a history of science-2-第34章

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　we　shall　find　that　the　course　of　the　moon　departs　from　a　straight　line　during　that　period　by　a　measurable　distancethat：　is　to　say；　the　moon　has　been　virtually　pulled　towards　the　earth　by　an　amount　that　is　represented　by　the　difference　between　its　actual　position　at　the　end　of　the　minute　under　observation　and　the　position　it　would　occupy　had　its　course　been　tangential；　as；　according　to　the　first　law　of　motion；　it　must　have　been　had　not　some　force　deflected　it　towards　the　earth。　Measuring　the　deflection　in　questionwhich　is　equivalent　to　the　so…called　versed　sine　of　the　arc　traversedwe　have　a　basis　for　determining　the　strength　of　the　deflecting　force。　Newton　constructed　such　a　diagram；　and；　measuring　the　amount　of　the　moon's　departure　from　a　tangential　rectilinear　course　in　one　minute；　determined　this　to　be；　by　his　calculation；　thirteen　feet。　Obviously；　then；　the　force　acting　upon　the　moon　is　one　that　would　cause　that　body　to　fall　towards　the　earth　to　the　distance　of　thirteen　feet　in　the　first　minute　of　its　fall。　Would　such　be　the　force　of　gravitation　acting　at　the　distance　of　the　moon　if　the　power　of　gravitation　varies　inversely　as　the　square　of　the　distance？　That　was　the　tangible　form　in　which　the　problem　presented　itself　to　Newton。　The　mathematical　solution　of　the　problem　was　simple　enough。　It　is　based　on　a　comparison　of　the　moon's　distance　with　the　length　of　the　earth's　radius。　On　making　this　calculation；　Newton　found　that　the　pull　of　gravitationif　that　were　really　the　force　that　controls　the　moongives　that　body　a　fall　of　slightly　over　fifteen　feet　in　the　first　minute；　instead　of　thirteen　feet。　Here　was　surely　a　suggestive　approximation；　yet；　on　the　other　band；　the　discrepancy　seemed　to　be　too　great　to　warrant　him　in　the　supposition　that　he　had　found　the　true　solution。　He　therefore　dismissed　the　matter　from　his　mind　for　the　time　being；　nor　did　he　return　to　it　definitely　for　some　years。　｛illustration　caption　=　　DIAGRAM　TO　ILLUSTRATE　NEWTON'S　LAW　OF　GRAVITATION　（E　represents　the　earth　and　A　the　moon。　Were　the　earth's　pull　on　the　moon　to　cease；　the　moon's　inertia　would　cause　it　to　take　the　tangential　course；　AB。　On　the　other　hand；　were　the　moon's　motion　to　be　stopped　for　an　instant；　the　moon　would　fall　directly　towards　the　earth；　along　the　line　AD。　The　moon's　actual　orbit；　resulting　from　these　component　forces；　is　AC。　Let　AC　represent　the　actual　flight　of　the　moon　in　one　minute。　Then　BC；　which　is　obviously　equal　to　AD；　represents　the　distance　which　the　moon　virtually　falls　towards　the　earth　in　one　minute。　Actual　computation；　based　on　measurements　of　the　moon's　orbit；　showed　this　distance　to　be　about　fifteen　feet。　Another　computation　showed　that　this　is　the　distance　that　the　moon　would　fall　towards　the　earth　under　the　influence　of　gravity；　on　the　supposition　that　the　force　of　gravity　decreases　inversely　with　the　square　of　the　distance；　the　basis　of　comparison　being　furnished　by　falling　bodies　at　the　surface　of　the　earth。　Theory　and　observations　thus　coinciding；　Newton　was　justified　in　declaring　that　the　force　that　pulls　the　moon　towards　the　earth　and　keeps　it　in　its　orbit；　is　the　familiar　force　of　gravity；　and　that　this　varies　inversely　as　the　square　of　the　distance。）｝　It　was　to　appear　in　due　time　that　Newton's　hypothesis　was　perfectly　valid　and　that　his　method　of　attempted　demonstration　was　equally　so。　The　difficulty　was　that　the　earth's　proper　dimensions　were　not　at　that　time　known。　A　wrong　estimate　of　the　earth's　size　vitiated　all　the　other　calculations　involved；　since　the　measurement　of　the　moon's　distance　depends　upon　the　observation　of　the　parallax；　which　cannot　lead　to　a　correct　computation　unless　the　length　of　the　earth's　radius　is　accurately　known。　Newton's　first　calculation　was　made　as　early　as　1666；　and　it　was　not　until　1682　that　his　attention　was　called　to　a　new　and　apparently　accurate　measurement　of　a　degree　of　the　earth's　meridian　made　by　the　French　astronomer　Picard。　The　new　measurement　made　a　degree　of　the　earth's　surface　69。10　miles；　instead　of　sixty　miles。　Learning　of　this　materially　altered　calculation　as　to　the　earth's　size；　Newton　was　led　to　take　up　again　his　problem　of　the　falling　moon。　As　he　proceeded　with　his　computation；　it　became　more　and　more　certain　that　this　time　the　result　was　to　harmonize　with　the　observed　facts。　As　the　story　goes；　he　was　so　completely　overwhelmed　with　emotion　that　he　was　forced　to　ask　a　friend　to　complete　the　simple　calculation。　That　story　may　well　be　true；　for；　simple　though　the　computation　was；　its　result　was　perhaps　the　most　wonderful　demonstration　hitherto　achieved　in　the　entire　field　of　science。　Now　at　last　it　was　known　that　the　force　of　gravitation　operates　at　the　distance　of　the　moon；　and　holds　that　body　in　its　elliptical　orbit；　and　it　required　but　a　slight　effort　of　the　imagination　to　assume　that　the　force　which　operates　through　such　a　reach　of　space　extends　its　influence　yet　more　widely。　That　such　is　really　the　case　was　demonstrated　presently　through　calculations　as　to　the　moons　of　Jupiter　and　by　similar　computations　regarding　the　orbital　motions　of　the　various　planets。　All　results　harmonizing；　Newton　was　justified　in　reaching　the　conclusion　that　gravitation　is　a　universal　property　of　matter。　It　remained；　as　we　shall　see；　for　nineteenth…century　scientists　to　prove　that　the　same　force　actually　operates　upon　the　stars；　though　it　should　be　added　that　this　demonstration　merely　fortified　a　belief　that　had　already　found　full　acceptance。　Having　thus　epitomized　Newton's　discovery；　we　must　now　take　up　the　steps　of　his　progress　somewhat　in　detail；　and　state　his　theories　and　their　demonstration　in　his　own　words。　Proposition　IV。；　theorem　4；　of　his　Principia　is　as　follows：　〃That　the　moon　gravitates　towards　the　earth　and　by　the　force　of　gravity　is　continually　drawn　off　from　a　rectilinear　motion　and　retained　in　its　orbit。　〃The　mean　distance　of　the　moon　from　the　earth；　in　the　syzygies　in　semi…diameters　of　the　earth；　is；　according　to　Ptolemy　and　most　astronomers；　59；　according　to　Vendelin　and　Huygens；　60；　to　Copernicus；　60　1/3；　to　Street；　60　2/3；　and　to　Tycho；　56　1/2。　But　Tycho；　and　all　that　follow　his　tables　of　refractions；　making　the　refractions　of　the　sun　and　moon　（altogether　against　the　nature　of　light）　to　exceed　the　refractions　of　the　fixed　stars；　and　that　by　four　or　five　minutes　NEAR　THE　HORIZON；　did　thereby　increase　the　moon's　HORIZONTAL　parallax　by　a　like　number　of　minutes；　that　is；　by　a　twelfth　or　fifteenth　part　of　the　whole　parallax。　Correct　this　error　and　the　distance　will　become　about　60　1/2　semi…diameters　of　the　earth；　near　to　what　others　have　assigned。　Let　us　assume　the　mean　distance　of　60　diameters　in　the　syzygies；　and　suppose　one　revolution　of　the　moon；　in　respect　to　the　fixed　stars；　to　be　completed　in　27d。　7h。　43'；　as　astronomers　have　determined；　and　the　circumference　of　the　earth　to　amount　to　123；249；600　Paris　feet；　as　the　French　have　found　by　mensuration。　And　now；　if　we　imagine　the　moon；　deprived　of　all　motion；　to　be　let　go；　so　as　to　descend　towards　the　earth　with　the　impulse　of　all　that　force　by　which　（by　Cor。　Prop。　iii。）　it　is　retained　in　its　orb；　it　will　in　the　space　of　one　minute　of　time　describe　in　its　fall　15　1/12　Paris　feet。　For　the　versed　sine　of　that　arc　which　the　moon；　in　the　space　of　one　minute　of　time；　would　by　its　mean　motion　describe　at　the　distance　of　sixty　semi…diameters　of　the　earth；　is　nearly　15　1/12　Paris　feet；　or　more　accurately　15　feet；　1　inch；　1　line　4/9。　Wherefore；　since　that　force；　in　approaching　the　earth；　increases　in　the　reciprocal…duplicate　proportion　of　the　distance；　and　upon　that　account；　at　the　surface　of　the　earth；　is　60　x　60　times　greater　than　at　the　moon；　a　body　in　our　regions；　falling　with　that　force；　ought　in　the　space　of　one　minute　of　time　to　describe　60　x　60　x　15　1/12　Paris　feet；　and　in　the　space　of　one　second　of　time；　to　describe　15　1/12　of　those　feet；　or　more　accurately；　15　feet；　1　inch；　1　line　4/9。　And　with　this　very　force　we　actually　find　that　bodies　here　upon　earth　do　really　descend；　for　a　pendulum　oscillating　seconds　in　the　latitude　of　Paris　will　be　3　Paris　feet；　and　8　lines　1/2　in　length；　as　Mr。　Huygens　has　observed。　And　the　space　which　a　heavy　body　describes　by　falling　in　one　second　of　time　is　to　half　the　length　of　the　pendulum　in　the　duplicate　ratio　of　the　circumference　of　a　circle　to　its　diameter　（as　Mr。　Huygens　has　also　shown）；　and　is　therefore　15　Paris　feet；　1　inch；　1　line　4/9。　And　therefore　the　force　by　which　the　moon　is　retained　in　its　orbit　is　that　very　same　force　which　we　commonly　call　gravity；　for；　were　gravity　another　force　different　from　that；　then　bodies　descending　to　the　earth　with　the　joint　impulse　of　both　forces　would　fall　with　a　double　velocity；　and　in　the　space　of　one　second　of　time　would　describe　30　1/6　Paris　feet；　altogether　against　experience。〃＇1＇　All　this　is　beautifully　clear；　and　its　validity　has　never　in　recent　generations　been　called　in　question；　yet　it　should　be　explained　that　the　argument　does　not　amount　to　an　actually　indisputable　demonstration。　It　is　at　least　possible　that　the　coincidence　between　the　observed　and　computed　motion　of　the　moon　may　be　a　mere　coincidence　and　nothing　more。　This　probability；　however；　is　so　remote　that　Newton　is　fully　justified　in　disregarding　it；　and；　as　has　been　said；　all　subsequent　generations　have　accepted　the　computation　as　demonstrative。　Let　us　produce　now　Newton's　further　computations　as　to　the　other　planetary　bodies；　passing　on　to　his　final　conclusion　that　gravity　is　a　universal　force。　　　　　　　　　　　〃PROPOSITION　V。；　THEOREM　V。　〃That　the　circumjovial　planets　gravitate　towards　Jupiter；　the　circumsaturnal　towards　Saturn；　the　circumsolar　towards　the　sun；　and　by　the　forces　of　their　gravity　are　drawn　off　from　rectilinear　motions；　and　retained　in　curvilinear　orbits。

〃For　the　revolutions　of　the　circumjovial　planets　about　Jupiter；　of　the　circumsaturnal　about　Saturn；　and　of　Mercury　and　Venus　and　the　other　circumsolar　planets　about　the　sun；　are　appearances　of　the　same　sort　with　the　revolution　of　the　moon　about　the　earth；　and　therefore；　by　Rule　ii。；　must　be　owing　to　the　same　sort　of　causes；　especially　since　it　has　been　demonstrated　that　the　forces　upon　which　those　revolutions　depend　tend　to　the　centres　of　Jupiter；　of　Saturn；　and　of　the　sun；　and　that　those　forces；　in　receding　from　Jupiter；　from　Saturn；　and　from　the　sun；　decrease　in　the　same　proportion；　and　according　to　the　same　law；　as　the　force　of　gravity　does　in　receding　from　the　earth。　〃COR。　1。There　is；　therefore；　a　power　of　gravity　tending　to　all　the　planets；　for　doubtless　Venus；　Mercury；　and　the　rest　are　bodies　of　the　same　sort　with　Jupiter　and　Saturn。　And　since　all　attraction　（by　Law　iii。）　is　mutual；　Jupiter　will　therefore　gravitate　towards　all　his　own　satellites；　Saturn　towards　his；　the　earth　towards　the　moon；　and　the　sun　towards　all　the　primary　planets。　〃COR。　2。The　force　of　gravity　which　tends　to　any　one　planet　is　reciprocally　as　the　square　of　the　distance　of　places　from　the　planet's　centre。　〃COR。　3。All　the　planets　do　mutually　gravitate　towards　one　another；　by　Cor。　1　and　2；　and　hence　it　is　that　Jupiter　and　Saturn；　when　near　their　conjunction；　by　their　mutual　attractions　sensibly　disturb　each　other's　motions。　So　the　sun　disturbs　the　motions　of　the　moon；　and　both　sun　and　moon　disturb　our　sea；　as　we　shall　hereafter　explain。　　　　　　　　　　　〃SCHOLIU