a history of science-1-第37章

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some　precision。　Fortunately；　the　writings　of　Archimedes　himself　are　still　extant；　in　which　the　results　of　his　remarkable　experiments　are　related；　so　we　may　present　the　results　in　the　words　of　the　discoverer。　Here　they　are：　〃First：　The　surface　of　every　coherent　liquid　in　a　state　of　rest　is　spherical；　and　the　centre　of　the　sphere　coincides　with　the　centre　of　the　earth。　Second：　A　solid　body　which；　bulk　for　bulk；　is　of　the　same　weight　as　a　liquid；　if　immersed　in　the　liquid　will　sink　so　that　the　surface　of　the　body　is　even　with　the　surface　of　the　liquid；　but　will　not　sink　deeper。　Third：　Any　solid　body　which　is　lighter；　bulk　for　bulk；　than　a　liquid；　if　placed　in　the　liquid　will　sink　so　deep　as　to　displace　the　mass　of　liquid　equal　in　weight　to　another　body。　Fourth：　If　a　body　which　is　lighter　than　a　liquid　is　forcibly　immersed　in　the　liquid；　it　will　be　pressed　upward　with　a　force　corresponding　to　the　weight　of　a　like　volume　of　water；　less　the　weight　of　the　body　itself。　Fifth：　Solid　bodies　which；　bulk　for　bulk；　are　heavier　than　a　liquid；　when　immersed　in　the　liquid　sink　to　the　bottom；　but　become　in　the　liquid　as　much　lighter　as　the　weight　of　the　displaced　water　itself　differs　from　the　weight　of　the　solid。〃　These　propositions　are　not　difficult　to　demonstrate；　once　they　are　conceived；　but　their　discovery；　combined　with　the　discovery　of　the　laws　of　statics　already　referred　to；　may　justly　be　considered　as　proving　Archimedes　the　most　inventive　experimenter　of　antiquity。　Curiously　enough；　the　discovery　which　Archimedes　himself　is　said　to　have　considered　the　most　important　of　all　his　innovations　is　one　that　seems　much　less　striking。　It　is　the　answer　to　the　question；　What　is　the　relation　in　bulk　between　a　sphere　and　its　circumscribing　cylinder？　Archimedes　finds　that　the　ratio　is　simply　two　to　three。　We　are　not　informed　as　to　how　he　reached　his　conclusion；　but　an　obvious　method　would　be　to　immerse　a　ball　in　a　cylindrical　cup。　The　experiment　is　one　which　any　one　can　make　for　himself；　with　approximate　accuracy；　with　the　aid　of　a　tumbler　and　a　solid　rubber　ball　or　a　billiard…ball　of　just　the　right　size。　Another　geometrical　problem　which　Archimedes　solved　was　the　problem　as　to　the　size　of　a　triangle　which　has　equal　area　with　a　circle；　the　answer　being；　a　triangle　having　for　its　base　the　circumference　of　the　circle　and　for　its　altitude　the　radius。　Archimedes　solved　also　the　problem　of　the　relation　of　the　diameter　of　the　circle　to　its　circumference；　his　answer　being　a　close　approximation　to　the　familiar　3。1416；　which　every　tyro　in　geometry　will　recall　as　the　equivalent　of　pi。　Numerous　other　of　the　studies　of　Archimedes　having　reference　to　conic　sections；　properties　of　curves　and　spirals；　and　the　like；　are　too　technical　to　be　detailed　here。　The　extent　of　his　mathematical　knowledge；　however；　is　suggested　by　the　fact　that　he　computed　in　great　detail　the　number　of　grains　of　sand　that　would　be　required　to　cover　the　sphere　of　the　sun's　orbit；　making　certain　hypothetical　assumptions　as　to　the　size　of　the　earth　and　the　distance　of　the　sun　for　the　purposes　of　argument。　Mathematicians　find　his　computation　peculiarly　interesting　because　it　evidences　a　crude　conception　of　the　idea　of　logarithms。　From　our　present　stand…point；　the　paper　in　which　this　calculation　is　contained　has　considerable　interest　because　of　its　assumptions　as　to　celestial　mechanics。　Thus　Archimedes　starts　out　with　the　preliminary　assumption　that　the　circumference　of　the　earth　is　less　than　three　million　stadia。　It　must　be　understood　that　this　assumption　is　purely　for　the　sake　of　argument。　Archimedes　expressly　states　that　he　takes　this　number　because　it　is　〃ten　times　as　large　as　the　earth　has　been　supposed　to　be　by　certain　investigators。〃　Here；　perhaps；　the　reference　is　to　Eratosthenes；　whose　measurement　of　the　earth　we　shall　have　occasion　to　revert　to　in　a　moment。　Continuing；　Archimedes　asserts　that　the　sun　is　larger　than　the　earth；　and　the　earth　larger　than　the　moon。　In　this　assumption；　he　says；　he　is　following　the　opinion　of　the　majority　of　astronomers。　In　the　third　place；　Archimedes　assumes　that　the　diameter　of　the　sun　is　not　more　than　thirty　times　greater　than　that　of　the　moon。　Here　he　is　probably　basing　his　argument　upon　another　set　of　measurements　of　Aristarchus；　to　which；　also；　we　shall　presently　refer　more　at　length。　In　reality；　his　assumption　is　very　far　from　the　truth；　since　the　actual　diameter　of　the　sun；　as　we　now　know；　is　something　like　four　hundred　times　that　of　the　moon。　Fourth；　the　circumference　of　the　sun　is　greater　than　one　side　of　the　thousand…　faced　figure　inscribed　in　its　orbit。　The　measurement；　it　is　expressly　stated；　is　based　on　the　measurements　of　Aristarchus；　who　makes　the　diameter　of　the　sun　1/170　of　its　orbit。　Archimedes　adds；　however；　that　he　himself　has　measured　the　angle　and　that　it　appears　to　him　to　be　less　than　1/164；　and　greater　than　1/200　part　of　the　orbit。　That　is　to　say；　reduced　to　modern　terminology；　he　places　the　limit　of　the　sun's　apparent　size　between　thirty…three　minutes　and　twenty…seven　minutes　of　arc。　As　the　real　diameter　is　thirty…two　minutes；　this　calculation　is　surprisingly　exact；　considering　the　implements　then　at　command。　But　the　honor　of　first　making　it　must　be　given　to　Aristarchus　and　not　to　Archimedes。　We　need　not　follow　Archimedes　to　the　limits　of　his　incomprehensible　numbers　of　sand…grains。　The　calculation　is　chiefly　remarkable　because　it　was　made　before　the　introduction　of　the　so…called　Arabic　numerals　had　simplified　mathematical　calculations。　It　will　be　recalled　that　the　Greeks　used　letters　for　numerals；　and；　having　no　cipher；　they　soon　found　themselves　in　difficulties　when　large　numbers　were　involved。　The　Roman　system　of　numerals　simplified　the　matter　somewhat；　but　the　beautiful　simplicity　of　the　decimal　system　did　not　come　into　vogue　until　the　Middle　Ages；　as　we　shall　see。　Notwithstanding　the　difficulties；　however；　Archimedes　followed　out　his　calculations　to　the　piling　up　of　bewildering　numbers；　which　the　modern　mathematician　finds　to　be　the　consistent　outcome　of　the　problem　he　had　set　himself。　But　it　remains　to　notice　the　most　interesting　feature　of　this　document　in　which　the　calculation　of　the　sand…　grains　is　contained。　〃It　was　known　to　me；〃　says　Archimedes；　〃that　most　astronomers　understand　by　the　expression　'world'　（universe）　a　ball　of　which　the　centre　is　the　middle　point　of　the　earth；　and　of　which　the　radius　is　a　straight　line　between　the　centre　of　the　earth　and　the　sun。〃　Archimedes　himself　appears　to　accept　this　opinion　of　the　majority；it　at　least　serves　as　well　as　the　contrary　hypothesis　for　the　purpose　of　his　calculation；but　he　goes　on　to　say：　〃Aristarchus　of　Samos；　in　his　writing　against　the　astronomers；　seeks　to　establish　the　fact　that　the　world　is　really　very　different　from　this。　He　holds　the　opinion　that　the　fixed　stars　and　the　sun　are　immovable　and　that　the　earth　revolves　in　a　circular　line　about　the　sun；　the　sun　being　at　the　centre　of　this　circle。〃　This　remarkable　bit　of　testimony　establishes　beyond　question　the　position　of　Aristarchus　of　Samos　as　the　Copernicus　of　antiquity。　We　must　make　further　inquiry　as　to　the　teachings　of　the　man　who　had　gained　such　a　remarkable　insight　into　the　true　system　of　the　heavens。

ARISTARCHUS　OF　SAMOS；　THE　COPERNICUS　OF　ANTIQUITY　It　appears　that　Aristarchus　was　a　contemporary　of　Archimedes；　but　the　exact　dates　of　his　life　are　not　known。　He　was　actively　engaged　in　making　astronomical　observations　in　Samos　somewhat　before　the　middle　of　the　third　century　B。C。；　in　other　words；　just　at　the　time　when　the　activities　of　the　Alexandrian　school　were　at　their　height。　Hipparchus；　at　a　later　day；　was　enabled　to　compare　his　own　observations　with　those　made　by　Aristarchus；　and；　as　we　have　just　seen；　his　work　was　well　known　to　so　distant　a　contemporary　as　Archimedes。　Yet　the　facts　of　his　life　are　almost　a　blank　for　us；　and　of　his　writings　only　a　single　one　has　been　preserved。　That　one；　however；　is　a　most　important　and　interesting　paper　on　the　measurements　of　the　sun　and　the　moon。　Unfortunately；　this　paper　gives　us　no　direct　clew　as　to　the　opinions　of　Aristarchus　concerning　the　relative　positions　of　the　earth　and　sun。　But　the　testimony　of　Archimedes　as　to　this　is　unequivocal；　and　this　testimony　is　supported　by　other　rumors　in　themselves　less　authoritative。　In　contemplating　this　astronomer　of　Samos；　then；　we　are　in　the　presence　of　a　man　who　had　solved　in　its　essentials　the　problem　of　the　mechanism　of　the　solar　system。　It　appears　from　the　words　of　Archimedes　that　Aristarchus；　had　propounded　his　theory　in　explicit　writings。　Unquestionably；　then；　he　held　to　it　as　a　positive　doctrine；　not　as　a　mere　vague　guess。　We　shall　show；　in　a　moment；　on　what　grounds　he　based　his　opinion。　Had　his　teaching　found　vogue；　the　story　of　science　would　be　very　different　from　what　it　is。　We　should　then　have　no　tale　to　tell　of　a　Copernicus　coming　upon　the　scene　fully　seventeen　hundred　years　later　with　the　revolutionary　doctrine　that　our　world　is　not　the　centre　of　the　universe。　We　should　not　have　to　tell　of　the　persecution　of　a　Bruno　or　of　a　Galileo　for　teaching　this　doctrine　in　the　seventeenth　century　of　an　era　which　did　not　begin　till　two　hundred　years　after　the　death　of　Aristarchus。　But；　as　we　know；　the　teaching　of　the　astronomer　of　Samos　did　no