a history of science-1-第32章

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is　eighteen　times　as　distant　as　the　moon。　Now；　by　comparing　the　apparent　size　of　the　sun　with　the　apparent　size　of　the　moonwhich；　as　we　have　seen；　Aristarchus　has　already　measuredhe　is　able　to　tell　us　that；　the　sun　is　〃more　than　5832　times；　and　less　than　8000〃　times　larger　than　the　moon；　though　his　measurements；　taken　by　themselves；　give　no　clew　to　the　actual　bulk　of　either　body。　These　conclusions；　be　it　understood；　are　absolutely　valid　inferencesnay；　demonstrationsfrom　the　measurements　involved；　provided　only　that　these　measurements　have　been　correct。　Unfortunately；　the　angle　of　the　triangle　we　have　just　seen　measured　is　exceedingly　difficult　to　determine　with　accuracy；　while　at　the　same　time；　as　a　moment's　reflection　will　show；　it　is　so　large　an　angle　that　a　very　slight　deviation　from　the　truth　will　greatly　affect　the　distance　at　which　its　line　joins　the　other　side　of　the　triangle。　Then　again；　it　is　virtually　impossible　to　tell　the　precise　moment　when　the　moon　is　at　half；　as　the　line　it　gives　is　not　so　sharp　that　we　can　fix　it　with　absolute　accuracy。　There　is；　moreover；　another　element　of　error　due　to　the　refraction　of　light　by　the　earth's　atmosphere。　The　experiment　was　probably　made　when　the　sun　was　near　the　horizon；　at　which　time；　as　we　now　know；　but　as　Aristarchus　probably　did　not　suspect；　the　apparent　displacement　of　the　sun's　position　is　considerable；　and　this　displacement；　it　will　be　observed；　is　in　the　direction　to　lessen　the　angle　in　question。　In　point　of　fact；　Aristarchus　estimated　the　angle　at　eighty…seven　degrees。　Had　his　instrument　been　more　precise；　and　had　he　been　able　to　take　account　of　all　the　elements　of　error；　he　would　have　found　it　eighty…seven　degrees　and　fifty…two　minutes。　The　difference　of　measurement　seems　slight；　but　it　sufficed　to　make　the　computations　differ　absurdly　from　the　truth。　The　sun　is　really　not　merely　eighteen　times　but　more　than　two　hundred　times　the　distance　of　the　moon；　as　Wendelein　discovered　on　repeating　the　experiment　of　Aristarchus　about　two　thousand　years　later。　Yet　this　discrepancy　does　not　in　the　least　take　away　from　the　validity　of　the　method　which　Aristarchus　employed。　Moreover；　his　conclusion；　stated　in　general　terms；　was　perfectly　correct：　the　sun　is　many　times　more　distant　than　the　moon　and　vastly　larger　than　that　body。　Granted；　then；　that　the　moon　is；　as　Aristarchus　correctly　believed；　considerably　less　in　size　than　the　earth；　the　sun　must　be　enormously　larger　than　the　earth；　and　this　is　the　vital　inference　which；　more　than　any　other；　must　have　seemed　to　Aristarchus　to　confirm　the　suspicion　that　the　sun　and　not　the　earth　is　the　centre　of　the　planetary　system。　It　seemed　to　him　inherently　improbable　that　an　enormously　large　body　like　the　sun　should　revolve　about　a　small　one　such　as　the　earth。　And　again；　it　seemed　inconceivable　that　a　body　so　distant　as　the　sun　should　whirl　through　space　so　rapidly　as　to　make　the　circuit　of　its　orbit　in　twenty…　four　hours。　But；　on　the　other　hand；　that　a　small　body　like　the　earth　should　revolve　about　the　gigantic　sun　seemed　inherently　probable。　This　proposition　granted；　the　rotation　of　the　earth　on　its　axis　follows　as　a　necessary　consequence　in　explanation　of　the　seeming　motion　of　the　stars。　Here；　then；　was　the　heliocentric　doctrine　reduced　to　a　virtual　demonstration　by　Aristarchus　of　Samos；　somewhere　about　the　middle　of　the　third　century　B。C。　It　must　be　understood　that　in　following　out　the；　steps　of　reasoning　by　which　we　suppose　Aristarchus　to　have　reached　so　remarkable　a　conclusion；　we　have　to　some　extent　guessed　at　the　processes　of　thought…　development；　for　no　line　of　explication　written　by　the　astronomer　himself　on　this　particular　point　has　come　down　to　us。　There　does　exist；　however；　as　we　have　already　stated；　a　very　remarkable　treatise　by　Aristarchus　on　the　Size　and　Distance　of　the　Sun　and　the　Moon；　which　so　clearly　suggests　the　methods　of　reasoning　of　the　great　astronomer；　and　so　explicitly　cites　the　results　of　his　measurements；　that　we　cannot　well　pass　it　by　without　quoting　from　it　at　some　length。　It　is　certainly　one　of　the　most　remarkable　scientific　documents　of　antiquity。　As　already　noted；　the　heliocentric　doctrine　is　not　expressly　stated　here。　It　seems　to　be　tacitly　implied　throughout；　but　it　is　not　a　necessary　consequence　of　any　of　the　propositions　expressly　stated。　These　propositions　have　to　do　with　certain　observations　and　measurements　and　what　Aristarchus　believes　to　be　inevitable　deductions　from　them；　and　he　perhaps　did　not　wish　to　have　these　deductions　challenged　through　associating　them　with　a　theory　which　his　contemporaries　did　not　accept。　In　a　word；　the　paper　of　Aristarchus　is　a　rigidly　scientific　document　unvitiated　by　association　with　any　theorizings　that　are　not　directly　germane　to　its　central　theme。　The　treatise　opens　with　certain　hypotheses　as　follows：　〃First。　The　moon　receives　its　light　from　the　sun。　〃Second。　The　earth　may　be　considered　as　a　point　and　as　the　centre　of　the　orbit　of　the　moon。　〃Third。　When　the　moon　appears　to　us　dichotomized　it　offers　to　our　view　a　great　circle　＇or　actual　meridian＇　of　its　circumference　which　divides　the　illuminated　part　from　the　dark　part。　〃Fourth。　When　the　moon　appears　dichotomized　its　distance　from　the　sun　is　less　than　a　quarter　of　the　circumference　＇of　its　orbit＇　by　a　thirtieth　part　of　that　quarter。〃　That　is　to　say；　in　modern　terminology；　the　moon　at　this　time　lacks　three　degrees　（one　thirtieth　of　ninety　degrees）　of　being　at　right　angles　with　the　line　of　the　sun　as　viewed　from　the　earth；　or；　stated　otherwise；　the　angular　distance　of　the　moon　from　the　sun　as　viewed　from　the　earth　is　at　this　time　eighty…seven　degreesthis　being；　as　we　have　already　observed；　the　fundamental　measurement　upon　which　so　much　depends。　We　may　fairly　suppose　that　some　previous　paper　of　Aristarchus's　has　detailed　the　measurement　which　here　is　taken　for　granted；　yet　which　of　course　could　depend　solely　on　observation。　〃Fifth。　The　diameter　of　the　shadow　＇cast　by　the　earth　at　the　point　where　the　moon's　orbit　cuts　that　shadow　when　the　moon　is　eclipsed＇　is　double　the　diameter　of　the　moon。〃　Here　again　a　knowledge　of　previously　established　measurements　is　taken　for　granted；　but；　indeed；　this　is　the　case　throughout　the　treatise。　〃Sixth。　The　arc　subtended　in　the　sky　by　the　moon　is　a　fifteenth　part　of　a　sign〃　of　the　zodiac；　that　is　to　say；　since　there　are　twenty…four；　signs　in　the　zodiac；　one…fifteenth　of　one　twenty…fourth；　or　in　modern　terminology；　one　degree　of　arc。　This　is　Aristarchus's　measurement　of　the　moon　to　which　we　have　already　referred　when　speaking　of　the　measurements　of　Archimedes。　〃If　we　admit　these　six　hypotheses；〃　Aristarchus　continues；　〃it　follows　that　the　sun　is　more　than　eighteen　times　more　distant　from　the　earth　than　is　the　moon；　and　that　it　is　less　than　twenty　times　more　distant；　and　that　the　diameter　of　the　sun　bears　a　corresponding　relation　to　the　diameter　of　the　moon；　which　is　proved　by　the　position　of　the　moon　when　dichotomized。　But　the　ratio　of　the　diameter　of　the　sun　to　that　of　the　earth　is　greater　than　nineteen　to　three　and　less　than　forty…three　to　six。　This　is　demonstrated　by　the　relation　of　the　distances；　by　the　position　＇of　the　moon＇　in　relation　to　the　earth's　shadow；　and　by　the　fact　that　the　arc　subtended　by　the　moon　is　a　fifteenth　part　of　a　sign。〃　Aristarchus　follows　with　nineteen　propositions　intended　to　elucidate　his　hypotheses　and　to　demonstrate　his　various　contentions。　These　show　a　singularly　clear　grasp　of　geometrical　problems　and　an　altogether　correct　conception　of　the　general　relations　as　to　size　and　position　of　the　earth；　the　moon；　and　the　sun。　His　reasoning　has　to　do　largely　with　the　shadow　cast　by　the　earth　and　by　the　moon；　and　it　presupposes　a　considerable　knowledge　of　the　phenomena　of　eclipses。　His　first　proposition　is　that　〃two　equal　spheres　may　always　be　circumscribed　in　a　cylinder；　two　unequal　spheres　in　a　cone　of　which　the　apex　is　found　on　the　side　of　the　smaller　sphere；　and　a　straight　line　joining　the　centres　of　these　spheres　is　perpendicular　to　each　of　the　two　circles　made　by　the　contact　of　the　surface　of　the　cylinder　or　of　the　cone　with　the　spheres。〃　It　will　be　observed　that　Aristarchus　has　in　mind　here　the　moon；　the　earth；　and　the　sun　as　spheres　to　be　circumscribed　within　a　cone；　which　cone　is　made　tangible　and　measurable　by　the　shadows　cast　by　the　non…luminous　bodies；　since；　continuing；　he　clearly　states　in　proposition　nine；　that　〃when　the　sun　is　totally　eclipsed；　an　observer　on　the　earth's　surface　is　at　an　apex　of　a　cone　comprising　the　moon　and　the　sun。〃　Various　propositions　deal　with　other　relations　of　the　shadows　which　need　not　detain　us　since　they　are　not　fundamentally　important；　and　we　may　pass　to　the　final　conclusions　of　Aristarchus；　as　reached　in　his　propositions　ten　to　nineteen。　Now；　since　（proposition　ten）　〃the　diameter　of　the　sun　is　more　than　eighteen　times　and　less　than　twenty　times　greater　than　that　of　the　moon；〃　it　follows　（proposition　eleven）　〃that　the　bulk　of　the　sun　is　to　that　of　the　moon　in　ratio；　greater　than　5832　to　1；　and　less　than　8000　to　1。〃　〃Proposition　sixteen。　The　diameter　of　the　sun　is　to　the　diameter　of　the　earth　in　greater　proportion　than　nineteen　to　three；　and　less　than　forty…three　to　six。　〃Proposition　seventeen。　The　bulk　of　the　sun　is　to　that　of　the　earth　in　greater　proportion　than　6859　to　27；　and　less　than　79；507　to　216。　〃Proposition　eighteen。　The　diameter　of　the　earth　is　to　the　diameter　of　the　moon　in　greater　proportion　than　108　to　43　and　less　than　60　to　19。　〃Proposition　nineteen。　The　bulk　of　the　earth　is　to　that　of　the　moon　in　greater　proportion　than　1；259；712　to　79；507　and　less　than　20；000　to　6859。〃　Such　then　are　the　more　important　conclusions　of　this　very　remarkable　papera　paper　which　seems　to　have　interest　to　the　successors　of　Aristarchus　generation　after　generation；　since　this　alone　of　all　the　writings　of　the　great　astronomer　has　been　preserved。　How　widely　the　exact　results　of　the　measurements　of　Aristarchus；　differ　from　the　truth；　we　have　pointed　out　as　we　progressed。　But　let　it　be　repeated　that　this　detracts　little　from　the　credit　of　the　astronomer　who　had　such　clear　and　correct　conceptions　of　the　relations　of　the　heavenly　bodies　and　who　invented　such　correct　methods　of　measurement。　Let　it　be　particularly　observed；　however；　that　all　the　conclusions　of　Aristarchus　are　stated　in　relative　terms。　He　nowhere　attempts　to　estimate　the　precise　size　of　the　earth；　of　the　moon；　or　of　the　sun；　or　the　actual　distance　of　one　of　these　bodies　from　another。　The　obvious　reason　for　this　is　that　no　data　were　at　hand　from　which　to　make　such　precise　measurements。　Had　Aristarchus　known　the　size　of　any　one　of　the　bodies　in　question；　he　might　readily；　of　course；　have　determined　the　size　of　the　others　by　the　mere　application　of　his　relative　scale；　but　he　had　no　means　of　determining　the　size　of　the　earth；　and　to　this　extent　his　system　of　measurements　remained　imperfect。　Where　Aristarchus　halted；　however；　another　worker　of　the　same　period　took　the　task　in　hand　and　by　an　altogether　wonderful　measurement　determined　the　size　of　the　earth；　and　thus　brought　the　scientific　theories　of　cosmology　to　their　climax。　This　worthy　supplementor　of　the　work　of　Aristarchus　was　Eratosthenes　of　Alexandria。

ERATOSTHENES；　〃THE　SURVEYOR　OF　THE　WORLD〃　An　altogether　remarkable　man　was　this　native　of　Cy