attribute，butfromtheirmakingtheoneaprinciple-andaprinciple
　　inthesenseofanelement-andgeneratingnumberfromtheone。The
　　oldpoetsagreewiththisinasmuchastheysaythatnotthosewho
　　arefirstintime，e。g。NightandHeavenorChaosorOcean，reign
　　andrule，butZeus。Thesepoets，however，areledtospeakthusonly
　　becausetheythinkoftherulersoftheworldaschanging；forthose
　　ofthemwhocombinethetwocharactersinthattheydonotuse
　　mythicallanguagethroughout，e。g。Pherecydesandsomeothers，make
　　theoriginalgeneratingagenttheBest，andsodotheMagi，andsome
　　ofthelatersagesalso，e。g。bothEmpedoclesandAnaxagoras，of
　　whomonemadeloveanelement，andtheothermadereasona
　　principle。Ofthosewhomaintaintheexistenceoftheunchangeable
　　substancessomesaytheOneitselfisthegooditself；butthey
　　thoughtitssubstancelaymainlyinitsunity。
　　This，then，istheproblem，-whichofthetwowaysofspeakingis
　　right。Itwouldbestrangeiftothatwhichisprimaryandeternaland
　　mostself-sufficientthisveryquality——self-sufficiencyand
　　self-maintenance——belongsprimarilyinsomeotherwaythanasa
　　good。Butindeeditcanbefornootherreasonindestructibleor
　　self-sufficientthanbecauseitsnatureisgood。Thereforetosaythat
　　thefirstprincipleisgoodisprobablycorrect；butthatthis
　　principleshouldbetheOneor，ifnotthat，atleastanelement，
　　andanelementofnumbers，isimpossible。Powerfulobjectionsarise，
　　toavoidwhichsomehavegivenupthetheoryviz。thosewhoagree
　　thattheOneisafirstprincipleandelement，butonlyof
　　mathematicalnumber。Foronthisviewalltheunitsbecome
　　identicalwithspeciesofgood，andthereisagreatprofusionof
　　goods。Again，iftheFormsarenumbers，alltheFormsareidentical
　　withspeciesofgood。ButletamanassumeIdeasofanythinghe
　　pleases。IftheseareIdeasonlyofgoods，theIdeaswillnotbe
　　substances；butiftheIdeasarealsoIdeasofsubstances，allanimals
　　andplantsandallindividualsthatshareinIdeaswillbegood。
　　Theseabsurditiesfollow，anditalsofollowsthatthecontrary
　　element，whetheritispluralityortheunequal，i。e。thegreatand
　　small，isthebad-itself。Henceonethinkeravoidedattachingthe
　　goodtotheOne，becauseitwouldnecessarilyfollow，sincegeneration
　　isfromcontraries，thatbadnessisthefundamentalnatureof
　　plurality；whileotherssayinequalityisthenatureofthebad。It
　　follows，then，thatallthingspartakeofthebadexceptone——the
　　Oneitself，andthatnumberspartakeofitinamoreundilutedform
　　thanspatialmagnitudes，andthatthebadisthespaceinwhichthe
　　goodisrealized，andthatitpartakesinanddesiresthatwhichtends
　　todestroyit；forcontrarytendstodestroycontrary。Andif，aswe
　　weresaying，thematteristhatwhichispotentiallyeachthing，
　　e。g。thatofactualfireisthatwhichispotentiallyfire，thebad
　　willbejustthepotentiallygood。
　　Alltheseobjections，then，follow，partlybecausetheymakeevery
　　principleanelement，partlybecausetheymakecontraries
　　principles，partlybecausetheymaketheOneaprinciple，partly
　　becausetheytreatthenumbersasthefirstsubstances，andascapable
　　ofexistingapart，andasForms。
　　If，then，itisequallyimpossiblenottoputthegoodamongthe
　　firstprinciplesandtoputitamongtheminthisway，evidentlythe
　　principlesarenotbeingcorrectlydescribed，norarethefirst
　　substances。Nordoesanyoneconceivethemattercorrectlyifhe
　　comparestheprinciplesoftheuniversetothatofanimalsandplants，
　　onthegroundthatthemorecompletealwayscomesfromthe
　　indefiniteandincomplete-whichiswhatleadsthisthinkertosaythat
　　thisisalsotrueofthefirstprinciplesofreality，sothatthe
　　Oneitselfisnotevenanexistingthing。Thisisincorrect，for
　　eveninthisworldofanimalsandplantstheprinciplesfromwhich
　　thesecomearecomplete；foritisamanthatproducesaman，and
　　theseedisnotfirst。
　　Itisoutofplace，also，togenerateplacesimultaneouslywith
　　themathematicalsolidsforplaceispeculiartotheindividual
　　things，andhencetheyareseparateinplace；butmathematicalobjects
　　arenowhere，andtosaythattheymustbesomewhere，butnotsaywhat
　　kindofthingtheirplaceis。
　　Thosewhosaythatexistingthingscomefromelementsandthatthe
　　firstofexistingthingsarethenumbers，shouldhavefirst
　　distinguishedthesensesinwhichonethingcomesfromanother，and
　　thensaidinwhichsensenumbercomesfromitsfirstprinciples。
　　Byintermixture？But1noteverythingiscapableof
　　intermixture，and2thatwhichisproducedbyitisdifferentfrom
　　itselements，andonthisviewtheonewillnotremainseparateora
　　distinctentity；buttheywantittobeso。
　　Byjuxtaposition，likeasyllable？Butthen1theelements
　　musthaveposition；and2hewhothinksofnumberwillbeableto
　　thinkoftheunityandthepluralityapart；numberthenwillbethis-a
　　unitandplurality，ortheoneandtheunequal。
　　Again，comingfromcertainthingsmeansinonesensethatthese
　　arestilltobefoundintheproduct，andinanotherthattheyare
　　not；whichsensedoesnumbercomefromtheseelements？Onlythings
　　thataregeneratedcancomefromelementswhicharepresentinthem。
　　Doesnumbercome，then，fromitselementsasfromseed？Butnothing
　　canbeexcretedfromthatwhichisindivisible。Doesitcomefrom
　　itscontrary，itscontrarynotpersisting？Butallthingsthatcomein
　　thiswaycomealsofromsomethingelsewhichdoespersist。Since，
　　then，onethinkerplacesthe1ascontrarytoplurality，andanother
　　placesitascontrarytotheunequal，treatingthe1asequal，
　　numbermustbebeingtreatedascomingfromcontraries。Thereis，
　　then，somethingelsethatpersists，fromwhichandfromonecontrary
　　thecompoundisorhascometobe。Again，whyintheworlddothe
　　otherthingsthatcomefromcontraries，orthathavecontraries，
　　perishevenwhenallofthecontraryisusedtoproducethem，
　　whilenumberdoesnot？Nothingissaidaboutthis。Yetwhetherpresent
　　ornotpresentinthecompoundthecontrarydestroysit，e。g。’strife’
　　destroysthe’mixture’yetitshouldnot；foritisnottothat
　　thatiscontrary。
　　Oncemore，ithasnotbeendeterminedatallinwhichway
　　numbersarethecausesofsubstancesandofbeing-whether1as
　　boundariesaspointsareofspatialmagnitudes。Thisishow
　　Eurytusdecidedwhatwasthenumberofwhate。g。oneofmanand
　　anotherofhorse，viz。byimitatingthefiguresoflivingthingswith
　　pebbles，assomepeoplebringnumbersintotheformsoftriangleand
　　square。Or2isitbecauseharmonyisaratioofnumbers，andso
　　ismanandeverythingelse？Buthowaretheattributes-whiteandsweet
　　andhot-numbers？Evidentlyitisnotthenumbersthatarethe
　　essenceorthecausesoftheform；fortheratioistheessence，while
　　thenumberthecausesoftheform；fortheratioistheessence，while
　　thenumberisthematter。E。g。theessenceoffleshorboneis
　　numberonlyinthisway，’threepartsoffireandtwoofearth’。Anda
　　number，whatevernumberitis，isalwaysanumberofcertainthings，
　　eitherofpartsoffireorearthorofunits；buttheessenceis
　　thatthereissomuchofonethingtosomuchofanotherinthe
　　mixture；andthisisnolongeranumberbutaratioofmixtureof
　　numbers，whetherthesearecorporealorofanyotherkind。
　　Number，then，whetheritbenumberingeneralorthenumber
　　whichconsistsofabstractunits，isneitherthecauseasagent，nor
　　thematter，northeratioandformofthings。Nor，ofcourse，isit
　　thefinalcause。
　　Onemightalsoraisethequestionwhatthegoodisthatthingsget
　　fromnumbersbecausetheircompositionisexpressiblebyanumber，
　　eitherbyonewhichiseasilycalculableorbyanoddnumber。Forin
　　facthoney-waterisnomorewholesomeifitismixedintheproportion
　　ofthreetimesthree，butitwoulddomoregoodifitwereinno
　　particularratiobutwelldilutedthanifitwerenumerically
　　expressiblebutstrong。Again，theratiosofmixturesareexpressedby
　　theaddingofnumbers，notbymerenumbers；e。g。itis’threepartsto
　　two’，not’threetimestwo’。Forinanymultiplicationthegenusof
　　thethingsmultipliedmustbethesame；thereforetheproduct1X2X3
　　mustbemeasurableby1，and4X5X6by4andthereforeallproducts
　　intowhichthesamefactorentersmustbemeasurablebythatfactor。
　　Thenumberoffire，then，cannotbe2X5X3X6andatthesametime
　　thatofwater2X3。
　　Ifallthingsmustshareinnumber，itmustfollowthatmany
　　thingsarethesame，andthesamenumbermustbelongtoonething
　　andtoanother。Isnumberthecause，then，anddoesthethingexist
　　becauseofitsnumber，oristhisnotcertain？E。g。themotionsofthe
　　sunhaveanumber，andagainthoseofthemoon，-yes，andthelife
　　andprimeofeachanimal。Why，then，shouldnotsomeofthese
　　numbersbesquares，somecubes，andsomeequal，othersdouble？There
　　isnoreasonwhytheyshouldnot，andindeedtheymustmovewithin
　　theselimits，sinceallthingswereassumedtoshareinnumber。Andit
　　wasassumedthatthingsthatdifferedmightfallunderthesame
　　number。Thereforeifthesamenumberhadbelongedtocertainthings，
　　thesewouldhavebeenthesameasoneanother，sincetheywouldhave
　　hadthesameformofnumber；e。g。sunandmoonwouldhavebeenthe
　　same。Butwhyneedthesenumbersbecauses？Therearesevenvowels，
　　thescaleconsistsofsevenstrings，thePleiadesareseven，at
　　sevenanimalslosetheirteethatleastsomedo，thoughsomedonot，
　　andthechampionswhofoughtagainstThebeswereseven。Isitthen
　　becausethenumberisthekindofnumberitis，thatthechampions
　　weresevenorthePleiadconsistsofsevenstars？Surelythechampions
　　weresevenbecausethereweresevengatesorforsomeotherreason，
　　andthePleiadwecountasseven，aswecounttheBearastwelve，
　　whileotherpeoplescountmorestarsinboth。Naytheyevensaythat
　　X，PsandZareconcordsandthatbecausetherearethreeconcords，
　　thedoubleconsonantsalsoarethree。Theyquiteneglectthefactthat
　　theremightbeathousandsuchletters；foronesymbolmightbe
　　assignedtoGP。Butiftheysaythateachofthesethreeisequalto
　　twooftheotherletters，andnootherisso，andifthecauseisthat
　　therearethreepartsofthemouthandoneletterisineachapplied
　　tosigma，itisforthisreasonthatthereareonlythree，notbecause
　　theconcordsarethree；sinceasamatteroffacttheconcordsare
　　morethanthree，butofdoubleconsonantstherecannotbemore。
　　Thesepeopleareliketheold-fashionedHomericscholars，who
　　seesmallresemblancesbutneglectgreatones。Somesaythatthereare
　　manysuchcases，e。g。thatthemiddlestringsarerepresentedby
　　nineandeight，andthattheepicversehasseventeensyllables，which
　　isequalinnumbertothetwostrings，andthatthescansionis，in
　　therighthalfofthelineninesyllables，andinthelefteight。
　　Andtheysaythatthedistanceinthelettersfromalphatoomegais
　　equaltothatfromthelowestnoteoftheflutetothehighest，and
　　thatthenumberofthisnoteisequaltothatofthewholechoirof
　　heaven。Itmaybesuspectedthatnoonecouldfinddifficultyeither
　　instatingsuchanalogiesorinfindingthemineternalthings，
　　sincetheycanbefoundeveninperishablethings。
　　Butthelaudedcharacteristicsofnumbers，andthecontrariesof
　　these，andgenerallythemathematicalrelations，assomedescribe
　　them，makingthemcausesofnature，seem，whenweinspecttheminthis
　　way，tovanish；fornoneofthemisacauseinanyofthesenses
　　thathavebeendistinguishedinreferencetothefirstprinciples。
　　Inasense，however，theymakeitplainthatgoodnessbelongsto
　　numbers，andthattheodd，thestraight，thesquare，thepotencies
　　ofcertainnumbers，areinthecolumnofthebeautiful。Forthe
　　seasonsandaparticularkindofnumbergotogether；andtheother
　　agreementsthattheycollectfromthetheoremsofmathematicsallhave
　　thismeaning。Hencetheyarelikecoincidences。Fortheyare
　　accidents，butthethingsthatagreeareallappropriatetoone
　　another，andonebyanalogy。Forineachcategoryofbeingan
　　analogoustermisfound-asthestraightisinlength，soisthe
　　levelinsurface，perhapstheoddinnumber，andthewhiteincolour。
　　Again，itisnottheidealnumbersthatarethecausesof
　　musicalphenomenaandthelikeforequalidealnumbersdifferfrom
　　oneanotherinform；foreventheunitsdo；sothatweneednot
　　assumeIdeasforthisreasonatleast。
　　These，then，aretheresultsofthetheory，andyetmoremight
　　bebroughttogether。Thefactthatouropponntshavemuchtroublewith
　　thegenerationofnumbersandcaninnowaymakeasystemofthem，
　　seemstoindicatethattheobjectsofmathematicsarenotseparable
　　fromsensiblethings，assomesay，andthattheyarenotthefirst
　　principles-
　　THEEND-