第5章

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Sinceeverythingthatchangeschangesfromsomethingtosomething，thatwhichhaschangedmustatthemomentwhenithasfirstchangedbeinthattowhichithaschanged。Forthatwhichchangesretiresfromorleavesthatfromwhichitchanges:andleaving，ifnotidenticalwithchanging，isatanyrateaconsequenceofit。Andifleavingisaconsequenceofchanging，havingleftisaconsequenceofhavingchanged:forthereisalikerelationbetweenthetwoineachcase。

Onekindofchange，then，beingchangeinarelationofcontradiction，whereathinghaschangedfromnot-beingtobeingithasleftnot-being。Thereforeitwillbeinbeing:foreverythingmusteitherbeornotbe。Itisevident，then，thatincontradictorychangethatwhichhaschangedmustbeinthattowhichithaschanged。Andifthisistrueinthiskindofchange，itwillbetrueinallotherkindsaswell:forinthismatterwhatholdsgoodinthecaseofonewillholdgoodlikewiseinthecaseoftherest。

Moreover，ifwetakeeachkindofchangeseparately，thetruthofourconclusionwillbeequallyevident，onthegroundthatthatthatwhichhaschangedmustbesomewhereorinsomething。For，sinceithasleftthatfromwhichithaschangedandmustbesomewhere，itmustbeeitherinthattowhichithaschangedorinsomethingelse。If，then，thatwhichhaschangedtoBisinsomethingotherthanB，sayG，itmustagainbechangingfromGtoB:foritcannotbeassumedthatthereisnointervalbetweenGandB，sincechangeiscontinuous。Thuswehavetheresultthatthethingthathaschanged，atthemomentwhenithaschanged，ischangingtothattowhichithaschanged，whichisimpossible:thatwhichhaschanged，therefore，mustbeinthattowhichithaschanged。Soitisevidentlikewisethatthatthatwhichhascometobe，atthemomentwhenithascometobe，willbe，andthatwhichhasceasedtobewillnot-be:forwhatwehavesaidappliesuniversallytoeverykindofchange，anditstruthismostobviousinthecaseofcontradictorychange。Itisclear，then，thatthatwhichhaschanged，atthemomentwhenithasfirstchanged，isinthattowhichithaschanged。

Wewillnowshowthatthe"primarywhen"inwhichthatwhichhaschangedeffectedthecompletionofitschangemustbeindivisible，whereby"primary"Imeanpossessingthecharacteristicsinquestionofitselfandnotinvirtueofthepossessionofthembysomethingelsebelongingtoit。ForletAGbedivisible，andletitbedividedatB。IfthenthecompletionofchangehasbeeneffectedinABoragaininBG，AGcannotbetheprimarythinginwhichthecompletionofchangehasbeeneffected。If，ontheotherhand，ithasbeenchanginginbothABandBG(foritmusteitherhavechangedorbechangingineachofthem)，itmusthavebeenchanginginthewholeAG:

butourassumptionwasthatAGcontainsonlythecompletionofthechange。ItisequallyimpossibletosupposethatonepartofAG

containstheprocessandtheotherthecompletionofthechange:forthenweshallhavesomethingpriortowhatisprimary。Sothatinwhichthecompletionofchangehasbeeneffectedmustbeindivisible。Itisalsoevident，therefore，thatthatthatinwhichthatwhichhasceasedtobehasceasedtobeandthatinwhichthatwhichhascometobehascometobeareindivisible。

Buttherearetwosensesoftheexpression"theprimarywheninwhichsomethinghaschanged"。Ontheonehanditmaymeantheprimarywhencontainingthecompletionoftheprocessofchange-themomentwhenitiscorrecttosay"ithaschanged":ontheotherhanditmaymeantheprimarywhencontainingthebeginningoftheprocessofchange。Nowtheprimarywhenthathasreferencetotheendofthechangeissomethingreallyexistent:forachangemayreallybecompleted，andthereissuchathingasanendofchange，whichwehaveinfactshowntobeindivisiblebecauseitisalimit。Butthatwhichhasreferencetothebeginningisnotexistentatall:forthereisnosuchthingasabeginningofaprocessofchange，andthetimeoccupiedbythechangedoesnotcontainanyprimarywheninwhichthechangebegan。ForsupposethatADissuchaprimarywhen。Thenitcannotbeindivisible:for，ifitwere，themomentimmediatelyprecedingthechangeandthemomentinwhichthechangebeginswouldbeconsecutive(andmomentscannotbeconsecutive)。Again，ifthechangingthingisatrestinthewholeprecedingtimeGA(forwemaysupposethatitisatrest)，itisatrestinAalso:soifADiswithoutparts，itwillsimultaneouslybeatrestandhavechanged:foritisatrestinAandhaschangedinD。SincethenADisnotwithoutparts，itmustbedivisible，andthechangingthingmusthavechangedineverypartofit(forifithaschangedinneitherofthetwopartsintowhichADisdivided，ithasnotchangedinthewholeeither:if，ontheotherhand，itisinprocessofchangeinbothparts，itislikewiseinprocessofchangeinthewhole:andif，again，ithaschangedinoneofthetwoparts，thewholeisnottheprimarywheninwhichithaschanged:itmustthereforehavechangedineverypart)。Itisevident，then，thatwithreferencetothebeginningofchangethereisnoprimarywheninwhichchangehasbeeneffected:forthedivisionsareinfinite。

So，too，ofthatwhichhaschangedthereisnoprimarypartthathaschanged。ForsupposethatofAEtheprimarypartthathaschangedisAZ(everythingthatchangeshavingbeenshowntobedivisible):andletOIbethetimeinwhichDZhaschanged。If，then，inthewholetimeDZhaschanged，inhalfthetimetherewillbeapartthathaschanged，lessthanandthereforepriortoDZ:andagaintherewillbeanotherpartpriortothis，andyetanother，andsoontoinfinity。

Thusofthatwhichchangestherecannotbeanyprimarypartthathaschanged。Itisevident，then，fromwhathasbeensaid，thatneitherofthatwhichchangesnorofthetimeinwhichitchangesisthereanyprimarypart。

Withregard，however，totheactualsubjectofchange-thatistosaythatinrespectofwhichathingchanges-thereisadifferencetobeobserved。Forinaprocessofchangewemaydistinguishthreeterms-thatwhichchanges，thatinwhichitchanges，andtheactualsubjectofchange，e。g。theman，thetime，andthefaircomplexion。Ofthesethemanandthetimearedivisible:butwiththefaircomplexionitisotherwise(thoughtheyarealldivisibleaccidentally，forthatinwhichthefaircomplexionoranyotherqualityisanaccidentisdivisible)。Forofactualsubjectsofchangeitwillbeseenthatthosewhichareclassedasessentially，notaccidentally，divisiblehavenoprimarypart。Takethecaseofmagnitudes:letABbeamagnitude，andsupposethatithasmovedfromBtoaprimary"where"

G。ThenifBGistakentobeindivisible，twothingswithoutpartswillhavetobecontiguous(whichisimpossible):ifontheotherhanditistakentobedivisible，therewillbesomethingpriortoGtowhichthemagnitudehaschanged，andsomethingelseagainpriortothat，andsoontoinfinity，becausetheprocessofdivisionmaybecontinuedwithoutend。Thustherecanbenoprimary"where"towhichathinghaschanged。Andifwetakethecaseofquantitativechange，weshallgetalikeresult，forheretoothechangeisinsomethingcontinuous。Itisevident，then，thatonlyinqualitativemotioncantherebeanythingessentiallyindivisible。

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Noweverythingthatchangeschangestime，andthatintwosenses:

forthetimeinwhichathingissaidtochangemaybetheprimarytime，orontheotherhanditmayhaveanextendedreference，ase。g。whenwesaythatathingchangesinaparticularyearbecauseitchangesinaparticularday。Thatbeingso，thatwhichchangesmustbechanginginanypartoftheprimarytimeinwhichitchanges。

Thisisclearfromourdefinitionof"primary"，inwhichthewordissaidtoexpressjustthis:itmayalso，however，bemadeevidentbythefollowingargument。LetChRhbetheprimarytimeinwhichthatwhichisinmotionisinmotion:and(asalltimeisdivisible)letitbedividedatK。NowinthetimeChKiteitherisinmotionorisnotinmotion，andthesameislikewisetrueofthetimeKRh。Thenifitisinmotioninneitherofthetwoparts，itwillbeatrestinthewhole:foritisimpossiblethatitshouldbeinmotioninatimeinnopartofwhichitisinmotion。Ifontheotherhanditisinmotioninonlyoneofthetwopartsofthetime，ChRhcannotbetheprimarytimeinwhichitisinmotion:foritsmotionwillhavereferencetoatimeotherthanChRh。Itmust，then，havebeeninmotioninanypartofChRh。

Andnowthatthishasbeenproved，itisevidentthateverythingthatisinmotionmusthavebeeninmotionbefore。ForifthatwhichisinmotionhastraversedthedistanceKLintheprimarytimeChRh，inhalfthetimeathingthatisinmotionwithequalvelocityandbeganitsmotionatthesametimewillhavetraversedhalfthedistance。Butifthissecondthingwhosevelocityisequalhastraversedacertaindistanceinacertaintime，theoriginalthingthatisinmotionmusthavetraversedthesamedistanceinthesametime。Hencethatwhichisinmotionmusthavebeeninmotionbefore。

Again，ifbytakingtheextrememomentofthetime-foritisthemomentthatdefinesthetime，andtimeisthatwhichisintermediatebetweenmoments-weareenabledtosaythatmotionhastakenplaceinthewholetimeChRhorinfactinanyperiodofit，motionmaylikewisebesaidtohavetakenplaceineveryothersuchperiod。Buthalfthetimefindsanextremeinthepointofdivision。Thereforemotionwillhavetakenplaceinhalfthetimeandinfactinanypartofit:forassoonasanydivisionismadethereisalwaysatimedefinedbymoments。If，then，alltimeisdivisible，andthatwhichisintermediatebetweenmomentsistime，everythingthatischangingmusthavecompletedaninfinitenumberofchanges。

Again，sinceathingthatchangescontinuouslyandhasnotperishedorceasedfromitschangemusteitherbechangingorhavechangedinanypartofthetimeofitschange，andsinceitcannotbechanginginamoment，itfollowsthatitmusthavechangedateverymomentinthetime:consequently，sincethemomentsareinfiniteinnumber，everythingthatischangingmusthavecompletedaninfinitenumberofchanges。

Andnotonlymustthatwhichischanginghavechanged，butthatwhichhaschangedmustalsopreviouslyhavebeenchanging，sinceeverythingthathaschangedfromsomethingtosomethinghaschangedinaperiodoftime。ForsupposethatathinghaschangedfromAtoB

inamoment。NowthemomentinwhichithaschangedcannotbethesameasthatinwhichitisatA(sinceinthatcaseitwouldbeinAandB

atonce):forwehaveshownabovethatthatthatwhichhaschanged，whenithaschanged，isnotinthatfromwhichithaschanged。If，ontheotherhand，itisadifferentmoment，therewillbeaperiodoftimeintermediatebetweenthetwo:for，aswesaw，momentsarenotconsecutive。Since，then，ithaschangedinaperiodoftime，andalltimeisdivisible，inhalfthetimeitwillhavecompletedanotherchange，inaquarteranother，andsoontoinfinity:consequentlywhenithaschanged，itmusthavepreviouslybeenchanging。

Moreover，thetruthofwhathasbeensaidismoreevidentinthecaseofmagnitude，becausethemagnitudeoverwhichwhatischangingchangesiscontinuous。ForsupposethatathinghaschangedfromG

toD。ThenifGDisindivisible，twothingswithoutpartswillbeconsecutive。Butsincethisisimpossible，thatwhichisintermediatebetweenthemmustbeamagnitudeanddivisibleintoaninfinitenumberofsegments:consequently，beforethechangeiscompleted，thethingchangestothosesegments。Everythingthathaschanged，therefore，mustpreviouslyhavebeenchanging:forthesameproofalsoholdsgoodofchangewithrespecttowhatisnotcontinuous，changes，thatistosay，betweencontrariesandbetweencontradictories。Insuchcaseswehaveonlytotakethetimeinwhichathinghaschangedandagainapplythesamereasoning。Sothatwhichhaschangedmusthavebeenchangingandthatwhichischangingmusthavechanged，andaprocessofchangeisprecededbyacompletionofchangeandacompletionbyaprocess:andwecannevertakeanystageandsaythatitisabsolutelythefirst。Thereasonofthisisthatnotwothingswithoutpartscanbecontiguous，andthereforeinchangetheprocessofdivisionisinfinite，justaslinesmaybeinfinitelydividedsothatonepartiscontinuallyincreasingandtheothercontinuallydecreasing。

Soitisevidentalsothatthatthatwhichhasbecomemustpreviouslyhavebeeninprocessofbecoming，andthatwhichisinprocessofbecomingmustpreviouslyhavebecome，everything(thatis)thatisdivisibleandcontinuous:thoughitisnotalwaystheactualthingthatisinprocessofbecomingofwhichthisistrue:

sometimesitissomethingelse，thatistosay，somepartofthethinginquestion，e。g。thefoundation-stoneofahouse。So，too，inthecaseofthatwhichisperishingandthatwhichhasperished:forthatwhichbecomesandthatwhichperishesmustcontainanelementofinfinitenessasanimmediateconsequenceofthefactthattheyarecontinuousthings:andsoathingcannotbeinprocessofbecomingwithouthavingbecomeorhavebecomewithouthavingbeeninprocessofbecoming。So，too，inthecaseofperishingandhavingperished:

perishingmustbeprecededbyhavingperished，andhavingperishedmustbeprecededbyperishing。Itisevident，then，thatthatwhichhasbecomemustpreviouslyhavebeeninprocessofbecoming，andthatwhichisinprocessofbecomingmustpreviouslyhavebecome:

forallmagnitudesandallperiodsoftimeareinfinitelydivisible。

Consequentlynoabsolutelyfirststageofchangecanberepresentedbyanyparticularpartofspaceortimewhichthechangingthingmayoccupy。

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Nowsincethemotionofeverythingthatisinmotionoccupiesaperiodoftime，andagreatermagnitudeistraversedinalongertime，itisimpossiblethatathingshouldundergoafinitemotioninaninfinitetime，ifthisisunderstoodtomeannotthatthesamemotionorapartofitiscontinuallyrepeated，butthatthewholeinfinitetimeisoccupiedbythewholefinitemotion。Inallcaseswhereathingisinmotionwithuniformvelocityitisclearthatthefinitemagnitudeistraversedinafinitetime。Forifwetakeapartofthemotionwhichshallbeameasureofthewhole，thewholemotioniscompletedinasmanyequalperiodsofthetimeastherearepartsofthemotion。Consequently，sincethesepartsarefinite，bothinsizeindividuallyandinnumbercollectively，thewholetimemustalsobefinite:foritwillbeamultipleoftheportion，equaltothetimeoccupiedincompletingtheaforesaidpartmultipliedbythenumberoftheparts。

Butitmakesnodifferenceevenifthevelocityisnotuniform。

ForletussupposethatthelineABrepresentsafinitestretchoverwhichathinghasbeenmovedinthegiventime，andletGDbetheinfinitetime。Nowifonepartofthestretchmusthavebeentraversedbeforeanotherpart(thisisclear，thatintheearlierandinthelaterpartofthetimeadifferentpartofthestretchhasbeentraversed:forasthetimelengthensadifferentpartofthemotionwillalwaysbecompletedinit，whetherthethinginmotionchangeswithuniformvelocityornot:andwhethertherateofmotionincreasesordiminishesorremainsstationarythisisnonethelessso)，letusthentakeAEapartofthewholestretchofmotionABwhichshallbeameasureofAB。Nowthispartofthemotionoccupiesacertainperiodoftheinfinitetime:itcannotitselfoccupyaninfinitetime，forweareassumingthatthatisoccupiedbythewholeAB。AndifagainItakeanotherpartequaltoAE，thatalsomustoccupyafinitetimeinconsequenceofthesameassumption。AndifIgoontakingpartsinthisway，ontheonehandthereisnopartwhichwillbeameasureoftheinfinitetime(fortheinfinitecannotbecomposedoffinitepartswhetherequalorunequal，becausetheremustbesomeunitywhichwillbeameasureofthingsfiniteinmultitudeorinmagnitude，which，whethertheyareequalorunequal，arenonethelesslimitedinmagnitude)；whileontheotherhandthefinitestretchofmotionABisacertainmultipleofAE:

consequentlythemotionABmustbeaccomplishedinafinitetime。

Moreoveritisthesamewithcomingtorestaswithmotion。Andsoitisimpossibleforoneandthesamethingtobeinfinitelyinprocessofbecomingorofperishing。Thereasoninghewillprovethatinafinitetimetherecannotbeaninfiniteextentofmotionorofcomingtorest，whetherthemotionisregularorirregular。

Forifwetakeapartwhichshallbeameasureofthewholetime，inthispartacertainfraction，notthewhole，ofthemagnitudewillbetraversed，becauseweassumethatthetraversingofthewholeoccupiesallthetime。Again，inanotherequalpartofthetimeanotherpartofthemagnitudewillbetraversed:andsimilarlyineachpartofthetimethatwetake，whetherequalorunequaltothepartoriginallytaken。Itmakesnodifferencewhetherthepartsareequalornot，ifonlyeachisfinite:foritisclearthatwhilethetimeisexhaustedbythesubtractionofitsparts，theinfinitemagnitudewillnotbethusexhausted，sincetheprocessofsubtractionisfinitebothinrespectofthequantitysubtractedandofthenumberoftimesasubtractionismade。Consequentlytheinfinitemagnitudewillnotbetraversedinfinitetime:anditmakesnodifferencewhetherthemagnitudeisinfiniteinonlyonedirectionorinboth:forthesamereasoningwillholdgood。

Thishavingbeenproved，itisevidentthatneithercanafinitemagnitudetraverseaninfinitemagnitudeinafinitetime，thereasonbeingthesameasthatgivenabove:inpartofthetimeitwilltraverseafinitemagnitudeandineachseveralpartlikewise，sothatinthewholetimeitwilltraverseafinitemagnitude。

Andsinceafinitemagnitudewillnottraverseaninfiniteinafinitetime，itisclearthatneitherwillaninfinitetraverseafiniteinafinitetime。Foriftheinfinitecouldtraversethefinite，thefinitecouldtraversetheinfinite；foritmakesnodifferencewhichofthetwoisthethinginmotion；eithercaseinvolvesthetraversingoftheinfinitebythefinite。ForwhentheinfinitemagnitudeAisinmotionapartofit，sayGD，willoccupythefiniteandthenanother，andthenanother，andsoontoinfinity。Thusthetworesultswillcoincide:theinfinitewillhavecompletedamotionoverthefiniteandthefinitewillhavetraversedtheinfinite:foritwouldseemtobeimpossibleforthemotionoftheinfiniteoverthefinitetooccurinanywayotherthanbythefinitetraversingtheinfiniteeitherbylocomotionoveritorbymeasuringit。Therefore，sincethisisimpossible，theinfinitecannottraversethefinite。

Noragainwilltheinfinitetraversetheinfiniteinafinitetime。Otherwiseitwouldalsotraversethefinite，fortheinfiniteincludesthefinite。Wecanfurtherprovethisinthesamewaybytakingthetimeasourstarting-point。

Since，then，itisestablishedthatinafinitetimeneitherwillthefinitetraversetheinfinite，northeinfinitethefinite，northeinfinitetheinfinite，itisevidentalsothatinafinitetimetherecannotbeinfinitemotion:forwhatdifferencedoesitmakewhetherwetakethemotionorthemagnitudetobeinfinite?Ifeitherofthetwoisinfinite，theothermustbesolikewise:foralllocomotionisinspace。

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Sinceeverythingtowhichmotionorrestisnaturalisinmotionoratrestinthenaturaltime，place，andmanner，thatwhichiscomingtoastand，whenitiscomingtoastand，mustbeinmotion:

forifitisnotinmotionitmustbeatrest:butthatwhichisatrestcannotbecomingtorest。Fromthisitevidentlyfollowsthatcomingtoastandmustoccupyaperiodoftime:forthemotionofthatwhichisinmotionoccupiesaperiodoftime，andthatwhichiscomingtoastandhasbeenshowntobeinmotion:consequentlycomingtoastandmustoccupyaperiodoftime。

Again，sincetheterms"quicker"and"slower"areusedonlyofthatwhichoccupiesaperiodoftime，andtheprocessofcomingtoastandmaybequickerorslower，thesameconclusionfollows。

Andthatwhichiscomingtoastandmustbecomingtoastandinanypartoftheprimarytimeinwhichitiscomingtoastand。Forifitiscomingtoastandinneitheroftwopartsintowhichthetimemaybedivided，itcannotbecomingtoastandinthewholetime，withtheresultthatthatthatwhichiscomingtoastandwillnotbecomingtoastand。Ifontheotherhanditiscomingtoastandinonlyoneofthetwopartsofthetime，thewholecannotbetheprimarytimeinwhichitiscomingtoastand:foritiscomingtoastandinthewholetimenotprimarilybutinvirtueofsomethingdistinctfromitself，theargumentbeingthesameasthatwhichweusedaboveaboutthingsinmotion。

Andjustasthereisnoprimarytimeinwhichthatwhichisinmotionisinmotion，sotoothereisnoprimarytimeinwhichthatwhichiscomingtoastandiscomingtoastand，therebeingnoprimarystageeitherofbeinginmotionorofcomingtoastand。ForletABbetheprimarytimeinwhichathingiscomingtoastand。

NowABcannotbewithoutparts:fortherecannotbemotioninthatwhichiswithoutparts，becausethemovingthingwouldnecessarilyhavebeenalreadymovedforpartofthetimeofitsmovement:andthatwhichiscomingtoastandhasbeenshowntobeinmotion。ButsinceABisthereforedivisible，thethingiscomingtoastandineveryoneofthepartsofAB:forwehaveshownabovethatitiscomingtoastandineveryoneofthepartsinwhichitisprimarilycomingtoastand。Sincethen，thatinwhichprimarilyathingiscomingtoastandmustbeaperiodoftimeandnotsomethingindivisible，andsincealltimeisinfinitelydivisible，therecannotbeanythinginwhichprimarilyitiscomingtoastand。

Noragaincantherebeaprimarytimeatwhichthebeingatrestofthatwhichisatrestoccurred:foritcannothaveoccurredinthatwhichhasnoparts，becausetherecannotbemotioninthatwhichisindivisible，andthatinwhichresttakesplaceisthesameasthatinwhichmotiontakesplace:forwedefinedastateofresttobethestateofathingtowhichmotionisnaturalbutwhichisnotinmotionwhen(thatistosayinthatinwhich)motionwouldbenaturaltoit。Again，ouruseofthephrase"beingatrest"alsoimpliesthatthepreviousstateofathingisstillunaltered，notonepointonlybuttwoatleastbeingthusneededtodetermineitspresence:consequentlythatinwhichathingisatrestcannotbewithoutparts。Since，thenitisdivisible，itmustbeaperiodoftime，andthethingmustbeatrestineveryoneofitsparts，asmaybeshownbythesamemethodasthatusedaboveinsimilardemonstrations。

Sotherecanbenoprimarypartofthetime:andthereasonisthatrestandmotionarealwaysinaperiodoftime，andaperiodoftimehasnoprimarypartanymorethanamagnitudeorinfactanythingcontinuous:foreverythingcontinuousisdivisibleintoaninfinitenumberofparts。

Andsinceeverythingthatisinmotionisinmotioninaperiodoftimeandchangesfromsomethingtosomething，whenitsmotioniscomprisedwithinaparticularperiodoftimeessentially-thatistosaywhenitfillsthewholeandnotmerelyapartofthetimeinquestion-itisimpossiblethatinthattimethatwhichisinmotionshouldbeoveragainstsomeparticularthingprimarily。Forifathing-itselfandeachofitsparts-occupiesthesamespaceforadefiniteperiodoftime，itisatrest:foritisinjustthesecircumstancesthatweusetheterm"beingatrest"-whenatonemomentafteranotheritcanbesaidwithtruththatathing，itselfanditsparts，occupiesthesamespace。Soifthisisbeingatrestitisimpossibleforthatwhichischangingtobeasawhole，atthetimewhenitisprimarilychanging，overagainstanyparticularthing(forthewholeperiodoftimeisdivisible)，sothatinonepartofitafteranotheritwillbetruetosaythatthething，itselfanditsparts，occupiesthesamespace。Ifthisisnotsoandtheaforesaidpropositionistrueonlyatasinglemoment，thenthethingwillbeoveragainstaparticularthingnotforanyperiodoftimebutonlyatamomentthatlimitsthetime。Itistruethatatanymomentitisalwaysoveragainstsomethingstationary:butitisnotatrest:foratamomentitisnotpossibleforanythingtobeeitherinmotionoratrest。Sowhileitistruetosaythatthatwhichisinmotionisatamomentnotinmotionandisoppositesomeparticularthing，itcannotinaperiodoftimebeoveragainstthatwhichisatrest:

forthatwouldinvolvetheconclusionthatthatwhichisinlocomotionisatrest。

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Zeno"sreasoning，however，isfallacious，whenhesaysthatifeverythingwhenitoccupiesanequalspaceisatrest，andifthatwhichisinlocomotionisalwaysoccupyingsuchaspaceatanymoment，theflyingarrowisthereforemotionless。Thisisfalse，fortimeisnotcomposedofindivisiblemomentsanymorethananyothermagnitudeiscomposedofindivisibles。

Zeno"sargumentsaboutmotion，whichcausesomuchdisquietudetothosewhotrytosolvetheproblemsthattheypresent，arefourinnumber。Thefirstassertsthenon-existenceofmotiononthegroundthatthatwhichisinlocomotionmustarriveatthehalf-waystagebeforeitarrivesatthegoal。Thiswehavediscussedabove。

Thesecondistheso-called"Achilles"，anditamountstothis，thatinaracethequickestrunnercanneverovertaketheslowest，sincethepursuermustfirstreachthepointwhencethepursuedstarted，sothattheslowermustalwaysholdalead。Thisargumentisthesameinprincipleasthatwhichdependsonbisection，thoughitdiffersfromitinthatthespaceswithwhichwesuccessivelyhavetodealarenotdividedintohalves。Theresultoftheargumentisthattheslowerisnotovertaken:butitproceedsalongthesamelinesasthebisection-argument(forinbothadivisionofthespaceinacertainwayleadstotheresultthatthegoalisnotreached，thoughthe"Achilles"goesfurtherinthatitaffirmsthateventhequickestrunnerinlegendarytraditionmustfailinhispursuitoftheslowest)，sothatthesolutionmustbethesame。Andtheaxiomthatthatwhichholdsaleadisneverovertakenisfalse:itisnotovertaken，itistrue，whileitholdsalead:butitisovertakenneverthelessifitisgrantedthatittraversesthefinitedistanceprescribed。Thesethenaretwoofhisarguments。

Thethirdisthatalreadygivenabove，totheeffectthattheflyingarrowisatrest，whichresultfollowsfromtheassumptionthattimeiscomposedofmoments:ifthisassumptionisnotgranted，theconclusionwillnotfollow。

Thefourthargumentisthatconcerningthetworowsofbodies，eachrowbeingcomposedofanequalnumberofbodiesofequalsize，passingeachotheronarace-courseastheyproceedwithequalvelocityinoppositedirections，theoneroworiginallyoccupyingthespacebetweenthegoalandthemiddlepointofthecourseandtheotherthatbetweenthemiddlepointandthestarting-post。This，hethinks，involvestheconclusionthathalfagiventimeisequaltodoublethattime。Thefallacyofthereasoningliesintheassumptionthatabodyoccupiesanequaltimeinpassingwithequalvelocityabodythatisinmotionandabodyofequalsizethatisatrest；whichisfalse。Forinstance(sorunstheargument)，letA，A……bethestationarybodiesofequalsize，B，B……thebodies，equalinnumberandinsizetoA，A……，originallyoccupyingthehalfofthecoursefromthestarting-posttothemiddleoftheA"s，andG，G……thoseoriginallyoccupyingtheotherhalffromthegoaltothemiddleoftheA"s，equalinnumber，size，andvelocitytoB，B……Thenthreeconsequencesfollow:

First，astheB"sandtheG"spassoneanother，thefirstB

reachesthelastGatthesamemomentasthefirstGreachesthelastB。SecondlyatthismomentthefirstGhaspassedalltheA"s，whereasthefirstBhaspassedonlyhalftheA"s，andhasconsequentlyoccupiedonlyhalfthetimeoccupiedbythefirstG，sinceeachofthetwooccupiesanequaltimeinpassingeachA。Thirdly，atthesamemomentalltheB"shavepassedalltheG"s:forthefirstGandthefirstBwillsimultaneouslyreachtheoppositeendsofthecourse，since(sosaysZeno)thetimeoccupiedbythefirstGinpassingeachoftheB"sisequaltothatoccupiedbyitinpassingeachoftheA"s，becauseanequaltimeisoccupiedbyboththefirstBandthefirstGinpassingalltheA"s。Thisistheargument，butitpresupposedtheaforesaidfallaciousassumption。

Norinreferencetocontradictorychangeshallwefindanythingunanswerableintheargumentthatifathingischangingfromnot-white，say，towhite，andisinneithercondition，thenitwillbeneitherwhitenornot-white:forthefactthatitisnotwhollyineitherconditionwillnotprecludeusfromcallingitwhiteornot-white。Wecallathingwhiteornot-whitenotnecessarilybecauseitisbeoneortheother，butcausemostofitspartsorthemostessentialpartsofitareso:notbeinginacertainconditionisdifferentfromnotbeingwhollyinthatcondition。So，too，inthecaseofbeingandnot-beingandallotherconditionswhichstandinacontradictoryrelation:whilethechangingthingmustofnecessitybeinoneofthetwoopposites，itisneverwhollyineither。

Again，inthecaseofcirclesandspheresandeverythingwhosemotionisconfinedwithinthespacethatitoccupies，itisnottruetosaythemotioncanbenothingbutrest，onthegroundthatsuchthingsinmotion，themselvesandtheirparts，willoccupythesamepositionforaperiodoftime，andthatthereforetheywillbeatonceatrestandinmotion。Forinthefirstplacethepartsdonotoccupythesamepositionforanyperiodoftime:andinthesecondplacethewholealsoisalwayschangingtoadifferentposition:forifwetaketheorbitasdescribedfromapointAonacircumference，itwillnotbethesameastheorbitasdescribedfromBorGoranyotherpointonthesamecircumferenceexceptinanaccidentalsense，thesensethatistosayinwhichamusicalmanisthesameasaman。Thusoneorbitisalwayschangingintoanother，andthethingwillneverbeatrest。Anditisthesamewiththesphereandeverythingelsewhosemotionisconfinedwithinthespacethatitoccupies。

10

Ournextpointisthatthatwhichiswithoutpartscannotbeinmotionexceptaccidentally:i。e。itcanbeinmotiononlyinsofarasthebodyorthemagnitudeisinmotionandthepartlessisinmotionbyinclusiontherein，justasthatwhichisinaboatmaybeinmotioninconsequenceofthelocomotionoftheboat，orapartmaybeinmotioninvirtueofthemotionofthewhole。(Itmustberemembered，however，thatby"thatwhichiswithoutparts"Imeanthatwhichisquantitativelyindivisible(andthatthecaseofthemotionofapartisnotexactlyparallel):forpartshavemotionsbelongingessentiallyandseverallytothemselvesdistinctfromthemotionofthewhole。Thedistinctionmaybeseenmostclearlyinthecaseofarevolvingsphere，inwhichthevelocitiesofthepartsnearthecentreandofthoseonthesurfacearedifferentfromoneanotherandfromthatofthewhole；thisimpliesthatthereisnotonemotionbutmany)。Aswehavesaid，then，thatwhichiswithoutpartscanbeinmotioninthesenseinwhichamansittinginaboatisinmotionwhentheboatistravelling，butitcannotbeinmotionofitself。ForsupposethatitischangingfromABtoBG-eitherfromonemagnitudetoanother，orfromoneformtoanother，orfromsomestatetoitscontradictory-andletDbetheprimarytimeinwhichitundergoesthechange。TheninthetimeinwhichitischangingitmustbeeitherinABorinBGorpartlyinoneandpartlyintheother:forthis，aswesaw，istrueofeverythingthatischanging。Nowitcannotbepartlyineachofthetwo:forthenitwouldbedivisibleintoparts。NoragaincanitbeinBG:forthenitwillhavecompletedthechange，whereastheassumptionisthatthechangeisinprocess。

Itremains，then，thatinthetimeinwhichitischanging，itisinAB。Thatbeingso，itwillbeatrest:for，aswesaw，tobeinthesameconditionforaperiodoftimeistobeatrest。Soitisnotpossibleforthatwhichhasnopartstobeinmotionortochangeinanyway:foronlyoneconditioncouldhavemadeitpossibleforittohavemotion，viz。thattimeshouldbecomposedofmoments，inwhichcaseatanymomentitwouldhavecompletedamotionorachange，sothatitwouldneverbeinmotion，butwouldalwayshavebeeninmotion。Butthiswehavealreadyshownabovetobeimpossible:timeisnotcomposedofmoments，justasalineisnotcomposedofpoints，andmotionisnotcomposedofstarts:forthistheorysimplymakesmotionconsistofindivisiblesinexactlythesamewayastimeismadetoconsistofmomentsoralengthofpoints。

Again，itmaybeshowninthefollowingwaythattherecanbenomotionofapointorofanyotherindivisible。Thatwhichisinmotioncannevertraverseaspacegreaterthanitselfwithoutfirsttraversingaspaceequaltoorlessthanitself。Thatbeingso，itisevidentthatthepointalsomustfirsttraverseaspaceequaltoorlessthanitself。Butsinceitisindivisible，therecanbenospacelessthanitselfforittotraversefirst:soitwillhavetotraverseadistanceequaltoitself。Thusthelinewillbecomposedofpoints，forthepoint，asitcontinuallytraversesadistanceequaltoitself，willbeameasureofthewholeline。Butsincethisisimpossible，itislikewiseimpossiblefortheindivisibletobeinmotion。

Again，sincemotionisalwaysinaperiodoftimeandneverinamoment，andalltimeisdivisible，foreverythingthatisinmotiontheremustbeatimelessthanthatinwhichittraversesadistanceasgreatasitself。Forthatinwhichitisinmotionwillbeatime，becauseallmotionisinaperiodoftime；andalltimehasbeenshownabovetobedivisible。Therefore，ifapointisinmotion，theremustbeatimelessthanthatinwhichithasitselftraversedanydistance。Butthisisimpossible，forinlesstimeitmusttraverselessdistance，andthustheindivisiblewillbedivisibleintosomethinglessthanitself，justasthetimeissodivisible:thefactbeingthattheonlyconditionunderwhichthatwhichiswithoutpartsandindivisiblecouldbeinmotionwouldhavebeenthepossibilityoftheinfinitelysmallbeinginmotioninamoment:forinthetwoquestions-thatofmotioninamomentandthatofmotionofsomethingindivisible-thesameprincipleisinvolved。

Ournextpointisthatnoprocessofchangeisinfinite:foreverychange，whetherbetweencontradictoriesorbetweencontraries，isachangefromsomethingtosomething。Thusincontradictorychangesthepositiveorthenegative，asthecasemaybe，isthelimit，e。g。

beingisthelimitofcomingtobeandnot-beingisthelimitofceasingtobe:andincontrarychangestheparticularcontrariesarethelimits，sincethesearetheextremepointsofanysuchprocessofchange，andconsequentlyofeveryprocessofalteration:foralterationisalwaysdependentuponsomecontraries。Similarlycontrariesaretheextremepointsofprocessesofincreaseanddecrease:thelimitofincreaseistobefoundinthecompletemagnitudepropertothepeculiarnatureofthethingthatisincreasing，whilethelimitofdecreaseisthecompletelossofsuchmagnitude。Locomotion，itistrue，wecannotshowtobefiniteinthisway，sinceitisnotalwaysbetweencontraries。Butsincethatwhichcannotbecut(inthesensethatitisinconceivablethatitshouldbecut，theterm"cannot"beingusedinseveralsenses)-sinceitisinconceivablethatthatwhichinthissensecannotbecutshouldbeinprocessofbeingcut，andgenerallythatthatwhichcannotcometobeshouldbeinprocessofcomingtobe，itfollowsthatitisinconceivablethatthatwhichcannotcompleteachangeshouldbeinprocessofchangingtothattowhichitcannotcompleteachange。

If，then，itistobeassumedthatthatwhichisinlocomotionisinprocessofchanging，itmustbecapableofcompletingthechange。

Consequentlyitsmotionisnotinfinite，anditwillnotbeinlocomotionoveraninfinitedistance，foritcannottraversesuchadistance。

Itisevident，then，thataprocessofchangecannotbeinfiniteinthesensethatitisnotdefinedbylimits。Butitremainstobeconsideredwhetheritispossibleinthesensethatoneandthesameprocessofchangemaybeinfiniteinrespectofthetimewhichitoccupies。Ifitisnotoneprocess，itwouldseemthatthereisnothingtopreventitsbeinginfiniteinthissense；e。g。ifaprocessoflocomotionbesucceededbyaprocessofalterationandthatbyaprocessofincreaseandthatagainbyaprocessofcomingtobe:inthiswaytheremaybemotionforeversofarasthetimeisconcerned，butitwillnotbeonemotion，becauseallthesemotionsdonotcomposeone。Ifitistobeoneprocess，nomotioncanbeinfiniteinrespectofthetimethatitoccupies，withthesingleexceptionofrotatorylocomotion。

BookVII

1

EVERYTHINGthatisinmotionmustbemovedbysomething。Forifithasnotthesourceofitsmotioninitselfitisevidentthatitismovedbysomethingotherthanitself，fortheremustbesomethingelsethatmovesit。Ifontheotherhandithasthesourceofitsmotioninitself，letABbetakentorepresentthatwhichisinmotionessentiallyofitselfandnotinvirtueofthefactthatsomethingbelongingtoitisinmotion。NowinthefirstplacetoassumethatAB，becauseitisinmotionasawholeandisnotmovedbyanythingexternaltoitself，isthereforemovedbyitself-thisisjustasif，supposingthatKLismovingLMandisalsoitselfinmotion，weweretodenythatKMismovedbyanythingonthegroundthatitisnotevidentwhichisthepartthatismovingitandwhichthepartthatismoved。Inthesecondplacethatwhichisinmotionwithoutbeingmovedbyanythingdoesnotnecessarilyceasefromitsmotionbecausesomethingelseisatrest，butathingmustbemovedbysomethingifthefactofsomethingelsehavingceasedfromitsmotioncausesittobeatrest。Thus，ifthisisaccepted，everythingthatisinmotionmustbemovedbysomething。ForAB，whichhasbeentakentorepresentthatwhichisinmotion，mustbedivisiblesinceeverythingthatisinmotionisdivisible。Letitbedivided，then，atG。NowifGBisnotinmotion，thenABwillnotbeinmotion:forifitis，itisclearthatAGwouldbeinmotionwhileBGisatrest，andthusABcannotbeinmotionessentiallyandprimarily。ButexhypothesiABisinmotionessentiallyandprimarily。ThereforeifGB

isnotinmotionABwillbeatrest。Butwehaveagreedthatthatwhichisatrestifsomethingelseisnotinmotionmustbemovedbysomething。Consequently，everythingthatisinmotionmustbemovedbysomething:forthatwhichisinmotionwillalwaysbedivisible，andifapartofitisnotinmotionthewholemustbeatrest。

Sinceeverythingthatisinmotionmustbemovedbysomething，letustakethecaseinwhichathingisinlocomotionandismovedbysomethingthatisitselfinmotion，andthatagainismovedbysomethingelsethatisinmotion，andthatbysomethingelse，andsooncontinually:thentheseriescannotgoontoinfinity，buttheremustbesomefirstmovent。Forletussupposethatthisisnotsoandtaketheseriestobeinfinite。LetAthenbemovedbyB，BbyG，GbyD，andsoon，eachmemberoftheseriesbeingmovedbythatwhichcomesnexttoit。Thensinceexhypothesithemoventwhilecausingmotionisalsoitselfinmotion，andthemotionofthemovedandthemotionofthemoventmustproceedsimultaneously(forthemoventiscausingmotionandthemovedisbeingmovedsimultaneously)itisevidentthattherespectivemotionsofA，B，G，andeachoftheothermovedmoventsaresimultaneous。LetustakethemotionofeachseparatelyandletEbethemotionofA，ZofB，andHandOrespectivelythemotionsofGandD:forthoughtheyareallmovedseverallyonebyanother，yetwemaystilltakethemotionofeachasnumericallyone，sinceeverymotionisfromsomethingtosomethingandisnotinfiniteinrespectofitsextremepoints。ByamotionthatisnumericallyoneImeanamotionthatproceedsfromsomethingnumericallyoneandthesametosomethingnumericallyoneandthesameinaperiodoftimenumericallyoneandthesame:foramotionmaybethesamegenerically，specifically，ornumerically:itisgenericallythesameifitbelongstothesamecategory，e。g。

substanceorquality:itisspecificallythesameifitproceedsfromsomethingspecificallythesametosomethingspecificallythesame，e。g。fromwhitetoblackorfromgoodtobad，whichisnotofakindspecificallydistinct:itisnumericallythesameifitproceedsfromsomethingnumericallyonetosomethingnumericallyoneinthesameperiodoftime，e。g。fromaparticularwhitetoaparticularblack，orfromaparticularplacetoaparticularplace，inaparticularperiodoftime:foriftheperiodoftimewerenotoneandthesame，themotionwouldnolongerbenumericallyonethoughitwouldstillbespecificallyone。

Wehavedealtwiththisquestionabove。NowletusfurthertakethetimeinwhichAhascompleteditsmotion，andletitberepresentedbyK。ThensincethemotionofAisfinitethetimewillalsobefinite。Butsincethemoventsandthethingsmovedareinfinite，themotionEZHO，i。e。themotionthatiscomposedofalltheindividualmotions，mustbeinfinite。ForthemotionsofA，B，andtheothersmaybeequal，orthemotionsoftheothersmaybegreater:

butassumingwhatisconceivable，wefindthatwhethertheyareequalorsomearegreater，inbothcasesthewholemotionisinfinite。

AndsincethemotionofAandthatofeachoftheothersaresimultaneous，thewholemotionmustoccupythesametimeasthemotionofA:butthetimeoccupiedbythemotionofAisfinite:consequentlythemotionwillbeinfiniteinafinitetime，whichisimpossible。

Itmightbethoughtthatwhatwesetouttoprovehasthusbeenshown，butourargumentsofardoesnotproveit，becauseitdoesnotyetprovethatanythingimpossibleresultsfromthecontrarysupposition:forinafinitetimetheremaybeaninfinitemotion，thoughnotofonething，butofmany:andinthecasethatweareconsideringthisisso:foreachthingaccomplishesitsownmotion，andthereisnoimpossibilityinmanythingsbeinginmotionsimultaneously。Butif(asweseetobeuniversallythecase)thatwhichprimarilyismovedlocallyandcorporeallymustbeeitherincontactwithorcontinuouswiththatwhichmovesit，thethingsmovedandthemoventsmustbecontinuousorincontactwithoneanother，sothattogethertheyallformasingleunity:whetherthisunityisfiniteorinfinitemakesnodifferencetoourpresentargument；forinanycasesincethethingsinmotionareinfiniteinnumberthewholemotionwillbeinfinite，if，asistheoreticallypossible，eachmotioniseitherequaltoorgreaterthanthatwhichfollowsitintheseries:forweshalltakeasactualthatwhichistheoreticallypossible。If，then，A，B，G，DformaninfinitemagnitudethatpassesthroughthemotionEZHOinthefinitetimeK，thisinvolvestheconclusionthataninfinitemotionispassedthroughinafinitetime:andwhetherthemagnitudeinquestionisfiniteorinfinitethisisineithercaseimpossible。Thereforetheseriesmustcometoanend，andtheremustbeafirstmoventandafirstmoved:forthefactthatthisimpossibilityresultsonlyfromtheassumptionofaparticularcaseisimmaterial，sincethecaseassumedistheoreticallypossible，andtheassumptionofatheoreticallypossiblecaseoughtnottogiverisetoanyimpossibleresult。

2

Thatwhichisthefirstmovementofathing-inthesensethatitsuppliesnot"thatforthesakeofwhich"butthesourceofthemotion-isalwaystogetherwiththatwhichismovedbyitby"together"

Imeanthatthereisnothingintermediatebetweenthem)。Thisisuniversallytruewhereveronethingismovedbyanother。Andsincetherearethreekindsofmotion，local，qualitative，andquantitative，theremustalsobethreekindsofmovent，thatwhichcauseslocomotion，thatwhichcausesalteration，andthatwhichcausesincreaseordecrease。

Letusbeginwithlocomotion，forthisistheprimarymotion。

Everythingthatisinlocomotionismovedeitherbyitselforbysomethingelse。Inthecaseofthingsthataremovedbythemselvesitisevidentthatthemovedandthemoventaretogether:fortheycontainwithinthemselvestheirfirstmovent，sothatthereisnothinginbetween。Themotionofthingsthataremovedbysomethingelsemustproceedinoneoffourways:fortherearefourkindsoflocomotioncausedbysomethingotherthanthatwhichisinmotion，viz。

pulling，pushing，carrying，andtwirling。Allformsoflocomotionarereducibletothese。Thuspushingonisaformofpushinginwhichthatwhichiscausingmotionawayfromitselffollowsupthatwhichitpushesandcontinuestopushit:pushingoffoccurswhenthemoventdoesnotfollowupthethingthatithasmoved:throwingwhenthemoventcausesamotionawayfromitselfmoreviolentthanthenaturallocomotionofthethingmoved，whichcontinuesitscoursesolongasitiscontrolledbythemotionimpartedtoit。Again，pushingapartandpushingtogetherareformsrespectivelyofpushingoffandpulling:pushingapartispushingoff，whichmaybeamotioneitherawayfromthepusherorawayfromsomethingelse，whilepushingtogetherispulling，whichmaybeamotiontowardssomethingelseaswellasthepuller。Wemaysimilarlyclassifyallthevarietiesoftheselasttwo，e。g。packingandcombing:theformerisaformofpushingtogether，thelatteraformofpushingapart。Thesameistrueoftheotherprocessesofcombinationandseparation(theywillallbefoundtobeformsofpushingapartorofpushingtogether)，exceptsuchasareinvolvedintheprocessesofbecomingandperishing。(Atsametimeitisevidentthatthereisnootherkindofmotionbutcombinationandseparation:fortheymayallbeapportionedtooneorotherofthosealreadymentioned。)Again，inhalingisaformofpulling，exhalingaformofpushing:andthesameistrueofspittingandofallothermotionsthatproceedthroughthebody，whethersecretiveorassimilative，theassimilativebeingformsofpulling，thesecretiveofpushingoff。Allotherkindsoflocomotionmustbesimilarlyreduced，fortheyallfallunderoneorotherofourfourheads。Andagain，ofthesefour，carryingandtwirlingaretopullingandpushing。Forcarryingalwaysfollowsoneoftheotherthreemethods，forthatwhichiscarriedisinmotionaccidentally，becauseitisinoruponsomethingthatisinmotion，andthatwhichcarriesitisindoingsobeingeitherpulledorpushedortwirled；

thuscarryingbelongstoalltheotherthreekindsofmotionincommon。Andtwirlingisacompoundofpullingandpushing，forthatwhichistwirlingathingmustbepullingonepartofthethingandpushinganotherpart，sinceitimpelsonepartawayfromitselfandanotherparttowardsitself。If，therefore，itcanbeshownthatthatwhichispushingandthatwhichispushingandpullingareadjacentrespectivelytothatwhichisbeingpushedandthatwhichisbeingpulled，itwillbeevidentthatinalllocomotionthereisnothingintermediatebetweenmovedandmovent。Buttheformerfactisclearevenfromthedefinitionsofpushingandpulling，forpushingismotiontosomethingelsefromoneselforfromsomethingelse，andpullingismotionfromsomethingelsetooneselfortosomethingelse，whenthemotionofthatwhichispullingisquickerthanthemotionthatwouldseparatefromoneanotherthetwothingsthatarecontinuous:foritisthisthatcausesonethingtobepulledonalongwiththeother。(Itmightindeedbethoughtthatthereisaformofpullingthatarisesinanotherway:thatwood，e。g。pullsfireinamannerdifferentfromthatdescribedabove。Butitmakesnodifferencewhetherthatwhichpullsisinmotionorisstationarywhenitispulling:inthelattercaseitpullstotheplacewhereitis，whileintheformeritpullstotheplacewhereitwas。)Nowitisimpossibletomoveanythingeitherfromoneselftosomethingelseorsomethingelsetooneselfwithoutbeingincontactwithit:itisevident，therefore，thatinalllocomotionthereisnothingintermediatebetweenmovedandmovent。

Noragainisthereanythingintermediatebetweenthatwhichundergoesandthatwhichcausesalteration:thiscanbeprovedbyinduction:forineverycasewefindthattherespectiveextremitiesofthatwhichcausesandthatwhichundergoesalterationareadjacent。

Forourassumptionisthatthingsthatareundergoingalterationarealteredinvirtueoftheirbeingaffectedinrespectoftheirso-calledaffectivequalities，sincethatwhichisofacertainqualityisalteredinsofarasitissensible，andthecharacteristicsinwhichbodiesdifferfromoneanotheraresensiblecharacteristics:foreverybodydiffersfromanotherinpossessingagreaterorlessernumberofsensiblecharacteristicsorinpossessingthesamesensiblecharacteristicsinagreaterorlesserdegree。Butthealterationofthatwhichundergoesalterationisalsocausedbytheabove-mentionedcharacteristics，whichareaffectionsofsomeparticularunderlyingquality。Thuswesaythatathingisalteredbybecominghotorsweetorthickordryorwhite:

andwemaketheseassertionsalikeofwhatisinanimateandofwhatisanimate，andfurther，whereanimatethingsareinquestion，wemakethembothofthepartsthathavenopowerofsense-perceptionandofthesensesthemselves。Forinawayeventhesensesundergoalteration，sincetheactivesenseisamotionthroughthebodyinthecourseofwhichthesenseisaffectedinacertainway。Wesee，then，thattheanimateiscapableofeverykindofalterationofwhichtheinanimateiscapable:buttheinanimateisnotcapableofeverykindofalterationofwhichtheanimateiscapable，sinceitisnotcapableofalterationinrespectofthesenses:moreovertheinanimateisunconsciousofbeingaffectedbyalteration，whereastheanimateisconsciousofit，thoughthereisnothingtopreventtheanimatealsobeingunconsciousofitwhentheprocessofthealterationdoesnotconcernthesenses。Since，then，thealterationofthatwhichundergoesalterationiscausedbysensiblethings，ineverycaseofsuchalterationitisevidentthattherespectiveextremitiesofthatwhichcausesandthatwhichundergoesalterationareadjacent。

Thustheairiscontinuouswiththatwhichcausesthealteration，andthebodythatundergoesalterationiscontinuouswiththeair。

Again，thecolouriscontinuouswiththelightandthelightwiththesight。Andthesameistrueofhearingandsmelling:fortheprimarymoventinrespecttothemovedistheair。Similarly，inthecaseoftasting，theflavourisadjacenttothesenseoftaste。Anditisjustthesameinthecaseofthingsthatareinanimateandincapableofsense-perception。Thustherecanbenothingintermediatebetweenthatwhichundergoesandthatwhichcausesalteration。

Nor，again，cantherebeanythingintermediatebetweenthatwhichsuffersandthatwhichcausesincrease:forthepartofthelatterthatstartstheincreasedoessobybecomingattachedinsuchawaytotheformerthatthewholebecomesone。Again，thedecreaseofthatwhichsuffersdecreaseiscausedbyapartofthethingbecomingdetached。Sothatwhichcausesincreaseandthatwhichcausesdecreasemustbecontinuouswiththatwhichsuffersincreaseandthatwhichsuffersdecreaserespectively:andiftwothingsarecontinuouswithoneanothertherecanbenothingintermediatebetweenthem。

Itisevident，therefore，thatbetweentheextremitiesofthemovedandthemoventthatarerespectivelyfirstandlastinreferencetothemovedthereisnothingintermediate。

3

Everything，wesay，thatundergoesalterationisalteredbysensiblecauses，andthereisalterationonlyinthingsthataresaidtobeessentiallyaffectedbysensiblethings。Thetruthofthisistobeseenfromthefollowingconsiderations。Ofallotherthingsitwouldbemostnaturaltosupposethatthereisalterationinfiguresandshapes，andinacquiredstatesandintheprocessesofacquiringandlosingthese:butasamatteroffactinneitherofthesetwoclassesofthingsistherealteration。

Inthefirstplace，whenaparticularformationofathingiscompleted，wedonotcallitbythenameofitsmaterial:e。g。wedonotcallthestatue"bronze"orthepyramid"wax"orthebed"wood"，butweuseaderivedexpressionandcallthem"ofbronze"，"waxen"，and"wooden"respectively。Butwhenathinghasbeenaffectedandalteredinanywaywestillcallitbytheoriginalname:thuswespeakofthebronzeorthewaxbeingdryorfluidorhardorhot。

Andnotonlyso:wealsospeakoftheparticularfluidorhotsubstanceasbeingbronze，givingthematerialthesamenameasthatwhichweusetodescribetheaffection。

Since，therefore，havingregardtothefigureorshapeofathingwenolongercallthatwhichhasbecomeofacertainfigurebythenameofthematerialthatexhibitsthefigure，whereashavingregardtoathing"saffectionsoralterationswestillcallitbythenameofitsmaterial，itisevidentthatbecomingsoftheformerkindcannotbealterations。

Moreoveritwouldseemabsurdeventospeakinthisway，tospeak，thatistosay，ofamanorhouseoranythingelsethathascomeintoexistenceashavingbeenaltered。Thoughitmaybetruethateverysuchbecomingisnecessarilytheresultofsomething"sbeingaltered，theresult，e。g。ofthematerial"sbeingcondensedorrarefiedorheatedorcooled，neverthelessitisnotthethingsthatarecomingintoexistencethatarealtered，andtheirbecomingisnotanalteration。

Again，acquiredstates，whetherofthebodyorofthesoul，arenotalterations。Forsomeareexcellencesandothersaredefects，andneitherexcellencenordefectisanalteration:excellenceisaperfection(forwhenanythingacquiresitsproperexcellencewecallitperfect，sinceitisthenifeverthatwehaveathinginitsnaturalstate:e。g。wehaveaperfectcirclewhenwehaveoneasgoodaspossible)，whiledefectisaperishingofordeparturefromthiscondition。Soaswhenspeakingofahousewedonotcallitsarrivalatperfectionanalteration(foritwouldbeabsurdtosupposethatthecopingorthetilingisanalterationorthatinreceivingitscopingoritstilingahouseisalteredandnotperfected)，thesamealsoholdsgoodinthecaseofexcellencesanddefectsandofthepersonsorthingsthatpossessoracquirethem:forexcellencesareperfectionsofathing"snatureanddefectsaredeparturesfromit:

consequentlytheyarenotalterations。

Further，wesaythatallexcellencesdependuponparticularrelations。Thusbodilyexcellencessuchashealthandagoodstateofbodyweregardasconsistinginablendingofhotandcoldelementswithinthebodyindueproportion，inrelationeithertooneanotherortothesurroundingatmosphere:andinlikemannerweregardbeauty，strength，andalltheotherbodilyexcellencesanddefects。Eachofthemexistsinvirtueofaparticularrelationandputsthatwhichpossessesitinagoodorbadconditionwithregardtoitsproperaffections，whereby"proper"affectionsImeanthoseinfluencesthatfromthenaturalconstitutionofathingtendtopromoteordestroyitsexistence。Sincethen，relativesareneitherthemselvesalterationsnorthesubjectsofalterationorofbecomingorinfactofanychangewhatever，itisevidentthatneitherstatesnortheprocessesoflosingandacquiringstatesarealterations，thoughitmaybetruethattheirbecomingorperishingisnecessarily，likethebecomingorperishingofaspecificcharacterorform，theresultofthealterationofcertainotherthings，e。g。hotandcoldordryandwetelementsortheelements，whatevertheymaybe，onwhichthestatesprimarilydepend。Foreachseveralbodilydefectorexcellenceinvolvesarelationwiththosethingsfromwhichthepossessorofthedefectorexcellenceisnaturallysubjecttoalteration:thusexcellencedisposesitspossessortobeunaffectedbytheseinfluencesortobeaffectedbythoseofthemthatoughttobeadmitted，whiledefectdisposesitspossessortobeaffectedbythemortobeunaffectedbythoseofthemthatoughttobeadmitted。

Andthecaseissimilarinregardtothestatesofthesoul，allofwhich(likethoseofbody)existinvirtueofparticularrelations，theexcellencesbeingperfectionsofnatureandthedefectsdeparturesfromit:moreover，excellenceputsitspossessoringoodcondition，whiledefectputsitspossessorinabadcondition，tomeethisproperaffections。Consequentlythesecannotanymorethanthebodilystatesbealterations，norcantheprocessesoflosingandacquiringthembeso，thoughtheirbecomingisnecessarilytheresultofanalterationofthesensitivepartofthesoul，andthisisalteredbysensibleobjects:forallmoralexcellenceisconcernedwithbodilypleasuresandpains，whichagaindependeitheruponactingoruponrememberingoruponanticipating。Nowthosethatdependuponactionaredeterminedbysense-perception，i。e。theyarestimulatedbysomethingsensible:andthosethatdependuponmemoryoranticipationarelikewisetobetracedtosense-perception，forinthesecasespleasureisfelteitherinrememberingwhatonehasexperiencedorinanticipatingwhatoneisgoingtoexperience。Thusallpleasureofthiskindmustbeproducedbysensiblethings:andsincethepresenceinanyoneofmoraldefectorexcellenceinvolvesthepresenceinhimofpleasureorpain(withwhichmoralexcellenceanddefectarealwaysconcerned)，andthesepleasuresandpainsarealterationsofthesensitivepart，itisevidentthatthelossandacquisitionofthesestatesnolessthanthelossandacquisitionofthestatesofthebodymustbetheresultofthealterationofsomethingelse。Consequently，thoughtheirbecomingisaccompaniedbyanalteration，theyarenotthemselvesalterations。

Again，thestatesoftheintellectualpartofthesoularenotalterations，noristhereanybecomingofthem。Inthefirstplaceitismuchmoretrueofthepossessionofknowledgethatitdependsuponaparticularrelation。Andfurther，itisevidentthatthereisnobecomingofthesestates。Forthatwhichispotentiallypossessedofknowledgebecomesactuallypossessedofitnotbybeingsetinmotionatallitselfbutbyreasonofthepresenceofsomethingelse:i。e。itiswhenitmeetswiththeparticularobjectthatitknowsinamannertheparticularthroughitsknowledgeoftheuniversal。(Again，thereisnobecomingoftheactualuseandactivityofthesestates，unlessitisthoughtthatthereisabecomingofvisionandtouchingandthattheactivityinquestionissimilartothese。)Andtheoriginalacquisitionofknowledgeisnotabecomingoranalteration:fortheterms"knowing"and"understanding"implythattheintellecthasreachedastateofrestandcometoastandstill，andthereisnobecomingthatleadstoastateofrest，since，aswehavesaidabove，changeatallcanhaveabecoming。

Moreover，justastosay，whenanyonehaspassedfromastateofintoxicationorsleepordiseasetothecontrarystate，thathehasbecomepossessedofknowledgeagainisincorrectinspiteofthefactthathewaspreviouslyincapableofusinghisknowledge，so，too，whenanyoneoriginallyacquiresthestate，itisincorrecttosaythathebecomespossessedofknowledge:forthepossessionofunderstandingandknowledgeisproducedbythesoul"ssettlingdownoutoftherestlessnessnaturaltoit。Hence，too，inlearningandinformingjudgementsonmattersrelatingtotheirsense-perceptionschildrenareinferiortoadultsowingtothegreatamountofrestlessnessandmotionintheirsouls。Natureitselfcausesthesoultosettledownandcometoastateofrestfortheperformanceofsomeofitsfunctions，whilefortheperformanceofothersotherthingsdoso:butineithercasetheresultisbroughtaboutthroughthealterationofsomethinginthebody，asweseeinthecaseoftheuseandactivityoftheintellectarisingfromaman"sbecomingsoberorbeingawakened。Itisevident，then，fromtheprecedingargumentthatalterationandbeingalteredoccurinsensiblethingsandinthesensitivepartofthesoul，and，exceptaccidentally，innothingelse。

4

Adifficultymayberaisedastowhethereverymotioniscommensurablewitheveryotherornot。Nowiftheyareallcommensurableandiftwothingstohavethesamevelocitymustaccomplishanequalmotioninanequaltime，thenwemayhaveacircumferenceequaltoastraightline，or，ofcourse，theonemaybegreaterorlessthantheother。Further，ifonethingaltersandanotheraccomplishesalocomotioninanequaltime，wemayhaveanalterationandalocomotionequaltooneanother:thusanaffectionwillbeequaltoalength，whichisimpossible。Butisitnotonlywhenanequalmotionisaccomplishedbytwothingsinanequaltimethatthevelocitiesofthetwoareequal?Nowanaffectioncannotbeequaltoalength。Thereforetherecannotbeanalterationequaltoorlessthanalocomotion:andconsequentlyitisnotthecasethateverymotioniscommensurablewitheveryother。

Buthowwillourconclusionworkoutinthecaseofthecircleandthestraightline?Itwouldbeabsurdtosupposethatthemotionofoneinacircleandofanotherinastraightlinecannotbesimilar，butthattheonemustinevitablymovemorequicklyormoreslowlythantheother，justasifthecourseofoneweredownhillandoftheotheruphill。Moreoveritdoesnotasamatteroffactmakeanydifferencetotheargumenttosaythattheonemotionmustinevitablybequickerorslowerthantheother:forthenthecircumferencecanbegreaterorlessthanthestraightline；andifsoitispossibleforthetwotobeequal。ForifinthetimeAthequicker(B)passesoverthedistanceB"andtheslower(G)passesoverthedistanceG"，B"willbegreaterthanG":forthisiswhatwetook"quicker"tomean:andsoquickermotionalsoimpliesthatonethingtraversesanequaldistanceinlesstimethananother:consequentlytherewillbeapartofAinwhichBwillpassoverapartofthecircleequaltoG"，whileGwilloccupythewholeofAinpassingoverG"。Nonetheless，ifthetwomotionsarecommensurable，weareconfrontedwiththeconsequencestatedabove，viz。thattheremaybeastraightlineequaltoacircle。Butthesearenotcommensurable:andsothecorrespondingmotionsarenotcommensurableeither。

Butmaywesaythatthingsarealwayscommensurableifthesametermsareappliedtothemwithoutequivocation?e。g。apen，awine，andthehighestnoteinascalearenotcommensurable:wecannotsaywhetheranyoneofthemissharperthananyother:andwhyisthis?

theyareincommensurablebecauseitisonlyequivocallythatthesameterm"sharp"isappliedtothem:whereasthehighestnoteinascaleiscommensurablewiththeleading-note，becausetheterm"sharp"

hasthesamemeaningasappliedtoboth。Canitbe，then，thattheterm"quick"hasnotthesamemeaningasappliedtostraightmotionandtocircularmotionrespectively?Ifso，farlesswillithavethesamemeaningasappliedtoalterationandtolocomotion。

Orshallweinthefirstplacedenythatthingsarealwayscommensurableifthesametermsareappliedtothemwithoutequivocation?Fortheterm"much"hasthesamemeaningwhetherappliedtowaterortoair，yetwaterandairarenotcommensurableinrespectofit:or，ifthisillustrationisnotconsideredsatisfactory，"double"atanyratewouldseemtohavethesamemeaningasappliedtoeach(denotingineachcasetheproportionoftwotoone)，yetwaterandairarenotcommensurableinrespectofit。Buthereagainmaywenottakeupthesamepositionandsaythattheterm"much"isequivocal?Infacttherearesometermsofwhicheventhedefinitionsareequivocal；e。g。if"much"weredefinedas"somuchandmore"，"somuch"wouldmeansomethingdifferentindifferentcases:

"equal"issimilarlyequivocal；and"one"againisperhapsinevitablyanequivocalterm；andif"one"isequivocal，sois"two"。Otherwisewhyisitthatsomethingsarecommensurablewhileothersarenot，ifthenatureoftheattributeinthetwocasesisreallyoneandthesame?

Canitbethattheincommensurabilityoftwothingsinrespectofanyattributeisduetoadifferenceinthatwhichisprimarilycapableofcarryingtheattribute?Thushorseanddogaresocommensurablethatwemaysaywhichisthewhiter，sincethatwhichprimarilycontainsthewhitenessisthesameinboth，viz。thesurface:andsimilarlytheyarecommensurableinrespectofsize。

Butwaterandspeecharenotcommensurableinrespectofclearness，sincethatwhichprimarilycontainstheattributeisdifferentinthetwocases。Itwouldseem，howeverthatwemustrejectthissolution，sinceclearlywecouldthusmakeallequivocalattributesunivocalandsaymerelythatthatcontainseachofthemisdifferentindifferentcases:thus"equality"，"sweetness"，and"whiteness"willseverallyalwaysbethesame，thoughthatwhichcontainsthemisdifferentindifferentcases。Moreover，itisnotanycasualthingthatiscapableofcarryinganyattribute:eachsingleattributecanbecarriedprimarilyonlybyonesinglething。

Mustwethensaythat，iftwothingsaretobecommensurableinrespectofanyattribute，notonlymusttheattributeinquestionbeapplicabletobothwithoutequivocation，buttheremustalsobenospecificdifferenceseitherintheattributeitselforinthatwhichcontainstheattribute-thatthese，Imean，mustnotbedivisibleinthewayinwhichcolourisdividedintokinds?Thusinthisrespectonethingwillnotbecommensurablewithanother，i。e。wecannotsaythatoneismorecolouredthantheotherwhereonlycolouringeneralandnotanyparticularcolourismeant；buttheyarecommensurableinrespectofwhiteness。

Similarlyinthecaseofmotion:twothingsareofthesamevelocityiftheyoccupyanequaltimeinaccomplishingacertainequalamountofmotion。Suppose，then，thatinacertaintimeanalterationisundergonebyonehalfofabody"slengthandalocomotionisaccomplishedtheotherhalf:canbesaythatinthiscasethealterationisequaltothelocomotionandofthesamevelocity?Thatwouldbeabsurd，andthereasonisthattherearedifferentspeciesofmotion。Andifinconsequenceofthiswemustsaythattwothingsareofequalvelocityiftheyaccomplishlocomotionoveranequaldistanceinanequaltime，wehavetoadmittheequalityofastraightlineandacircumference。What，then，isthereasonofthis?Isitthatlocomotionisagenusorthatlineisagenus?(Wemayleavethetimeoutofaccount，sincethatisoneandthesame。)Ifthelinesarespecificallydifferent，thelocomotionsalsodifferspecificallyfromoneanother:forlocomotionisspecificallydifferentiatedaccordingtothespecificdifferentiationofthatoverwhichittakesplace。(Itisalsosimilarlydifferentiated，itwouldseem，accordinglyastheinstrumentofthelocomotionisdifferent:thusiffeetaretheinstrument，itiswalking，ifwingsitisflying；

butperhapsweshouldrathersaythatthisisnotso，andthatinthiscasethedifferencesinthelocomotionaremerelydifferencesofpostureinthatwhichisinmotion。)Wemaysay，therefore，thatthingsareofequalvelocityinanequaltimetheytraversethesamemagnitude:andwhenIcallit"thesame"Imeanthatitcontainsnospecificdifferenceandthereforenodifferenceinthemotionthattakesplaceoverit。Sowehavenowtoconsiderhowmotionisdifferentiated:andthisdiscussionservestoshowthatthegenusisnotaunitybutcontainsapluralitylatentinitanddistinctfromit，andthatinthecaseofequivocaltermssometimesthedifferentsensesinwhichtheyareusedarefarremovedfromoneanother，whilesometimesthereisacertainlikenessbetweenthem，andsometimesagaintheyarenearlyrelatedeithergenericallyoranalogically，withtheresultthattheyseemnottobeequivocalthoughtheyreallyare。

When，then，isthereadifferenceofspecies?Isanattributespecificallydifferentifthesubjectisdifferentwhiletheattributeisthesame，ormusttheattributeitselfbedifferentaswell?Andhowarewetodefinethelimitsofaspecies?Whatwillenableustodecidethatparticularinstancesofwhitenessorsweetnessarethesameordifferent?Isitenoughthatitappearsdifferentinonesubjectfromwhatappearsinanother?Ormusttherebenosamenessatall?Andfurther，wherealterationisinquestion，howisonealterationtobeofequalvelocitywithanother?Onepersonmaybecuredquicklyandanotherslowly，andcuresmayalsobesimultaneous:sothat，recoveryofhealthbeinganalteration，wehaveherealterationsofequalvelocity，sinceeachalterationoccupiesanequaltime。Butwhatalteration?Wecannotherespeakofan"equal"

alteration:whatcorrespondsinthecategoryofqualitytoequalityinthecategoryofquantityis"likeness"。However，letussaythatthereisequalvelocitywherethesamechangeisaccomplishedinanequaltime。Arewe，then，tofindthecommensurabilityinthesubjectoftheaffectionorintheaffectionitself?Inthecasethatwehavejustbeenconsideringitisthefactthathealthisoneandthesamethatenablesustoarriveattheconclusionthattheonealterationisneithermorenorlessthantheother，butthatbotharealike。Ifontheotherhandtheaffectionisdifferentinthetwocases，e。g。

whenthealterationstaketheformofbecomingwhiteandbecominghealthyrespectively，herethereisnosamenessorequalityorlikenessinasmuchasthedifferenceintheaffectionsatoncemakesthealterationsspecificallydifferent，andthereisnounityofalterationanymorethantherewouldbeunityoflocomotionunderlikeconditions。Sowemustfindouthowmanyspeciesthereareofalterationandoflocomotionrespectively。Nowifthethingsthatareinmotion-thatistosay，thethingstowhichthemotionsbelongessentiallyandnotaccidentally-differspecifically，thentheirrespectivemotionswillalsodifferspecifically:ifontheotherhandtheydiffergenericallyornumerically，themotionsalsowilldiffergenericallyornumericallyasthecasemaybe。Buttherestillremainsthequestionwhether，supposingthattwoalterationsareofequalvelocity，weoughttolookforthisequalityinthesameness(orlikeness)oftheaffections，orinthethingsaltered，toseee。g。

whetheracertainquantityofeachhasbecomewhite。Oroughtwenotrathertolookforitinboth?Thatistosay，thealterationsarethesameordifferentaccordingastheaffectionsarethesameordifferent，whiletheyareequalorunequalaccordingasthethingsalteredareequalorunequal。

Andnowwemustconsiderthesamequestioninthecaseofbecomingandperishing:howisonebecomingofequalvelocitywithanother?

Theyareofequalvelocityifinanequaltimethereareproducedtwothingsthatarethesameandspecificallyinseparable，e。g。twomen(notmerelygenericallyinseparablease。g。twoanimals)。

Similarlyoneisquickerthantheotherifinanequaltimetheproductisdifferentinthetwocases。Istateitthusbecausewehavenopairoftermsthatwillconveythis"difference"inthewayinwhichunlikenessisconveyed。Ifweadoptthetheorythatitisnumberthatconstitutesbeing，wemayindeedspeakofa"greaternumber"

anda"lessernumber"withinthesamespecies，butthereisnocommontermthatwillincludebothrelations，noraretheretermstoexpresseachofthemseparatelyinthesamewayasweindicateahigherdegreeorpreponderanceofanaffectionby"more"，ofaquantityby"greater。"

5

Nowsincewhereverthereisamovent，itsmotionalwaysactsuponsomething，isalwaysinsomething，andalwaysextendstosomething(by"isalwaysinsomething"Imeanthatitoccupiesatime:andby"extendstosomething"Imeanthatitinvolvesthetraversingofacertainamountofdistance:foratanymomentwhenathingiscausingmotion，italsohascausedmotion，sothattheremustalwaysbeacertainamountofdistancethathasbeentraversedandacertainamountoftimethathasbeenoccupied)。then，AthemovementhavemovedBadistanceGinatimeD，theninthesametimethesameforceAwillmove1/2BtwicethedistanceG，andin1/2Ditwillmove1/2BthewholedistanceforG:thustherulesofproportionwillbeobserved。Againifagivenforcemoveagivenweightacertaindistanceinacertaintimeandhalfthedistanceinhalfthetime，halfthemotivepowerwillmovehalftheweightthesamedistanceinthesametime。LetErepresenthalfthemotivepowerAandZhalftheweightB:thentheratiobetweenthemotivepowerandtheweightintheonecaseissimilarandproportionatetotheratiointheother，sothateachforcewillcausethesamedistancetobetraversedinthesametime。ButifEmoveZadistanceGinatimeD，itdoesnotnecessarilyfollowthatEcanmovetwiceZhalfthedistanceGinthesametime。If，then，AmoveBadistanceGinatimeD，itdoesnotfollowthatE，beinghalfofA，willinthetimeDorinanyfractionofitcauseBtotraverseapartofGtheratiobetweenwhichandthewholeofGisproportionatetothatbetweenAandE(whateverfractionofAEmaybe):infactitmightwellbethatitwillcausenomotionatall；foritdoesnotfollowthat，ifagivenmotivepowercausesacertainamountofmotion，halfthatpowerwillcausemotioneitherofanyparticularamountorinanylengthoftime:otherwiseonemanmightmoveaship，sinceboththemotivepoweroftheship-haulersandthedistancethattheyallcausetheshiptotraversearedivisibleintoasmanypartsastherearemen。HenceZeno"sreasoningisfalsewhenhearguesthatthereisnopartofthemilletthatdoesnotmakeasound:forthereisnoreasonwhyanysuchpartshouldnotinanylengthoftimefailtomovetheairthatthewholebushelmovesinfalling。Infactitdoesnotofitselfmoveevensuchaquantityoftheairasitwouldmoveifthispartwerebyitself:fornopartevenexistsotherwisethanpotentially。

Ifontheotherhandwehavetwoforceseachofwhichseparatelymovesoneoftwoweightsagivendistanceinagiventime，thentheforcesincombinationwillmovethecombinedweightsanequaldistanceinanequaltime:forinthiscasetherulesofproportionapply。

Thendoesthisholdgoodofalterationandofincreasealso?

Surelyitdoes，forinanygivencasewehaveadefinitethingthatcauseincreaseandadefinitethingthatsuffersincrease，andtheonecausesandtheothersuffersacertainamountofincreaseinacertainamountoftime。Similarlywehaveadefinitethingthatcausesalterationandadefinitethingthatundergoesalteration，andacertainamount，orratherdegree，ofalterationiscompletedinacertainamountoftime:thusintwiceasmuchtimetwiceasmuchalterationwillbecompletedandconverselytwiceasmuchalterationwilloccupytwiceasmuchtime:andthealterationofhalfofitsobjectwilloccupyhalfasmuchtimeandinhalfasmuchtimehalfoftheobjectwillbealtered:oragain，inthesameamountoftimeitwillbealteredtwiceasmuch。

Ontheotherhandifthatwhichcausesalterationorincreasecausesacertainamountofincreaseoralterationrespectivelyinacertainamountoftime，itdoesnotnecessarilyfollowthathalftheforcewilloccupytwicethetimeinalteringorincreasingtheobject，orthatintwicethetimethealterationorincreasewillbecompletedbyit:itmayhappenthattherewillbenoalterationorincreaseatall，thecasebeingthesameaswiththeweight。

BookVIII