starting-point？Ashasbeensaid，therightangleisthoughttobe
　　priortotheacute，andtheacutetotheright，andeachisone。
　　Accordinglytheymake1thestarting-pointinbothways。Butthisis
　　impossible。Fortheuniversalisoneasformorsubstance，whilethe
　　elementisoneasapartorasmatter。Foreachofthetwoisina
　　senseone-intrutheachofthetwounitsexistspotentiallyat
　　leastifthenumberisaunityandnotlikeaheap，i。e。if
　　differentnumbersconsistofdifferentiatedunits，astheysay，but
　　notincompletereality；andthecauseoftheerrortheyfellinto
　　isthattheywereconductingtheirinquiryatthesametimefromthe
　　standpointofmathematicsandfromthatofuniversaldefinitions，so
　　that1fromtheformerstandpointtheytreatedunity，theirfirst
　　principle，asapoint；fortheunitisapointwithoutposition。
　　Theyputthingstogetheroutofthesmallestparts，assomeothers
　　alsohavedone。Thereforetheunitbecomesthematterofnumbersand
　　atthesametimepriorto2；andagainposterior，2beingtreatedasa
　　whole，aunity，andaform。But2becausetheywereseekingthe
　　universaltheytreatedtheunitywhichcanbepredicatedofa
　　number，asinthissensealsoapartofthenumber。Butthese
　　characteristicscannotbelongatthesametimetothesamething。
　　Ifthe1-itselfmustbeunitaryforitdiffersinnothingfrom
　　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。Forigeometrical
　　classesareseveredfromoneanother，unlesstheprinciplesofthese
　　areimpliedinoneanotherinsuchawaythatthe’broadandnarrow’
　　isalso’longandshort’butifthisisso，theplanewillbeline
　　andthesolidaplane；again，howwillanglesandfiguresandsuch
　　thingsbeexplained？。Andiithesamehappensasinregardto
　　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；foraeachofthepartsmustbeindivisibleorit
　　willbeapluralityandtheunitwillbedivisibleandtheelements
　　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
　　distancethepoint-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
　　conclusionswehavereachedaresufficientforwhilehewhois
　　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，arealthingswill
　　bejustofthesamenumberastheelements，andbtheelements
　　willnotbeknowable。Foraletthesyllablesinspeechbe
　　substances，andtheirelementselementsofsubstances；thentheremust
　　beonlyone’ba’andoneofeachofthesyllables，sincetheyare
　　notuniversalandthesameinformbuteachisoneinnumberanda
　　’this’andnotakindpossessedofacommonnameandagainthey
　　supposethatthe’justwhatathingis’isineachcaseone。Andif
　　thesyllablesareunique，sotooarethepartsofwhichthey
　　consist；therewillnot，then，bemorea’sthanone，normorethanone
　　ofanyoftheotherelements，onthesameprincipleonwhichan
　　identicalsyllablecannotexistinthepluralnumber。Butifthisis
　　so，therewillnotbeotherthingsexistingbesidestheelements，
　　butonlytheelements。
　　bAgain，theelementswillnotbeevenknowable；fortheyare
　　notuniversal，andknowledgeisofuniversals。Thisisclearfrom
　　demonstrationsandfromdefinitions；forwedonotconcludethat
　　thistrianglehasitsanglesequaltotworightangles，unlessevery
　　trianglehasitsanglesequaltotworightangles，northatthisman
　　isananimal，unlesseverymanisananimal。
　　Butiftheprinciplesareuniversal，eitherthesubstances