WhenwepasstothatthirdgreatAthenianteacher，Aristotle，thecaseisfardifferent。HerewasamanwhosenamewastobereceivedasalmostasynonymforGreekscienceformorethanathousandyearsafterhisdeath。AllthroughtheMiddleAgeshiswritingsweretobeacceptedasvirtuallythelastwordregardingtheproblemsofnature。Weshallseethathisfollowersactuallypreferredhismandatetothetestimonyoftheirownsenses。Weshallsee，further，thatmodernscienceprogressedsomewhatinproportionasitoverthrewtheAristoteliandogmas。Butthetraditionsofseventeenoreighteencenturiesarenoteasilysetaside，anditisperhapsnottoomuchtosaythatthenameofAristotlestands，eveninourowntime，asvaguelyrepresentativeinthepopularmindofallthatwashighestandbestinthescienceofantiquity。Yet，perhaps，itwouldnotbegoingtoofartoassertthatsomethinglikeareversalofthisjudgmentwouldbenearerthetruth。Aristotledid，indeed，bringtogetheragreatmassoffactsregardinganimalsinhisworkonnaturalhistory，which，beingpreserved，hasbeendeemedtoentitleitsauthortobecalledthe"fatherofzoology。"Butthereisnoreasontosupposethatanyconsiderableportionofthisworkcontainedmatterthatwasnovel，orrecordedobservationsthatwereoriginalwithAristotle；andtheclassificationsthereoutlinedareatbestbutavagueforeshadowingoftheelaborationofthescience。Suchasitis，however，thenaturalhistorystandstothecreditoftheStagirite。Hemustbecredited，too，withaclearenunciationofonemostimportantscientificdoctrine——namely，thedoctrineofthesphericalfigureoftheearth。WehavealreadyseenthatthistheoryoriginatedwiththePythagoreanphilosophersoutinItaly。Wehaveseen，too，thatthedoctrinehadnotmadeitswayinAtticainthetimeofAnaxagoras。Butintheinterveningcenturyithadgainedwidecurrency，elsesoessentiallyconservativeathinkerasAristotlewouldscarcelyhaveacceptedit。Hedidacceptit，however，andgavethedoctrineclearestandmostpreciseexpression。Herearehiswords:[2]
"Astothefigureoftheearthitmustnecessarilybespherical……Ifitwerenotso，theeclipsesofthemoonwouldnothavesuchsectionsastheyhave。Forintheconfigurationsinthecourseofamonththedeficientparttakesalldifferentshapes；itisstraight，andconcave，andconvex；butineclipsesitalwayshasthelineofdivisionsconvex；wherefore，sincethemooniseclipsedinconsequenceoftheinterpositionoftheearth，theperipheryoftheearthmustbethecauseofthisbyhavingasphericalform。Andagain，fromtheappearanceofthestarsitisclear，notonlythattheearthisround，butthatitssizeisnotverylarge；forwhenwemakeasmallremovaltothesouthorthenorth，thecircleofthehorizonbecomespalpablydifferent，sothatthestarsoverheadundergoagreatchange，andarenotthesametothosethattravelinthenorthandtothesouth。ForsomestarsareseeninEgyptoratCyprus，butarenotseeninthecountriestothenorthofthese；andthestarsthatinthenortharevisiblewhiletheymakeacompletecircuit，thereundergoasetting。Sothatfromthisitismanifest，notonlythattheformoftheearthisround，butalsothatitisapartofanotverylargesphere；forotherwisethedifferencewouldnotbesoobvioustopersonsmakingsosmallachangeofplace。WhereforewemayjudgethatthosepersonswhoconnecttheregionintheneighborhoodofthepillarsofHerculeswiththattowardsIndia，andwhoassertthatinthiswaytheseaisone，donotassertthingsveryimprobable。Theyconfirmthisconjecturemoreoverbytheelephants，whicharesaidtobeofthesamespeciestowardseachextreme；asifthiscircumstancewasaconsequenceoftheconjunctionoftheextremes。Themathematicianswhotrytocalculatethemeasureofthecircumference，makeitamounttofourhundredthousandstadia；
whencewecollectthattheearthisnotonlyspherical，butisnotlargecomparedwiththemagnitudeoftheotherstars。"
ButingivingfullmeedofpraisetoAristotleforthepromulgationofthisdoctrineofthesphericityoftheearth，itmustunfortunatelybeaddedthattheconservativephilosopherpausedwithouttakingoneotherimportantstep。Hecouldnotaccept，but，onthecontrary，heexpresslyrepudiated，thedoctrineoftheearth’smotion。WehaveseenthatthisideaalsowasapartofthePythagoreandoctrine，andweshallhaveoccasiontodwellmoreatlengthonthispointinasucceedingchapter。IthasevenbeencontendedbysomecriticsthatitwastheadverseconvictionofthePeripateticphilosopherwhich，morethananyothersingleinfluence，tendedtoretardtheprogressofthetruedoctrineregardingthemechanismoftheheavens。
Aristotleacceptedthesphericityoftheearth，andthatdoctrinebecameacommonplaceofscientificknowledge，andsocontinuedthroughoutclassicalantiquity。ButAristotlerejectedthedoctrineoftheearth’smotion，andthatdoctrine，thoughpromulgatedactivelybyafewcontemporariesandimmediatesuccessorsoftheStagirite，wasthendoomedtosinkoutofviewformorethanathousandyears。IfitbeacorrectassumptionthattheinfluenceofAristotlewas，inalargemeasure，responsibleforthisresult，thenweshallperhapsnotbefarastrayinassumingthatthegreatfounderofthePeripateticschoolwas，onthewhole，moreinstrumentalinretardingtheprogressofastronomicalsciencethatanyotheronemanthateverlived。
ThefieldofscienceinwhichAristotlewaspre-eminentlyapathfinderiszoology。Hiswritingsonnaturalhistoryhavelargelybeenpreserved，andtheyconstitutebyfarthemostimportantcontributiontothesubjectthathascomedowntousfromantiquity。TheyshowusthatAristotlehadgainedpossessionofthewidestrangeoffactsregardingtheanimalkingdom，and，whatisfarmoreimportant，hadattemptedtoclassifythesefacts。Insodoinghebecamethefounderofsystematiczoology。
Aristotle’sclassificationoftheanimalkingdomwasknownandstudiedthroughouttheMiddleAges，and，infact，remainedinvogueuntilsupersededbythatofCuvierinthenineteenthcentury。ItisnottobesupposedthatallthetermsofAristotle’sclassificationoriginatedwithhim。Someofthedivisionsaretoopatenttohaveescapedtheobservationofhispredecessors。Thus，forexample，thedistinctionbetweenbirdsandfishesasseparateclassesofanimalsissoobviousthatitmustappealtoachildortoasavage。ButtheeffortsofAristotleextended，asweshallsee，tolesspatentgeneralizations。Attheveryoutset，hisgranddivisionoftheanimalkingdomintoblood-bearingandbloodlessanimalsimpliesaverybroadandphilosophicalconceptionoftheentireanimalkingdom。Themodernphysiologistdoesnotaccepttheclassification，inasmuchasitisnowknownthatcolorlessfluidsperformthefunctionsofbloodforallthelowerorganisms。ButthefactremainsthatAristotle’sgranddivisionscorrespondtothegranddivisionsoftheLamarckiansystem——vertebratesandinvertebrates——whicheveryonenowaccepts。Aristotle，aswehavesaid，basedhisclassificationuponobservationoftheblood；Lamarckwasguidedbyastudyoftheskeleton。Thefactthatsuchdiversepointsofviewcoulddirecttheobservertowardsthesameresultgives，inferentially，asuggestivelessoninwhatthemodernphysiologistcallsthehomologiesofpartsoftheorganism。
Aristotledivideshisso-calledblood-bearinganimalsintofiveclasses:1Four-footedanimalsthatbringforththeiryoungalive；2birds；3egg-layingfour-footedanimalsincludingwhatmodernnaturalistscallreptilesandamphibians；4whalesandtheirallies；5fishes。Thisclassification，aswillbeobserved，isnotsoveryfarafieldfromthemoderndivisionsintomammals，birds，reptiles，amphibians，andfishes。ThatAristotleshouldhaverecognizedthefundamentaldistinctionbetweenfishesandthefish-likewhales，dolphins，andporpoisesprovesthefarfromsuperficialcharacterofhisstudies。
Aristotleknewthattheseanimalsbreathebymeansoflungsandthattheyproducelivingyoung。Herecognized，therefore，theiraffinitywithhisfirstclassofanimals，evenifhedidnot，likethemodernnaturalist，considertheseaffinitiescloseenoughtojustifybringingthetwotypestogetherintoasingleclass。
ThebloodlessanimalswerealsodividedbyAristotleintofiveclasses——namely:1Cephalopodatheoctopus，cuttle-fish，etc。；2weak-shelledanimalscrabs，etc。；3insectsandtheiralliesincludingvariousforms，suchasspidersandcentipedes，whichthemodernclassifierpreferstoplacebythemselves；4hard-shelledanimalsclams，oysters，snails，etc。；5aconglomerategroupofmarineforms，includingstar-fish，sea-urchins，andvariousanomalousformsthatwereregardedaslinkingtheanimaltothevegetableworlds。ThisclassificationofthelowerformsofanimallifecontinuedinvogueuntilCuviersubstitutedforithisfamousgroupingintoarticulates，mollusks，andradiates；whichgroupinginturnwasinpartsupersededlaterinthenineteenthcentury。
WhatAristotledidfortheanimalkingdomhispupil，Theophrastus，didinsomemeasureforthevegetablekingdom。
Theophrastus，however，wasmuchlessaclassifierthanhismaster，andhisworkonbotany，calledTheNaturalHistoryofDevelopment，payscomparativelyslightattentiontotheoreticalquestions。Itdealslargelywithsuchpracticalitiesasthemakingofcharcoal，ofpitch，andofresin，andtheeffectsofvariousplantsontheanimalorganismwhentakenasfoodsorasmedicines。InthisregardtheworkofTheophrastus，ismorenearlyakintothenaturalhistoryofthefamousRomancompiler，Pliny。Itremained，however，throughoutantiquityasthemostimportantworkonitssubject，anditentitlesTheophrastustobecalledthe"fatherofbotany。"Theophrastusdealsalsowiththemineralkingdomaftermuchthesamefashion，andhereagainhisworkisthemostnotablethatwasproducedinantiquity。
Weareenteringnowuponthemostimportantscientificepochofantiquity。WhenAristotleandTheophrastuspassedfromthescene，Athensceasedtobeinanysensethescientificcentreoftheworld。Thatcitystillretaineditsreminiscentglory，andcannotbeignoredinthehistoryofculture，butnogreatscientificleaderwaseveragaintobebornortotakeuphispermanentabodewithintheconfinesofGreeceproper。Withalmostcataclysmicsuddenness，anewintellectualcentreappearedonthesouthshoreoftheMediterranean。ThiswasthecityofAlexandria，acitywhichAlexandertheGreathadfoundedduringhisbriefvisittoEgypt，andwhichbecamethecapitalofPtolemySoterwhenhechoseEgyptashisportionofthedismemberedempireofthegreatMacedonian。PtolemyhadbeenwithhismasterintheEast，andwaswithhiminBabyloniawhenhedied。HehadthereforecomepersonallyincontactwithBabyloniancivilization，andwecannotdoubtthatthishadamostimportantinfluenceuponhislife，andthroughhimuponthenewcivilizationoftheWest。Inpointofculture，AlexandriamustberegardedasthesuccessorofBabylon，scarcelylessdirectlythanofGreece。FollowingtheBabylonianmodel，Ptolemyerectedagreatmuseumandbegancollectingalibrary。Beforehisdeathitwassaidthathehadcollectednofewerthantwohundredthousandmanuscripts。Hehadgatheredalsoacompanyofgreatteachersandfoundedaschoolofsciencewhich，ashasjustbeensaid，madeAlexandriatheculture-centreoftheworld。
Athensinthedayofherprimehadknownnothingquitelikethis。
SuchprivatecitizensasAristotleareknowntohavehadlibraries，buttherewerenogreatpubliccollectionsofbooksinAthens，orinanyotherpartoftheGreekdomain，untilPtolemyfoundedhisfamouslibrary。Asiswellknown，suchlibrarieshadexistedinBabyloniaforthousandsofyears。ThecharacterwhichthePtolemaicepochtookonwasnodoubtduetoBabylonianinfluence，butquiteasmuchtothepersonalexperienceofPtolemyhimselfasanexplorerintheFarEast。ThemarvellousconqueringjourneyofAlexanderhadenormouslywidenedthehorizonoftheGreekgeographer，andstimulatedtheimaginationofallranksofthepeople，Itwasbutnatural，then，thatgeographyanditsparentscienceastronomyshouldoccupytheattentionofthebestmindsinthissucceedingepoch。Inpointoffact，suchacompanyofstar-gazersandearth-measurerscameuponthesceneinthisthirdcenturyB。C。ashadneverbeforeexistedanywhereintheworld。Thewholetrendofthetimewastowardsmechanics。Itwasasifthegreatestthinkershadsquarelyfacedaboutfromtheattitudeofthemysticalphilosophersoftheprecedingcentury，andhadsetthemselvesthetaskofsolvingallthemechanicalriddlesoftheuniverse，Theynolongertroubledthemselvesaboutproblemsof"being"and"becoming"；theygavebutlittleheedtometaphysicalsubtleties；theydemandedthattheirthoughtsshouldbegaugedbyobjectiverealities。Hencetherearoseasuccessionofgreatgeometers，andtheirconceptionswereappliedtotheconstructionofnewmechanicalcontrivancesontheonehand，andtotheelaborationoftheoriesofsiderealmechanicsontheother。
ThewonderfulcompanyofmenwhoperformedthefeatsthatareabouttoberecordeddidnotallfindtheirhomeinAlexandria，tobesure；buttheyallcamemoreorlessundertheAlexandrianinfluence。Weshallseethattherearetwootherimportantcentres；oneoutinSicily，almostattheconfinesoftheGreekterritoryinthewest；theotherinAsiaMinor，notablyontheislandofSamos——theislandwhich，itwillberecalled，wasatanearlierdaythebirthplaceofPythagoras。ButwhereasinthepreviouscenturycolonistsfromtheconfinesofthecivilizedworldcametoAthens，nowalleyesturnedtowardsAlexandria，andsoimprovedwerethefacilitiesforcommunicationthatnodoubtthediscoveriesofonecoterieofworkerswereknowntoalltheothersmuchmorequicklythanhadeverbeenpossiblebefore。Welearn，forexample，thatthestudiesofAristarchusofSamosweredefinitelyknowntoArchimedesofSyracuse，outinSicily。
Indeed，asweshallsee，itisthroughachancereferencepreservedinoneofthewritingsofArchimedesthatoneofthemostimportantspeculationsofAristarchusismadeknowntous。
ThisillustratessufficientlytheintercommunicationthroughwhichthethoughtoftheAlexandrianepochwasbroughtintoasinglechannel。Wenolonger，asinthedayoftheearlierschoolsofGreekphilosophy，haveisolatedgroupsofthinkers。
Thescientificdramaisnowplayedoutuponasinglestage；andifwepass，asweshallinthepresentchapter，fromAlexandriatoSyracuseandfromSyracusetoSamos，theshiftofscenesdoesnoviolencetothedramaticunities。
NotwithstandingthenumberofgreatworkerswhowerenotproperlyAlexandrians，nonethelesstheepochiswithproprietytermedAlexandrian。NotmerelyinthethirdcenturyB。C。，butthroughoutthelapseofatleastfoursucceedingcenturies，thecityofAlexanderandthePtolemiescontinuedtoholditsplaceastheundisputedculture-centreoftheworld。DuringthatperiodRomerosetoitspinnacleofgloryandbegantodecline，withouteverchallengingtheintellectualsupremacyoftheEgyptiancity。Weshallsee，inalaterchapter，thattheAlexandrianinfluenceswerepassedontotheMohammedanconquerors，andeveryoneisawarethatwhenAlexandriawasfinallyoverthrownitsplacewastakenbyanotherGreekcity，ByzantiumorConstantinople。ButthattransferdidnotoccuruntilAlexandriahadenjoyedalongerperiodofsupremacyasanintellectualcentrethanhadperhapseverbeforebeengrantedtoanycity，withthepossibleexceptionofBabylon。
EUCLIDABOUT300B。C。
OurpresentconcerniswiththatfirstwonderfuldevelopmentofscientificactivitywhichbeganunderthefirstPtolemy，andwhichpresents，inthecourseofthefirstcenturyofAlexandrianinfluence，themostremarkablecoterieofscientificworkersandthinkersthatantiquityproduced。Theearliestgroupofthesenewleadersinsciencehadatitsheadamanwhosenamehasbeenahouseholdwordeversince。ThiswasEuclid，thefatherofsystematicgeometry。Traditionhaspreservedtousbutlittleofthepersonalityofthisremarkableteacher；but，ontheotherhand，hismostimportantworkhascomedowntousinitsentirety。TheElementsofGeometry，withwhichthenameofEuclidisassociatedinthemindofeveryschool-boy，presentedthechiefpropositionsofitssubjectinsosimpleandlogicalaformthattheworkremainedatextbookeverywhereformorethantwothousandyears。Indeeditisonlynowbeginningtobesuperseded。
ItisnottwentyyearssinceEnglishmathematicianscoulddeplorethefactthat，despitecertainratherobviousdefectsoftheworkofEuclid，nobettertextbookthanthiswasavailable。Euclid’swork，ofcourse，givesexpressiontomuchknowledgethatdidnotoriginatewithhim。WehavealreadyseenthatseveralimportantpropositionsofgeometryhadbeendevelopedbyThales，andonebyPythagoras，andthattherudimentsofthesubjectwereatleastasoldasEgyptiancivilization。PreciselyhowmuchEuclidaddedthroughhisowninvestigationscannotbeascertained。Itseemsprobablethathewasadiffuserofknowledgeratherthananoriginator，butasagreatteacherhisfameissecure。Heiscreditedwithanepigramwhichinitselfmightinsurehimperpetuityoffame:"Thereisnoroyalroadtogeometry，"washisanswertoPtolemywhenthatrulerhadquestionedwhethertheElementsmightnotbesimplified。Doubtlessthis，likemostsimilargoodsayings，isapocryphal；butwhoeverinventedithasmadetheworldhisdebtor。
ThecatholicityofPtolemy’stastesledhim，naturallyenough，tocultivatethebiologicalnolessthanthephysicalsciences。Inparticularhisinfluencepermittedanepochaladvanceinthefieldofmedicine。Twoanatomistsbecamefamousthroughtheinvestigationstheywerepermittedtomakeunderthepatronageoftheenlightenedruler。TheseearliestofreallyscientificinvestigatorsofthemechanismofthehumanbodywerenamedHerophilusandErasistratus。Thesetwoanatomistsgainedtheirknowledgebythedissectionofhumanbodiestheirsarethefirstrecordsthatwehaveofsuchpractices，andKingPtolemyhimselfissaidtohavebeenpresentatsomeofthesedissections。Theywerethefirsttodiscoverthatthenerve-trunkshavetheirorigininthebrainandspinalcord，andtheyarecreditedalsowiththediscoverythatthesenerve-trunksareoftwodifferentkinds——onetoconveymotor，andtheothersensoryimpulses。Theydiscovered，described，andnamedthecoveringsofthebrain。ThenameofHerophilusisstillappliedbyanatomists，inhonorofthediscoverer，tooneofthesinusesorlargecanalsthatconveythevenousbloodfromthehead。Herophilusalsonoticedanddescribedfourcavitiesorventriclesinthebrain，andreachedtheconclusionthatoneoftheseventricleswastheseatofthesoul——abeliefshareduntilcomparativelyrecenttimesbymanyphysiologists。Hemadealsoacarefulandfairlyaccuratestudyoftheanatomyoftheeye，agreatlyimprovedtheoldoperationforcataract。
Withtheincreasedknowledgeofanatomycamealsocorrespondingadvancesinsurgery，andmanyexperimentaloperationsaresaidtohavebeenperformeduponcondemnedcriminalswhowerehandedovertothesurgeonsbythePtolemies。Whilemanymodernwritershaveattemptedtodiscredittheseassertions，itisnotimprobablethatsuchoperationswereperformed。Inanagewhenhumanlifewasheldsocheap，andamongapeopleaccustomedtotorturingcondemnedprisonersforcomparativelyslightoffences，itisnotunlikelythatthesurgeonswereallowedtoinflictperhapslesspainfultorturesinthecauseofscience。Furthermore，weknowthatcondemnedcriminalsweresometimeshandedovertothemedicalprofessiontobe"operateduponandkilledinwhateverwaytheythoughtbest"evenaslateasthesixteenthcentury。
Tertullian[1]probablyexaggerates，however，whenheputsthenumberofsuchvictimsinAlexandriaatsixhundred。
HadHerophilusandErasistratusbeenashappyintheirdeductionsastothefunctionsoftheorgansastheywereintheirknowledgeofanatomy，thescienceofmedicinewouldhavebeenplaceduponaveryhighplaneevenintheirtime。Unfortunately，however，theynotonlydrewerroneousinferencesastothefunctionsoftheorgans，butalsodisagreedradicallyastowhatfunctionscertainorgansperformed，andhowdiseasesshouldbetreated，evenwhenagreeingperfectlyonthesubjectofanatomyitself。Theircontributiontotheknowledgeofthescientifictreatmentofdiseasesholdsnosuchplace，therefore，astheiranatomicalinvestigations。
HalfacenturyafterthetimeofHerophilusthereappearedaGreekphysician，Heraclides，whosereputationintheuseofdrugsfarsurpassesthatoftheanatomistsoftheAlexandrianschool。
Hisreputationhasbeenhandeddownthroughthecenturiesasthatofaphysician，ratherthanasurgeon，althoughinhisowntimehewasconsideredoneofthegreatsurgeonsoftheperiod。
Heraclidesbelongedtothe"Empiric"school，whichrejectedanatomyasuseless，dependingentirelyontheuseofdrugs。Heisthoughttohavebeenthefirstphysiciantopointoutthevalueofopiumincertainpainfuldiseases。Hisprescriptionofthisdrugforcertaincasesof"sleeplessness，spasm，cholera，andcolic，"showsthathisuseofitwasnotunlikethatofthemodernphysicianincertaincases；andhistreatmentoffevers，bykeepingthepatient’sheadcoolandfacilitatingthesecretionsofthebody，isstillrecognizedas"goodpractice。"
Headvocatedafreeuseofliquidsinquenchingthefeverpatient’sthirst——arecognizedtherapeuticmeasureto-day，butonethatwaswidelycondemnedacenturyago。
WedonotknowjustwhenEucliddied，butashewasattheheightofhisfameinthetimeofPtolemyI。，whosereignendedintheyear285B。C。，itishardlyprobablethathewasstilllivingwhenayoungmannamedArchimedescametoAlexandriatostudy。
ArchimedeswasbornintheGreekcolonyofSyracuse，ontheislandofSicily，intheyear287B。C。WhenhevisitedAlexandriaheprobablyfoundApolloniusofPerga，thepupilofEuclid，attheheadofthemathematicalschoolthere。JusthowlongArchimedesremainedatAlexandriaisnotknown。Whenhehadsatisfiedhiscuriosityorcompletedhisstudies，hereturnedtoSyracuseandspenthislifethere，chieflyunderthepatronageofKingHiero，whoseemsfullytohaveappreciatedhisabilities。
Archimedeswasprimarilyamathematician。Lefttohisowndevices，hewouldprobablyhavedevotedhisentiretimetothestudyofgeometricalproblems。ButKingHierohaddiscoveredthathisprotegehadwonderfulmechanicalingenuity，andhemadegooduseofthisdiscovery。Understressoftheking’surgings，thephilosopherwasledtoinventagreatvarietyofmechanicalcontrivances，someofthemmostcuriousones。Antiquitycreditedhimwiththeinventionofmorethanfortymachines，anditisthese，ratherthanhispurelymathematicaldiscoveries，thatgavehisnamepopularvoguebothamonghiscontemporariesandwithposterity。EveryonehasheardofthescrewofArchimedes，throughwhichtheparadoxicaleffectwasproducedofmakingwaterseemtoflowuphill。Thebestideaofthiscuriousmechanismisobtainedifonewilltakeinhandanordinarycorkscrew，andimaginethisinstrumenttobechangedintoahollowtube，retainingpreciselythesameshapebutincreasedtosomefeetinlengthandtoaproportionatediameter。Ifonewillholdthecorkscrewinaslantingdirectionandturnitslowlytotheright，supposingthatthepointdipsupaportionofwatereachtimeitrevolves，onecaninimaginationfollowtheflowofthatportionofwaterfromspiraltospiral，thewateralwaysrunningdownward，ofcourse，yetparadoxicallybeingliftedhigherandhighertowardsthebaseofthecorkscrew，untilfinallyitpoursoutintheactualArchimedes’tubeatthetop。Thereisanotherformofthescrewinwhicharevolvingspiralbladeoperateswithinacylinder，buttheprincipleispreciselythesame。Witheitherformwatermaybelifted，bythemereturningofthescrew，toanydesiredheight。TheingeniousmechanismexcitedthewonderofthecontemporariesofArchimedes，aswellitmight。
Moreefficientdeviceshavesupersededitinmoderntimes，butitstillexcitestheadmirationofallwhoexamineit，anditseffectsseemasparadoxicalasever。
SomeotherofthemechanismsofArchimedeshavebeenmadeknowntosuccessivegenerationsofreadersthroughthepagesofPolybiusandPlutarch。ThesearethedevicesthroughwhichArchimedesaidedKingHierotowardofftheattacksoftheRomangeneralMarcellus，whointhecourseofthesecondPunicwarlaidsiegetoSyracuse。
Plutarch，inhislifeofMarcellus，describestheRoman’sattackandArchimedes’defenceinmuchdetail。IncidentallyhetellsusalsohowArchimedescametomakethedevicesthatrenderedthesiegesofamous:
"Marcellushimself，withthreescoregalleysoffiverowersateverybank，wellarmedandfullofallsortsofartilleryandfireworks，didassaultbysea，androwedhardtothewall，havingmadeagreatengineanddeviceofbattery，uponeightgalleyschainedtogether，tobatterthewall:trustinginthegreatmultitudeofhisenginesofbattery，andtoallsuchothernecessaryprovisionashehadforwars，asalsoinhisownreputation。ButArchimedesmadelightaccountofallhisdevices，asindeedtheywerenothingcomparabletotheengineshimselfhadinvented。Thisinventivearttoframeinstrumentsandengineswhicharecalledmechanical，ororganical，sohighlycommendedandesteemedofallsortsofpeoplewasfirstsetforthbyArchitas，andbyEudoxus:partlytobeautifyalittlethescienceofgeometrybythisfineness，andpartlytoproveandconfirmbymaterialexamplesandsensibleinstruments，certaingeometricalconclusions，whereofamancannotfindouttheconceivabledemonstrationsbyenforcedreasonsandproofs。Asthatconclusionwhichinstructethonetosearchouttwolinesmeanproportional，whichcannotbeprovedbyreasondemonstrative，andyetnotwithstandingisaprincipleandanacceptedgroundformanythingswhicharecontainedintheartofportraiture。Bothofthemhavefashionedittotheworkmanshipofcertaininstruments，calledmesolabesormesographs，whichservetofindthesemeanlinesproportional，bydrawingcertaincurvelines，andoverthwartandobliquesections。ButafterthatPlatowasoffendedwiththem，andmaintainedagainstthem，thattheydidutterlycorruptanddisgrace，theworthinessandexcellenceofgeometry，makingittodescendfromthingsnotcomprehensibleandwithoutbody，untothingssensibleandmaterial，andtobringittoapalpablesubstance，wherethevileandbasehandiworkofmanistobeemployed:sincethattime，Isay，handicraft，ortheartofengines，cametobeseparatedfromgeometry，andbeinglongtimedespisedbythephilosophers，itcametobeoneofthewarlikearts。
"ButArchimedeshavingtoldKingHiero，hiskinsmanandfriend，thatitwaspossibletoremoveasgreataweightashewould，withaslittlestrengthashelistedtoputtoit:andboastinghimselfthusastheyreportofhimandtrustingtotheforceofhisreasons，wherewithheprovedthisconclusion，thatiftherewereanotherglobeofearth，hewasabletoremovethisofours，andpassitovertotheother:KingHierowonderingtohearhim，requiredhimtoputhisdeviceinexecution，andtomakehimseebyexperience，somegreatorheavyweightremoved，bylittleforce。SoArchimedescaughtholdwithabookofoneofthegreatestcarects，orhulksofthekingthattodrawittotheshoreoutofthewaterrequiredamarvellousnumberofpeopletogoaboutit，andwashardlytobedonesoandputagreatnumberofmenmoreintoher，thanherordinaryburden:andhehimselfsittingaloneathiseasefaroff，withoutanystrainingatall，drawingtheendofanenginewithmanywheelsandpulleys，fairandsoftlywithhishand，madeitcomeasgentlyandsmoothlytohim，asithadfloatedinthesea。Thekingwonderingtoseethesight，andknowingbyproofthegreatnessofhisart；beprayedhimtomakehimsomeengines，bothtoassaultanddefend，inallmannerofsiegesandassaults。SoArchimedesmadehimmanyengines，butKingHieroneveroccupiedanyofthem，becausehereignedthemostpartofhistimeinpeacewithoutanywars。Butthisprovisionandmunitionofengines，servedtheSyracusan’sturnmarvellouslyatthattime:andnotonlytheprovisionoftheenginesreadymade，butalsotheengineerandwork-masterhimself，thathadinventedthem。
"NowtheSyracusans，seeingthemselvesassaultedbytheRomans，bothbyseaandbyland，weremarvellouslyperplexed，andcouldnottellwhattosay，theyweresoafraid:imaginingitwasimpossibleforthemtowithstandsogreatanarmy。ButwhenArchimedesfelltohandlinghisengines，andtosetthematliberty，thereflewintheairinfinitekindsofshot，andmarvellousgreatstones，withanincrediblenoiseandforceonthesudden，uponthefootmenthatcametoassaultthecitybyland，bearingdown，andtearinginpiecesallthosewhichcameagainstthem，orinwhatplacesoevertheylighted，noearthlybodybeingabletoresisttheviolenceofsoheavyaweight:sothatalltheirranksweremarvellouslydisordered。Andasforthegalleysthatgaveassaultbysea，someweresunkwithlongpiecesoftimberlikeuntotheyardsofships，wheretotheyfastentheirsails，whichweresuddenlyblownoverthewallswithforceoftheirenginesintotheirgalleys，andsosunkthembytheirovergreatweight。"
Polybiusdescribeswhatwasperhapsthemostimportantofthesecontrivances，whichwas，hetellsus，"abandofiron，hangingbyachainfromthebeakofamachine，whichwasusedinthefollowingmanner。Thepersonwho，likeapilot，guidedthebeak，havingletfallthehand，andcatchedholdoftheprowofanyvessel，drewdowntheoppositeendofthemachinethatwasontheinsideofthewalls。Andwhenthevesselwasthusraisederectuponitsstem，themachineitselfwasheldimmovable；but，thechainbeingsuddenlyloosenedfromthebeakbythemeansofpulleys，someofthevesselswerethrownupontheirsides，othersturnedwiththebottomupwards；andthegreatestpart，astheprowswereplungedfromaconsiderableheightintothesea，werefilledwithwater，andallthatwereonboardthrownintotumultanddisorder。
"Marcelluswasinnosmalldegreeembarrassed，"Polybiuscontinues，"whenhefoundhimselfencounteredineveryattemptbysuchresistance。Heperceivedthatallhiseffortsweredefeatedwithloss；andwereevenderidedbytheenemy。But，amidstalltheanxietythathesuffered，hecouldnothelpjestingupontheinventionsofArchimedes。Thisman，saidhe，employsourshipsasbucketstodrawwater:andboxingaboutoursackbuts，asiftheywereunworthytobeassociatedwithhim，drivesthemfromhiscompanywithdisgrace。Suchwasthesuccessofthesiegeonthesideofthesea。"
Subsequently，however，Marcellustookthecitybystrategy，andArchimedeswaskilled，contrary，itissaid，totheexpressordersofMarcellus。"Syracusebeingtaken，"saysPlutarch，"nothinggrievedMarcellusmorethanthelossofArchimedes。Who，beinginhisstudywhenthecitywastaken，busilyseekingoutbyhimselfthedemonstrationofsomegeometricalpropositionwhichhehaddrawninfigure，andsoearnestlyoccupiedtherein，asheneithersawnorheardanynoiseofenemiesthatranupanddownthecity，andmuchlessknewitwastaken:hewonderedwhenhesawasoldierbyhim，thatbadehimgowithhimtoMarcellus。
Notwithstanding，hespaketothesoldier，andbadehimtarryuntilhehaddonehisconclusion，andbroughtittodemonstration:butthesoldierbeingangrywithhisanswer，drewouthisswordandkilledhim。Otherssay，thattheRomansoldierwhenhecame，offeredthesword’spointtohim，tokillhim:andthatArchimedeswhenhesawhim，prayedhimtoholdhishandalittle，thathemightnotleavethematterhelookedforimperfect，withoutdemonstration。Butthesoldiermakingnoreckoningofhisspeculation，killedhimpresently。Itisreportedathirdwayalso，sayingthatcertainsoldiersmethiminthestreetsgoingtoMarcellus，carryingcertainmathematicalinstrumentsinalittleprettycoffer，asdialsforthesun，spheres，andangles，wherewiththeymeasurethegreatnessofthebodyofthesunbyview:andtheysupposinghehadcarriedsomegoldorsilver，orotherpreciousjewelsinthatlittlecoffer，slewhimforit。ButitismostcertainthatMarcelluswasmarvellouslysorryforhisdeath，andeverafterhatedthevillainthatslewhim，asacursedandexecrableperson:andhowhehadmadealsomarvellousmuchafterwardsofArchimedes’
kinsmenforhissake。"
WearefurtherindebtedtoPlutarchforasummaryofthecharacterandinfluenceofArchimedes，andforaninterestingsuggestionastotheestimatewhichthegreatphilosopherputupontherelativeimportanceofhisowndiscoveries。
"NotwithstandingArchimedeshadsuchagreatmind，andwassoprofoundlylearned，havinghiddeninhimtheonlytreasureandsecretsofgeometricalinventions:asbewouldneversetforthanybookhowtomakeallthesewarlikeengines，whichwonhimatthattimethefameandglory，notofman’sknowledge，butratherofdivinewisdom。Butheesteemingallkindofhandicraftandinventiontomakeengines，andgenerallyallmannerofsciencesbringingcommoncommoditybytheuseofthem，tobebutvile，beggarly，andmercenarydross:employedhiswitandstudyonlytowritethings，thebeautyandsubtletywhereofwerenotmingledanythingatallwithnecessity。Forallthathehathwritten，aregeometricalpropositions，whicharewithoutcomparisonofanyotherwritingswhatsoever:becausethesubjectwhereoftheytreat，dothappearbydemonstration，themakergivesthemthegraceandthegreatness，andthedemonstrationprovingitsoexquisitely，withwonderfulreasonandfacility，asitisnotrepugnable。Forinallgeometryarenottobefoundmoreprofoundanddifficultmatterswritten，inmoreplainandsimpleterms，andbymoreeasyprinciples，thanthosewhichhehathinvented。
Nowsomedoimputethis，tothesharpnessofhiswitandunderstanding，whichwasanaturalgiftinhim:othersdoreferittotheextremepainshetook，whichmadethesethingscomesoeasilyfromhim，thattheyseemedasiftheyhadbeennotroubletohimatall。Fornomanlivingofhimselfcandevisethedemonstrationofhispropositions，whatpainssoeverhetaketoseekit:andyetstraightsosoonashecomethtodeclareandopenit，everymanthenimaginethwithhimselfhecouldhavefounditoutwellenough，hecanthensoplainlymakedemonstrationofthethinghemeanethtoshow。Andthereforethatmethinksislikelytobetrue，whichtheywriteofhim:thathewassoravishedanddrunkwiththesweetenticementsofthissiren，whichasitwerelaycontinuallywithhim，asheforgothismeatanddrink，andwascarelessotherwiseofhimself，thatoftentimeshisservantsgothimagainsthiswilltothebathstowashandanointhim:andyetbeingthere，hewouldeverbedrawingoutofthegeometricalfigures，evenintheveryimbersofthechimney。Andwhiletheywereanointingofhimwithoilsandsweetsavours，withhisfingerhediddrawlinesuponhisnakedbody:sofarwashetakenfromhimself，andbroughtintoanecstasyortrance，withthedelighthehadinthestudyofgeometry，andtrulyravishedwiththeloveoftheMuses。Butamongstmanynotablethingshedevised，itappeareth，thathemostesteemedthedemonstrationoftheproportionbetweenthecylindertowit，theroundcolumnandthesphereorglobecontainedinthesame:forheprayedhiskinsmenandfriends，thatafterhisdeaththeywouldputacylinderuponhistomb，containingamassysphere，withaninscriptionoftheproportion，whereofthecontinentexceedeththethingcontained。"[2]
ItshouldbeobservedthatneitherPolybiusnorPlutarchmentionstheuseofburning-glassesinconnectionwiththesiegeofSyracuse，norindeedarethesereferredtobyanyotherancientwriterofauthority。Nevertheless，astorygainedcredencedowntoalatedaytotheeffectthatArchimedeshadsetfiretothefleetoftheenemywiththeaidofconcavemirrors。AnexperimentwasmadebySirIsaacNewtontoshowthepossibilityofaphenomenonsowellinaccordwiththegeniusofArchimedes，butthesilenceofalltheearlyauthoritiesmakesitmorethandoubtfulwhetheranysuchexpedientwasreallyadopted。
Itwillbeobservedthatthechiefprincipleinvolvedinallthesemechanismswasacapacitytotransmitgreatpowerthroughleversandpulleys，andthisbringsustothemostimportantfieldoftheSyracusanphilosopher’sactivity。ItwasasastudentoftheleverandthepulleythatArchimedeswasledtosomeofhisgreatestmechanicaldiscoveries。Heisevencreditedwithbeingthediscovererofthecompoundpulley。Morelikelyhewasitsdeveloperonly，sincetheprincipleofthepulleywasknowntotheoldBabylonians，astheirsculpturestestify。ButthereisnoreasontodoubtthegeneraloutlinesofthestorythatArchimedesastoundedKingHierobyprovingthat，withtheaidofmultiplepulleys，thestrengthofonemancouldsufficetodragthelargestshipfromitsmoorings。
Thepropertyofthelever，fromitsfundamentalprinciple，wasstudiedbyhim，beginningwiththeself-evidentfactthat"equalbodiesattheendsoftheequalarmsofarod，supportedonitsmiddlepoint，willbalanceeachother"；or，whatamountstothesamethingstatedinanotherway，aregularcylinderofuniformmatterwillbalanceatitsmiddlepoint。Fromthisstarting-pointheelaboratedthesubjectonsuchclearandsatisfactoryprinciplesthattheystandto-daypracticallyunchangedandwithfewadditions。Fromallhisstudiesandexperimentshefinallyformulatedtheprinciplethat"bodieswillbeinequilibriowhentheirdistancefromthefulcrumorpointofsupportisinverselyastheirweight。"Heiscreditedwithhavingsummeduphisestimateofthecapabilitiesoftheleverwiththewell-knownexpression，"Givemeafulcrumonwhichtorestoraplaceonwhichtostand，andIwillmovetheearth。"
Butperhapsthefeatofallothersthatmostappealedtotheimaginationofhiscontemporaries，andpossiblyalsotheonethathadthegreatestbearinguponthepositionofArchimedesasascientificdiscoverer，wastheonemadefamiliarthroughthetaleofthecrownofHiero。Thiscrown，sothestorygoes，wassupposedtobemadeofsolidgold，butKingHieroforsomereasonsuspectedthehonestyofthejeweller，anddesiredtoknowifArchimedescoulddeviseawayoftestingthequestionwithoutinjuringthecrown。Greekimaginationseldomspoiledastoryinthetelling，andinthiscasethetalewasallowedtotakeonthemostpicturesqueofphases。Thephilosopher，weareassured，ponderedtheproblemforalongtimewithoutsucceeding，butonedayashesteppedintoabath，hisattentionwasattractedbytheoverflowofwater。Anewtrainofideaswasstartedinhisever-receptivebrain。Wildwithenthusiasmhesprangfromthebath，and，forgettinghisrobe，dashedalongthestreetsofSyracuse，shouting:"Eureka！Eureka！"Ihavefoundit！Thethoughtthathadcomeintohismindwasthis:Thatanyheavysubstancemusthaveabulkproportionatetoitsweight；thatgoldandsilverdifferinweight，bulkforbulk，andthatthewaytotestthebulkofsuchanirregularobjectasacrownwastoimmerseitinwater。Theexperimentwasmade。Alumpofpuregoldoftheweightofthecrownwasimmersedinacertainreceptaclefilledwithwater，andtheoverflownoted。Thenalumpofpuresilverofthesameweightwassimilarlyimmersed；lastlythecrownitselfwasimmersed，andofcourse——forthestorymustnotlackitsdramaticsequel——wasfoundbulkierthanitsweightofpuregold。Thusthegeniusthatcouldbalkwarriorsandarmiescouldalsofoilthewilesofthesilversmith。
Whateverthetruthofthispicturesquenarrative，thefactremainsthatsome，suchexperimentsasthesemusthavepavedthewayforperhapsthegreatestofallthestudiesofArchimedes——thosethatrelatetothebuoyancyofwater。Leavingthefieldoffable，wemustnowexaminethesewithsomeprecision。Fortunately，thewritingsofArchimedeshimselfarestillextant，inwhichtheresultsofhisremarkableexperimentsarerelated，sowemaypresenttheresultsinthewordsofthediscoverer。
Heretheyare:"First:Thesurfaceofeverycoherentliquidinastateofrestisspherical，andthecentreofthespherecoincideswiththecentreoftheearth。Second:Asolidbodywhich，bulkforbulk，isofthesameweightasaliquid，ifimmersedintheliquidwillsinksothatthesurfaceofthebodyisevenwiththesurfaceoftheliquid，butwillnotsinkdeeper。
Third:Anysolidbodywhichislighter，bulkforbulk，thanaliquid，ifplacedintheliquidwillsinksodeepastodisplacethemassofliquidequalinweighttoanotherbody。Fourth:Ifabodywhichislighterthanaliquidisforciblyimmersedintheliquid，itwillbepressedupwardwithaforcecorrespondingtotheweightofalikevolumeofwater，lesstheweightofthebodyitself。Fifth:Solidbodieswhich，bulkforbulk，areheavierthanaliquid，whenimmersedintheliquidsinktothebottom，butbecomeintheliquidasmuchlighterastheweightofthedisplacedwateritselfdiffersfromtheweightofthesolid。"
Thesepropositionsarenotdifficulttodemonstrate，oncetheyareconceived，buttheirdiscovery，combinedwiththediscoveryofthelawsofstaticsalreadyreferredto，mayjustlybeconsideredasprovingArchimedesthemostinventiveexperimenterofantiquity。
Curiouslyenough，thediscoverywhichArchimedeshimselfissaidtohaveconsideredthemostimportantofallhisinnovationsisonethatseemsmuchlessstriking。Itistheanswertothequestion，Whatistherelationinbulkbetweenasphereanditscircumscribingcylinder？Archimedesfindsthattheratioissimplytwotothree。Wearenotinformedastohowhereachedhisconclusion，butanobviousmethodwouldbetoimmerseaballinacylindricalcup。Theexperimentisonewhichanyonecanmakeforhimself，withapproximateaccuracy，withtheaidofatumblerandasolidrubberballorabilliard-ballofjusttherightsize。
AnothergeometricalproblemwhichArchimedessolvedwastheproblemastothesizeofatrianglewhichhasequalareawithacircle；theanswerbeing，atrianglehavingforitsbasethecircumferenceofthecircleandforitsaltitudetheradius。
Archimedessolvedalsotheproblemoftherelationofthediameterofthecircletoitscircumference；hisanswerbeingacloseapproximationtothefamiliar3。1416，whicheverytyroingeometrywillrecallastheequivalentofpi。
NumerousotherofthestudiesofArchimedeshavingreferencetoconicsections，propertiesofcurvesandspirals，andthelike，aretootechnicaltobedetailedhere。Theextentofhismathematicalknowledge，however，issuggestedbythefactthathecomputedingreatdetailthenumberofgrainsofsandthatwouldberequiredtocoverthesphereofthesun’sorbit，makingcertainhypotheticalassumptionsastothesizeoftheearthandthedistanceofthesunforthepurposesofargument。
Mathematiciansfindhiscomputationpeculiarlyinterestingbecauseitevidencesacrudeconceptionoftheideaoflogarithms。Fromourpresentstand-point，thepaperinwhichthiscalculationiscontainedhasconsiderableinterestbecauseofitsassumptionsastocelestialmechanics。ThusArchimedesstartsoutwiththepreliminaryassumptionthatthecircumferenceoftheearthislessthanthreemillionstadia。Itmustbeunderstoodthatthisassumptionispurelyforthesakeofargument。
Archimedesexpresslystatesthathetakesthisnumberbecauseitis"tentimesaslargeastheearthhasbeensupposedtobebycertaininvestigators。"Here，perhaps，thereferenceistoEratosthenes，whosemeasurementoftheearthweshallhaveoccasiontoreverttoinamoment。Continuing，Archimedesassertsthatthesunislargerthantheearth，andtheearthlargerthanthemoon。Inthisassumption，hesays，heisfollowingtheopinionofthemajorityofastronomers。Inthethirdplace，Archimedesassumesthatthediameterofthesunisnotmorethanthirtytimesgreaterthanthatofthemoon。HereheisprobablybasinghisargumentuponanothersetofmeasurementsofAristarchus，towhich，also，weshallpresentlyrefermoreatlength。Inreality，hisassumptionisveryfarfromthetruth，sincetheactualdiameterofthesun，aswenowknow，issomethinglikefourhundredtimesthatofthemoon。Fourth，thecircumferenceofthesunisgreaterthanonesideofthethousand-facedfigureinscribedinitsorbit。Themeasurement，itisexpresslystated，isbasedonthemeasurementsofAristarchus，whomakesthediameterofthesun1/170ofitsorbit。Archimedesadds，however，thathehimselfhasmeasuredtheangleandthatitappearstohimtobelessthan1/164，andgreaterthan1/200partoftheorbit。Thatistosay，reducedtomodernterminology，heplacesthelimitofthesun’sapparentsizebetweenthirty-threeminutesandtwenty-sevenminutesofarc。Astherealdiameteristhirty-twominutes，thiscalculationissurprisinglyexact，consideringtheimplementsthenatcommand。ButthehonoroffirstmakingitmustbegiventoAristarchusandnottoArchimedes。
WeneednotfollowArchimedestothelimitsofhisincomprehensiblenumbersofsand-grains。Thecalculationischieflyremarkablebecauseitwasmadebeforetheintroductionoftheso-calledArabicnumeralshadsimplifiedmathematicalcalculations。ItwillberecalledthattheGreeksusedlettersfornumerals，and，havingnocipher，theysoonfoundthemselvesindifficultieswhenlargenumberswereinvolved。TheRomansystemofnumeralssimplifiedthemattersomewhat，butthebeautifulsimplicityofthedecimalsystemdidnotcomeintovogueuntiltheMiddleAges，asweshallsee。Notwithstandingthedifficulties，however，Archimedesfollowedouthiscalculationstothepilingupofbewilderingnumbers，whichthemodernmathematicianfindstobetheconsistentoutcomeoftheproblemhehadsethimself。
Butitremainstonoticethemostinterestingfeatureofthisdocumentinwhichthecalculationofthesand-grainsiscontained。"Itwasknowntome，"saysArchimedes，"thatmostastronomersunderstandbytheexpression’world’universeaballofwhichthecentreisthemiddlepointoftheearth，andofwhichtheradiusisastraightlinebetweenthecentreoftheearthandthesun。"Archimedeshimselfappearstoacceptthisopinionofthemajority，——itatleastservesaswellasthecontraryhypothesisforthepurposeofhiscalculation，——buthegoesontosay:"AristarchusofSamos，inhiswritingagainsttheastronomers，seekstoestablishthefactthattheworldisreallyverydifferentfromthis。Heholdstheopinionthatthefixedstarsandthesunareimmovableandthattheearthrevolvesinacircularlineaboutthesun，thesunbeingatthecentreofthiscircle。"ThisremarkablebitoftestimonyestablishesbeyondquestionthepositionofAristarchusofSamosastheCopernicusofantiquity。Wemustmakefurtherinquiryastotheteachingsofthemanwhohadgainedsucharemarkableinsightintothetruesystemoftheheavens。
ItappearsthatAristarchuswasacontemporaryofArchimedes，buttheexactdatesofhislifearenotknown。HewasactivelyengagedinmakingastronomicalobservationsinSamossomewhatbeforethemiddleofthethirdcenturyB。C。；inotherwords，justatthetimewhentheactivitiesoftheAlexandrianschoolwereattheirheight。Hipparchus，atalaterday，wasenabledtocomparehisownobservationswiththosemadebyAristarchus，and，aswehavejustseen，hisworkwaswellknowntosodistantacontemporaryasArchimedes。Yetthefactsofhislifearealmostablankforus，andofhiswritingsonlyasingleonehasbeenpreserved。Thatone，however，isamostimportantandinterestingpaperonthemeasurementsofthesunandthemoon。Unfortunately，thispapergivesusnodirectclewastotheopinionsofAristarchusconcerningtherelativepositionsoftheearthandsun。ButthetestimonyofArchimedesastothisisunequivocal，andthistestimonyissupportedbyotherrumorsinthemselveslessauthoritative。
IncontemplatingthisastronomerofSamos，then，weareinthepresenceofamanwhohadsolvedinitsessentialstheproblemofthemechanismofthesolarsystem。ItappearsfromthewordsofArchimedesthatAristarchus；hadpropoundedhistheoryinexplicitwritings。Unquestionably，then，heheldtoitasapositivedoctrine，notasamerevagueguess。Weshallshow，inamoment，onwhatgroundshebasedhisopinion。Hadhisteachingfoundvogue，thestoryofsciencewouldbeverydifferentfromwhatitis。WeshouldthenhavenotaletotellofaCopernicuscominguponthescenefullyseventeenhundredyearslaterwiththerevolutionarydoctrinethatourworldisnotthecentreoftheuniverse。WeshouldnothavetotellofthepersecutionofaBrunoorofaGalileoforteachingthisdoctrineintheseventeenthcenturyofanerawhichdidnotbegintilltwohundredyearsafterthedeathofAristarchus。But，asweknow，theteachingoftheastronomerofSamosdidnotwinitsway。Theoldconservativegeocentricdoctrine，seeminglysomuchmoreinaccordancewiththeevery-dayobservationsofmankind，supportedbythemajorityofastronomerswiththePeripateticphilosophersattheirhead，helditsplace。ItfoundfreshsupporterspresentlyamongthelaterAlexandrians，andsofullyeclipsedtheheliocentricviewthatweshouldscarcelyknowthatviewhadevenfoundanadvocatewereitnotforhereandtheresuchachancerecordasthephraseswehavejustquotedfromArchimedes。Yet，aswenowsee，theheliocentricdoctrine，whichweknowtobetrue，hadbeenthoughtoutandadvocatedasthecorrecttheoryofcelestialmechanicsbyatleastoneworkerofthethirdcenturyB。C。Suchanidea，wemaybesure，didnotspringintothemindofitsoriginatorexceptastheculminationofalongseriesofobservationsandinferences。Theprecisecharacteroftheevolutionweperhapscannottrace，butitsbroaderoutlinesareopentoourobservation，andwemaynotleavesoimportantatopicwithoutatleastbrieflynotingthem。
FullytounderstandthetheoryofAristarchus，wemustgobackacenturyortwoandrecallthataslongagoasthetimeofthatothergreatnativeofSamos，Pythagoras，theconceptionhadbeenreachedthattheearthisinmotion。Wesaw，indealingwithPythagoras，thatwecouldnotbesureastopreciselywhathehimselftaught，butthereisnoquestionthattheideaoftheworld’smotionbecamefromanearlydayaso-calledPythagoreandoctrine。Whilealltheotherphilosophers，sofarasweknow，stillbelievedthattheworldwasflat，thePythagoreansoutinItalytaughtthattheworldisasphereandthattheapparentmotionsoftheheavenlybodiesarereallyduetotheactualmotionoftheearthitself。Theydidnot，however，vaulttotheconclusionthatthistruemotionoftheearthtakesplaceintheformofacircuitaboutthesun。Insteadofthat，theyconceivedthecentralbodyoftheuniversetobeagreatfire，invisiblefromtheearth，becausetheinhabitedsideoftheterrestrialballwasturnedawayfromit。Thesun，itwasheld，isbutagreatmirror，whichreflectsthelightfromthecentralfire。Sunandearthalikerevolveaboutthisgreatfire，eachinitsownorbit。Betweentheearthandthecentralfiretherewas，curiouslyenough，supposedtobeaninvisibleearthlikebodywhichwasgiventhenameofAnticthon，orcounter-earth。Thisbody，itselfrevolvingaboutthecentralfire，wassupposedtoshutoffthecentrallightnowandagainfromthesunorfromthemoon，andthustoaccountforcertaineclipsesforwhichtheshadowoftheearthdidnotseemresponsible。Itwas，perhaps，largelytoaccountforsucheclipsesthatthecounter-earthwasinvented。Butitissupposedthattherewasanotherreason。ThePythagoreansheldthatthereisapeculiarsacrednessinthenumberten。JustastheBabyloniansoftheearlydayandtheHegelianphilosophersofamorerecentepochsawasacredconnectionbetweenthenumbersevenandthenumberofplanetarybodies，sothePythagoreansthoughtthattheuniversemustbearrangedinaccordancewiththenumberten。Theircountoftheheavenlybodies，includingthesphereofthefixedstars，seemedtoshownine，andthecounter-earthsuppliedthemissingbody。
Theprecisegenesisanddevelopmentofthisideacannotnowbefollowed，butthatitwasprevalentaboutthefifthcenturyB。C。
asaPythagoreandoctrinecannotbequestioned。Anaxagorasalsoissaidtohavetakenaccountofthehypotheticalcounter-earthinhisexplanationofeclipses；though，aswehaveseen，heprobablydidnotacceptthatpartofthedoctrinewhichheldtheearthtobeasphere。ThenamesofPhilolausandHeraclideshavebeenlinkedwithcertainofthesePythagoreandoctrines。Eudoxus，too，who，liketheothers，livedinAsiaMinorinthefourthcenturyB。C。，washeldtohavemadespecialstudiesoftheheavenlyspheresandperhapstohavetaughtthattheearthmoves。
So，too，Nicetasmustbenamedamongthosewhomrumorcreditedwithhavingtaughtthattheworldisinmotion。Inaword，theevidence，sofaraswecangarneritfromtheremainingfragments，tendstoshowthatallalong，fromthetimeoftheearlyPythagoreans，therehadbeenanundercurrentofopinioninthephilosophicalworldwhichquestionedthefixityoftheearth；
anditwouldseemthattheschoolofthinkerswhotendedtoaccepttherevolutionaryviewcentredinAsiaMinor，notfarfromtheearlyhomeofthefounderofthePythagoreandoctrines。Itwasnotstrange，then，thatthemanwhowasfinallytocarrythesenewopinionstotheirlogicalconclusionshouldhailfromSamos。
Butwhatwasthesupportwhichobservationcouldgivetothisnew，strangeconceptionthattheheavenlybodiesdonotinrealitymoveastheyseemtomove，butthattheirapparentmotionisduetotheactualrevolutionoftheearth？Itisextremelydifficultforanyonenowadaystoputhimselfinamentalpositiontoanswerthisquestion。Wearesoaccustomedtoconceivethesolarsystemasweknowittobe，thatwearewonttoforgethowverydifferentitisfromwhatitseems。Yetoneneedsbuttoglanceupatthesky，andthentoglanceaboutoneatthesolidearth，togrant，onamoment’sreflection，thatthegeocentricideaisofallothersthemostnatural；andthattoconceivethesunastheactualCentreofthesolarsystemisanideawhichmustlookforsupporttosomeotherevidencethanthatwhichordinaryobservationcangive。Suchwastheviewofmostoftheancientphilosophers，andsuchcontinuedtobetheopinionofthemajorityofmankindlongafterthetimeofCopernicus。WemustnotforgetthatevensogreatanobservingastronomerasTychoBrahe，solateastheseventeenthcentury，declinedtoaccepttheheliocentrictheory，thoughadmittingthatalltheplanetsexcepttheearthrevolveaboutthesun。WeshallseethatbeforetheAlexandrianschoollostitsinfluenceageocentricschemehadbeenevolvedwhichfullyexplainedalltheapparentmotionsoftheheavenlybodies。Allthis，then，makesusbutwonderthemorethatthegeniusofanAristarchuscouldgiveprecedencetoscientificinductionasagainsttheseeminglyclearevidenceofthesenses。
What，then，wasthelineofscientificinductionthatledAristarchustothiswonderfulgoal？Fortunately，weareabletoanswerthatquery，atleastinpart。Aristarchusgainedhisevidencethroughsomewonderfulmeasurements。First，hemeasuredthedisksofthesunandthemoon。This，ofcourse，couldinitselfgivehimnoclewtothedistanceofthesebodies，andthereforenoclewastotheirrelativesize；butinattemptingtoobtainsuchaclewhehituponawonderfulyetaltogethersimpleexperiment。Itoccurredtohimthatwhenthemoonispreciselydichotomized——thatistosay，preciselyatthehalf-thelineofvisionfromtheearthtothemoonmustbepreciselyatrightangleswiththelineoflightpassingfromthesuntothemoon。
Atthismoment，then，theimaginarylinesjoiningthesun，themoon，andtheearth，makearightangletriangle。Butthepropertiesoftheright-angletrianglehadlongbeenstudiedandwerewellunderstood。Oneacuteangleofsuchatriangledeterminesthefigureofthetriangleitself。WehavealreadyseenthatThales，theveryearliestoftheGreekphilosophers，measuredthedistanceofashipatseabytheapplicationofthisprinciple。NowAristarchussightsthesuninplaceofThales’
ship，and，sightingthemoonatthesametime，measurestheangleandestablishestheshapeofhisright-angletriangle。Thisdoesnottellhimthedistanceofthesun，tobesure，forhedoesnotknowthelengthofhisbase-line——thatistosay，ofthelinebetweenthemoonandtheearth。Butitdoesestablishtherelationofthatbase-linetotheotherlinesofthetriangle；inotherwords，ittellshimthedistanceofthesunintermsofthemoon’sdistance。AsAristarchusstrikestheangle，itshowsthatthesuniseighteentimesasdistantasthemoon。Now，bycomparingtheapparentsizeofthesunwiththeapparentsizeofthemoon——which，aswehaveseen，Aristarchushasalreadymeasured——heisabletotellusthat，thesunis"morethan5832
times，andlessthan8000"timeslargerthanthemoon；thoughhismeasurements，takenbythemselves，givenoclewtotheactualbulkofeitherbody。Theseconclusions，beitunderstood，areabsolutelyvalidinferences——nay，demonstrations——fromthemeasurementsinvolved，providedonlythatthesemeasurementshavebeencorrect。Unfortunately，theangleofthetrianglewehavejustseenmeasuredisexceedinglydifficulttodeterminewithaccuracy，whileatthesametime，asamoment’sreflectionwillshow，itissolargeananglethataveryslightdeviationfromthetruthwillgreatlyaffectthedistanceatwhichitslinejoinstheothersideofthetriangle。Thenagain，itisvirtuallyimpossibletotelltheprecisemomentwhenthemoonisathalf，asthelineitgivesisnotsosharpthatwecanfixitwithabsoluteaccuracy。Thereis，moreover，anotherelementoferrorduetotherefractionoflightbytheearth’satmosphere。Theexperimentwasprobablymadewhenthesunwasnearthehorizon，atwhichtime，aswenowknow，butasAristarchusprobablydidnotsuspect，theapparentdisplacementofthesun’spositionisconsiderable；andthisdisplacement，itwillbeobserved，isinthedirectiontolessentheangleinquestion。
Inpointoffact，Aristarchusestimatedtheangleateighty-sevendegrees。Hadhisinstrumentbeenmoreprecise，andhadhebeenabletotakeaccountofalltheelementsoferror，hewouldhavefounditeighty-sevendegreesandfifty-twominutes。Thedifferenceofmeasurementseemsslight；butitsufficedtomakethecomputationsdifferabsurdlyfromthetruth。Thesunisreallynotmerelyeighteentimesbutmorethantwohundredtimesthedistanceofthemoon，asWendeleindiscoveredonrepeatingtheexperimentofAristarchusabouttwothousandyearslater。YetthisdiscrepancydoesnotintheleasttakeawayfromthevalidityofthemethodwhichAristarchusemployed。Moreover，hisconclusion，statedingeneralterms，wasperfectlycorrect:thesunismanytimesmoredistantthanthemoonandvastlylargerthanthatbody。Granted，then，thatthemoonis，asAristarchuscorrectlybelieved，considerablylessinsizethantheearth，thesunmustbeenormouslylargerthantheearth；andthisisthevitalinferencewhich，morethananyother，musthaveseemedtoAristarchustoconfirmthesuspicionthatthesunandnottheearthisthecentreoftheplanetarysystem。Itseemedtohiminherentlyimprobablethatanenormouslylargebodylikethesunshouldrevolveaboutasmallonesuchastheearth。Andagain，itseemedinconceivablethatabodysodistantasthesunshouldwhirlthroughspacesorapidlyastomakethecircuitofitsorbitintwenty-fourhours。But，ontheotherhand，thatasmallbodyliketheearthshouldrevolveaboutthegiganticsunseemedinherentlyprobable。Thispropositiongranted，therotationoftheearthonitsaxisfollowsasanecessaryconsequenceinexplanationoftheseemingmotionofthestars。Here，then，wastheheliocentricdoctrinereducedtoavirtualdemonstrationbyAristarchusofSamos，somewhereaboutthemiddleofthethirdcenturyB。C。
Itmustbeunderstoodthatinfollowingoutthe，stepsofreasoningbywhichwesupposeAristarchustohavereachedsoremarkableaconclusion，wehavetosomeextentguessedattheprocessesofthought-development；fornolineofexplicationwrittenbytheastronomerhimselfonthisparticularpointhascomedowntous。Theredoesexist，however，aswehavealreadystated，averyremarkabletreatisebyAristarchusontheSizeandDistanceoftheSunandtheMoon，whichsoclearlysuggeststhemethodsofreasoningofthegreatastronomer，andsoexplicitlycitestheresultsofhismeasurements，thatwecannotwellpassitbywithoutquotingfromitatsomelength。Itiscertainlyoneofthemostremarkablescientificdocumentsofantiquity。Asalreadynoted，theheliocentricdoctrineisnotexpresslystatedhere。Itseemstobetacitlyimpliedthroughout，butitisnotanecessaryconsequenceofanyofthepropositionsexpresslystated。ThesepropositionshavetodowithcertainobservationsandmeasurementsandwhatAristarchusbelievestobeinevitabledeductionsfromthem，andheperhapsdidnotwishtohavethesedeductionschallengedthroughassociatingthemwithatheorywhichhiscontemporariesdidnotaccept。Inaword，thepaperofAristarchusisarigidlyscientificdocumentunvitiatedbyassociationwithanytheorizingsthatarenotdirectlygermanetoitscentraltheme。Thetreatiseopenswithcertainhypothesesasfollows:
"First。Themoonreceivesitslightfromthesun。
"Second。Theearthmaybeconsideredasapointandasthecentreoftheorbitofthemoon。
"Third。Whenthemoonappearstousdichotomizeditofferstoourviewagreatcircle[oractualmeridian]ofitscircumferencewhichdividestheilluminatedpartfromthedarkpart。
"Fourth。Whenthemoonappearsdichotomizeditsdistancefromthesunislessthanaquarterofthecircumference[ofitsorbit]byathirtiethpartofthatquarter。"
Thatistosay，inmodernterminology，themoonatthistimelacksthreedegreesonethirtiethofninetydegreesofbeingatrightangleswiththelineofthesunasviewedfromtheearth；
or，statedotherwise，theangulardistanceofthemoonfromthesunasviewedfromtheearthisatthistimeeighty-sevendegrees——thisbeing，aswehavealreadyobserved，thefundamentalmeasurementuponwhichsomuchdepends。WemayfairlysupposethatsomepreviouspaperofAristarchus’shasdetailedthemeasurementwhichhereistakenforgranted，yetwhichofcoursecoulddependsolelyonobservation。
"Fifth。Thediameteroftheshadow[castbytheearthatthepointwherethemoon’sorbitcutsthatshadowwhenthemooniseclipsed]isdoublethediameterofthemoon。"
Hereagainaknowledgeofpreviouslyestablishedmeasurementsistakenforgranted；but，indeed，thisisthecasethroughoutthetreatise。
"Sixth。Thearcsubtendedintheskybythemoonisafifteenthpartofasign"ofthezodiac；thatistosay，sincetherearetwenty-four，signsinthezodiac，one-fifteenthofonetwenty-fourth，orinmodernterminology，onedegreeofarc。ThisisAristarchus’smeasurementofthemoontowhichwehavealreadyreferredwhenspeakingofthemeasurementsofArchimedes。
"Ifweadmitthesesixhypotheses，"Aristarchuscontinues，"itfollowsthatthesunismorethaneighteentimesmoredistantfromtheearththanisthemoon，andthatitislessthantwentytimesmoredistant，andthatthediameterofthesunbearsacorrespondingrelationtothediameterofthemoon；whichisprovedbythepositionofthemoonwhendichotomized。Buttheratioofthediameterofthesuntothatoftheearthisgreaterthannineteentothreeandlessthanforty-threetosix。Thisisdemonstratedbytherelationofthedistances，bytheposition[ofthemoon]inrelationtotheearth’sshadow，andbythefactthatthearcsubtendedbythemoonisafifteenthpartofasign。"
Aristarchusfollowswithnineteenpropositionsintendedtoelucidatehishypothesesandtodemonstratehisvariouscontentions。Theseshowasingularlycleargraspofgeometricalproblemsandanaltogethercorrectconceptionofthegeneralrelationsastosizeandpositionoftheearth，themoon，andthesun。Hisreasoninghastodolargelywiththeshadowcastbytheearthandbythemoon，anditpresupposesaconsiderableknowledgeofthephenomenaofeclipses。Hisfirstpropositionisthat"twoequalspheresmayalwaysbecircumscribedinacylinder；twounequalspheresinaconeofwhichtheapexisfoundonthesideofthesmallersphere；andastraightlinejoiningthecentresofthesespheresisperpendiculartoeachofthetwocirclesmadebythecontactofthesurfaceofthecylinderoroftheconewiththespheres。"
ItwillbeobservedthatAristarchushasinmindherethemoon，theearth，andthesunasspherestobecircumscribedwithinacone，whichconeismadetangibleandmeasurablebytheshadowscastbythenon-luminousbodies；since，continuing，heclearlystatesinpropositionnine，that"whenthesunistotallyeclipsed，anobserverontheearth’ssurfaceisatanapexofaconecomprisingthemoonandthesun。"Variouspropositionsdealwithotherrelationsoftheshadowswhichneednotdetainussincetheyarenotfundamentallyimportant，andwemaypasstothefinalconclusionsofAristarchus，asreachedinhispropositionstentonineteen。
Now，sincepropositionten"thediameterofthesunismorethaneighteentimesandlessthantwentytimesgreaterthanthatofthemoon，"itfollowspropositioneleven"thatthebulkofthesunistothatofthemooninratio，greaterthan5832to1，andlessthan8000to1。"
"Propositionsixteen。Thediameterofthesunistothediameteroftheearthingreaterproportionthannineteentothree，andlessthanforty-threetosix。
"Propositionseventeen。Thebulkofthesunistothatoftheearthingreaterproportionthan6859to27，andlessthan79，507
to216。
"Propositioneighteen。Thediameteroftheearthistothediameterofthemooningreaterproportionthan108to43andlessthan60to19。
"Propositionnineteen。Thebulkoftheearthistothatofthemooningreaterproportionthan1，259，712to79，507andlessthan20，000to6859。"
Suchthenarethemoreimportantconclusionsofthisveryremarkablepaper——apaperwhichseemstohaveinteresttothesuccessorsofAristarchusgenerationaftergeneration，sincethisaloneofallthewritingsofthegreatastronomerhasbeenpreserved。HowwidelytheexactresultsofthemeasurementsofAristarchus，differfromthetruth，wehavepointedoutasweprogressed。Butletitberepeatedthatthisdetractslittlefromthecreditoftheastronomerwhohadsuchclearandcorrectconceptionsoftherelationsoftheheavenlybodiesandwhoinventedsuchcorrectmethodsofmeasurement。Letitbeparticularlyobserved，however，thatalltheconclusionsofAristarchusarestatedinrelativeterms。Henowhereattemptstoestimatetheprecisesizeoftheearth，ofthemoon，orofthesun，ortheactualdistanceofoneofthesebodiesfromanother。
Theobviousreasonforthisisthatnodatawereathandfromwhichtomakesuchprecisemeasurements。HadAristarchusknownthesizeofanyoneofthebodiesinquestion，hemightreadily，ofcourse，havedeterminedthesizeoftheothersbythemereapplicationofhisrelativescale；buthehadnomeansofdeterminingthesizeoftheearth，andtothisextenthissystemofmeasurementsremainedimperfect。WhereAristarchushalted，however，anotherworkerofthesameperiodtookthetaskinhandandbyanaltogetherwonderfulmeasurementdeterminedthesizeoftheearth，andthusbroughtthescientifictheoriesofcosmologytotheirclimax。ThisworthysupplementoroftheworkofAristarchuswasEratosthenesofAlexandria。