othersthinktheyareofthelattercharacter。Platoandthe
　　Pythagoreansthoughtbeingandunitywerenothingelse，butthiswas
　　theirnature，theiressencebeingjustunityandbeing。Butthe
　　naturalphilosopherstakeadifferentline；e。g。Empedocles-as
　　thoughreducingtosomethingmoreintelligible-sayswhatunityis；for
　　hewouldseemtosayitislove:atleast，thisisforallthings
　　thecauseoftheirbeingone。Otherssaythisunityandbeing，of
　　whichthingsconsistandhavebeenmade，isfire，andotherssayitis
　　air。Asimilarviewisexpressedbythosewhomaketheelementsmore
　　thanone；forthesealsomustsaythatunityandbeingareprecisely
　　allthethingswhichtheysayareprinciples。
　　AIfwedonotsupposeunityandbeingtobesubstances，it
　　followsthatnoneoftheotheruniversalsisasubstance；forthese
　　aremostuniversalofall，andifthereisnounityitselfor
　　being-itself，therewillscarcelybeinanyothercaseanything
　　apartfromwhatarecalledtheindividuals。Further，ifunityisnota
　　substance，evidentlynumberalsowillnotexistasanentity
　　separatefromtheindividualthings；fornumberisunits，andtheunit
　　ispreciselyacertainkindofone。
　　ButBifthereisaunity-itselfandabeingitself，unityand
　　beingmustbetheirsubstance；foritisnotsomethingelsethatis
　　predicateduniversallyofthethingsthatareandareone，butjust
　　unityandbeing。Butifthereistobeabeing-itselfanda
　　unity-itself，thereismuchdifficultyinseeinghowtherewillbe
　　anythingelsebesidesthese，-Imean，howthingswillbemorethan
　　oneinnumber。Forwhatisdifferentfrombeingdoesnotexist，so
　　thatitnecessarilyfollows，accordingtotheargumentof
　　Parmenides，thatallthingsthatareareoneandthisisbeing。
　　Thereareobjectionstobothviews。Forwhetherunityisnota
　　substanceorthereisaunity-itself，numbercannotbeasubstance。We
　　havealreadysaidwhythisresultfollowsifunityisnotasubstance；
　　andifitis，thesamedifficultyarisesasarosewithregardto
　　being。Forwhenceistheretobeanotheronebesidesunity-itself？
　　Itmustbenot-one；butallthingsareeitheroneormany，andof
　　themanyeachisone。
　　Further，ifunity-itselfisindivisible，accordingtoZeno’s
　　postulateitwillbenothing。Forthatwhichneitherwhenadded
　　makesathinggreaternorwhensubtractedmakesitless，heassertsto
　　havenobeing，evidentlyassumingthatwhateverhasbeingisaspatial
　　magnitude。Andifitisamagnitude，itiscorporeal；forthe
　　corporealhasbeingineverydimension，whiletheotherobjectsof
　　mathematics，e。g。aplaneoraline，addedinonewaywillincrease
　　whattheyareaddedto，butinanotherwaywillnotdoso，andapoint
　　oraunitdoessoinnoway。But，sincehistheoryisofalow
　　order，andanindivisiblethingcanexistinsuchawayastohavea
　　defenceevenagainsthimfortheindivisiblewhenaddedwillmakethe
　　number，thoughnotthesize，greater，-yethowcanamagnitudeproceed
　　fromonesuchindivisibleorfrommany？Itislikesayingthatthe
　　lineismadeoutofpoints。
　　Buteveniforesupposesthecasetobesuchthat，assomesay，
　　numberproceedsfromunity-itselfandsomethingelsewhichisnotone，
　　nonethelesswemustinquirewhyandhowtheproductwillbe
　　sometimesanumberandsometimesamagnitude，ifthenot-onewas
　　inequalityandwasthesameprincipleineithercase。Foritisnot
　　evidenthowmagnitudescouldproceedeitherfromtheoneandthis
　　principle，orfromsomenumberandthisprinciple。
　　14Aquestionconnectedwiththeseiswhethernumbersandbodies
　　andplanesandpointsaresubstancesofakind，ornot。Iftheyare
　　not，itbafflesustosaywhatbeingisandwhatthesubstancesof
　　thingsare。Formodificationsandmovementsandrelationsand
　　dispositionsandratiosdonotseemtoindicatethesubstanceof
　　anything；forallarepredicatedofasubject，andnoneisa’this’。
　　Andastothethingswhichmightseemmostofalltoindicate
　　substance，waterandearthandfireandair，ofwhichcompositebodies
　　consist，heatandcoldandthelikearemodificationsofthese，not
　　substances，andthebodywhichisthusmodifiedalonepersistsas
　　somethingrealandasasubstance。But，ontheotherhand，thebodyis
　　surelylessofasubstancethanthesurface，andthesurfacethan
　　theline，andthelinethantheunitandthepoint。Forthebodyis
　　boundedbythese；andtheyarethoughttobecapableofexisting
　　withoutbody，butbodyincapableofexistingwithoutthese。Thisis
　　why，whilemostofthephilosophersandtheearlieramongthemthought
　　thatsubstanceandbeingwereidenticalwithbody，andthatall
　　otherthingsweremodificationsofthis，sothatthefirst
　　principlesofthebodieswerethefirstprinciplesofbeing，the
　　morerecentandthosewhowereheldtobewiserthoughtnumberswere
　　thefirstprinciples。Aswesaid，then，ifthesearenotsubstance，
　　thereisnosubstanceandnobeingatall；fortheaccidentsof
　　theseitcannotberighttocallbeings。
　　Butifthisisadmitted，thatlinesandpointsaresubstance
　　morethanbodies，butwedonotseetowhatsortofbodiesthesecould
　　belongfortheycannotbeinperceptiblebodies，therecanbeno
　　substance-Further，theseareallevidentlydivisionsofbody，-one
　　inbreadth，anotherindepth，anotherinlength。Besidesthis，nosort
　　ofshapeispresentinthesolidmorethananyother；sothatifthe
　　Hermesisnotinthestone，neitheristhehalfofthecubeinthe
　　cubeassomethingdeterminate；thereforethesurfaceisnotinit
　　either；forifanysortofsurfacewereinit，thesurfacewhichmarks
　　offthehalfofthecubewouldbeinittoo。Andthesameaccount
　　appliestothelineandtothepointandtheunit。Therefore，ifon
　　theonehandbodyisinthehighestdegreesubstance，andontheother
　　handthesethingsaresomorethanbody，butthesearenoteven
　　instancesofsubstance，itbafflesustosaywhatbeingisandwhat
　　thesubstanceofthingsis-Forbesideswhathasbeensaid，the
　　questionsofgenerationandinstructionconfrontuswithfurther
　　paradoxes。Forifsubstance，nothavingexistedbefore，nowexists，or
　　havingexistedbefore，afterwardsdoesnotexist，thischangeis
　　thoughttobeaccompaniedbyaprocessofbecomingorperishing；but
　　pointsandlinesandsurfacescannotbeinprocesseitherof
　　becomingorofperishing，whentheyatonetimeexistandatanother
　　donot。Forwhenbodiescomeintocontactoraredivided，their
　　boundariessimultaneouslybecomeoneintheonecasewhenthey
　　touch，andtwointheother-whentheyaredivided；sothatwhenthey
　　havebeenputtogetheroneboundarydoesnotexistbuthasperished，
　　andwhentheyhavebeendividedtheboundariesexistwhichbefore
　　didnotexistforitcannotbesaidthatthepoint，whichis
　　indivisible，wasdividedintotwo。Andiftheboundariescomeinto
　　beingandceasetobe，fromwhatdotheycomeintobeing？Asimilar
　　accountmayalsobegivenofthe’now’intime；forthisalsocannot
　　beinprocessofcomingintobeingorofceasingtobe，butyet
　　seemstobealwaysdifferent，whichshowsthatitisnota
　　substance。Andevidentlythesameistrueofpointsandlinesand
　　planes；forthesameargumentapplies，sincetheyareallalikeeither
　　limitsordivisions。
　　Ingeneralonemightraisethequestionwhyafterall，besides
　　perceptiblethingsandtheintermediates，wehavetolookfor
　　anotherclassofthings，i。e。theFormswhichweposit。Ifitisfor
　　thisreason，becausetheobjectsofmathematics，whiletheydiffer
　　fromthethingsinthisworldinsomeotherrespect，differnotatall
　　inthattherearemanyofthesamekind，sothattheirfirst
　　principlescannotbelimitedinnumberjustastheelementsofall
　　thelanguageinthissensibleworldarenotlimitedinnumber，but
　　inkind，unlessonetakestheelementsofthisindividualsyllable
　　orofthisindividualarticulatesound-whoseelementswillbe
　　limitedeveninnumber；soisitalsointhecaseofthe
　　intermediates；fortherealsothemembersofthesamekindare
　　infiniteinnumber，sothatiftherearenot-besidesperceptible
　　andmathematicalobjects-otherssuchassomemaintaintheFormstobe，
　　therewillbenosubstancewhichisoneinnumber，butonlyinkind，
　　norwillthefirstprinciplesofthingsbedeterminateinnumber，
　　butonlyinkind:-ifthenthismustbeso，theFormsalsomust
　　thereforebeheldtoexist。Evenifthosewhosupportthisviewdonot
　　expressitarticulately，stillthisiswhattheymean，andtheymust
　　bemaintainingtheFormsjustbecauseeachoftheFormsisasubstance
　　andnoneisbyaccident。
　　ButifwearetosupposeboththattheFormsexistandthatthe
　　principlesareoneinnumber，notinkind，wehavementionedthe
　　impossibleresultsthatnecessarilyfollow。
　　13Closelyconnectedwiththisisthequestionwhetherthe
　　elementsexistpotentiallyorinsomeothermanner。Ifinsomeother
　　way，therewillbesomethingelsepriortothefirstprinciples；for
　　thepotencyispriortotheactualcause，anditisnotnecessary
　　foreverythingpotentialtobeactual-Butiftheelementsexist
　　potentially，itispossiblethateverythingthatisshouldnotbe。For
　　eventhatwhichisnotyetiscapableofbeing；forthatwhichis
　　notcomestobe，butnothingthatisincapableofbeingcomestobe。
　　12Wemustnotonlyraisethesequestionsaboutthefirst
　　principles，butalsoaskwhethertheyareuniversalorwhatwecall
　　individuals。Iftheyareuniversal，theywillnotbesubstances；for
　　everythingthatiscommonindicatesnota’this’buta’such’，but
　　substanceisa’this’。Andifwearetobeallowedtolayitdownthat
　　acommonpredicateisa’this’andasinglething，Socrateswillbe
　　severalanimals-himselfand’man’and’animal’，ifeachofthese
　　indicatesa’this’andasinglething。
　　If，then，theprinciplesareuniversals，theseuniversal。
　　Thereforeifthereistoberesultsfollow；iftheyarenotuniversals
　　butofknowledgeoftheprinciplestheremustbethenatureof
　　individuals，theywillnotbeotherprinciplespriortothem，namely
　　thoseknowable；fortheknowledgeofanythingisthatare
　　universallypredicatedofthem。
　　THEREisasciencewhichinvestigatesbeingasbeingandthe
　　attributeswhichbelongtothisinvirtueofitsownnature。Now
　　thisisnotthesameasanyoftheso-calledspecialsciences；for
　　noneoftheseotherstreatsuniversallyofbeingasbeing。Theycut
　　offapartofbeingandinvestigatetheattributeofthispart；this
　　iswhatthemathematicalsciencesforinstancedo。Nowsinceweare
　　seekingthefirstprinciplesandthehighestcauses，clearlythere
　　mustbesomethingtowhichthesebelonginvirtueofitsown
　　nature。Ifthenthosewhosoughttheelementsofexistingthings
　　wereseekingthesesameprinciples，itisnecessarythatthe
　　elementsmustbeelementsofbeingnotbyaccidentbutjustbecauseit
　　isbeing。Thereforeitisofbeingasbeingthatwealsomustgrasp
　　thefirstcauses。