a history of science-1-第30章

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ls　and　sweet　savours；　with　his　finger　he　did　draw　lines　upon　his　naked　body：　so　far　was　he　taken　from　himself；　and　brought　into　an　ecstasy　or　trance；　with　the　delight　he　had　in　the　study　of　geometry；　and　truly　ravished　with　the　love　of　the　Muses。　But　amongst　many　notable　things　he　devised；　it　appeareth；　that　he　most　esteemed　the　demonstration　of　the　proportion　between　the　cylinder　（to　wit；　the　round　column）　and　the　sphere　or　globe　contained　in　the　same：　for　he　prayed　his　kinsmen　and　friends；　that　after　his　death　they　would　put　a　cylinder　upon　his　tomb；　containing　a　massy　sphere；　with　an　inscription　of　the　proportion；　whereof　the　continent　exceedeth　the　thing　contained。〃＇2＇　It　should　be　observed　that　neither　Polybius　nor　Plutarch　mentions　the　use　of　burning…glasses　in　connection　with　the　siege　of　Syracuse；　nor　indeed　are　these　referred　to　by　any　other　ancient　writer　of　authority。　Nevertheless；　a　story　gained　credence　down　to　a　late　day　to　the　effect　that　Archimedes　had　set　fire　to　the　fleet　of　the　enemy　with　the　aid　of　concave　mirrors。　An　experiment　was　made　by　Sir　Isaac　Newton　to　show　the　possibility　of　a　phenomenon　so　well　in　accord　with　the　genius　of　Archimedes；　but　the　silence　of　all　the　early　authorities　makes　it　more　than　doubtful　whether　any　such　expedient　was　really　adopted。　It　will　be　observed　that　the　chief　principle　involved　in　all　these　mechanisms　was　a　capacity　to　transmit　great　power　through　levers　and　pulleys；　and　this　brings　us　to　the　most　important　field　of　the　Syracusan　philosopher's　activity。　It　was　as　a　student　of　the　lever　and　the　pulley　that　Archimedes　was　led　to　some　of　his　greatest　mechanical　discoveries。　He　is　even　credited　with　being　the　discoverer　of　the　compound　pulley。　More　likely　he　was　its　developer　only；　since　the　principle　of　the　pulley　was　known　to　the　old　Babylonians；　as　their　sculptures　testify。　But　there　is　no　reason　to　doubt　the　general　outlines　of　the　story　that　Archimedes　astounded　King　Hiero　by　proving　that；　with　the　aid　of　multiple　pulleys；　the　strength　of　one　man　could　suffice　to　drag　the　largest　ship　from　its　moorings。　The　property　of　the　lever；　from　its　fundamental　principle；　was　studied　by　him；　beginning　with　the　self…　evident　fact　that　〃equal　bodies　at　the　ends　of　the　equal　arms　of　a　rod；　supported　on　its　middle　point；　will　balance　each　other〃；　or；　what　amounts　to　the　same　thing　stated　in　another　way；　a　regular　cylinder　of　uniform　matter　will　balance　at　its　middle　point。　From　this　starting…point　he　elaborated　the　subject　on　such　clear　and　satisfactory　principles　that　they　stand　to…day　practically　unchanged　and　with　few　additions。　From　all　his　studies　and　experiments　he　finally　formulated　the　principle　that　〃bodies　will　be　in　equilibrio　when　their　distance　from　the　fulcrum　or　point　of　support　is　inversely　as　their　weight。〃　He　is　credited　with　having　summed　up　his　estimate　of　the　capabilities　of　the　lever　with　the　well…known　expression；　〃Give　me　a　fulcrum　on　which　to　rest　or　a　place　on　which　to　stand；　and　I　will　move　the　earth。〃　But　perhaps　the　feat　of　all　others　that　most　appealed　to　the　imagination　of　his　contemporaries；　and　possibly　also　the　one　that　had　the　greatest　bearing　upon　the　position　of　Archimedes　as　a　scientific　discoverer；　was　the　one　made　familiar　through　the　tale　of　the　crown　of　Hiero。　This　crown；　so　the　story　goes；　was　supposed　to　be　made　of　solid　gold；　but　King　Hiero　for　some　reason　suspected　the　honesty　of　the　jeweller；　and　desired　to　know　if　Archimedes　could　devise　a　way　of　testing　the　question　without　injuring　the　crown。　Greek　imagination　seldom　spoiled　a　story　in　the　telling；　and　in　this　case　the　tale　was　allowed　to　take　on　the　most　picturesque　of　phases。　The　philosopher；　we　are　assured；　pondered　the　problem　for　a　long　time　without　succeeding；　but　one　day　as　he　stepped　into　a　bath；　his　attention　was　attracted　by　the　overflow　of　water。　A　new　train　of　ideas　was　started　in　his　ever…receptive　brain。　Wild　with　enthusiasm　he　sprang　from　the　bath；　and；　forgetting　his　robe；　dashed　along　the　streets　of　Syracuse；　shouting：　〃Eureka！　Eureka！〃　（I　have　found　it！）　The　thought　that　had　come　into　his　mind　was　this：　That　any　heavy　substance　must　have　a　bulk　proportionate　to　its　weight；　that　gold　and　silver　differ　in　weight；　bulk　for　bulk；　and　that　the　way　to　test　the　bulk　of　such　an　irregular　object　as　a　crown　was　to　immerse　it　in　water。　The　experiment　was　made。　A　lump　of　pure　gold　of　the　weight　of　the　crown　was　immersed　in　a　certain　receptacle　filled　with　water；　and　the　overflow　noted。　Then　a　lump　of　pure　silver　of　the　same　weight　was　similarly　immersed；　lastly　the　crown　itself　was　immersed；　and　of　coursefor　the　story　must　not　lack　its　dramatic　sequelwas　found　bulkier　than　its　weight　of　pure　gold。　Thus　the　genius　that　could　balk　warriors　and　armies　could　also　foil　the　wiles　of　the　silversmith。　Whatever　the　truth　of　this　picturesque　narrative；　the　fact　remains　that　some；　such　experiments　as　these　must　have　paved　the　way　for　perhaps　the　greatest　of　all　the　studies　of　Archimedesthose　that　relate　to　the　buoyancy　of　water。　Leaving　the　field　of　fable；　we　must　now　examine　these　with　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