5
Firstthenweobservethatsomethingsalwayscometopassinthesameway，andothersforthemostpart。Itisclearlyofneitherofthesethatchanceissaidtobethecause，norcanthe\'effectofchance\'beidentifiedwithanyofthethingsthatcometopassbynecessityandalways，orforthemostpart。Butasthereisathirdclassofeventsbesidesthesetwo-eventswhichallsayare\'bychance\'-itisplainthatthereissuchathingaschanceandspontaneity；forweknowthatthingsofthiskindareduetochanceandthatthingsduetochanceareofthiskind。
But，secondly，someeventsareforthesakeofsomething，othersnot。Again，someoftheformerclassareinaccordancewithdeliberateintention，othersnot，butbothareintheclassofthingswhichareforthesakeofsomething。Henceitisclearthatevenamongthethingswhichareoutsidethenecessaryandthenormal，therearesomeinconnexionwithwhichthephrase\'forthesakeofsomething\'
isapplicable。Eventsthatareforthesakeofsomethingincludewhatevermaybedoneasaresultofthoughtorofnature。Thingsofthiskind，then，whentheycometopassincidentalaresaidtobe\'bychance\'。Forjustasathingissomethingeitherinvirtueofitselforincidentally，somayitbeacause。Forinstance，thehousebuildingfacultyisinvirtueofitselfthecauseofahouse，whereasthepaleorthemusicalistheincidentalcause。Thatwhichispersecauseoftheeffectisdeterminate，buttheincidentalcauseisindeterminable，forthepossibleattributesofanindividualareinnumerable。Toresumethen；whenathingofthiskindcomestopassamongeventswhichareforthesakeofsomething，itissaidtobespontaneousorbychance。Thedistinctionbetweenthetwomustbemadelater-forthepresentitissufficientifitisplainthatbothareinthesphereofthingsdoneforthesakeofsomething。
Example:Amanisengagedincollectingsubscriptionsforafeast。
Hewouldhavegonetosuchandsuchaplaceforthepurposeofgettingthemoney，ifhehadknown。Heactuallywentthereforanotherpurposeanditwasonlyincidentallythathegothismoneybygoingthere；andthiswasnotduetothefactthathewentthereasaruleornecessarily，noristheendeffectedgettingthemoneyacausepresentinhimself-itbelongstotheclassofthingsthatareintentionalandtheresultofintelligentdeliberation。Itiswhentheseconditionsaresatisfiedthatthemanissaidtohavegone\'bychance\'。Ifhehadgoneofdeliberatepurposeandforthesakeofthis-ifhealwaysornormallywenttherewhenhewascollectingpayments-hewouldnotbesaidtohavegone\'bychance\'。
Itisclearthenthatchanceisanincidentalcauseinthesphereofthoseactionsforthesakeofsomethingwhichinvolvepurpose。
Intelligentreflection，then，andchanceareinthesamesphere，forpurposeimpliesintelligentreflection。
Itisnecessary，nodoubt，thatthecausesofwhatcomestopassbychancebeindefinite；andthatiswhychanceissupposedtobelongtotheclassoftheindefiniteandtobeinscrutabletoman，andwhyitmightbethoughtthat，inaway，nothingoccursbychance。Forallthesestatementsarecorrect，becausetheyarewellgrounded。Thingsdo，inaway，occurbychance，fortheyoccurincidentallyandchanceisanincidentalcause。Butstrictlyitisnotthecause-withoutqualification-ofanything；forinstance，ahousebuilderisthecauseofahouse；incidentally，afluteplayermaybeso。
Andthecausesoftheman\'scomingandgettingthemoneywhenhedidnotcomeforthesakeofthatareinnumerable。Hemayhavewishedtoseesomebodyorbeenfollowingsomebodyoravoidingsomebody，ormayhavegonetoseeaspectacle。Thustosaythatchanceisathingcontrarytoruleiscorrect。For\'rule\'appliestowhatisalwaystrueortrueforthemostpart，whereaschancebelongstoathirdtypeofevent。Hence，toconclude，sincecausesofthiskindareindefinite，chancetooisindefinite。Yetinsomecasesonemightraisethequestionwhetheranyincidentalfactmightbethecauseofthechanceoccurrence，e。g。ofhealththefreshairorthesun\'sheatmaybethecause，buthavinghadone\'shaircutcannot；forsomeincidentalcausesaremorerelevanttotheeffectthanothers。
Chanceorfortuneiscalled\'good\'whentheresultisgood，\'evil\'
whenitisevil。Theterms\'goodfortune\'and\'illfortune\'areusedwheneitherresultisofconsiderablemagnitude。Thusonewhocomeswithinanaceofsomegreatevilorgreatgoodissaidtobefortunateorunfortunate。Themindaffirmstheessenceoftheattribute，ignoringthehair\'sbreadthofdifference。Further，itiswithreasonthatgoodfortuneisregardedasunstable；forchanceisunstable，asnoneofthethingswhichresultfromitcanbeinvariableornormal。
Botharethen，asIhavesaid，incidentalcauses-bothchanceandspontaneity-inthesphereofthingswhicharecapableofcomingtopassnotnecessarily，nornormally，andwithreferencetosuchoftheseasmightcometopassforthesakeofsomething。
6
Theydifferinthat\'spontaneity\'isthewiderterm。Everyresultofchanceisfromwhatisspontaneous，butnoteverythingthatisfromwhatisspontaneousisfromchance。
Chanceandwhatresultsfromchanceareappropriatetoagentsthatarecapableofgoodfortuneandofmoralactiongenerally。Thereforenecessarilychanceisinthesphereofmoralactions。Thisisindicatedbythefactthatgoodfortuneisthoughttobethesame，ornearlythesame，ashappiness，andhappinesstobeakindofmoralaction，sinceitiswell-doing。Hencewhatisnotcapableofmoralactioncannotdoanythingbychance。Thusaninanimatethingoraloweranimalorachildcannotdoanythingbychance，becauseitisincapableofdeliberateintention；norcan\'goodfortune\'or\'illfortune\'beascribedtothem，exceptmetaphorically，asProtarchus，forexample，saidthatthestonesofwhichaltarsaremadearefortunatebecausetheyareheldinhonour，whiletheirfellowsaretroddenunderfoot。Eventhesethings，however，caninawaybeaffectedbychance，whenonewhoisdealingwiththemdoessomethingtothembychance，butnototherwise。
Thespontaneousontheotherhandisfoundbothintheloweranimalsandinmanyinanimateobjects。Wesay，forexample，thatthehorsecame\'spontaneously\'，because，thoughhiscomingsavedhim，hedidnotcomeforthesakeofsafety。Again，thetripodfell\'ofitself\'，because，thoughwhenitfellitstoodonitsfeetsoastoserveforaseat，itdidnotfallforthesakeofthat。
Henceitisclearthateventswhich1belongtothegeneralclassofthingsthatmaycometopassforthesakeofsomething，2
donotcometopassforthesakeofwhatactuallyresults，and3
haveanexternalcause，maybedescribedbythephrase\'fromspontaneity\'。These\'spontaneous\'eventsaresaidtobe\'fromchance\'iftheyhavethefurthercharacteristicsofbeingtheobjectsofdeliberateintentionandduetoagentscapableofthatmodeofaction。Thisisindicatedbythephrase\'invain\'，whichisusedwhenAwhichisforthesakeofB，doesnotresultinB。Forinstance，takingawalkisforthesakeofevacuationofthebowels；ifthisdoesnotfollowafterwalking，wesaythatwehavewalked\'invain\'
andthatthewalkingwas\'vain\'。Thisimpliesthatwhatisnaturallythemeanstoanendis\'invain\'，whenitdoesnoteffecttheendtowardswhichitwasthenaturalmeans-foritwouldbeabsurdforamantosaythathehadbathedinvainbecausethesunwasnoteclipsed，sincetheonewasnotdonewithaviewtotheother。Thusthespontaneousisevenaccordingtoitsderivationthecaseinwhichthethingitselfhappensinvain。Thestonethatstruckthemandidnotfallforthepurposeofstrikinghim；thereforeitfellspontaneously，becauseitmighthavefallenbytheactionofanagentandforthepurposeofstriking。Thedifferencebetweenspontaneityandwhatresultsbychanceisgreatestinthingsthatcometobebynature；forwhenanythingcomestobecontrarytonature，wedonotsaythatitcametobebychance，butbyspontaneity。Yetstrictlythistooisdifferentfromthespontaneousproper；forthecauseofthelatterisexternal，thatoftheformerinternal。
Wehavenowexplainedwhatchanceisandwhatspontaneityis，andinwhattheydifferfromeachother。Bothbelongtothemodeofcausation\'sourceofchange\'，foreithersomenaturalorsomeintelligentagentisalwaysthecause；butinthissortofcausationthenumberofpossiblecausesisinfinite。
Spontaneityandchancearecausesofeffectswhichthoughtheymightresultfromintelligenceornature，haveinfactbeencausedbysomethingincidentally。Nowsincenothingwhichisincidentalispriortowhatisperse，itisclearthatnoincidentalcausecanbepriortoacauseperse。Spontaneityandchance，therefore，areposteriortointelligenceandnature。Hence，howevertrueitmaybethattheheavensareduetospontaneity，itwillstillbetruethatintelligenceandnaturewillbepriorcausesofthisAllandofmanythingsinitbesides。
7
Itisclearthenthattherearecauses，andthatthenumberofthemiswhatwehavestated。Thenumberisthesameasthatofthethingscomprehendedunderthequestion\'why\'。The\'why\'isreferredultimatelyeither1，inthingswhichdonotinvolvemotion，e。g。
inmathematics，tothe\'what\'tothedefinitionof\'straightline\'or\'commensurable\'，&c。，or2towhatinitiatedamotion，e。g。\'whydidtheygotowar?-becausetherehadbeenaraid\'；or3weareinquiring\'forthesakeofwhat?\'-\'thattheymayrule\'；or4，inthecaseofthingsthatcomeintobeing，wearelookingforthematter。
Thecauses，therefore，aretheseandsomanyinnumber。
Now，thecausesbeingfour，itisthebusinessofthephysicisttoknowaboutthemall，andifherefershisproblemsbacktoallofthem，hewillassignthe\'why\'inthewaypropertohisscience-thematter，theform，themover，\'thatforthesakeofwhich\'。Thelastthreeoftencoincide；forthe\'what\'and\'thatforthesakeofwhich\'areone，whiletheprimarysourceofmotionisthesameinspeciesastheseformangeneratesman，andsotoo，ingeneral，areallthingswhichcausemovementbybeingthemselvesmoved；andsuchasarenotofthiskindarenolongerinsidetheprovinceofphysics，fortheycausemotionnotbypossessingmotionorasourceofmotioninthemselves，butbeingthemselvesincapableofmotion。
Hencetherearethreebranchesofstudy，oneofthingswhichareincapableofmotion，thesecondofthingsinmotion，butindestructible，thethirdofdestructiblethings。
Thequestion\'why\'，then，isansweredbyreferencetothematter，totheform，andtotheprimarymovingcause。Forinrespectofcomingtobeitismostlyinthislastwaythatcausesareinvestigated-\'whatcomestobeafterwhat?whatwastheprimaryagentorpatient?\'andsoateachstepoftheseries。
Nowtheprincipleswhichcausemotioninaphysicalwayaretwo，ofwhichoneisnotphysical，asithasnoprincipleofmotioninitself。Ofthiskindiswhatevercausesmovement，notbeingitselfmoved，suchas1thatwhichiscompletelyunchangeable，theprimaryreality，and2theessenceofthatwhichiscomingtobe，i。e。theform；forthisistheendor\'thatforthesakeofwhich\'。
Hencesincenatureisforthesakeofsomething，wemustknowthiscausealso。Wemustexplainthe\'why\'inallthesensesoftheterm，namely，1thatfromthisthatwillnecessarilyresult\'fromthis\'
eitherwithoutqualificationorinmostcases；2that\'thismustbesoifthatistobeso\'astheconclusionpresupposesthepremisses；
3thatthiswastheessenceofthething；and4becauseitisbetterthusnotwithoutqualification，butwithreferencetotheessentialnatureineachcase。
8
Wemustexplainthen1thatNaturebelongstotheclassofcauseswhichactforthesakeofsomething；2aboutthenecessaryanditsplaceinphysicalproblems，forallwritersascribethingstothiscause，arguingthatsincethehotandthecold，&c。，areofsuchandsuchakind，thereforecertainthingsnecessarilyareandcometobe-andiftheymentionanyothercauseonehis\'friendshipandstrife\'，anotherhis\'mind\'，itisonlytotouchonit，andthengood-byetoit。
Adifficultypresentsitself:whyshouldnotnaturework，notforthesakeofsomething，norbecauseitisbetterso，butjustastheskyrains，notinordertomakethecorngrow，butofnecessity?
Whatisdrawnupmustcool，andwhathasbeencooledmustbecomewateranddescend，theresultofthisbeingthatthecorngrows。Similarlyifaman\'scropisspoiledonthethreshing-floor，theraindidnotfallforthesakeofthis-inorderthatthecropmightbespoiled-butthatresultjustfollowed。Whythenshoulditnotbethesamewiththepartsinnature，e。g。thatourteethshouldcomeupofnecessity-thefrontteethsharp，fittedfortearing，themolarsbroadandusefulforgrindingdownthefood-sincetheydidnotariseforthisend，butitwasmerelyacoincidentresult；andsowithallotherpartsinwhichwesupposethatthereispurpose?Whereverthenallthepartscameaboutjustwhattheywouldhavebeeniftheyhadcomebeforanend，suchthingssurvived，beingorganizedspontaneouslyinafittingway；whereasthosewhichgrewotherwiseperishedandcontinuetoperish，asEmpedoclessayshis\'man-facedox-progeny\'did。
Sucharetheargumentsandothersofthekindwhichmaycausedifficultyonthispoint。Yetitisimpossiblethatthisshouldbethetrueview。Forteethandallothernaturalthingseitherinvariablyornormallycomeaboutinagivenway；butofnotoneoftheresultsofchanceorspontaneityisthistrue。Wedonotascribetochanceormerecoincidencethefrequencyofraininwinter，butfrequentraininsummerwedo；norheatinthedog-days，butonlyifwehaveitinwinter。Ifthen，itisagreedthatthingsareeithertheresultofcoincidenceorforanend，andthesecannotbetheresultofcoincidenceorspontaneity，itfollowsthattheymustbeforanend；
andthatsuchthingsareallduetonatureeventhechampionsofthetheorywhichisbeforeuswouldagree。Thereforeactionforanendispresentinthingswhichcometobeandarebynature。
Further，whereaserieshasacompletion，alltheprecedingstepsareforthesakeofthat。Nowsurelyasinintelligentaction，soinnature；andasinnature，soitisineachaction，ifnothinginterferes。Nowintelligentactionisforthesakeofanend；
thereforethenatureofthingsalsoisso。Thusifahouse，e。g。hadbeenathingmadebynature，itwouldhavebeenmadeinthesamewayasitisnowbyart；andifthingsmadebynatureweremadealsobyart，theywouldcometobeinthesamewayasbynature。Eachstepthenintheseriesisforthesakeofthenext；andgenerallyartpartlycompleteswhatnaturecannotbringtoafinish，andpartlyimitatesher。If，therefore，artificialproductsareforthesakeofanend，soclearlyalsoarenaturalproducts。Therelationofthelatertotheearliertermsoftheseriesisthesameinboth。Thisismostobviousintheanimalsotherthanman:theymakethingsneitherbyartnorafterinquiryordeliberation。Whereforepeoplediscusswhetheritisbyintelligenceorbysomeotherfacultythatthesecreatureswork，spiders，ants，andthelike。Bygradualadvanceinthisdirectionwecometoseeclearlythatinplantstoothatisproducedwhichisconducivetotheend-leaves，e。g。growtoprovideshadeforthefruit。Ifthenitisbothbynatureandforanendthattheswallowmakesitsnestandthespideritsweb，andplantsgrowleavesforthesakeofthefruitandsendtheirrootsdownnotupforthesakeofnourishment，itisplainthatthiskindofcauseisoperativeinthingswhichcometobeandarebynature。Andsince\'nature\'meanstwothings，thematterandtheform，ofwhichthelatteristheend，andsincealltherestisforthesakeoftheend，theformmustbethecauseinthesenseof\'thatforthesakeofwhich\'。
Nowmistakescometopassevenintheoperationsofart:thegrammarianmakesamistakeinwritingandthedoctorpoursoutthewrongdose。Henceclearlymistakesarepossibleintheoperationsofnaturealso。Iftheninarttherearecasesinwhichwhatisrightlyproducedservesapurpose，andifwheremistakesoccurtherewasapurposeinwhatwasattempted，onlyitwasnotattained，somustitbealsoinnaturalproducts，andmonstrositieswillbefailuresinthepurposiveeffort。Thusintheoriginalcombinationsthe\'ox-progeny\'
iftheyfailedtoreachadeterminateendmusthavearisenthroughthecorruptionofsomeprinciplecorrespondingtowhatisnowtheseed。
Further，seedmusthavecomeintobeingfirst，andnotstraightwaytheanimals:thewords\'whole-naturedfirst……\'musthavemeantseed。
Again，inplantstoowefindtherelationofmeanstoend，thoughthedegreeoforganizationisless。Weretheretheninplantsalso\'olive-headedvine-progeny\'，likethe\'man-headedox-progeny\'，ornot?
Anabsurdsuggestion；yettheremusthavebeen，ifthereweresuchthingsamonganimals。
Moreover，amongtheseedsanythingmusthavecometobeatrandom。
Butthepersonwhoassertsthisentirelydoesawaywith\'nature\'andwhatexists\'bynature\'。Forthosethingsarenaturalwhich，byacontinuousmovementoriginatedfromaninternalprinciple，arriveatsomecompletion:thesamecompletionisnotreachedfromeveryprinciple；noranychancecompletion，butalwaysthetendencyineachistowardsthesameend，ifthereisnoimpediment。
Theendandthemeanstowardsitmaycomeaboutbychance。Wesay，forinstance，thatastrangerhascomebychance，paidtheransom，andgoneaway，whenhedoessoasifhehadcomeforthatpurpose，thoughitwasnotforthatthathecame。Thisisincidental，forchanceisanincidentalcause，asIremarkedbefore。Butwhenaneventtakesplacealwaysorforthemostpart，itisnotincidentalorbychance。Innaturalproductsthesequenceisinvariable，ifthereisnoimpediment。
Itisabsurdtosupposethatpurposeisnotpresentbecausewedonotobservetheagentdeliberating。Artdoesnotdeliberate。Iftheship-buildingartwereinthewood，itwouldproducethesameresultsbynature。If，therefore，purposeispresentinart，itispresentalsoinnature。Thebestillustrationisadoctordoctoringhimself:natureislikethat。
Itisplainthenthatnatureisacause，acausethatoperatesforapurpose。
9
Asregardswhatis\'ofnecessity\'，wemustaskwhetherthenecessityis\'hypothetical\'，or\'simple\'aswell。Thecurrentviewplaceswhatisofnecessityintheprocessofproduction，justasifoneweretosupposethatthewallofahousenecessarilycomestobebecausewhatisheavyisnaturallycarrieddownwardsandwhatislighttothetop，whereforethestonesandfoundationstakethelowestplace，withearthabovebecauseitislighter，andwoodatthetopofallasbeingthelightest。Whereas，thoughthewalldoesnotcometobewithoutthese，itisnotduetothese，exceptasitsmaterialcause:
itcomestobeforthesakeofshelteringandguardingcertainthings。
Similarlyinallotherthingswhichinvolveproductionforanend；theproductcannotcometobewithoutthingswhichhaveanecessarynature，butitisnotduetotheseexceptasitsmaterial；itcomestobeforanend。Forinstance，whyisasawsuchasitis?Toeffectso-and-soandforthesakeofso-and-so。Thisend，however，cannotberealizedunlessthesawismadeofiron。Itis，therefore，necessaryforittobeofiron，itwearetohaveasawandperformtheoperationofsawing。Whatisnecessarythen，isnecessaryonahypothesis；itisnotaresultnecessarilydeterminedbyantecedents。Necessityisinthematter，while\'thatforthesakeofwhich\'isinthedefinition。
Necessityinmathematicsisinawaysimilartonecessityinthingswhichcometobethroughtheoperationofnature。Sinceastraightlineiswhatitis，itisnecessarythattheanglesofatriangleshouldequaltworightangles。Butnotconversely；thoughiftheanglesarenotequaltotworightangles，thenthestraightlineisnotwhatitiseither。Butinthingswhichcometobeforanend，thereverseistrue。Iftheendistoexistordoesexist，thatalsowhichprecedesitwillexistordoesexist；otherwisejustasthere，if-theconclusionisnottrue，thepremisswillnotbetrue，soheretheendor\'thatforthesakeofwhich\'willnotexist。Forthistooisitselfastarting-point，butofthereasoning，notoftheaction；whileinmathematicsthestarting-pointisthestarting-pointofthereasoningonly，asthereisnoaction。Ifthenthereistobeahouse，such-and-suchthingsmustbemadeorbetherealreadyorexist，orgenerallythematterrelativetotheend，bricksandstonesifitisahouse。Buttheendisnotduetotheseexceptasthematter，norwillitcometoexistbecauseofthem。Yetiftheydonotexistatall，neitherwillthehouse，orthesaw-theformerintheabsenceofstones，thelatterintheabsenceofiron-justasintheothercasethepremisseswillnotbetrue，iftheanglesofthetrianglearenotequaltotworightangles。
Thenecessaryinnature，then，isplainlywhatwecallbythenameofmatter，andthechangesinit。Bothcausesmustbestatedbythephysicist，butespeciallytheend；forthatisthecauseofthematter，notviceversa；andtheendis\'thatforthesakeofwhich\'，andthebeginningstartsfromthedefinitionoressence；asinartificialproducts，sinceahouseisofsuch-and-suchakind，certainthingsmustnecessarilycometobeorbetherealready，orsincehealthisthis，thesethingsmustnecessarilycometobeorbetherealready。Similarlyifmanisthis，thenthese；ifthese，thenthose。
Perhapsthenecessaryispresentalsointhedefinition。Forifonedefinestheoperationofsawingasbeingacertainkindofdividing，thenthiscannotcomeaboutunlessthesawhasteethofacertainkind；andthesecannotbeunlessitisofiron。Forinthedefinitiontootherearesomepartsthatare，asitwere，itsmatter。
BookIII
1
NATUREhasbeendefinedasa\'principleofmotionandchange\'，anditisthesubjectofourinquiry。Wemustthereforeseethatweunderstandthemeaningof\'motion\'；forifitwereunknown，themeaningof\'nature\'toowouldbeunknown。
Whenwehavedeterminedthenatureofmotion，ournexttaskwillbetoattackinthesamewaythetermswhichareinvolvedinit。Nowmotionissupposedtobelongtotheclassofthingswhicharecontinuous；andtheinfinitepresentsitselffirstinthecontinuous-thatishowitcomesaboutthat\'infinite\'isoftenusedindefinitionsofthecontinuous\'whatisinfinitelydivisibleiscontinuous\'。Besidesthese，place，void，andtimearethoughttobenecessaryconditionsofmotion。
Clearly，then，forthesereasonsandalsobecausetheattributesmentionedarecommonto，andcoextensivewith，alltheobjectsofourscience，wemustfirsttakeeachoftheminhandanddiscussit。
Fortheinvestigationofspecialattributescomesafterthatofthecommonattributes。
Tobeginthen，aswesaid，withmotion。
Wemaystartbydistinguishing1whatexistsinastateoffulfilmentonly，2whatexistsaspotential，3whatexistsaspotentialandalsoinfulfilment-onebeinga\'this\'，another\'somuch\'，athird\'such\'，andsimilarlyineachoftheothermodesofthepredicationofbeing。
Further，theword\'relative\'isusedwithreferenceto1excessanddefect，2agentandpatientandgenerallywhatcanmoveandwhatcanbemoved。For\'whatcancausemovement\'isrelativeto\'whatcanbemoved\'，andviceversa。
Again，thereisnosuchthingasmotionoverandabovethethings。
Itisalwayswithrespecttosubstanceortoquantityortoqualityortoplacethatwhatchangeschanges。Butitisimpossible，asweassert，tofindanythingcommontothesewhichisneither\'this\'norquantumnorqualenoranyoftheotherpredicates。Henceneitherwillmotionandchangehavereferencetosomethingoverandabovethethingsmentioned，forthereisnothingoverandabovethem。
Noweachofthesebelongstoallitssubjectsineitheroftwoways:
namely1substance-theoneispositiveform，theotherprivation；
2inquality，whiteandblack；3inquantity，completeandincomplete；4inrespectoflocomotion，upwardsanddownwardsorlightandheavy。Hencethereareasmanytypesofmotionorchangeastherearemeaningsoftheword\'is\'。
Wehavenowbeforeusthedistinctionsinthevariousclassesofbeingbetweenwhatisfullrealandwhatispotential。
Def。Thefulfilmentofwhatexistspotentially，insofarasitexistspotentially，ismotion-namely，ofwhatisalterablequaalterable，alteration:ofwhatcanbeincreasedanditsoppositewhatcanbedecreasedthereisnocommonname，increaseanddecrease:ofwhatcancometobeandcanpassaway，comingtoheandpassingaway:ofwhatcanbecarriedalong，locomotion。
Exampleswillelucidatethisdefinitionofmotion。Whenthebuildable，insofarasitisjustthat，isfullyreal，itisbeingbuilt，andthisisbuilding。Similarly，learning，doctoring，rolling，leaping，ripening，ageing。
Thesamething，ifitisofacertainkind，canbebothpotentialandfullyreal，notindeedatthesametimeornotinthesamerespect，bute。g。potentiallyhotandactuallycold。Henceatoncesuchthingswillactandbeactedonbyoneanotherinmanyways:eachofthemwillbecapableatthesametimeofcausingalterationandofbeingaltered。Hence，too，whateffectsmotionasaphysicalagentcanbemoved:whenathingofthiskindcausesmotion，itisitselfalsomoved。This，indeed，hasledsomepeopletosupposethateverymoverismoved。Butthisquestiondependsonanothersetofarguments，andthetruthwillbemadeclearlater。ispossibleforathingtocausemotion，thoughitisitselfincapableofbeingmoved。
Itisthefulfilmentofwhatispotentialwhenitisalreadyfullyrealandoperatesnotasitselfbutasmovable，thatismotion。WhatI
meanby\'as\'isthis:Bronzeispotentiallyastatue。Butitisnotthefulfilmentofbronzeasbronzewhichismotion。For\'tobebronze\'
and\'tobeacertainpotentiality\'arenotthesame。
Iftheywereidenticalwithoutqualification，i。e。indefinition，thefulfilmentofbronzeasbronzewouldhavebeenmotion。Buttheyarenotthesame，ashasbeensaid。Thisisobviousincontraries。
\'Tobecapableofhealth\'and\'tobecapableofillness\'arenotthesame，foriftheyweretherewouldbenodifferencebetweenbeingillandbeingwell。Yetthesubjectbothofhealthandofsickness-whetheritishumourorblood-isoneandthesame。
Wecandistinguish，then，betweenthetwo-justas，togiveanotherexample，\'colour\'andvisible\'aredifferent-andclearlyitisthefulfilmentofwhatispotentialaspotentialthatismotion。Sothis，precisely，ismotion。
Furtheritisevidentthatmotionisanattributeofathingjustwhenitisfullyrealinthisway，andneitherbeforenorafter。Foreachthingofthiskindiscapableofbeingatonetimeactual，atanothernot。Takeforinstancethebuildableasbuildable。Theactualityofthebuildableasbuildableistheprocessofbuilding。
Fortheactualityofthebuildablemustbeeitherthisorthehouse。
Butwhenthereisahouse，thebuildableisnolongerbuildable。Ontheotherhand，itisthebuildablewhichisbeingbuilt。TheprocessthenofbeingbuiltmustbethekindofactualityrequiredButbuildingisakindofmotion，andthesameaccountwillapplytotheotherkindsalso。
2
Thesoundnessofthisdefinitionisevidentbothwhenweconsidertheaccountsofmotionthattheothershavegiven，andalsofromthedifficultyofdefiningitotherwise。
Onecouldnoteasilyputmotionandchangeinanothergenus-thisisplainifweconsiderwheresomepeopleputit；theyidentifymotionwithor\'inequality\'or\'notbeing\'；butsuchthingsarenotnecessarilymoved，whethertheyare\'different\'or\'unequal\'or\'non-existent\'；Norischangeeithertoorfromtheseratherthantoorfromtheiropposites。
Thereasonwhytheyputmotionintothesegeneraisthatitisthoughttobesomethingindefinite，andtheprinciplesinthesecondcolumnareindefinitebecausetheyareprivative:noneofthemiseither\'this\'or\'such\'orcomesunderanyoftheothermodesofpredication。Thereasoninturnwhymotionisthoughttobeindefiniteisthatitcannotbeclassedsimplyasapotentialityorasanactuality-athingthatismerelycapableofhavingacertainsizeisnotundergoingchange，noryetathingthatisactuallyofacertainsize，andmotionisthoughttobeasortofactuality，butincomplete，thereasonforthisviewbeingthatthepotentialwhoseactualityitisisincomplete。Thisiswhyitishardtograspwhatmotionis。Itisnecessarytoclassitwithprivationorwithpotentialityorwithsheeractuality，yetnoneoftheseseemspossible。Thereremainsthenthesuggestedmodeofdefinition，namelythatitisasortofactuality，oractualityofthekinddescribed，hardtograsp，butnotincapableofexisting。
Themovertooismoved，ashasbeensaid-everymover，thatis，whichiscapableofmotion，andwhoseimmobilityisrest-whenathingissubjecttomotionitsimmobilityisrest。Fortoactonthemovableassuchisjusttomoveit。Butthisitdoesbycontact，sothatatthesametimeitisalsoactedon。Hencewecandefinemotionasthefulfilmentofthemovablequamovable，thecauseoftheattributebeingcontactwithwhatcanmovesothatthemoverisalsoactedon。
Themoveroragentwillalwaysbethevehicleofaform，eithera\'this\'or\'such\'，which，whenitacts，willbethesourceandcauseofthechange，e。g。thefull-formedmanbegetsmanfromwhatispotentiallyman。
3
Thesolutionofthedifficultythatisraisedaboutthemotion-whetheritisinthemovable-isplain。Itisthefulfilmentofthispotentiality，andbytheactionofthatwhichhasthepowerofcausingmotion；andtheactualityofthatwhichhasthepowerofcausingmotionisnototherthantheactualityofthemovable，foritmustbethefulfilmentofboth。Athingiscapableofcausingmotionbecauseitcandothis，itisamoverbecauseitactuallydoesit。Butitisonthemovablethatitiscapableofacting。
Hencethereisasingleactualityofbothalike，justasonetotwoandtwotoonearethesameinterval，andthesteepascentandthesteepdescentareone-fortheseareoneandthesame，althoughtheycanbedescribedindifferentways。Soitiswiththemoverandthemoved。
Thisviewhasadialecticaldifficulty。Perhapsitisnecessarythattheactualityoftheagentandthatofthepatientshouldnotbethesame。Theoneis\'agency\'andtheother\'patiency\'；andtheoutcomeandcompletionoftheoneisan\'action\'，thatoftheothera\'passion\'。Sincethentheyarebothmotions，wemayask:inwhatarethey，iftheyaredifferent?Eitherabothareinwhatisactedonandmoved，orbtheagencyisintheagentandthepatiencyinthepatient。Ifweoughttocallthelatteralso\'agency\'，thewordwouldbeusedintwosenses。
Now，inalternativeb，themotionwillbeinthemover，forthesamestatementwillholdof\'mover\'and\'moved\'。Henceeithereverymoverwillbemoved，or，thoughhavingmotion，itwillnotbemoved。
Ifontheotherhandabothareinwhatismovedandactedon-boththeagencyandthepatiencye。g。bothteachingandlearning，thoughtheyaretwo，inthelearner，then，first，theactualityofeachwillnotbepresentineach，and，asecondabsurdity，athingwillhavetwomotionsatthesametime。Howwilltherebetwoalterationsofqualityinonesubjecttowardsonedefinitequality?Thethingisimpossible:theactualizationwillbeone。
Butsomeonewillsayitiscontrarytoreasontosupposethatthereshouldbeoneidenticalactualizationoftwothingswhicharedifferentinkind。Yettherewillbe，ifteachingandlearningarethesame，andagencyandpatiency。Toteachwillbethesameastolearn，andtoactthesameastobeactedon-theteacherwillnecessarilybelearningeverythingthatheteaches，andtheagentwillbeactedon。Onemayreply:
1Itisnotabsurdthattheactualizationofonethingshouldbeinanother。Teachingistheactivityofapersonwhocanteach，yettheoperationisperformedonsomepatient-itisnotcutadriftfromasubject，butisofAonB。
2Thereisnothingtopreventtwothingshavingoneandthesameactualization，providedtheactualizationsarenotdescribedinthesameway，butarerelatedaswhatcanacttowhatisacting。
3Norisitnecessarythattheteachershouldlearn，eveniftoactandtobeactedonareoneandthesame，providedtheyarenotthesameindefinitionas\'raiment\'and\'dress\'，butarethesamemerelyinthesenseinwhichtheroadfromThebestoAthensandtheroadfromAthenstoThebesarethesame，ashasbeenexplainedabove。Foritisnotthingswhichareinawaythesamethathavealltheirattributesthesame，butonlysuchashavethesamedefinition。Butindeeditbynomeansfollowsfromthefactthatteachingisthesameaslearning，thattolearnisthesameastoteach，anymorethanitfollowsfromthefactthatthereisonedistancebetweentwothingswhichareatadistancefromeachother，thatthetwovectorsABandBA，areoneandthesame。Togeneralize，teachingisnotthesameaslearning，oragencyaspatiency，inthefullsense，thoughtheybelongtothesamesubject，themotion；forthe\'actualizationofXinY\'andthe\'actualizationofYthroughtheactionofX\'differindefinition。
WhatthenMotionis，hasbeenstatedbothgenerallyandparticularly。Itisnotdifficulttoseehoweachofitstypeswillbedefined-alterationisthefulfillmentofthealterablequaalterableor，morescientifically，thefulfilmentofwhatcanactandwhatcanbeactedon，assuch-generallyandagainineachparticularcase，building，healing，&c。Asimilardefinitionwillapplytoeachoftheotherkindsofmotion。
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Thescienceofnatureisconcernedwithspatialmagnitudesandmotionandtime，andeachoftheseatleastisnecessarilyinfiniteorfinite，evenifsomethingsdealtwithbythesciencearenot，e。g。
aqualityorapoint-itisnotnecessaryperhapsthatsuchthingsshouldbeputundereitherhead。Henceitisincumbentonthepersonwhospecializesinphysicstodiscusstheinfiniteandtoinquirewhetherthereissuchathingornot，and，ifthereis，whatitis。
Theappropriatenesstothescienceofthisproblemisclearlyindicated。Allwhohavetouchedonthiskindofscienceinawayworthconsideringhaveformulatedviewsabouttheinfinite，andindeed，toaman，makeitaprincipleofthings。
1Some，asthePythagoreansandPlato，maketheinfiniteaprincipleinthesenseofaself-subsistentsubstance，andnotasamereattributeofsomeotherthing。OnlythePythagoreansplacetheinfiniteamongtheobjectsofsensetheydonotregardnumberasseparablefromthese，andassertthatwhatisoutsidetheheavenisinfinite。Plato，ontheotherhand，holdsthatthereisnobodyoutsidetheFormsarenotoutsidebecausetheyarenowhere，yetthattheinfiniteispresentnotonlyintheobjectsofsensebutintheFormsalso。
Further，thePythagoreansidentifytheinfinitewiththeeven。Forthis，theysay，whenitiscutoffandshutinbytheodd，providesthingswiththeelementofinfinity。Anindicationofthisiswhathappenswithnumbers。Ifthegnomonsareplacedroundtheone，andwithouttheone，intheoneconstructionthefigurethatresultsisalwaysdifferent，intheotheritisalwaysthesame。ButPlatohastwoinfinites，theGreatandtheSmall。
Thephysicists，ontheotherhand，allofthem，alwaysregardtheinfiniteasanattributeofasubstancewhichisdifferentfromitandbelongstotheclassoftheso-calledelements-waterorairorwhatisintermediatebetweenthem。Thosewhomakethemlimitedinnumbernevermaketheminfiniteinamount。Butthosewhomaketheelementsinfiniteinnumber，asAnaxagorasandDemocritusdo，saythattheinfiniteiscontinuousbycontact-compoundedofthehomogeneouspartsaccordingtotheone，oftheseed-massoftheatomicshapesaccordingtotheother。
Further，AnaxagorasheldthatanypartisamixtureinthesamewayastheAll，onthegroundoftheobservedfactthatanythingcomesoutofanything。Foritisprobablyforthisreasonthathemaintainsthatonceuponatimeallthingsweretogether。Thisfleshandthisboneweretogether，andsoofanything:thereforeallthings:andatthesametimetoo。Forthereisabeginningofseparation，notonlyforeachthing，butforall。Eachthingthatcomestobecomesfromasimilarbody，andthereisacomingtobeofallthings，thoughnot，itistrue，atthesametime。Hencetheremustalsobeanoriginofcomingtobe。OnesuchsourcethereiswhichhecallsMind，andMindbeginsitsworkofthinkingfromsomestarting-point。Sonecessarilyallthingsmusthavebeentogetheratacertaintime，andmusthavebeguntobemovedatacertaintime。
Democritus，forhispart，assertsthecontrary，namelythatnoelementarisesfromanotherelement。Neverthelessforhimthecommonbodyisasourceofallthings，differingfromparttopartinsizeandinshape。
Itisclearthenfromtheseconsiderationsthattheinquiryconcernsthephysicist。Norisitwithoutreasonthattheyallmakeitaprincipleorsource。Wecannotsaythattheinfinitehasnoeffect，andtheonlyeffectivenesswhichwecanascribetoitisthatofaprinciple。Everythingiseitherasourceorderivedfromasource。Buttherecannotbeasourceoftheinfiniteorlimitless，forthatwouldbealimitofit。Further，asitisabeginning，itisbothuncreatableandindestructible。Fortheremustbeapointatwhichwhathascometobereachescompletion，andalsoaterminationofallpassingaway。Thatiswhy，aswesay，thereisnoprincipleofthis，butitisthiswhichisheldtobetheprincipleofotherthings，andtoencompassallandtosteerall，asthoseassertwhodonotrecognize，alongsidetheinfinite，othercauses，suchasMindorFriendship。FurthertheyidentifyitwiththeDivine，foritis\'deathlessandimperishable\'asAnaximandersays，withthemajorityofthephysicists。
Beliefintheexistenceoftheinfinitecomesmainlyfromfiveconsiderations:
1Fromthenatureoftime-foritisinfinite。
2Fromthedivisionofmagnitudes-forthemathematiciansalsousethenotionoftheinfinite。
3Ifcomingtobeandpassingawaydonotgiveout，itisonlybecausethatfromwhichthingscometobeisinfinite。
4Becausethelimitedalwaysfindsitslimitinsomething，sothattheremustbenolimit，ifeverythingisalwayslimitedbysomethingdifferentfromitself。
5Mostofall，areasonwhichispeculiarlyappropriateandpresentsthedifficultythatisfeltbyeverybody-notonlynumberbutalsomathematicalmagnitudesandwhatisoutsidetheheavenaresupposedtobeinfinitebecausetheynevergiveoutinourthought。
Thelastfactthatwhatisoutsideisinfiniteleadspeopletosupposethatbodyalsoisinfinite，andthatthereisaninfinitenumberofworlds。Whyshouldtherebebodyinonepartofthevoidratherthaninanother?Grantonlythatmassisanywhereanditfollowsthatitmustbeeverywhere。Also，ifvoidandplaceareinfinite，theremustbeinfinitebodytoo，forinthecaseofeternalthingswhatmaybemustbe。Buttheproblemoftheinfiniteisdifficult:manycontradictionsresultwhetherwesupposeittoexistornottoexist。Ifitexists，wehavestilltoaskhowitexists；
asasubstanceorastheessentialattributeofsomeentity?Orinneitherway，yetnonethelessistheresomethingwhichisinfiniteorsomethingswhichareinfinitelymany?
Theproblem，however，whichspeciallybelongstothephysicististoinvestigatewhetherthereisasensiblemagnitudewhichisinfinite。
Wemustbeginbydistinguishingthevarioussensesinwhichtheterm\'infinite\'isused。
1Whatisincapableofbeinggonethrough，becauseitisnotinitsnaturetobegonethroughthesenseinwhichthevoiceis\'invisible\'。
2Whatadmitsofbeinggonethrough，theprocesshoweverhavingnotermination，orwhatscarcelyadmitsofbeinggonethrough。
3Whatnaturallyadmitsofbeinggonethrough，butisnotactuallygonethroughordoesnotactuallyreachanend。
Further，everythingthatisinfinitemaybesoinrespectofadditionordivisionorboth。
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Nowitisimpossiblethattheinfiniteshouldbeathingwhichisitselfinfinite，separablefromsensibleobjects。Iftheinfiniteisneitheramagnitudenoranaggregate，butisitselfasubstanceandnotanattribute，itwillbeindivisible；forthedivisiblemustbeeitheramagnitudeoranaggregate。Butifindivisible，thennotinfinite，exceptinthesense1inwhichthevoiceis\'invisible\'。
Butthisisnotthesenseinwhichitisusedbythosewhosaythattheinfiniteexists，northatinwhichweareinvestigatingit，namelyas2\'thatwhichcannotbegonethrough\'。Butiftheinfiniteexistsasanattribute，itwouldnotbe，quainfiniteanelementinsubstances，anymorethantheinvisiblewouldbeanelementofspeech，thoughthevoiceisinvisible。
Further，howcantheinfinitebeitselfanything，unlessbothnumberandmagnitude，ofwhichitisanessentialattribute，existinthatway?Iftheyarenotsubstances，afortioritheinfiniteisnot。
Itisplain，too，thattheinfinitecannotbeanactualthingandasubstanceandprinciple。Foranypartofitthatistakenwillbeinfinite，ifithasparts:for\'tobeinfinite\'and\'theinfinite\'arethesame，ifitisasubstanceandnotpredicatedofasubject。
Henceitwillbeeitherindivisibleordivisibleintoinfinites。Butthesamethingcannotbemanyinfinites。Yetjustaspartofairisair，soapartoftheinfinitewouldbeinfinite，ifitissupposedtobeasubstanceandprinciple。Thereforetheinfinitemustbewithoutpartsandindivisible。Butthiscannotbetrueofwhatisinfiniteinfullcompletion:foritmustbeadefinitequantity。
Supposethenthatinfinitybelongstosubstanceasanattribute。
But，ifso，itcannot，aswehavesaid，bedescribedasaprinciple，butratherthatofwhichitisanattribute-theairortheevennumber。
ThustheviewofthosewhospeakafterthemannerofthePythagoreansisabsurd。Withthesamebreaththeytreattheinfiniteassubstance，anddivideitintoparts。
Thisdiscussion，however，involvesthemoregeneralquestionwhethertheinfinitecanbepresentinmathematicalobjectsandthingswhichareintelligibleanddonothaveextension，aswellasamongsensibleobjects。Ourinquiryasphysicistsislimitedtoitsspecialsubject-matter，theobjectsofsense，andwehavetoaskwhetherthereisorisnotamongthemabodywhichisinfiniteinthedirectionofincrease。
Wemaybeginwithadialecticalargumentandshowasfollowsthatthereisnosuchthing。If\'boundedbyasurface\'isthedefinitionofbodytherecannotbeaninfinitebodyeitherintelligibleorsensible。
Norcannumbertakeninabstractionbeinfinite，fornumberorthatwhichhasnumberisnumerable。Ifthenthenumerablecanbenumbered，itwouldalsobepossibletogothroughtheinfinite。
If，ontheotherhand，weinvestigatethequestionmoreinaccordancewithprinciplesappropriatetophysics，weareledasfollowstothesameresult。
Theinfinitebodymustbeeither1compound，or2simple；yetneitheralternativeispossible。
1Compoundtheinfinitebodywillnotbe，iftheelementsarefiniteinnumber。Fortheymustbemorethanone，andthecontrariesmustalwaysbalance，andnooneofthemcanbeinfinite。Ifoneofthebodiesfallsinanydegreeshortoftheotherinpotency-supposefireisfiniteinamountwhileairisinfiniteandagivenquantityoffireexceedsinpowerthesameamountofairinanyratioprovideditisnumericallydefinite-theinfinitebodywillobviouslyprevailoverandannihilatethefinitebody。Ontheotherhand，itisimpossiblethateachshouldbeinfinite。\'Body\'iswhathasextensioninalldirectionsandtheinfiniteiswhatisboundlesslyextended，sothattheinfinitebodywouldbeextendedinalldirectionsadinfinitum。
Nor2cantheinfinitebodybeoneandsimple，whetheritis，assomehold，athingoverandabovetheelementsfromwhichtheygeneratetheelementsorisnotthusqualified。
aWemustconsidertheformeralternative；fortherearesomepeoplewhomakethistheinfinite，andnotairorwater，inorderthattheotherelementsmaynotbeannihilatedbytheelementwhichisinfinite。Theyhavecontrarietywitheachother-airiscold，watermoist，firehot；ifonewereinfinite，theothersbynowwouldhaveceasedtobe。Asitis，theysay，theinfiniteisdifferentfromthemandistheirsource。
Itisimpossible，however，thatthereshouldbesuchabody；notbecauseitisinfiniteonthatpointageneralproofcanbegivenwhichappliesequallytoall，air，water，oranythingelse-butsimplybecausethereis，asamatteroffact，nosuchsensiblebody，alongsidetheso-calledelements。Everythingcanberesolvedintotheelementsofwhichitiscomposed。Hencethebodyinquestionwouldhavebeenpresentinourworldhere，alongsideairandfireandearthandwater:butnothingofthekindisobserved。
bNorcanfireoranyotheroftheelementsbeinfinite。Forgenerally，andapartfromthequestionofhowanyofthemcouldbeinfinite，theAll，evenifitwerelimited，cannoteitherbeorbecomeoneofthem，asHeraclitussaysthatatsometimeallthingsbecomefire。Thesameargumentappliesalsototheonewhichthephysicistssupposetoexistalongsidetheelements:foreverythingchangesfromcontrarytocontrary，e。g。fromhottocold。
Theprecedingconsiderationofthevariouscasesservestoshowuswhetheritisorisnotpossiblethatthereshouldbeaninfinitesensiblebody。Thefollowingargumentsgiveageneraldemonstrationthatitisnotpossible。
Itisthenatureofeverykindofsensiblebodytobesomewhere，andthereisaplaceappropriatetoeach，thesameforthepartandforthewhole，e。g。forthewholeearthandforasingleclod，andforfireandforaspark。
Supposeathattheinfinitesensiblebodyishomogeneous。Theneachpartwillbeeitherimmovableoralwaysbeingcarriedalong。
Yetneitherispossible。Forwhydownwardsratherthanupwardsorinanyotherdirection?Imean，e。g，ifyoutakeaclod，wherewillitbemovedorwherewillitbeatrest?Forexhypothesitheplaceofthebodyakintoitisinfinite。Willitoccupythewholeplace，then?Andhow?Whatthenwillbethenatureofitsrestandofitsmovement，orwherewilltheybe?Itwilleitherbeathomeeverywhere-thenitwillnotbemoved；oritwillbemovedeverywhere-thenitwillnotcometorest。
ButifbtheAllhasdissimilarparts，theproperplacesofthepartswillbedissimilaralso，andthebodyoftheAllwillhavenounityexceptthatofcontact。Then，further，thepartswillbeeitherfiniteorinfiniteinvarietyofkind。iFinitetheycannotbe，foriftheAllistobeinfinite，someofthemwouldhavetobeinfinite，whiletheotherswerenot，e。g。fireorwaterwillbeinfinite。But，aswehaveseenbefore，suchanelementwoulddestroywhatiscontrarytoit。Thisindeedisthereasonwhynoneofthephysicistsmadefireorearththeoneinfinitebody，buteitherwaterorairorwhatisintermediatebetweenthem，becausetheabodeofeachofthetwowasplainlydeterminate，whiletheothershaveanambiguousplacebetweenupanddown。
Butiiifthepartsareinfiniteinnumberandsimple，theirproperplacestoowillbeinfiniteinnumber，andthesamewillbetrueoftheelementsthemselves。Ifthatisimpossible，andtheplacesarefinite，thewholetoomustbefinite；fortheplaceandthebodycannotbutfiteachother。Neitheristhewholeplacelargerthanwhatcanbefilledbythebodyandthenthebodywouldnolongerbeinfinite，noristhebodylargerthantheplace；foreithertherewouldbeanemptyspaceorabodywhosenatureitistobenowhere。
Anaxagorasgivesanabsurdaccountofwhytheinfiniteisatrest。
Hesaysthattheinfiniteitselfisthecauseofitsbeingfixed。Thisbecauseitisinitself，sincenothingelsecontainsit-ontheassumptionthatwhereveranythingis，itistherebyitsownnature。
Butthisisnottrue:athingcouldbesomewherebycompulsion，andnotwhereitisitsnaturetobe。
Evenifitistrueastruecanbethatthewholeisnotmovedforwhatisfixedbyitselfandisinitselfmustbeimmovable，yetwemustexplainwhyitisnotitsnaturetobemoved。Itisnotenoughjusttomakethisstatementandthendecamp。Anythingelsemightbeinastateofrest，butthereisnoreasonwhyitshouldnotbeitsnaturetobemoved。Theearthisnotcarriedalong，andwouldnotbecarriedalongifitwereinfinite，provideditisheldtogetherbythecentre。Butitwouldnotbebecausetherewasnootherregioninwhichitcouldbecarriedalongthatitwouldremainatthecentre，butbecausethisisitsnature。Yetinthiscasealsowemaysaythatitfixesitself。Iftheninthecaseoftheearth，supposedtobeinfinite，itisatrest，notbecauseitisinfinite，butbecauseithasweightandwhatisheavyrestsatthecentreandtheearthisatthecentre，similarlytheinfinitealsowouldrestinitself，notbecauseitisinfiniteandfixesitself，butowingtosomeothercause。
Anotherdifficultyemergesatthesametime。Anypartoftheinfinitebodyoughttoremainatrest。Justastheinfiniteremainsatrestinitselfbecauseitfixesitself，sotooanypartofityoumaytakewillremaininitself。Theappropriateplacesofthewholeandofthepartarealike，e。g。ofthewholeearthandofaclodtheappropriateplaceisthelowerregion；offireasawholeandofaspark，theupperregion。If，therefore，tobeinitselfistheplaceoftheinfinite，thatalsowillbeappropriatetothepart。
Thereforeitwillremaininitself。
Ingeneral，theviewthatthereisaninfinitebodyisplainlyincompatiblewiththedoctrinethatthereisnecessarilyaproperplaceforeachkindofbody，ifeverysensiblebodyhaseitherweightorlightness，andifabodyhasanaturallocomotiontowardsthecentreifitisheavy，andupwardsifitislight。Thiswouldneedtobetrueoftheinfinitealso。Butneithercharactercanbelongtoit:itcannotbeeitherasawhole，norcanitbehalftheoneandhalftheother。Forhowshouldyoudivideit?orhowcantheinfinitehavetheonepartupandtheotherdown，oranextremityandacentre?
Further，everysensiblebodyisinplace，andthekindsordifferencesofplaceareup-down，before-behind，right-left；andthesedistinctionsholdnotonlyinrelationtousandbyarbitraryagreement，butalsointhewholeitself。Butintheinfinitebodytheycannotexist。Ingeneral，ifitisimpossiblethatthereshouldbeaninfiniteplace，andifeverybodyisinplace，therecannotbeaninfinitebody。
Surelywhatisinaspecialplaceisinplace，andwhatisinplaceisinaspecialplace。Just，then，astheinfinitecannotbequantity-thatwouldimplythatithasaparticularquantity，e，g，twoorthreecubits；quantityjustmeansthese-soathing\'sbeinginplacemeansthatitissomewhere，andthatiseitherupordownorinsomeotherofthesixdifferencesofposition:buteachoftheseisalimit。
Itisplainfromtheseargumentsthatthereisnobodywhichisactuallyinfinite。
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Butontheotherhandtosupposethattheinfinitedoesnotexistinanywayleadsobviouslytomanyimpossibleconsequences:therewillbeabeginningandanendoftime，amagnitudewillnotbedivisibleintomagnitudes，numberwillnotbeinfinite。If，then，inviewoftheaboveconsiderations，neitheralternativeseemspossible，anarbitermustbecalledin；andclearlythereisasenseinwhichtheinfiniteexistsandanotherinwhichitdoesnot。
Wemustkeepinmindthattheword\'is\'meanseitherwhatpotentiallyisorwhatfullyis。Further，athingisinfiniteeitherbyadditionorbydivision。
Now，aswehaveseen，magnitudeisnotactuallyinfinite。Butbydivisionitisinfinite。Thereisnodifficultyinrefutingthetheoryofindivisiblelines。Thealternativethenremainsthattheinfinitehasapotentialexistence。
Butthephrase\'potentialexistence\'isambiguous。Whenwespeakofthepotentialexistenceofastatuewemeanthattherewillbeanactualstatue。Itisnotsowiththeinfinite。Therewillnotbeanactualinfinite。Theword\'is\'hasmanysenses，andwesaythattheinfinite\'is\'inthesenseinwhichwesay\'itisday\'or\'itisthegames\'，becauseonethingafteranotherisalwayscomingintoexistence。Forofthesethingstoothedistinctionbetweenpotentialandactualexistenceholds。WesaythatthereareOlympicgames，bothinthesensethattheymayoccurandthattheyareactuallyoccurring。
Theinfiniteexhibitsitselfindifferentways-intime，inthegenerationsofman，andinthedivisionofmagnitudes。Forgenerallytheinfinitehasthismodeofexistence:onethingisalwaysbeingtakenafteranother，andeachthingthatistakenisalwaysfinite，butalwaysdifferent。Again，\'being\'hasmorethanonesense，sothatwemustnotregardtheinfiniteasa\'this\'，suchasamanorahorse，butmustsupposeittoexistinthesenseinwhichwespeakofthedayorthegamesasexistingthingswhosebeinghasnotcometothemlikethatofasubstance，butconsistsinaprocessofcomingtobeorpassingaway；definiteifyoulikeateachstage，yetalwaysdifferent。
Butwhenthistakesplaceinspatialmagnitudes，whatistakenperists，whileinthesuccessionoftimeandofmenittakesplacebythepassingawayoftheseinsuchawaythatthesourceofsupplynevergivesout。
Inawaytheinfinitebyadditionisthesamethingastheinfinitebydivision。Inafinitemagnitude，theinfinitebyadditioncomesaboutinawayinversetothatoftheother。Forinproportionasweseedivisiongoingon，inthesameproportionweseeadditionbeingmadetowhatisalreadymarkedoff。Forifwetakeadeterminatepartofafinitemagnitudeandaddanotherpartdeterminedbythesamerationottakinginthesameamountoftheoriginalwhole，andsoon，weshallnottraversethegivenmagnitude。
Butifweincreasetheratioofthepart，soasalwaystotakeinthesameamount，weshalltraversethemagnitude，foreveryfinitemagnitudeisexhaustedbymeansofanydeterminatequantityhoweversmall。
Theinfinite，then，existsinnootherway，butinthiswayitdoesexist，potentiallyandbyreduction。Itexistsfullyinthesenseinwhichwesay\'itisday\'or\'itisthegames\'；andpotentiallyasmatterexists，notindependentlyaswhatisfinitedoes。
Byadditionthen，also，thereispotentiallyaninfinite，namely，whatwehavedescribedasbeinginasensethesameastheinfiniteinrespectofdivision。Foritwillalwaysbepossibletotakesomethingahextra。Yetthesumofthepartstakenwillnotexceedeverydeterminatemagnitude，justasinthedirectionofdivisioneverydeterminatemagnitudeissurpassedinsmallnessandtherewillbeasmallerpart。
Butinrespectofadditiontherecannotbeaninfinitewhichevenpotentiallyexceedseveryassignablemagnitude，unlessithastheattributeofbeingactuallyinfinite，asthephysicistsholdtobetrueofthebodywhichisoutsidetheworld，whoseessentialnatureisairorsomethingofthekind。Butiftherecannotbeinthiswayasensiblebodywhichisinfiniteinthefullsense，evidentlytherecannomorebeabodywhichispotentiallyinfiniteinrespectofaddition，exceptastheinverseoftheinfinitebydivision，aswehavesaid。ItisforthisreasonthatPlatoalsomadetheinfinitestwoinnumber，becauseitissupposedtobepossibletoexceedalllimitsandtoproceedadinfinituminthedirectionbothofincreaseandofreduction。Yetthoughhemakestheinfinitestwo，hedoesnotusethem。Forinthenumberstheinfiniteinthedirectionofreductionisnotpresent，asthemonadisthesmallest；noristheinfiniteinthedirectionofincrease，forthepartsnumberonlyuptothedecad。
Theinfiniteturnsouttobethecontraryofwhatitissaidtobe。Itisnotwhathasnothingoutsideitthatisinfinite，butwhatalwayshassomethingoutsideit。Thisisindicatedbythefactthatringsalsothathavenobezelaredescribedas\'endless\'，becauseitisalwayspossibletotakeapartwhichisoutsideagivenpart。Thedescriptiondependsonacertainsimilarity，butitisnottrueinthefullsenseoftheword。Thisconditionaloneisnotsufficient:itisnecessaryalsothatthenextpartwhichistakenshouldneverbethesame。Inthecircle，thelatterconditionisnotsatisfied:itisonlytheadjacentpartfromwhichthenewpartisdifferent。
Ourdefinitionthenisasfollows:
Aquantityisinfiniteifitissuchthatwecanalwaystakeapartoutsidewhathasbeenalreadytaken。Ontheotherhand，whathasnothingoutsideitiscompleteandwhole。Forthuswedefinethewhole-thatfromwhichnothingiswanting，asawholemanorawholebox。Whatistrueofeachparticularistrueofthewholeassuch-thewholeisthatofwhichnothingisoutside。Ontheotherhandthatfromwhichsomethingisabsentandoutside，howeversmallthatmaybe，isnot\'all\'。\'Whole\'and\'complete\'areeitherquiteidenticalorcloselyakin。Nothingiscompleteteleionwhichhasnoendtelos；andtheendisalimit。
HenceParmenidesmustbethoughttohavespokenbetterthanMelissus。Thelattersaysthatthewholeisinfinite，buttheformerdescribesitaslimited，\'equallybalancedfromthemiddle\'。Fortoconnecttheinfinitewiththeallandthewholeisnotlikejoiningtwopiecesofstring；foritisfromthistheygetthedignitytheyascribetotheinfinite-itscontainingallthingsandholdingtheallinitself-fromitshavingacertainsimilaritytothewhole。Itisinfactthematterofthecompletenesswhichbelongstosize，andwhatispotentiallyawhole，thoughnotinthefullsense。Itisdivisiblebothinthedirectionofreductionandoftheinverseaddition。Itisawholeandlimited；not，however，invirtueofitsownnature，butinvirtueofwhatisotherthanit。Itdoesnotcontain，but，insofarasitisinfinite，iscontained。Consequently，also，itisunknowable，quainfinite；forthematterhasnoform。
Henceitisplainthattheinfinitestandsintherelationofpartratherthanofwhole。Forthematterispartofthewhole，asthebronzeisofthebronzestatue。Ifitcontainsinthecaseofsensiblethings，inthecaseofintelligiblethingsthegreatandthesmalloughttocontainthem。Butitisabsurdandimpossibletosupposethattheunknowableandindeterminateshouldcontainanddetermine。
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Itisreasonablethatthereshouldnotbeheldtobeaninfiniteinrespectofadditionsuchastosurpasseverymagnitude，butthatthereshouldbethoughttobesuchaninfiniteinthedirectionofdivision。Forthematterandtheinfinitearecontainedinsidewhatcontainsthem，whileitistheformwhichcontains。Itisnaturaltootosupposethatinnumberthereisalimitinthedirectionoftheminimum，andthatintheotherdirectioneveryassignednumberissurpassed。Inmagnitude，onthecontrary，everyassignedmagnitudeissurpassedinthedirectionofsmallness，whileintheotherdirectionthereisnoinfinitemagnitude。Thereasonisthatwhatisoneisindivisiblewhateveritmaybe，e。g。amanisoneman，notmany。Numberontheotherhandisapluralityof\'ones\'andacertainquantityofthem。Hencenumbermuststopattheindivisible:
for\'two\'and\'three\'aremerelyderivativeterms，andsowitheachoftheothernumbers。Butinthedirectionoflargenessitisalwayspossibletothinkofalargernumber:forthenumberoftimesamagnitudecanbebisectedisinfinite。Hencethisinfiniteispotential，neveractual:thenumberofpartsthatcanbetakenalwayssurpassesanyassignednumber。Butthisnumberisnotseparablefromtheprocessofbisection，anditsinfinityisnotapermanentactualitybutconsistsinaprocessofcomingtobe，liketimeandthenumberoftime。
Withmagnitudesthecontraryholds。Whatiscontinuousisdividedadinfinitum，butthereisnoinfiniteinthedirectionofincrease。
Forthesizewhichitcanpotentiallybe，itcanalsoactuallybe。
Hencesincenosensiblemagnitudeisinfinite，itisimpossibletoexceedeveryassignedmagnitude；forifitwerepossibletherewouldbesomethingbiggerthantheheavens。
Theinfiniteisnotthesameinmagnitudeandmovementandtime，inthesenseofasinglenature，butitssecondarysensedependsonitsprimarysense，i。e。movementiscalledinfiniteinvirtueofthemagnitudecoveredbythemovementoralterationorgrowth，andtimebecauseofthemovement。Iusethesetermsforthemoment。LaterIshallexplainwhateachofthemmeans，andalsowhyeverymagnitudeisdivisibleintomagnitudes。
Ouraccountdoesnotrobthemathematiciansoftheirscience，bydisprovingtheactualexistenceoftheinfiniteinthedirectionofincrease，inthesenseoftheuntraversable。Inpointoffacttheydonotneedtheinfiniteanddonotuseit。Theypostulateonlythatthefinitestraightlinemaybeproducedasfarastheywish。Itispossibletohavedividedinthesameratioasthelargestquantityanothermagnitudeofanysizeyoulike。Hence，forthepurposesofproof，itwillmakenodifferencetothemtohavesuchaninfiniteinstead，whileitsexistencewillbeinthesphereofrealmagnitudes。
Inthefourfoldschemeofcauses，itisplainthattheinfiniteisacauseinthesenseofmatter，andthatitsessenceisprivation，thesubjectassuchbeingwhatiscontinuousandsensible。Alltheotherthinkers，too，evidentlytreattheinfiniteasmatter-thatiswhyitisinconsistentinthemtomakeitwhatcontains，andnotwhatiscontained。
8
Itremainstodisposeoftheargumentswhicharesupposedtosupporttheviewthattheinfiniteexistsnotonlypotentiallybutasaseparatething。Somehavenocogency；otherscanbemetbyfreshobjectionsthatarevalid。
1Inorderthatcomingtobeshouldnotfail，itisnotnecessarythatthereshouldbeasensiblebodywhichisactuallyinfinite。Thepassingawayofonethingmaybethecomingtobeofanother，theAllbeinglimited。
2Thereisadifferencebetweentouchingandbeinglimited。Theformerisrelativetosomethingandisthetouchingofsomethingforeverythingthattouchestouchessomething，andfurtherisanattributeofsomeoneofthethingswhicharelimited。Ontheotherhand，whatislimitedisnotlimitedinrelationtoanything。Again，contactisnotnecessarilypossiblebetweenanytwothingstakenatrandom。
3Torelyonmerethinkingisabsurd，forthentheexcessordefectisnotinthethingbutinthethought。Onemightthinkthatoneofusisbiggerthanheisandmagnifyhimadinfinitum。Butitdoesnotfollowthatheisbiggerthanthesizeweare，justbecausesomeonethinksheis，butonlybecauseheisthesizeheis。Thethoughtisanaccident。
aTimeindeedandmovementareinfinite，andalsothinking，inthesensethateachpartthatistakenpassesinsuccessionoutofexistence。
bMagnitudeisnotinfiniteeitherinthewayofreductionorofmagnificationinthought。
Thisconcludesmyaccountofthewayinwhichtheinfiniteexists，andofthewayinwhichitdoesnotexist，andofwhatitis。
BookIV
1
THEphysicistmusthaveaknowledgeofPlace，too，aswellasoftheinfinite-namely，whetherthereissuchathingornot，andthemannerofitsexistenceandwhatitis-bothbecauseallsupposethatthingswhichexistaresomewherethenon-existentisnowhere——whereisthegoat-stagorthesphinx?，andbecause\'motion\'initsmostgeneralandprimarysenseischangeofplace，whichwecall\'locomotion\'。
Thequestion，whatisplace?presentsmanydifficulties。Anexaminationofalltherelevantfactsseemstoleadtodivergentconclusions。Moreover，wehaveinheritednothingfrompreviousthinkers，whetherinthewayofastatementofdifficultiesorofasolution。
Theexistenceofplaceisheldtobeobviousfromthefactofmutualreplacement。Wherewaternowis，thereinturn，whenthewaterhasgoneoutasfromavessel，airispresent。Whenthereforeanotherbodyoccupiesthissameplace，theplaceisthoughttobedifferentfromallthebodieswhichcometobeinitandreplaceoneanother。Whatnowcontainsairformerlycontainedwater，sothatclearlytheplaceorspaceintowhichandoutofwhichtheypassedwassomethingdifferentfromboth。
Further，thetypicallocomotionsoftheelementarynaturalbodies-namely，fire，earth，andthelike-shownotonlythatplaceissomething，butalsothatitexertsacertaininfluence。Eachiscarriedtoitsownplace，ifitisnothindered，theoneup，theotherdown。Nowtheseareregionsorkindsofplace-upanddownandtherestofthesixdirections。Nordosuchdistinctionsupanddownandrightandleft，&c。holdonlyinrelationtous。Toustheyarenotalwaysthesamebutchangewiththedirectioninwhichweareturned:thatiswhythesamethingmaybebothrightandleft，upanddown，beforeandbehind。Butinnatureeachisdistinct，takenapartbyitself。Itisnoteverychancedirectionwhichis\'up\'，butwherefireandwhatislightarecarried；similarly，too，\'down\'isnotanychancedirectionbutwherewhathasweightandwhatismadeofeartharecarried-theimplicationbeingthattheseplacesdonotdiffermerelyinrelativeposition，butalsoaspossessingdistinctpotencies。Thisismadeplainalsobytheobjectsstudiedbymathematics。Thoughtheyhavenorealplace，theynevertheless，inrespectoftheirpositionrelativelytous，havearightandleftasattributesascribedtothemonlyinconsequenceoftheirrelativeposition，nothavingbynaturethesevariouscharacteristics。Again，thetheorythatthevoidexistsinvolvestheexistenceofplace:foronewoulddefinevoidasplacebereftofbody。
Theseconsiderationsthenwouldleadustosupposethatplaceissomethingdistinctfrombodies，andthateverysensiblebodyisinplace。Hesiodtoomightbeheldtohavegivenacorrectaccountofitwhenhemadechaosfirst。Atleasthesays:
\'Firstofallthingscamechaostobeing，thenbroad-breastedearth，\'
implyingthatthingsneedtohavespacefirst，becausehethought，withmostpeople，thateverythingissomewhereandinplace。Ifthisisitsnature，thepotencyofplacemustbeamarvellousthing，andtakeprecedenceofallotherthings。Forthatwithoutwhichnothingelsecanexist，whileitcanexistwithouttheothers，mustneedsbefirst；forplacedoesnotpassoutofexistencewhenthethingsinitareannihilated。
True，butevenifwesupposeitsexistencesettled，thequestionofitsnaturepresentsdifficulty-whetheritissomesortof\'bulk\'ofbodyorsomeentityotherthanthat，forwemustfirstdetermineitsgenus。
1Nowithasthreedimensions，length，breadth，depth，thedimensionsbywhichallbodyalsoisbounded。Buttheplacecannotbebody；forifitweretherewouldbetwobodiesinthesameplace。
2Further，ifbodyhasaplaceandspace，clearlysotoohavesurfaceandtheotherlimitsofbody；forthesamestatementwillapplytothem:wheretheboundingplanesofthewaterwere，thereinturnwillbethoseoftheair。Butwhenwecometoapointwecannotmakeadistinctionbetweenitanditsplace。Henceiftheplaceofapointisnotdifferentfromthepoint，nomorewillthatofanyoftheothersbedifferent，andplacewillnotbesomethingdifferentfromeachofthem。
3Whatintheworldthenarewetosupposeplacetobe?Ifithasthesortofnaturedescribed，itcannotbeanelementorcomposedofelements，whetherthesebecorporealorincorporeal:forwhileithassize，ithasnotbody。Buttheelementsofsensiblebodiesarebodies，whilenothingthathassizeresultsfromacombinationofintelligibleelements。
4Alsowemayask:ofwhatinthingsisspacethecause?Noneofthefourmodesofcausationcanbeascribedtoit。Itisneitherinthesenseofthematterofexistentsfornothingiscomposedofit，norastheformanddefinitionofthings，norasend，nordoesitmoveexistents。
5Further，too，ifitisitselfanexistent，wherewillitbe?
Zeno\'sdifficultydemandsanexplanation:forifeverythingthatexistshasaplace，placetoowillhaveaplace，andsoonadinfinitum。
6Again，justaseverybodyisinplace，so，too，everyplacehasabodyinit。Whatthenshallwesayaboutgrowingthings?Itfollowsfromthesepremissesthattheirplacemustgrowwiththem，iftheirplaceisneitherlessnorgreaterthantheyare。
Byaskingthesequestions，then，wemustraisethewholeproblemaboutplace-notonlyastowhatitis，butevenwhetherthereissuchathing。