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。