on the heavens-第3部分

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is　an　impossibility。　Yet　our　eyes　tell　us　that　the　heavens　revolve　in　a　circle；　and　by　argument　also　we　have　determined　that　there　is　something　to　which　circular　movement　belongs。　　　（2）　Again；　if　from　a　finite　time　a　finite　time　be　subtracted；　what　remains　must　be　finite　and　have　a　beginning。　And　if　the　time　of　a　journey　has　a　beginning；　there　must　be　a　beginning　also　of　the　movement；　and　consequently　also　of　the　distance　traversed。　This　applies　universally。　Take　a　line；　ACE；　infinite　in　one　direction；　E；　and　another　line；　BB；　infinite　in　both　directions。　Let　ACE　describe　a　circle；　revolving　upon　C　as　centre。　In　its　movement　it　will　cut　BB　continuously　for　a　certain　time。　This　will　be　a　finite　time；　since　the　total　time　is　finite　in　which　the　heavens　complete　their　circular　orbit；　and　consequently　the　time　subtracted　from　it；　during　which　the　one　line　in　its　motion　cuts　the　other；　is　also　finite。　Therefore　there　will　be　a　point　at　which　ACE　began　for　the　first　time　to　cut　BB。　This；　however；　is　impossible。　The　infinite；　then；　cannot　revolve　in　a　circle；　nor　could　the　world；　if　it　were　infinite。　　　（3）　That　the　infinite　cannot　move　may　also　be　shown　as　follows。　Let　A　be　a　finite　line　moving　past　the　finite　line；　B。　Of　necessity　A　will　pass　clear　of　B　and　B　of　A　at　the　same　moment；　for　each　overlaps　the　other　to　precisely　the　same　extent。　Now　if　the　two　were　both　moving；　and　moving　in　contrary　directions；　they　would　pass　clear　of　one　another　more　rapidly；　if　one　were　still　and　the　other　moving　past　it；　less　rapidly；　provided　that　the　speed　of　the　latter　were　the　same　in　both　cases。　This；　however；　is　clear：　that　it　is　impossible　to　traverse　an　infinite　line　in　a　finite　time。　Infinite　time；　then；　would　be　required。　（This　we　demonstrated　above　in　the　discussion　of　movement。）　And　it　makes　no　difference　whether　a　finite　is　passing　by　an　infinite　or　an　infinite　by　a　finite。　For　when　A　is　passing　B；　then　B　overlaps　A　and　it　makes　no　difference　whether　B　is　moved　or　unmoved；　except　that；　if　both　move；　they　pass　clear　of　one　another　more　quickly。　It　is；　however；　quite　possible　that　a　moving　line　should　in　certain　cases　pass　one　which　is　stationary　quicker　than　it　passes　one　moving　in　an　opposite　direction。　One　has　only　to　imagine　the　movement　to　be　slow　where　both　move　and　much　faster　where　one　is　stationary。　To　suppose　one　line　stationary；　then；　makes　no　difficulty　for　our　argument；　since　it　is　quite　possible　for　A　to　pass　B　at　a　slower　rate　when　both　are　moving　than　when　only　one　is。　If；　therefore；　the　time　which　the　finite　moving　line　takes　to　pass　the　other　is　infinite；　then　necessarily　the　time　occupied　by　the　motion　of　the　infinite　past　the　finite　is　also　infinite。　For　the　infinite　to　move　at　all　is　thus　absolutely　impossible；　since　the　very　smallest　movement　conceivable　must　take　an　infinity　of　time。　Moreover　the　heavens　certainly　revolve；　and　they　complete　their　circular　orbit　in　a　finite　time；　so　that　they　pass　round　the　whole　extent　of　any　line　within　their　orbit；　such　as　the　finite　line　AB。　The　revolving　body；　therefore；　cannot　be　infinite。　　　（4）　Again；　as　a　line　which　has　a　limit　cannot　be　infinite；　or；　if　it　is　infinite；　is　so　only　in　length；　so　a　surface　cannot　be　infinite　in　that　respect　in　which　it　has　a　limit；　or；　indeed；　if　it　is　completely　determinate；　in　any　respect　whatever。　Whether　it　be　a　square　or　a　circle　or　a　sphere；　it　cannot　be　infinite；　any　more　than　a　foot…rule　can。　There　is　then　no　such　thing　as　an　infinite　sphere　or　square　or　circle；　and　where　there　is　no　circle　there　can　be　no　circular　movement；　and　similarly　where　there　is　no　infinite　at　all　there　can　be　no　infinite　movement；　and　from　this　it　follows　that；　an　infinite　circle　being　itself　an　impossibility；　there　can　be　no　circular　motion　of　an　infinite　body。　　　（5）　Again；　take　a　centre　C；　an　infinite　line；　AB；　another　infinite　line　at　right　angles　to　it；　E；　and　a　moving　radius；　CD。　CD　will　never　cease　contact　with　E；　but　the　position　will　always　be　something　like　CE；　CD　cutting　E　at　F。　The　infinite　line；　therefore；　refuses　to　complete　the　circle。　　　（6）　Again；　if　the　heaven　is　infinite　and　moves　in　a　circle；　we　shall　have　to　admit　that　in　a　finite　time　it　has　traversed　the　infinite。　For　suppose　the　fixed　heaven　infinite；　and　that　which　moves　within　it　equal　to　it。　It　results　that　when　the　infinite　body　has　completed　its　revolution；　it　has　traversed　an　infinite　equal　to　itself　in　a　finite　time。　But　that　we　know　to　be　impossible。　　　（7）　It　can　also　be　shown；　conversely；　that　if　the　time　of　revolution　is　finite；　the　area　traversed　must　also　be　finite；　but　the　area　traversed　was　equal　to　itself；　therefore；　it　is　itself　finite。　　　We　have　now　shown　that　the　body　which　moves　in　a　circle　is　not　endless　or　infinite；　but　has　its　limit。

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　　Further；　neither　that　which　moves　towards　nor　that　which　moves　away　from　the　centre　can　be　infinite。　For　the　upward　and　downward　motions　are　contraries　and　are　therefore　motions　towards　contrary　places。　But　if　one　of　a　pair　of　contraries　is　determinate；　the　other　must　be　determinate　also。　Now　the　centre　is　determined；　for；　from　whatever　point　the　body　which　sinks　to　the　bottom　starts　its　downward　motion；　it　cannot　go　farther　than　the　centre。　The　centre；　therefore；　being　determinate；　the　upper　place　must　also　be　determinate。　But　if　these　two　places　are　determined　and　finite；　the　corresponding　bodies　must　also　be　finite。　Further；　if　up　and　down　are　determinate；　the　intermediate　place　is　also　necessarily　determinate。　For；　if　it　is　indeterminate；　the　movement　within　it　will　be　infinite；　and　that　we　have　already　shown　to　be　an　impossibility。　The　middle　region　then　is　determinate；　and　consequently　any　body　which　either　is　in　it；　or　might　be　in　it；　is　determinate。　But　the　bodies　which　move　up　and　down　may　be　in　it；　since　the　one　moves　naturally　away　from　the　centre　and　the　other　towards　it。　　　From　this　alone　it　is　clear　that　an　infinite　body　is　an　impossibility；　but　there　is　a　further　point。　If　there　is　no　such　thing　as　infinite　weight；　then　it　follows　that　none　of　these　bodies　can　be　infinite。　For　the　supposed　infinite　body　would　have　to　be　infinite　in　weight。　（The　same　argument　applies　to　lightness：　for　as　the　one　supposition　involves　infinite　weight；　so　the　infinity　of　the　body　which　rises　to　the　surface　involves　infinite　lightness。）　This　is　proved　as　follows。　Assume　the　weight　to　be　finite；　and　take　an　infinite　body；　AB；　of　the　weight　C。　Subtract　from　the　infinite　body　a　finite　mass；　BD；　the　weight　of　which　shall　be　E。　E　then　is　less　than　C；　since　it　is　the　weight　of　a　lesser　mass。　Suppose　then　that　the　smaller　goes　into　the　greater　a　certain　number　of　times；　and　take　BF　bearing　the　same　proportion　to　BD　which　the　greater　weight　bears　to　the　smaller。　For　you　may　subtract　as　much　as　you　please　from　an　infinite。　If　now　the　masses　are　proportionate　to　the　weights；　and　the　lesser　weight　is　that　of　the　lesser　mass；　the　greater　must　be　that　of　the　greater。　The　weights；　therefore；　of　the　finite　and　of　the　infinite　body　are　equal。　Again；　if　the　weight　of　a　greater　body　is　greater　than　that　of　a　less；　the　weight　of　GB　will　be　greater　than　that　of　FB；　and　thus　the　weight　of　the　finite　body　is　greater　than　that　of　the　infinite。　And；　further；　the　weight　of　unequal　masses　will　be　the　same；　since　the　infinite　and　the　finite　cannot　be　equal。　It　does　not　matter　whether　the　weights　are　commensurable　or　not。　If　（a）　they　are　incommensurable　the　same　reasoning　holds。　For　instance；　suppose　E　multiplied　by　three　is　rather　more　than　C：　the　weight　of　three　masses　of　the　full　size　of　BD　will　be　greater　than　C。　We　thus　arrive　at　the　same　impossibility　as　before。　Again　（b）　we　may　assume　weights　which　are　commensurate；　for　it　makes　no　difference　whether　we　begin　with　the　weight　or　with　the　mass。　For　example；　assume　the　weight　E　to　be　commensurate　with　C；　and　take　from　the　infinite　mass　a　part　BD　of　weight　E。　Then　let　a　mass　BF　be　taken　having　the　same　proportion　to　BD　which　the　two　weights　have　to　one　another。　（For　the　mass　being　infinite　you　may　subtract　from　it　as　much　as　you　please。）　These　assumed　bodies　will　be　commensurate　in　mass　and　in　weight　alike。　Nor　again　does　it　make　any　difference　to　our　demonstration　whether　the　total　mass　has　its　weight　equally　or　unequally　distributed。　For　it　must　always　be　Possible　to　take　from　the　infinite　mass　a　body　of　equal　weight　to　BD　by　diminishing　or　increasing　the　size　of　the　section　to　the　necessary　extent。　　　From　what　we　have　said；　then；　it　is　clear　that　the　weight　of　the　infinite　body　cannot　be　finite。　It　must　then　be　infinite。　We　have　therefore　only　to　show　this　to　be　impossible　in　order　to　prove　an　infinite　body　impossible。　But　the　impossibility　of　infinite　weight　can　be　shown　in　the　following　way。　A　given　weight　moves　a　given　distance　in　a　given　time；　a　weight　which　is　as　great　and　more　moves　the　same　distance　in　a　less　time；　the　times　being　in　inverse　proportion　to　the　weights。　For　instance；　if　one　weight　is　twice　another；　it　will　take　half　as　long　over　a　given　movement。　Further；　a　finite　weight　traverses　any　finite　distance　in　a　finite　time。　It　necessarily　follows　from　this　that　infinite　weight；　if　there　is　such　a　thing；　being；　on　the　one　hand；　as　great　and　more　than　as　great　as　the　finite；　will　move　accordingly；　but　being；　on　the　other　hand；　compelled　to　move　in　a　time　inversely　proportionate　to　its　greatness；　cannot　move　at　all。　The　time　should　be　less　in　proportion　as　the　weight　is　greater。　But　there　is　no　proportion　between　the　infinite　and　the　finite：　proportion　can　only　hold　between　a　less　and　a　greater　finite　time。　And　though　you　may　say　that　the　time　of　the　movement　can　be　continually　diminished；　yet　there　is　no　minimum。　Nor；　if　there　were；　would　it　help　us。　For　some　finite　body　could　have　been　found　greater　than　the　given　finite　in　the　same　proportion　which　is　supposed　to　hold　between　the　infinite　and　the　given　finite；　so　that　an　infinite　and　a　finite　weight　must　have　traversed　an　equal　distance　in　equal　time。　But　that　is　impossible。　Again；　whatever　the　time；　so　long　as　it　is　finite；　in　which　the　infinite　performs　the　motion；　a　finite　weight　must　necessarily　move　a　certain　finite　distance　in　that　same　time。　Infinite　weight　is　therefore　impossible；　and　the　same　reasoning　applies　also