WEhavesaidpreviously，inourdistinctionofthevarious
　　meaningsofwords，that’one’hasseveralmeanings；thethingsthat
　　aredirectlyandoftheirownnatureandnotaccidentallycalledone
　　maybesummarizedunderfourheads，thoughthewordisusedinmore
　　senses。1Thereisthecontinuous，eitheringeneral，or
　　especiallythatwhichiscontinuousbynatureandnotbycontactnor
　　bybeingtogether；andofthese，thathasmoreunityandisprior，
　　whosemovementismoreindivisibleandsimpler。2Thatwhichisa
　　wholeandhasacertainshapeandformisoneinastillhigher
　　degree；andespeciallyifathingisofthissortbynature，andnot
　　byforcelikethethingswhichareunifiedbyglueornailsorby
　　beingtiedtogether，i。e。ifithasinitselfthecauseofits
　　continuity。Athingisofthissortbecauseitsmovementisoneand
　　indivisibleinplaceandtime；sothatevidentlyifathinghasby
　　natureaprincipleofmovementthatisofthefirstkindi。e。local
　　movementandthefirstinthatkindi。e。circularmovement，thisis
　　intheprimarysenseoneextendedthing。Somethings，then，areonein
　　thisway，quacontinuousorwhole，andtheotherthingsthatareone
　　arethosewhosedefinitionisone。Ofthissortarethethingsthe
　　thoughtofwhichisone，i。e。thosethethoughtofwhichis
　　indivisible；anditisindivisibleifthethingisindivisibleinkind
　　orinnumber。3Innumber，then，theindividualisindivisible，
　　and4inkind，thatwhichinintelligibilityandinknowledgeis
　　indivisible，sothatthatwhichcausessubstancestobeonemustbe
　　oneintheprimarysense。’One’，then，hasallthesemeanings-the
　　naturallycontinuousandthewhole，andtheindividualandthe
　　universal。Andalltheseareonebecauseinsomecasesthemovement，
　　inothersthethoughtorthedefinitionisindivisible。
　　Butitmustbeobservedthatthequestions，whatsortofthings
　　aresaidtobeone，andwhatitistobeoneandwhatisthe
　　definitionofit，shouldnotbeassumedtobethesame。’One’has
　　allthesemeanings，andeachofthethingstowhichoneofthesekinds
　　ofunitybelongswillbeone；but’tobeone’willsometimesmean
　　beingoneofthesethings，andsometimesbeingsomethingelsewhichis
　　evennearertothemeaningoftheword’one’whiletheseother
　　thingsapproximatetoitsapplication。Thisisalsotrueof
　　’element’or’cause’，ifonehadbothtospecifythethingsofwhich
　　itispredicableandtorenderthedefinitionoftheword。Forina
　　sensefireisanelementanddoubtlessalso’theindefinite’or
　　somethingelseofthesortisbyitsownnaturetheelement，butina
　　senseitisnot；foritisnotthesamethingtobefireandtobe
　　anelement，butwhileasaparticularthingwithanatureofitsown
　　fireisanelement，thename’element’meansthatithasthis
　　attribute，thatthereissomethingwhichismadeofitasaprimary
　　constituent。Andsowith’cause’and’one’andallsuchterms。For
　　thisreason，too，’tobeone’means’tobeindivisible，being
　　essentiallyonemeansa“this“andcapableofbeingisolatedeitherin
　　place，orinformorthought’；orperhaps’tobewholeand
　　indivisible’；butitmeansespecially’tobethefirstmeasureofa
　　kind’，andmoststrictlyofquantity；foritisfromthisthatit
　　hasbeenextendedtotheothercategories。Formeasureisthatby
　　whichquantityisknown；andquantityquaquantityisknowneither
　　bya’one’orbyanumber，andallnumberisknownbya’one’。
　　Thereforeallquantityquaquantityisknownbytheone，andthatby
　　whichquantitiesareprimarilyknownistheoneitself；andsotheone
　　isthestarting-pointofnumberquanumber。Andhenceintheother
　　classestoo’measure’meansthatbywhicheachisfirstknown，andthe
　　measureofeachisaunit-inlength，inbreadth，indepth，in
　　weight，inspeed。Thewords’weight’and’speed’arecommontoboth
　　contraries；foreachofthemhastwomeanings-’weight’meansboththat
　　whichhasanyamountofgravityandthatwhichhasanexcessof
　　gravity，and’speed’boththatwhichhasanyamountofmovementand
　　thatwhichhasanexcessofmovement；foreventheslowhasa
　　certainspeedandthecomparativelylightacertainweight。
　　Inallthese，then，themeasureandstarting-pointissomething
　　oneandindivisible，sinceeveninlineswetreatasindivisiblethe
　　lineafootlong。Foreverywhereweseekasthemeasuresomething
　　oneandindivisible；andthisisthatwhichissimpleeitherin
　　qualityorinquantity。Nowwhereitisthoughtimpossibletotake
　　awayortoadd，therethemeasureisexacthencethatofnumberis
　　mostexact；forweposittheunitasindivisibleineveryrespect；
　　butinallothercasesweimitatethissortofmeasure。Forinthe
　　caseofafurlongoratalentorofanythingcomparativelylargeany
　　additionorsubtractionmightmoreeasilyescapeournoticethanin
　　thecaseofsomethingsmaller；sothatthefirstthingfromwhich，
　　asfarasourperceptiongoes，nothingcanbesubtracted，allmenmake
　　themeasure，whetherofliquidsorofsolids，whetherofweightor
　　ofsize；andtheythinktheyknowthequantitywhentheyknowitby
　　meansofthismeasure。Andindeedtheyknowmovementtoobythesimple
　　movementandthequickest；forthisoccupiesleasttime。Andsoin
　　astronomya’one’ofthissortisthestarting-pointandmeasure
　　fortheyassumethemovementoftheheavenstobeuniformandthe
　　quickest，andjudgetheothersbyreferencetoit，andinmusicthe
　　quarter-tonebecauseitistheleastinterval，andinspeechthe
　　letter。Andalltheseareonesinthissense——notthat’one’is
　　somethingpredicableinthesamesenseofallofthese，butinthe
　　sensewehavementioned。
　　Butthemeasureisnotalwaysoneinnumber——sometimesthereare
　　several；e。g。thequarter-tonesnottotheear，butasdetermined
　　bytheratiosaretwo，andthearticulatesoundsbywhichwe
　　measurearemorethanone，andthediagonalofthesquareanditsside
　　aremeasuredbytwoquantities，andallspatialmagnitudesreveal
　　similarvarietiesofunit。Thus，then，theoneisthemeasureofall
　　things，becausewecometoknowtheelementsinthesubstanceby
　　dividingthethingseitherinrespectofquantityorinrespectof
　　kind。Andtheoneisindivisiblejustbecausethefirstofeach
　　classofthingsisindivisible。Butitisnotinthesamewaythat
　　every’one’isindivisiblee。g。afootandaunit；thelatteris
　　indivisibleineveryrespect，whiletheformermustbeplacedamong
　　thingswhichareundividedtoperception，ashasbeensaid
　　already-onlytoperception，fordoubtlesseverycontinuousthingis
　　divisible。
　　Themeasureisalwayshomogeneouswiththethingmeasured；the
　　measureofspatialmagnitudesisaspatialmagnitude，andin
　　particularthatoflengthisalength，thatofbreadthabreadth，that
　　ofarticulatesoundanarticulatesound，thatofweightaweight，that
　　ofunitsaunit。Forwemuststatethematterso，andnotsaythat
　　themeasureofnumbersisanumber；weoughtindeedtosaythisif
　　weweretousethecorrespondingformofwords，buttheclaimdoesnot
　　reallycorrespond-itisasifoneclaimedthatthemeasureofunitsis
　　unitsandnotaunit；numberisapluralityofunits。
　　Knowledge，also，andperception，wecallthemeasureofthingsfor
　　thesamereason，becausewecometoknowsomethingbythem-whileas
　　amatteroffacttheyaremeasuredratherthanmeasureotherthings。
　　Butitiswithusasifsomeoneelsemeasuredusandwecameto
　　knowhowbigwearebyseeingthatheappliedthecubit-measureto
　　suchandsuchafractionofus。ButProtagorassays’manisthe
　　measureofallthings’，asifhehadsaid’themanwhoknows’or
　　’themanwhoperceives’；andthesebecausetheyhaverespectively
　　knowledgeandperception，whichwesayarethemeasuresofobjects。
　　Suchthinkersaresayingnothing，then，whiletheyappeartobesaying
　　somethingremarkable。
　　Evidently，then，unityinthestrictestsense，ifwedefineit
　　accordingtothemeaningoftheword，isameasure，andmost
　　properlyofquantity，andsecondlyofquality。Andsomethingswillbe
　　oneiftheyareindivisibleinquantity，andothersiftheyare
　　indivisibleinquality；andsothatwhichisoneisindivisible，
　　eitherabsolutelyorquaone。
　　Withregardtothesubstanceandnatureoftheonewemustask
　　inwhichoftwowaysitexists。Thisistheveryquestionthatwe
　　reviewedinourdiscussionofproblems，viz。whattheoneisandhow
　　wemustconceiveofit，whetherwemusttaketheoneitselfasbeinga
　　substanceasboththePythagoreanssayinearlierandPlatoin
　　latertimes，orthereis，rather，anunderlyingnatureandtheone
　　shouldbedescribedmoreintelligiblyandmoreinthemannerofthe
　　physicalphilosophers，ofwhomonesaystheoneislove，another
　　saysitisair，andanothertheindefinite。
　　If，then，nouniversalcanbeasubstance，ashasbeensaidour
　　discussionofsubstanceandbeing，andifbeingitselfcannotbea
　　substanceinthesenseofaoneapartfromthemanyforitis
　　commontothemany，butisonlyapredicate，clearlyunityalso
　　cannotbeasubstance；forbeingandunityarethemostuniversalof
　　allpredicates。Therefore，ontheonehand，generaarenotcertain
　　entitiesandsubstancesseparablefromotherthings；andonthe
　　otherhandtheonecannotbeagenus，forthesamereasonsforwhich
　　beingandsubstancecannotbegenera。
　　Further，thepositionmustbesimilarinallthekindsofunity。
　　Now’unity’hasjustasmanymeaningsas’being’；sothatsinceinthe
　　sphereofqualitiestheoneissomethingdefinite-someparticularkind
　　ofthing-andsimilarlyinthesphereofquantities，clearlywemustin
　　everycategoryaskwhattheoneis，aswemustaskwhattheexistent
　　is，sinceitisnotenoughtosaythatitsnatureisjusttobeoneor
　　existent。Butincolourstheoneisacolour，e。g。white，andthenthe
　　othercoloursareobservedtobeproducedoutofthisandblack，and
　　blackistheprivationofwhite，asdarknessoflight。Thereforeif
　　allexistentthingswerecolours，existentthingswouldhavebeena
　　number，indeed，butofwhat？Clearlyofcolours；andthe’one’would
　　havebeenaparticular’one’，i。e。white。Andsimilarlyifall
　　existingthingsweretunes，theywouldhavebeenanumber，buta
　　numberofquarter-tones，andtheiressencewouldnothavebeennumber；
　　andtheonewouldhavebeensomethingwhosesubstancewasnottobe
　　onebuttobethequarter-tone。Andsimilarlyifallexistentthings
　　hadbeenarticulatesounds，theywouldhavebeenanumberof
　　letters，andtheonewouldhavebeenavowel。Andifallexistent
　　thingswererectilinearfigures，theywouldhavebeenanumberof
　　figures，andtheonewouldhavebeenthetriangle。Andthesame
　　argumentappliestoallotherclasses。Since，therefore，whilethere
　　arenumbersandaonebothinaffectionsandinqualitiesandin
　　quantitiesandinmovement，inallcasesthenumberisanumberof
　　particularthingsandtheoneisonesomething，anditssubstanceis
　　notjusttobeone，thesamemustbetrueofsubstancesalso；forit
　　istrueofallcasesalike。
　　Thattheone，then，ineveryclassisadefinitething，andin
　　nocaseisitsnaturejustthis，unity，isevident；butasin
　　colourstheone-itselfwhichwemustseekisonecolour，sotooin
　　substancetheone-itselfisonesubstance。Thatinasenseunitymeans
　　thesameasbeingisclearfromthefactsthatitsmeaningscorrespond
　　tothecategoriesonetoone，anditisnotcomprisedwithinany
　　categorye。g。itiscomprisedneitherin’whatathingis’norin
　　quality，butisrelatedtothemjustasbeingis；thatin’oneman’
　　nothingmoreispredicatedthanin’man’justasbeingisnothing
　　apartfromsubstanceorqualityorquantity；andthattobeoneis
　　justtobeaparticularthing。