerrorisnotincludedinthepremisses。
　　Eachquestionwillbebestinvestigatedinthisway-bysetting
　　upbyanactofseparationwhatisnotseparate，asthe
　　arithmeticianandthegeometerdo。Foramanquamanisone
　　indivisiblething；andthearithmeticiansupposedoneindivisible
　　thing，andthenconsideredwhetheranyattributebelongstoaman
　　quaindivisible。Butthegeometertreatshimneitherquamannorqua
　　indivisible，butasasolid。Forevidentlythepropertieswhich
　　wouldhavebelongedtohimevenifperchancehehadnotbeen
　　indivisible，canbelongtohimevenapartfromtheseattributes。Thus，
　　then，geometersspeakcorrectly；theytalkaboutexistingthings，
　　andtheirsubjectsdoexist；forbeinghastwoforms-itexistsnot
　　onlyincompleterealitybutalsomaterially。
　　Nowsincethegoodandthebeautifularedifferentfortheformer
　　alwaysimpliesconductasitssubject，whilethebeautifulisfound
　　alsoinmotionlessthings，thosewhoassertthatthemathematical
　　sciencessaynothingofthebeautifulorthegoodareinerror。For
　　thesesciencessayandproveagreatdealaboutthem；iftheydonot
　　expresslymentionthem，butproveattributeswhicharetheirresults
　　ortheirdefinitions，itisnottruetosaythattheytellus
　　nothingaboutthem。Thechiefformsofbeautyareorderandsymmetry
　　anddefiniteness，whichthemathematicalsciencesdemonstrateina
　　specialdegree。Andsincethesee。g。orderanddefinitenessare
　　obviouslycausesofmanythings，evidentlythesesciencesmusttreat
　　thissortofcausativeprinciplealsoi。e。thebeautifulasin
　　somesenseacause。Butweshallspeakmoreplainlyelsewhereabout
　　thesematters。
　　Somuchthenfortheobjectsofmathematics；wehavesaidthat
　　theyexistandinwhatsensetheyexist，andinwhatsensetheyare
　　priorandinwhatsensenotprior。Now，regardingtheIdeas，wemust
　　firstexaminetheidealtheoryitself，notconnectingitinanyway
　　withthenatureofnumbers，buttreatingitintheforminwhichit
　　wasoriginallyunderstoodbythosewhofirstmaintainedthe
　　existenceoftheIdeas。Thesupportersoftheidealtheorywereledto
　　itbecauseonthequestionaboutthetruthofthingstheyacceptedthe
　　Heracliteansayingswhichdescribeallsensiblethingsaseverpassing
　　away，sothatifknowledgeorthoughtistohaveanobject，theremust
　　besomeotherandpermanententities，apartfromthosewhichare
　　sensible；fortherecouldbenoknowledgeofthingswhichwereina
　　stateofflux。ButwhenSocrateswasoccupyinghimselfwiththe
　　excellencesofcharacter，andinconnexionwiththembecamethe
　　firsttoraisetheproblemofuniversaldefinitionforofthe
　　physicistsDemocritusonlytouchedonthesubjecttoasmallextent，
　　anddefined，afterafashion，thehotandthecold；whilethe
　　Pythagoreanshadbeforethistreatedofafewthings，whose
　　definitions-e。g。thoseofopportunity，justice，ormarriage-they
　　connectedwithnumbers；butitwasnaturalthatSocratesshouldbe
　　seekingtheessence，forhewasseekingtosyllogize，and’whata
　　thingis’isthestarting-pointofsyllogisms；fortherewasasyet
　　noneofthedialecticalpowerwhichenablespeopleevenwithout
　　knowledgeoftheessencetospeculateaboutcontrariesandinquire
　　whetherthesamesciencedealswithcontraries；fortwothingsmay
　　befairlyascribedtoSocrates-inductiveargumentsanduniversal
　　definition，bothofwhichareconcernedwiththestarting-pointof
　　science:-butSocratesdidnotmaketheuniversalsorthe
　　definitionsexistapart:they，however，gavethemseparate
　　existence，andthiswasthekindofthingtheycalledIdeas。Therefore
　　itfollowedforthem，almostbythesameargument，thattheremust
　　beIdeasofallthingsthatarespokenofuniversally，anditwas
　　almostasifamanwishedtocountcertainthings，andwhiletheywere
　　fewthoughthewouldnotbeabletocountthem，butmademoreof
　　themandthencountedthem；fortheFormsare，onemaysay，more
　　numerousthantheparticularsensiblethings，yetitwasinseeking
　　thecausesofthesethattheyproceededfromthemtotheForms。Forto
　　eachthingthereanswersanentitywhichhasthesamenameand
　　existsapartfromthesubstances，andsoalsointhecaseofallother
　　groupsthereisaoneovermany，whetherthesebeofthisworldor
　　eternal。
　　Again，ofthewaysinwhichitisprovedthattheFormsexist，
　　noneisconvincing；forfromsomenoinferencenecessarilyfollows，
　　andfromsomeariseFormsevenofthingsofwhichtheythinkthereare
　　noForms。Foraccordingtotheargumentsfromthesciencesthere
　　willbeFormsofallthingsofwhichtherearesciences，andaccording
　　totheargumentofthe’oneovermany’therewillbeFormsevenof
　　negations，andaccordingtotheargumentthatthoughthasanobject
　　whentheindividualobjecthasperished，therewillbeFormsof
　　perishablethings；forwehaveanimageofthese。Again，ofthemost
　　accuratearguments，someleadtoIdeasofrelations，ofwhichtheysay
　　thereisnoindependentclass，andothersintroducethe’thirdman’。
　　AndingeneraltheargumentsfortheFormsdestroythingsfor
　　whoseexistencethebelieversinFormsaremorezealousthanforthe
　　existenceoftheIdeas；foritfollowsthatnotthedyadbutnumberis
　　first，andthatpriortonumberistherelative，andthatthisis
　　priortotheabsolute-besidesalltheotherpointsonwhichcertain
　　people，byfollowingouttheopinionsheldabouttheForms，came
　　intoconflictwiththeprinciplesofthetheory。
　　Again，accordingtotheassumptiononthebeliefintheIdeas
　　rests，therewillbeFormsnotonlyofsubstancesbutalsoofmany
　　otherthings；fortheconceptissinglenotonlyinthecaseof
　　substances，butalsointhatofnon-substances，andtherearesciences
　　ofotherthingsthansubstance；andathousandothersuchdifficulties
　　confrontthem。Butaccordingtothenecessitiesofthecaseandthe
　　opinionsabouttheForms，iftheycanbesharedintheremustbeIdeas
　　ofsubstancesonly。Fortheyarenotsharedinincidentally，but
　　eachFormmustbesharedinassomethingnotpredicatedofa
　　subject。By’beingsharedinincidentally’Imeanthatifathing
　　sharesin’doubleitself’，itsharesalsoin’eternal’，but
　　incidentally；for’thedouble’happenstobeeternal。Thereforethe
　　Formswillbesubstance。Butthesamenamesindicatesubstanceinthis
　　andintheidealworldorwhatwillbethemeaningofsayingthat
　　thereissomethingapartfromtheparticulars-theoneovermany？。And
　　iftheIdeasandthethingsthatshareinthemhavethesameform，
　　therewillbesomethingcommon:forwhyshould’2’beoneandthesame
　　intheperishable2’s，orinthe2’swhicharemanybuteternal，and
　　notthesameinthe’2itself’asintheindividual2？Butifthey
　　havenotthesameform，theywillhaveonlythenameincommon，andit
　　isasifoneweretocallbothCalliasandapieceofwooda’man’，
　　withoutobservinganycommunitybetweenthem。
　　Butifwearetosupposethatinotherrespectsthecommon
　　definitionsapplytotheForms，e。g。that’planefigure’andtheother
　　partsofthedefinitionapplytothecircleitself，but’whatreally
　　is’hastobeadded，wemustinquirewhetherthisisnotabsolutely
　　meaningless。Fortowhatisthistobeadded？To’centre’orto
　　’plane’ortoallthepartsofthedefinition？Foralltheelementsin
　　theessenceareIdeas，e。g。’animal’and’two-footed’。Further，
　　theremustbesomeIdealansweringto’plane’above，somenaturewhich
　　willbepresentinalltheFormsastheirgenus。
　　Aboveallonemightdiscussthequestionwhatintheworldthe
　　Formscontributetosensiblethings，eithertothosethatare
　　eternalortothosethatcomeintobeingandceasetobe；forthey
　　causeneithermovementnoranychangeinthem。Butagaintheyhelp
　　innowiseeithertowardstheknowledgeofotherthingsforthey
　　arenoteventhesubstanceofthese，elsetheywouldhavebeenin
　　them，ortowardstheirbeing，iftheyarenotintheindividuals
　　whichshareinthem；thoughiftheywere，theymightbethoughtto
　　becauses，aswhitecauseswhitenessinawhiteobjectbyentering
　　intoitscomposition。Butthisargument，whichwasusedfirstby
　　Anaxagoras，andlaterbyEudoxusinhisdiscussionofdifficultiesand
　　bycertainothers，isveryeasilyupset；foritiseasytocollect
　　manyandinsuperableobjectionstosuchaview。