Further，whyshouldtherealwaysbebecoming，andwhatisthe
　　causeofbecoming？-thisnoonetellsus。Andthosewhosupposetwo
　　principlesmustsupposeanother，asuperiorprinciple，andsomust
　　thosewhobelieveintheForms；forwhydidthingscometo
　　participate，orwhydotheyparticipate，intheForms？Andallother
　　thinkersareconfrontedbythenecessaryconsequencethatthereis
　　somethingcontrarytoWisdom，i。e。tothehighestknowledge；butwe
　　arenot。Forthereisnothingcontrarytothatwhichisprimary；for
　　allcontrarieshavematter，andthingsthathavematterexistonly
　　potentially；andtheignorancewhichiscontrarytoanyknowledge
　　leadstoanobjectcontrarytotheobjectoftheknowledge；butwhat
　　isprimaryhasnocontrary。
　　Again，ifbesidessensiblethingsnoothersexist，therewillbe
　　nofirstprinciple，noorder，nobecoming，noheavenlybodies，but
　　eachprinciplewillhaveaprinciplebeforeit，asintheaccounts
　　ofthetheologiansandallthenaturalphilosophers。Butifthe
　　Formsorthenumbersaretoexist，theywillbecausesofnothing；
　　orifnotthat，atleastnotofmovement。Further，howisextension，
　　i。e。acontinuum，tobeproducedoutofunextendedparts？Fornumber
　　willnot，eitherasmoverorasform，produceacontinuum。Butagain
　　therecannotbeanycontrarythatisalsoessentiallyaproductive
　　ormovingprinciple；foritwouldbepossibleforitnottobe。Or
　　atleastitsactionwouldbeposteriortoitspotency。Theworld，
　　then，wouldnotbeeternal。Butitis；oneofthesepremisses，then，
　　mustbedenied。Andwehavesaidhowthismustbedone。Further，in
　　virtueofwhatthenumbers，orthesoulandthebody，oringeneral
　　theformandthething，areone-ofthisnoonetellsusanything；
　　norcananyonetell，unlesshesays，aswedo，thatthemovermakes
　　themone。Andthosewhosaymathematicalnumberisfirstandgoon
　　togenerateonekindofsubstanceafteranotherandgivedifferent
　　principlesforeach，makethesubstanceoftheuniverseamere
　　seriesofepisodesforonesubstancehasnoinfluenceonanotherby
　　itsexistenceornonexistence，andtheygiveusmanygoverning
　　principles；buttheworldrefusestobegovernedbadly。
　　’Theruleofmanyisnotgood；onerulerlettherebe。’
　　WEhavestatedwhatisthesubstanceofsensiblethings，dealing
　　inthetreatiseonphysicswithmatter，andlaterwiththesubstance
　　whichhasactualexistence。Nowsinceourinquiryiswhetherthere
　　isorisnotbesidesthesensiblesubstancesanywhichisimmovable
　　andeternal，and，ifthereis，whatitis，wemustfirstconsiderwhat
　　issaidbyothers，sothat，ifthereisanythingwhichtheysay
　　wrongly，wemaynotbeliabletothesameobjections，while，if
　　thereisanyopinioncommontothemandus，weshallhavenoprivate
　　grievanceagainstourselvesonthataccount；foronemustbecontent
　　tostatesomepointsbetterthanone’spredecessors，andothersno
　　worse。
　　Twoopinionsareheldonthissubject；itissaidthattheobjects
　　ofmathematics-i。e。numbersandlinesandthelike-aresubstances，and
　　againthattheIdeasaresubstances。And1sincesomerecognize
　　theseastwodifferentclasses-theIdeasandthemathematicalnumbers，
　　and2somerecognizebothashavingonenature，while3some
　　otherssaythatthemathematicalsubstancesaretheonlysubstances，
　　wemustconsiderfirsttheobjectsofmathematics，notqualifyingthem
　　byanyothercharacteristic-notasking，forinstance，whethertheyare
　　infactIdeasornot，orwhethertheyaretheprinciplesand
　　substancesofexistingthingsornot，butonlywhetherasobjectsof
　　mathematicstheyexistornot，andiftheyexist，howtheyexist。Then
　　afterthiswemustseparatelyconsidertheIdeasthemselvesina
　　generalway，andonlyasfarastheacceptedmodeoftreatment
　　demands；formostofthepointshavebeenrepeatedlymadeevenby
　　thediscussionsoutsideourschool，and，further，thegreaterpart
　　ofouraccountmustfinishbythrowinglightonthatinquiry，viz。
　　whenweexaminewhetherthesubstancesandtheprinciplesof
　　existingthingsarenumbersandIdeas；forafterthediscussionofthe
　　Ideasthisremansasathirdinquiry。
　　Iftheobjectsofmathematicsexist，theymustexisteitherin
　　sensibleobjects，assomesay，orseparatefromsensibleobjects
　　andthisalsoissaidbysome；oriftheyexistinneitherof
　　theseways，eithertheydonotexist，ortheyexistonlyinsome
　　specialsense。Sothatthesubjectofourdiscussionwillbenot
　　whethertheyexistbuthowtheyexist。
　　Thatitisimpossibleformathematicalobjectstoexistin
　　sensiblethings，andatthesametimethatthedoctrineinquestionis
　　anartificialone，hasbeensaidalreadyinourdiscussionof
　　difficultieswehavepointedoutthatitisimpossiblefortwo
　　solidstobeinthesameplace，andalsothataccordingtothesame
　　argumenttheotherpowersandcharacteristicsalsoshouldexistin
　　sensiblethingsandnoneofthemseparately。Thiswehavesaid
　　already。But，further，itisobviousthatonthistheoryitis
　　impossibleforanybodywhatevertobedivided；foritwouldhaveto
　　bedividedataplane，andtheplaneataline，andthelineata
　　point，sothatifthepointcannotbedivided，neithercantheline，
　　andifthelinecannot，neithercantheplanenorthesolid。What
　　difference，then，doesitmakewhethersensiblethingsaresuch
　　indivisibleentities，or，withoutbeingsothemselves，have
　　indivisibleentitiesinthem？Theresultwillbethesame；ifthe
　　sensibleentitiesaredividedtheotherswillbedividedtoo，or
　　elsenoteventhesensibleentitiescanbedivided。
　　But，again，itisnotpossiblethatsuchentitiesshouldexist
　　separately。Forifbesidesthesensiblesolidstherearetobeother
　　solidswhichareseparatefromthemandpriortothesensible
　　solids，itisplainthatbesidestheplanesalsotheremustbeother
　　andseparateplanesandpointsandlines；forconsistencyrequires
　　this。Butiftheseexist，againbesidestheplanesandlinesand
　　pointsofthemathematicalsolidtheremustbeotherswhichare
　　separate。Forincompositesarepriortocompounds；andifthere
　　are，priortothesensiblebodies，bodieswhicharenotsensible，by
　　thesameargumenttheplaneswhichexistbythemselvesmustbeprior
　　tothosewhichareinthemotionlesssolids。Thereforethesewillbe
　　planesandlinesotherthanthosethatexistalongwiththe
　　mathematicalsolidstowhichthesethinkersassignseparateexistence；
　　forthelatterexistalongwiththemathematicalsolids，whilethe
　　othersarepriortothemathematicalsolids。Again，therefore，
　　therewillbe，belongingtotheseplanes，lines，andpriortothem
　　therewillhavetobe，bythesameargument，otherlinesandpoints；
　　andpriortothesepointsinthepriorlinestherewillhavetobe
　　otherpoints，thoughtherewillbenootherspriortothese。Now1
　　theaccumulationbecomesabsurd；forwefindourselveswithonesetof
　　solidsapartfromthesensiblesolids；threesetsofplanesapartfrom
　　thesensibleplanes-thosewhichexistapartfromthesensible
　　planes，andthoseinthemathematicalsolids，andthosewhichexist
　　apartfromthoseinthemathematicalsolids；foursetsoflines，and
　　fivesetsofpoints。Withwhichofthese，then，willthe
　　mathematicalsciencesdeal？Certainlynotwiththeplanesandlines
　　andpointsinthemotionlesssolid；forsciencealwaysdealswithwhat
　　isprior。Andthesameaccountwillapplyalsotonumbers；for
　　therewillbeadifferentsetofunitsapartfromeachsetof
　　points，andalsoapartfromeachsetofrealities，fromtheobjectsof
　　senseandagainfromthoseofthought；sothattherewillbevarious
　　classesofmathematicalnumbers。
　　Again，howisitpossibletosolvethequestionswhichwehave
　　alreadyenumeratedinourdiscussionofdifficulties？Forthe
　　objectsofastronomywillexistapartfromsensiblethingsjustasthe
　　objectsofgeometrywill；buthowisitpossiblethataheavenandits
　　parts-oranythingelsewhichhasmovement-shouldexistapart？
　　Similarlyalsotheobjectsofopticsandofharmonicswillexist
　　apart；fortherewillbebothvoiceandsightbesidesthesensible
　　orindividualvoicesandsights。Thereforeitisplainthatthe
　　othersensesaswell，andtheotherobjectsofsense，willexist
　　apart；forwhyshouldonesetofthemdosoandanothernot？Andif
　　thisisso，therewillalsobeanimalsexistingapart，sincethere
　　willbesenses。
　　Again，therearecertainmathematicaltheoremsthatareuniversal，
　　extendingbeyondthesesubstances。Herethenweshallhaveanother
　　intermediatesubstanceseparatebothfromtheIdeasandfromthe
　　intermediates，-asubstancewhichisneithernumbernorpointsnor
　　spatialmagnitudenortime。Andifthisisimpossible，plainlyitis
　　alsoimpossiblethattheformerentitiesshouldexistseparatefrom
　　sensiblethings。
　　And，ingeneral，conclusioncontraryaliketothetruthandtothe
　　usualviewsfollow，ifoneistosupposetheobjectsofmathematicsto
　　existthusasseparateentities。Forbecausetheyexistthustheymust
　　bepriortosensiblespatialmagnitudes，butintruththeymustbe
　　posterior；fortheincompletespatialmagnitudeisintheorderof
　　generationprior，butintheorderofsubstanceposterior，asthe
　　lifelessistotheliving。
　　Again，byvirtueofwhat，andwhen，willmathematicalmagnitudes
　　beone？Forthingsinourperceptibleworldareoneinvirtueofsoul，
　　orofapartofsoul，orofsomethingelsethatisreasonable
　　enough；whenthesearenotpresent，thethingisaplurality，and
　　splitsupintoparts。Butinthecaseofthesubjectsof
　　mathematics，whicharedivisibleandarequantities，whatisthecause
　　oftheirbeingoneandholdingtogether？
　　Again，themodesofgenerationoftheobjectsofmathematics
　　showthatweareright。Forthedimensionfirstgeneratedislength，
　　thencomesbreadth，lastlydepth，andtheprocessiscomplete。If，
　　then，thatwhichisposteriorintheorderofgenerationispriorin
　　theorderofsubstantiality，thesolidwillbepriortotheplane
　　andtheline。Andinthiswayalsoitisbothmorecompleteandmore
　　whole，becauseitcanbecomeanimate。How，ontheotherhand，could
　　alineoraplanebeanimate？Thesuppositionpassesthepowerof
　　oursenses。
　　Again，thesolidisasortofsubstance；foritalreadyhasina
　　sensecompleteness。Buthowcanlinesbesubstances？Neitherasaform
　　orshape，asthesoulperhapsis，norasmatter，likethesolid；for
　　wehavenoexperienceofanythingthatcanbeputtogetheroutof
　　linesorplanesorpoints，whileifthesehadbeenasortof
　　materialsubstance，weshouldhaveobservedthingswhichcouldbe
　　puttogetheroutofthem。
　　Grant，then，thattheyarepriorindefinition。Stillnotall
　　thingsthatarepriorindefinitionarealsopriorin
　　substantiality。Forthosethingsarepriorinsubstantialitywhich
　　whenseparatedfromotherthingssurpasstheminthepowerof
　　independentexistence，butthingsarepriorindefinitiontothose
　　whosedefinitionsarecompoundedoutoftheirdefinitions；andthese
　　twopropertiesarenotcoextensive。Forifattributesdonotexist
　　apartfromthesubstancese。g。a’mobile’orapale’，paleis
　　priortothepalemanindefinition，butnotinsubstantiality。Forit
　　cannotexistseparately，butisalwaysalongwiththeconcrete
　　thing；andbytheconcretethingImeanthepaleman。Thereforeit
　　isplainthatneitheristheresultofabstractionpriornorthat
　　whichisproducedbyaddingdeterminantsposterior；foritisby
　　addingadeterminanttopalethatwespeakofthepaleman。
　　Ithas，then，beensufficientlypointedoutthattheobjectsof
　　mathematicsarenotsubstancesinahigherdegreethanbodiesare，and
　　thattheyarenotpriortosensiblesinbeing，butonlyindefinition，
　　andthattheycannotexistsomewhereapart。Butsinceitwasnot
　　possibleforthemtoexistinsensibleseither，itisplainthat
　　theyeitherdonotexistatallorexistinaspecialsenseand
　　thereforedonot’exist’withoutqualification。For’exist’hasmany
　　senses。