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，
toavoidwhichsomehavegivenupthetheory(viz。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
themathematicalsolids(forplaceispeculiartotheindividual
things，andhencetheyareseparateinplace；butmathematicalobjects
arenowhere)，andtosaythattheymustbesomewhere，butnotsaywhat
kindofthingtheirplaceis。
Thosewhosaythatexistingthingscomefromelementsandthatthe
firstofexistingthingsarethenumbers，shouldhavefirst
distinguishedthesensesinwhichonethingcomesfromanother，and
thensaidinwhichsensenumbercomesfromitsfirstprinciples。
Byintermixture？But(1)noteverythingiscapableof
intermixture，and(2)thatwhichisproducedbyitisdifferentfrom
itselements，andonthisviewtheonewillnotremainseparateora
distinctentity；buttheywantittobeso。
Byjuxtaposition，likeasyllable？Butthen(1)theelements
musthaveposition；and(2)hewhothinksofnumberwillbeableto
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，
perish(evenwhenallofthecontraryisusedtoproducethem)，
whilenumberdoesnot？Nothingissaidaboutthis。Yetwhetherpresent
ornotpresentinthecompoundthecontrarydestroysit，e。g。’strife’
destroysthe’mixture’(yetitshouldnot；foritisnottothat
thatiscontrary)。
Oncemore，ithasnotbeendeterminedatallinwhichway
numbersarethecausesofsubstancesandofbeing-whether(1)as
boundaries(aspointsareofspatialmagnitudes)。Thisishow
Eurytusdecidedwhatwasthenumberofwhat(e。g。oneofmanand
anotherofhorse)，viz。byimitatingthefiguresoflivingthingswith
pebbles，assomepeoplebringnumbersintotheformsoftriangleand
square。Or(2)isitbecauseharmonyisaratioofnumbers，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
sevenanimalslosetheirteeth(atleastsomedo，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
musicalphenomenaandthelike(forequalidealnumbersdifferfrom
oneanotherinform；foreventheunitsdo)；sothatweneednot
assumeIdeasforthisreasonatleast。
These，then，aretheresultsofthetheory，andyetmoremight
bebroughttogether。Thefactthatouropponntshavemuchtroublewith
thegenerationofnumbersandcaninnowaymakeasystemofthem，
seemstoindicatethattheobjectsofmathematicsarenotseparable
fromsensiblethings，assomesay，andthattheyarenotthefirst
principles-
THEEND-
