f　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　to　infinite　lightness。　Bodies　then　of　infinite　weight　and　of　infinite　lightness　are　equally　impossible。　　　That　there　is　no　infinite　body　may　be　shown；　as　we　have　shown　it；　by　a　detailed　consideration　of　the　various　cases。　But　it　may　also　be　shown　universally；　not　only　by　such　reasoning　as　we　advanced　in　our　discussion　of　principles　（though　in　that　passage　we　have　already　determined　universally　the　sense　in　which　the　existence　of　an　infinite　is　to　be　asserted　or　denied）；　but　also　suitably　to　our　present　purpose　in　the　following　way。　That　will　lead　us　to　a　further　question。　Even　if　the　total　mass　is　not　infinite；　it　may　yet　be　great　enough　to　admit　a　plurality　of　universes。　The　question　might　possibly　be　raised　whether　there　is　any　obstacle　to　our　believing　that　there　are　other　universes　composed　on　the　pattern　of　our　own；　more　than　one；　though　stopping　short　of　infinity。　First；　however；　let　us　treat　of　the　infinite　universally。
  7
  Every　body　must　necessarily　be　either　finite　or　infinite；　and　if　infinite；　either　of　similar　or　of　dissimilar　parts。　If　its　parts　are　dissimilar；　they　must　represent　either　a　finite　or　an　infinite　number　of　kinds。　That　the　kinds　cannot　be　infinite　is　evident；　if　our　original　presuppositions　remain　unchallenged。　For　the　primary　movements　being　finite　in　number；　the　kinds　of　simple　body　are　necessarily　also　finite；　since　the　movement　of　a　simple　body　is　simple；　and　the　simple　movements　are　finite；　and　every　natural　body　must　always　have　its　proper　motion。　Now　if　the　infinite　body　is　to　be　composed　of　a　finite　number　of　kinds；　then　each　of　its　parts　must　necessarily　be　infinite　in　quantity；　that　is　to　say；　the　water；　fire；　&c。；　which　compose　it。　But　this　is　impossible；　because；　as　we　have　already　shown；　infinite　weight　and　lightness　do　not　exist。　Moreover　it　would　be　necessary　also　that　their　places　should　be　infinite　in　extent；　so　that　the　movements　too　of　all　these　bodies　would　be　infinite。　But　this　is　not　possible；　if　we　are　to　hold　to　the　truth　of　our　original　presuppositions　and　to　the　view　that　neither　that　which　moves　downward；　nor；　by　the　same　reasoning；　that　which　moves　upward；　can　prolong　its　movement　to　infinity。　For　it　is　true　in　regard　to　quality；　quantity；　and　place　alike　that　any　process　of　change　is　impossible　which　can　have　no　end。　I　mean　that　if　it　is　impossible　for　a　thing　to　have　come　to　be　white；　or　a　cubit　long；　or　in　Egypt；　it　is　also　impossible　for　it　to　be　in　process　of　coming　to　be　any　of　these。　It　is　thus　impossible　for　a　thing　to　be　moving　to　a　place　at　which　in　its　motion　it　can　never　by　any　possibility　arrive。　Again；　suppose　the　body　to　exist　in　dispersion；　it　may　be　maintained　none　the　less　that　the　total　of　all　these　scattered　particles；　say；　of　fire；　is　infinite。　But　body　we　saw　to　be　that　which　has　extension　every　way。　How　can　there　be　several　dissimilar　elements；　each　infinite？　Each　would　have　to　be　infinitely　extended　every　way。　　　It　is　no　more　conceivable；　again；　that　the　infinite　should　exist　as　a　whole　of　similar　parts。　For；　in　the　first　place；　there　is　no　other　（straight）　movement　beyond　those　mentioned：　we　must　therefore　give　it　one　of　them。　And　if　so；　we　shall　have　to　admit　either　infinite　weight　or　infinite　lightness。　Nor；　secondly；　could　the　body　whose　movement　is　circular　be　infinite；　since　it　is　impossible　for　the　infinite　to　move　in　a　circle。　This；　indeed；　would　be　as　good　as　saying　that　the　heavens　are　infinite；　which　we　have　shown　to　be　impossible。　　　Moreover；　in　general；　it　is　impossible　that　the　infinite　should　move　at　all。　If　it　did；　it　would　move　either　naturally　or　by　constraint：　and　if　by　constraint；　it　possesses　also　a　natural　motion；　that　is　to　say；　there　is　another　place；　infinite　like　itself；　to　which　it　will　move。　But　that　is　impossible。　　　That　in　general　it　is　impossible　for　the　infinite　to　be　acted　upon　by　the　finite　or　to　act　upon　it　may　be　shown　as　follows。　　　（1。　The　infinite　cannot　be　acted　upon　by　the　finite。）　Let　A　be　an　infinite；　B　a　finite；　C　the　time　of　a　given　movement　produced　by　one　in　the　other。　Suppose；　then；　that　A　was　heated；　or　impelled；　or　modified　in　any　way；　or　caused　to　undergo　any　sort　of　movement　whatever；　by　in　the　time　C。　Let　D　be　less　than　B；　and；　assuming　that　a　lesser　agent　moves　a　lesser　patient　in　an　equal　time；　call　the　quantity　thus　modified　by　D；　E。　Then；　as　D　is　to　B；　so　is　E　to　some　finite　quantum。　We　assume　that　the　alteration　of　equal　by　equal　takes　equal　time；　and　the　alteration　of　less　by　less　or　of　greater　by　greater　takes　the　same　time；　if　the　quantity　of　the　patient　is　such　as　to　keep　the　proportion　which　obtains　between　the　agents；　greater　and　less。　If　so；　no　movement　can　be　caused　in　the　infinite　by　any　finite　agent　in　any　time　whatever。　For　a　less　agent　will　produce　that　movement　in　a　less　patient　in　an　equal　time；　and　the　proportionate　equivalent　of　that　patient　will　be　a　finite　quantity；　since　no　proportion　holds　between　finite　and　infinite。　　　　　（2。　The　infinite　cannot　act　upon　the　finite。）　Nor；　again；　can　the　infinite　produce　a　movement　in　the　finite　in　any　time　whatever。　Let　A　be　an　infinite；　B　a　finite；　C　the　time　of　action。　In　the　time　C；　D　will　produce　that　motion　in　a　patient　less　than　B；　say　F。　Then　take　E；　bearing　the　same　proportion　to　D　as　the　whole　BF　bears　to　F。　E　will　produce　the　motion　in　BF　in　the　time　C。　Thus　the　finite　and　infinite　effect　the　same　alteration　in　equal　times。　But　this　is　impossible；　for　the　assumption　is　that　the　greater　effects　it　in　a　shorter　time。　It　will　be　the　same　with　any　time　that　can　be　taken；　so　that　there　will　no　time　in　which　the　infinite　can　effect　this　movement。　And；　as　to　infinite　time；　in　that　nothing　can　move　another　or　be　moved　by　it。　For　such　time　has　no　limit；　while　the　action　and　reaction　have。　　　（3。　There　is　no　interaction　between　infinites。）　Nor　can　infinite　be　acted　upon　in　any　way　by　infinite。　Let　A　and　B　be　infinites；　CD　being　the　time　of　the　action　A　of　upon　B。　Now　the　whole　B　was　modified　in　a　certain　time；　and　the　part　of　this　infinite；　E；　cannot　be　so　modified　in　the　same　time；　since　we　assume　that　a