f　it　is　really　contrary；　exists　also　naturally。　In　some　form　it　must　be　present；　since　the　matter　of　contraries　is　the　same。　Also；　the　positive　is　prior　to　its　privation　（warm；　for　instance；　to　cold）；　and　rest　and　heaviness　stand　for　the　privation　of　lightness　and　movement。　But　further；　if　fire　and　earth　exist；　the　intermediate　bodies　must　exist　also：　each　element　stands　in　a　contrary　relation　to　every　other。　（This；　again；　we　will　here　take　for　granted　and　try　later　to　explain。）　these　four　elements　generation　clearly　is　involved；　since　none　of　them　can　be　eternal：　for　contraries　interact　with　one　another　and　destroy　one　another。　Further；　it　is　inconceivable　that　a　movable　body　should　be　eternal；　if　its　movement　cannot　be　regarded　as　naturally　eternal：　and　these　bodies　we　know　to　possess　movement。　Thus　we　see　that　generation　is　necessarily　involved。　But　if　so；　there　must　be　at　least　one　other　circular　motion：　for　a　single　movement　of　the　whole　heaven　would　necessitate　an　identical　relation　of　the　elements　of　bodies　to　one　another。　This　matter　also　shall　be　cleared　up　in　what　follows：　but　for　the　present　so　much　is　clear；　that　the　reason　why　there　is　more　than　one　circular　body　is　the　necessity　of　generation；　which　follows　on　the　presence　of　fire；　which；　with　that　of　the　other　bodies；　follows　on　that　of　earth；　and　earth　is　required　because　eternal　movement　in　one　body　necessitates　eternal　rest　in　another。
  4
  The　shape　of　the　heaven　is　of　necessity　spherical；　for　that　is　the　shape　most　appropriate　to　its　substance　and　also　by　nature　primary。　　　First；　let　us　consider　generally　which　shape　is　primary　among　planes　and　solids　alike。　Every　plane　figure　must　be　either　rectilinear　or　curvilinear。　Now　the　rectilinear　is　bounded　by　more　than　one　line；　the　curvilinear　by　one　only。　But　since　in　any　kind　the　one　is　naturally　prior　to　the　many　and　the　simple　to　the　complex；　the　circle　will　be　the　first　of　plane　figures。　Again；　if　by　complete；　as　previously　defined；　we　mean　a　thing　outside　which　no　part　of　itself　can　be　found；　and　if　addition　is　always　possible　to　the　straight　line　but　never　to　the　circular；　clearly　the　line　which　embraces　the　circle　is　complete。　If　then　the　complete　is　prior　to　the　incomplete；　it　follows　on　this　ground　also　that　the　circle　is　primary　among　figures。　And　the　sphere　holds　the　same　position　among　solids。　For　it　alone　is　embraced　by　a　single　surface；　while　rectilinear　solids　have　several。　The　sphere　is　among　solids　what　the　circle　is　among　plane　figures。　Further；　those　who　divide　bodies　into　planes　and　generate　them　out　of　planes　seem　to　bear　witness　to　the　truth　of　this。　Alone　among　solids　they　leave　the　sphere　undivided；　as　not　possessing　more　than　one　surface：　for　the　division　into　surfaces　is　not　just　dividing　a　whole　by　cutting　it　into　its　parts；　but　division　of　another　fashion　into　parts　different　in　form。　It　is　clear；　then；　that　the　sphere　is　first　of　solid　figures。　　　If；　again；　one　orders　figures　according　to　their　numbers；　it　is　most　natural　to　arrange　them　in　this　way。　The　circle　corresponds　to　the　number　one；　the　triangle；　being　the　sum　of　two　right　angles；　to　the　number　two。　But　if　one　is　assigned　to　the　triangle；　the　circle　will　not　be　a　figure　at　all。　　　Now　the　first　figure　belongs　to　the　first　body；　and　the　first　body　is　that　at　the　farthest　circumference。　It　follows　that　the　body　which　revolves　with　a　circular　movement　must　be　spherical。　The　same　then　will　be　true　of　the　body　continuous　with　it：　for　that　which　is　continuous　with　the　spherical　is　spherical。　The　same　again　holds　of　the　bodies　between　these　and　the　centre。　Bodies　which　are　bounded　by　the　spherical　and　in　contact　with　it　must　be；　as　wholes；　spherical；　and　the　bodies　below　the　sphere　of　the　planets　are　contiguous　with　the　sphere　above　them。　The　sphere　then　will　be　spherical　throughout；　for　every　body　within　it　is　contiguous　and　continuous　with　spheres。　　　Again；　since　the　whole　revolves；　palpably　and　by　assumption；　in　a　circle；　and　since　it　has　been　shown　that　outside　the　farthest　circumference　there　is　neither　void　nor　place；　from　these　grounds　also　it　will　follow　necessarily　that　the　heaven　is　spherical。　For　if　it　is　to　be　rectilinear　in　shape；　it　will　follow　that　there　is　place　and　body　and　void　without　it。　For　a　rectilinear　figure　as　it　revolves　never　continues　in　the　same　room；　but　where　formerly　was　body；　is　now　none；　and　where　now　is　none；　body　will　be　in　a　moment　because　of　the　projection　at　the　corners。　Similarly；　if　the　world　had　some　other　figure　with　unequal　radii；　if；　for　instance；　it　were　lentiform；　or　oviform；　in　every　case　we　should　have　to　admit　space　and　void　outside　the　moving　body；　because　the　whole　body　would　not　always　occupy　the　same　room。　　　Again；　if　the　motion　of　the　heaven　is　the　measure　of　all　movements　whatever　in　virtue　of　being　alone　continuous　and　regular　and　eternal；　and　if；　in　each　kind；　the　measure　is　the　minimum；　and　the　minimum　movement　is　the　swiftest；　then；　clearly；　the　movement　of　the　heaven　must　be　the　swiftest　of　all　movements。　Now　of　lines　which　return　upon　themselves　the　line　which　bounds　the　circle　is　the　shortest；　and　that　movement　is　the　swiftest　which　follows　the　shortest　line。　Therefore；　if　the　heaven　moves　in　a　circle　and　moves　more　swiftly　than　anything　else；　it　must　necessarily　be　spherical。　　　Corroborative　evidence　may　be　drawn　from　the　bodies　whose　position　is　about　the　centre。　If　earth　is　enclosed　by　water；　water　by　air；　air　by　fire；　and　these　similarly　by　the　upper　bodies…which　while　not　continuous　are　yet　contiguous　with　them…and　if　the　surface　of　water　is　spherical；　and　that　which　is　continuous　with　or　embraces　the　spherical　must　itself　be　spherical；　then　on　these　grounds　also　it　is　clear　that　the　heavens　are　spherical。　But　the　surface　of　water　is　seen　to　be　spherical　if　we　take　as　our　starting…point　the　fact　that　water　naturally　tends　to　collect　in　a　hollow　place…'hollow'　meaning　'nearer　the　centre'。　Draw　from　the　centre　the　lines　AB；　AC；　and　let　their　extremities　be　joined　by　the　straight　line　BC。　The　line　AD；　drawn　to　the　base　of　the　triangle；　will　be　shorter　than　either　of　the　radii。　Therefore　the　place　in　which　it　terminates　will　be　a　hollow　place。　The　water　then　will　collect　there　until　equality　is　established；　that　is　until　the　line　AE　is　equal　to　the　two　radii。　Thus　water　forces　its　way　to　the　ends　of　the　radii；　and　there　only　will　it　rest：　but　the　line　which　connects　the　extremities　of　the　radii　is　circular：　therefore　the　surface　of　the　water　BEC　is　spherical。　　　It　is　plain　from　the　foregoing　that　the　universe　is　spherical。　It　is　plain；　further；　that　it　is　turned　（so　to　speak）　with　a　finish　which　no　manufactured　thing　nor　anything　else　within　the　range　of　our　observation　can　even　approach。　For　the　matter　of　which　these　are　composed　does　not　admit　of　anything　like　the　same　regularity　and　finish　as　the　substance　of　the　enveloping　body；　since　with　each　step　away　from　earth　the　matter　manifestly　becomes　finer　in　the　same　proportion　as　water　is　finer　than　earth。
  5
  Now　there　are　two　ways　of　moving　along　a　circle；　from　A　to　B　or　from　A　to　C；　and　we　have　already　explained　that　these　movements　are　not　contrary　to　one　another。　But　nothing　which　concerns　the　eternal　can　be　a　matter　of　chance　or　spontaneity；　and　the　heaven　and　its　circular　motion　are　eternal。　We　must　therefore　ask　why　this　motion　takes　one　direction　and　not　the　other。　Either　this　is　itself　an　ultimate　fact　or　there　is　an　ultimate　fact　behind　it。　It　may　seem　evidence　of　excessive　folly　or　excessive　zeal　to　try　to　provide　an　explanation　of　some　things；　or　of　everything；　admitting　no　exception。　The　criticism；　however；　is　not　always　just：　one　should　first　consider　what　reason　there　is　for　speaking；　and　also　what　kind　of　certainty　is　looked　for；　whether　human　merely　or　of　a　more　cogent　kind。　When　any　one　shall　succeed　in　finding　proofs　of　greater　precision；　gratitude　will　be　due　to　him　for　the　discovery；　but　at　present　we　must　be　content　with　a　probable　solution。　If　nature　always　follows　the　best　course　possible；　and；　just　as　upward　movement　is　the　superior　form　of　rectilinear　movement；　since　the　upper　region　is　more　divine　than　the　lower；　so　forward　movement　is　superior　to　backward；　then　front　and　back　exhibits；　like　right　and　left；　as　we　said　before　and　as　the　difficulty　just　stated　itself　suggests；　the　distinction　of　prior　and　posterior；　which　provides　a　reason　and　so　solves　our　difficulty。　Supposing　that　nature　is　ordered　in　the　best　way　possible；　this　may　stand　as　the　reason　of　the　fact　mentioned。　For　it　is　best　to　move　with　a　movement　simple　and　unceasing；　and；　further；　in　the　superior　of　two　possible　directions。
  6
  We　have　next　to　show　that　the　movement　of　the　heaven　is　regular　and　not　irregular。　This　applies　only　to　the　first　heaven　and　the　first　movement；　for　the　lower　spheres　exhibit　a　composition　of　several　movements　into　one。　If　the　movement　is　uneven；　clearly　there　will　be　acceleration；　maximum　speed；　and　retardation；　since　these　appear　in　all　irregular　motions。　The　maximum　may　occur　either　at　the　starting…point　or　at　the　goal　or　between　the　two；　and　we　expect　natural　motion　to　reach　its　maximum　at　the