starting-point？Ashasbeensaid，therightangleisthoughttobe
priortotheacute，andtheacutetotheright，andeachisone。
Accordinglytheymake1thestarting-pointinbothways。Butthisis
impossible。Fortheuniversalisoneasformorsubstance，whilethe
elementisoneasapartorasmatter。Foreachofthetwoisina
senseone-intrutheachofthetwounitsexistspotentially(at
leastifthenumberisaunityandnotlikeaheap，i。e。if
differentnumbersconsistofdifferentiatedunits，astheysay)，but
notincompletereality；andthecauseoftheerrortheyfellinto
isthattheywereconductingtheirinquiryatthesametimefromthe
standpointofmathematicsandfromthatofuniversaldefinitions，so
that(1)fromtheformerstandpointtheytreatedunity，theirfirst
principle，asapoint；fortheunitisapointwithoutposition。
Theyputthingstogetheroutofthesmallestparts，assomeothers
alsohavedone。Thereforetheunitbecomesthematterofnumbersand
atthesametimepriorto2；andagainposterior，2beingtreatedasa
whole，aunity，andaform。But(2)becausetheywereseekingthe
universaltheytreatedtheunitywhichcanbepredicatedofa
number，asinthissensealsoapartofthenumber。Butthese
characteristicscannotbelongatthesametimetothesamething。
Ifthe1-itselfmustbeunitary(foritdiffersinnothingfrom
other1’sexceptthatitisthestarting-point)，andthe2is
divisiblebuttheunitisnot，theunitmustbelikerthe1-itself
thanthe2is。Butiftheunitislikerit，itmustbelikertothe
unitthantothe2；thereforeeachoftheunitsin2mustbeprior
tothe2。Buttheydenythis；atleasttheygeneratethe2first。
Again，ifthe2-itselfisaunityandthe3-itselfisonealso，both
forma2。Fromwhat，then，isthis2produced？
Sincethereisnotcontactinnumbers，butsuccession，viz。
betweentheunitsbetweenwhichthereisnothing，e。g。betweenthose
in2orin3onemightaskwhetherthesesucceedthe1-itselfor
not，andwhether，ofthetermsthatsucceedit，2oreitherofthe
unitsin2isprior。
Similardifficultiesoccurwithregardtotheclassesofthings
posteriortonumber，-theline，theplane，andthesolid。Forsome
constructtheseoutofthespeciesofthe’greatandsmall’；e。g。
linesfromthe’longandshort’，planesfromthe’broadandnarrow’，
massesfromthe’deepandshallow’；whicharespeciesofthe’great
andsmall’。Andtheoriginativeprincipleofsuchthingswhichanswers
tothe1differentthinkersdescribeindifferentways，Andinthese
alsotheimpossibilities，thefictions，andthecontradictionsof
allprobabilityareseentobeinnumerable。For(i)geometrical
classesareseveredfromoneanother，unlesstheprinciplesofthese
areimpliedinoneanotherinsuchawaythatthe’broadandnarrow’
isalso’longandshort’(butifthisisso，theplanewillbeline
andthesolidaplane；again，howwillanglesandfiguresandsuch
thingsbeexplained？)。And(ii)thesamehappensasinregardto
number；for’longandshort’，&c。，areattributesofmagnitude，but
magnitudedoesnotconsistofthese，anymorethanthelineconsists
of’straightandcurved’，orsolidsof’smoothandrough’。
(Alltheseviewsshareadifficultywhichoccurswithregardto
species-of-a-genus，whenonepositstheuniversals，viz。whetheritis
animal-itselforsomethingotherthananimal-itselfthatisinthe
particularanimal。True，iftheuniversalisnotseparablefrom
sensiblethings，thiswillpresentnodifficulty；butifthe1andthe
numbersareseparable，asthosewhoexpresstheseviewssay，itisnot
easytosolvethedifficulty，ifonemayapplythewords’noteasy’to
theimpossible。Forwhenweapprehendtheunityin2，oringeneralin
anumber，doweapprehendathing-itselforsomethingelse？)。
Some，then，generatespatialmagnitudesfrommatterofthis
sort，othersfromthepoint-andthepointisthoughtbythemtobe
not1butsomethinglike1-andfromothermatterlikeplurality，but
notidenticalwithit；aboutwhichprinciplesnonethelessthesame
difficultiesoccur。Forifthematterisone，lineandplane-and
soliwillbethesame；forfromthesameelementswillcomeoneand
thesamething。Butifthemattersaremorethanone，andthereisone
forthelineandasecondfortheplaneandanotherforthesolid，
theyeitherareimpliedinoneanotherornot，sothatthesame
resultswillfollowevenso；foreithertheplanewillnotcontaina
lineoritwillhealine。
Again，hownumbercanconsistoftheoneandplurality，they
makenoattempttoexplain；buthowevertheyexpressthemselves，the
sameobjectionsariseasconfrontthosewhoconstructnumberoutof
theoneandtheindefinitedyad。Fortheoneviewgeneratesnumber
fromtheuniversallypredicatedplurality，andnotfromaparticular
plurality；andtheothergeneratesitfromaparticularplurality，but
thefirst；for2issaidtobea’firstplurality’。Thereforethereis
practicallynodifference，butthesamedifficultieswillfollow，-is
itintermixtureorpositionorblendingorgeneration？andsoon。
Aboveallonemightpressthequestion’ifeachunitisone，whatdoes
itcomefrom？’Certainlyeachisnottheone-itself。Itmust，then，
comefromtheoneitselfandplurality，orapartofplurality。Tosay
thattheunitisapluralityisimpossible，foritisindivisible；and
togenerateitfromapartofpluralityinvolvesmanyother
objections；for(a)eachofthepartsmustbeindivisible(orit
willbeapluralityandtheunitwillbedivisible)andtheelements
willnotbetheoneandplurality；forthesingleunitsdonotcome
frompluralityandtheone。Again，(，theholderofthisviewdoes
nothingbutpresupposeanothernumber；forhispluralityof
indivisiblesisanumber。Again，wemustinquire，inviewofthis
theoryalso，whetherthenumberisinfiniteorfinite。Fortherewas
atfirst，asitseems，apluralitythatwasitselffinite，from
whichandfromtheonecomesthefinitenumberofunits。Andthere
isanotherpluralitythatisplurality-itselfandinfinite
plurality；whichsortofplurality，then，istheelementwhich
co-operateswiththeone？Onemightinquiresimilarlyaboutthepoint，
i。e。theelementoutofwhichtheymakespatialmagnitudes。Forsurely
thisisnottheoneandonlypoint；atanyrate，then，letthemsay
outofwhateachofthepointsisformed。Certainlynotofsome
distance+thepoint-itself。Noragaincantherebeindivisible
partsofadistance，astheelementsoutofwhichtheunitsaresaid
tobemadeareindivisiblepartsofplurality；fornumberconsists
ofindivisibles，butspatialmagnitudesdonot。
Alltheseobjections，then，andothersofthesortmakeitevident
thatnumberandspatialmagnitudescannotexistapartfromthings。
Again，thediscordaboutnumbersbetweenthevariousversionsisa
signthatitistheincorrectnessoftheallegedfactsthemselvesthat
bringsconfusionintothetheories。Forthosewhomaketheobjects
ofmathematicsaloneexistapartfromsensiblethings，seeingthe
difficultyabouttheFormsandtheirfictitiousness，abandonedideal
numberandpositedmathematical。Butthosewhowishedtomakethe
Formsatthesametimealsonumbers，butdidnotsee，ifoneassumed
theseprinciples，howmathematicalnumberwastoexistapartfrom
ideal，madeidealandmathematicalnumberthesame-inwords，since
infactmathematicalnumberhasbeendestroyed；fortheystate
hypothesespeculiartothemselvesandnotthoseofmathematics。Andhe
whofirstsupposedthattheFormsexistandthattheFormsarenumbers
andthattheobjectsofmathematicsexist，naturallyseparatedthe
two。Thereforeitturnsoutthatallofthemarerightinsome
respect，butonthewholenotright。Andtheythemselvesconfirmthis，
fortheirstatementsdonotagreebutconflict。Thecauseisthat
theirhypothesesandtheirprinciplesarefalse。Anditishardto
makeagoodcaseoutofbadmaterials，accordingtoEpicharmus:’as
soonas’tissaid，’tisseentobewrong。’
Butregardingnumbersthequestionswehaveraisedandthe
conclusionswehavereachedaresufficient(forwhilehewhois
alreadyconvincedmightbefurtherconvincedbyalongerdiscussion，
onenotyetconvincedwouldnotcomeanynearertoconviction)；
regardingthefirstprinciplesandthefirstcausesandelements，
theviewsexpressedbythosewhodiscussonlysensiblesubstance
havebeenpartlystatedinourworksonnature，andpartlydonot
belongtothepresentinquiry；buttheviewsofthosewhoassert
thatthereareothersubstancesbesidesthesensiblemustbe
considerednextafterthosewehavebeenmentioning。Since，then，some
saythattheIdeasandthenumbersaresuchsubstances，andthatthe
elementsoftheseareelementsandprinciplesofrealthings，we
mustinquireregardingthesewhattheysayandinwhatsensethey
sayit。
Thosewhopositnumbersonly，andthesemathematical，mustbe
consideredlater；butasregardsthosewhobelieveintheIdeasone
mightsurveyatthesametimetheirwayofthinkingandthedifficulty
intowhichtheyfall。FortheyatthesametimemaketheIdeas
universalandagaintreatthemasseparableandasindividuals。That
thisisnotpossiblehasbeenarguedbefore。Thereasonwhythose
whodescribedtheirsubstancesasuniversalcombinedthesetwo
characteristicsinonething，isthattheydidnotmakesubstances
identicalwithsensiblethings。Theythoughtthattheparticularsin
thesensibleworldwereastateoffluxandnoneofthemremained，but
thattheuniversalwasapartfromtheseandsomethingdifferent。And
Socratesgavetheimpulsetothistheory，aswesaidinourearlier
discussion，byreasonofhisdefinitions，buthedidnotseparate
universalsfromindividuals；andinthishethoughtrightly，innot
separatingthem。Thisisplainfromtheresults；forwithoutthe
universalitisnotpossibletogetknowledge，buttheseparationis
thecauseoftheobjectionsthatarisewithregardtotheIdeas。His
successors，however，treatingitasnecessary，iftherearetobe
anysubstancesbesidesthesensibleandtransientsubstances，that
theymustbeseparable，hadnoothers，butgaveseparateexistence
totheseuniversallypredicatedsubstances，sothatitfollowedthat
universalsandindividualswerealmostthesamesortofthing。Thisin
itself，then，wouldbeonedifficultyintheviewwehavementioned。
Letusnowmentionapointwhichpresentsacertaindifficulty
bothtothosewhobelieveintheIdeasandtothosewhodonot，and
whichwasstatedbefore，atthebeginning，amongtheproblems。Ifwe
donotsupposesubstancestobeseparate，andinthewayinwhich
individualthingsaresaidtobeseparate，weshalldestroy
substanceinthesenseinwhichweunderstand’substance’；butifwe
conceivesubstancestobeseparable，howarewetoconceivetheir
elementsandtheirprinciples？
Iftheyareindividualandnotuniversal，(a)realthingswill
bejustofthesamenumberastheelements，and(b)theelements
willnotbeknowable。For(a)letthesyllablesinspeechbe
substances，andtheirelementselementsofsubstances；thentheremust
beonlyone’ba’andoneofeachofthesyllables，sincetheyare
notuniversalandthesameinformbuteachisoneinnumberanda
’this’andnotakindpossessedofacommonname(andagainthey
supposethatthe’justwhatathingis’isineachcaseone)。Andif
thesyllablesareunique，sotooarethepartsofwhichthey
consist；therewillnot，then，bemorea’sthanone，normorethanone
ofanyoftheotherelements，onthesameprincipleonwhichan
identicalsyllablecannotexistinthepluralnumber。Butifthisis
so，therewillnotbeotherthingsexistingbesidestheelements，
butonlytheelements。
(b)Again，theelementswillnotbeevenknowable；fortheyare
notuniversal，andknowledgeisofuniversals。Thisisclearfrom
demonstrationsandfromdefinitions；forwedonotconcludethat
thistrianglehasitsanglesequaltotworightangles，unlessevery
trianglehasitsanglesequaltotworightangles，northatthisman
isananimal，unlesseverymanisananimal。
Butiftheprinciplesareuniversal，eitherthesubstances
