TWO NEW SCIENCESBY GALILEO

°_o°THE MACMILLAN COMPANY

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GALILEOGALILEI.Subterman'sportrait,paintedaboutI64O;nowin the Galler_adr' Pittiat Florence.

DIALOGUES

CONCERNING

TWO NEW SCIENCESBY

GALILEOGALILEI

Translatedfromthe Italianand Latin intoEnglishbyHENRY CREWAND ALFONSODE SALVIO

of Northwestern University

WITH AN INTRODUCTION BY

ANTONIOFAVARO

of the University of Padua.

"I think with your friend that it has been

of late too much the mode to slight thelearning of the ancients."

Benjamin Franklin, Phil. Trans.64,445.(I774.)

N_ _orkTHE MACMILLAN COMPANY

I914

All rightsreserved

COPX_IGBT.X914B_ THEMACMILLANCOMPANY

Set upande[ectrotyped. PublishedMay_I914.

"La Dynamiqueest lasciencedesforcesaccN6ra-tricesor retardatrices,et des mouvemensvariesqu'ellesdoiventproduire.Cette scienceest dueenti_rementauxmodernes,et Galileeest cduiquien a jet_les premiersfondemens."LagrangeMec.dad.I.zzl,

TRANSLATORS' PREFACE

OR morethana centuryEnglishspeakingstudentshavebeenplacedin theanomalouspositionof hearingGalileoconstantlyre-ferredto asthefounderofmodernphysicalscience,withouthavinganychanceto read,in theirownlanguage,whatGalileohimselfhas to say. Archimedeshas beenmadeavailablebyHeath;Huygens'Lighthasbeen

turnedinto Englishby Thompson,whileMotte has put thePrincipiaof Newtonbackintothe languagein whichit wasconceived.To renderthe Physicsof GalileoalsoaccessibletoEnglishandAmericanstudentsis the purposeofthefollowingtranslation.

The lastof the greatcreatorsof the Renaissancewasnot aprophetwithouthonorin hisowntime;for it wasonlyonegroupofhiscountry-menthat failedto appreciatehim. Evenduringhislifetime,hisMechanicshadbeenrenderedintoFrenchbyoneofthe leadingphysicistsoftheworld,Mersenne.

Withintwenty-fiveyearsafterthe deathofGalileo,hisDia-loguesonAstronomy,andthoseon TwoNewSciences,hadbeendoneinto Englishby ThomasSalusburyand wereworthilyprinted in two handsomequarto volumes. The TwoNewSciences,whichcontainspracticallyallthat Galileohastosayonthe subje&ofphysics,issuedfromthe Englishpressin I665.

vi TRANSLATORS'PREFACEIt issupposedthatmostofthecopiesweredestroyedinthegreatLondonfirewhichoccurredinihe yearfollowing.Weare notawareofanycopyinAmerica:eventhat belongingtotheBritishMuseumisan imperfecCtone.

Againin 173° the TwoNewScienceswasdoneintoEnglishby ThomasWeston;but thisbook,nownearlytwo centuriesold, is scarceand expensive.Moreover,the literalnesswithwhichthis translationwasmaderendersmanypassageseitherambiguousorunintelligibleto themodernreader. Otherthanthesetwo,noEnglishversionhasbeenmade.

Quite recentlyan eminentItalian scholar,after spendingthirtyofthebestyearsofhislifeuponthe sul_jecCt,hasbroughtto completionthe greatNationalEdition of the WorksofGalileo.Wereferto the twentysuperbvolumesin whichPro-fessorAntonioFavaroofPaduahasgivena definitivepresenta-tionofthelaborsofthe manwhocreatedthemodernscienceofphysics.

The followingrenditionincludesneitherLe MechanicheofGalileonorhispaperDeMotuAccelerato,sincethe formerofthesecontainslittlebut the StaticswhichwascurrentbeforethetimeofGalileo,andthelatter isessentiallyincludedin theDialogueoftheThirdDay. Dynamicswasthe onesubjecCttowhichundervariousforms,suchas Ballistics,Acoustics,As-tronomy,he consistentlyand persistentlydevotedhis wholelife. Into the one volumehere translatedhe seemsto havegathered,duringhis last years,pracCticallyall that is of valueeither to the engineeror the physicist. The historian,thephilosopher,and the astronomerwillfind the other volumesrepletewithinterestingmaterial.

tt ishardlynecessaryto addthatwehavestric°dyfollowedthetextofthe NationalEdition---essentiallythe Elzevireditionof1638.Allcommentsandannotationshavebeenomittedsavehereandtherea foot-noteintendedto economizethe reader'stime. To eachofthesefootnoteshasbeenattachedthe signa-ture[Trans.]inorderto preservetheoriginalasnearlyintacCtaspossible.

Muchofthevalueofanyhistoricaldocumentliesin the lan-guageemployed,andthis is doublytruewhenoneattemptsto

TRANSLATORS' PREFACE viitrace the rise and growthof any set of conceptssuch as thoseemployedin modem physics. We have thereforemade thistranslationasliteral as is consistentwithclearnessandmodem-ity. In caseswherethere is any importantdeviationfromthisrule, and in the caseof many technicaltermswherethere is nodeviationfrom it, we have giventhe originalItalian or Latinphrase in italics enclosedin square brackets. The intentionhere is to illustratethe great varietyof termsemployedby theearlyphysiciststo describea singledefiniteidea,and conversely,to illustratethe numeroussensesinwhich,then as now,a singlewordis used. For the fewexplanatoryEnglishwordswhichareplacedin squarebrackets without italics,the translatorsaloneare responsible.The pagingofthe NationalEditionis indicatedin squarebrackets inserted alongthe medianlineof the page.

The imperfecCtionsof the followingpageswouldhave beenmanymorebut for the aidof threeof our colleagues.ProfessorD. R. Curtiss was kind enoughto assist in the translationofthosepageswhichdiscussthe natureofInfinity:ProfessorO.H.Basquingavevaluablehelp in the renditionof the chapteronStrengthof Materials;and ProfessorO. F. Long clearedup themeaningofa numberof Latin phrases.

To ProfessorA. Favaroof the UniversityofPaduathe trans-lators share, with every reader, a feelingof sincereobligationfor his IntroducCtion.

H.C.A. DE S.

EVANSTON)ILLINOIS)x5February,I9x4.

INTRODUCTION..................'...............................RITINGto hisfaithfulfriendEliaDiodati,

_ Galileospeaksofthe"NewSciences"which

hehadin mindto printas being"superiorto everythingelseof minehithertopub-lished";elsewherehe says"they containresults-whichI considerthemostimportantofallmy studies";andthisopinionwhichhe expressedconcerninghisownworkhas

beenconfirmedby posterity:the "NewSciences"are,indeed,the masterpieceofGalileowhoat the timewhenhemadetheaboveremarkshadspentuponthemmorethanthirty laboriousyears.

Onewhowishesto tracethe historyofthisremarkableworkwillfindthat the greatphilosopherlaidits foundationsduringthe eighteenbest yearsof his lifc thosewhichhe spentatPadua. As welearnfromhis last scholar,VincenzioViviani,the numerousresultsatwhichGalileohadarrivedwhilein thiscity,awakenedintenseadmirationin the friendswhohadwit-nessedvariousexperimentsby meansof whichhe wasaccus-tomedto investigateinterestingquestionsinphysics.FraPaoloSarpiexclaimed:To giveus the ScienceofMotion,GodandNaturehavejoinedhandsandcreatedthe intellecCtofGalileo.Andwhenthe "NewSciences"camefromthe pressoneofhisforemostpupils,PaoloAproino,wrotethatthevolumecontainedmuchwhichhehad "alreadyheardfromhisownlips"duringstudentdaysat Padua.

Limitingourselvesto onlythe moreimportantdocumentswhichmightbecitedin supportofourstatement,it willsufficeto mentionthe letter,writtentoGuidobaldodelMonteon the29thofNovember,I6O2,concerningthedescentofheavybodies

x INTRODUCTIONalongthearcsofcirclesandthe chordssubtendedbythem;thatto Sarpi,datedI6thofOctober,I6o4,dealingwiththefreefallof heavybodies;the letterto Afltoniode'Medicion the IIthofFebruary,I6o9,in whichhestatesthat hehas"completedallthe theoremsanddemonstrationspertainingto forcesandre-sistancesof beamsof variouslengths,thicknessesand shapes,provingthat theyareweakerat the middlethanneartheends,that theycancarrya greaterloadwhenthat loadisdistributedthroughoutthe lengthof the beamthanwhenconcentratedatonepoint,demonstratingalsowhatshapeshouldbe givento abeamin orderthat it mayhavethe samebendingstrengthateverypoint,"andthat hewasnowengaged"uponsomeques-tionsdealingwiththemotionofprojeCtiles";andfinallyintheletter to BelisarioVinta,dated7th of May, x6IO,concerninghisreturnfromPaduato Florence,heenumeratesvariouspiecesofworkwhichwerestillto becompleted,mentioningexplicitlythreebooksonan entirelynewsciencedealingwiththe theoryofmotion. Althoughat varioustimesafterthe returnto hisnativestatehe devotedconsiderablethoughttotheworkwhich,evenat that date,he hadin mindas isshownby certainfrag-mentswhichclearlybelongto differentperiodsof his lifeandwhichhave,for the first time,beenpublishedin the NationalEdition;andalthoughthesestudieswerealwaysuppermostinhisthoughtit doesnotappearthat hegavehimselfseriouslytothemuntilafterthe publicationof the Dialogueandthe com-pletionof that trialwhichwasrightlydescribedas thedisgraceofthecentury. InfaCtaslateasOctober,x63o,he barelymen-tionsto Aggiuntihisdiscoveriesin the theoryof motion,andonlytwoyearslater,in alettertoMarsiliconcerningthemotionofprojeCtiles,hehintsat abooknearlyreadyforpublicationinwhichhewilltreat alsoofthis subject;andonlya yearafterthishewritesto Arrighettithat hehas inhandatreatiseontheresistanceofsolids.

But the workwasgivendefiniteformby Galileoduringhisenforcedresidenceat Siena:in thesefivemonthsspentquietlywith the Archbishophe himselfwritesthat he hascompleted"a treatiseona newbranchofmechanicsfullofinterestingandusefulideas";sothat a fewmonthslaterhe wasableto send

INTRODUCTION xiwordto Micanziothat the "workwasready";as soonashisfriendslearnedofthis,theyurgeditspublication.It was,how-ever,no easymatterto print the workofa manalreadycon-demnedbythe HolyOffice:andsinceGalileocouldnothopetoprintit eitherinFlorenceor inRome,heturnedto the faithfulMicanzioaskinghimto findoutwhetherthiswouldbepossibleinVenice,fromwhencehehadreceivedoffersto printthe Dia-logueonthePrincipalSystems,as soonasthenewshadreachedtherethat hewasencounteringdifficulties.At firsteverythingwentsmoothly;sothat GalileocommencedsendingtoMicanziosomeofthe manuscriptwhichwasreceivedby the latterwithan enthusiasmin whichhewassecondto noneof thewarmestadmirersof the greatphilosopher.But whenMicanziocon-sultedthe Inquisitor,he receivedthe answerthat therewasan expressorderprohibitingthe printingor reprintingof anyworkof Galileo,eitherin Veniceor in anyotherplace,hulloexcepto.

As soonasGalileoreceivedthisdiscouragingnewshebeganto lookwithmorefavoruponofferswhichhadcometohimfromGermanywherehisfriend,andperhapsalsohisscholar,Gio-vanniBattistaPieroni,wasin the serviceof the Emperor,asmilitaryengineer;consequentlyGalileogaveto PrinceMattiade'Mediciwhowasjust leavingforGermanythe firsttwoDia-loguesto be handedto Pieroniwhowasundecidedwhethertopublishthemat ViennaorPragueor at someplaceinMoravia;in themeantime,however,hehadobtainedpermissionto printbothat Viennaandat Olmtitz.But Galileorecognizeddangerat everypointwithinreachof the longarmof the CourtofRome;hence,availinghimselfoftheopportunityofferedby thearrivalof LouisElzevirin Italyin 1636,alsoofthe friendshipbetweenthe latter and Micanzio,not to mentiona visit atArcetri,he decidedto abandonallotherplansandentrusttothe Dutchpublisherthe printingof hisnewworkthe manu-scriptofwhich,althoughnotcomplete,Elzevirtookwithhimonhisreturnhome.

In the courseoftheyear1637,theprimingwasfinished,andat the beginningofthe followingyeartherewaslackingonlythe index,the title-pageand the dedication.This last had,

xii INTRODUCTIONthroughthegoodofficesofDiodati,beenofferedto theCountofNoailles,a formerscholarofGalileoat Padua,andsince1634ambassadorofFranceatRome,amanwhodidmuchto alleviatethe distressingconsequencesof the celebratedtrial; and theofferwasgratefullyaccepted.The phrasingof the dedicationdeservesbriefcomment.SinceGalileowasaware,onthe onehand,ofthe prohibitionagainstthe printingofhisworksandsince,on theotherhand,he didnotwishto irritatethe CourtofRomefromwhosehandshewasalwayshopingforcompletefreedom,he pretendsin the dedicatoryletter (where,probablythroughexcessof caution,he givesonlymainoutlines)that hehadnothingto dowiththeprintingofhisbook,assertingthathe willneveragainpublishanyof his researches,andwillatmostdistributehereandthere a manuscriptcopy. He evenexpressesgreatsurprisethat hisnewDialogueshavefallenintothe handsof the Elzevirsandweresoonto be published;sothat, havingbeenaskedtowritea dedication,hecouldthinkofno manmoreworthywhocouldalsoon this occasiondefendhimagainsthisenemies.

As to the title whichreads:Discoursesand MathematicalDemonstrationsconcerningTwoNew SciencespertainingtoMe-chanicsandLocalMotions,thisonlyisknown,namely,that thetitleisnot theonewhichGalileohaddevisedandsuggested;infac_he protestedagainstthe publisherstakingthe libertyofchangingit andsubstituting"a lowandcommontitle forthenobleanddignifiedonecarrieduponthetitle-page."

In reprintingthisworkin the NationalEdition,I havefol-lowedthe Leydentextof 1638faithfullybut notslavishly,be-causeI wishedto utilizethe largeamountofmanuscriptma-terialwhichhascomedownto us,forthepurposeof colTecctinga considerablenumberof errorsin this firstedition,and alsofor thesakeofinsertingcertainadditionsdesiredby the authorhimself.IntheLeydenEdition,thefourDialoguesarefollowedby an"Appendixcontainingsometheoremsandtheirproofs,deal-zngwithcentersof gravityof solidbodies,writtenby thesameAuthorat anearlierdate,"whichhasnoimmediateconnecCtionwiththesubjec°cstreatedintheDialogues;thesetheoremswerefoundby Galileo,as hetellsus, "at the ageoftwenty-twoand

INTRODUCTION _5iiaftertwoyearsstudyofgeometry"andwerehereinsertedonlyto savethemfromoblivion.

But it wasnot the intentionof Galileothat the DialoguesontheNewSciencesshouldcontainonlythefourDaysandtheabove-mentionedappendixwhichconstitutethe LeydenEdi-tion;while,on the onehand,the Elzevirswerehasteningtheprintingandstrivingto completeit at theearliestpossibledate,Galileo,on the otherhand,kepton speakingof anotherDay,besidesthe four,thusembarrassingandperplexingtheprinters.Fromthe correspondencewhichwenton betweenauthorandpublisher,it appearsthat thisFifthDaywasto havetreated"of the forceof percussionandthe useofthe catenary";,butas the typographicalworkapproachedcompletion,the pnnterbecameanxiousfor the bookto issuefromthe presswithoutfurtherdelay;and thus it cameto passthat the DiscorsieDimostrazioniappearedcontainingonlythe fourDaysandtheAppendix,in spiteof the fact that inApril,I638,Galileohadplungedmoredeeplythanever"into the profoundquestionofpercussion"and"had almostreacheda completesolution."

The "NewSciences"nowappearin an editionfollowingthetext whichI, afterthemostcarefulanddevotedstudy,deter-mineduponfor the NationalEdition. It appearsalsoin thatlanguageinwhich,aboveallothers,I havedesiredto seeit. Inthistranslation,thelastandripestworkofthegreatphilosophermakesits first appearancein the NewWorld:if towardthisimportantresultI mayhopetohavecontributedinsomemeas-ureI shallfeelamplyrewardedforhavinggivento thisfieldofresearchthe bestyearsofmylife. AwroNmFAv_a_o.UNIVERSITY OF PADUA_

2_ o]October,I9x3.

DISCOR, SIE

DIMOSTR.AZIONIMI.A.TEMA TI C H E,

intorno_duenuouefiiene_eAtxenentialla

M_-CANICA_ i Mo VIMENTI LOCALt;

delSignorGALILEO GALILEI LINCEO,

Filofofoe MatematicoprimariodelSereniilimoGrandDucadiTofcana.

c°o_z_a_ppendicedelceutrodigrauit_d'klc-_ni._olidi._

IN LEIDA,

ApprdrogliEl/_virii._.D.c,xxxv_.

' [431TO THE MOST ILLUSTRIOUS LORD

COUNT OF NOAILLESCounsellorofhisMostChristianMajesty,KnightoftheOrder

of theHolyGhost,FieldMarshaland Commander,SeneschalandGovernorofRouergue,andHis

Majesty'sLieutenantinAuvergne,myLordandFForshipfulPatron

OSTILLUSTRIOUSLORD:-In the pleasurewhichyou derivefrom

the possessionof thisworkof mineI rec-i]_/i[ ll_r_-_[_ognizeyourLordship'smagnanimity.Thei[lV/[]_-4_}disappointmentanddiscouragementI have

_'_[,_feltoverthe ill-fortunewhichhasfollowed__ myotherbooksare alreadyknownto you.

Indeed,I had decidednot to publishanymoreof my work. Andyet in order to save it fromcom-plete oblivion,it seemedto me wiseto leavea manuscriptcopyin someplacewhereit wouldbe availableat leasttothosewho followintelligentlythe subjet% whichI have treated.AccordinglyI chosefirstto placemyworkin yourLordship'shands,askingno moreworthydepository,andbelievingthat,onaccountofyouraffecCtionforme,youwouldhaveat heartthepreservationof mystudiesand labors.Therefore,whenyouwerereturninghomefromyourmissionto Rome,I cameto paymyrespecCtsinpersonasI hadalreadydonemanytimesbeforebyletter. At thismeetingI presentedto yourLordshipacopyofthesetwoworkswhichat thattimeIhappenedtohaveready.InthegraciousreceptionwhichyougavetheseI foundassuranceof

xviii TO THE COUNT OF NOAILLESoftheirpreservation.Thefacetofyourcarryingthemto Franceandshowingthemto friendsofyourswhoare skilledin thesesciencesgaveevidencethatmysilencewasnotto beinterpretedascompleteidleness.Alittlelater,justasI wasonthepointof

[44]sendingothercopiesto Germany,Flanders,England,Spainandpossiblyto someplacesin Italy,I wasnotifiedby theElzevirsthat theyhadtheseworksofminein pressandthat I oughttodecideupona dedicationandsendthema replyat once. ThissuddenandunexpecCtednewsledmeto thinkthat theeagernessof your Lordshipto reviveand spreadmy nameby passingtheseworkson to variousfriendswasthe real causeof theirfallingintothe handsofprinterswho,becausetheyhadalreadypublishedotherworksofmine,nowwishedto honormewithabeautifulandornateeditionofthiswork. Butthesewritingsofminemusthavereceivedadditionalvaluefromthe criticismofso excellenta judgeas yourLordship,whoby the unionofmanyvirtueshaswonthe admirationof all. Your desiretoenlargetherenownofmyworkshowsyourunparalleledgeneros-ity and your zeal for the publicwelfarewhichyou thoughtwouldthus be promoted. Under these circumstancesit iseminentlyfittingthat I should,in unmistakableterms,grate-fullyacknowledgethisgenerosityon thepartofyourLordship,whohasgivento myfamewingsthathavecarriedit intoregionsmoredistantthanI haddaredto hope. It is,therefore,properthat I dedicateto yourLordshipthischildof my brain. TothiscourseI amconstrainednotonlybythe weightof obliga-tion underwhichyou haveplacedme,but also,if I may sospeak,by the interestwhichI havein securingyourLordshipas the defenderofmyreputationagainstadversarieswhomayattackit whileI remainunderyourprotecCtion.

Andnow,advancingunderyourbanner,I paymy respecCtsto youbywishingthatyoumayberewardedforthesekindnessesbythe achievementofthe highesthappinessandgreatness.

I amyourLordship'sMost devotedServant,

GALILEOGALILEn_lrcetri,6March,I638.

THE PUBLISHER TO THE READERINCEsocietyisheldtogetherbythemutualserviceswhichmenrenderoneto another,andsinceto thisendthe artsandscienceshave largelycontributed,investigationsinthesefieldshavealwaysbeenheldin greatesteemandhavebeenhighlyregardedbyourwiseforefathers.Thelargertheutilityandexcellenceofthe inventions,the greaterhas

beenthehonorandpraisebestowedupontheinventors.Indeed,menhaveevendeifiedthemandhaveunitedin theattempttoperpetuatethe memoryof theirbenefafftorsbythe bestowalofthissupremehonor.

Praiseand admirationare likewisedueto thosecleverin-tellecCtswho, confiningtheir attentionto the known,havediscoveredand corre&edfallaciesand errorsin many andmany a propositionenunciatedby men of distincCtionandacceptedforagesasfacet.Althoughthesemenhaveonlypointedoutfalsehoodandhavenotreplacedit bytruth,theyarenever-thelessworthyof commendationwhenweconsiderthe well-knowndifficultyof discoveringfacet,a difficultywhichledtheprinceof oratorsto exclaim:Utinaratamfacilepossemverareperire,quamfalsa convincere.*And indeed,these latestcenturiesmeritthis praisebecauseit is duringthemthat theartsandsciences,discoveredbytheancients,havebeenreducedto so great and constantlyincreasingperle&ionthroughtheinvestigationsand experimentsof clear-seeingminds. Thisdevelopmentis particularlyevidentin the caseof the mathe-maticalsciences.Here,withoutmentioningvariousmenwhohaveachievedsuccess,wemustwithouthesitationandwiththe

*Cicero.deNaturaDeorum,I, 9I. [Trans.]

xx THE PUBLISHER TO THE READER

unanimousapprovalof scholarsassignthe first placeto GalileoGalilei,Memberof theAcademyoftheLincei. Thishedeservesnot only because he has effectivelydemonstratedfallaciesinmany of our current conclusions,as is amply shownby hispublishedworks, but also becauseby means of the telescope(inventedin this countrybut greatlyperfectedby him) he hasdiscoveredthe four satellitesof Jupiter, has shownus the truecharaCterof the MilkyWay, and has made us acquaintedwithspots on the Sun, with the rough and cloudyportionsof thelunar surface, with the threefold nature of Saturn, with thephases of Venus and with the physical charaCterof comets.Thesematterswereentirelyunknownto the ancientastronomersandphilosophers;so that wemay truly say that he has restoredto the worldthe scienceof astronomyand has presentedit in anewlight.

Rememberingthat the wisdomand powerand goodnessoftheCreator are nowhereexhibitedso well as in the heavensandcelestialbodies,we can easilyrecognizethe great merit of himwho has brought these bodies to our knowledgeand has, inspite of their almost infinite distance, rendered them easilyvisible. For, accordingto the commonsaying,sight can teachmoreandwith greatercertaintyina singleday than canprecepteven though repeateda thousand times; or, as another says,intuitiveknowledgekeepspacewithaccuratedefinition.

But the divine and natural gifts of this man are showntobest advantage in the present 'workwhere he is seen to havediscovered,though not w_6houtmany labors and long vigils,twoentirely new sciencesand to have demonstratedthem in arigid, that is, geometric,manner: and what is even more re-markablein this work is the facetthat one of the two sciencesdealswith a subjeCtof never-endinginterest,perhapsthe mostimportantin nature, onewhichhas engagedthe mindsof all thegreat philosophersand one concerningwhichan extraordinarynumber of bookshave been written. I refer to motion [motolocale],a phenomenonexhibitingvery many wonderfulproper-ties,noneofwhichhas hithertobeendiscoveredor demonstratedby any one. The other sciencewhichhehas alsodevelopedfrom

its

THE PUBLISHER TO THE READER xxiits very foundationsdealswith the resistancewhichsolidbodiesoffer to fracture by external forces_er violenza],a subjectofgreat utility, especiallyin the sciencesand mechanicalarts,and onealsoaboundinginpropertiesand theoremsnothithertoobserved.

In this volume one finds the first treatment of these twosciences,full of propositionsto which,as time goes on, ablethinkers willadd many more;alsoby meansof a largenumberof clear demonstrationsthe author points the way to manyother theoremsas will be readilyseenandunderstoodby all in-telligent readers.

TABLE OF CONTENTS

I PageFirstnewscience,treatingof theresistancewhichsolidbodies

offertofracture.FirstDay.......................... I

IIConcerningthecauseof cohesion.SecondDay............ lO9

IIISecondnewscience,treatingof motion[movimentilocah].

ThirdDay........................................ 153Uniformmotion...................................... 154Naturallyacceleratedmotion........................... 16o

IV

Violentmotions.ProjeFtiles.FourthDay............... 244

V

Appendix;theoremsanddemonstrationsconcerningthecentersof gravityofsolids.................................. 295

TWO NEW SCIENCESBYGALILEO

FIRST DAYINTERLOCUTORS: SALVIATI, SA-

GREDO AND SIMPLICIO:'..............:'..............................ALV.TheconstantactivitywhichyouVene-

_ tiansdisplayinyourfamousarsenalsuggests

tothestudiousminda largefieldforinvesti-gation,especiallythat part of the workwhichinvolvesmechanics;forinthisdepart-mentalltypesofinstrumentsandmachinesare constantlybeingconstrue'tedby manyartisans,amongwhomtheremustbesome

who,partlybyinheritedexperienceandpartlyby theirownob-servations,havebecomehighlyexpertandcleverin explanation.

SAc_.Youare quitefight. Indeed,I myself,beingcuriousby nature,frequentlyvisitthis placeforthe merepleasureofobservingtheworkofthosewho,onaccountoftheirsuperiorityoverotherartisans,wecall"firstrankmen." Conferencewiththemhasoftenhelpedmein the investigationofcertaineffec2sincludingnotonlythosewhicharestriking,butalsothosewhicharereconditeandalmostincredible.AttimesalsoI havebeenput to confusionanddrivento despairofeverexplainingsome-thingforwhichI couldnotaccount,butwhichmysensestoldmeto be true. Andnotwithstandingthefadtthat whatthe oldman told us a little whileago is proverbialand commonlyaccepted,yet it seemedto mealtogetherfalse,likemanyanothersayingwhichis currentamongthe ignorant;for I thinktheyintroducetheseexpressionsin orderto givethe appearanceofknowingsomethingaboutmatterswhichtheydonotunderstand.Salv.

2 THE TWO NEW SCIENCESOF GALILEO[3o]

SAJ_v.Yourefer,perhaps,to that lastremarkofhiswhenweaskedthe reasonwhy they employedstocks,scaffoldingandbracingoflargerdimensionsforlaunchingabigvesselthantheydofora smallone;andheansweredthat theydidthisinordertoavoidthe dangeroftheshippartingunderitsownheavyweight[vastamole],a dangertowhichsmallboatsarenotsubject?

SAcR.Yes,that iswhatI mean;andI referespeciallyto hislastassertionwhichI havealwaysregardedasa false,thoughcurrent,opinion;namely,that in speakingof theseandothersimilarmachinesonecannotarguefromthe smallto the large,becausemany deviceswhichsucceedon a smallscaledo notworkona largescale.Now,sincemechanicshasitsfoundationin geometry,wheremeresizecutsnofigure,I donotseethat thepropertiesof circles,triangles,cylinders,conesandothersolidfigureswillchangewiththeirsize. If, therefore,alargemachinebe construcCtedinsuchaway that itspartsbearto oneanotherthesameratioasinasmallerone,andifthesmallerissufficientlystrongfor the purposeforwhichit wasdesigned,I donot seewhythe largeralsoshouldnotbeableto withstandanysevereanddestrucCtiveteststowhichit maybesubjected.

SAT.v.Thecommonopinionishereabsolutelywrong.Indeed,it is sofar wrongthat preciselythe oppositeis true, namely,thatmanymachinescanbeconstrucCtedevenmoreperfecCtlyonalargescalethanonasmall;thus,forinstance,aclockwhichindi-catesandstrikesthehourcanbemademoreaccurateona largescalethan on a small. Thereare someintelligentpeoplewhomaintainthis sameopinion,but on morereasonablegrounds,whenthey cut loosefromgeometryand arguethat the betterperformanceofthe largemachineisowingto the imperfecCtionsandvariationsofthematerial.HereI trustyouwillnotcharge

Is1]mewith arroganceif I say that imperfectionsin the material,eventhosewhichare great enoughto invalidatethe clearestmathematicalproof,arenotsufficientto explainthe deviationsobservedbetweenmachinesinthe concreteandinthe abstra_.Yet I shallsayit andwillaffirmthat,evenif the imperfecCtions

did

FIRST DAY 3didnotexistandmatterwereabsolutelyperfecCt,unalterableandfreefrom all accidentalvariations,still the merefact that it ismatter makes the larger machine,built of the same materialand in the same proportionas the smaller,correspondwithexacCtnessto the smallerineveryrespecCtexceptthat it willnotbe so strong or so resistant against violent treatment; thelarger the machine,the greater its weakness. SinceI assumematter to be unchangeableand alwaysthe same,it is clearthatweare nolessable to treat this constantandinvariablepropertyin a rigidmannerthan ifit belongedto simpleand puremathe-matics. Therefore,Sagredo,you woulddo wellto changetheopinionwhich you, and perhapsalso many other studentsofmechanics,haveentertainedconcerningthe abilityof machinesand structures to resist external disturbances,thinkingthatwhenthey are builtof the samematerialand maintainthesameratio betweenparts, they are able equally,or rather propor-tionally, to resist or yield to such external disturbancesandblows. For we can demonstrateby geometrythat the largemachineis notproportionatelystrongerthan thesmall. Finally,we may say that, for every machineand strucCture,whetherartificialor natural, there is set a necessarylimitbeyondwhichneitherart nor nature canpass; it is hereunderstood,of course,that the material is the same and the proportionpreserved.

SAGI_.Mybrain alreadyreels. Mymind,likea cloudmomen-tarily illuminatedby a lightning-flash,is for an instant filledwithan unusuallight,whichnowbeckonsto meand whichnowsuddenly mingles and obscuresstrange, crude ideas. Fromwhat you have said it appears to me impossibleto build twosimilarstrucCturesofthe samematerial,but ofdifferentsizesandhave themproportionatelystrong;and if this wereso, it would

[52]not be possibleto tind twosinglepolesmade of the same-woodwhich shall be alike in strength and resistancebut unlikeinsize.

SALv.Soit is,Sagredo. _Andto makesurethat weunderstandeach other, I say that if we take a woodenrod of a certainlength and size, fitted, say, into a wall at right angles, i. e.,

parallel

4 THE TWO NEW SCIENCES OF GALILEOparallelto the horizon,it may be reducedto sucha lengththatit willjust support itself; so that if a ha_r'sbreadthbe addedtoits lengthit willbreakunder its ownweightandwillbe the onlyrodof the kind in the world.* Thus if,for instance,its lengthbea hundredtimesits breadth,youwillnotbe able to findanotherrodwhoselengthis alsoa hundred timesits breadthand which,like the former, is just able to sustain its own weight and nomore:all the largeroneswillbreakwhileall the shorteroneswillbe strong enough to support somethingmore than their ownweight. And thiswhichI havesaidabout the abilityto supportitselfmust beunderstoodto applyalsotoother tests; so that if apieceof scantling[corrente]willcarrythe weightoften similartoitself, a beam [trave]having the same proportionswill not beableto supportten similarbeams.

Please observe, gentlemen,how faCtswhich at first seemimprobablewill, even on scant explanation,drop the cloakwhich has hidden them and stand forth in naked and simplebeauty. Who doesnot knowthat a horsefallingfrom a heightof three or four cubitswill break his bones,whilea dog fallingfromthe sameheightor a cat froma heightof eightor ten cubitswill sufferno injury? Equally harmlesswouldbe the fall of agrasshopperfrom a toweror the fallof an ant fromthe distanceof the moon. Do not childrenfallwith impunityfromheightswhichwouldcosttheir eldersabroken legor perhapsa fraCturedskull? And just assmalleranimalsare proportionatelystrongerand more robust than the larger,soalsosmallerplantsare ableto stand upbetter than larger. I amcertainyoubothknowthatan oak two hundred cubits [braccia]highwouldnot be able tosustainits ownbranchesif they weredistributedas in a tree ofordinarysize;and that nature cannotproducea horseas largeastwenty ordinary horses or a giant ten times taller than an

' [53]ordinary man unless by miracle or by greatly altering theproportionsofhis limbsand especiallyofhis bones,whichwouldhave to be considerablyenlargedover the ordinary. Likewisethe currentbeliefthat, in the caseof artificialmachinesthevery

*The authorhereapparentlymeansthat the solutionis unique.[Trans.]

FIRST DAY 5largeand the smallare equallyfeasibleand lastingis a man_festerror. Thus, for example,a smallobeliskor columnor othersolidfigurecancertainlybe laiddownor setup without dangerofbreaking,whilethe very largeoneswillgotopiecesundertheslightestprovocation,and that purely on accountof their ownweight. AndhereI must relatea circumstancewhichis worthyofyourattention asindeedare alleventswhichhappencontraryto expecCtation,especiallywhen a precautionarymeasureturnsout to be a causeof disaster. A largemarblecolumnwas laidout so that its two ends rested each upon a pieceof beam; alittle later it occurredto a mechanicthat, inorderto be doublysureof its notbreakingin the middleby its ownweight,it wouldbe wise to lay a third support midway;this seemedto all anexcellentidea;but the sequelshowedthat it wasquite the oppo-site, for not many monthspassedbeforethe columnwas foundcrackedand brokenexadtlyabovethe newmiddlesupport.

Sn_P.A very remarkableand thoroughlyunexpectedacci-dent, especiallyif causedby placingthat new support in themiddle.

SALV.Surely this is the explanation,and the moment thecause is knownour surprisevanishes;for when the two piecesof the columnwereplacedon levelgroundit wasobservedthatone of the end beamshad, after a longwhile,becomedecayedand sunken,but that the middleone remainedhard and strong,thus causingone halfof the columnto projecCtin the air withoutany support. Under these circumstancesthe body thereforebehaveddifferentlyfrom what it wouldhavedoneif supportedonly upon the first beams; becauseno matter howmuch theymight have sunken the columnwouldhave gonewith them.This is an accidentwhichcouldnotpossiblyhavehappenedto asmallcolumn,eventhoughmadeofthe samestoneand havingalength correspondingto its thickness,i. e., preservingthe ratiobetweenthicknessand lengthfoundin the largepillar.

[541SAc_.I am quite convincedofthe fa_s of the case,but I do

not understandwhy the strength and resistanceare not multi-pliedin the sameproportionas the material;and I am the more

puzzled

6 THE TWO NEW SCIENCES OF GALILEO

puzzledbecause,on the contrary, I havenoticedin other casesthat the strength and resistanceagainstbreakingincreasein alargerratio than the amountof material. Thus, for instance,iftwo nails be driven into a wall, the one which is twice as bigas the other will supportnot only twiceas muchweightas theother,but threeor fourtimes asmuch.

SALv.Indeedyouwillnotbe far wrongifyousay eighttimesas much; nor doesthis phenomenoncontradicCtthe other eventhoughinappearancethey seemsodifferent.

SACR.Will you not then, Salviati,removethese difficultiesand clear awaythese obscuritiesif possible:for I imaginethatthis problemofresistanceopensup a fieldofbeautifuland usefulideas;and if youare pleasedtomake thisthe subjecCtof to-day'sdiscourseyou will placeSimplicioand me undermany obliga-tions.

SALV.I am at your serviceif only I cancall to mindwhat Ilearned from our Academician* who had thought much uponthis subjecCtand accordingto his custom had demonstratedeverything by geometricalmethods so that one might fairlycall this a new science. For, althoughsomeof his conclusionshad been reachedby others, first of all by Aristotle,these arenot the most beautiful and, what is more important,they hadnot beenproven ina rigidmannerfromfundamentalprinciples.Now,since I wish to convinceyou by demonstrativereasoningrather than to persuadeyou by mereprobabilities,I shall sup-posethat youare familiarwithpresent-daymechanicssofar asit is needed in our discussion. First of all it is necessary,toconsiderwhat happenswhena pieceofwoodor any other solid

. which coheres firmly is broken; for this is the fundamentalfacet,involvingthe firstand simpleprinciplewhichwemust takefor grantedas wellknown.

To grasp this more clearly,imaginea cylinderor prism,AB,made of wood or other solid coherent material. Fasten theupper end, A, so that the cylinderhangs vertically. To thelowerend, B, attach the weight C. It is clear that howevergreat they may be, the tenacity and coherence [tenacit_e

• I. e. Galileo:The authorfrequentlyrefersto himselfunderthisname. [Tran_r.]

FIRST DAY 7[55]

eoeren_]betweenthe partsofthissolid,so longastheyarenotinfinite,canbeovercomeby thepullofthe weightC, aweightwhichcanbeincreasedindefinitelyuntilfinallythe solidbreakslikea rope. Andas in the caseofthe ropewhosestrengthweknowto be derivedfroma multitudeof hempthreadswhichcomposeit, soin thecaseofthe wood,weobserveitsfibresandfilamentsrunlengthwiseand renderit muchstrongerthan ahempropeof the samethickness.But in thecaseof a stoneor metalliccylinderwherethe'coherenceseemsto be stillgreaterthe cementwhichholdsthe parts togethermustbe some-thingotherthan filamentsand fibres;and yeteventhiscanbebrokenbya strongpull.

Srme.If thismatterbe asyousay I canwellunderstandthat the fibresofthewood,beingaslongas thepieceofwooditself,renderit strongand resistant againstlargeforcestendingtobreakit. But howcan onemakea ropeonehundredcubitslongoutofhempenfibreswhicharenotmorethantwoor threecubitslong,andstillgiveit somuchstrength? Besides,I shouldbegladto hearyouropinionas to themannerinwhichthe parts of metal,stone,andotherma-terialsnot showinga filamentousstrucCtureare Fig.iput together;for, if I mistakenot, theyexhibitevengreatertenacity.

SALV.To solvetheproblemswhichyouraiseit willbeneces-saryto makea digressionintosubjecCtswhichhavelittlebearinguponourpresentpurpose.

SAcg.But if, by digressions,wecanreachnewtruth, whatharm is there in makingonenow,so that we maynot losethisknowledge,rememberingthat suchan opportunity,onceomitted,maynotreturn;rememberingalsothatwearenottieddowntoa fixedandbriefmethodbutthat wemeetsolelyforourownentertainment?Indeed,whoknowsbutthat wemaythus

[S6]frequently

8 THE TWO NEW SCIENCESOF GALILEOfrequentlydiscoversomethingmore interestingandbeautifulthanthe solutionoriginallysought._I begofyou,therefore,tograntthe requestof Simplicio,whichis alsomine;forI amnolesscuriousanddesirousthanhe to learnwhatis the bindingmaterialwhichholdstogetherthe partsof solidsso that theycanscarcelybe separated.Thisinformationis alsoneededtounderstandthe coherenceof the parts of fibresthemselvesofWhichsomesolidsarebuiltup.

SAJ_V.I am at yourservice,sinceyou desireit. The firstquestionis,Howare fibres,eachnot morethantwo or threecubitsin length,so tightlyboundtogetherinthe caseof aropeonehundredcubitslongthat greatforce[violent]isrequiredtobreakit?

Nowtellme,Simplicio,canyounotholda hempenfibresotightlybetweenyourfingersthat I, pullingby the otherend,wouldbreakit beforedrawingit awayfromyou? Certainlyyoucan. Andnowwhenthefibresofhempareheldnotonlyatthe ends,but aregraspedbythe surroundingmediumthrough-outtheirentirelengthisit notmanifestlymoredii_cultto tearthemloosefromwhatholdsthemthanto breakthem? Butinthe caseofthe ropethe veryacCtoftwistingcausesthe threadstobindoneanotherin suchawaythatwhentheropeisstretchedwitha greatforcethe fibresbreakratherthan separatefromeachother.

At the pointwherea ropepartsthe fibresare,as everyoneknows,veryshort,nothinglikea cubitlong,astheywoaldbeifthe partingof the ropeoccurred,not by the breakingof thefilaments,but bytheirslippingoneovertheother.

SAGR.In confirmationofthis it maybe remarkedthat ropessometimesbreak not by a lengthwisepull but by excessivetwisting.This,it seemsto me,isaconclusiveargumentbecausethe threadsbindoneanotherso tightlythat the compressingfibresdonotpermitthosewhicharecompressedto lengthenthespiralseventhat littlebit bywhichit is necessaryfor themtolengtheninorderto surroundtheropewhich,ontwisting,growsshorterandthicker.

SALv.Youarequiteright. Nowseehowonefa_ suggestsanother

FIRST DAY 9another. Thethreadheldbetweenthe fingersdoesnot yield

[ST]to onewhowishesto drawit awayevenwhenpulledwithcon-siderableforce,but resistsbecauseit is heldbackby a doublecompression,seeingthat the upperfingerpressesagainsttheloweras hamasthe loweragainsttheupper.Now,ifwecouldretainonlyoneof thesepressuresthereis no doubtthat onlyhalf the originalresistancewouldremain;but sincewe are_not able,by lifting,say, the upperfinger,to removeoneofthesepressureswithoutalsoremovingthe other,it becomesnecessaryto preserveoneof themby meansof a newdevicewhichcausesthe threadto pressitselfagainstthe fingeroragainstsomeothersolidbodyuponwhichit rests;andthusit isbroughtaboutthat theveryforcewhichpullsit in orderto snatchit awaycompressesitmoreand moreas the pullincreases.Thisis accomplishedby wrappingthe threadaroundthe solidin the mannerof a spiral; _I_andwillbebetterunderstoodbymeansofafigure.LetABandCDbetwocylindersbe-tweenwhichis stretchedthethreadEF:and _Ofor the sakeof greaterclearnesswewillim-agineit to be a smallcord. If these twocylindersbe pressedstronglytogether,thecordEF, whendrawnbytheendF,willun-doubtedlystanda considerablepullbeforeitslipsbetweenthe two compressingsolids.But ifweremoveoneofthesecylindersthecord,thoughremainingin contacCtwith theother, willnot therebybe preventedfromslippingfreely. On the other hand,if oneholdsthecordlooselyagainstthe topof the Fig.2cylinderA, windsit in the spiralformAFLOTR,and thenpullsit by the endR, it is evidentthat the cordwillbegintobindthe cylinder;the greaterthe numberof spiralsthe moretightlywillthe cordbe pressedagainstthe cylinderby anygivenpull. Thusasthe numberof turnsincreases,the lineofcontacCt

Io THE TWO NEW SCIENCESOF GALILEOcontactbecomeslongerandin consequencemoreresistant;sothat the cordslipsandyieldsto thetractiveforcewithincreas-ingdifficulty.

[58]Isitnotclearthatthisispreciselythekindofresistancewhich

onemeetsin thecaseofa thickhempropewherethefibresformthousandsandthousandsof similarspirals?And,indeed,the

qbindingeffecCtof theseturnsis sogreatthat a fewshortrusheswoventogetherintoa fewinterlacingspiralsformoneof thestrongestof ropeswhichI believetheycallpackrope[susta].

SAoR.Whatyou sayhasclearedup twopointswhichI didnotpreviouslyunderstand.Onefact is howtwo,or at mostthree,turnsofa ropearoundtheaxleofa windlasscannotonlyholdit fast,but canalsopreventit fromslippingwhenpulledby the immenseforceof the weight[forzadelpeso]whichitsustains;andmoreoverhow,byturningthewindlass,thissameaxle,by merefricCtionof the ropearoundit, canwindup and

lifthugestoneswhilea mereboy i'sableto handlethe slackof therope. TheotherfaCthasto dowithasimplebutcleverdevice,inventedbyayoungkins-manof mine,forthe purposeof descendingfromawindowby meansof a ropewithoutlaceratingthepalmsofhishands,ashadhappenedto himshortlybeforeandgreatlytohisdiscomfort.Asmallsketchwillmakethis clear. He took a woodencylinder,AB,aboutasthickasa walkingstickandaboutonespanlong:on thishe cut a spiralchannelof aboutoneturnanda half,andlargeenoughto just receivetheropewhichhewishedtouse. Havingintroducedtheropeat theendAandledit outagainat the end

BB, he enclosedboth the cylinderandthe ropein acaseofwoodor tin, hingedalongthe 81desothat it

Fig.3 couldbe easilyopenedand closed. After he hadIastenedtheropeto afirmsupportabove,he could,on grasp-ingandsqueezingthecasewith bothhands,hangbyhisarms.The pressureon therope,lyingbetweenthe caseand the cyl-inder,wassuchthat he could,at will,eithergraspthe case

more

FIRST DAY IImoretightlyand holdhimselffromslipping,or slackenhisholdanddescendasslowlyashewished.

IS9]SALV.A truly ingeniousdevice! I feel,however,that for

a completeexplanationotherconsiderationsmightwellenter;yet I mustnotnowdigressuponthisparticulartopicsinceyouarewaitingto hearwhatI thinkaboutthebreakingstrengthofother materialswhich,unlikeropesandmostwoods,do notshowa filamentousstructure. The coherenceof thesebodiesis,in my estimation,producedby othercauseswhichmaybegroupedundertwoheads. Oneis that much-talked-ofrepug-nancewhichnatureexhibitstowardsavacuum;but thishorrorof a vacuumnot beingsufficient,it is necessaryto introduceanothercausein theformof aglueyor viscoussubstancewhichbindsfirmlytogetherthecomponentpartsofthebody.

FirstI shallspeakofthe vacuum,demonstratingby definiteexperimentthe qualityandquantityofits force[o/rt_].If youtaketwohighlypolishedandsmoothplatesofmarble,metal,orglassandplacethemfaceto face,onewillslideoverthe otherwiththe greatestease,showingconclusivelythat thereisnoth-ingof aviscousnaturebetweenthem. Butwhenyouattemptto separatethemandkeepthemat a constantdistanceapart,youfindtheplatesexhibitsucha repugnanceto separationthatthe upperonewillcarrythe loweronewithit andkeepit liftedindefinitely,evenwhenthelatterisbigandheavy.

This experimentshowsthe aversionof nature for emptyspace,evenduringthebriefmomentrequiredfortheoutsideairto rush in andfillup theregionbetweenthe twoplates. It isalsoobservedthat if twoplatesare not thoroughlypolished,theircontactisimperfectsothat whenyouattemptto separatethem slowlythe onlyresistanceofferedis that of weight;if,however,the pullbe sudden,then the lowerplate rises,butquicklyfallsback,havingfollowedtheupperplateonlyforthatvery shortintervalof time requiredfor the expansionof thesmallamountof air remainingbetweenthe plates,in conse-quenceoftheirnotfitting,andfortheentranceofthesurround-ingair. This resistancewhichis exhibitedbetweenthe two

plates

Iz THE TWO NEW SCIENCESOF GALILEOplatesisdoubtlesslikewisepresentbetweenthepartsofa solid,and enters,at leastin par[, as a concomitantcauseof theircoherence.

[6o]SAGR.Allowme to interruptyoufora moment,please;for

I wantto speakof somethingwhichjustoccursto me,namely,whenI seehowthe lowerplatefollowsthe upperoneandhowrapidlyit is lifted,I feelsurethat, contraryto the opinionofmanyphilosophers,includingperhapsevenAristotlehimself,motionin a vacuumis notinstantaneous.If thiswereso thetwo platesmentionedabovewouldseparatewithoutany re-sistancewhatever,seeingthat the sameinstantof timewouldsufficefor theirseparationandfor the surroundingmediumtorushin andfillthe vacuumbetweenthem. The fa&that thelowerplatefollowsthe upperoneallowsus to infer,not onlythat motionin a vacuumis not instantaneous,but alsothat,betweenthe twoplates,a vacuumreallyexists,at leastforaveryshorttime,sufficientto allowthe surroundingmediumtorushin andfillthe vacuum;for iftherewerenovacuumtherewouldbenoneedofanymotioninthemedium.Onemustadmitthenthat a vacuumis sometimesproducedby violentmotion[violenza]or contraryto the lawsof nature,(althoughin myopinionnothingoccurscontraryto natureexcepttheimpossible,andthat neveroccurs).

But hereanotherdifficultyarises. Whileexperimentcon-vincesmeofthe correcCtnessof thisconclusion,mymindis notentirelysatisfiedas to the causeto whichthis effe&is to beattributed. For the separationof the platesprecedestheformationof the vacuumwhichis producedas a consequenceofthisseparation;andsinceit appearstomethat,in theorderofnature,the causemustprecedethe effe&,eventhoughit ap-pearsto followin pointoftime,andsinceeverypositiveeffecCtmusthavea positivecause,I do notseehowthe adhesionoftwoplatesandtheirresistanceto separation--acCrualfa_s---canbe referredto a vacuumas causewhenthisvacuumis yet tofollow.Accordingto the infalliblemaximof the Philosopher,thenon-existentcanproducenoeffe&.

Simp.

FIRST DAY 13Sire,. Seeingthat youacceptthis axiomofAristotle,I hardly

thinkyouwillreje_ anotherexcellentand reliablemaximof his,namely,Nature undertakesonly that which happenswithoutresistance;and in this saying,it appearsto me,youwillfindthesolutionof your difficulty. Sincenature abhorsa vacuum,shepreventsthat fromwhicha vacuumwouldfollowas a necessaryconsequence.Thus it happensthat naturepreventsthe separa-tion ofthe twoplates.

[6i]SACR.Nowadmittingthat what Simpliciosaysis an adequate

solutionof my difficulty,it seemsto me, ifI may be allowedtoresume my former argument, that this very resistanceto avacuumought to be sufficientto hold togetherthe parts eitherof stoneor of metalor the parts of any other solidwhichis knittogethermorestronglyandwhichismoreresistantto separation.If for one effe_ there be onlyone cause,or if, more beingas-signed,they canbe reducedto one, thenwhyis not this vacuumwhichreallyexistsa sufficientcausefor allkinds of resistance?

SALV.I do not wishjust nowto enter this discussionas towhether the vacuum alone is sufficientto hold together theseparateparts of a solidbody;but I assureyouthat the vacuumwhichacCtsas a sufficientcausein the caseofthetwoplatesisnotalone sufficientto bind togetherthe parts of a solidcylinderofmarble or metal which, when pulled violently,separates anddivides. Andnow if I finda methodof distinguishingthiswellknown resistance,dependingupon the vacuum, from everyother kind which might increasethe coherence,and if I showyou that the aforesaidresistancealone is not nearlysufficientfor such an effect, will you not grant that we are bound tointroduceanother cause._ Help him, Simplicio,since he doesnot knowwhat replytomake.

SIMP.Surely,Sagredo'shesitationmust be owingto anotherreason,for therecanbe nodoubtconcerninga conclusionwhichisat oncesoclearandlogical.SACra.Youhaveguessedrightly,Simplicio. I waswondering

whether, if a millionof gold each year from Spain were notsufficientto pay the army, it might not be necessary to

make

I4 THE TWO NEW SCIENCES OF GALILEOmake provisionother than small coin for the pay of thesoldiers.*

But go ahead,Salviati;assumethat I admityour conclusionand showusyourmethodof separatingtheacCtionof thevacuumfrom other causes;and by measuringit showus how it is notsufficientto producethe effectin question.

SALV.Your good angel assist you. I will tell you how toseparate the force of the vacuum from the others, and a{ter-wards how to measure it. For this purposelet us consideracontinuoussubstancewhoseparts lack all resistanceto separa-tion exceptthat derivedfroma vacuum,suchas is the casewithwater,a fact fullydemonstratedbyourAcademicianinoneof histreatises. Whenevera cylinderofwaterissubjectedto apull and

[62]offersa resistanceto the separationof itsparts this canbe attrib-

uted tonoothercausethantheresistanceof the

/k_j vacuum. In orderto try suchan experiment

I have invented a devicewhichI can betterexplainby meansof a sketch than by merewords. Let CABDrepresentthe crosssectionof a cylindereither of metal or, preferably,of glass,hollowinsideand accuratelyturned.

G I-I Into this is introduced a perfec°dyfittingiC . Dcylinderof wood,representedin crosssection

by EGHF, and capableof up-and-downmo-tion. Through the middleof this cylinderisboreda holeto receivean ironwire,carryinga hook at the end K, while the upper endof the wire, I, is providedwith a conicalhead. The wooden cylinder is countersunk

Fig.4 at the top so as to receive,witha perfectfit, the conicalhead I of the wire,IK,when pulleddown bytheendK.

NowinsertthewoodencylinderEH in the hollowcyllnderAD,soas not to touchtheupperend of the latterbut to leavefree aspaceof two or three finger-breadths;this spaceis to be filled

*ThebearingofthisremarkbecomesclearonreadingwhatSalviatisaysonp. 18below.[Trans.]

FIRST DAY 15withwaterbyholdingthevesselwiththemouthCD upwards,pushingdownonthestopperEH,andat thesametimekeepingtheconicalheadofthewire,I, awayfromthe hollowportionofthewoodencylinder.Theairisthusallowedtoescapealongsidethe ironwire(whichdoesnotmakea closefit) assoonas onepressesdownon the woodenstopper. The air havingbeenallowedto escapeand theironwirehavingbeendrawnbacksothat it fits snuglyagainstthe conicaldepressionin the wood,invertthevessel,bringingitmouthdownwards,andhangonthehookK a vesselwhichcanbe filledwith sandor anyheavymaterialin quantity sufficientto finallyseparatethe uppersurfaceofthe stopper,EF, fromthelowersurfaceofthewaterto whichit wasattachedonlyby theresistanceofthevacuum.Next weighthe stopperandwiretogetherwith the attachedvesseland its contents;we shallthenhave the forceof thevacuum[forzaddvacuo].Ifoneattachestoacylinderofmarble

[63]or glassa weightwhich,togetherwiththe weightofthemarbleor glassitself,is just equalto the sumof the weightsbeforementioned,andifbreakingoccursweshallthenbejustifiedinsayingthat thevacuumaloneholdsthepartsofthe marbleandglasstogether;but ifthisweightdoesnotsufficeandifbreakingoccursonlyafteradding,say,fourtimesthisweight,weshallthenbe compelledto say that the vacuumfurnishesonlyonefifthofthetotalresistance[resf._ema].

SLurP.No onecandoubtthe clevernessofthe device;yet itpresentsmanydifficultieswhichmakemedoubtits reliability.Forwhowillassureus thattheair doesnotcreepinbetweentheglassandstopperevenif it is wellpackedwith towor otheryieldingmaterial._I questionalsowhetheroilingwithwaxorturpentinewillsufficeto makethecone,I, fit snuglyonitsseat.Besides,maynot the parts of the waterexpandand dilate?Whymaynotthe airorexhalationsorsomeothermoresubtilesubstancespenetratetheporesofthewood,orevenoftheglassitself?

SAT.v.WithgreatskillindeedhasSimpliciolaidbeforeus thedifficulties;andhehasevenpartlysuggestedhowtopreventthe

air

x6 THE TWO NEW SCIENCESOF GALILEOairfrompenetratingthewoodor passingbetweenthewoodandthe glass. Butnowletmepoifitoutthat, asourexperiencein-creases,weshalllearnwhetheror nottheseallegeddifficultiesreallyexist. For if, as is the casewithair,wateris bynatureexpansible,althoughonlyunderseveretreatment,weshallseethe stopperdescend;and ifweput a smallexcavationin theupperpart ofthe glassvessel,suchas indicatedbyV, thentheair or anyother tenuousandgaseoussubstance,whichmightpenetratethe poresof glassor wood,wouldpassthroughthewaterandcolle&inthisreceptacleV. Butifthesethingsdonothappenwemayrestassuredthat ourexperknenthasbeenper-formedwithpropercaution;andweshalldiscoverthat waterdoesnot dilate and that glassdoesnot allowany material,howevertenuous,to penetrateit.

SAGmThanksto thisdiscussion,Ihavelearnedthecauseofacertaineffe&whichI havelongwonderedat anddespairedofunderstanding.I oncesawa cisternwhichhadbeenprovidedwith a pumpunderthe mistakenimpressionthat the watermightthusbedrawnwithlesseffortorin greaterquantitythanbymeansofthe ordinarybucket. Thestockofthe pumpcar-

[64]riedits suckerandvalveintheupperpartsothatthewaterwasliftedby attra&ionandnotbya pushasisthe casewithpumpsin whichthe suckerisplacedlowerdown.Thispumpworkedpeffedtlysolongasthewaterinthecisternstoodaboveacertainlevel;but belowthis levelthe pumpfailedto work. WhenIfirstnoticedthisphenomenonI thoughtthemachinewasoutoforder;but the workmanwhomI calledin to repairit toldmethe defecCtwasnot in the pumpbut in the waterwhichhadfallentoolowto beraisedthroughsucha height;andheaddedthat it wasnot possible,eitherby a pumpor by any othermachineworkingon the principleof attra&ion,to liftwaterahair's breadthaboveeighteencubits;whetherthe pump belargeor smallthis is the extremelimitof the lift. Up to thistimeI hadbeensothoughtlessthat,althoughI knewa rope,orrodofwood,orof iron,if sufficientlylong,wouldbreakby itsownweightwhenheldbytheupperend,it neveroccurredto me

" that

FIRST DAY 17that the samethingwouldhappen,onlymuchmoreeasily,to acolumnof water. And reallyis not that thing whichis at-tra_ed in the pumpa columnofwaterattachedat the upperendandstretchedmoreandmoreuntilfinallyapointisreachedwhereit breaks,likea rope,onaccountofits excessiveweight?

SALV.That ispreciselythe wayit works;thisfixedelevationofeighteencubitsistrueforanyquantityofwaterwhatever,bethe pumplargeor smallor evenasfineas a straw. Wemaythereforesay that, onweighingthe watercontainedin a tubeeighteencubitslong,no matterwhat the diameter,we shallobtainthevalueoftheresistanceofthevacuumina cylinderofanysolidmaterialhavinga boreof thissamediameter.Andhavinggonesofar, let us seehoweasyit is to findto whatlengthcylindersofmetal,stone,wood,glass,etc.,ofanydiam-eter canbe elongatedwithoutbreakingby theirownweight.

[6S]Takefor instancea copperwireofanylengthandthickness;

fixthe upperend andto the otherend attacha greaterandgreaterloaduntilfinallythewirebreaks;letthemaximumloadbe, say,fiftypounds. Then it is clearthat if fiftypoundsofcopper,in additionto the weightofthe wireitselfwhichmaybe, say,z/sounce,is drawnout intowireof thissamesizeweshallhavethegreatestlengthofthiskindofwirewhichcansus-tain its ownweight. Supposethewirewhichbreaksto beonecubitin lengthandI/souncein weight;thensinceit supports5olbs.in additionto its ownweight,i.e.,48ooeighths-of-an-ounce,it followsthat allcopperwires,independentofsize,cansustainthemselvesup to a lengthof48Olcubitsandnomore.Sincethen a copperrod can sustainits ownweightup to alengthof48Olcubitsit followsthat that part ofthe breakingstrength[resistenza]whichdependsuponthevacuum,comparingitwiththe remainingfacetorsofresistance,isequalto theweightofa rodofwater,eighteencubitslongandasthickasthecopperrod. If,forexample,copperisninetimesasheavyaswater,thebreakingstrength[resistenzaallostrappars.z]of anycopperrod,insofarasit dependsuponthevacuum,asequalto theweightof twocubitsof thissamerod. By a similarmethodonecan

find

I8 THE TWO NEW SCIENCESOF GALILEOfindthe maximumlengthofwireorrodofanymaterialwhichwilljust sustainits ownweight,andcanat the sametimedis-coverthe partwhichthevacuumplaysinits breakingstrength.

SACR.It stillremainsforyouto tellus uponwhatdependstheresistanceto breaking,otherthanthatofthevacuum;whatis the glueyor viscoussubstancewhichcementstogetherthepartsof the solid? For I cannotimaginea gluethat willnotburnup in a highlyheatedfurnacein twoor threemonths,orcertainlywithintenor a hundred.Forifgold,silverandglassare keptfora longwhilein the moltenstateandare removedfromthe furnace,theirparts,on cooling,immediatelyreuniteand bind themselvestogetheras before. Not only so, butwhateverdifficultyariseswithrespe_to thecementationofthepartsof the glassarisesalsowithregardto thepartsoftheglue;in otherwords,whatisthat whichholdsthesepartstogethersofirmly?

[661SALv.A littlewhileago,I expressedthehopethatyourgood

angelmightassistyou. I nowfindmyselfin the samestraits.Experimentleavesno doubtthat the reasonwhytwo platescannotbe separated,exceptwithviolenteffort,isthat theyareheldtogetherby the resistanceofthe vacuum;andthe samecanbe saidof two largepiecesof a marbleor bronzecolumn.Thisbeingso,I donotseewhythissamecausemaynotexplainthe coherenceofsmallerpartsandindeedof the verysmallestparticlesof thesematerials.Now,sinceeacheffe_musthaveonetrueandsufficientcauseandsinceI findnoothercement,amI notjustifiedintryingto discoverwhetherthevacuumisnot asufficientcause?

S_. But seeingthat youhavealreadyprovedthat the re-sistancewhichthe largevacuumoffersto the separationoftwolargepartsofa solidisreallyverysmallin comparisonwiththatcohesiveforcewhichbindstogetherthemostminuteparts,whydo you hesitateto regardthis latter as somethingverydifferentfromtheformer?

S_v. Sagredohasalready[p.I3 above]answeredthisques-tionwhenhe remarkedthat eachindividualsoldierwasbeing

paid

FIRST DAY 19paidfromcoincoiled-tedby a generaltaxofpenniesandfarth-ings,whileevena millionofgoldwouldnot sufficeto paytheentirearmy. Andwhoknowsbut that theremay be otherextremelyminutevacuawhichaffecCtthe smallestparticlessothat that whichbindstogetherthe contiguouspartsisthrough-outofthesamemintage? Letmetellyousomethingwhichhasjustoccurredto meandwhichI donotofferasanabsolutefacet,but ratheras a passingthought,stillimmatureandcallingformorecarefulconsideration.Youmaytakeofit whatyoulike;andjudgethe restasyouseefit. SometimeswhenI haveob-servedhowfirewindsits way in betweenthe mostminuteparticlesofthisorthatmetaland,eventhoughthesearesolidlycementedtogether,tearsthemapartandseparatesthem,andwhenI haveobservedthat,onremovingthefire,theseparticlesreunitewiththe sametenacityas at first,withoutany lossofquantityin the caseofgoldandwithlittlelossin the caseofothermetals,eventhoughthesepartshavebeenseparatedforalongwhile,I havethoughtthat theexplanationmightlieinthefact that the extremelyfineparticlesof fire,penetratingtheslenderporesof the metal(toosmallto admiteventhe finestparticlesof air or of manyother fluids),wouldfill the smallinterveningvacuaandwouldsetfreethesesmallparticlesfromthe attracCtionwhichthesesamevacuaexertuponthemandwhichpreventstheirseparation.Thustheparticlesareableto

[671movefreelysothat the mass[rnassa]becomesfluidandremainssoaslongastheparticlesoffireremaininside;butiftheydepartandleavethe formervacuathentheoriginalattraction[attraz-zione]returnsandthepartsareagaincementedtogether.

In replytothe questionraisedbySimplicio,onemaysaythatalthougheachparticularvacuumis exceedinglyminuteandthereforeeasilyovercome,yettheirnumberissoextraordinarilygreatthat theircombinedresistanceis,so to speak,multipledalmostwithoutlimit. The nature and the amountof force[forza]whichresults[risulta]fromaddingtogetheran immensenumberof smallforces[debolissimirnornent_]is clearlyillus-tratedbythefa_ that aweightofmillionsofpounds,suspended

by

20 THE TWO NEW SCIENCES OF GALILEOby great cables,is overcomeand lifted, when the south windcarries innumerableatoms of water, suspendedin thin mist,whichmovingthroughthe air penetratebetweenthefibresof thetense ropes in spite of the tremendousforce of the hangingweight. When these particles enter the narrow pores theyswell the ropes, thereby shorten them, and perforcelift theheavymass[mole].

SAcR.There canbe no doubt that any resistance,so longasit is not infinite,may be overcomeby a multitudeof minuteforces. Thus a vast numberof ants might carry ashorea shipladen with grain. And since experienceshowsus daily thatone ant caneasilycarry onegrain,it is clearthat the numberofrains in the ship is not infinite,but fallsbelowa certainlimit.you take anothernumber four or six times as great, and if

you set to work a correspondingnumberof ants theywillcarrythe grain ashoreand the boat also. It is true that thiswill callfor a prodigiousnumberof ants, but in my opinionthis is pre-ciselythe casewith the vacua which bind together the leastparticlesofa metal.

SALV.But even if this demandedan infinitenumberwouldyou still think it impossible?

SACR.Not if the mass [mole]of metal were infinite;other-wise....

[68]SAT.V.Otherwise what? Now since we have arrived at

paradoxeslet us see if we cannot provethat within a finiteex-tent it ispossibletodiscoveran infinitenumberofvacua. Atthesametimewe shallat least reacha solutionof the mostremark-able of all that list of problemswhichAristotle himselfcallswonderful;I referto his Questionsin Mechanics.This solutionmay be no lessclearand conclusivethan that whichhe himselfgivesand quitedifferentalsofromthat socleverlyexpoundedbythemostlearnedMonsignordiGuevara.*

First it is necessaryto considera proposition,not treated byothers,but uponwhichdependsthe solutionofthe problemandfrom which, if I mistake not, we shallderiveother new andremarkable facts. For the sake of clearnesslet us draw an

*BishopofTeano;b. x56x, d.I64I. [Trans.]

FIRST DAY 2Iaccuratefigure. AboutG as a centerdescribean equiangularandequilateralpolygonofanynumberofsides,saythehexagonABCDEF. Similarto this and concentricwith it, describeanothersmalleronewhichweshallcallHIIZT.MN.Prolongthe

F , .....ff"i............4 , , .T

-- ,,I ! t , I

( .":iim]B

Fig.5sideAB,of the largerhexagon,indefinitelytowardS; in likemannerprolongthe correspondingsideHI ofthe smallerhex-agon,in the samedirecCtion,so that the lineHT isparalleltoAS;andthroughthe centerdrawthe lineGVparallelto theothertwo. Thisdone,imaginethe largerpolygonto rollupon

[69]thelineAS,carryingwithit thesmallerpolygon.It isevidentthat,if thepointB,the endofthesideAB,remainsfixedat thebeginningofthe rotation,thepointAwillriseandthe pointCwillfalldescribingthearc CQuntilthesideBCcoincideswiththelineBQ,equaltoBC. ButduringthisrotationthepointI,onthe smallerpolygon,willriseabovethelineIT becauseIBisobliquetoAS;andit willnotagainreturnto thelineITuntilthepointC shallhavereachedthepositionQ. ThepointI, havingdescribedthe arcIO abovethe lineHT,willreachtheposition

Oat

2z THE TWO NEW SCIENCES OF GALILEO

0 at the sametime the sideIK assumesthe position0P; but inthe meantimethe centerG has traverseda path aboveGVanddoesnot return to it until it has _ompletedthe arc GC. Thisstephavingbeentaken,the largerpolygonhas beenbroughttorest with its sideBC coincidingwith the lineBQwhilethe sideIK of the smallerpolygonhas beenmade to coincidewith thelineOP,havingpassedoverthe portionI0 withouttouchingit;alsothe centerG willhavereachedthe positionC after havingtraversed all its courseabovethe parallellineGV. Andfinallythe entire figurewill assumea positionsimilarto the first, sothat ifwe continuethe rotationand cometo the next step, thesideDC of the largerpolygonwillcoincidewith the portionQX

, and the sideKL of the smallerpolygon,havingfirstskippedthearc PY,will fallon YZ, whilethe centerstillkeepingabovethelineGV will return to it at R after havingjumpedthe intervalCR. At the endofonecompleterotationthe largerpolygonwillhave traced upon the lineAS,without break, six linestogetherequal to its perimeter; the lesserpolygonwill likewisehaveimprintedsix linesequal to its perimeter,but separatedby theinterpositionof five arcs, whose chords represent the partsof HT not touchedby the polygon:the centerG neverreachesthe lineGV exceptat sixpoints. From this it is clearthat thespacetraversedby the smallerpolygonis almostequal to thattraversed by the larger, that is, the lineHT approximatesthelineAS,differingfrom it onlyby the lengthof one chordof oneofthese arcs,providedwe understandthe lineI-ITto includethefiveskippedarcs.

Now this expositionwhichI have givenin the caseof thesehexagonsmust be understoodto be applicable to all otherpolygons,whateverthe numberof sides,providedonlytheyare

[70]similar, concentric,and rigidly connecCted,so that when thegreateronerotates the lesserwillalsoturn howeversmallit maybe. Youmust alsounderstandthat the linesdescribedby thesetwo are nearlyequal providedwe includein the spacetraversedby the smallerone the intervalswhichare not touchedby anypart ofthe perimeterofthis smallerpolygon.

Let

FIRST DAY z3Let a largepolygonof, say, one thousandsides make one

completerotationand thus layoff a lineequal to its perimeter;at the sametime the smallonewillpassoveran approximatelyequal distance, made up of a thousand small portions eachequal to oneof its sides,but interruptedby a thousandspaceswhich,in contrastwith the portionsthat coincidewith the sidesof the polygon,we may call empty. So far the matter is freefromdifficultyor doubt.

But nowsupposethat about any center,say A, we describetwo concentricand rigidlyconneCtedcircles;and supposethatfrom the points C and B, on their radii, there are drawn thetangentsCEand BF and that throughthe centerAthe lineADis drawnparallel to them, then if the large circlemakes onecompleterotation alongthe lineBF, equal notonly to its cir-cumferencebut alsoto the other two linesCE andAD, tellmewhat the smallercirclewilldoand alsowhat the centerwilldo.Asto the center it will certainlytraverseand touch the entirelineAD whilethe circumferenceof the smallercirclewillhavemeasuredoffby its pointsof contaCtthe entire lineCE,just aswasdoneby theabovementionedpolygons.The onlydifferenceis that the lineI-ITwas not at everypoint in contactwith theperimeterof the smallerpolygon,but therewereleftuntouchedas manyvacant spacesas therewerespacescoincidingwith thesides. But herein the caseofthe circlesthe circumferenceofthesmalleroneneverleavesthe lineCE, sothat nopart of the latterisleftuntouched,noris thereevera timewhensomepointonthecircleis not in contaCtwith the straightline. Hownowcanthesmallercircletraverse a lengthgreater than its circumferenceunlessit goby jumps?

8AGmIt seemsto methat onemaysaythat just as thecenterofthe circle,by itself,carriedalongthe lineAD is constantlyincontac2with it, althoughit isonlya singlepoint,sothepointsonthe circumferenceof the smaller circle,carried alongby themotionof the largercircle,wouldslideover somesmallparts ofthe lineCE.

: [7I]: SALV.There are two reasonswhy this cannot happen. First

because

?

24 THE TWO NEW SCIENCESOF GALILEObecausethereis nogroundforthinkingthat onepointof con-taCt,suchas that at C, ratherthan another,shouldslipovercertainportionsofthe lineCE. Butif suchslidingsalongCEdidoccurtheywouldbe infinitein numbersincethe pointsofcontaCt(beingmerepoints)are infinitein number:an infinitenumberoffiniteslipswillhowevermakeaninfinitelylongline,whileasamatteroffaCtthelineCEisfinite. Theotherreasonisthat asthe greatercircle,in its rotation,changesitspointofcontactcontinuouslythe lessercirclemustdothe samebecauseBistheonlypointfromwhichastraightlinecanbedrawntoAandpassthroughC. Accordinglythesmallcirclemustchangeitspointofcontactwheneverthelargeonechanges:nopointofthe smallcircletouchesthe straightlineCE in morethanonepoint. Not onlyso,but evenin the rotationof the polygonstherewasnopointon theperimeterofthe smallerwhichcoin-cidedwithmorethanonepointonthe linetraversedby thatperimeter;this is at onceclearwhenyou rememberthat thelineIK isparallelto BCandthatthereforeIK willremainaboveIPuntilBCcoincideswithBQ,andthatIK willnotlieuponIPexceptat theveryinstantwhenBCoccupiesthepositionBQ;atthisinstanttheentirelineIK coincideswithOPandimmediatelyafterwardsrisesaboveit.

SAOl_.Thisisaveryintricatematter. I seenosolution.Prayexplainit to us.

SALV.Let usreturnto the considerationofthe abovemen-tionedpolygonswhosebehaviorwealreadyunderstand.Nowin the caseofpolygonswith IOOOOOsides,thelinetraversedbythe perimeterof the greater,i. e., the line laid downby itsIOOCXX)sidesoneafteranother,isequalto thelinetracedoutbythe IOCX:_sidesofthe smaller,providedweincludethe IO(Xx_vacantspacesinterspersed.Soin thecaseofthe circles,poly-gonshavingan infinitudeof sides,the linetraversedby thecontinuouslydistributed[continuamentedispostz]infinitudeofsidesisin the greatercircleequalto thelinelaiddownby theinfinitudeof sidesin the smallercirclebut withthe exceptionthat these latter alternatewith emptyspaces;and sincethesidesarenotfinitein number,butinfinite,soalsoaretheinter-

vening

FIRST DAY 25veningempty spacesnot finitebut infinite. The linetraversedby the largercircleconsiststhen of an infinitenumberof pointswhichcompletelyfillit; whilethat whichistracedbythe smallercircleconsistsof an infinitenumberof pointswhichleaveemptyspacesand only partly fill the line. And here I wishyou toobservethat after dividingand resolvinga line into a finitenumberof parts, that is,into a numberwhichcanbecounted,it

[72]is notpossibleto arrangethem again intoa greaterlengththanthat whichthey occupiedwhen they formeda continuum[con-tinuate]and were conne_ed without the interpositionof asmany empty spaces. But if we considerthe lineresolvedintoan infinitenumberof infinitelysmalland indivisibleparts, weshallbe ableto conceivethe lineextendedindefinitelyby theinterposition,not of a finite, but of an infinitenumberof in-finitelysmallindivisibleemptyspaces.

Nowthiswhichhasbeensaidconcerningsimplelinesmust beunderstoodto holdalsoin the caseof surfacesand solidbodies,it being assumedthat they are made up of an infinite,not afinite, number of atoms. Such a body once divided into afinitenumberofparts it is impossibletoreassemblethemsoas tooccupymore space than before unless we interpose a finitenumber of empty spaces,that is to say, spacesfree from thesubstanceof whichthe solidis made. But if we imaginethebody, by someextreme and final analysis, resolvedinto itsprimaryelements,infinitein number,then we shallbe able tothink of them as indefinitelyextended in space, not by theinterpositionof a finite, but of an infinitenumber of emptyspaces. Thus one can easilyimaginea smallball of goldex-panded into a very largespacewithout the introducCtionof afinite number of empty spaces,alwaysprovided the gold ismadeupof aninfinitenumberof indivisibleparts.

SIM1,.It seemsto me that you are travellingalongtowardthosevacuaadvocatedby a certainancientphilosopher.

SAzv.But youhavefailedto add,"whodeniedDivineProvi-dence,"an inapt remarkmade on a similaroccasionby a cer-tain antagonistofour Academician.

Simp.

26 THE TWO NEW SCIENCESOF GALILEOSr_P.I noticed,andnotwithoutindignation,the rancorof

this ill-naturedopponent;furtherreferencesto theseaffairsIomit,not onlyas a matterof goodform,but alsobecauseIknowhowunpleasanttheyareto the goodtemperedandwellorderedmindof oneso religiousand pious,soorthodoxandGod-fearingasyou.

Butto returnto our subject,yourpreviousdiscourseleaveswith me manydifficultieswhichI am unableto solve. Firstamongtheseis that,if the circumferencesofthe twocirclesareequalto the two straightlines,CE and BF, the latter con-sideredasa continuum,the formeras interruptedwith an in-finityofemptypoints,I donotseehowit ispossibleto saythatthelineADdescribedbythecenter,andmadeupofan infinityofpoints,isequalto thiscenterwhichisa singlepoint. Besides,thisbuildingupof linesoutofpoints,divisiblesoutofindivisi-bles,andfinitesoutofinfinites,offersmeanobstacledifficulttoavoid;andthe necessityof introducinga vacuum,soconclu-sivelyrefutedbyAristotle,presentsthesamedifficulty.

[73]SAr.V.Thesedifficultiesare real;andtheyare not the only

ones. But let us rememberthat weare dealingwithinfinitiesand indivisibles,both of whichtranscendour finiteunder-standing,the formeron accountoftheirmagnitude,the latterbecauseoftheirsmallness.In spiteofthis,mencannotrefrainfromdiscussingthem,eventhoughit mustbe donein a round-aboutway.

ThereforeI alsoshouldliketo takethelibertyto presentsomeof my ideaswhich,thoughnot necessarilyconvincing,would,onaccountof theirnovelty,at least,provesomewhatstartling.But sucha diversionmightperhapscarryus toofarawayfromthe subjectunderdiscussionandmightthereforeappearto youinopportuneandnotverypleasing.

SACR.Prayletusenjoythe advantagesandprivilegeswhichcomefromconversationbetweenfriends,especiallyuponsub-jects freelychosenand not forceduponus, a matter vastlydifferentfromdealingwithdeadbookswhichgiveriseto manydoubtsbutremovenone. Sharewithus,therefore,thethoughts

which

FIRST DAY z7whichourdiscussionhassuggestedto you;forsincewearefreefromurgentbusinesstherewillbe abundanttimeto pursuethetopics alreadymentioned;and in particularthe obje£tionsraisedbySimpliciooughtnotinanywisetobenegle&ed.

S_J_v.Granted,sinceyousodesire.Thefirstquestionwas,Howcana singlepointbe equalto a line? SinceI cannotdomoreat presentI shallattemptto remove,orat leastdiminish,oneimprobabilityby introducinga similaror a greaterone,justas sometimesawonderisdiminishedbyamiracle.*

AndthisI shalldoby showingyoutwo equalsurfaces,to-getherwithtwoequalsolidslocateduponthesesamesurfacesasbases,allfourofwhichdiminishcontinuouslyanduniformlyin sucha waythat theirremaindersalwayspreserveequalityamongthemselves,andfinallyboththe surfacesandthe solidsterminatetheirpreviousconstantequalityby degenerating,theonesolidandthe onesurfaceintoa verylongline,the othersolidand the other surfaceinto a singlepoint;that is, thelatterto onepoint,theformerto aninfinitenumberofpoints.

[74]SACR.This propositionappearsto me wonderfial,indeed;

butletusheartheexplanationanddemonstration.SALV.Sincethe proofis purelygeometricalwe shallneed

a figure. Let_FB be a semicirclewithcenterat C; aboutitdescribethe re&angleADEBand fromthe centerdrawthestraightlinesCDandCEto thepointsD andE. ImaginetheradiusCFto bedrawnperpendicularto eitherofthelinesABorDE, andtheentirefigureto rotateaboutthisradiusasanaxis.It isclearthatthere&angleADEBwillthusdescribeacylinder,thesemicircleAFBahemisphere,andthetriangleCDE,a cone.Nextletus removethe hemispherebut leavethe coneandtherestofthecylinder,which,onaccountofitsshape,wewillcalla"bowl." Firstweshallprovethat the bowlandthe coneareequal;thenweshallshowthataplanedrawnparalleltothecirclewhichformsthebaseofthebowlandwhichhasthelineDEfordiameterandF foracenterwaplanewhosetraceisGN---cutsthebowlinthepointsG,I, O,N,andtheconeinthepointsI-I,L,sothat thepartofthe coneindicatedbyCHLisalwaysequalto

*Cf.p.3obelow.[Trans.]

z8 THE TWO NEW SCIENCESOF GALILEOthepartofthebowlwhoseprofileisrepresentedbythetrianglesGAIandBON. Besidesthisweshallprovethat thebaseofthecone,i.e.,thecirclewhosediameterisHL,isequaltothecircular

A C 5 surfacewhichformsthebaseof

___ thisportionof the bowl,or as

onemightsay,equaltoa ribbonG Nwhosewidthis OI. (Noteby

the waythe natureof mathe-maticaldefinitionswhichcon-

. sistmerelyin the impositionofD F _ namesor,ifyouprefer,abbrevi-

Fig.6 ationsofspeechestablishedandintroducedin orderto avoidthe tediousdrudgerywhichyouand I now experiencesimplybecausewe have not agreedto call this surfacea "circularband" and that sharpsolidportionof the bowla "round razor.") Now call them by

[75]whatnameyouplease,itsufficesto understandthat theplane,drawnat any heightwhatever,so longas it is paralleltothe base,i.e., to the circlewhosediameterisDE, alwayscutsthetwosolidssothattheportionCHLoftheconeisequalto theupperportionofthebowl;likewisethetwoareaswhicharethebasesofthesesolids,namelythebandandthecircleI-IL,arealsoequal. Herewehavethemiraclementionedabove;asthe cut-tingplaneapproachesthe lineABthe portionsofthe solidscutoffarealwaysequal,soalsotheareasoftheirbases.Andasthecuttingplanecomesnearthe top,thetwosolids(alwaysequal)aswellastheirbases(areaswhicharealsoequal)finallyvanish,onepairofthemdegeneratingintothecircumferenceofa circle,theotherintoasinglepoint,namely,theupperedgeofthebowlandthe apexof the cone. Now,sinceas thesesolidsdiminishequalityismaintainedbetweenthemupto theverylast,wearejustifiedin sayingthat, at the extremeandfinalendof thisdiminution,theyare still equaland that oneis not infinitelygreaterthan the other. It appearsthereforethat we mayequatethecircumferenceofa largecircleto asinglepoint. Andthiswhichistrueofthe solidsistruealsoofthe surfaceswhich

form

FIRST DAY 29formtheirbases;forthese alsopreserveequalitybetweenthem-selvesthroughout their diminutionand in the end vanish, theone into the circumferenceof a circle,the other into a singlepoint. Shallwenot then callthemequalseeingthattheyarethelast tracesand remnantsof equalmagnitudes? Note alsothat,even if these vesselswere large enoughto contain immensecelestialhemispheres,both their upperedgesand the apexesofthe cones therein containedwould always remainequal andwouldvanish,the formerinto circleshavingthe dimensionsofthe largestcelestialorbits, the latter into singlepoints. Hencein conformitywith the precedingwe may say that all circum-ferencesof circles,howeverdifferent,are equal to each other,andareeachequalto a singlepoint.

SAtin.This presentationstrikes me as so clever and novelthat, even if I were able, I wouldnot be willingto opposeit;for to defacesobeautifula stru_ure by a bluntpedanticattackwouldbe nothingshortofsinful. But for our completesatisfac-

[76]tion pray give us this geometricalproof that there is alwaysequality between these solidsand between their bases; for itcannot,I think, fail to be very ingenious,seeinghow subtle isthe philosophicalargumentbaseduponthis result.

SAJ_v.The demonstrationis both short and easy. Referringto the precedingfigure,sinceIPC is a rightanglethe squareofthe radiusIC is equalto the sumofthe squareson thetwo sidesIP, PC; but the radiusIC is equalto ACand alsoto GP, whileCP isequalto PH. Hencethe squareof the lineGP is equaltothe sumof the squaresof IP andPH, ormuklplyingthroughby4,wehave thesquareof the diameterGN equalto the sumofthesquareson IO and HL. And, sincethe areasof circlesare toeachother as the squaresof their diameters,it followsthat theareaofthe circlewhosediameterisGN isequalto the sumoftheareasof circleshavingdiametersIO andI-i-L,sothat ifweremovethe commonarea of the circlehavingIO for diameterthe re-mainingarea of the circleGN willbe equal to the area of thecirclewhosediameterisHL. Somuchfor thefirstpart. Asforthe otherpart, weleaveitsdemonstrationforthe present,partly

because

30 THE TWO NEW SCIENCES OF GALILEObecausethosewhowishto followit willfindit in the twelfthpropositionofthesecondbookofDecentrogravitatissolidorumbytheArchimedesofourage,LucaValerio,*whomadeuseofitfora differentobjec°c,andpartlybecause,forourpurpose,itsufficesto have seenthat the above-mentionedsurfacesarealwaysequalandthat, as theykeepon diminishinguniformly,theydegenerate,the oneintoa singlepoint,theotherintothecircumferenceofa circlelargerthananyassignable;in thisfa&liesourmiracle.t

SACR.The demonstrationis ingeniousand the inferencesdrawnfromit areremarkable.Andnowletushearsomethingconcerningthe otherdifficultyraisedby Simplicio,ifyouhaveanythingspecialto say,which,however,seemsto me hardlypossible,sincethe matterhasalreadybeensothoroughlydis-cussed.

S_mv.ButI dohavesomethingspecialto say,andwillfirstof all repeatwhatI saida littlewhileago,namely,that in-finityandindivisibilityarein theirverynatureincomprehensi-bleto us; imaginethenwhattheyarewhencombined.Yet if

[77]wewishto buildup a lineout of indivisiblepoints,wemusttake an infinitenumberof them,andare,therefore,boundtounderstandboth the infiniteand the indivisibleat the sametime. ManyideashavepassedthroughmymindconcerningthissubjecCt,someofwhich,possiblythemoreimportant,I maynotbe ableto recallonthe spurof themoment;but in thecourseofourdiscussionit mayhappenthat I shallawakeninyou,andespeciallyin Simplicio,objecCtionsand difficultieswhichinturn willbringto memorythat which,withoutsuchstimulus,wouldhavelaindormantinmymind. Allowmethereforethecustomarylibertyofintroducingsomeofourhumanfancies,forindeedwemayso callthemin comparisonwith supernaturaltruth whichfurnishesthe onetrue andsaferecoursefordeci-sionin ourdiscussionsandwhichis an infallibleguidein thedarkanddubiouspathsofthought.

*DistinguishedItalianmathematician;bornat FerraraaboutI5S2;admittedtotheAccademiadeiLincelI612;diedI618.[Trans.]

JfCf.p.27above.[Trans.]

FIRST DAY 3IOneof the mainobjec°cionsurgedagainstthis buildingup

of continuousquantitiesout of indivisiblequantities[continuod' Cndivisibih]is that the additionof one indivisibleto an-othercannotproducea divisible,for if thiswereso it wouldrenderthe indivisibledivisible.Thusif twoindivisibles,saytwopoints,canbe unitedto forma quantity,saya divisibleline,thenan evenmoredivisiblelinemightbe formedby theunionofthree,five,seven,or anyotheroddnumberofpoints.Sincehowevertheselinescanbe cut intotwo equalparts,itbecomespossibleto cutthe indivisiblewhichliesexac°dyin themiddleoftheline. In answerto thisandotherobjec°donsofthesametypewereplythat a divisiblemagnitudecannotbe con-stru(tedoutoftwoortenorahundredorathousandindivisibleS,butrequiresaninfinitenumberofthem.

Sire,.Herea difficultypresentsitselfwhichappearsto meinsoluble.Sinceit is clearthat wemayhaveonelinegreaterthan another,eachcontainingan infinitenumberof points,we are forcedto admitthat, withinoneand the sameclass,wemayhavesomethinggreaterthaninfinity,becausethe in-finityof pointsin the longlineis greaterthan the infinityofpointsin theshortline. Thisassigningto an infinitequantityavaluegreaterthaninfinityisquitebeyondmycomprehension.

SALv.This is oneof the difficultieswhicharisewhenweattempt,withourfiniteminds,to discussthe infinite,assigningto itthosepropertieswhichwegivefothefiniteandlimited;but[78]thisI thinkiswrong,forwecannotspeakofinfinitequantitiesas beingtheonegreaterorlessthanorequalto another.ToprovethisI haveinmindanargumentwhich,forthe sakeofclearness,I shallput intheformofquestionsto Simpliciowhoraisedthisdifficulty.

I takeit forgrantedthatyouknowwhichofthenumbersaresquaresandwhicharenot.

Sn_P.I amquiteawarethatasquarednumberisonewhichre-sultsfromthemultiplicationof anothernumberbyitself;thus4,9,etc.,aresquarednumberswhichcomefrommultiplying2,3,etc.,bythemselves. Salv.

37 THE TWO NEW SCIENCES OF GALILEOSALV.Very well;and youalsoknowthat just as the products

are calledsquaresso the favors are calledsidesor roots;whileon the other hand those numberswhichdo not consistof twoequal facCtorsare not squares. Thereforeif I assert that allnumbers, includingboth squares and non-squares,are morethan the squaresalone, I shall speak the truth, shall I not?

Snvn,.Most certainly.SALV.If t shouldask furtherhowmanysquaresthereareone

might reply truly that there are as many as file correspondingnumberof roots,sinceeverysquarehas its own rootand everyroot its own square, whileno squarehas more than one rootand norootmorethan onesquare.

SIMP.Preciselyso.SALV.But if I -inquirehowmany roots there are, it cannot

be deniedthat there are as manyas there are numbersbecauseevery number is a root of some square. This beinggrantedwe must say that there are as many squaresas there are num-bers becausethey are just as numerousas their roots, and allthe numbers are roots. Yet at the outset we said there aremany more numbersthan squares, sincethe largerportion ofthem are not squares. Not only so, but the proportionatenumber of squares diminishesas we pass to larger numbers.Thusup to IoowehaveIOsquares,that is,the squaresconstituteI/IO part of all the numbers;up to IOOOO,we findonly I/IO0

[79]part to be squares;and up to a milliononly I/IOOOpart; on theotherhand in an infinitenumber,ifone couldconceiveof suchathing, he wouldbe forced to admit that there are as manysquaresas therearenumbersall taken together.

SAGR.What then must one concludeunder these circum-stances?

SALV.So far as I seewe can only infer that the totality ofall numbersis infinite,that the number of squares is infinite,and that the number of their roots is infinite; neither is thenumber of squaresless than the totality of all numbers,northe latter greater than the former; and finally the attributes"equal," "greater," and "less," are not applicableto infinite,

but

FIRST DAY 33but only to finite, quantkies. When thereforeSimpllc[oin-troducesseverallinesof differentlengthsand asksme how itis possible that the longerones do not contain more pointsthan the shorter,I answerhim that one linedoesnot containmore or lessor just as many points as another,but that eachline containsan infinitenumber. Or if I had repliedto himthat the pointsinone llnewereequalin numberto the squares;inanother,greaterthan thetotality ofnumbers;andin the littleone,asmany asthe numberof cubes,mightI not, indeed,havesatisfiedhim by thus placingmore points in one line than inanother and yet maintainingan infinitenumberin each? Somuchforthe firstdifficulty.

SAGg.Pray stop a momentand let me add to what has al-readybeen said an idea whichjust occursto me. If the pre-cedingbe true, it seemsto me impossibleto say either that oneinfinitenumberis greater than anotheror eventhat it isgreaterthan a finitenumber,becauseifthe infinitenumberweregreaterthan, say, a millionit wouldfollowthat on passingfrom themillionto higher and highernumberswe wouldbe approach-ing the infinite;but this is not so; on the contrary, the lar-ger the number to which we pass, the more we recedefrom[thispropertyof]infinity,becausethe greater the numbersthefewer [relatively]are the squarescontained in them; but thesquaresin infinity cannot be lessthan the totality of all thenumbers,as we havejust agreed;hencethe approachto greaterand greaternumbersmeansa departurefrominfinity.*

SAT.v.And thus fromyour ingeniousargumentwe areled to[8o]concludethat the attributes "larger," "smaller,"and "equal"have no placeeither in comparinginfinitequantitieswith eachother or in comparinginfinitewith finitequantities.

I pass now to another consideration. Sincelines and allcontinuousquantitiesare divisibleinto parts whichare them-selvesdivisiblewithout end, I do not see how it is possible

*Acertainconfusionofthoughtappearstobeintroducedherethrougha failuretodistinguishbetweenthenumbern andtheclassofthefirstnnumbers;andlikewisefroma failuretodistinguishinfinityasa numberfrominfinityastheclassofallnumbers.[Trans.]

34 THE TWO NEW SCIENCESOF GALILEOto avoidthe conclusionthat theselinesare builtup of an in-finitenumberofindivisiblequantitiesbecausea divisionandasubdivisionwhichcan be carriedon indefinitelypresupposesthat theparts areinfinitein number,otherwisethe subdivisionwouldreachan end;andif thepartsareinfinitein number,wemustconcludethat theyare not finitein size,becausean in-finitenumberoffinitequantitieswouldgivean infinitemagni-tude. Andthuswehavea continuousquantitybuiltupof aninfinitenumberof indivisibles.

Shay.But if wecan carryon indefinitelythe divisionintofiniteparts what necessityis therethen for the introductionofnon-finlteparts?

SALV.The very facetthat oneis ableto continue,withoutend,the divisionintofiniteparts[inpattiquante]makesit nec-essaryto regardthe quantityas composedof an infinitenum-ber of immeasurablysmallelements[di infinitinon quanta].Nowin orderto settlethismatterI shallaskyou to tellmewhether,in youropinion,a continuumis madeup of a finiteorofaninfinitenumberoffiniteparts[partiquante].

SIMI,.My answeris that their numberis both infiniteandfinite;potentiallyinfinitebut afftuallyfinite[infinite,in po..tenza;efinite,in atto];that is to say,potentiallyinfinitebeforedivisionandactuallyfiniteafterdivision;becausepartscannotbe saidto existin a bodywhichis notyet dividedor at leastmarkedout;ifthisisnotdonewesaythattheyexistpotentially.

SALV.So that a linewhichis, for instance,twentyspanslongis notsaidto containafftuallytwentylineseachonespanin lengthexceptafterdivisionintotwentyequalparts; beforedivisionit is saidto containthemonlypotentially.Supposethefacetsareasyousay;tellmethenwhether,whenthedivisionis oncemade,the sizeof the originalquantityis therebyin-creased,diminished,orunaffecCted.

SIMV.It neitherincreasesnordiminishes.SALV.That is my opinionalso. Thereforethe finiteparts

[pattiquante]in a continuum,whethera&uallyor potentiallypresent,donotmakethe quantityeitherlargeror smaller;butit is perfecCtlyclearthat, if thenumberoffiniteparts aCtually

contained

FIRST DAY 35containedin thewholeisinfinitein number,theywillmakethemagnitudeinfinite.Hencethenumberoffiniteparts,althoughexistingonlypotentially,cannotbeinfiniteunlessthemagnitudecontainingthembeinfinite;andconverselyif themagnitudeis

finiteit cannotcontainan infinitenumberoffinitepartseitheractuallyor potentially.

SAGe.Howthenisit possibleto dividea continuumwithoutlimitintopartswhicharethemselvesalwayscapableofsubdivi-sion?

SAT.V.ThisdistinCtionofyoursbetweenactualandpotentialappearsto rendereasybyonemethodwhatwouldbeimpossibleby another. But I shallendeavorto reconcilethesemattersin anotherway;and as to the querywhetherthe finitepartsof a limitedcontinuum[continuoterminato]are finiteor in-finitein numberI will,contraryto the opinionof Simplicio,answerthattheyareneitherfinitenorinfinite.

SIMP.Thisanswerwouldneverhaveoccurredto mesinceIdidnotthinkthat thereexistedanyintermediatestepbetweenthe finiteandthe infinite,so that the classificationor distinc-tionwhichassumesthat a thingmustbeeitherfiniteor infiniteisfaultyanddefective.

SALv.Soit seemstome. Andifweconsiderdiscretequanti-ties I thinkthereis, betweenfiniteandinfinitequantities,athird intermediatetermwhichcorrespondsto everyassignednumber;so that if asked,as in the presentcase,whetherthefinitepartsof a continuumarefiniteor infinitein numberthebest replyisthat theyareneitherfinitenorinfinitebut corre-spondto everyassignednumber. In orderthat thismaybepossible,it isnecessarythat thosepartsshouldnotbeincludedwithina limitednumber,forin that casetheywouldnotcorre-spondto a numberwhichisgreater;norcantheybeinfiniteinnumbersincenoassignednumberis infinite;andthus at thepleasureofthe questionerwemay,to anygivenline,assignahundredfiniteparts,a thousand,ahundredthousand,or indeedanynumberwemaypleasesolongasit benotinfinite.I grant,therefore,to the philosophers_that the continuumcontainsas

many

36 THE TWO NEW SCIENCESOF GALILEOmanyfinitepartsas theypleaseandI concedealsothat it con-tainsthem,eitheraCtuallyor potentially,astheymaylike;butI mustaddthat justasalineten fathomsJeanne]inlengthcon-tainsten lineseachof onefathomandfortylineseachof onecubit[braccia]andeightylineseachof halfa cubit,etc.,soitcontainsan infinitenumberof points;callthemaCtualor po-tential,asyoulike,foras to thisdetail,Simplicio,Ideferto youropinionandto yourjudgment.

[821SL_P.I cannothelp admiringyour discussion;but I fear

that this parallelismbetweenthe pointsand the finitepartscontainedina linewillnotprovesatisfaCtory,andthat youwillnotfindit soeasyto dividea givenlineintoan infinitenum-berofpointsasthe philosophersdoto cut it intotenfathomsorfortycubits;notonlyso,butsucha divisionis quiteimpossibleto realizein praCtice,sothat thiswillbe oneof thosepoten-tialitieswhichcannotbereducedto actuality.

SALV.Thefact that somethingcanbe doneonlywitheffortordiligenceorwithgreatexpenditureoftimedoesnotrenderitimpossible;for I thinkthat youyourselfcouldnoteasilydividea lineinto a thousandparts,andmuchlessif the numberofparts were937or any other largeprimenumber. But if Iwereto accomplishthisdivisionwhichyoudeemimpossibleasreadilyas anotherpersonwoulddividethe lineintofortypartswouldyouthenbemorewilling,inourdiscussion,toconcedethepossibilityofsuchadivision?

Snvn,.In generalI enjoygreatlyyourmethod;andreplyingto yourquery,I answerthat it wouldbe morethansufficientif it provenotmoredifficultto resolvea lineintopointsthantodivideit intoa thousandparts.

SALv.I willnowsaysomethingwhichmayperhapsastonishyou; it refersto the possibilityof dividinga lineinto its in-finitelysmallelementsby followingthe sameorderwhichoneemploysin dividingthesamelineintoforty,sixty,ora hundredparts,that is,bydividingit intotwo,four,etc. Hewhothinksthat, byfollowingthismethod,hecanreachan infinitenumberofpointsisgreatlymistaken;forif thisprocesswerefollowedto

etemiw/

7

FIRST DAY 37eternity there wouldstill remain finiteparts whichwere un-divided.

Indeedby such a methodone is very far from reachingthegoal of indivisibility;on the contrary he recedesfrom it andwhilehe thinksthat, by continuingthis divisionand by multi-plying the multitude of parts, he will approachinfinity,he is,inmy opinion,getting farther and farther awayfromit. Myreasonis this. In the precedingdiscussionwe concludedthat,inan infinitenumber,it is necessarythat the squaresand cubesshouldbe as numerousas the totality of the natural numbers[tuttii numerz],becauseboth of these are as numerousas theirroots which constitute the totality of the natural numbers.Nextwe sawthat thelargerthe numberstakenthemoresparselydistributedwerethe squares,and stillmore sparselythe cubes;thereforeit is clearthat the largerthe numbersto whichwepassthe farther we recedefromthe infinitenumber;henceit follows

[8S]that, sincethis processcarriesus fartherand fartherfromtheendsought, if on turningback we shallfindthat any numbercan be said to be infinite,it mustbe unity. Here indeedaresatisfiedall those conditionswhichare requisiteforan infinitenumber;I meanthat unity containsin itselfasmanysquaresastherearecubesandnaturalnumbers[tuttii numen].

SIMP.I donotquitegraspthemeaningofthis.SALV.There is no difficultyin the matter becauseunity is at

once a square, a cube, a squareof a squareand all the otherpowers[dignity];noris thereany essentialpeculiarityinsquaresor cubeswhichdoesnot belongto unity; as, for example,thepropertyof twosquarenumbersthat they havebetweenthemamean proportional;take any squarenumberyou pleaseas thefirst term and unity for the other, then youwill alwaysfind anumberwhichis a meanproportional. Considerthe twosquarenumbers,9 and 4; then 3 is the mean proportionalbetween9 and I ;while2is ameanproportionalbetween4and I; between9 and 4 we have6 as a mean proportional.A propertyof cubesis that they must have betweenthem two mean proportionalnumbers; take 8 and 27; betweenthem lie IZ and 18;while

between

38 THE TWO NEW SCIENCESOF-GALILEObetweenIand8wehave2 and4 intervening;andbetweenI and27therelie3 and9. Thereforeweconcludethat unity is theonlyinfinitenumber.Thesearesomeofthemarvelswhichourimaginationcannotgraspandwhichshouldwarnusagainsttheseriouserror of thosewhoattempt to discussthe infinitebyassigningto it the samepropertieswhichweemployfor thefinite,thenaturesofthetwohavingnothingincommon.

Withregardto thissubjecCtI musttellyouof a remarkablepropertywhichjustnowoccursto me andwhichwillexplainthevastalterationandchangeofcharacCterwhicha finitequan-tity wouldundergoin passingto infinity.Let us drawthestraightlineABofarbitrarylengthandlet the pointC divideit into twounequalparts;thenI saythat,ifpairsoflinesbedrawn,onefromeachof the terminalpointsA andB, andiftheratiobetweenthelengthsoftheselinesisthe sameas thatbetweenACandCB,theirpointsofinterse&ionwillalllieuponthe circumferenceof oneandthe samecircle. Thus,for ex-

[84]ample,ALandBLdrawnfromAandB,meetingat thepointL,beatingto oneanotherthe sameratioasACto BC, andthe

pair AK and BK

meetingat K alsobeatingto one an-otherthesameratio,andlikewisethepairs

A 6 _c B-ii"-"""_ EM, BI,AH,BH,AG,BG, AF, BF, AE,BE,havetheirpointsofintersec°donL, K,I,H,G,F,E,allly-

Fig.7 inguponthecircum-ferenceofoneandthesamecircle. AccordinglyifweimaginethepointCtomovecontinuouslyinsuchamannerthatthelinesdrawnfromittothefixedterminalpoints,AandB,alwaysmain-tainthesameratiobetweentheirlengthsasexistsbetweentheoriginalparts,ACandCB,thenthepointC will,as I shallpres-entlyprove,describeacircle.And thecirclethusdescribedwill

_crcase

r

FIRST DAY 39increasein sizewithoutlimitas the pointC approachesthemid-dlepoint whichwe may callO; but it willdiminishinsizeas Capproachesthe endB. So'thatthe infinitenumberof pointslo-catedinthe lineOBwill,if the motionbeasexplainedabove,de-scribecirclesofeverysize,somesmallerthan thepupilofthe eyeofa flea,otherslargerthanthe celestialequator. Nowifwemoveanyof the pointslying betweenthe two endsO and B they willall describecircles,thosenearestO, immensecircles;but if wemove the point O itself,and continueto moveit accordingtothe aforesaidlaw,namely,that the linesdrawnfrom O to theterminalpoints,A andB, maintainthesameratioasthe originallinesAOandOB,what kindof a linewillbe produced? A circlewillbe drawnlargerthan the largestof the others,a circlewhichisthereforeinfinite. But fromthepointOa straightlinewillalsobe drawnperpendicularto BA and extendingto infinitywith-out ever turning, as did the others, to join its lastend with itsfirst; for the point C, with its limitedmotion,havingdescribed

[85]the upper semi-circle,CHE, proceedsto describe the lowersemicircleEMC, thus returning to the startingpoint. But thepoint O havingstarted to describeits circle,as didall the otherpoints in the lineAB, (for the points in the other portionOAdescribetheir circlesalso, the largest beingthose nearest thepoint O) is unable to return to its startingpoint becausethecircleit describes,beingthe largestof all, is infinite;in fact, itdescribesan infinitestraight lineas circumferenceof its infinitecircle. Thinknowwhat a differencethereisbetweena finiteandan infinitecirclesince the latter changescharacter in such amannerthat it losesnotonly its existencebut alsoits possibilityof existence;indeed,we alreadyclearlyunderstandthat therecanbe nosuch thingas an infinitecircle;similarlytherecanbeno infinitesphere, no infinitebody, and no infinitesurfaceofanyshape. Nowwhat shallwesayconcerningthismetamorpho-sisin the transition fromfiniteto infinite? Andwhy shouldwefeel greater repugnance,seeing that, in our search after theinfiniteamongnumberswe foundit in unity? Having brokenup a solidinto many parts, havingreducedit to the finestof

powder

4o THE TWO NEW SCIENCESOF GALILEOpowderandhavingresolvedit intoitsinfinitelysmallindivisibleatomswhymaywenotsaythat thissolidhasbeenreducedto asinglecontinuum[unsolocontinuo]perhapsa fluidlikewaterormercuryor evena liquifledmetal? Anddowenotseestonesmeltintoglassandtheglassitselfunderstrongheatbecomemorefluidthanwater?

SAtin.Arewethento believethat substancesbecomefluidinvirtueof beingresolvedinto their infinitelysmallindivisiblecomponents?

SALv.I amnot ableto findanybettermeansofaccountingforcertainphenomenaofwhichthe followingisone. WhenItakea hardsubstancesuchasstoneormetalandwhenI reduceit by meansof a hammeror finefileto the mostminuteandimpalpablepowder,it is clearthat itsfinestparticles,althoughwhentakenoneby oneare,onaccountoftheirsmallness,im-perceptibleto our sightandtouch,are neverthelessfiniteinsize,possessshape,andcapabilityof*beingcounted. It is alsotruethatwhenonceheapeduptheyremainin aheap;andifanexcavationbemadewithinlimitsthecavitywillremainandthesurroundingparticleswillnot rushin to fill it; if shakentheparticlescometo restimmediatelyaftertheexternaldisturbingagentis removed;the sameeffecCtsareobservedin allpilesof

[86]largerandlargerparticles,ofanyshape,evenif spherical,asisthecasewithpilesofmillet,wheat,leadshot,andeveryothermaterial. But if weattempt to discoversuchpropertiesinwaterwedonotfindthem;forwhenonceheapedup it imme-diatelyflattensoutunlessheldupbysomevesselorotherexter-nalretainingbody;whenhollowedout it quicklyrushesintofillthe cavity;and whendisturbedit flu&uatesfora longtimeandsendsoutitswavesthroughgreatdistances.

Seeingthat waterhas less firmness[consistenza]than thefinestofpowder,infa_ hasnoconsistencewhatever,wemay,it seemsto me, very reasonablyconcludethat the smallestparticlesintowhlchit canbe resolvedarequitedifferentfromfiniteanddivisibleparticles;indeedthe onlydifferenceI amabletodiscoveristhattheformerareindivisible.Theexquisite

transparency

FIRST DAY 4Itransparencyofwateralsofavorsthisview;forthemosttrans-parentcrystalwhenbrokenandgroundandreducedto powderlosesitstransparency;thefinerthegrindingthegreatertheloss;but in the caseof waterwherethe attritionis of the highestdegreewehaveextremetransparency.Goldandsilverwhenpulverizedwithacids[acquefortz]morefinelythanis possiblewith anyfilestill remainpowders,*anddonotbecomefluidsuntilthe finestparticles[gl'indivisibih]offireorofthe raysofthesundissolvethem,asI think,intotheirultimate,indivisible,andinfinitelysmallcomponents.

SACR.Thisphenomenonof lightwhichyoumentionis onewhichI havemanyti.mesremarkedwithastonishment.I have,for instance,seenleadmeltedinstantlybymeansofa concavemirroronly threehands[palmz]in diameter.HenceI thinkthat if themirrorwereverylarge,well-polishedandofa para-bolicfigure,it wouldjustas readilyandquicklymeltanyothermetal,seeingthat thesmallmirror,whichwasnotwellpolishedandhadonlya sphericalshape,wasablesoenergeticallyto meltlead andburn everycombustiblesubstance.Sucheffe&sastheserendercredibleto me the marvelsaccomplishedby themirrorsofArchimedes.

SALV.Speakingof the effe&sproducedby the mirrorsofArchimedes,it washisownbooks(whichI hadalreadyreadandstudiedwithinfiniteastonishment)that renderedcredibletomeallthemiraclesdescribedbyvariouswriters.Andifanydoubthadremainedthe bookwhichFatherBuonaventuraCavalierit

[87]has recentlypublishedon the subjec°cof the burningglass[speccMoustorfo]andwhichI havereadwithadmirationwouldhaveremovedthelastdifficulty.

SAGR.I alsohaveseenthis treatiseand havereadit with* It is not clear what Galileohere meansby sayingthat gold and

silverwhentreated withacidsstillremainpowders. [Trans.]OneofthemostactiveinvestigatorsamongGalileo'scontemporaries;

bornatMilanI598;diedatBologna_647;aJesuitfather,firsttointro-ducetheuseoflogarithmsintoItalyandfirsttoderivetheexpressionforthefocallengthofalenshavingunequalradiiofcurvature.His"methodofindivisibles"is to bereckonedasa precursorof theinfinitesimalcalculus. [Trans.]

42 THE TWO NEW SCIENCES OF GALILEOpleasureand astonishment;and knowingthe author I was con-firmed in the opinionwhichI had alreadyformedof him thathe was destinedto becomeone of the leadingmathematiciansof our age. But now, with regard to the surprisingeffect ofsolarrays inmeltingmetals,must webelievethat sucha furiousadtionis devoidofmotionor that it is accompaniedbythe mostrapid ofmotions?

SALv.We observethat other combustionsand resolutionsareaccompaniedby motion,and that, the most rapid; note the ac-tion of lightningand of powderas used in minesand petards;note alsohowthe charcoalflame,mixedas it is with heavy andimpurevapors, increasesits powerto liquify metalswheneverquickenedby a pair ofbellows. HenceI donotunderstandhowthe action oflight, althoughverypure, can be devoidof motionand that ofthe swiftesttype.

SAGR.But of what kindand howgreatmust we considerthisspeedof lightto be? Is it instantaneousor momentaryor doesit likeother motionsrequiretime? Can we not decidethis byexperiment?

Sn_P.Everyday experienceshowsthat the propagationoflight is instantaneous;for whenwe seea pieceof artilleryfired,at great distance,the flash reachesour eyes without lapse oftime; but the sound reachesthe ear only after a noticeableinterval.

SAGR.Well,Simplicio,the only thing I am able to inferfromthis familiarbit of experienceis that sound, in reachingourear,travelsmoreslowlythanlight;it doesnot informmewhetherthe comingof the light is instantaneousor whether, althoughextremelyrapid, it still occupiestime. An observationof thiskind tells us nothingmorethan one in whichit is claimedthat"As soonas the sun reachesthe horizonits light reachesoureyes"; but whowill assureme that these rays had not reachedthis limitearlierthan they reachedour vision?

SAT.v.The small conclusivenessof these and other similarobservationsonceledmeto devisea methodbywhichonemightaccuratelyascertainwhetherillumination,i. e.,the propagationof light, is really instantaneous. The fac2that the speed of

sound

FIRST DAY 43[88]

soundis as high as it is, assuresus that the motionof lightcannotfailto beextraordinarilyswift.TheexperimentwhichIdevisedwasasfollows:

Leteachoftwopersonstakea lightcontainedina lantern,orotherreceptacle,suchthatbythe interpositionofthehand,theonecanshutoffor admitthelightto the visionof theother.Nextlet themstandoppositeeachotherat a distanceofa fewcubitsandprac°dceuntiltheyacquiresuchskillinuncoveringandoccultingtheirlightsthattheinstantoneseesthelightofhiscompanionhe willuncoverhisown. Aftera fewtrialstheresponsewillbe sopromptthat withoutsensibleerror[svario]theuncoveringofonelightis immediatelyfollowedby theun-coveringoftheother,sothatassoonasoneexposeshislighthewillinstantlyseethatoftheother. Havingacquiredskillatthisshortdistancelet the two experimenters,equippedasbefore,takeuppositionsseparatedby a distanceoftwoor threemilesandlet themperformthesameexperimentatnight,notingcare-fullywhethertheexposuresandoccultationsoccurinthe samemannerasat shortdistances;iftheydo,wemaysafelyconcludethat the propagationof lightis instantaneous;butif timeisrequiredat a distanceof threemileswhich,consideringthegoingofonelightandthe comingoftheother,reallyamountsto six,thenthe delayoughtto be easilyobservable.If theexperimentis to be madeat stillgreaterdistances,sayeightor ten miles,telescopesmaybe employed,eachobserverad-justingoneforhimselfat the placewhereheis to maketheexperimentat night;thenalthoughthelightsarenotlargeandarethereforeinvisibleto the nakedeyeat sogreata distance,theycanreadilybecoveredanduncoveredsincebyaidofthetelescopes,onceadjustedandfixed,they willbecomeeasilyvisible.

SAGa.Thisexperimentstrikesmeasa cleverandreliablein-vention.But tellus whatyouconcludefromtheresults.

SALV.In fadt I havetriedtheexperimentonlyat a shortdistance,lessthan a mile,fromwhichI havenotbeenabletoascertainwith certaintywhetherthe appearanceof the o19_

posite

44 THE TWO NEW SCIENCES OF GALILEOix)sitelight was instantaneousor not; but if not instantaneousit is extraordinarilyrapid--I shoialdcall it momentary;and forthe present I shouldcompareit to motionwhichwe see in thelightningflashbetweencloudseightor ten milesdistant fromus.We see the beginningof this light--I might say its head and

[89]source--locatedat a particular placeamongthe clouds;but itimmediatelyspreadsto the surroundingones,whichseemsto bean argumentthat at least sometime is requiredforpropagation;for if the illuminationwere instantaneousand not gradual,weshould not be able to distinguishits orlgin--its center,so tospeak--from its outlyingportions. What a seawe are grad-ually slippinginto without knowingit! With vacua and in-finities and indivisiblesand instantaneousmotions, shall weeverbe able,evenby meansof a thousanddiscussions,to reachdry land?

SAGR.Really these matters lie far beyondour grasp. Justthink; whenwe seek the infiniteamongnumberswe find it inunity; that whichis ever divisibleis derivedfrom indivisibles;the vacuum is found inseparablyconnectedwith the plenum;indeedthe viewscommonlyheld concerningthe natureof thesematters are soreversedthat eventhe circumferenceof a circleturns out to be an infinitestraight line, a fact which, if mymemory servesme correctly,you, Salviati,were intending todemonstrategeometrically. Please thereforeproceedwithoutfurtherdigression.

SAr.V.I am at your service;but for the sakeof greater clear-nesslet mefirstdemonstratethe followingproblem:

Given a straight linedividedinto unequalparts whichbearto eachother any ratio whatever,to describea circlesuchthat two straight lines drawn from the ends of the givenline to any point on the circumferencewill bear to eachother the sameratio asthe twoparts of the givenline,thusmakingthoselineswhichare drawnfromthe sameterminalpointshomologous.

LetABrepresentthe givenstraight linedividedinto any twounequalparts by the point C; the problemis to describea circle

such

FIRST DAY 45suchthat twostraightlinesdrawnfromthe terminalpoints,A andB, to anypointon thecircumferencewillbearto eachotherthesameratioasthepartACbearsto BC,sothat linesdrawnfromthesameterminalpointsarehomologous.AboutC as centerdescribea circlehavingthe shorterpartCBofthegivenline,asradius.ThroughAdrawa straightlineADwhich

[90]shallbe tangentto the circleat D andindefinitelyprolongedtowardE. Drawthe radiusCDwhichwillbe perpendicularto AE. At ]3erecta perpendicularto AB;thisperpendicularwillintersectAE atsomepointsincetheangleat A is acute;callthis pointof in-tersecCtionE, andfromit drawa per- (Ipendicular to AEwhichwillintersecCtAB prolongedin F.NowI say the twostraightlinesFEandFC are equal. For _I""_Lifwejoin E and C,we shall have two Fig.8triangles,DECand BEC,in whichthe twosidesof the one,DE and EC, are equal to the two sidesof the other,BEandEC, both DE andEB beingtangentsto the circleDBwhilethe basesDC and CB are likewiseequal;hencethetwo angles,DEC and BEC, willbe equal. NowsincetheangleBCEdiffersfroma rightanglebytheangleCEB,andtheangleCEFalsod_ffersfromarightanglebytheangleCED,andsincethesedifferencesareequal,it followsthat the angleFCEis equalto CEF;consequentlythesidesFE andFCare equal.If wedescribea circlewithF as centerandFE asradiusit willpassthroughthepointC;letCEGbesuchacircle.Thisisthecirclesought,forifwedrawlinesfromtheterminalpointsAandB to anypointonitscircumferencetheywillbearto eachotherthe

46 THE TWO NEW SCIENCESOF GALILEOthesameratioasthetwoportionsACandBCwhichmeetat thepointC. Thisismanifestin the caseof the twolinesAEandBE,meetingat thepointE, becausetheangleEofthe triangleAEBisbise_edbythelineCE,andthereforeAC:CB=AE:BE.ThesamemaybeprovedofthetwolinesAGandBGterminat-ingin thepointG. For sincethe trianglesAFEandEFBaresimilar,wehaveAF:FE=EF:FB, or AF:FC=CF:FB, anddividendoAC:CF=CB:BF, or AC:FG=CB:BF; alsocom-ponendowehavebothAB:BG=CB:BFandAG:GB=CF:FB=AE:EB=AC:BC. Q.E.D.

[9_]Take now any other point in the circumference,say H,

wherethe twolinesAH andBH intersect;in likemannerweshallhaveAC:CB=AH:I-lB.ProlongHB untilit meetsthecircumferenceat I and join IF; and sincewe havealreadyfoundthat AB:BG=CB:BF it followsthat the recCtangleAB.BFis equalto therecCtangleCB.BGorIB.BH. HenceAB:BH--IB:BF. But the anglesat B are equalandthereforeAH:HB=IF:FB=EF:FB=AE:EB.

Besides,I mayadd,that it isimpossibleforlineswhichmain-tain this sameratioandwhichare drawnfromthe terminalpoints,AandB,to meetat anypointeitherinsideoroutsidethecircle,CEG. Forsupposethiswerepossible;letALandBLbetwo suchlinesintersec°dngat the pointL outsidethe circle:prolongLBtill it meetsthe circumferenceat M andjoinMF.If AL:BL=AC:BC=MF:FB,then we shallhave two tri-anglesAI.R and MFB whichhave the sidesaboutthe twoanglesproportional,the anglesat the vertex,B,equal,andthetwo remainingangles,FMB andLAB,lessthanrightangles(becausetherightangleatMhasforitsbasetheentirediameterCGandnotmerelya partBF: andtheotherangleat thepointAisacutebecausethelineAL,thehomologueofAC,isgreaterthanBL,the homologueofBC). Fromthisit followsthat thetrianglesABLandMBF are similarandthereforeAB:BL=MB:BF, makingthe recCtangleAB.BF=MB.BL;but it hasbeendemonstratedthat therecCtangleAB.BFisequaltoCB.BG;whenceit wouldfollowthat thereeCtangleMB.BLisequalto the

rectangle

FIRST DAY 47recCtangleCB.BGwhichisimpossible;thereforetheintersecCtioncannotfalloutsidethecircle.Andinlikemannerwecanshowthatit cannotfallinside;hencealltheseintersectionsfallonthecircumference.

But nowit istimeforus to gobackandgranttherequestofSimplicioby showinghimthat it is notonlynotimpossibletoresolvea lineintoan infinitenumberofpointsbut that thisisquiteaseasyasto divideit intoitsfiniteparts. ThisI willdounderthe followingconditionwhichI am sure,Simplicio,youwillnot denyme,namely,that youwillnot requiremeto sep-aratethe points,onefromthe other,andshowthemto you,[92]oneby one,on this paper;for I shouldbe contentthat you,withoutseparatingthe fouror sixpartsof a linefromonean-other,shouldshowmethemarkeddivisionsorat mostthatyoushouldfoldthemat anglesforminga squareor a hexagon:for,then, I am certainyou wouldconsiderthe divisiondistin_lyandactuallyaccomplished.

S_P. I certainlyshould.SALv.If nowthechangewhichtakesplacewhenyoubenda

lineat anglessoasto formnowa square,nowanoctagon,nowapolygonof forty,a hundredor a thousandangles,is sufficientto bring into acCtualitythe four,eight, forty,hundred,andthousandpartswhich,accordingto you,existedat firstonlypotentiallyin thestraightline,mayI notsay,withequalright,that,whenI havebentthestraightlineintoa polygonhavinganinfinitenumberof sides,i. e., intoa circle,I havereducedtoactualitythat infinitenumberofpartswhichyouclaimed,whileitwasstraight,werecontainedin it onlypotentially? Norcanonedenythat thedivisionintoan infinitenumberofpointsisjustastrulyaccomplishedas theoneintofourpartswhenthesquareisformedor intoa thousandpartswhenthemillagonisformed;forinsuchadivisionthesameconditionsaresatisfiedasin the caseof a polygonofa thousandor a hundredthousandsides. Sucha polygonlaidupona straightlinetouchesitwithoneof its sides,i. e.,withoneof its hundredthousandparts;whilethecirclewhichisapolygonof aninfinitenumberofsidestouches

48 THE TWO NEW SCIENCES OF GALILEOtouches the same straight linewith one of its sideswhich is asinglepoint differentfrom all its neighborsand thereforesep-arate and distincCtinno lessdegreethan is onesideof a polygonfromthe other sides. Andjust as a polygon,whenrolledalonga plane,marks out upon this plane,by the successivecontac%of its sides,a straight lineequal to its perimeter,so the circlerolleduponsuch a plane alsotracesby its infinitesuccessionofcontadtsa straight lineequal in lengthto its owncircumference.I amwilling,Sirnplicio,at the outset,to grantto thePeripateticsthe truth of their opinionthat a continuousquantity [if con-tinuo]is divisibleonlyinto parts whichare still further divisibleso that howeverfar the divisionand subdivisionbe continuednoend will be reached;but I am not so certain that they willconcedeto me that noneof these divisionsof theirs can be afinalone, as is surely the facCt,becausethere alwaysremains"another"; the final and ultimate divisionis rather one whichresolvesa continuousquantity into an infinitenumber of in-divisiblequantities,a resultwhichI grant canneverbe reachedby successivedivisioninto an ever-increasingnumber of parts.But if they employthe methodwhichI proposefor separating

[931and resolvingthe wholeof infinity[tuttala infini_], at a singlestroke (an artificewhich surely ought not to be deniedme),I think that theywouldbe contentedto admitthat a continuousquantity is built up out of absolutelyindivisibleatoms, es-pecially since this method, perhaps better than any other,enablesus to avoidmany intricate labyrinths,suchas cohesionin solids,alreadymentioned,and the questionof expansionandcontra_ion,withoutforcinguponus the objectionableadmissionofemptyspaces[insolids]whichcarrieswith it the penetrabilityof bodies. BothoftheseobjecCfions,it appearstome, areavoidedif we accept the above-mentionedview of indivisiblecon-stituents.

Stop. I hardly knowwhat the Peripateticswouldsay sincethe viewsadvancedby youwouldstrikethemasmostlynew,andas suchwemust considerthem. It ishowevernot unlikelythatthey wouldfindanswersand solutionsfor theseproblemswhich

I,

FIRST DAY 49I, forwantoftimeandcriticalability,amat presentunabletosolve. Leavingthis to onesideforthemoment,I shouldliketo hearhowthe introductionof these indivisiblequantitieshelpsus to understandcontractionandexpansionavoidingatthesametimethevacuumandthepenetrabilityofbodies.

Sacg.I alsoshalllistenwithkeeninteresttothissamematterwhichis farfromclearin mymind;providedI amallowedtohearwhat,amomentago,Simpliciosuggestedweomit,namely,the reasonswhichAristotleoffersagainstthe existenceof thevacuumandtheargumentswhichyoumustadvanceinrebuttal.

SALv.I willdoboth. Andfirst,just as,forthe productionof expansion,weemploythe linedescribedby the smallcircleduringonerotationof the largeone--alinegreaterthan thecircumferenceof the smallcircle--so,in orderto explaincon-traction,wepointout that,duringeachrotationofthe smallercircle,the largeronedescribesa straightlinewhichis shorterthanitscircumference.

For thebetterunderstandingof thisweproceedto the con-siderationofwhathappensin thecaseofpolygons.Employing

[941a figuresimilarto the earlierone,construcCtthetwohexagons,ABCandHIK, aboutthe commoncenterL, andlet themrollalongtheparallellinesHOMandABc. NowholdingthevertexI fixed,allowthe smallerpolygonto rotateuntilthe sideIKliesuponthe parallel,duringwhichmotionthe pointK willdescribethe arcKM, andthe sideKI willcoincidewithIM.Let us seewhat,in the meantime,the sideCBof the largerPheOlygonhasbeendoing.SincetherotationisaboutthepointI,

terminalpointB, of the lineIB, movingbackwards,willdescribethearcBbunderneaththeparallelcAsothat whenthesideKI coincideswiththelineMI,thesideBCwillcoincidewithbc,havingadvancedonlythroughthe distanceBc,but havingretreatedthrougha portionofthe lineBAwhichsubtendsthearcBb.Ifweallowtherotationofthesmallerpolygonto goonit willtraverseanddescribealongits parallela lineequalto itsperimeter;whilethe largeronewilltraverseanddescribea linelessthanitsperimeterbyasmanytimesthelengthbBasthereare

5o THE TWO NEW SCIENCES OF GALILEOare sides lessone; this line is approximatelyequal to that de-scribedby the smallerpolygonexceedingit onlyby the distancebB. Here now we see, without any difficulty,why the larger

polygon,whencarriedbythe smaller, d oe s notmeasureoffwith its sidesa line longer than thattraversed by the smallerone;this isbecausea por-tion ofeach sideis super-posed upon its immedi-ately precedingneighbor.

......" " ' Let us next considertwocircles,havinga com-moncenter at A, ly-and

I_ I ingupontheir respedtive

parallels,the smallerbe-ing tangentto its parallel

2k at the point B; the larger,at thepoint C.Herewhenthe smallcirclecommen-ces to roll the point B

does not remainat rest{:: "for a whileso as to allow

Fig.9 BC to move backwardand carry with it the point C, as happenedin the caseof thepolygons,where the point I remainedfixeduntil the sideKIcoincidedwithMI and the lineIB carriedthe terminalpoint Bbackwardas far asb, so that the sideBCfelluponbc,thus super-posinguponthe lineBA, the portionBb,and advancingby anamount Bc, equal to MI, that is, to one sideof the smallerpolygon. On account of these superpositions,which are theexcessesof the sidesof the largerover the smallerpolygon,eachnet advanceis equalto onesideof the smallerpolygonand,dur-ingone completerotation, theseamountto a straight lineequalin lengthto the perimeterof the smallerpolygon.

But

FIRST DAY 5IBut nowreasoningin the samewayconcerningthe circles,

wemustobservethat whereasthenumberofsidesinanypoly-gonis comprisedwithina certainlimit,thenumberofsidesinacircleis infinite;the formerare finiteanddivisible;the latterinfiniteandindivisible.In thecaseofthepolygon,theverticesremainat restduringan intervalof timewhichbearsto theperiodofonecompleterotationthe sameratiowhichonesidebearsto the perimeter;likewise,in the caseof the circles,thedelayof eachof the infinitenumberof verticesis merelyin-stantaneous,becausean instantis sucha fracCtionof a finiteintervalasa pointisofa linewhichcontainsaninfinitenumberofpoints.Theretrogressionofthesidesofthelargerpolygonisnot equalto the lengthofoneof its sidesbut merelyto theexcessof sucha sideoveronesideof the smallerpolygon,thenetadvancebeingequalto thissmallerside;butinthecircle,thepointorsideC,duringtheinstantaneousrestofB,recedesbyanamountequalto itsexcessoverthesideB,makinganetprogressequalto B itself. In shortthe infinitenumberof indivisiblesidesofthegreatercirclewiththeirinfinitenumberofindivisibleretrogressions,madeduringtheinfinitenumberofinstantaneousdelaysof the infinitenumberof verticesof the smallercircle,togetherwiththe infinitenumberof progressions,equalto theinfinitenumberof sidesin the smallercircle•all these,I say,addup to.a lineequalto that describedby the smallercircle,a line whichcontainsan infinitenumberof infinitelysmallsuperpositions,thusbringingabouta thickeningor contracCtionwithoutany overlappingor interpenetrationof finiteparts.Thisresultcouldnot beobtainedin thecaseofa linedivided

[96]intofinitepartssuchas istheperimeterofanypolygon,whichwhenlaldoutin a straightlinecannotbe shortenedexceptbytheoverlappingandinterpenetrationofitssides.Thiscontrac-tionofan infinitenumberofinfinitelysmallpartswithouttheinterpenetrationoroverlappingoffinitepartsandthepreviouslymentioned[p.7o,Nat.Ed.]expansionof an infinitenumberofindivisiblepartsby the interpositionof indivisiblevacuais,inmyopinion,themostthat canbesaidconcerningthecontrac°donand

52 THE TWO NEW SCIENCES OF GALILEOand rarefacCtionof bodies,unlesswe giveup the impenetrabilityofmatter and introduceemptyspacesof finitesize. If youfindanything herethat you considerworthwhile,pray use it; ifnotregard it, together with my remarks, as idle talk; but thisremember,we are dealingwith the infiniteand the indivisible.

SAGR.I franklyconfessthat your idea is subtle and that itimpressesme as new and strange; but whether, as a matter offacet,nature actually behaves accordingto such a law I amunable to determine;however,until I find a more satisfacCtoryexplanationI shall hold fast to this one. Perhaps Sirnpliciocan tell us somethingwhichI havenot yet heard, namely,howto explainthe explanationwhich the philosophershave givenof this abstruse matter; for, indeed,all that I have hithertoread concerningcontracCtionis so denseand that concerningex-pansionso thin that my poor brain canneither penetrate theformernorgraspthe latter.

SIMP.I am all at sea and finddiflicultlesin followingeitherpath, especiallythis newone; becauseaccordingto this theoryan ounceof goldmight be rarefiedand expandeduntil its sizewouldexceedthat of theearth, whilethe earth, in turn, mightbecondensedand reduceduntil it wouldbecomesmallerthan awalnut, somethingwhichI donot believe;nor doI believethatyoubelieve it. The argumentsand demonstrationswhichyouhave advancedare mathematical,abstracCt,and far removedfromconcretematter; and I do notbelievethat whenappliedtothe physicaland naturalworldtheselawswillhold.

SALV.I am not able to render the invisiblevisible,nor doI think that youwillask this. But nowthat youmentiongold,donot our sensestell us that that metal canbe immenselyex-panded? I donot knowwhetheryouhaveobservedthe method

[97]employedby thosewhoare skilledindrawinggoldwire,ofwhichreallyonly the surfaceis gold,the insidematerialbeing silver.The way they drawit is as follows:they take a cylinderor, ifyou please,a rod of silver,about half a cubit longand three orfour times as wide as one's _daumb;this rod they coverwithgold-leafwhichis so thin that it almostfloatsin air, puttingon

not

FIRST DAY 53notmorethaneightorten thicknesses.Oncegildedtheybeginto pullit, withgreatforce,throughtheholesofa draw-plate;againandagainit ismadeto passthroughsmallerandsmallerholes,until, aftervery many passages,it is reducedto thefinenessofa lady'shair,or perhapsevenfiner;yet the surfaceremainsgilded.Imaginenowhowthesubstanceofthisgoldhasbeenexpandedandtowhatfinenessit hasbeenreduced.

Sire,.I do not seethat this processwouldproduce,as aconsequence,that marvellousthinningof the substanceofthegoldwhichyousuggest:first,becausetheoriginalgildingcon-sistingoftenlayersofgold-leafhasasensiblethickness;secondly,becausein drawingoutthe silverit growsin lengthbut at thesametime diminishesproportionallyin thickness;and, sinceonedimensionthuscompensatestheother,theareawillnotbesoincreasedasto makeitnecessaryduringtheprocessofgildingto reducethe thinnessof the goldbeyondthat oftheoriginalleaves.

SALV.You are greatlymistaken,Simplicio,becausethe sur-faceincreasesdirectlyas the squarerootof the length,a facCtwhichI candemonstrategeometrically.

SAGR.Pleasegiveus the demonstrationnotonlyformyownsakebut alsoforSimplicioprovidedyou thinkwecanunder-standit.

SALV.I'llseeif I can recallit on the spurof the moment..Attheoutset,it is clearthat theoriginalthickrodofsilverandthe wiredrawnout to an enormouslengtharetwocylindersofthe samevolume,sincetheyare the samebodyof silver. So

[981that,if I determinetheratiobetweenthesurfacesofcylindersofthe samevolume,theproblemwillbesolved.I saythen,

The areasof cylindersof equalvolumes,neglectingthebases,bearto eachothera ratiowhichisthesquarerootoftheratiooftheirlengths.

Taketwo cylindersof equalvolumehavingthe altitudesABandCD,betweenwhichthelineEisameanproportional.ThenI claimthat,omittingthebasesofeachcylinder,the surfaceofthe cylinderABis to that ofthecylinderCDasthelengthABis

54 THE TWO NEW SCIENCES OF GALILEOis to the lineE, that is,as the squarerootofAB isto the squareroot of CD. Now cut off the cylinderAB at F sothat the alti-tude AF is equal to CD. Then sincethe basesof cylindersofequal volume bear to one another the inverseratio of theirheights, it followsthat the area of the circular base of thecylinderCD willbe to the area of the circularbaseof ABas thealtitudeBAistoDC:moreover,sincecirclesare to one anotheras the squares of their diameters,the said squareswill be toeachother as BAis to CD. But BAis to CD as the squareof

E BA is to the squareof E: and, therefore,these_-_ 1 four squareswill form a proportion;and like-I wise their sides; so the lineAB is to E as the

diameterof circle C is to the diameter of thecircleA. But the diametersare proportionalto the circumferencesand the circumferencesare proportionalto the areas of cylinders of

"-F equalheight; hencethe lineAB is to E as thesurfaceof the cylinderCD is to the surfaceofthe cylinderAF. Nowsincethe heightAF is toAB asthe surfaceofAF is to the surfaceof AB;and sincethe heightAB is to the lineE as thesurfaceCD is to AF, it follows,ex _equaliin

Fig.io proportioneperturbata,*that the height AF isto E as the surfaceCD is to the surfaceAB, and con$ertendo,the surfaceof the cylinderAB is to the surfaceof the cyl-inder CD as the line E is to AF, i. e., to CD, or as AB is toE which is the squareroot of the ratio of AB to CD. q.E.D.

If nowwe apply these resultsto the casein hand,and assumethat the silver cylinderat the time of gildinghad a length ofonly half a cubit and a thicknessthree or four times that of

[99]one's thumb,we shallfindthat, whenthe wirehasbeenreducedto the finenessof a hair and has been drawnout to a lengthoftwenty thousand cubits (and perhaps more), the area of itssurface will have been increasednot less than two hundredtimes. Consequentlythe ten leavesof goldwhichwerelaidon

*SeeEuclid,BookV,Def.2o.,Todhunter'sEd.,p.I37(London,I877.)[Trans.]

FIRST DAY 55have beenextendedover a surfacetwohundred timesgreater,assuringus that the thicknessofthe goldwhichnowcoversthesurfaceof somany cubitsof wirecannot be greater than onetwentieth that of an ordinary leaf of beaten gold. Considernowwhat degreeof finenessit musthaveandwhetherone couldconceiveit to happenin any other way than by enormousex-pansionof parts; consideralsowhetherthisexperimentdoesnotsuggest that physicalbodies [rnateriefisiche]are composedofinfinitelysmall indivisibleparticles,a viewwhichis supportedbyothermorestrikingand conclusiveexamples.

SACR.Thisdemonstrationis sobeautifulthat, evenif it doesnot have the cogencyoriginally intended,--althoughto mymind, it is very forceful--the short time devoted to it hasneverthelessbeenmosthappilyspent.

SAT.v.Sinceyou are so fondof thesegeometricaldemonstra-tions, which carry with them distinctgain, I will give you acompaniontheorem which answersan extremelyinterestingquery. Wehaveseenabovewhat relationsholdbetweenequalcylindersof differentheightor length;let usnowseewhat holdswhen the cylindersare equal in area but unequal in height,understandingarea to includethe curvedsurface,but not theupperandlowerbases. The theoremis:

The volumesof right cylindershavingequal curvedsur-facesareinverselyproportionalto theiraltitudes.

Let the surfacesof the twocylinders,AlEand CF, be equalbutlet the height of the latter, CD, be greater than that of theformer,AB: then I say that the volumeof the cylinderAE isto that of the cylinderCF as the heightCD is to AB. Nowsince the surfaceof CF is equal to the surfaceof AE, it fol-lowsthat the volumeof CF is lessthan that of_ALE;for, if theywereequal, the surfaceof CF would,by the precedingproposi-tion, exceedthat of AE, and the excesswouldbe so muchthegreaterif the volumeof the cylinderCF weregreaterthan that[ oolofAE. Let usnowtake a cylinderID havinga volumeequaltothat o_?_; then, accordingto the precedingtheorem,the sur-face of the cylinder'ID is to the surfaceof _ as the altitudeIF

56 THE TWO NEW SCIENCESOF GALILEOIF isto themeanproportionalbetweenIF andAB. Butsinceonedatumof the problemis that the surfaceof AEis equalto that of CF, andsincethe surfaceID is to the surfaceCFasthe altitudeIF isto thealtitudeCD,it followsthat CDisa

meanproportionalbetweenIF andAB.""i Not onlyso,but sincethe volumeof thecylinderID is equalto that ofAE, each

willbearthe sameratioto the volumeofthe cylinderCF;but the volumeID is tothe volumeCF asthealtitudeIF is to thealtitudeCD; hencethe volumeof AE iscto the volumeof CF as the lengthIF isto thelengthCD,thatis,asthelengthCD

A is to thelengthAB. q.E.D.Thisexplainsaphenomenonuponwhich

Ithe commonpeople alwayslook withIwonder,namely,ifwehavea pieceofstuff!whichhasone sidelongerthanthe other,!wecanmakefromit a cornsack,usingthe

E B ]customarywoodenbase,whichwiltholdI . .m Fimorewhenthe shortsldeof the clothis

usedfortheheightofthesackandthelongFig.n sideis wrappedaroundthe woodenbase,

thanwith the alternativearrangement.Sothat, forinstance,froma pieceofclothwhichissixcubitsononesideandtwelveontheother,a sackcanbemadewhichwillholdmorewhenthesideoftwelvecubitsis wrappedaroundthewoodenbase,leav-ing the sacksix cubitshighthan whenthe sixcubitside isputaroundthebasemakingthe sacktwelvecubitshigh. Fromwhathasbeenprovenabovewelearnnotonlythegeneralfa_that onesackholdsmorethantheother,butwealsogetspecificand particularinformationas to how muchmore,namely,just in proportionas the altitudeof the sackdiminishesthecontentsincreaseand viceversa. Thus if weuse the figuresgivenwhichmakethe clothtwiceaslongaswideandifweusethe longsideforthe seam,the volumeof the sackwillbejustone-halfasgreatas withthe oppositearrangement.Likewise

if

FIRST DAY 57[ioI]

if we havea pieceof mattingwhichmeasures7 x 25cubitsandmakefromit a basket,the contentsof thebasketwill,whentheseamislengthwise,be sevenas comparedwith twenty-fivewhenthe seamrunsendwise.

SAGI_.It is with great pleasurethat we continuethus to ac-quire new and usefulinformation. But as regardsthe subjecCtjust discussed,I reallybelievethat, amongthosewho are notalreadyfamiliarwithgeometry,youwouldscarcelyfindfourper-sonsina hundredwhowouldnot,at firstsight,makethemistakeofbelievingthat bodieshavingequalsurfaceswouldbe equalinother respecCts.Speakingof areas,the sameerror ismadewhenoneattempts,as oftenhappens,to determinethe sizesofvariouscities by measuringtheir boundary lines, forgettingthat thecircuit of one may be equalto the circuitof anotherwhilethearea of the one is much greater than that of the other. Andthis is true not only in the caseof irregular,but alsoof regularsurfaces,,wherethe polygonhavingthe greaternumberof sidesalwayscontainsa largerarea than the onewith the lessnumberof sides,so that finallythe circlewhich is a polygonof an in-finitenumberof sidescontainsthe largestarea ofallpolygonsofequal perimeter. I rememberwith particularpleasurehavingseen this demonstrationwhen I was studying the sphere ofSacrobosco*with the aidofa learnedcommentary.

SALV.Very true! I too cameacrossthe samepassagewhichsuggestedto me a method of showinghow, by a singleshortdemonstration,one can prove that the circlehas the largestcontent of all regular isoperimetricfigures;and that, of other[io ]figures,the onewhichhasthe largernumberof sidescontainsagreater areathan that whichhasthe smallernumber.

SAGR.Beingexceedinglyfondofchoiceanduncommonpropo-sitions,I beseechyouto letushaveyourdemonstration.

SALV. I cando this in a fewwordsby provingthe followingtheorem:

The area of a circleis a mean proportionalbetweenany• Seeinterestingbiographicalnoteon Sacrobosco[JohnHolywood]

inEncy.,grit.,Ilth Ed. [Trans.]

58 THE TWO NEW SCIENCES OF GALILEOtwo regularand similarpolygonsof whichonecircum-scribesit andtheotherisisoperimetricwithit. Inaddition,the areaofthecircleislessthanthat ofanycircumscribedpolygonandgreaterthanthat ofanyisoperimetricpolygon.Andfurther,ofthesecircumscribedpolygons,theonewhichhasthegreaternumberofsidesissmallerthantheonewhichhas a lessnumber;but, on the otherhand,that isoperi-metricpolygonwhichhas the greaternumberof sidesisthe larger.

LetA andB betwosimilarpolygonsofwhichA circumscribesthegivencircleandB isisoperimetricwithit. Theareaofthecirclewillthenbeameanproportionalbetweenthe areasofthepolygons.For ifweindicatetheradiusofthecirclebyAC andifwerememberthat the areaof thecircleisequaltothat ofaright-angledtrianglein whichoneof the sidesaboutthe rightangleis equalto the radius,AC,andthe otherto the circum-ference;andif likewisewerememberthat theareaofthe poly-gonA is equalto the area of a right-angledtriangleoneof

[io31whosesidesabouttherightanglehasthesamelengthasACandtheotherisequalto theperimeterofthepolygonitself;it isthen

1

c oFig.IZ

manifestthat the circumscribedpolygonbearsto the circlethesameratiowhichitsperimeterbearsto the circumferenceofthecircle,or to theperimeterofthepolygonB whichis,byhypoth-esis,equalto the circumferenceof the circle. But sincethepolygonsAandB aresimilartheirareasareto eachotherasthesquaresoftheirperimeters;hencethe areaofthe circleA is a

mean

FIRST DAY 59meanproportionalbetweenthe areasofthe twopolygonsA andB. And sincethe area of the polygonA is greaterthan that ofthe circleA, it is clearthat the area of the circleA is greaterthan that of the isoperimetricpolygonB, and is thereforethegreatest of all regular polygonshavingthe same perimeterasthe circle.

We now demonstratethe remainingportionof the theorem,whichis to prove that, in the caseof polygonscircumscribinga givencircle,the one havingthe smallernumberof sideshasa larger area than one having a greaternumberof sides;butthat on the other hand, in the caseof isoperimetricpolygons,the one having the more sideshas a larger area than the onewith lesssides. To the circlewhich has O for center and OAfor radius draw the tangent AD; and on this tangent lay off,say,AD whichshallrepresentone-halfof the sideof a circum-scribedpentagonand AC whichshall representone-halfof thesideofa heptagon;drawthe straight linesOC_ and OFD; thenwithO as a center and OC as radius drawthe arcECI. Nowsincethe triangleDOCis greaterthan the sectorEOCand sincethe sectorCOI is greaterthan the triangleCOAL,it followsthatthe triangleDOCbearsto the triangleCO.A_a greaterratiothanthe sectorEOCbearsto the secCtorCOI,that is, than thesector

- FOG bears to the secCtorGOA. Hence, componendoet per-mutando,the triangleDOA bears to the secCtorFOAa greaterratio than that which the triangle COA bears to the sectorGOA,and also IOsuch trianglesDOA bear to IOsuch secCtorsFOAa greaterratio than 14suchtrianglesCOAbear to 14suchsectorsGOA, that is to say, the circumscribedpentagonbearsto the circlea greater ratio than doesthe heptagon. Hencethepentagonexceedsthe heptagoninarea.

But nowlet us assumethat both the heptagoilandthe penta-gonhave the sameperimeteras that of a givencircle. Then Isay the heptagonwillcontaina largerarea than the pentagon.For sincethe area of the circleis a mean proportionalbetweenareas of the circumscribedand of the isoperimetricpentagons,

[ o4]and since likewiseit is a mean proportionalbetween the cir-

cumscribed

60 THE TWO NEW SCIENCESOF GALILEOcumscribedandisoperimetricheptagons,andsincealsowehaveprovedthat the circumscribedpentagonis largerthan thecircumscribedheptagon,it followsthat thiscircumscribedpen-tagonbearsto the circlea largerratiothandoesthe heptagon,that is, the circlewill bear to its isoperimetricpentagonagreaterratio than to its isoperimetricheptagon. Hencethepentagonis smallerthan its isoperimetricheptagon.Q._. D.

SAGR.Averycleverandelegantdemonstration!Buthowdidwecometo plungeintogeometrywhilediscussingtheobjectionsurgedbySimplicio,objecCtionsofgreatmoment,especiallythatonereferringtodensitywhichstrikesmeasparticularlydifficult?

SALV.If contracCtionand expansion[condensazionee rare-fa_ione]consistin contrarymotions,oneoughtto findforeachgreatexpansiona correspondinglylargecontracCtion.Butoursurpriseis increasedwhen,everyday,weseeenormousexpan-sions taking pLce almostinstantaneously.Think what atremendousexpansionoccurswhena smallquantityof gun-powderflaresup intoa vastvolumeoffire! Thinktooofthealmost limitlessexpansionof the light whichit produces!ImaginethecontracCtionwhichwouldtakeplaceif thisfireandthis lightwereto reunite,which,indeed,isnot impossiblesinceonlya lktle whileagotheywerelocatedtogetherin thissmallspace. Youwillfind,uponobservation,athousandsuchexpan-sionsfor theyare moreobviousthan contracCtionssincedensematterismorepalpableandaccessibleto our senses.Wecantakewoodandseeit goup infireandlight,but wedonotsee

[IoSlthemrecombineto formwood;weseefruitsandflowersandathousandothersolidbodiesdissolvelargelyintoodors,but wedo not observethesefragrantatomscomingtogetherto formfragrantsolids. But wherethe sensesfailus reasonmuststepin; forit willenableusto understandthemotioninvolvedinthecondensationof extremelyrarefiedandtenuoussubstancesjustas clearlyas that involvedin the expansionanddissolutiono_solids.Moreoverwearetryingto findouthowit ispossibletoproduceexpansionandcontracCtionin bodieswhicharecapableof suchchangeswithoutintroducingvacuaandwithoutgiving

up

FIRST DAY 6Iup the impenetrabilityofmatter;but thisdoesnotexcludethepossibilityoftherebeingmaterialswhichpossessnosuchprop-ertiesand do not, therefore,carry with them consequenceswhichyou call inconvenientand impossible.And finally,Simplicio,I have,forthe sakeofyouphilosophers,takenpainsto findan explanationof howexpansionandcontra_ioncantakeplacewithoutouradmittingthepenetrabilityofmatterandintroducingvacua,propertieswhichyou denyand dislike;ifyouwereto admitthem,I shouldnotopposeyousovigorously.NoweitheradmitthesediflScultiesor acceptmyviewsor sug-gestsomethingbetter.

SACR.I quite agreewith the peripateticphilosopherskndenyingthe penetrabilityofmatter. Asto thevacuaI shouldliketo heara thoroughdiscussionof Aristotle'sdemonstrationinwhichheopposesthem,andwhatyou,Salviati,haveto sayinreply. I begof you, Simplicio,that you giveus the preciseproofof the Philosopherand that you, Salviati,giveus thereply.

SnaP.So far as I remember,Aristotleinveighsagainsttheancientviewthat a vacuumis a necessaryprerequisiteformotionandthat the lattercouldnotoccurwithouttheformer.In oppositionto thisviewAristotleshowsthat it is preciselythe phenomenonof motion,as we shall see,whichrendersuntenablethe ideaof a vacuum. Hismethodisto dividetheargumentintotwoparts. Hefirstsupposesbodiesofdifferentweightstomoveinthesamemedium;thensupposes,oneandthesamebodyto movein differentmedia. In the first case,he

[_o61supposesbodiesofdifferentweightto moveinoneandthesamemediumwithdifferentspeedswhichstandto oneanotherin thesameratioastheweights;sothat,forexample,abodywhichistentimesasheavyas anotherwillmovetentimesasrapidlyastheother. In thesecondcaseheassumesthatthe speedsofoneand the samebodymovingin differentmediaare in inverseratioto the densitiesof thesemedia;thus,for instance,if thedensityofwaterweretentimesthatofair,thespeedinairwouldbe ten timesgreaterthanin water. Fromthissecondsupposi-

tion,

62 THE TWO NEW SCIENCESOF GALILEOti0n,he showsthat, sincethe tenuityof a vacuumdiffersin-finitelyfromthat of anymediumfilledwith matterhoweverrare, any bodywhichmovesin a plenumthrougha certainspacein a certaintimeoughtto movethrougha vacuumin-stantaneously;but instantaneousmotionis an impossibility;it isthereforeimpossiblethat avacuumshouldbeproducedbymotion.

SALV.Theargumentis,asyousee,adhorninem,that is,it isdirectedagainstthosewhothoughtthe vacuuma prerequisiteformotion.Nowif I admittheargumentto be conclusiveandconcedealsothat motioncannottakeplacein a vacuum,theassumptionof a vacuumconsideredabsolutelyand not withreferencetomotion,isriottherebyinvalidated.But to tellyouwhatthe ancientsmightpossiblyhaverepliedandin ordertobetterunderstandjust howconclusiveAristotle'sdemonstra-tionis,wemay,in my opinion,denybothofhisassumptions.Andasto thefirst,I greatlydoubtthatAristotleevertestedbyexperimentwhetherit betruethat twostones,oneweighingtentimesas muchas the other,ff allowedto fall,at the samein-stant,froma heightof,say,Ioocubits,wouldsodifferin speedthat whenthe heavierhadreachedtheground,theotherwouldnothavefallenmorethanIOcubits.

SIMI'.Hislanguagewouldseemto indicatethat hehadtriedthe experiment,becausehe says:Weseetheheavier;nowthewordseeshowsthat hehadmadethe experiment.

SACR.But I, Simplicio,whohavemadethe test canassure[lO7]

youthat a cannonballweighingoneortwohundredpounds,orevenmore,willnotreachthegroundbyasmuchasaspanaheadofa musketballweighingonlyhalfa pound,providedbotharedroppedfromaheightof2oocubits.

SALV.But, evenwithoutfurtherexperiment,it ispossibletoproveclearly,by meansof a shortand conclusiveargument,that a heavierbodydoesnotmovemorerapidlythana lighteroneprovidedbothbodiesareof thesamematerialandin shortsuchas thosementionedby Aristotle.But tellme,Simplicio,whetheryou admitthat eachfallingbodyacquiresa definite

speed

FIRST DAY 63speed fixedby nature, a velocitywhichcannot be increasedordiminishedexcept by the use of force[violenza]or resistance.

SIMP.Therecanbe nodoubtbut that oneand the samebodymovingin a singlemediumhas a fixedvelocitywhichis deter-minedby nature and whichcannot be increasedexceptby theadditionof momentum [impeto]or diminishedexceptby someresistancewhichretardsit.

SALV.If then we take two bodieswhosenaturalspeedsaredifferent,it is clearthat onunitingthe two,the morerapidonewill be partly retardedby the slower,and the slowerwill besomewhathastenedby the swifter. Do you not agreewithmein this opinion?

SIMP.You are unquestionablyright.SALV.But if this is true, and if a largestone moveswith a

speedof, say,eightwhilea smallermoveswitha speedof four,then when theyare united, the systemwillmovewith a speedless than eight;but the two stoneswhen tied togethermake astonelargerthan that whichbeforemovedwitha speedofeight.Hencethe heavierbodymoveswith lessspeedthan the lighter;an effe_ whichis contraryto yoursupposition. Thus you see

[ oS]how,from your assumptionthat the heavierbody movesmorerapidlythan the lighterone,I inferthat the heavierbody movesmoreslowly.

SIM1,.I am allat seabecauseit appearsto methat the smallerstonewhenaddedtothe largerincreasesitsweightandby addingweightI do not seehow it canfail to increaseits speedor, atleast, not to diminishit.

SaLv.Here again you are in error, Simplicio,becauseit isnottrue that the smallerstoneaddsweightto the larger.

SIM1,.This is, indeed,quitebeyondmy comprehension.SALV.It willnotbe beyondyouwhenI haveonceshownyou

the mistake under which you are laboring. Note that it isnecessaryto distinguishbetweenheavybodiesinmotionand thesamebodiesat rest. A largestoneplacedina balancenotonlyacquiresadditionalweightby havinganotherstoneplaceduponit, but evenby the additionof a handfulof hemp its weightis

augmented

64 THE TWO NEW SCIENCESOF GALILEOaugmentedsixto tenouncesaccordingto thequantityofhemp.But if you tie the hempto the stoneandallowthemto fallfreelyfromsomeheight,do you believethat the hempwillpressdownuponthe stoneandthusaccelerateitsmotionor doyou think the motionwillbe retardedby a partialupwardpressure? Onealwaysfeelsthe pressureuponhis shoulderswhenhepreventsthemotionofa loadrestinguponhim;but ifonedescendsjust asrapidlyasthe loadwouldfallhowcan itgravitateorpressuponhim? Doyounotseethat thiswouldbethesameastryingto strikeamanwitha lancewhenheisrun-ningawayfromyou witha speedwhichis equalto, or evengreater,thanthatwithwhichyouarefollowinghim? Youmustthereforeconcludethat, duringfreeandnaturalfall,the smallstonedoesnotpressuponthe largerandconsequentlydoesnotincreaseitsweightasit doeswhenatrest.

Sn_P.But what if we shouldplacethe largerstoneuponthesmaller?[109]

SALV.Itsweightwouldbeincreasedifthelargerstonemovedmorerapidly;but wehavealreadyconcludedthat whenthesmallstonemovesmoreslowlyit retardsto someextentthespeedofthelarger,sothat thecombinationofthe two,whichisaheavierbodythanthelargerofthetwostones,wouldmovelessrapidly,a conclusionwhichis contraryto your hypothesis.Weinferthereforethat largeand smallbodiesmovewiththesamespeedprovidedtheyareofthesamespecificgravity.

SIMP.Yourdiscussionisreallyadmirable;yet I donotfinditeasyto believethat a bird-shotfallsas swiftlyasa cannonball.

SAT.v.Whynotsayagrainofsandasrapidlyasa grindstone?But,Simplicio,I trustyouwillnotfollowtheexampleofmanyotherswhodivertthe discussionfromitsmainintentandfastenuponsomestatementofminewhichlacksa hair's-breadthofthetruthand,underthishair,hidethe faultofanotherwhichis asbigas a ship'scable. Aristotlesaysthat "an ironballof onehundredpoundsfallingfroma heightof onehundredcubitsreachesthe groundbeforea one-poundballhas fallena singlecubit." I saythat theyarriveat thesametime. Youfind,on

making

FIRST DAY 65makingthe experiment,that the largeroutstripsthe smallerbytwo finger-breadths,that is, when the larger has reached theground,the other is shortof it by twofinger-breadths;nowyouwouldnothidebehindthesetwofingersthe ninety-ninecubitsofAristotle,norwouldyoumentionmysmallerrorandat the sametime passover in silencehis very largeone. Aristotledeclaresthat bodiesof differentweights, in the same medium,travel(in so far as their motion dependsupon gravity) with speedswhichareproportionalto theirweights;thishe illustratesbyuseof bodiesin which it is possibleto perceivethe pure and un-adulterated effectof gravity, eliminatingother considerations,for example,figureas being of small importance[minimirno-znentz],influenceswhicharegreatlydependentuponthe mediumwhichmodifiesthe singleeffec°cof gravity alone. Thus weob-servethat gold,the densestof all substances,when beatenoutinto a very thin leaf, goesfloatingthrough the air; the samething happenswith stonewhengroundinto a very finepowder.But if you wish to maintain the generalpropositionyou willhave to showthat the sameratio of speedsis preservedin the

[ Io]caseof all heavy bodies, and that a stone of twenty poundsmovesten times as rapidlyas one of two;but I claimthat thisis falseand that, if they fallfrom a heightof fiftyor a hundredcubits,theywillreachthe earthat the samemoment.

SnuP.Perhaps the result wouldbe differentif the fall tookplacenot from a fewcubitsbut fromsomethousandsof cubits.

S_v. If this were what Aristotlemeant you wouldburdenhim with another error which wouldamount to a falsehood;because,sincethereis no suchsheerheightavailableon earth, itis clearthat Aristotle couldnot havemade the experiment;yethe wishesto giveus the impressionof his havingperformeditwhenhe speaksof suchan effecCtasonewhichwe see.

S_rP.In fact, Aristotle doesnot employthis principle,butusesthe other onewhichis not, I believe,subjecCtto thesesamedifficulties.

SALv.But the oneis as falseas the other;and I amsurprisedthat you yourselfdo not see the fallacyand that you do not

perceive

66 THE TWO NEW SCIENCESOF GALILEOperceivethat ifit weretruethat,in mediaofdifferentdensitiesand differentresistances,suchas waterand air, oneandthesamebodymovedin air morerapidlythaninwater,inpropor-tionas the densityofwateris greaterthanthat ofair,thenitwouldfollowthat anybodywhichfallsthroughair oughtalsoto fallthroughwater. But thisconclusionisfalseinasmuchasmanybodieswhichdescendin air not onlydo notdescendinwater,but agtuallyrise.

Sn_P.I do not understandthe necessityof yourinference;andin additionI willsay that Aristotlediscussesonly thosebodieswhichfallin bothmedia,not thosewhichfallin air butriseinwater.

SAT.V.The argumentswhichyou advancefor the Philos-opheraresuchashehimselfwouldhavecertainlyavoidedsoasnotto aggravatehisfirstmistake.Buttellmenowwhetherthedensity[corpulenza]of the water,or whateverit maybe that

[III]retardsthe motion,bearsa definiteratioto the densityof airwhichis lessretardative;andif so fixa valuefor it at yourpleasure.

SIv_P.Sucha ratiodoesexist;letusassumeitto beten;then,forabodywhichfallsinboththesemedia,thespeedinwaterwillbetentimesslowerthanin air.

SALV.I shallnowtakeoneofthosebodieswhichfallin airbutnotinwater,sayawoodenball,andI shallaskyouto assigntoit anyspeedyoupleaseforitsdescentthroughair.

SIMP.Letussupposeit moveswitha speedoftwenty.SAT.V.Verywell. Then it is clearthat thisspeedbearsto

somesmallerspeedthe sameratioasthedensityofwaterbearsto that of air; andthe valueof thissmallerspeedis two. Sothat reallyif wefollowexa_lythe assumptionofAristotleweoughtto inferthat the woodenball whichfallsin air, a sub-stancetentimesless-resistingthanwater,withaspeedoftwentywouldfallinwaterwithaspeedoftwo,insteadofcomingto thesurfacefromthe bottomasit does;unlessperhapsyouwishtoreply,whichI donotbelieveyouwill,that therisingofthewoodthroughthewateristhesameasits fallingwitha speedoftwo.

But

FIRST DAY 67Butsincethewoodenballdoesnotgotothebottom,I thinkyouwillagreewithme that wecanfinda ballofanothermaterial,notwood,whichdoesfallinwaterwitha speedoftwo.

SIMP.Undoubtedlywecan;but it mustbe of a substanceconsiderablyheavierthanwood.

SAT.V.Thatisit exa_ly. But ifthissecondballfallsinwaterwitha speedof two,whatwillbe ks speedofdescentin air?If youholdto the ruleofAristotleyoumustreplythat it willmoveat the rateoftwenty;buttwentyisthe speedwhichyouyourselfhavealreadyassignedto the woodenball;hencethisandtheotherheavierballwilleachmovethroughairwiththesamespeed. But nowhowdoesthe Philosopherharmonizethisresultwithhisother,namely,that bodiesofdifferentweightmovethroughthe samemediumwithdifferentspeeds--speedswhichareproportionalto theirweights?But withoutgoinginto the mattermoredeeply,howhave thesecommonand

[II2]

obviouspropertiesescapedyournotice?Haveyounotobservedthat twobodieswhichfallin water,onewitha speeda hundredtimesasgreatasthatOftheother,willfallinairwithspeedssonearlyequalthat onewillnotsurpasstheotherby asmuchasonehundredthpart? Thus,forexample,aneggmadeofmarblewilldescendin wateronehundredtimesmorerapidlythanahen'segg,whilein airfallingfroma heightoftwentycubitstheonewillfallshortof theotherby lessthanfourfinger-breadths.In short,aheavybodywhichsinksthroughtencubitsofwaterin threehourswilltraversetencubitsofair inoneortwopulse-beats;and if the heavybodybe a ball of leadit willeasilytraversethe ten cubitsof waterin lessthan doublethe timerequiredfor tencubitsofair. Andhere,I amsure,Simplicio,youfindno groundfordifferenceor objecCtion.Weconclude,therefore,that theargumentdoesnotbearagainsttheexistenceofa vacuum;but if itdid,itwouldonlydoawaywithvacuaofconsiderablesizewhichneitherI nor,inmyopinion,theancientseverbelievedto existinnature,althoughtheymightpossiblybeproducedbyforce[_olenza]asmaybegatheredfromvariousex-perimentswhosedescriptionwouldhereoccupytoomuchtime.

Sagr.

68 THE TWO NEW SCIENCES OF GALILEOSAGR.Seeingthat Simplicioissilent,I willtaketheopportu-

nityofsayingsomething.Sinceyou'haveclearlydemonstratedthat bodiesofdifferentweightsdonotmoveinoneandthesamemediumwithvelocitiesproportionalto theirweights,but thatthey allmovewith the samespeed,understandingof coursethat theyare of the samesubstanceor at leastof the samespecificgravity;certainlynotofdifferentspecificgravities,forIhardlythink youwouldhaveus believea ballof corkmoves

[II3lwiththe samespeedas oneoflead;andagainsinceyouhaveclearlydemonstratedthat one and the same body movingthroughdifferentlyresistingmediadoesnotacquirespeedswhichare inverselyproportionalto the resistances,I am curioustolearnwhataretheratiosa_uallyobservedin thesecases.

SAT.V.Theseare interestingquestionsand I have thoughtmuchconcerningthem. Iwillgiveyouthemethodofapproachandthe resultwhichI finallyreached.Havingonceestablishedthefalsityofthepropositionthatoneandthesamebodymovingthroughdifferentlyresistingmediaacquiresspeedswhichareinverselyproportionalto the resistancesof thesemedia,andhavingalsodisprovedthe statementthat in the samemediumbodiesofdifferentweightacquirevelocitiesproportionalto theirweights(understandingthat this appliesalsoto bodieswhichdiffermerelyinspecificgravity),I thenbeganto combinethesetwofa_s andto considerwhatwouldhappenifbodiesofdiffer-entweightwereplacedinmediaofdifferentresistances;andIfoundthat the differencesin speedweregreaterin thosemediawhichweremoreresistant,thatis,lessyielding.Thisdifferencewassuchthat twobodieswhichdifferedscarcelyat allin theirspeedthroughair would,inwater,falltheonewitha speedtentimesas greatas that ofthe other. Further,thereare bodieswhichwillfallrapidlyin air,whereasifplacedinwaternotonlywillnotsinkbut willremainat restor willevenriseto the top:for it ispossibleto findsomekindsofwood,suchasknotsandroots,whichremainat restin waterbut fallrapidlyin air.

SAcg.I haveoftentried with the utmostpatienceto addgrainsofsandto a ballofwaxuntilit shouldacquirethe same

specific

FIRST DAY 69specificgravityas water and wouldthereforeremainat rest inthis medium. But with allmy care I was neverable to accom-plish this. Indeed, I do not knowwhetherthere is any solidsubstancewhosespecificgravity is, by nature, so nearlyequalto that ofwater that ifplacedanywhereinwater it willremainat rest.

SALV.In this, as in a thousand other operations,men aresurpassedby animals. In this problemof yoursonemay learnmuch fromthe fishwhich are veryskillfulin maintainingtheirequilibriumnot only in one kind of water, but also in waterswhichare notably differenteither by their own nature or by

[II41someaccidentalmuddinessor through salinity,each of whichproducesa markedchange. So perfectlyindeedcan fishkeeptheir equilibriumthat they are ableto remainmotionlessin anyposition. This they accomplish,I believe, by means of anapparatus especiallyprovided by nature, namely, a bladderlocated in the body and communicatingwith the mouth bymeansof a narrowtube throughwhichthey are able,at will,toexpela portionof the air containedin the bladder:by risingtothe surfacetheycantake inmoreair; thus theymakethemselvesheavieror lighterthan water at will and maintain equilibrium.

SAOl<.By meansofanotherdeviceI was ableto deceivesomeMendsto whomI had boasted that I couldmake up a ball ofwax that wouldbe in equilibriuminwater. In the bottom of avesselI placedsomesaltwater and uponthis somefreshwater;then I showedthem that the ball stoppedin the middleof thewater, and that, when pushed to the bottom or lifted to thetop, wouldnot remainineitherof theseplacesbut wouldreturnto the middle.

SALV.This experimentis not withoutusefulness. For whenphysiciansare testing the variousqualitiesof waters,especiallytheir specificgravities,they employa ball of this kind so ad-justed that, in certainwater, it willneitherrise nor fall. Thenin testing another water, differingever so slightlyin specificgravity [peso],the ballwill sink ifthis waterbe lighterand riseif it be heavier. And soexact is this experimentthat the addi-

tion

70 THE NEW TWO SCIENCESOF GALILEOtionoftwograinsof saltto sixpoundsofwaterissufficienttomaketheballriseto thesurfacefromthebottomtowhichit hadfallen.Toillustratetheprecisionofthisexperimentandalsotoclearlydemonstratethe non-resistanceof water to division,I wishto addthat thisnotabledifferencein specificgravitycanbe producednot onlyby solutionof someheaviersubstance,but alsobymerelyheatingor cooling;andsosensitiveis waterto thisprocessthat bysimplyaddingfourdropsofanotherwaterwhichis slightlywarmeror coolerthanthe sixpoundsonecancausetheballto sinkorrise;itwillsinkwhenthewarmwaterispouredinandwillriseupontheadditionofcoldwater. Nowyou

[II5]canseehowmistakenarethosephilosopherswhoascribeto waterviscosityorsomeothercoherenceofpartswhichoffersresistanceto separationofpartsandto penetration.

SAoR.Withregardto thisquestionI havefoundmanycon-vincingargumentsinatreatisebyourAcademician;butthereisonegreatdifficultyofwhichI havenotbeenableto ridmyself,namely,iftherebenotenacityorcoherencebetweentheparticlesofwaterhowisit possibleforthoselargedropsofwaterto standoutinreliefuponcabbageleaveswithoutscatteringorspreadingout?

SALV.Althoughthosewhoarein possessionofthe truthareableto solveallobje_ionsraised,I wouldnotarrogateto myselfsuchpower;neverthelessmyinabilityshouldnotbe allowedtobecloudthe truth. To beginwithletme confessthat I donotunderstandhowtheselargeglobulesofwaterstandoutandholdthemselvesup, althoughI knowfora certainty,that it is notowingto any internaltenacityactingbetweenthe particlesofwater;whenceit must followthat the causeof this effe_ isexternal.Besidetheexperimentsalreadyshownto provethatthe causeis not internal,I canofferanotherwhichisverycon-vincing.If theparticlesofwaterwhichsustainthemselvesinaheap,whilesurroundedby air, didso in virtueof an internalcausethentheywouldsustainthemselvesmuchmoreeasilywhensurroundedbya mediumin whichtheyexhibitlesstendencytofall than they do in air; sucha mediumwouldbe any fluid

heavier&

FIRST DAY 71heavierthanair, as, for instance,wine:and thereforeif somewinebepouredaboutsucha dropofwater,thewinemightriseuntil the dropwasentirelycovered,withoutthe particlesofwater,heldtogetherby this internalcoherence,everpartingcompany. But this is not the fa_; foras soonas the winetouchesthe water,the latter withoutwaitingto be coveredscattersandspreadsoutunderneaththe wineifit be red. Thecauseof thiseffed'tis thereforeexternaland is possiblyto befoundin the surroundingair. Indeedthereappearsto be aconsiderableantagonismbetweenair andwateras I haveob-servedin thefollowingexperiment.Havingtakena glassglobewhichhada mouthof aboutthe samediameteras a straw,Ifilledit withwaterandturnedit mouthdownwards;neverthe-

[II6lless,thewater,althoughquiteheavyandproneto descend,andthe air, whichis very lightand disposedto risethroughthewater,refused,the one to descendand the other to ascendthroughtheopening,but both remainedstubbornanddefiant.On the otherhand,as soonas I applyto thisopeninga glassof redwine,whichis almostinappreciablylighterthanwater,redstreaksareimmediatelyobservedto ascendslowlythroughthewaterwhilethewaterwithequalslownessdescendsthroughthe winewithoutmixing,untilfinallythe globeis completelyfilledwithwineandthewaterhasallgonedownintothevesselbelow.Whatthencanwesayexceptthat thereexists,betweenwaterandair,a certainincompatibilitywhichI donotunder-stand,but perhaps....

S_P. I feelalmostlikelaughingat thegreatantipathywhichSalviatiexhibitsagainsttheuseof thewordantipathy;andyetit isexcellentlyadaptedto explainthe difficulty.

8_v. Alright,if it pleaseSimplido,let thiswordantipathybethesolutionofourdifficulty.Returningfromthisdigressmn,let usagaintakeupourproblem.Wehavealreadyseenthatthe differenceof speedbetweenbodiesof differentspecificgravitiesis mostmarkedin thosemediawhicharethe mostresistant:thus, in a mediumof quicksilver,goldnotmerelysinksto the bottommorerapidlythanleadbutit is the only

substance

72 THE TWO NEW SCIENCESOF GALILEOsubstancethat willdescendatall;allothermetalsandstonesriseto the surfaceandfloat. On the otherhandthe variationofspeedin airbetweenballsof gold,lead,copper,porphyry,andotherheavymaterialsis soslightthat in a fallof IOOcubitsaballofgoldwouldsurelynotoutstriponeofcopperby asmuchasfourfingers.HavingobservedthisI cameto the conclusionthat in a mediumtotallydevoidofresistanceallbodieswouldfallwiththesamespeed.

Smm.Thisis a remarkablestatement,Salvlati.But I shallneverbelievethat evenin a vacuum,ifmotionin sucha placewerepossible,a lockofwoolanda bit ofleadcanfallwith thesamevelocity.

SAT.v.Alittlemoreslowly,Simplicio.Yourdifficultyis notso reconditenor am I so imprudentas to warrant you inbelievingthat I havenot alreadyconsideredthismatterandfoundthe propersolution. Hencefor my justificationand

[117]foryourenlightenmenthearwhatI haveto say. Ourproblemisto findoutwhathappensto bodiesofdifferentweightmovingin a mediumdevoidofresistance,so that theonlydifferenceinspeedisthat whicharisesfrominequalityofweight. Sincenomediumexceptoneentirelyfreefromairandotherbodies,be iteverso tenuousandyielding,can furnishour senseswith theevidencewe are lookingfor,and sincesucha mediumis notavailable,we shallobservewhat happensin the rarest andleastresistantmediaas comparedwithwhathappensin denserandmoreresistantmedia.Becauseifwefindasa facetthat thevariationofspeedamongbodiesofdifferentspecificgravitiesislessandlessaccordingasthe mediumbecomesmoreandmoreyielding,andif finallyin a mediumof extremetenuity,thoughnota perfectvacuum,wefindthat, inspiteofgreatdiversityofspecificgravity[peso],the differencein speedisverysmallandalmostinappreciable,thenwearejustifiedin believingit highlyprobablethat in a vacuumallbodieswouldfallwith the samespeed. Letus, inviewofthis,considerwhattakesplacein air,whereforthesakeofa definitefigureandlightmaterialimagineaninflatedbladder.Theairinthisbladderwhensurroundedby

air

FIRST DAY 73airwillweighlittleornothing,sinceit canbeonlyslightlycom-pressed;its weightthenis smallbeingmerelythat of the skinwhichdoesnot amountto the thousandthpart of a massofleadhavingthe samesizeas the inflatedbladder.Now,Sim-plicio,ifweallowthesetwobodiestofallfroma heightoffouror sixcubits,by what distancedoyou imaginethe lead willanticipatethe bladder?Youmaybe surethat theleadwillnottravel three times,or eventwice,as swiftlyas the bladder,althoughyou wouldhavemadeit movea thousandtimesasrapidly.

Sire,.It maybe asyousayduringthefirstfourorsixcubitsofthe fall;but afterthe motionhascontinueda longwhile,Ibelievethat the leadwillhaveleftthe bladderbehindnotonlysix out of twelveparts of the distancebut eveneightorten.

SAzv.I quiteagreewith you and doubtnot that,in verylongdistances,theleadmightcoveronehundredmileswhilethe

bladderwastraversingone;but,mydearSimpliclo,thisphenom-enonwhichyouadduceagainstmypropositionis preciselytheone whichconfirmsit. Let me oncemoreexplainthat thevariationofspeedobservedinbodiesofdifferentspecificgravi-ties is not causedby the differenceof specificgravitybut de-pendsuponexternalcircumstancesand,in particular,upontheresistanceof the medium,sothat if this is removedallbodieswouldfallwith the samevelocity;and this resultI deducemainlyfromthefacetwhichyouhavejustadmittedandwhichisvery true, namely,that, in the caseof bodieswhichdifferwidelyin weight,theirvelocitiesdiffermoreandmoreas thespacestraversedincrease,somethingwhichwouldnotoccurifthe effe_ dependedupondifferencesof specificgravity. Forsincethesespecificgravitiesremainconstant,the ratiobetweenthe distancestraversedoughtto remainconstantwhereasthefacetis that this ratiokeepson increasingas the motioncon-tinues. Thusa veryheavybodyin a fallofonecubitwillnotanticipatea verylightonebysomuchasthetenthpartofthisspace;but in a falloftwelvecubitstheheavybodywouldout-

strip

74 THE TWO NEW SCIENCESOF GALILEOstriptheotherbyone-third,andin.afallofonehundredcubitsby9o/Ioo,etc.

Sn_. Verywell:but,followingyourownlineofargument,ifdifferencesofweightin bodiesofdifferentspecificgravitiescannotproducea changein theratiooftheirspeeds,on thegroundthat theirspecificgravitiesdo not change,howis itpossibleforthemedium,whichalsowesupposeto remaincon-stant,tobringaboutanychangeintheratioofthesevelocities?

S_v. Thisobjedtionwithwhichyouopposemystatementisclever;andI mustmeetit. I beginbysayingthata heavybodyhasaninherenttendencyto movewitha constantlyanduniformlyacceleratedmotiontowardthe commoncenterofgravity,thatis,towardthecenterofourearth,sothatduringequalintervalsoftimeitreceivesequalincrementsofmomentumandvelocity.This,youmustunderstand,holdswheneverallexternalandaccidentalhindranceshavebeenremoved;butofthesethereis onewhichwecanneverremove,namely,themediumwhichmustbe penetratedandthrustasideby thefallingbody.Thisquiet,yielding,fluidmediumopposesmotion

[II9]throughitwitharesistancewhichisproportionaltotherapiditywithwhichthemediummustgivewayto thepassageof thebody;whichbody,as I havesaid,isby naturecontinuouslyacceleratedsothat itmeetswithmoreandmoreresistanceinthemediumandhencea diminutionin itsrateofgainofspeeduntilfinallythespeedreachessucha pointandtheresistanceofthemediumbecomessogreatthat,balancingeachother,theypreventanyfurtheraccelerationandreducethemotionofthebodytoonewhichisuniformandwhichwillthereaftermaintaina constantvalue.Thereis,therefore,anincreaseintheresist-anceofthemedium,notonaccountofanychangeinitsessentialproperties,butonaccountofthechangeinrapiditywithwhichitmustyieldandgivewaylaterallytothepassageofthefallingbodywhichisbeingconstantlyaccelerated.

Nowseeinghowgreatistheresistancewhichtheairofferstotheslightmomentum[momento]of thebladderandhowsmallthatwhichit offersto the largeweight[peso]of the lead,I

am

FIRST DAY 75am convincedthat, if the mediumwereentirelyremoved,theadvantagereceivedby the bladderwouldbe sogreatandthatcomingto theleadsosmallthat theirspeedswouldbeequalized.Assumingthis principle,that all fallingbodiesacquireequalspeedsina mediumwhich,onaccountofavacuumorsomethingelse,offersnoresistanceto the speedofthe motion,weshallbeableaccordinglyto determinethe ratiosof the speedsof bothsimilaranddissimilarbodiesmovingeitherthroughoneandthesamemediumor throughdifferentspace-filling,andthereforeresistant,media. Thisresultwemayobtainbyobservinghowmuchtheweightofthemedilmldetractsfromtheweightofthemovingbody,whichweightisthemeansemployedbythefallingbodytoopena pathforitselfandto pushasidethepartsofthemedium,somethingwhichdoesnothappenina vacuumwhere,therefore,no difference[ofspeed]is to be expectedfromadifferenceof specificgravity. Andsinceit is knownthat theeffecCtofthemediumistodiminishtheweightof.thebodybytheweightofthemediumdisplaced,wemayaccomplishourpurposebydiminishingin just thisproportionthe speedsof the fallingbodies,whichina non-resistingmediumwehaveassumedto beequal.

Thus,forexample,imagineleadto beten thousandtimesasheavyas air whileebonyisonlyonethousandtimesasheavy.

[12o]Herewehavetwosubstanceswhosespeedsoffallin a mediumdevoidofresistanceareequal:but,whenairis themedium,itwillsubtra_fromthe speedoftheleadonepartintenthousand,and fromthe speedofthe ebonyonepart inonethousand,i.e.ten parts in ten thousand. Whilethereforeleadand ebonywouldfall fromanygivenheightin the sameintervalof time,providedthe retardingeffe_of the airwereremoved,the leadwill,inair,loseinspeedonepartintenthousand;andtheebony,tenpartsintenthousand.Inotherwords,iftheelevationfromwhichthe bodiesstartbe dividedintoten thousandparts,theleadwillreachthe groundleavingtheebonybehindby asmuchasten,orat leastnine,oftheseparts. Is it notclearthenthat aleadenball allowedto fall froma towertwo hundredcubits

high

76 THE TWO NEW SCIENCES OF GALILEOhighwilloutstrip'anebonyballby,lessthanfourinches? Nowebonyweighsa thousandtimesasmuchasair but thisinflatedbladderonlyfourtimesas much;thereforeair diminishestheinherentandnaturalspeedofebonybyonepart in athousand;whilethat of the bladderwhich,if freefromhindrance,wouldbe the same,experiencesa diminutionin air amountingto onepart in four. So that whenthe ebonyball,fallingfrom thetower,has reachedthe earth, the bladderwillhave traversedonlythree-quartersof this distance.Leadis twelvetimesasheavyaswater;but ivoryisonlytwiceasheavy. Thespeedsofthesetwosubstanceswhich,whenentirelyunhindered,areequalwillbe diminishedin water,that of leadbyonepart in twelve,that of ivoryby half. Accordinglywhenthe lead has fallenthroughelevencubitsofwatertheivorywillhavefallenthroughonlysix. Employingthisprincipleweshall,I believe,findamuchcloseragreementofexperimentwithourcomputationthanwiththat ofAristotle.

In a similarmannerwemayfindtheratioofthespeedsofoneandthesamebodyindifferentfluidmedia,notbycomparingthedifferentresistancesofthemedia,but byconsideringthe excessof the specificgravityof the bodyabovethoseof the media.Thus, forexample,fin is onethousandtimesheavierthan airandten timesheavierthanwater;hence,if wedivideits un-hinderedspeedinto IOOOparts,air willrob it of oneof thesepartsso that it willfallwitha speedof999,whilein wateritsspeedwillbe9oo,seeingthat waterdiminishesitsweightbyonepartintenwhileairbyonlyonepartin athousand.

Againtakea solida littleheavierthanwater,suchasoak,aball ofwhichwillweighlet us say IOOOdrachms;supposean

[i2i]equalvolumeofwaterto weigh95o,andanequalvolumeofair,2;thenit isclearthat if theunhinderedspeedoftheballisIooo,its speedin airwillbe998,but inwateronly5o,seeingthat thewaterremoves950of the IOOOparts whichthe bodyweighs,leavingonly50.

Sucha solidwouldthereforemovealmosttwentytimesasfast in air asin water,sinceits specificgravityexceedsthat of

water

FIRST DAY 77waterby onepart in twenty. Andherewemustconsiderthefa_ that onlythosesubstanceswhichhavea specificgravitygreaterthanwatercanfallthroughit--substanceswhichmust,therefore,behundredsoftimesheavierthanair;hencewhenwetry to obtainthe ratioofthe speedin air to that in water,wemay,withoutappreciableerror,assumethatairdoesnot,to anyconsiderableextent,diminishthe freeweight[assolutagravita],andconsequentlytheunhinderedspeed[assolutavelocitY]ofsuchsubstances.Havingthuseasilyfoundtheexcessoftheweightofthesesubstancesoverthat ofwater,wecansaythat theirspeedinairisto theirspeedinwaterastheirfreeweight[totalegravitY]isto the excessofthisweightoverthat ofwater. Forexample,a ball of ivoryweighs20 ounces;an equalvolumeof waterweighs17ounces;hencethe speedof ivoryin air bearsto itsspeedinwatertheapproximateratioof20:3.

SAGR.I havemadeagreatstepforwardinthistrulyinterest-ingsubjecCtuponwhichI have longlaboredin vain. In orderto put these theoriesinto practicewe needonlydiscoveramethodof determiningthe specificgravityofair withreferenceto waterandhencewith referenceto otherheavysubstances.

Sr_P.But if we find that air has levityinsteadof gravitywhat then shallwe sayof the foregoingdiscussionwhich,inotherrespecCts,isveryclever?

SAT.v.I shouldsay that it was empty,vain,and trifling.Butcanyoudoubtthat airhasweightwhenyouhavethecleartestimonyofAristotleaffirmingthatalltheelementshaveweightincludingair, and exceptingonlyfire? As evidenceof thishecitesthe fa_ that a leatherbottleweighsmorewheninflatedthan whencollapsed.

Snvn,.I am inclinedto believethat the increaseof weightobservedintheinflatedleatherbottleorbladderarises,notfromthe gravityofthe air,but fromthemanythickvaporsmingledwith it in theselowerregions.To this I wouldattributetheincreaseofweightintheleatherbottle.

S_mv.I wouldnot haveyousaythis,andmuchlessattributeit to Aristotle;because,ifspeakingoftheelements,hewishedto

persuade

78 THE TWO NEW SCIENCES OF GALILEOpersuademeby experimentthat air hasweightandwereto sayto me:"Takea leatherbottle,filliiwithheavyvaporsandob-servehowits weightincreases,"I wouldreplythat the bottlewouldweighstillmoreif filledwithbran;andwouldthenaddthat thismerelyprovesthat branandthickvaporsareheavy,but in regardto air I shouldstillremainin the samedoubtasbefore.However,the experimentofAristotleis goodandthepropositionistrue. ButI cannotsayasmuchofacertainotherconsideration,takenat facevalue;this considerationwasof-feredbyaphilosopherwhosenameslipsme;but I knowI havereadhisargumentwhichisthat airexhibitsgreatergravitythanlevity,becauseit carriesheavybodiesdownwardmoreeasilythanit doeslightonesupward.

SAcR.Fineindeed! Soaccordingto thistheoryair ismuchheavierthanwater,sinceallheavybodiesarecarrieddownwardmoreeasilythroughair thanthroughwater,andalllightbodiesbuoyedup moreeasilythroughwaterthanthroughair;furtherthereis an infinitenumberof heavybodieswhichfallthroughair but ascendinwaterandthereis an infinitenumberof sub-stanceswhichrisein waterandfallin air. But,Simplicio,thequestionasto whetherthe weightofthe leatherbottleis owingto thickvaporsor to pureair doesnotaffecCtourproblemwhichisto discoverhowbodiesmovethroughthisvapor-ladenatmos-phereofours. Returningnowto thequestionwhichinterestsmemore,I shouldlike,forthesakeofmorecompleteandthoroughknowledgeof thismatter,notonly to be strengthenedin mybeliefthat airhasweightbut alsoto learn,ifpossible,howgreat

.itsspecificgravityis. Therefore,Salviati,ifyoucansatisfymycuriosityonthispointpraydoso.

SALv.The experimentwith the inflatedleatherbottle ofAristotleprovesconclusivelythat air possessespositivegravityandnot, as somehavebelieved,levity,a propertypossessedpossiblyby nosubstancewhatever;for if air didpossessthisqualityofabsoluteandpositivelevity,it shouldoncompression

[I 31exhibitgreaterlevityand,hence,a greatertendencyto rise;butexperimentshowspreciselytheopposite.

As

FIRSTDAY 79As to the otherquestion,namely,howto determinethe

specificgravityofair,I haveemployedthefollowingmethod.I tooka ratherlargeglassbottlewitha narrowneckandat-tachedtoit aleathercover,bindingit tightlyabouttheneckofthebottle:inthetopofthiscoverI insertedandfirmlyfastenedthevalveofa leatherbottle,throughwhichI forcedintotheglassbottle,by meansof a syringe,a largequantityof air.Andsinceair iseasilycondensedonecanpumpintothebottletwoorthreetimesitsownvolumeofair. AfterthisI tookanaccuratebalanceandweighedthisbottleofcompressedairwiththeutmostprecision,adjustingtheweightwithfinesand. Inextopenedthevalveandallowedthecompressedairtoescape;thenreplacedtheflaskuponthe balanceandfoundit per-ceptiblylighter:fromthesandwhichhadbeenusedasa counter-weightI nowremovedandlaidasideasmuchaswasnecessaryto againsecurebalance.Undertheseconditionstherecanbenodoubtbutthattheweightofthesandthuslaidasiderepresentstheweightoftheairwhichhadbeenforcedintotheflaskandhadafterwardsescaped.But afterall this experimenttellsmemerelythattheweightofthecompressedairisthesameasthatofthesand'removedfromthebalance;whenhoweveritcomestoknowingcertainlyanddefinitelytheweightofairascomparedwiththatofwateroranyotherheavysubstancethisI cannothopeto do withoutfirstmeasuringthevolume[quantitY]ofcompressedair;forthismeasurementI havedevisedthe twofollowingmethods.

Accordingtothefirstmethodonetakesabottlewithanarrownecksimilartothepreviousone;overthemouthofthisbottleisslippeda leathertubewhichisboundtightlyabouttheneckoftheflask;theotherendofthistubeembracesthevalveattachedto thefirstflaskandis tightlyboundaboutit. Thissecondflaskis providedwitha holeinthebottomthroughwhichanironrodcanbeplacedsoasto open,at will,thevalveabovementionedandthuspermitthesurplusairofthefirsttoescapeafterit hasoncebeenweighed:buthissecondbottlemustbefilledwithwater.Havingpreparedeverythingin themanner

[I241above

8o THE TWO NEW SCIENCES OF GALILEOabovedescribed,openthe valvewiththe rod; the air willrushintothe flaskcontainingthewaterandwilldriveit throughtheholeat thebottom,it beingclearthat the volume[quanti_]ofwaterthusdisplacedis equalto the volume[moleequanti_]ofair escapedfromthe othervessel. Havingset asidethis dis-placedwater,weighthe vesselfromwhichthe air hasescaped(whichis supposedto have beenweighedpreviouslywhilecontainingthe compressedair),andremovethe surplusofsandas describedabove;it is thenmanifestthat the weightof thissandispreciselythe weightofavolume[mole]ofair equalto thevolumeofwaterdisplacedandsetaside;thiswaterwecanweighandfindhowmanytimesits weightcontainsthe weightof theremovedsand, thus determiningdefinitelyhowmany timesheavierwater is than air; andweshallfind,contraryto theopinionofAristotle,that thisisnotIOtimes,but,asourexperi-mentshows,morenearly40otimes.

The secondmethodis moreexpeditiousandcanbe carriedout witha singlevesselfittedup asthe firstwas. Herenoairis addedto that whichthevesselnaturallycontainsbut waterisforcedintoit withoutallowinganyair to escape;thewaterthusintroducednecessarilycompressesthe air. Havingforcedintothevesselasmuchwateraspossible,fillingit,say,three-fourthsMll,whichdoesnot requireanyextraordinaryeffort,placeituponthe balanceandweighit accurately;nextholdthe vesselmouthup, openthe valve,and allowthe air to escape;thevolumeof theair thusescapingispreciselyequalto the volumeof watercontainedin the flask. Againweighthevesselwhichwillhavediminishedin weighton accountofthe escapedair;this lossinweightrepresentstheweightofavolumeofairequalto thevolumeofwatercontainedinthevessel.

Sm_,.Noonecandenythe clevernessandingenuityof yourdevices;but whilethey appearto givecompleteintellecCtualsatisfacCtiontheyconfusemein anotherdirecCtion.Forsinceit isundoubtedlytruethat theelementswhenintheirproperplaceshaveneitherweightnor levity,I cannotunderstandhowit ispossiblefor that portionofair, whichappearedto weigh,say,4 drachmsofsand,shouldreallyhavesucha weightin airasthe

sand

FIRST DAY 81sandwhichcounterbalancesit. It seemsto me,therefore,thattheexperimentshouldbecarriedout,notinair,butinamedium

[I25]in whichthe air couldexhibititspropertyofweightif suchitreallyhas.

SALV.TheobjectionofSimplicioiscertainlyto thepointandmustthereforeeitherbe unanswerableor demandan equallyclearsolution.It isperfectlyevidentthatthatairwhich,undercompression,weighedasmuchas the sand,losesthisweightwhenonceallowedto escapeintoitsownelement,while,indeed,the sandretainsits weight. Hencefor this experimentit be-comesnecessaryto selecta placewhereairaswellassandcangravitate;because,as hasbeenoftenremarked,themediumdiminishestheweightof anysubstanceimmersedin it by anamountequalto theweightof the displacedmedium;so thatairinairlosesallitsweight. Ifthereforethisexperimentistobemadewithaccuracyit shouldbeperformedina vacuumwhereeveryheavybodyexhibitsits momentumwithouttheslightestdiminution.If then,Simplicio,wewereto weigha portionofairina vacuumwouldyouthenbesatisfiedandassuredofthefact?

Stop.Yes truly:but this is to wishor ask the impossible.Sm_v.Your obligationwillthen be very great if, foryour

sake,I accomplishtheimpossible.ButIdonotwantto sellyousomethingwhichI havealreadygivenyou;for in the previousexperimentweweighedtheairin vacuumandnotinairorothermedium. The fact that any fluidmediumdiminishestheweightof amassimmersedin it, is due,Simplicio,to theresist-ancewhichthis mediumoffersto its beingopenedup,drivenaside,andfinallyliftedup. Theevidenceforthis isseenin thereadinesswith whichthe fluidrushesto fillup any spacefor-merlyoccupiedbythe mass;if themediumwerenot affectedbysuchan immersionthenitwouldnotreactagainsttheimmersedbody. Tellmenow,whenyouhaveaflask,in air,filledwithitsnaturalamountof air andthenproceedto pumpintothevesselmoreair,doesthisextrachargein anywayseparateor divideorchangethecircumambientair? Doesthevesselperhapsexpand

SO

82 THE TWO NEW SCIENCESOF GALILEOso that the surroundingmediumi_ displacedin orderto givemoreroom?Certainlynot. Thereforeoneis ableto say that

[126]this extrachargeof air is not immersedin the surroundingmediumfor it occupiesno spacein it, but is,as it were,in avacuum. Indeed,itis reallyinavacuum;forit diffusesintothevacuitieswhicharenot completelyfilledby the originalanduncondensedair. In facetI donot seeanydifferencebetweenthe enclosedand the surroundingmedia:for the surroundingmediumdoesnotpressupontheenclosedmediumand,viceversa,theenclosedmediumexertsnopressureagainstthesurroundingone;thissamerelationshipexistsin thecaseofanymatterin avacuum,as wellas in the caseof the extrachargeof aircom-pressedinto the flask. The weightof this condensedair isthereforethe sameas that whichit wouldhaveif setfreein avacuum. It istrueofcoursethattheweightofthesandusedasa counterpoisewouldbe alittlegreaterin vacuothaninfreeair.Wemust,then,saythatthe airis slightlylighterthanthe sandrequiredtocounterbalanceit, thatisto say,by anamountequaltotheweightin vacuoofa volumeofairequalto thevolumeofthesand.

Atthis pointinanannotatedcopyoftheoriginaleditionthefollowingnotebyGalileois found.

[SAcraAverycleverdiscussion,solvinga wonderfulproblem,because -it demonstratesbrieflyand conciselythe manner in whichone mayfind the weightof a body in vacuoby simplyweighingit in air. Theexplanationisas follows:whenaheavybodyis immersedinairit losesinweightan amountequalto theweightofavolume[mole]ofairequivalentto the volume[mole]of the body itself. Hence if oneadds to a body,withoutexpandingit, a quantity of air equalto that whichit displacesand weighsit, hewillobtainits absoluteweightin vacuo,since,withoutincreasingit insize,hehas increaseditsweightby just theamountwhichit lostthroughimmersionin air.

Whenthereforewe forcea quantityof waterinto a vesselwhichal-readycontainsits normalamountof air, withoutallowingany of thisair to escapeit isclearthat thisnormalquantityofairwillbecompressedand condensedintoa smallerspacein orderto make roomfor the waterwhichis forcedin: it is also clear that the volumeof air thus com-pressedis equalto the volumeof wateradded. If now the vesselbe

weighed

FIRST DAY 83weighed{n air in this condition,it is manifestthat the weightof thewaterwillbeincreasedbythat ofanequalvolumeofair; the total weightof waterand air thus obtainedisequalto the weightofthe wateralone_n _(ZCUO.

Now recordthe weightof the entire vesselandthen allowthe com-pressedair to escape;weighthe remainder;the differenceof thesetwoweightswillbe the weightof the compressedair which, in volume,isequalto that ofthe water. Next findthe weightof thewateraloneandaddto it that of the compressedair; weshallthen have the wateralonein vacuo. To find the weightof the waterwe shallhave to removeitfromthe vesseland weighthe vesselalone; subtra&this weightfromthat ofthe vesselandwatertogether. It isclearthat theremainderwillbetheweightof thewateraloneinair.]

[I27]SzMv.The previousexperiments,in my opinion,left some_

thingtobedesired:butnowI amfullysatisfied.SAT.v.The facts set forth by me up to this pohntand, in

' particular,the onewhichshowsthat differenceofweight,evenwhenverygreat,iswithouteffeCtinchangingthespeedoffallingbodies,so that as far asweightis concernedtheyallfallwithequalspeed:this ideais, I say, sonew,andat firstglancesoremotefromfaCt,that ifwedonothavethemeansofmakingitjust as clearas sunlight,it hadbetternot be mentioned;buthavingoncealloweditto passmy lipsI mustnegleCtnoexperi-mentorargumentto establishit.

SAGR.Not onlythisbut alsomanyotherof yourviewsareso far removedfrom the commonlyacceptedopinionsanddoCtrinesthat if youwereto publishthemyouwouldstirupa largenumberof antagonists;forhumannatureis suchthatmendonot lookwithfavorupondiscoveries--eitheroftruthorfallacy--intheir ownfield,whenmadeby othersthan them-selves. Theycallhiman innovatorof doCtrine,an unpleasanttitle,bywhichtheyhopeto cutthoseknotswhichtheycannotuntie,and by subterraneanminesthey seekto destroystruc-tureswhichpatientartisanshavebuiltwith customarytools.[Iz8]But as for ourselveswhohaveno suchthoughts,the experi-mentsand argumentswhichyou have thus far adducedarefullysatisfaCtory;howeverifyouhaveanyexperimentswhichare

84 THE TWO NEW SCIENCES OF GALILEOaremoredirector any argumentswhichare moreconvincingwewillhearthemwithpleasure.

SALV.Theexperimentmadeto ascertainwhethertwobodies,differinggreatlyin weightwillfallfromagivenheightwiththesamespeedofferssomedifficulty;because,if the heightis con-siderable,the retardingeffecCtof the medium,whichmustbepenetratedandthrustasidebythe fallingbody,willbe greaterin the caseof the smallmomentumofthe verylightbodythanin the caseof the greatforce[violenza]of the heavybody;sothat,in a longdistance,thelightbodywillbe leftbehind;if theheightbe small,one maywelldoubt whetherthere is anydifference;andif therebe a differenceit willbe inappreciable.

It occurredto me thereforeto repeatmanytimesthe fallthrougha smallheightin sucha waythat I mightaccumulateall thosesmallintervalsoftimethat elapsebetweenthe arrivalof the heavy and lightbodiesrespecCtivelyat their commonterminus,so that this summakesan intervalof timewhichisnotonlyobservable,but easilyobservable.In orderto employtheslowestspeedspossibleandthusreducethechangewhichtheresistingmediumproducesuponthe simpleeffecCtofgravityitoccurredto meto allowthebodiesto fallalonga planeslightlyinclinedto thehorizontal.Forinsuchaplane,justaswellasinaverticalplane,onemaydiscoverhowbodiesofdifferentweightbehave:and besidesthis, I alsowishedto rid myselfof theresistancewhichmightarisefromcontactof the movingbodywith the aforesaidinclinedplane. AccordinglyI took twoballs,oneofleadandoneofcork,the formermorethana hun-dredtimesheavierthanthelatter,andsuspendedthembymeansoftwoequalfinethreads,eachfourorfivecubitslong. Pullingeachballasidefromtheperpendicular,I letthemgoat thesameinstant,and they, fallingalongthe circumferencesof circleshavingtheseequalstringsforseml-diameters,passedbeyondtheperpendicularand returnedalongthe samepath. This freevibration[perlorrnedesimele and.atee le tornate]repeatedahundredtimesshowedclearlythat theheavybodymaintainsso[Iz9]nearlythe periodof the lightbodythat neitherin a hundred

swings

FIRST DAY 85swingsnorevenin a thousandwillthe formeranticipatethelatter by as muchas a singlemoment[minirnornomento],soperfec°dydotheykeepstep. Wecanalsoobservethe etTecCtofthemediumwhich,bythe resistancewhichit offersto motion,diminishesthevibrationofthecorkmorethanthat ofthe lead,butwithoutalteringthe frequencyofeither;evenwhenthe arctraversedbythecorkdidnotexceedfiveorsixdegreeswhilethatofthe leadwasfiftyor sixty,theswingswereperformedinequaltimes.

S_rP.If this be so,whyis notthe speedofthe leadgreaterthanthat of thecork,seeingthat theformertraversessixtyde-greesin thesameintervalinwhichthelattercoversscarcelysix?

S_mv.But whatwouldyou say, Simplicio,if both coveredtheirpathsinthesametimewhenthecork,drawnasidethroughthirty degrees,traversesan arc of sixty,whilethe leadpulledasideonlytwo degreestraversesan arc of four? Wouldnotthenthe corkbe proportionatelyswifter?Andyet suchis theexperimentalfa_. But observethis:havingpulledasidethependulumoflead,saythroughanarcoffiftydegrees,andsetitfree, it swingsbeyondthe perpendicularalmostfiftydegrees,thus describingan arc of nearlyonehundreddegrees;on thereturnswingit describesa littlesmallerarc;andaftera largenumberofsuchvibrationsit finallycomesto rest. Eachvibra-tion, whetherof ninety, fifty, twenty,ten, or four degreesoccupiesthe sametime:accordinglythe speedof the movingbodykeepson diminishingsincein equalintervalsof time,ittraversesarcswhichgrowsmallerandsmaller.

Preciselythesamethingshappenwiththependulumofcork,suspendedby a stringof equallength,exceptthat a smallernumberof vibrationsis requiredto bringit to rest, sinceonaccountofitslightnessit islessableto overcomethe resistanceoftheair;neverthelessthevibrations,whetherlargeorsmall,areallperformedin time-intervalswhicharenotonlyequalamongthemselves,but alsoequalto theperiodof the leadpendulum.Henceit is true that, ifwhilethe leadis traversingan arc offifty degreesthe corkcoversoneof onlyten, the corkmovesmoreslowlythanthelead;but ontheotherhandit isalsotruethat

86 THE TWO NEW SCIENCES OF GALILEO[ 3o]

that thecorkmaycoveranarcoffiftywhiletheleadpassesoveroneofonlyten or six;thus, at differenttimes,wehavenowthecork,nowthe lead,movingmorerapidly.But if thesesamebodiestraverseequalarcsin equaltimeswemayrestassuredthattheirspeedsareequal.

Snvn_.I hesitateto admitthe conclusivenessofthisargumentbecauseof the confusionwhicharisesfromyourmakingbothbodiesmovenowrapidly,nowslowlyand nowvery slowly,whichleavesme in doubtas to whethertheir velocitiesarealwaysequal.

SAOR.Allowme, if you please,Salviati,to say just a fewwords. Nowtell me, Simpliclo,whetheryou admitthat onecansaywithcertaintythat the speedsofthe corkandtheleadareequalwheneverboth,startingfromrestat thesamemomentanddescendingthe sameslopes,alwaystraverseequalspacesin equaltimes._

S_P. Thiscanneitherbedoubtednorgainsaid.SAoR.Nowithappens,inthecaseofthependulums,that each

ofthemtraversesnowanarcofsixtydegrees,nowoneoffifty,orthirtyor ten oreightor fouror two,etc.;andwhentheybothswingthroughan arcofsixtydegreestheydosoin equalinter-valsoftime;thesamethinghappenswhenthearcisfiftydegreesor thirtyortenor anyothernumber;andthereforeweconcludethat thespeedoftheleadinanarcofsixtydegreesisequalto thespeedofthe corkwhenthe latteralsoswingsthroughan arcofsixtydegrees;in the caseof a fifty-degreearc thesespeedsarealsoequal to eachother; so also in the caseof other arcs.But this isnot sayingthat the speedwhichoccursin an arc ofsixty is thesameas that whichoccursin an arcoffifty;nor isthe speedinanarcoffiftyequalto that inoneofthirty,etc.;butthe smallerthearcs,thesmallerthe speeds;thefactobservedisthat oneandthe samemovingbodyrequiresthe sametimefortraversinga largearcof sixtydegreesasfora smallarcoffiftyorevenaverysmallarcoften;allthesearcs,indeed,arecoveredin the sameintervaloftime. It is truethereforethat the lead

[I3I]and

FIRST DAY 87andthe corkeachdiminishtheirspeed[rnoto]in proportionastheirarcsdiminish;but this doesnot contradicCtthe fac°cthattheymaintainequalspeedsinequalarcs.

My reasonforsayingthesethingshasbeenratherbecauseIwantedto learnwhetherI had correcCtlyunderstoodSalviati,thanbecauseI thoughtSimpliciohadanyneedofa clearerex-planationthanthat givenbySalviafiwhichlikeeverythingelseofhis is extremelylucid,so lucid,indeed,that whenhe solvesquestionswhichare di_cult not merelyin appearance,but inrealityand in fact, he doessowith reasons,observationsandexperimentswhicharecommonandfamiliarto everyone.

In thismannerhehas,asI havelearnedfromvarioussources,givenoccasionto a highlyesteemedprofessorforundervaluinghisdiscoverieson the groundthat theyare commonplace,andestablishedupona meanandvulgarbasis;as if it werenot amost admirableand praiseworthyfeature of demonstrativesciencethat it springsfromandgrowsout of principleswell-known,understoodandconcededbyall.

But let us continuewith this lightdiet;and if Simplicioissatisfiedto understandand admit that the gravityinherent[internagravith]in variousfallingbodieshasnothingto dowiththe differenceof speedobservedamongthem,and that allbodies,in sofar as their speedsdependuponit, wouldmovewiththe samevelocity,praytellus, Salviati,howyouexplainthe appreciableand evidentinequalityof motion;pleasereplyalsoto theobjecCtionurgedbySimplicio--anobjecCtioninwhichI concur--namely,that a cannonballfallsmorerapidlythanabird-shot.Frommypointofview,onemightexpecCtthediffer-enceof speedto besmallin thecaseofbodiesofthe samesub-stancemovingthroughany singlemedium,whereasthe largeroneswilldescend,duringa singlepulse-beat,a distancewhichthesmalleroneswillnottraversein anhour,orin four,orevenin twentyhours;as for instancein the caseof stonesandfinesandandespeciallythat veryfinesandwhichproducesmuddywaterandwhichinmanyhourswillnotfallthroughasmuchastwocubits,a distancewhichstonesnotmuchlargerwilltraverse

in a singlepulse-beat. Salv.

88 THE TWO NEW SCIENCES OF GALILEOSALV.The at°donof the mediumin producinga greater

retardationuponthosebodieswhichhavealessspecificgravityhasalreadybeenexplainedby showingthat theyexperienceadiminutionofweight. But to explainhowone andthe same[I3 ]mediumproducessuchdifferentretardationsin bodieswhichare madeof the samematerialandhave the sameshape,butdifferonlyin size,requiresa discussionmorecleverthan thatby whichoneexplainshowa moreexpandedshapeor an op-posingmotionof the mediumretardsthe speedof the movingbody. The solutionofthe presentproblemlies,I think,in theroughnessandporositywhichare generallyandalmostneces-sarilyfoundin thesurfacesofsolidbodies.Whenthebodyis inmotiontheseroughplacesstrikethe air or other ambientme-dium. The evidencefor this is foundin the hummingwhichaccompaniesthe rapidmotionofa bodythroughair,evenwhenthat bodyisasroundaspossible.Onehearsnotonlyhumming,butalsohissingandwhistling,wheneverthereisanyappreciablecavityor elevationuponthe body. We observealsothat aroundsolidbodyrotatingin a latheproducesa currentofair.Butwhatmoredoweneed?Whena topspinsonthegroundatits greatestspeeddo wenot hear a distinctbuzzingof highpitch? Thissibilantnotediminishesin pitchas the speedofrotationslackens,whichis evidencethat thesesmallrugositiesonthesurfacemeetresistanceintheair. Therecanbenodoubt,therefore,that in the motionof fallingbodiestheserugositiesstrikethe surroundingfluidandretardthespeed;andthis theydosomuchthemoreinproportionasthesurfaceislarger,whichisthecaseofsmallbodiesascomparedwithgreater.

Sire,.Stopamomentplease,I amgettingconfused.Foral-thoughIunderstandandadmitthat frictionofthemediumuponthe surfaceof the bodyretardsits motionandthat, if otherthingsare the same,the largersurfacesuffersgreaterretarda-tion,I donotseeonwhatgroundyousaythatthe surfaceofthesmallerbodyis larger. Besidesif,as yousay,thelargersurfacesuffersgreaterretardationthe largersolidshouldmovemoreslowly,whichis notthe fac2. But thisobjedtioncanbe easily

met

FIRST DAY 89met by sayingthat, althoughthe largerbodyhasa largersur-face,it hasalsoagreaterweight,in comparisonwithwhichtheresistanceofthelargersurfaceisnomorethantheresistanceofthe smallsurfacein comparisonwithitssmallerweight;sothatthe speedofthe largersoliddoesnotbecomeless. I thereforeseenoreasonforexpec°dnganydifferenceofspeedsolongasthedrivingweight[gravit_movente]diminishesin the samepropor-

[i33]tionas the retardingpower[facol_ritardante]of the surface.

SAr.v.I shallanswerallyourobje&ionsat once. Youwilladmit,ofcourse,Simplicio,thatifonetakestwoequalbodies,ofthe samematerialandsamefigure,bodieswhichwouldthereforefallwithequalspeeds,andifhediminishestheweightofoneofthem in the sameproportionas its surface(maintainingthesimilarityofshape)hewouldnottherebydiminishthespeedofthisbody.

S_v. Thisinferenceseemsto beinharmonywithyourtheorywhichstates that the weightof a bodyhasnoeffe&in eitheracceleratingorretardingitsmotion.

SALV.I quiteagreewithyou in thisopinionfromwhichitappearsto followthat, if the weightof a bodyis diminishedingreaterproportionthan its surface,themotionis retardedto acertainextent;and this retardationis greaterand greaterinproportionasthe diminutionofweightexceedsthat ofthe sur-face.

Sma,.ThisI admitwithouthesitation.SALV.Nowyoumustknow,Simplicio,that it is notpossible

to diminishthe surfaceofa solidbodyin the sameratioastheweight,and at the sametime maintainsimilarityof figure.For sinceit is clearthat in the caseofa diminishingsolidtheweightgrowslessin proportionto the volume,and sincethevolumealwaysdiminishesmorerapidlythanthe surface,whenthesameshapeismaintained,theweightmustthereforedimin-ishmorerapidlythanthe surface. But geometryteachesusthat, in the caseof similarsolids,the ratioof twovolumesisgreaterthanthe ratioof theirsurfaces;which,forthe sakeofbetterunderstanding,I shallillustratebya particularcase.

Take,

90 THE TWO NEW SCIENCES OF GALILEOTake,forexample,a cubetwoinchesona sideso that each

facehasan areaof foursquareir/chesand the total area,i. e.,thesumofthe sixfaces,amountsto twenty-foursquareinches;nowknaginethiscubeto be sawedthroughthreetimessoastodivideit intoeightsmallercubes,eachoneinchontheside,eachfaceone inchsquare,and the total surfaceof eachcubesixsquareinchesinsteadoftwenty-fouras in the caseofthe larger

[I34]cube. It isevidentthereforethat the surfaceofthelittlecubeisonlyone-fourththat of the larger,namely,the ratioof six totwenty-four;but thevolumeofthe solidcubeitselfisonlyone-eighth;thevolume,andhencealsotheweight,diminishesthere-foremuchmorerapidlythanthesurface.If weagaindividethelittlecubeintoeightothersweshallhave,for the total surfaceofoneof these,oneandone-halfsquareinches,whichis one-sixteenthofthe surfaceofthe originalcube;but ks volumeisonlyone-sixty-fourthpart. Thus,by twodivisions,youseethatthe volumeis diminishedfour timesas muchas the surface.And,if the subdivisionbe continueduntilthe originalsolidbereducedto afinepowder,weshallfindthat theweightofoneofthesesmallestparticleshasdiminishedhundredsandhundredsoftimesasmuchasits surface.AndthiswhichI haveillustratedin the caseof cubesholdsalsoin the caseof allsimilarsolids,wherethe volumesstandinsesquialteralratioto theirsurfaces.Observethenhowmuchgreatertheresistance,arisingfromcon-ta_ ofthe surfaceofthe movingbodywiththe medium,in thecaseof smallbodiesthan in the caseof large;andwhenoneconsidersthat the rugositiesonthe verysmallsurfacesof finedustparticlesareperhapsnosmallerthanthoseonthe surfacesof largersolidswhichhavebeencarefullypolished,hewillseehowknportantit is that the mediumshouldbeveryfluidandoffernoresistancetobeingthrustaside,easilyyieldingto asmallforce. Yousee,therefore,Shnplicio,that I wasnot mAstakenwhen,not longago,I said that the surfaceof a smallsolidiscomparativelygreaterthanthat ofa largeone.

Snw. I amquiteconvinced;and,believeme,if I wereagainbeginningmystudies,I shouldfollowthe adviceof Plato and

start

FIRST DAY 9Istartwithmathematics,asciencewhichproceedsverycautiouslyandadmitsnothingasestablisheduntilithasbeenrigidlydem-onstrated.

SAG_.This discussionhas affordedme greatpleasure;butbeforeproceedingfurtherI shouldliketo heartheexplanationofaphraseofyourswhichisnewto me,namely,thatsimilarsolidsareto eachotherin the sesquialteralratiooftheirsurfaces;foralthoughI haveseenandunderstoodthepropositioninwhichitis demonstratedthat the surfacesof similarsolidsare in the

[I35]duplicateratio of their sidesand also the propositionwhichprovesthat thevolumesarein thetriplicateratiooftheirsides,yet I havenot so muchas heardmentionedthe ratioof thevolumeofasolidto itssurface.

S_v. Youyourselfhavesuggestedthe answerto yourques-tion and haveremovedeverydoubt. For if onequantityisthe cubeof somethingofwhichanotherquantityis the squaredoesitnotfollowthatthecubeisthesesquialteralofthesquare?Surely. Now if the surfacevariesas the squareof its lineardimensionswhilethe volumevariesasthecubeofthesedimen-sionsmaywenot say that the volumestandsin sesquialteralratioto thesurface?

SACR.Quiteso. Andnowalthoughthereare stillsomede-tails, in connectionwith the subjeCtunderdiscussion,con-cemingwhichI mightaskquestionsyet,ifwekeepmakingonedigressionafter another,it willbe longbeforewe reachthemaintopicwhichhasto dowiththevarietyofpropertiesfoundin the resistancewhichsolidbodiesofferto fraCture;and,therefore,ifyouplease,let us returnto the subjeCtwhichweoriginallyproposedto discuss.

SaLV.Verywell;but the questionswhichwehavealreadyconsideredare sonumerousandsovaried,andhavetakenupsomuchtime that thereis not muchof thisdayleftto spenduponourmaintopicwhichaboundsin geometricaldemonstra-tionscallingforcarefulconsideration.MayI, therefore,suggestthat wepostponethe meetinguntilto-morrow,notonlyforthereasonjustmentionedbut alsoinorderthat I maybringwithme

,9z THE TWO NEW SCIENCES OF GALILEOmesomepapersin whichI havesetdowninan orderlywaythetheoremsand propositionsdealingwiththe variousphasesofthissubjec_t,matterswhich,frommemoryalone,I couldnotpresentintheproperorder.

SAog.I fullyconcurin youropinionandall the morewill-inglybecausethiswillleavetime to-dayto take up someofmy difficultieswith the subjeCtwhichwehavejust beendis-cussing.One questionis whetherwe are to considerthe re-sistanceofthe mediumas sufficientto destroythe accelerationof a body of very heavymaterial,very largevolume,and

[I36]sphericalfigure.I saysphericalinorderto selecCtavolumewhichis containedwithina minimumsurfaceandthereforelesssub-jec°cto retardation.

Anotherquestiondealswith the vibrationsof pendulumswhichmay be regardedfrom severalviewpoints;the first iswhetherallvibrations,large,medium,andsmall,areperformedin exactlyandpreciselyequaltimes:anotheristo findtheratioofthe timesofvibrationofpendulumssupportedbythreadsofunequallength.

SALV.Theseareinterestingquestions:but I fearthat here,asin thecaseofallotherfacets,ifwetakeupfordiscussionanyoneofthem,itwillcarryinitswakesomanyotherfactsandcuriousconsequencesthat timewillnotremainto-dayforthe discussionofall.

SAOR.If theseareasfullof interestasthe foregoing,I wouldgladlyspendas manydaysas thereremainhoursbetweennowand nightfall;and I dare say that Simpliciowouldnot beweariedbythesediscussions.

Sn_P.Certainlynot; especiallywhenthe questionspertainto naturalscienceandhavenot beentreatedby otherphilos-ophers.

SAr.V.Nowtakingup thefirstquestion,I canassertwithouthesitationthat there is no sphereso large,or composedofmaterialsodensebut that the resistanceof the medium,al-thoughveryslight,wouldcheckits accelerationandwould,intime reduceits motionto uniformity;a statementwhichis

strongly

FIRST DAY 93stronglysupportedby experiment.For if a fallingbody,astimegoeson,wereto acquirea speedasgreatasyouplease,nosuchspeed,impressedbyexternalforces[motoreesterno],canbesogreatbutthat thebodywillfirstacquireitandthen,owingtothe resistingmedium,loseit. Thus,for instance,if a cannonball,havingfallena distanceoffourcubitsthroughtheair andhavingacquireda speedof,say,ten units[gradz]wereto strikethesurfaceofthewater,andifthe resistanceofthewaterwerenot ableto checkthe momentum[irnpeto]ofthe shot,it wouldeitherincreasein speedormaintaina uniformmotionuntilthebottomwerereached:butsuchisnot theobservedfacet;onthecontrary,the waterwhenonlya fewcubitsdeephindersanddiminishesthe motionin sucha way that the shotdeliversto the bedofthe riveror lakea very slightimpulse.Clearly

[ 371then ifa shortfall throughthe wateris sufficientto depriveacannonballofits speed,thisspeedcannotbe regainedbya fallofevenathousandcubits. Howcouldabodyacquire,in afallofa thousandcubits,thatwhichit losesina falloffour?Butwhatmoreisneeded? Dowenotobservethat theenormousmomen-tum,deliveredto a shotbya cannon,is sodeadenedby passingthrougha fewcubitsofwaterthat theball,sofarfrominjuringthe ship,barelystrikesit? Eventhe air,althoughaveryyield-ingmedium,canalsodiminishthe speedof a fallingbody,asmaybe easilyunderstoodfromsimilarexperiments.For if agunbe fireddownwardsfromthe topof a veryhightowertheshot willmakea smallerimpressionuponthe groundthan ifthe gun had beenfiredfroman elevationof onlyfouror sixcubits;this is clearevidencethat the momentumofthe ball,firedfromthe top of the tower,diminishescontinuallyfromthe instant it leavesthe barreluntil it reachesthe ground.Thereforea fallfromeversogreatan altitudewillnotsufficetogiveto a bodythat momentumwhichit hasoncelostthroughtheresistanceoftheair,nomatterhowitwasoriginallyacquired.In likemanner,thedestructiveeffectproduceduponawallbyashotfiredfroma gunat a distanceoftwentycubitscannotbeduplicatedby the fallof the sameshotfromanyaltitudehow-

ever

94 THE TWO NEW SCIENCES OF GALILEOevergreat. My opinionis, therefore,that underthe circum-stanceswhichoccurin nature,theaccelerationofanybodyfall-ing fromrest reachesan end and that the resistanceof themediumfinallyreducesits speedto a constantvaluewhichisthereaftermaintained.

SAQR.These experimentsare in my opinionmuchto thepurpose;the onlyquestionis whetheran opponentmightnotmakeboldto denythe factinthe caseofbodies[moh]whichareverylargeandheavyorto assertthat a cannonball,fallingfromthe distanceofthemoonorfromtheupperregionsoftheatmos-phere,woulddelivera heavierblowthan if just leavingthemuzzleofthegun.

Sazv.No doubtmanyobje&ionsmaybe raisednot all ofwhichcanberefutedby experiment:howeverin thisparticular

[3s]casethe followingconsiderationmustbe takeninto account,namely,thatit is verylikelythat a heavybodyfallingfromaheightwill,onreachingtheground,haveacquiredjustasmuchmomentumaswasnecessaryto carryit to thatheight;as maybe clearlyseenin thecaseof a ratherheavypendulumwhich,whenpulledasidefiftyor sixtydegreesfromthe vertical,willacquirepreclselythat speedandforcewhichare sufficienttocarryit to anequalelevationsaveonlythatsmallportionwhichit losesthroughffi&iononthe air. In orderto placea cannonballat suchaheightasmightsufficeto giveit just thatmomen-tum whichthe powderimpartedto it on leavingthe gun weneedonlyfireit verticallyupwardsfromthe samegun;andwecanthenobservewhetheronfallingbackit deliversablowequalto that ofthegunfiredatcloserange;in myopinionit wouldbemuchweaker. The resistanceof the air would,therefore,Ithink,prevent the muzzlevelocityfrombeingequalledby anaturalfallfromrestat anyheightwhatsoever.

Wecomenowto the otherquestions,relatingto pendulums,a subjec°cwhichmay appearto many exceedinglyarid, es-peciallyto thosephilosopherswho are continuallyoccupiedwiththe moreprofoundquestionsofnature. Nevertheless,theproblemisonewhichI donot scorn. I am encouragedby the

e_ample

FIRST DAY 95exampleofAristotlewhomI admireespeciallybecausehe didnotfailto discusseverysubjecCtwhichhethoughtinany degreeworthyofconsideration.

Impelledby yourqueriesI maygiveyousomeof myideasconcerningcertainproblemsin music,a splendidsubjecCt,uponwhichso many eminentmenhavewritten:amongthese isAristotlehimselfwhohasdiscussednumerousinterestingacous-ticalquestions.Accordingly,if onthe basisofsomeeasyandtangibleexperiments,I shallexplainsomestrikingphenomenain the domainofsound,I trustmyexplanationswillmeetyourapproval.

SAGR.I shallreceivethemnot onlygratefullybut eagerly.For,althoughI takepleasurein everykindofmusicalinstru-

[x391mentandhavepaidconsiderableattentionto harmony,I haveneverbeenableto fullyunderstandwhysomecombinationsoftonesare morepleasingthanothers,or whycertaincombina-tions not only fail to pleasebut are evenhighlyoffensive.Thenthereisthe oldproblemoftwostretchedstringsinunison;whenoneof themis sounded,the otherbeginsto vibrateandto emit its note; nordo I understandthe differentratiosofharmony[formeddleconsonanze]andsomeotherdetails.

SALv.Letusseewhetherwecannotderivefromthependuluma safisfafftorysolutionofall thesedifficulties.Andfirst,astothe questionwhetheroneand the samependulumreallyper-formsits vibrations,large,medium,and small,all in exac°dythesametime,I shallrelyuponwhatI havealreadyheardfromour Academician.He has clearlyshownthat the time ofdescentis the samealongallchords,whateverthe arcswhichsubtendthem,as wellalongan arc of I8o° (i.e., the wholediameter)as alongoneof IOO0,6o°, IO0,20,_/40,or 4'. It isunderstood,of course,that thesearcs all terminateat thelowestpointofthecircle,whereit touchesthehorizontalplane.

If nowweconsiderdescentalongarcsinsteadoftheirchordsthen,providedthesedonot exceed9o°,experimentshowsthattheyarealltraversedinequaltimes;butthesetimesaregreaterfor the chordthan forthe arc,an effe_whichis all the more

remarkable

96 THE TWO NEW SCIENCES OF GALILEOremarkablebecauseat firstglanceone wouldthink just theoppositeto be true. For sincethe terminalpointsof the twomotionsare the sameand sincethe straightline includedbe-tweenthesetwopointsistheshortestdistancebetweenthem,itwouldseemreasonablethat motionalongthis lineshouldbeexecutedin the shortesttime;but this is not the case,for theshortesttimc and thereforethe most rapidmotion--isthatemployedalongthe arc ofwhichthisstraightlineisthe chord.

Asto thetimesofvibrationofbodiessuspendedbythreadsofdifferentlengths,theybearto eachotherthesameproportionasthe squarerootsof the lengthsof the thread;or onemightsaythelengthsareto eachotherasthesquaresofthe times;sothatifonewishesto makethe vibration-timeofonependulumtwicethat ofanother,hemustmakeits suspensionfourtimesaslong.In likemanner,ifonependulumhasa suspensionninetimesas

[14o]longas another,thissecondpendulumwillexecutethreevibra-tionsduringeachoneofthe first;fromwhichit followsthat thelengthsof the suspendingcordsbearto eachother the [inverse]ratioofthesquaresofthenumberofvibrationsperformedin thesametime.

SAog.Then,if I understandyoucorreCtly,I caneasilymeas-urethe lengthof a stringwhoseupperendis attachedat anyheightwhateverevenif thisendwereinvisibleandI couldseeonlythelowerextremity.Forif I attachto thelowerendofthisstringa ratherheavyweightandgiveit a to-and-fromotion,andifI askafriendto countanumberofits vibrations,whileI,duringthe sametime-interval,countthe numberof vibrationsofapendulumwhichisexactlyonecubitinlength,thenknowingthe numberof vibrationswhicheachpendulummakesin thegivenintervalof time one can determinethe lengthof thestring. Suppose,forexample,that myfriendcounts2ovibra-tionsof the longcordduringthe sametime in whichI count24oofmystringwhichisonecubitinlength;takingthesquaresofthetwonumbers,2oand24o,namely4ooand576oo,then,Isay,the longstringcontains576o0units of suchlengththatmypendulumwillcontain400ofthem;andsincethe lengthof

my

FIRST DAY 97mystringisonecubit,I shalldivide57600by40oandthusob-tain 144. AccordinglyI shallcallthe lengthof the string144cubits.

SALv.NOrwillyoumissit by asmuchasa hand'sbreadth,especiallyifyouobservealargenumberofvibrations.

SAGI_.Yougiveme frequentoccasionto admirethe wealthand profusionof naturewhen,fromsuchcommonand eventrivialphenomena,youderivefaCtswhicharenotonlystrikingandnewbutwhichare oftenfarremovedfromwhatwewouldhaveimagined.ThousandsoftimesI haveobservedvibrationsespeciallyin churcheswherelamps,suspendedby longcords,hadbeeninadvertentlyset intomotion;but the mostwhichIcouldinferfromtheseobservationswasthat the viewofthosewhothinkthat suchvibrationsaremaintainedbythe mediumis highlyimprobable:for,in that case,the airmustneedshaveconsiderablejudgmentandlittleelseto dobutkilltimebypush-ingto andfroapendentweightwithperfectregularity.ButIneverdreamedof learningthat oneandthe samebody,when

[I4I]suspendedfroma stringahundredcubitslongandpulledasidethroughan arcof9°0orevenI°or i/_o,wouldemploythesametimein passingthroughtheleastasthroughthelargestofthesearcs;and, indeed,it still strikesme as somewhatunlikely.NowIamwaitingtohearhowthesesamesimplephenomenacanfurnishsolutionsforthoseacousticalproblems---solutionswhichwillbeat leastpartlysatisfaCtory.

SALV.Firstof allonemustobservethat eachpendulumhasits own time of vibrationso definiteand determinatethatit is not possibleto makeit movewith anyotherperiod[altroperiodo]thanthat whichnaturehasgivenit. Forlet anyonetake in hishandthe cordto whichthe weightis attachedandtry.,asmuchashepleases,to increaseordiminishthefrequency[frequenza]of its vibrations;it willbe time wasted. On theotherhand,onecanconfermotionuponevenaheavypendulumwhichisat restbysimplyblowingagainstit;by repeatingtheseblastswitha frequencywhichis thesameasthat of thependu-lumonecanimpartconsiderablemotion. Supposethat bythefirst

98 THE TWO NEW SCIENCES OF GALILEOfirstpuffwehavedisplacedthependulumfromthe verticalby,say,halfaninch;thenif,afterthependulumhasreturnedandisaboutto beginthe secondvibration,weadda secondpuff,weshallimpartadditionalmotion;and so on with other blastsprovidedtheyareappliedat therightinstant,andnotwhenthependulumiscomingtowardussincein thiscasetheblastwouldimpederather than aid the motion. Continuingthus withmany impulses[impulsz]we impart to the pendulumsuchmomentum[impeto]that agreaterimpulse[fore.a]thanthat ofasingleblastwillbeneededto stopit.

SAGR.Even as a boy,I observedthat oneman alonebygivingtheseimpulsesat the right instantwasableto ringabell so largethat whenfour,or evensix,menseizedthe ropeand tried to stop it they wereliftedfromthe ground,allofthemtogetherbeingunableto counterbalancethe momentumwhicha singleman,by properly-timedpulls,hadgivenit.

SALV.Yourillustrationmakesmymeaningclearandisquiteas wellfitted,aswhatI havejustsaid,to explainthewonderfulphenomenonofthe stringsofthe cittem[cetera]orofthe spinet

[ 42][cimbalo],namely,the fa& that a vibratingstringwill setanotherstringinmotionandcauseitto soundnotonlywhenthelatterisinunisonbut evenwhenit differsfromtheformerbyanoctaveor a fifth. A stringwhichhas beenstruckbeginstovibrate and continuesthe motionas longas onehears thesound[risonanza];thesevibrationscausethe immediatelysur-roundingairto vibrateandquiver;thentheseripplesin the airexpandfar intospaceandstrikenotonlyall the stringsof thesameinstrumentbut eventhoseof neighboringinstruments.Sincethat stringwhichis tunedto unisonwiththe onepluckedis capableof vibratingwith the samefrequency,it acquires,at the first impulse,a slightoscillation;after receivingtwo,three,twenty,or moreimpulses,deliveredat properintervals,it finallyaccumulatesa vibratorymotionequalto that of thepluckedstring,as is clearlyshownby equalityofamplitudeintheirvibrations.Thisundulationexpandsthroughthe air andsets intovibrationnotonly strings,but alsoany other bodywhich

FIRST DAY 99whichhappensto havethe sameperiodasthat ofthe pluckedstring. Accordinglyifwe attachto the sideof an instrumentsmallpiecesofbristleorotherflexiblebodies,weshallobservethat, whena spinetis sounded,onlythosepiecesrespondthathavethe sameperiodas the stringwhichhasbeenstruck;theremainingpiecesdonotvibratein responseto this string,nordothe formerpiecesrespondto anyothertone.

If onebowsthe base stringon a violarather smartlyandbringsnearit a gobletoffine,thin glasshavingthe sametone[tuono]asthat ofthestring,thisgobletwillvibrateandaudiblyresound. That the undulationsof the mediumare widelydispersedaboutthe soundingbodyis evincedbythe facetthat aglassofwatermaybemadetoemita tonemerelybythefri&ionof thefinger-tipuponthe rimofthe glass;for in thiswaterisproduceda seriesof regularwaves. The samephenomenonisobservedto betteradvantageby fixingthe baseof the gobletuponthebottomofa ratherlargevesselofwaterfillednearlytothe edgeof the goblet;forif, asbefore,wesoundthe glassbyfricCtionof the finger,we shallseeripplesspreadingwiththeutmostregularityandwithhighspeedto largedistancesaboutthe glass. I haveoften remarked,in thus soundinga rather

[1431' largeglassnearlyfullofwater,that at firstthewavesarespaced

with greatuniformity,and when,as sometimeshappens,thetoneof the glassjumpsan o&avehigherI havenotedthat atthismomenteachof the aforesaidwavesdMdesinto two; aphenomenonwhichshowsclearlythat the ratioinvolvedin theo&ave[formadell'ottava]istwo.

SAGP.MorethanoncehaveI observedthissamething,muchto mydelightandalsoto myprofit. For a longtimeI havebeenperplexedabout thesedifferentharmoniessincethe ex-planationshithertogivenby thoselearnedin musicimpressmeas not sufficientlyconclusive.Theytellus that the diapa-son,i. e. theocCtave,involvestheratiooftwo,that thediapentewhichwecall the fifthinvolvesa ratioof 3:2,etc.;becauseifthe openstringof a monochordbe soundedandafterwardsabridgebe placedin the middleandthe halflengthbe sounded

•one

Ioo THE TWO NEW SCIENCESOF GALILEOonehearstheoctave;andifthe bridgebeplacedat 1/3thelengthof the string,thenon pluckingfirst theopenstringandafter-wards2/3ofits lengththe fifthis given;forthisreasontheysaythat the ocCtavedependsupon the ratio of two to one [con-tenutatra'lduee l'uno]andthe fifthuponthe ratioof threetotwo. This explanationdoesnot impressme as sufficienttoestablish2and3/2asthenaturalratiosoftheoctaveandthefifth;andmy reasonfor thinkingsois as follows.Thereare threedifferentwaysin whichthe toneof a stringmaybe sharpened,namely,by shorteningit, by stretchingit and by makingitthinner. If the tensionandsizeof the stringremainconstantoneobtainstheoc2avebyshorteningittoone-half,i.e.,bysound-ingfirsttheopenstringandthenone-halfofit; butiflengthandsizeremainconstantandoneattemptstoproducetheoc°cavebystretchinghe willfindthat it doesnot sufficeto doublethestretchingweight;it mustbe quadrupled;sothat, if the funda-mentalnoteis producedby a weightof onepound,fourwillberequiredto bringouttheoc°cave.

Andfinallyif the lengthand tensionremainconstant,whileonechangesthe size* ofthe stringhewillfindthat in ordertoproducethe ortavethe sizemustbe reducedto _ that whichgavethe fundamental.Andwhat I havesaidconcerningtheoctave,namely,that its ratioas derivedfromthe tensionandsizeofthe stringis the squareof that derivedfromthelength,appliesequallywell to all other musicalintervals[intemalli

[x44]musicz].Thusifonewishesto producea fifthby changingthelengthhefindsthat theratioofthelengthsmustbesesquialteral,in otherwordshe soundsfirst the openstring,thentwo-thirdsofit;butifhewishestoproducethissameresultbystretchingorthinningthe stringthen it becomesnecessaryto squaretheratio3/2that is bytaking9/4 [duplasesguiquarta];accordingly,if the fundamentalrequiresa weightof 4 pounds,the highernotewillbe producednot by 6, but by 9 pounds;the sameistruein regardto size,the stringwhichgivesthe fundamentalislargerthan that whichyieldsthe fifth in the ratio of 9 to 4.

In viewofthesefacets,I seenoreasonwhythosewisephilos-• Fortheexactmeaningof"size"seep.io3below.[Trans.]

FIRST DAY IOIophersshouldadopt2 ratherthan4 as the ratioof theo_ave,orwhyin thecaseofthefifththeyshouldemploythesesquialt-eralratio,3/2,ratherthanthat of9/4- Sinceit isimpossibletocountthevibrationsofa soundingstringonaccountof itshighfrequency,I shouldstillhavebeenin doubtas to whetherastring,emittingthe upperochre, madetwiceas manyvibra_tionsin the sametime as onegivingthe fundamental,had itnotbeenforthefollowingfa_, namely,thatat theinstantwhenthetonejumpsto the oC%ave,thewaveswhichconstantlyac-companythe vibratingglassdivideup intosmalleroneswhicharepreciselyhalfaslongastheformer.

SALV.This is a beautifulexperimentenablingus to distin-guishindividuallythe waveswhichareproducedby thevibra-tionsofasonorousbody,whichspreadthroughtheair,bringingtothetympanumoftheearastimuluswhichthemindtranslatesintosound.Butsincethesewavesinthewaterlastonlysolongas the fri_°donof the fingercontinuesandare,eventhen,notconstantbut arealwaysforminganddisappearing,wouldit notbe a finething ifonehad the abilityto producewaveswhichwouldpersistfora longwhile,evenmonthsandyears,soas toeasilymeasureandcountthem?

SAcR.Suchan inventionwould,I assureyou,commandmyadmiration.

SALv.ThedeviceisonewhichI hituponbyaccident;mypartconsistsmerelyin theobservationofit andin the appreciationofitsvalueas a confirmationofsomethingto whichI hadgivenprofoundconsideration;andyet the deviceis, in itself,rathercommon.As I wasscrapinga brassplatewitha sharpiron

.[145]chiselin orderto removesomespotsfromit andwasrunningthe chiselratherrapidlyoverit, I onceor twice,duringmanystrokes,heardtheplateemitaratherstrongandclearwhistlingsound;onlookingat the platemorecarefully,I noticeda longrowoffinestreaksparalleland equidistantfromoneanother.Scrapingwith the chiseloverandoveragain,I noticedthat itwasonlywhenthe plate emittedthis hissingnoisethat anymarkswereleftuponit;whenthescrapingwasnotaccompanied

by

io2 THE TWO NEW SCIENCESOF GALII.F.Obythissibilantnotetherewasnotthe leasttraceofsuchmarks.Repeatingthe trickseveraltimesaridmakingthe stroke,nowwithgreaternowwithlessspeed,thewhistlingfollowedwithapitchwhichwascorrespondinglyhigherand lower. I notedalso that the marksmadewhenthe toneswerehigherwereclosertogether;but whenthe toneswere deeper,they werefartherapart. I alsoobservedthat when,duringasinglestroke,the speedincreasedtowardthe endthe soundbecamesharperandthestreaksgrewclosertogether,but alwaysin suchawayasto remainsharplydefinedand equidistant.Besideswheneverthe strokewasaccompaniedby hissingI felt thechiseltremblein mygraspandasortofshiverrun throughmyhand. In shortweseeandhearin the caseof the chiselpreciselythat whichis seenandheardin the caseof a whisperfollowedby a loudvoice;for,whenthe breathis emittedwithoutthe produ_ionof a tone,onedoesnot feeleitherin the throator mouthanymotionto speakof in comparisonwiththat whichis feltin thelarynxand upperpart of the throatwhenthe voiceis used,especiallywhenthetonesemployedarelowandstrong.

At timesI havealsoobservedamongthe stringsofthe spinettwowhichwerein unisonwithtwoofthe tonesproducedbytheaforesaidscraping;and amongthosewhichdifferedmost inpitchI foundtwo whichwereseparatedby an intervalof aperfe_ fifth. Uponmeasuringthe distancebetweenthe mark-ingsproducedbythe twoscrapingsk wasfoundthat the spacewhichcontained45ofonecontained3° of the other,whichispreciselythe ratioassignedto thefifth.

But nowbeforeproceedingany fartherI want to callyourattentionto thefar that,of thethreemethodsforsharpeningatone,the onewhichyou referto as the finenessof the stringshouldbe attributedto its weight. Solongas the materialof

[i46]the stringis unchanged,the sizeandweightvary in the sameratio. Thusin the caseofgut-strings,weobtainthe oc_r_vebymakingonestring4 timesas largeas the other;soalsoin thecaseof brassonewiremusthave4 timesthe sizeof the other;butifnowwewishto obtainthe odtaveofa gnt-string,byuseof

brass

FIRSTDAY Io3brasswire,wemustmakeit, not fourtimesas large,but fourtimesas heavyas the gut-string:as regardssizethereforethemetalstringis not fourtimesas bigbut fourtimesas heavy.Thewiremaythereforebe eventhinnerthanthegut notwith-standingthefaCtthat the lattergivesthehighernote. Henceiftwospinetsarestrung,onewithgoldwiretheotherwithbrass,and if the correspondingstringseachhave the samelength,diameter,andtensionit followsthat the instrumentstrungwithgoldwillhavea pitchaboutone-fifthlowerthanthe otherbe-causegoldhasa densityalmosttwicethat ofbrass. Andhereit is to be notedthat it is the weightratherthanthe sizeofamovingbodywhichoffersresistanceto changeofmotion[_elocithdelmoto]contraryto whatonemightat firstglancethink. Foritseemsreasonabletobelievethat abodywhichislargeandlightshouldsuffergreaterretardationofmotioninthrustingasidethemediumthan wouldonewhichis thin and heavy;yet hereexactlytheoppositeistrue.

Returningnowto the originalsubje&ofdiscussion,I assertthat the ratioof a musicalintervalis not immediatelydeter-minedeitherby the length,size,or tensionof the stringsbutratherby theratiooftheirfrequencies,that is,by the numberof pulsesof air waveswhichstrikethe tympanumof the ear,causingit alsoto vibratewiththe samefrequency.This faCtestablished,wemaypossiblyexplainwhycertainpairsofnotes,differingin pitch producea pleasingsensation,othersa lesspleasanteffect,andstillothersa disagreeablesensation.Suchan explanationwouldbe tantamountto an explanationof themoreor lessperfectconsonancesandof dissonances.Theun-pleasantsensationproducedby the latterarises,I think,fromthediscordantvibrationsoftwodifferenttoneswhichstriketheear out of time [sproporzionatamente].Especiallyharshis thedissonancebetweennotes whosefrequenciesare incommen-surable;sucha caseoccurswhenonehas twostringsin unisonandsoundsoneofthemopen,togetherwitha part oftheother

[471whichbearsthe sameratioto itswholelengthas the sideof asquarebearsto the diagonal;thisyieldsa dissonancesimilar

to

Io4 THE TWONEW SCIENCESOF GALIT.F.Oto the augmentedfourthor diminishedfifth[tr/tonoo semi-diapente].

Agreeableconsonancesarepairsof toneswhichstriketheearwitha certainregularity;this regularityconsistsin thefa&that thepulsesdeliveredbythe twotones,in thesameintervaloftime,shallbecommensurableinnumber,soasnottokeeptheeardruminperpetualtorment,bendingintwodifferentdire&ionsin orderto yieldto the ever-discordantimpulses.

The firstand mostpleasingconsonanceis, therefore,theo&avesince,foreverypulsegivento the tympanumbythelowerstring,thesharpstringdeliverstwo;accordinglyateveryothervibrationof the upperstringbothpulsesaredeliveredsimultaneouslysothatone-halftheentirenumberofpulsesaredeliveredin unison.Butwhentwostringsareinunisontheirvibrationsalwayscoincideandthe effe&is that of a singlestring;hencewedonotreferto it asconsonance.Thefifthisalsoa pleasingintervalsinceforeverytwovibrationsofthelowerstringtheupperonegivesthree,sothatconsideringtheentirenumberofpulsesfromtheupperstringone-thirdofthemwillstrikeinunison,i.e.,betweeneachpairofconcordantvibra-tionsthereintervenetwosinglevibrations;andwhenthe in-tervalisa fourth,threesinglevibrationsintervene.Incasetheintervalisa secondwheretheratiois9/8it isonlyeveryninthvibrationof theupperstringwhichreachesthe earsimulta-neouslywithoneofthelower;alltheothersarediscordantandproducea harsheffe&upontherecipientearwhichinterpretsthemasdissonances.

S_v. Won'tyoubegoodenoughto explainthisargumentalittlemoreclearly?

SAr.v.LetAB denotethe lengthof a wave[lospazioe ladilatazioned'unavibrazione]emittedby the lowerstringandCDthatofa higherstringwhichisemittingtheoctaveofA.B;divideABin themiddleat E. If the twostringsbegintheirmotionsatAandC,itisclearthatwhenthesharpvibrationhasreachedtheendD, theothervibrationwillhavetravelledonlyasfarasE,which,notbeingaterminalpoint,willemitnopulse;bu_thereisa blowdeliveredatD. Accordinglywhentheone

wave

FIRST DAY Io5,: wavecomesbackfromD to C,theotherpassesonfromE to ]3;! hencethe twopulsesfromB andC strikethe drumof the eari simultaneously.Seeingthat these vibrationsare repeated

againandagainin the samemanner,wecon-i dude that eachalternatepulsefromCD falls&, E B

in unisonwith onefromAB. But eachofthe - ,[I48] ¢ "

pulsationsat the terminalpoints,A and]3,is A _ o sconstantlyaccompaniedbyonewhichleavesal- t , , ,waysfromC or alwaysfromD. Thisis clear; • :becauseifwesupposethewavesto reachAandO DC at the sameinstant,then,whileonewave Fig.13travelsfromA to B, the otherwillproceedfromC to D andback to C, so that wavesstrikeat C andB simultaneously;duringthepassageofthewavefromBbackto.&thedisturbanceat C goesto D and againreturnsto C,sothat oncemorethepulsesatAandCaresimultaneous.

Nextlet the vibrationsABandCD be separatedby an in-tervalofafifth,that is,bya ratioof3/2;choosethepointsE andOsuchthat theywilldividethewavelengthofthelowerstringintothreeequalpartsandimaginethevibrationsto startat thesameinstantfromeachoftheterminalsA andC. It isevidentthat whenthe pulsehasbeendeliveredat the terminalD, thewaveinABhastravelledonlyasfar asO;the drumofthe earreceives,therefore,onlythe pulsefromD. Then duringthereturnoftheonevibrationfromDto C,theotherwillpassfromOto B andthenbackto O,producinganisolatedpulseat B--apulsewhichis out of timebut onewhichmustbe takenintoconsideration.

Nowsincewehaveassumedthat the firstpulsationsstartedfromthe terminalsAandC at the sameinstant,it followsthatthe secondpulsation,isolatedatD,occurredafteran intervaloftimeequalto that requiredforpassagefromCto D or,whatisthe samething,fromAto O;butthe nextpulsation,theoneatB, is separatedfromthe precedingby onlyhalfthis interval,namely,thetimerequiredforpassagefrom0 to B. NextwhiletheonevibrationtravelsfromOtoA,theothertravelsfromCto

D,

106 THE TWO NEW SCIENCESOF GALILEOD,theresultofwhichisthattwopulsationsoccursimulta-neouslyatA andD. Cyclesofthiskindfollowoneafteran-other,i.e.,onesolitarypulseofthelowerstrknginterposedbe-tweentwo solitarypulsesof the upperstring. Let us nowimaginetime to be dividedinto very smallequal intervals;thenifweassumethat, duringthe firsttwooftheseintervals,thedisturbanceswhichoccurredsimultaneouslyatAandChavetravelledasfarasOandD andhaveproducedapulseat D; andif we assumethat duringthe third and fourthintervalsonedisturbancereturnsfromD to C,producinga pulseat C,whilethe other,passingonfromO to B andbackto O,producesapulseat B;andif finally,duringthefifthandsixthintervals,thedisturbancestravelfromO and C to A andD, producingapulseat eachofthe lattertwo,thenthe sequencein whichthepulsesstrikethe earwillbe suchthat, ifwebeginto counttimefromany instantwheretwo pulsesare simultaneous,the eardrumwill,afterthe lapseoftwoofthe saidintervals,receiveasolitarypulse;at theendofthe thirdinterval,anothersolitary

[I491puIse;so alsoat the endof the fourthinterval;andtwo in-tervalslater,i. e.,at theendofthe sixthinterval,willbeheardtwopulsesin unison.Hereendsthe cycle--theanomaly,so tospeak--whichrepeatsitselfoverandoveragain.

SAGR.I cannolongerremainsilent;forImustexpressto youthegreatpleasureI haveinhearingsucha completeexplanationof phenomenawith regardto whichI haveso longbeenindarkness.NowI understandwhyunisondoesnotdifferfromasingletone;I understandwhythe ocCtaveis the principalhar-mony,but solikeunisonasoftento bemistakenfor it andalsowhyit occurswith the other harmonies.It resemblesunisonbecausethepulsationsofstringsinunisonalwaysoccursimulta-neously,andthoseofthe lowerstringofthe o&aveare alwaysaccompaniedby thoseoftheupperstring;andamongthelatteris interposeda solitarypulseat equalintervalsandin suchamannerasto producenodisturbance;the resultisthat suchaharmonyis rathertoomuchsoftenedandlacksfire. But thefifthischaracterizedbyitsdisplacedbeatsandbytheinterposi-

tion

FIRST DAY lO7tionof twosolitarybeatsof theupperstringandonesolitarybeat of the lowerstringbetweeneachpairof simultaneouspulses;thesethreesolitarypulsesareseparatedby intervalsoftime equalto half the intervalwhichseparateseachpair ofsimultaneousbeatsfromthe solitarybeatsofthe upperstring.ThustheeffecCtof the fifthis to producea ticklingofthe eardrumsuchthatits softnessismodifiedwithsprightliness,givingat thesamemomenttheimpressionofagentlekissandofabite.

SALv.Seeingthat youhavederivedsomuchpleasurefromthesenovelties,Imustshowyouamethodbywhichtheeyemayenjoythe samegameastheear. Suspendthreeballsoflead,orother heavymaterial,by meansof stringsof differentlengthsuchthat whilethe longestmakestwovibrationsthe shortestwillmakefourandthe mediumthree;thiswilltakeplacewhenthe longeststringmeasures16,eitherinhandbreadthsor inanyotherunit,the medium9 andtheshortest4,allmeasuredinthesameunit.

Nowpullall thesependulumsasidefromthe perpendicularand releasethemat the sameinstant;you willseea curiousinterplayof the threadspassingeachotherin variousmannersbut suchthat at the completionof everyfourthvibrationofthe longestpendulum,all threewillarrivesimultaneouslyatthe sameterminus,whencetheystartoveragainto repeatthesamecycle.Thiscombinationofvibrations,whenproducedonstringsis preciselythat whichyieldsthe intervalof the ocCtaveandthe intermediatefifth. If weemploythe samedisposition

[ 5o1of apparatusbut changethe lengthsof the threads,alwayshoweverin sucha waythattheirvibrationscorrespondto thoseofagreeablemusicalintervals,weshallseea differentcrossingofthesethreadsbut alwayssuchthat,afteradefiniteintervaloftimeandaftera definitenumberofvibrations,allthe threads,whetherthreeor four,willreachthesameterminusat the sameinstant,andthenbegina repetitionofthecycle.

If howeverthe vibrationsoftwoor morestringsareincom-mensurableso that theynevercompletea definitenumberofvibrationsat the sameinstant_or ifcommensurabletheyreturn

only

Io8 THETWONEWSCWNCESOFGALILVOonlyaftera longintervalof timeandaftera largenumberofvibrations,thentheeyeis confusedby thedisorderlysuccessionof crossedthreads. In like mannerthe ear is painedby anirregularsequenceofairwaveswhichstrikethetympanumwith-outanyfixedorder.

But,gentlemen,whitherhavewedriftedduringthesemanyhoursluredonbyvariousproblemsandunexpecCteddigressions?The day is alreadyendedand wehavescarcelytouchedthesubjecCtproposedfordiscussion.Indeedwehavedeviatedsofarthat I rememberonlywithdifficultyourearlyintroducCtionandthe littleprogressmadeinthe wayofhypothesesandprinciplesforuseinlaterdemonstrations.

SAng.Letus thenadjournforto-dayin orderthat ourmindsmay find refreshmentin sleepand that we may return to-morrow,if sopleaseyou,andresumethe discussionofthe mainquestion.

SALV.I shallnot failto be hereto-morrowat thesamehour,hopingnot only to ,renderyou servicebut alsoto enjoyyourcompany.

END OF THI_FIRST DAY.

[I5I]

SECOND DAY• ._,AGR.WhileSimplicioandI wereawaiting

yourarrivalweweretryingtorecallthatlastconsiderationwhichyouadvancedasaprin-

;{_cipleandbasisfortheresultsyouintendedto,_lobtain;this considerationdealt with the_[_resistancewhichallsolidsofferto fracCture

anddependeduponacertaincementwhich_]_heldthe partsgluedtogetherso that they

wouldyieldandseparateonlyunderconsiderablepull [potenteattrazzione].Later we tried to find the explanationof thiscoherence,seekingit mainlyin the vacuum;thiswasthe occa-sior_ofourmanydigressionswhichoccupiedthe entiredayandledus far afieldfromthe originalquestionwhich,as I havealreadystated,wastheconsiderationoftheresistance[resistenza]that solidsoffertofracCture.

SALV.I rememberit allverywell. Resumingthe threadofourdiscourse,whateverthenatureofthisresistancewhichsolidsofferto largetrafftiveforces[_olentaattrazzione]therecanatleastbe nodoubtof itsexistence;andthoughthisresistanceisverygreatinthecaseofa direcCtpull,it isfound,asa rule,to belessin the caseof bendingforces[nel_iolentargliper traverso].Thus,forexample,a rodofsteelorofglasswillsustaina longi-tudinalpullofa thousandpoundswhileaweightoffiftypoundswouldbe quitesufficientto breakit if the rodwerefastenedatrightanglesintoa verticalwall. It is this secondtypeof re-sistancewhichwemust consider,seekingto discoverin what

[Iszlproportion

im THE TWO NEW SCIENCESOF GALTT.V.Oproportionit is foundin prismsand cylindersof the samematerial,whetheralikeor unlikein shape,length,and thick-ness. In thisdiscussionI shalltakeforgrantedthewell-knownmechanicalprinciplewhichhas been shownto governthebehaviorof a bar,whichwecalla lever,namely,that theforcebearsto the resistancethe inverseratioofthe distanceswhichseparatethefulcrumfromtheforceandresistancerespecCtively.

Snvw.This wasdemonstratedfirstof allby Aristotle,in hisMechanics.

SALV.Yes,I amwillingto concedehimpriorityin pointoftime;but as regardsrigorofdemonstrationthefirstplacemustbegivento Archimedes,sinceupona singlepropositionprovedin hisbookon Equilibrium* dependsnot onlythe lawof theleverbutalsothoseofmostothermechanicaldevices.

SA_R.Sincenowthis principleis fundamentalto all thedemonstrationswhichyouproposeto setforthwouldit notbeadvisableto giveus a completeand thoroughproofof thispropositionunlesspossiblyit wouldtaketoomuchtime?

SAzv.Yes, that wouldbe quiteproper,but it is better Ithinkto approachour sub]e&in a mannersomewhatdifferentfromthat employedby Archimedes,namely,by firstassumingmerelythat equalweightsplacedin a balanceofequalarmswillproduceequilibrium--aprinciplealsoassumedbyArchimedes--and then provingthat it is no lesstrue that unequalweightsproduceequilibriumwhen the arms of the steelyardhavelengthsinverselyproportionalto the weightssuspendedfromthem;in other words,it amountsto the samethingwhetheroneplacesequalweightsat equaldistancesorunequalweightsat distanceswhichbearto eachotherthe inverseratioof theweights.

In orderto makethismatterclearimaginea prismor solidcylinder,AB,suspendedat eachendto the rod [linealHI, andsupportedby twothreadsHA andI_B;it is evidentthat if Iattachathread,C,at themiddlepointofthebalancebeamHI,theentireprismABwill,accordingtotheprincipleassumed,hangin equilibriumsinceone-halfits weightliesononeside,andtheotherhalfontheotherside,ofthepointofsuspensionC. Now

• WorksofArchlmedes.Trans.byT.L.Heath,pp.I89-22o.[Trans.]

%

[_ SECONDDAY I II

[_ supposethe prismto be dividedintounequalpartsby a plane[ s3]_ throughthe lineD, andletthe partDAbe the largerandDBthe smaller:this divisionhavingbeenmade,imaginea thread_.; ED, attachedat the pointE andsupportingthe partsADand

- DB,in orderthat thesepartsmayremainin the sameposition- relativeto lineHI: andsincethe relativepositionof the prismandthe beamHI remainsunchanged,therecanbe no doubtbutthat theprismwillmaintainitsformerstateofequilibrium.

i H GC_B F I

Fig. 14

But circumstanceswouldremainthe sameif that part of theprismwhichisnowheldup,at theends,bythe threadsAHandDE weresupportedat the middleby a singlethreadGL;andlikewisethe otherpart DB wouldnot changepositionif heldby a threadFMplacedat its middlepoint. SupposenowthethreadsHA,ED, andIB to be removed,leavingonlythe twoGL andFM, thenthe sameequilibriumwillbe maintainedsolongasthesuspensionisatC. NowletusconsiderthatwehaveheretwoheavybodiesADandDBhungattheendsGandF,ofa balancebeamGF in equilibriumaboutthe pointC, so thatthe lineCG is the distancefromC to the pointof suspensionofthe heavybodyAD,whileCF is the distanceat whichtheotherheavybody,DB, is supported.It remainsnowonlytoshowthat thesedistancesbearto eachotherthe inverseratioof the weightsthemselves,that is,the distanceGC is to thedistanceCFastheprismDBistotheprismDA--apropositionwhichweshallproveas follows:Sincethe lineGEis thehalfofEH,andsinceEF isthehalfofEI, thewholelengthGFwillbe

half

iiz THE TWO NEW SCIENCESOF GALTLVDhalfofthe entirelineHI, andthereforeequalto CI: ifnowwesubtra&thecommonpartCFtheremainderGCwillbeequaltotheremainderFI, that is,to FE,andif to eachoftheseweaddCE we shallhaveGE equal to CF: henceGE_F=FC:CG.But GEandEF bearthe sameratioto eachotheras do theirdoublesHE andEI, that is,the sameratioasthe prismADtoDB. Therefore,by equatingratioswehave,convertendo,thedistanceGCis to the distanceCF as the weightBE)is to theweightDA,whichiswhatI desiredtoprove.

[,54]If whatprecedesis clear,youwillnot hesitate,I think, to

admitthat the twoprismsADandDBareinequilibriumaboutthe pointC sinceone-halfof the wholebodyAB lieson therightofthe suspensionC andtheotherhalfontheleft;inotherwords,thisarrangementisequivalentto twoequalweightsdis-posedatequaldistances.I donotseehowanyonecandoubt,ifthetwoprismsADandDBweretransformedintocubes,spheres,orintoanyotherfigurewhateverandifGandFwereretainedaspointsof suspension,that they wouldremainin equilibriumaboutthepointC,forit isonlytooevidentthat changeoffiguredoesnotproducechangeofweightsolongasthe mass[quantit_di materialdoesnotvary. Fromthiswemayderivethegeneralconclusionthat any two heavybodiesare ha equilibriumatdistanceswbAchare inverselyproportionalto their weights.

This principleestablished,I desire,beforepassingto anyothersubje&,to callyourattentionto thefa&that theseforces,resistances,moments,figures,etc.,maybe consideredeitherinthe abstra&,dissociatedfrommatter,or in the concrete,asso-ciatedwith matter. Hence the propertieswhichbelongtofiguresthat are merelygeometricalandnon-materialmust bemodifiedwhenwefill thesefigureswith matter andthereforegivethem weight. Take, for example,the leverBA which,restinguponthe supportE, is usedto lift a heavystoneD.Theprinciplejust demonstratedmakesit clearthat a forceap-plied at the extremityB willjust sufficeto equilibratetheresistanceofferedby the heavy bodyD providedthis force[momento]bearsto theforce[momento]atD the sameratioasthe

distance

SECOND DAY II3distanceACbearsto thedistanceCB;andthisistruesolongasweconsideronlythemomentsofthesingleforceat B andoftheresistanceat D, treatingtheleverasanimmaterialbodydevoidofweight. But ifwetake intoaccounttheweightofthe leveritselfmaninstrumentwhichmaybe madeeitherofwoodor ofiron--itismanifestthat,whenthisweighthasbeenaddedto the

[Iss]forceat B, the ratiowillbe changedandmust thereforebeexpressedin differentterms. Hencebeforegoingfurtherlet

Fig. 15us agreeto distinguishbetweenthesetwopointsofview;whenweconsideran instrumentin the abstra&,i. e.,apartfromtheweightof its ownmaterial,weshallspeakof "takingit in anabsolutesense"[prendereassolutamente];butifwefilloneofthesesimpleandabsolutefigureswithmatterandthusgiveitweight,we shall referto sucha materialfigureas a "moment"or"compoundforce"[momentooforzacomposta].

SAoR.I mustbreakmyresolutionaboutnot leadingyouoffintoa digression;forI cannotconcentratemy attentionuponwhat is to followuntila certaindoubtis removedfrommymind,namely,you seemto comparethe forceat B withthetotal weightof the stoneD, a part of whichmpossiblythegreaterpart--rests uponthe horizontalplane:so that

SALV.I understandpede&ly:youneedgonofurther.How"everpleaseobservethat I havenotmentionedthetotalweightofthe stone;I spokeonlyofits force[momento]at the pointA,theextremityofthe leverBA,whichforceis alwayslessthanthe total weightof the stone,andvarieswithits shapeandelevation.

SAc_.Good:but thereoccursto me-anotherquestionaboutwhich

I I4 THE TWO NEW SCIENCESOF GALTI.F.OwhichI am curious. For a completeunderstandingof thismatter, I shouldlikeyou to showme,ifpossible,howonecandeterminewhat part of the total weightis supportedby theunderlyingplaneand what part by the endA of the lever.

SALV.The explanationwillnot delayus long and I shallthereforehavepleasurein grantingyourrequest. In theaccom-panyingfigure,let us understandthat the weighthavingitscenterofgravityat A restswiththe end]3uponthehorizontalplaneand with the other enduponthe leverCG. LetN bethefulcrumofalevertowhichtheforce[potenza]isappliedat G.Let falltheperpendiculars,AOandCF,fromthe centerAandthe endC. Then I say,the magnitude[momento]of the entireweightbears to the magnitudeof the force [momentodellapo_enza]at G a ratiocompoundedofthe ratiobetweenthe two

Fig.16distancesGN and NC and the ratio betweenFB and BO.LayoffadistanceXsuchthat itsratioto NCisthesameasthatofBOto FB; then,sincethe total weightA is counterbalancedbythetwoforcesat BandatC,itfollowsthat theforceat Bistothat at C as the distanceFO is to the distanceOB. Hence,

[ 56]compone_o,the sumofthe forcesat B andC,that is,the totalweightA [momentodi tutto'l pesod], isto theforceat C asthelineFB is to the lineBO,that is,as NCis to X: but the force[momentodellapotenza]appliedat C is to the forceappliedatG as the distanceGN is to the distanceNC; henceit follows,excequaliin l_roportioneperturbata,*that the entireweightA isto the forceappliedat G as thedistanceGNis to X. But theratioofGNtoX iscompoundedoftheratioofGNtoNC andofNC toX, that is,ofI_Bto B0; hencetheweightAbearsto the

*FordefinitionofperturbataseeTodhunter'sEuclid.BookV,Def.2o.[Trans.]

SECONDDAY _15equilibratingforceat G a ratiocompoundedof that ofGN toNCandofFBto BO:whichwastobeproved.

Letusnowreturnto ouroriginalsubjecCt;then, ifwhathashithertobeensaid is clear,it willbe easilyunderstoodthat,

PROPOSITIONI

A prismor solidcylinderof glass,steel,woodorotherbreak-ablematerialwhichiscapableofsustainingaveryheavyweightwhenappliedlongitudinallyis, as previouslyremarked,easilybrokenbythe transverseapplicationof aweightwhichmaybemuchsmallerinproportionasthe lengthofthecylinderexceedsits thickness.

Let us imaginea solidprismABCDfastenedintoa wallattheendAB,andsupportingaweightE at theotherend;under-standalsothat thewallisverticalandthat theprismorcylinderis fastenedat rightanglesto thewall. It is clearthat, if thecylinderbreaks,fracturewilloccurat the pointB wheretheedgeofthemortiseacCtsasa fulcrumforthe leverBC,to whichtheforceisapplied;thethicknessofthesolidBAistheotherannoftheleveralongwhichislocatedtheresistance.Thisresistanceopposesthe separationof the part BD,lyingoutsidethewall,fromthat portionlyinginside. Fromthe preceding,it followsthat themagnitude[momento]oftheforceappliedat Cbearstothemagnitude[momento]oftheresistance,foundinthethicknessofthe prism,i.e.,in theattachmentofthe baseBAto its con-tiguousparts,the sameratiowhichthe lengthCBbearsto halfthe lengthBA;if nowwedefineabsoluteresistanceto fracCture

[i57]asthat offeredto a longitudinalpull(inwhichcasethestretch-ingforceacCtsin the samedirec°donasthat throughwhichthebodyismoved),thenit followsthat the absoluteresistanceofthe prismBDis to the breakingloadplacedat the endof theleverBCin thesameratioasthelengthBCisto thehalfofABin the caseof a prism,or the semidiameterin the caseof acylinder.Thisisour firstproposition.*Observethat in what

*Theonefundamentalerrorwhichisimplicitlyintroducedintothispropositionand which is carried throughthe entire discussionof the

II6 THE TWO NEW SCIENCESOF GALII.VOhasherebeensaidthe weightof the solidBE)itselfhasbeenleftoutof consideration,or rather,the prismhasbeenassumedto bedevoidofweight. But if theweightofthe prismis to betakenaccountofin conjuncCtionwiththeweightE,wemustadd

to theweightEonehalf that of theprismBD: so thatif, forexample,thelatter weighstwopounds and theweightE is tenpounds we musttreat the weightEas if it wereeleven

D pounds.SL_P.Why not

twelve?SALv.Theweight

E,my dearSimp-licio,hangingattheextremeendC acCtsupontheleverBCwithits_allmo-mentoftenpounds:so alsowould the

Fig.17 solid BD if sus-pendedat the samepoint exertitsfilllmomentof twopounds;but, as you know,this solidis uniformlydistributedthrough-SecondDay consistsin a failureto seethat, in sucha beam,there mustbe equilibriumbetweenthe forcesof tensionand compressionoveranycross-section.The correctpoint of viewseemsfirstto have beenfoundby E. Mariotte in I68Oand by A. Parent in I7I3. Fortunately thiserror does not vitiate the conclusionsof the subsequentpropositionswhich deal only with proportlons--not actual strength--of beams.FollowingK. Pearson(Todhunter'sHistoryof Elasticity)onemightsaythat Galileo'smistakelay in supposingthefibresofthe strainedbeamtobe inextensible.Or, confessingthe anachronism,onemightsay that theerrorconsistedin taking the lowestfibreof thebeamas the neutralaxis.

[Tra_;.]

SECOND DAY ii 7out its entirelength,BC,so that the partswhichlie neartheendB arelesseffeCtivethanthosemoreremote.

Accordinglyif we strike a balancebetweenthe two, theweightof the entireprismmaybe consideredas concentratedat its centerof gravitywhichliesmidwayof the leverBC.But a weighthungat the extremityC exertsa momenttwiceas greatas it wouldif suspendedfromthe middle:therefore

[is8]ifweconsiderthe momentsofbothaslocatedat theendC wemustaddtotheweightEone-halfthatoftheprism.

Sn_P.I understandperfeCtly;andmoreover,ifI mistakenot,the forceofthe twoweightsBDandE, thusdisposed,wouldexertthesamemomentaswouldtheentireweightBIDtogetherwith twicethe weightE suspendedat the middleof the leverBC.

S_J_v.Preciselyso, and a faCtworth remembering.Nowwecanreadilyunderstand

PROPOSITIONIIHowandin whatproportiona rod,or rathera prism,whose

widthis greaterthan its thicknessoffersmoreresistancetofracCturewhenthe

appliedin a__forceisthe direCtionof its cV___-_r-_I] JbreadththaninthedireCtionof it s _b-__ ___t_

thickness. __:_-'_ --For the sakeof

clearness,take a

ruler ad whose _ _width is ac andwhose thickness, Fig.i8cb,is muchlessthan its width. Thequestionnowis whywillthe ruler,if stoodon edge,as in the firstfigure,withstandagreatweightT, while,whenlaidflat,as in the secondfigure,it willnot supportthe weightX whichis lessthanT. Theansweris evidentwhenwe rememberthat in the onecase

the

Iz8 THE TWO NEW SCIENCESOF GAL!!.F.Othe fulcrumis at the line bc, and in the other caseat ca,whilethe distanceat whichthe forceis appliedis the sameinboth cases,namely,the length be/:but in the first casethedistanceofthe resistancefromthe fulcrum--halfthe lineca--is greaterthan in the othercasewhereit is onlyhalf of bc.Thereforethe weightT is greaterthanX in the sameratio ashalfthe widthcaisgreaterthanhalfthe thicknessbc,sincetheformerarts as a leverarmforca,andthe latterforcb,againstthe sameresistance,namely,the strengthofall thefibresin thecross-sectionab. Weconclude,therefore,that anygivenruler,or prism,whosewidthexceedsits thickness,willoffergreaterresistanceto fracturewhenstandingon edgethanwhenlyingflat,andthisin theratioofthewidthto thethickness.

PRoPosrrmr¢IIIConsideringnowthecaseofaprismorcylindergrowinglonger

in a horizontaldirection,wemustfindout in what ratiothemomentof its ownweightincreasesin comparisonwith itsresistanceto fracture. ThismomentI findincreasesinpropor-

[I59]tionto the squareofthe length. In orderto provethis letADbeaprismorcylinderlyinghorizontalwithitsendA firmlyfixedin awall. Let thelengthoftheprismbe increasedbythe addi-tion of the portionBE. It is clearthat merelychangingthelengthoftheleverfromAB toACwill,ifwedisregarditsweight,increasethemomentofthe force[attheend]tendingto producefracCtureat A in the ratioofCA to BA. But, besidesthis, theweightofthe solidportionBE,addedto theweightofthe solidABincreasesthe momentof thetotal weightin theratiooftheweightof the prismAEto that of the prism_A_B,whichis thesameastheratioofthelengthAC toAB.

It follows,therefore,that, whenthe lengthandweightaresimultaneouslyincreasedin anygivenproportion,the moment,whichistheproductofthesetwo,isincreasedina ratiowhichisthe squareofthe precedingproportion.Theconclusionis thenthat the bendingmomentsdueto the weightof prismsandcylinderswhichhavethe samethicknessbut differentlengths,

bear

i SECOND DAY 119bearto eachothera ratiowhichis the squareof the ratiooftheirlengths,or,whatisthesamething,theratioofthesquaresoftheirlengths.We shallnextshowin what ratiothe resistanceto frac_re

B C

Fig. 19

[bendingstrength],in prismsandcylinders,increaseswith in-[ 6o]

creaseof thicknesswhilethe lengthremainsunchanged.HereI saythat

PIIOPOSlTIONIV

In prismsand cylindersof equallength,but of unequalthicknesses,theresistanceto fra_ureincreasesinthesameratioas thecubeofthe diameterof the thickness,i.e., ofthebase.

LetAandB betwocylindersofequallengthsDG,FH;lettheirbasesbecircularbutunequal,havingthe diametersCDandEF.ThenI saythatthe resistancetofradtureofferedbythe cylinderB

i2o THE TWO NEW SCIENCESOF GALII,V,OB isto that offeredbyAasthe cubeofthe diameterFEistothecubeof thediameterDC. For, ifweconsiderthe resistancetofragtureby longitudinalpullasdependentuponthe bases,i. e.,uponthecirclesEFandDC,noonecandoubtthat the strength[resistenza]of the cylinderB is greaterthan that of A in the

* sameproportioninwhichthe areaofthe circleEFexceedsthatof CD;becauseit is preciselyin this ratiothat the numberoffibresbindingthe partsofthe solidtogetherinthe onecylinderexceedsthat intheothercylinder.

But in the caseof a forceacCtingtransverselyit mustbe re-memberedthat weareemployingtwoleversinwhichthe forces

C areappliedatdistancesDG,FH, and the fulcrumsarelocatedat the pointsD and

C D F; but the resistancesare1_appliedat distanceswhich

........... areeqiaalto the radiiofthe

_ circlesDC and EF, sincethe fibresdistributedoverl-I I_these entire cross-secCtions

Fig.2o ac°casif concentratedat thecenters. Rememberingthis and rememberingalsothat thearms,DGandFH, throughwhichthe forcesG and H acCtareequal,we can understandthat the resistance,locatedat thecenterof the baseEF, actingagainstthe forceat H, is moreeffecCtive[maggiore]than the resistanceat the centerof thebase CD opposingthe forceG, in the ratioof the radiusFEto the radiusDC. Accordinglythe resistanceto fracCtureof-feredby the cylinderB isgreaterthan that of the cylinderAin a ratiowhichiscompoundedofthat of theareaofthe circlesEF andDCandthat oftheirradii,i. e.,oftheirdiameters;butthe areasofcirclesareasthe squaresoftheirdiameters.There-forethe ratioof the resistances,beingthe produc°cof the twoprecedingratios,isthesameasthat ofthecubesofthediameters.ThisiswhatI setoutto prove.Alsosincethevolumeofa cube

[I6I]variesas the thirdpowerof its edgewemaysay that the re-

sistance

SECONDDAY IZIsistance[strength]ofa cylinderwhoselengthremainsconstantvariesasthethirdpowerofitsdiameter.

Fromtheprecedingweareableto concludethatCORO_L__RY

The resistance[strength]of a prismor cylinderof constantlengthvariesinthesesquialteralratioofitsvolume.

Thisis evidentbecausethe volumeofa prismor cylinderofconstantaltitudevariesdirecCtlyasthe areaof itsbase,i.e.,asthe squareofasideordiameterofthisbase;but,asjustdemon-strated,the resistance[strength]variesasthe cubeofthissamesideordiameter.Hencetheresistancevariesinthesesquialteralratioof the volume consequentlyalsoof the weight--ofthesoliditself.

S_v. BeforeproceedingfurtherI shouldliketo haveoneofmydifficultiesremoved.Up to thispointyouhavenot takeninto considerationa certainother kindof resistancewhich,itappearsto me,diminishesas thesolidgrowslonger,andthis isquiteastruein the caseofbendingas in pulling;it ispreciselythusthat in thecaseofa ropeweobservethat a verylongoneislessableto supporta largeweightthana shortone. Whence,Ibelieve,ashortrodofwoodorironwillsupporta greaterweightthanif it werelong,providedthe forcebe alwaysappliedlongi-tudinallyandnottransversely,andprovidedalsothat wetakeintoaccounttheweightoftheropeitselfwhichincreaseswithitslength.

SALV.I fear,Simplicio,if I correc°dycatchyourmeaning,that inthisparticularyouaremakingthesamemistakeasmanyothers;that isifyoumeanto saythata longrope,oneofperhaps4° cubits,cannotholdup sogreata weightasa shorterlength,sayoneortwocubits,ofthesamerope.

SIMV.ThatiswhatI meant,andasfarasI seethepropositionishighlyprobable.

SALV.On the contrary,I considerit notmerelyimprobablebut false;andI think I caneasilyconvinceyouof yourerror.LetABrepresenttherope,fastenedat theupperendA: at thelowe.rendattach a weightC whoseforceis just sufficienttobreak

122 THE TWO NEW SCIENCESOF GALILEObreaktherope. Now,Simplicio,pointouttheexactplacewhereyouthinkthebreakoughttooceur.

[1621Sn_. Let ussayD.S_v. Andwhyat D?SrMe.Becauseat thispointthe ropeisnot strongenoughto

support,say,IOOpounds,madeupoftheportionoftheropeDBandthestoneC.

SALV.Accordinglywheneverthe ropeisstretched[violentata]withtheweightofIOOpoundsat D it willbreakthere.

Sna_.I thinkso.S_v. But tell me,if insteadof attachingthe weightat the

endof the rope,B,one fastensit at a pointnearerD, say, at E: or if, insteadof fixingthe upperendofthe ropeat A,onefastensit at somepointF, just

A. aboveD,willnottherope,at thepointD,be subjeCtto thesamepullofIOOpounds?

Sn_P.It would,providedyou includewith theP stoneCtheportionofropeF__.B.IB SALV.Let us thereforesupposethat the rope is

stretchedatthepointD witha weightofIOOpounds,then accordingto yourownadmissionit willbreak;but FE isonly a smallportionofAB;howcanyouthereforemaintainthat the longropeisweakerthanthe shortone? Give up then this erroneousview

B whichyousharewithmanyvery intelligentpeople,and let usproceed.

Now havingdemonstratedthat, in the caseof[uniformlyloaded]prismsand cylindersof constantthickness,the momentof forcetendingto produce

Fig.2I fracture[momentosoprale proprieresistenze]variesas the squareof the length;and havinglikewiseshownthat,whenthe lengthisconstantandthe thicknessvaries,the resist-anceto fraCturevariesas the cubeof the side,or diameter,ofthe base,letuspassto theinvestigationofthe caseof solidswhichsimultaneouslyvaryinbothlengthandthickness.HereIobservethat,

'i SECONDDAY xz3,¢

PROPOSITIONV

_ Prismsand cylinderswhichdifferin both length andthicknessofferresistancesto fraffture[i.e.,cansupportattheir endsloads]whichare directlyproportionalto thecubesofthe diametersoftheirbasesandinverselypropor-tionalto theirlengths.

[I65]LetABCandDEF betwosuchcylinders;thenthe resistance[bendingstrength]of thecylinderACbearsto the resistanceofthe cylinderDF a ratiowhichisthe produCtofthecubeofthediameterABdividedbythecubeofthediameterDE,andofthelengthEF dividedby the Alength BC. Make EG _--_equal to BC:let H be a Bthird proportionalto the ClinesAB and DE; let I D

be a fourth proportional,__[AB/DE=H/I]: and let - _:- ---, "....I :S=EF:BC. G FNowsincetheresistanceA_ B

of the cylinderAC is to D'. 'Ethat of the cylinderDG t_as thecubeofABisto thecubeofDE, that is,asthe I _lengthABis to the length $'I; andsincethe resistance Fig.zzofthe cylinderDG isto that of the cylinderDF asthe lengthFEisto EG,that is,asI is to S, it followsthat the lengthABis to S as the resistanceof the cylinderAC is to that of thecylinderDF. Butthe lineAB bearsto S a ratiowhichis theproduCtof AB/I and I/S. Hence the resistance[bendingstrength]ofthe cylinderACbearstothe resistanceof the cyl-inderDF a ratiowhichis the produCtof AB/I(that is,AB3/DE8)and of I/S (that is,EF/BC):whichis whatI meanttoprove.

This propositionhavingbeen demonstrated,let us nextconsider

_z4 THE TWO NEW SCIENCESOF GALILEOconsiderthe caseof prismsand cylinderswhichare similar.Concerningtheseweshallshowthat,

PROPOSITIONVI

Inthecaseofsimilarcylindersandprisms,themoments[stretchingforces]whichresultfrommultiplyingtogethertheirweightandlength[i.e.,fromthemomentsproducedby their ownweightand length],whichlatter acCtsas alever-arm,bearto eachother a ratiowhichis the sesqui-alteralofthe ratiobetweenthe resistancesoftheirbases.

In orderto provethis letusindicatethe twosimilarcylindersbyABandCD:thenthemagnitudeoftheforce[momento]inthecylinderAB,opposingthe resistanceof itsbaseB, bearsto themagnitude[momento]oftheforceat CD,opposingtheresistanceof its base D, a ratiowhichis the sesquialteralof the ratio

[I641betweenthe resistanceofthe baseB andthe resistanceof the

baseD. AndsincetheA BSolidsABandCD,are

effe&ivein opposingthe resistancesoftheirbasesB andD, in pro-

C m..... =-_D portiontotheirweightsandto the mechanical

Fig.23 advantages[forze]oftheirleverarmsrespectively,andsincetheadvantage[forza]oftheleverarmABisequalto the advantage[forza]ofthe leverarmCD (thisis true becausein virtueof the similarityof thecylindersthe lengthAB is to the radiusof the baseB asthelengthCDisto theradiusofthebaseD),it followsthat thetotalforce[momento]ofthecylinderABisto thetotalforce[mornento]ofthe cylinderCD astheweightaloneofthe cylinderABis tothe weightaloneof the cylinderGD,that is,as thevolumeofthe cylinderAB [l'istessocilindroAB] is to the volumeCD[all'istessoCD]:but these are as the cubesof the diametersoftheirbasesB andD; andthe resistancesof the bases,being

to

SECONDDAY I25to eachother as their areas,are to eachother consequentlyasthesquaresoftheirdiameters.Thereforetheforces[moment,]ofthecylindersareto eachotherinthesesquialteralratiooftheresistanceoftheirbases.*

SLMP.Thispropositionstrikesmeasbothnewandsurprising:at firstglanceit is very differentfrom'anythingwhichI my-self shouldhave guessed:for sincethesefiguresare similarin all other respects,I shouldhave certainlythoughtthatthe forces[moment,]andtheresistancesofthesecylinderswouldhavebornetoeachotherthesameratio.

SAG_.Thisistheproofofthepropositionto whichI referred,at theverybeginningofourdiscussion,as oneimperfectlyun-derstoodbyme.

SALv.Fora while,Simplicio,I usedto think,asyoudo,thatthe resistancesofsimilarsolidsweresimilar;but acertaincasualobservationshowedme that similarsolidsdo not exhibitastrengthwhichisproportionalto theirsize,thelargeronesbeinglessfittedto undergoroughusagejustastallmenaremoreaptthan smallchildrento be injuredby a fall. And,as we re-markedat the outset,a largebeamor columnfallingfroma

[x651givenheightwillgotopieceswhenunderthesamecircumstancesasmallscantlingorsmallmarblecylinderwillnotbreak. It wasthisobservationwhichledme to the investigationof the factwhichI amaboutto demonstratetoyou:itisa veryremarkablethingthat,amongthe infinitevarietyofsolidswhicharesimilaroneto another,therearenotwoofwhichthe forces[moment,],andtheresistancesof thesesolidsarerelatedinthe sameratio.

SLuP.YouremindmenowofapassageinArstofle'sQuestions* The precedingparagraphbeginningwith Prop.VI is of more than

usualinterest asillustratingthe confusionof terminologycurrent in thetime of Galileo. The translationgivenis literal exceptin the caseofthosewordsfor whichthe Italian is supplied. The factswhichGalileohas in mind are soevident that it is difficultto see howone can hereinterpret "mo_nent"to mean the force "opposingthe resistanceof itsbase,"unless"theforceof theleverarradg" betakento mean"themechanicaladvantageof the levermadeup of AB and theradiusof thebaseB"; and similarlyfor "theforceof theleverarmCD." [Trans.]

i26 THE TWO NEW SCIENCESOF GALILEOinMechanicsinwhichhetriesto explainwhyit isthat awoodenbeambecomesweakerandcanbemoreeasilybentas it growslonger,notwithstandingthe faCtthat the shorterbeamis thin-ner andthe longeronethicker:and, if I remembercorrectly,heexplainsit intermsofthe simplelever.

SALv.Verytrue: but, sincethis solutionseemedto leaveroomfordoubt,BishopdiGuevara,*whosetruly learnedcom-mentarieshave greatlyenrichedand illuminatedthis work,indulgesin additionalcleverspeculationswiththehopeofthusovercomingall difficulties;neverthelessevenhe is confusedasregardsthisparticularpoint,namely,whether,whenthe lengthandthicknessof thesesolidfiguresincreasein the sameratio,theirstrengthandresistanceto fraCture,aswellasto bending,remainconstant. Aftermuchthoughtuponthissubjeft,Ihavereachedthe followingresult. FirstI shallshowthat,

PROPOSITIONVII

Amongheavyprismsandcylindersof similarfigure,thereis oneand only one whichunderthe stressof its ownweightlies just on the limitbetweenbreakingand notbreaking:so that everylargeroneis unableto carrytheloadofitsownweightandbreaks;whileeverysmalleroneisabletowithstandsomeadditionalforcetendingtobreakit.

LetABbe a heavyprism,the longestpossiblethat willjustsustainitsownweight,sothat if it belengthenedthe leastbit itwillbreak. Then,I say,thisprismisuniqueamongallsimilarprisms--infinitein number--inoccupyingthat boundarylinebetweenbreakingand not breaking;so that everylargerone

[i661willbreakunderits ownweight,andeverysmalleronewillnotbreak,but willbe ableto withstandsomeforcein additionto itsownweight.

Let the prismCE be similarto, but largerthan,AB: then,I say, it willnot remaininta_ but willbreakunderits ownweight. Layoffthe portionCD,equalinlengthtoAB. And,since,theresistance[bendingstrength]ofCDisto that ofAB as

* BishopofTeano;b. 156I;d. I64L [Trans.]

SECONDDAY 127the cubeofthethicknessofCDistothe cubeofthethicknessof:_AB,that is,astheprismCEisto thesimilarprismAB,it followsthat the weightof CEis theutmostloadwhicha prismofthelengthCDcansustain;but the lengthofCE isgreater;there-forethe prismCE willbreak.ANowtake anotherprismFG _B P_--"-_"G _twhichis smallerthan AB.

LetFH equalAB,thenit can __,___beshownin a similarmannerC ............ Ethat the resistance[bending Dstrength]of FG is to that of Fig.24AB as the prismFG is to the prismAB providedthe dis-tanceABthat is FH, is equalto the distanceFG; but ABis greaterthan FG, and thereforethe momentof the prismFG appliedat G isnot sufficientto breaktheprismFG.

SAG_.Thedemonstrationisshortandclear;whiletheproposi-tion which,at firstglance,appearedimprobableis nowseento be bothtrueandinevitable.In orderthereforeto bringthisprisminto that limitingconditionwhichseparatesbreakingfromnot breaking,it wouldbe necessaryto changethe ratiobetweenthicknessandlengtheitherbyincreasingthethicknessorby diminishingthe length. Aninvestigationof thislimitingstatewill,I believe,demandequalingenuity.

SALV.Nay,evenmore;forthequestionismoredifficult;thisI knowbecauseI spentnosmallamountoftimein itsdiscoverywhichInowwishto sharewithyou.

PRoPosmo_VIIIGivena cylinderor prismof the greatestlengthconsist-entwithitsnot breakingunderitsownweight;andhavinggivena greaterlength,to find the diameterof anothercylinderor prismofthisgreaterlengthwhichshallbetheonlyandlargestonecapableofwithstandingitsownweight.

LetBCbe the largestcylindercapableofsustainingits ownweight;andletDEbe alengthgreaterthanAC:theproblemisto findthe diameterof the cylinderwhich,havingthe length

[I67]DE,

Tz8 THE TWO NEW SCIENCESOF GALILEODE, shallbe the largestonejust ableto withstandits ownweight. Let I be a thirdproportionalto the lengthsDE andAC;let the diameterFDbe to the diameterBAasDE is to I;drawthe cylinderFE; then, amongall cylindershavingthesameproportions,this is the largestandonlyonejust capableofsustainingitsownweight.

Let M be a thirdproportionalto DE and I: alsolet Obe afourthproportionalto DE, I, andM; layoffFG equalto AC.NowsincethediameterFD isto the diameterABasthelengthDE isto I, andsinceOisa fourthproportionalto DE,I andM,it followsthat FD3:BAS=DE:O.But the resistance[bending_. C strength]of the cylinderDG is

to the resistanceof the cylinder]_BCasthe cubeof FD isto the

cubeofBA:hencethe resistanceP ofthe cylinderDG isto that of:t, G cylinderBCasthe lengthDEis_a,J, ' ' !o: : to O. And sincethe moment

Fig.25 of the cylinderBC is held inequilibriumby [?equalealla]its resistance,weshallaccomplishourend(whichisto provethat themomentofthe cylinderFEis equalto the resistancelocatedat FD), if weshowthat themomentofthecylinderYEisto themomentofthecylinderBCasthe resistanceDF isto the resistanceBA,that is,asthe cubeof FD isto the cubeofBA,or as the lengthDE isto O. Themomentof the cylinderFE is to the momentof the cylinderDGasthe squareofDE isto the squareofAC,that is, asthelengthDE is to I; but the momentofthe cylinderDGisto themomentofthe cylinderBC,asthe squareofDFis to thesquareofBA,that is,asthesquareofDEisto thesquareofI, or asthesquareofI is to the squareofM, or,asI isto O. Thereforebyequatingratios,it resultsthat the momentofthe cylinderFE isto themomentofthecylinderBCasthelengthDE isto O,thatis,asthecubeofDFisto the cubeofBA,orastheresistanceofthe baseDFisto the resistanceofthe baseBA;whichwastobeproven.

SAGI_.This demonstration,Salviati,is ratherlongand diffi-cult

SECONDDAY x29_ cult to keepin mind from a singlehearing. Will you not,

therefore,begoodenoughto repeatit?SALV.Asyoulike;butI wouldsuggestinsteadamoredire_

anda shorterproof:this will,however,necessitatea different_o figure._-_ [x68];" SAGR.The favorwillbe that muchgreater:neverthelessI_ hopeyouwillobligemeby puttingintowrittenformthe argu-si mentjustgivensothat Imaystudyit atmyleisure.

SALV.I shallgladlydoso. LetA denotea cylinderofdiam-' eter DC andthe largestcapableofsustainingits ownweight:; the problemis to determinea largercylinderwhichshallbe

at oncethe maximumandtheuniqueonecapableofsustainingitsownweight.

LetE be sucha cylinder,similarto A,havingthe assignedlength,andhavinga diameterKL. LetMNbea thirdpropor-tionalto thetwolengthsDCandKL: n Alet MN alsobe the diameterof an- c_other cylinder,X, havingthe samelengthasE: then,I say,X isthe cyl-inder sought.Now sincethe resist- Manceofthe baseDC is to the resist-anceofthe baseRE as the squareofDCis to the squareofI{_L,that is,asthe squareof KL is to the squareof Fig.26MN,or,as the cylinderEis to the cylinderX, that is, as themomentE is to the momentX; and sincealsothe resistance[bendingstrength]of the baseKL is to the resistanceof thebaseMN as the cubeof KLis to the cubeof MN,that is,asthe cubeofDC isto thecubeofKL,or,asthecylinderAisto the cylinderE. that is,asthemolfientofAisto themomentof E; henceit follows,ex cecualiin proportioneperturbata,*that themomentofAisto themomentofX astheresistanceofthe baseDC is to the resistanceof the baseMN; thereforemomentand resistanceare relatedto eachotherin prismXpreciselyastheyareinprismA.

* For definitionof perturbataseeTodhunter'sEuclid,BookV,De{.20.[Trans.]

I3o THE TWO NEW SCIENCESOF GALILEOLet us now generalizethe problem;then it willread as

follows:Givena cylinderAC in whichmomentand resistance[bendingstrength]are relatedin anymannerwhatsoever;let DE be the lengthof anothercylinder;thendeterminewhatits thicknessmust be in orderthat the relationbe-tweenits momentand resistanceshallbe identicalwiththat ofthecylinderAC.

UsingFig.25in the samemanneras above,wemaysaythat,sincethe momentof the cylinderFE is to the momentof theportionDGas thesquareofED is to thesquareofFG,that is,as thelengthDEis to I; andsincethemomentofthe cylinderFGis to themomentofthe cylinderAC asthesquareofFD isto thesquareofA_B,or,asthesquareofEDis to the squareofI, or, asthe squareof I is to the squareofM, that is, as thelengthI is to O; it follows,exaequali,that themomentof the

[I69]cylinderFE is to themomentofthe cylinderACas thelengthDE is to O, that is,as the cubeof DE isto the cubeof I, or,asthe cubeofFD isto thecubeofAB,that is,astheresistanceofthebaseFD isto theresistanceofthebaseAB;whichwastobeproven.

Fromwhathas a/readybeendemonstrated,youcanplainlyseethe impossibilityof increasingthe sizeofstructuresto vastdimensionseitherin art or in nature;likewisetheimpossibilityof buildingships,palaces,or templesofenormoussizeinsuchawaythat theiroars,yards,beams,iron-bolts,and,in short,alltheirother parts willhold together;nor can natureproducetreesof extraordinarysizebecausethe brancheswouldbreakdownundertheirownweight;soalsoit wouldbe impossibletobuildupthebonystruCturesofmen,horses,orotheranimalssoasto holdtogetherandperformtheirnormalfunCtionsif theseanimalswereto be increasedenormouslyin height;for thisincreasein heightcanbe accomplishedonlybyemploy_gamaterialwhichisharderandstrongerthanusual,orbyenlargingthe sizeofthe bones,thus changingtheirshapeuntilthe formandappearanceof the animalssuggesta monstrosity.This is

perhaps

SECOND DAY x3Iperhapswhat our wisePoet had in mind,whenhe says,indescribingahugegiant:

"Impossibleit isto reckonhisheight"Sobeyondmeasureishissize."*

To illustratebriefly,I havesketcheda bonewhosenaturallength has been increasedthree timesand whosethicknesshasbeenmultiplieduntil,for a correspondinglylargeanimal,it wouldperformthe same_ncCtionwhichthe smallboneper-formsfor its smallanimal. Fromthe figureshereshownyoucan see howout of proportionthe enlargedbone appears.Clearlythenifonewishesto maintainina greatgiantthesameproportionoflimbasthat _ _t_foundinanordinarymanhe must either find aharderandstrongerma-terial for making the

[17o]bones,orhe mustadmita diminutionof strengthin comparisonwithmenofmediumstature;forifhis heightbe increased Fig. 27inordinatelyhewillfallandbecrushedunderhisownweight.Whereas,if the sizeof a bodybe diminished,the strengthofthat bodyisnotdiminishedin the sameproportion;indeedthesmallerthe bodythe greaterits relativestrength. Thus asmalldogcouldprobablycarryon hisbacktwoor threedogsof hisownsize;but I believethat a horsecouldnot carryevenoneofhisownsize.

SrM1,.Thismaybeso;butI amledto doubtit onaccountofthe enormoussizereachedby certainfish,suchas the whalewhich,I understand,is ten timesas largeas an elephant;yettheyallsupportthemselves.

SALv.Your question,Simplicio,suggestsanotherprinciple,• Non si pu_ compartirquantosia lungo,

Si smisuratamentek tuttogrosso.Ariosto'sOrlandoFurioso,XVII,3o[Trans.]

_3z THE TWO NEW SCIENCESOF GALILEOonewhichhadhithertoescapedmyattentionandwhichenablesgiants and other animalsof vast sizeto supportthemselvesand to moveaboutaswellas smalleranimalsdo. This resultmaybe securedeitherby increasingthe strengthof the bonesandotherpartsintendedto carrynotonlytheirweightbut alsothe superincumbentload;or, keepingthe proportionsof thebonystructureconstant,the skeletonwillholdtogetherin thesamemanneror evenmoreeasily,providedone diminishes,in the properproportion,the weightof the bony material,of the flesh,andof anythingelsewhichthe skeletonhas tocarry. It is thissecondprinciplewhichisemployedby naturein the strucCtureof fish,makingtheir bonesand musclesnotmerelylightbutentirelydevoidofweight.

SzMP.The trend of your argument,Salviatl,is evident.Sincefishliveinwaterwhichonaccountofitsdensity[corpulenza]or, as otherswould say, heaviness[gravitY]diminishestheweight[peso]of bodiesimmersedin it, youmeanto saythat,for thisreason,thebodiesoffishwillbe devoidofweightandwillbesupportedwithoutinjuryto theirbones. But this isnotall;foralthoughthe remainderof the bodyofthe fishmaybewithoutweight,therecan be noquestionbut that theirboneshaveweight.Takethe caseofa whale'srib,havingthe dimen-sionsofa beam;whocandenyitsgreatweightoritstendencytogoto thebottomwhenplacedinwater? Onewould,therefore,

[I7I]hardlyexpectthesegreatmassesto sustainthemselves.

SAT.V.AveryshrewdobjecCtion!Andnow,in reply,tellmewhetheryouhaveeverseenfishstandmotionlessat willunderwater,neitherdescendingto the bottomnor risingto the top,withouttheexertionofforcebyswimming?

Sire,.Thisis awell-knownphenomenon.SALv.The facetthenthat fishare ableto remainmotionless

underwaterisa conclusivereasonforthinkingthat thematerialof theirbodieshas the samespecificgravityas that ofwater;accordingly,if in theirmake-upthereare certainparts whichare heavierthanwatertheremustbe otherswhicharelighter,forotherwisetheywouldnotproduceequilibrium.

Hence

C-

SECOND DAY I33Hence,if the bonesareheavier,it isnecessarythat themus-

clesorotherconstituentsofthebodyshouldbelighterin orderthat their buoyancymay counterbalancethe weightof thebones. In aquaticanimalsthereforecircumstancesare justreversedfromwhattheyarewithlandanimalsinasmuchas, inthe latter,the bonessustainnotonlytheirownweightbut alsothat ofthe flesh,whilein the formerit is the fleshwhichsup-portsnot onlyits ownweightbut alsothat ofthe bones. Wemust thereforeceaseto wonderwhy theseenormouslylargeanimalsinhabitthe waterratherthan theland,that isto say,theair.

Snvn,.I am convincedandI onlywishto addthatwhatwecall landanimalsoughtreallyto be calledair animals,seeingthat theylivein theair, aresurroundedby air,andbreatheair.

SAG_.I haveenjoyedSimplicio'sdiscussionincludingboththe questionraisedand its answer. MoreoverI can easilyunderstandthat oneofthesegiantfish,ifpulledashore,wouldnotperhapssustainitselfforanygreatlengthoftime,butwouldbe crushedunderits ownmass as soonas the connecCtionsbetweenthebonesgaveway.

SALV.I am inclinedto youropinion;and, indeed,I almostthink that the samethingwouldhappenin the caseof a verybigshipwhichfloatson the seawithoutgoingto piecesunder

[I72]its loadofmerchandiseandarmament,but whichondry landandin air wouldprobablyfall apart. But let usproceedandshowhow:

Givena prismor cylinder,also its ownweightand themaximumloadwhichit can carry,it is then possibletofindamaximumlengthbeyondwhichthe cylindercannotbe prolongedwithoutbreakingunderitsownweight.

Let AC indicateboth the prismand its ownweight;alsolet D representthe maximumloadwhichthe prismcan carryat the endC withoutfracCture;it is requiredto findthe max-imumtowhichthelengthofthesaidprismcanbeincreasedwithoutbreaking.DrawAHof sucha lengththat theweightof the prismACis to the sumofACandtwicethe weightD

as

134 THE TWO NEW SCIENCESOF GALILEOasthe lengthCAisto AH;andletAGbe a meanproportionalbetweenCA and AH; then, I say,AGis the lengthsought.Sincethe momentofthe weight[momentogravante]D attachedat thepointCisequalto themomentofaweighttwiceaslargeasDplacedat themiddlepointAC,throughwhichtheweightof

....... the prismAC a&s,it fol-__ .... -x'[-_.......... l-ttlowsthat the momentofA the resistanceof the prismk _]__ AClocatedat A isequiva-

lentto twicethe weightDplustheweightofAC,bothacCtingthroughthe middle

Fig.28 pointofAC. Andsincewehave agreedthat the momentof the weightsthus located,namely,twiceD plusAC,bearsto the momentofACthesameratiowhichthelengthHAbearsto CAandsinceAGisameanproportionalbetweenthesetwolengths,it followsthat themo-mentoftwiceD plusACisto themomentofACas the squareofGAis to the squareof CA. But the momentarisingfromtheweight[momentopremente]oftheprismC_ isto themomentofACasthe squareofGAis to the squareofCA;thenceAGisthemaximumlengthsought,that is,the lengthupto whichtheprismACmaybeprolongedandstillsupportitself,but beyondwhichit willbreak.

Hithertowehave consideredthe momentsand resistancesof prismsand solidcylindersfixedat one endwith a weightappliedat the other end;threecaseswerediscussed,namely,that in whichthe appliedforcewasthe onlyoneacCting,thatin whichthe weightof the prismitselfis alsotakeninto con-sideration,and that in whichthe weightof the prismaloneistaken into consideration.Let us now considerthese same

[_73]prismsandcylinderswhensupportedat bothendsor at a singlepointplacedsomewherebetweenthe ends. In the firstplace,I remarkthat acylindercarryingonlyitsownweightandhavingthe maximumlength,beyondwhichit willbreak,will,whensupportedeitherin the middleor at both ends,havetwicethe

length

¢

i SECONDDAY 135lengthofonewhichis mortisedintoa wallandsupportedonlyat oneend. This is very evidentbecause,if wedenotethe

_: cylinderbyABCand ifweassumethat one-halfof it, AB, is!_ the greatestpossiblelength capableof supportingits own

weightwithoneendfixedat B,then,forthesamereason,if the; cylinderiscarriedonthe pointG, thefirsthalfwillbe counter-! balancedbytheotherhalfBC. Soalsointhecaseofthecylinder

DEF,if itslengthbesuchthat it willsupportonlyone-halfthisB

Fig.29lengthwhenthe endD isheldfixed,orthe otherhaftwhentheendF isfixed,thenit is evidentthat whensupports,suchasHandI, areplacedundertheendsD andF respecCtivelythemo-mentofanyadditionalforceorweightplacedatE willproducefracCtureat thispoint.

A more intricateand difficultproblemis the following:neglectthe weightof a solidsuchas the precedingandfindwhetherthe sameforceorweightwhichproducesfra&urewhenappliedat the middleofa cylinder,supportedat bothends,willalsobreakthe cylinderwhenappliedat someotherpointneareroneendthantheother.

Thus,forexample,ifonewishedto breaka stickby holdingit with onehand at eachend andapplyinghis kneeat themiddle,wouldthe sameforceberequiredtobreakit inthesamemannerifthe kneewereapplied,notat themiddle,but at somepointnearerto oneend?

SAGR.This problem,I believe,hasbeentoucheduponbyAristotleinhisQuestionsinMechanics.

Salv.

I36 THE TWO NEW SCIENCESOF GALTLEO[I74]

S_v. Hisinquiryhoweverisnotquitethe same;forhe seeksmerelyto discoverwhyit is that a stickmaybe moreeasilybrokenbytakinghold,onehandat eachendofthe stick,thatis, far removedfromthe knee,than if the handswereclosertogether. He givesa generalexplanation,referringit to thelengthenedleverarmswhicharesecuredbyplacingthehandsatthe endsof the stick. Ourinquirycallsforsomethingmore:whatwewantto knowiswhether,whenthehandsareretainedat the endsof the stick,the sameforceis requiredto breakitwhereverthe kneebe placed.

SAGR.At firstglancethiswouldappearto be so,becausethetwoleverarmsexert,in acertainway,the samemoment,seeingthat asonegrowsshortertheothergrowscorrespondinglylonger.

SALV.Nowyouseehowreadilyonefallsintoerrorandwhatcautionandcircumspec°donarerequiredto avoidit. Whatyouhave just saidappearsat firstglancehighlyprobable,but oncloserexaminationit provesto be quitefar fromtrue;as willbe seenfromthe fac°cthatwhethertheknec thefulcrumofthetwoleverswbeplacedin the middleornotmakessucha differ-encethat, if fractureis to be producedat anyotherpointthanthemiddle,thebreakingforceat themiddle,evenwhenmulti-plied ]:our,ten, a hundred,or a thousandtimeswouldnotsuffice.Tobeginwithweshalloffersomegeneralconsiderationsand then passto the determinationof the ratioin whichthebreakingforcemustchangein orderto producefrac°cureat onepointratherthananother.

Let AB denotea woodencylinderwhichis to be brokeninthe middle,overthe supportingpointC, andletDE representan identicalcylinderwhichis to be brokenjust overthe sup-portingpoint F whichis not in the middle. First of all it isclearthat, sincethe distancesACandCBare equal,the forcesappliedat theextremitiesBandAmustalsobeequal. Secondlysincethe distanceDF is lessthanthe distanceACthemomentof anyforceac°dngat D is lessthan the momento]_the sameforceatA, that is,appliedat thedistanceCA;andthemomentsare lessin the ratioof thelengthDF to AC;consequentlyit is

necessary

¢

_ SECOND DAY x37necessaryto increasethe force[momento]at D in orderto over-come,or evento balance,the resistanceat F; but incomparison

i with the length AC the distanceDF can be diminishedin-definitely:inorderthereforeto counterbalancethe resistanceatF it will be necessaryto increaseindefinitelythe force [forza]applied at D. On the other A C Bhand, inproportionas we in-

creasethe distanceFE over 5_that of CB,we must diminishthe force at E in order to I_ P _.x:ounterbalancethe resistance_--_._.__i_-7-._at F; but the distance FE, _____ .....measured in terms of CB,__ .....<--_---_<-_'_'_cannot be increased indefi- Fig.3onitelyby slidingthe fulcrumF towardthe endD; indeed,it can-notevenbe madedoublethe lengthCB. Thereforethe forcere-quired at E to balancethe resistanceat F will alwaysbe morethan halfthat requiredat B. It isclearthen that, asthe fulcrumF approachesthe end D, we must of necessityindefinitelyin-creasethe sum of the forcesapplied at E and D in order tobalance,or overcome,the resistanceat F.

SAGR.What shallwe say, Simplicio? Must we not confessthat geometry is the most powerfulof all instruments forsharpeningthe wit and training the mind to think correc°cly?Was not Plato perfecCtlyright when he wishedthat his pupilsshould be first of all well groundedin mathematics? As formyself,I quite understoodthe propertyof the leverand how,by increasingor diminishingits length, one can increaseordiminishthe moment of forceand of resistance;and yet, inthe solutionofthe presentproblemI wasnot slightly,butgreatly,deceived.

Sire,. IndeedI beginto understandthat whilelogicis an ex-cellentguidein discourse,it doesnot, as regardsstimulationtodiscovery,comparewith the powerof sharp distinCtionwhichbelongsto geometry.

SA¢I_.Logic, it appears to me, teachesus how to test theconclusiveness

I38 THE TWO NEW SCIENCESOF GALILEOconclusivenessof any argumentor demonstrationalreadydis-coveredandcompleted;but I donot believethat it teachesus to discovercorreCtargumentsanddemonstrations.But itwouldbe betterif Salviatiwereto showus in justwhat pro-portiontheforcesmustbeincreasedinorderto producefractureas the fulcrumis movedfromonepointto anotheralongoneandthesamewoodenrod.

[176]SaLv.The ratiowhichyou desireis determinedasfollows:

If upon a cylinderonemarkstwo pointsat whichfrac-ture is to be produced,then theresistancesat thesetwopointswillbear to eachother the inverseratio of thereCtanglesformedby the distancesfromthe respeCtivepointsto the endsofthe cylinder.

LetA andB denotethe leastforceswhichwillbringaboutfraCtureof the cylinderat C; likewiseE and F the smallestforceswhichwillbreakitat D. Then,I say,that thesumoftheforcesA and]3is to the sumofthe forcesE andF as the areaofthe reCtangleAD.DBisto theareaofthe reCtangleAC.CB.Becausethe sumoftheforcesAand]3bearsto the sumoftheforcesEandF aratiowhichistheproduCtofthethreefollowingratios,namely,(A+B)/B,]3/F,andF/(F+E); but the lengthBAisto thelengthCAasthesumoftheforcesAand]3isto the

AI. _D _c _:force ]3; and,as the lengthDB

is to the lengthCB, so is theforce]3 to the forceF; alsoas

E the lengthADis to AB,so is theforceF to the sumof theforcesF

Fig.31 andE.Henceit followsthat the sumof the forcesA andB bears

to the sumofthe forcesE andF a ratiowhichis the produCtof the three followingratios,namely,BA/CA,BD/BC,andAD/AB.ButDA/CAis the produCtof DA/BAandBA/CA.Thereforethe sumoftheforcesA andB bearsto thesumoftheforcesE andF a ratiowhichis the produCtofDA:CAandDB:CB. But the refftangleAD.DBbearsto the reCtangleAC.CBa ratiowhichis the produCtof DA/CAandDB/CB.

Accordingly

SECONDDAY I39Accordinglythe sumofthe forcesA andB isto thesumoftheforcesE and F as the rec°cangleAD.DBis to the red'tangleAC.CB,that is,theresistanceto fracCtureatCistotheresistanceto fracCtureat D as the recCtangleAD.DBis to the rec°cangleAC.CB. Q.E.D.

[I77]Anotherratherinterestingproblemmaybe solvedas a con-

sequenced thistheorem,namely,Giventhe maximumweightwhicha cylinderor prismcansupportat its middle-pointwherethe resistanceisa mini-mum, andgivenalsoa largerweight,find that point inthe cylinderforwhichthis largerweightis the maximumloadthatcanbesupported.

Let that oneof the givenweightswhichis largerthan themaximumweightsupportedat the middleof the cylinderABbearto thismaximumweightthe sameratiowhichthe lengthE bearsto the lengthF. The problemis to findthat pointin the cylinderat whichthis largerweightbecomesthe max-imumthat canbe supported.LetG be a meanproportionalbetweenthe lengthsE andF. DrawADand S sothat theybearto eachotherthesameratioasE to G; accordinglySwillbelessthanAD.

LetADbe the diameterofa semicircleAHD,inwhichtakeAHequalto S;jointhepointsH andD andlayoffDRequaltoHD. Then,I say,R is thepointsought,namely,the pointatwhichthe givenweight,greaterthanthe maximumsupportedat the middleof the cylinderD,wouldbecomethe maximumload.

OnABas diameterdrawthesemicircle_ANB:erec°cthe per-pendicularRN andjointhe pointsN andD. Nowsincethesumof the squareson NR andRD is equalto the squareofND,that is,to thesquareofAD,orto thesumofthe squaresofAHandHD;and,sincethesquareofliD isequalto thesquareofDR, it followsthat thesquareofNR, that is,the recCtangleAR.RB,is equalto the squareof AH,alsothereforeto thesquareof S;but the squareofS is to the squareofADasthelengthF is to the lengthE, that is, as the maximumweight

supported

I4o THE TWO NEW SCIENCESOF GALILEOsupportedat D istothe largerofthetwogivenweights.Hencethelatterwillbethemaximumloadwhichcanbecarriedat thepointR; whichisthesolutionsought.

SACR.NowIunderstandthoroughly;andI amthinkingthat,sincetheprismABgrowsconstantlystrongerandmoreresistant

to the pressureof its load atpointswhicharemoreandmore

l_I removedfrom the middle,wecouldin the caseof largeheavybeamscut awaya considerable

A _ portion near the ends whichwouldnotablylessenthe weight,

Z, ' ' andwhich,in the beamworkoflargerooms,wouldproveto be

It: : ofgreatutilityandconvenience.Fig.32 [_781

It wouldbe afinethingifonecoulddiscoverthepropershapeto givea solidin orderto makeit equallyresistantat everypoint, in whichcasea loadplacedat the middlewouldnotproducefra&uremore easily than if placedat any otherpoint.*

SAT.v.I wasjust on the point of mentioningan interestingandremarkablefa& conne&edwith thisveryquestion.Mymeaningwillbe clearerif I drawa figure. Let DB representa prism;then,aswehavealreadyshown,its resistanceto frac-ture[bendingstrength]at theendAD,owingtoa loadplacedattheendB,willbelessthantheresistanceat CIintheratioofthelengthCB to AB. Nowimaginethis sameprismto be cutthroughdiagonallyalongthe lineFB sothat the oppositefaceswillbe triangular;thesidefacinguswillbeFAB. Sucha solid

*Thereaderwillnoticethat two differentproblemsarehereinvolved.That which is suggestedin the last remark of Sagredois the fol-lowing:

To find a beam whosemaximumstresshas the same value whena constantloadmovesfromoneend of the beamto the other.

ThesecondproblemwtheonewhichSalviatiproceedsto solve--isthefollowing:

To find a beam in all cross-secCtionsof which the maximumstressis thesamefora constantloadinafixedposition. [Trans.]

SECONDDAY I4Iwillhavepropertiesdifferentfromthoseof the prism;for,ifthe loadremainat B, the resistanceagainstfracCture[bendingstrength]at C willbe lessthan that at A in the ratioof thelengthCBto the lengthAB. Thisiseasilyproved:forifCNOrepresentsa cross-secCtionparallelto AFD,thenthelengthFAbearsto the lengthCN, in the triangleFAB,the sameratiowhichthe lengthAB bearsto D I

the lengthCB. Therefore,ifl_._.._weimagineA andC to be the ] .-i...

pointsat whichthe fulcrumis [ pI.......... __'-'l'--,Jplaced,the leverarms in thetwocasesBA,AF andBC,CN.4,''_' L ¢Vwill be proportional[simih]. Fig.33HencethemomentofanyforceappliedatB andacCtingthrough.the armBA,againsta resistanceplacedat a distanceAFwillbe equalto that of thesameforceat B acCtingthroughthearmBCagainstthe sameresistancelocatedat adistanceCN. Butnow,if theforcestillbeappliedat B,the resistancetobe over-comewhenthe fulcrumisat C,a&ingthroughthe armCN,islessthanthe resistancewiththefulcrumat A inthe samepro-portionas the recCtangularcross-seCtionCO is lessthan therecCtangularcross-secCtionAD,that is,as the lengthCN islessthanAF,orCBthanBA.

Consequentlythe resistanceto fracCtureat C,offeredby theportionOBC,is lessthantheresistanceto fracCtureatA,offeredby theentireblockDAB,in thesameproportionasthe lengthCBissmallerthanthelengthAB.

By thls diagonalsaw-cutwe havenowremovedfromthebeam,or prismDB,a portion,i. e., a half,andhave left thewedge,or triangularprism,FBA. We thus have two solids

[I791possessingoppositeproperties;one body growsstrongerasit is shortenedwhilethe othergrowsweaker. This beingsoit wouldseemnotmerelyreasonable,but inevitable,that thereexistsa lineofsecCtionsuchthat,whenthesuperfluousmaterialhasbeenremoved,therewillremaina solidofsuchfigurethatitwillofferthesameresistance[strength]atallpoints.

Simp.

I4_ THE TWO NEW SCIENCESOF GALII,FOSnaP.Evidentlyonemust,in passingfromgreaterto less,

encounterequality.SAcx.But nowthe questionis whatpath the sawshould

followinmakingthecut.Sr_. Itseemstomethatthisoughtnotto bea difficulttask:

forifby sawingtheprismalongthediagonallineandremovinghalfofthematerial,the remainderacquiresapropertyjust theoppositetothatoftheentireprism,sothat at everypointwherethe latter gainsstrengththe formerbecomesweaker,then itseemsto methat by takingamiddlepath,i.e.,byremovinghalftheformerhall orone-quarterofthewhole,thestrengthoftheremainingfigurewillbeconstantat allthosepointswhere,inthetwopreviousfigures,thegainin onewasequalto thelossintheother.

SALV.Youhavemissedthe mark,Simplicio.For,asI shallpresentlyshowyou,theamountwhichyoucanremovefromtheprismwithoutweakeningit is not a quarterbut a third. Itnowremains,as suggestedby Sagredo,to discoverthe pathalongwhichthesawmusttravel:this,asI shallprove,mustbeaparabola.But it is firstnecessarytodemonstratethe fol!owinglemma:

If thefulcrumsare soplacedundertwoleversorbalancesthatthearmsthroughwhichtheforcesacCtareto eachotherinthe sameratioas thesquaresofthe armsthroughwhichtheresistancesacCt,andif theseresistancesareto eachotherin thesameratioasthearmsthroughwhichtheyact,thenthe forceswillbe equal.

Let AB and CD representtwo leverswhoselengthsareA ,. , . , _l_dividedby their fulcrumsin

I_ _ -such a wayasto makethe dis-_ tanceEB bearto the distance

D FDaratiowhichisequalto theFig.34 squareof theratiobetweenthe

distancesEAand FC. Letthe resistanceslocatedat AandC[ 8o]

be to eachotherasEAisto FC. Then,I say, theforceswhichmustbe appliedat B andD in orderto holdin equilibriumthe

resistances

SECONDDAY I43resistancesat A andC areequal. Let EGbea meanpropor-tionalbetweenEB and FD. ThenweshallhaveBE-.EG=EG:FD=AE:CF.But this last ratiois preciselythat whichwehaveassumedto existbetweenthe resistancesat A andC.AndsinceEG:FD=AE:CF,it follows,permutando,that EG:AE=FD:CF.SeeingthatthedistancesDCandGAaredividedin thesameratiobythepointsF andE, it followsthat thesameforcewhich,whenappliedat D, willequilibratethe resistanceat C,wouldifappliedatGequilibrateatAa resistanceequaltothat foundatC.

Butonedatumoftheproblemisthatthe resistanceat AistotheresistanceatCasthedistanceAEisto thedistanceCF,orasBEisto EG. Thereforetheforceappliedat G,orratherat D,will,whenappliedatB,justbalancetheresis_ncelocatedatA.

Q.E.D.

ThisbeingcleardrawtheparabolaFNBin thefaceFBoftheprismDB. Let the prismbesawedalongthisparabolawhosevertexis at B. Theportionofthe solidwhichremainswillbeincludedbetweenthe baseAD,the recCtangularplaneAG,thestraightlineBG andthe surfaceDGBF,whosecurvatureisidenticalwiththat oftheparabolaFNB. Thissolidwillhave,I say,the samestrengthat everypoint. Let the solidbe cutby a planeCOparallelto D......:..................... .the planeAD. Imagine ., ...............io (_f I :......4..the pointsA andC to be _ _o.. J..Ithe fulcrumsof twolevers [...fl. .b[_

....... ° ........... _ ........ _ ....ofwhichonewillhavethearmsBAandAF;theother_ cBCandCN. Thensincein Fig.35theparabolaFBA,wehaveBA:BC=AF_:CN2,it isclearthatthearmBAofoneleveristothearmBCoftheotherleverasthesquareof the armAF is to the squareof the otherarmCN.Sincethe resistanceto be balancedby the leverBAis to theresistanceto be balancedbythe leverBCin the sameratioasthe recCtangleDAisto the recCtangleOC,that is as the lengthAFistothelengthCN,whichtwolengthsaretheotherarmsofthe levers,it 'follows,by the lemmajust demonstrated,thatthe

I44 THE TWO NEW SCIENCESOF GALILEOthe sameforcewhich,whenappliedat BGwillequilibratetheresistanceat DA,willalsobalancethe resistanceat CO. The

[iSi]"sameis true for any other secCfion.Thereforethisparabolicsolidisequallystrongthroughout.

It cannowbe shownthat, if the prismbe sawedalongthelineoftheparabolaFN-B,one-thirdpartof it willbe removed;becausethe rectangleFB andthe surfaceFNBAboundedbytheparabolaare the basesof twosolidsincludedbetweentwoparallelplanes,i. e.,betweenthe rectanglesFB andDG; con-sequentlythe volumesof thesetwosolidsbearto eachotherthe sameratioas theirbases. But the areaof the red-tangleis oneand a halftimesas largeas the areaFNBAundertheparabola;henceby cuttingtheprismalongthe parabolawere-moveone-thirdof the volume. It is thus seenhowonecandiminishthe weightofa beamby asmuchas thirty-threepercentwithoutdiminishingits strength;a fact ofnosmallutilityinthe construcCtionoflargevessels,andespeciallyin supportingthe decks,sincein suchstructureslightnessis of primeim-portance.

SAcraTheadvantagesderivedfromthisfactaresonumerousthat it wouldbe both wearisomeand impossibleto mentionthemall;but leavingthismatterto oneside,I shouldliketolearnjusthowit happensthat diminutionofweightis possibleinthe ratioabovestated. I canreadilyunderstandthat,whena sectionis madealongthe diagonal,one-halfthe weightisremoved;but, as for the parabolicsectionremovingone-thirdofthe prism,thisI canonlyacceptonthewordofSalviatiwhoisalwaysreliable;howeverI preferfirst-handknowledgeto thewordofanother.

SALe.Youwouldlikethena demonstrationof the factthatthe excessof the volumeof a prismoverthe volumeofwhatwe havecalled.theparabolicsolidis one-thirdof the entireprism. This I havealreadygivenyouon apreviousoccasion;howeverI shallnowtry to recallthe demonstrationin whichI rememberhavinguseda certainlemmafromArchimedes'bookOn Spirals,*namely,Givenany numberof lines,differingin

* For demonstrationof the theoremhere cited, see "Works of Arch-

SECONDDAY 145lengthonefromanotherbya commondifferencewhichasequalto the shortestoftheselines;andgivenalsoanequalnumberof lineseachof whichhasthe samelengthas the longestofthe first-mentionedseries;thenthe sumofthe squaresof thelinesofthissecondgroupwillbelessthanthreetimesthesumofthesquaresofthelinesinthefirstgroup.Butthesumofthesquaresofthesecondgroupwillbegreaterthanthreetimesthesumofthesquaresofallexceptingthelongestofthefirstgroup.

[I82]Assumingthis,inscribeinthe re&angleACBPtheparabola

AB. Wehavenowto provethatthemixedtriangleBAPwhosesidesareBP andPA,andwhosebaseistheparabolaBA,is athirdpartoftheentirere(tangleCP. If thisisnottrueit willbeeithergreaterorlessthana third. Supposeitto belessbyanareawhichisrepresentedbyX. Bydrawinglinesparallelto thesidesBPandCA,wecandividethe re&angleCP intoequalparts;andif the processbecontinuedweshallfinallyreachadivisionintopartssosmallthat eachof themwillbesmallerthan the areaX; let the rec-s vtangleOBrepresentoneofthese T oparts and, throughthe points _ swherethe otherparallelscutthe "_'_ Rparabola,drawlinesparallelto .... r .'2,, _I,AP. Letusnowdescribeabout ... E\our "mixedtriangle" a figure _ \ AKmadeup of re&anglessuchas ¢ D,BO,IN, HM,FL,EK,andGA; ] x ]this figurewillalsobe lessthan Fig.36a thirdpartofthe re&angleCPbecausetheexcessofthisfigureabovetheareaofthe "mixedtriangle"ismuchsmallerthanthere&angleBOwhichwehavealreadymadesmallerthanX.

SAGtt.Moreslowly,please;_[orI donotseehowtheexcessofthisfiguredescribedaboutthe"mixedtriangle"ismuchsmallerthanthere&angleBO.

SALV.Doesnotthere&angleBOhavean areawhichisequalto thesumoftheareasofallthelittlere&anglesthroughwhichimede._"translatedbyT.L.Heath(Camb.Univ.PressI897)p. Io7andp. I6_, [Trans.]

i46 THE TWO NEW SCIF_CES OF GALII.FD

the parabolapasses?I meanthe re&anglesBI, IH, I-IF,FE,EG, andGAofwhichonlya partliesoutsidethe "mixedtri-angle." Havewenot takenthe re&angleBOsmallerthantheareaX? Thereforeif,as ouropponentmightsay,the triangleplusX is equalto a thirdpart of this re&angleCP, the cir-cumscribedfigure,wtfichaddsto the trianglean arealessthanX, willstillremainsmallerthana thirdpart of the re&angle,CP. But thiscannotbe, becausethis circumscribedfigureislargerthan a thirdof the area. Hencek is not truethat our"mixedtriangle"islessthana thirdofthere&angle.

[ s3]SAGR.Youhaveclearedupmydifficulty;but it stillremains

to be shownthat thecircumscribedfigureislargerthana thirdpartofthe re&angleCP,a taskwhichwillnot, I believe,provesoeasy.

S_v. Thereis nothingverydifficultaboutit. Sincein theparabolaDE'_'=DA:AZ= re&angleKE: re&angleAG,seeingthatthealtitudesofthesetwore&angles,_A_KandKL,areequal, it followsthat E--D_:__2:_*=re&angle KE:re&angleKZ. In preciselythe samemannerit maybe shownthat theotherre&anglesLF,MH,NI,0]3,standto oneanotherin the sameratioasthe squaresof thelinesMA,NA,OA,PA.

Let us nowconsiderthe circumscribedfigure,composedofareaswhichbearto eachotherthesameratioasthesquaresofaseriesoflineswhosecommondifferencein lengthisequalto theshortestonein the series;notealsothat the re&angleCP ismadeup of anequalnumberofareaseachequalto the largestandeachequalto there&angleOB. Consequently,accordingtothelemmaofArchimedes,thecircumscribedfigureislargerthanathirdpartofthere&angleCP;butit wasalsosmaller,whichisimpossible.Hencethe "mixedtriangle"is not lessthan athirdpartofthere&angleCP.

Likewise,I say,it cannotbegreater. For,letussupposethatitisgreaterthanathirdpartofthere&angleCPandlettheareaX representthe excessofthe triangleoverthethirdpartof there&angleCP; subdividethe rectangleintoequalre&anglesandcontinuethe processuntiloneof thesesubdivisionsis smaller

than

SECONDDAY x47thantheareaX. LetBOrepresentsucha rec°_nglesmallerthanX. Usingthe abovefigure,wehavein the"mixedtriangle"aninscribedfigure,madeup of the red-tanglesVO,TiM,SM,RL,andQK,whichwillnot be lessthana thirdpart ofthe largerectangleCP.

For the "m_ed triangle"exceedsthe inscribedfigureby aquantitylessthan that bywhichit exceedsthe thirdpart ofthe recCtangleCP; to seethat this is truewehaveonlyto re-memberthat theexcessofthetriangleoverthethirdpartoftherectangleCP is equalto the areaX, whichis lessthan therecetangleBO,whichin turn ismuchlessthanthe excessofthetriangleover the inscribedfigure. For the rectangleBO is

[_84]madeupofthesmallredt_mglesAG,GE,EF,FH,HI,andIB;andthe excessof the triangleoverthe inscribedfigureis lessthan half the sumof theselittle reCtangles.ThussincethetriangleexceedsthethirdpartoftherectangieCPbyanamountX, whichis moretbnn that by whichit exceedsthe inscribedfigure,thelatterwillalsoexceedthethirdpartofthe rectangle,CP. But,bythe lemmawhichwehaveassumed,it issmaller.For the rectangleCP,beingthe sumofthe largestrectangles,bearsto the componentrectanglesof the inscribedfigurethesameratiowhichthe sumofall the squaresofthe linesequalto the longestbearsto the squaresof the lineswhichhaveacommondifference,after the squareof the longesthasbeensubtracCted.

Therefore,as in the caseof squares,the sumtotal of thelargesttee°tangles,i. e.,the rectangleCP,is greaterthanthreetimesthe sumtotalof thosehavinga commondifferenceminusthelargest;but theselastmakeupthe inscribedfigure.Hencethe "mixedtriangle"is neithergreaternor lessthanthe thirdpartofrectangleCP;itisthereforeequalto it.

SACR.Afine,cleverdemonstration;andall the moresobe-causeit givesusthequadratureoftheparabola,provingitto befour-thirdsofthe inscribed*triangle,a faCtwhichArchimedesdemonstratesbymeansoftwodifferent,butadmirable,seriesof

• Distinguishcarefullybetweenthis triangleand the "mixed tri-angle" abovementioned.[Trans.]

x48 THE TWO NEW-SCIENCESOF GALII.F.Omanypropositions.This sametheoremhasalsobeenrecentlyestablishedby LucaValerio,*the Archimedesof our age;hisdemonstrationistobefoundinhisbookdealingwiththecenters

•ofgravityofsolids.S_v. Abookwhich,indeed,isnotto beplacedsecondto any

producedby the mosteminentgeometerseitherof thepresentor ofthe past;a bookwhich,assoonasit fellintothehandsofourAcademician,ledhimto abandonhisownresearchesalongtheselines;forhe sawhowhappilyeverythinghadbeentreatedanddemonstratedbyValerio.

[i8s1SAoR.WhenI wasinformedofthiseventbytheAcademician

himself,I beggedofhimto showthe demonstrationswhichhehad discoveredbeforeseeingValerio'sbook;but in thisI didnotsucceed.

SAT.V.I havea copyofthemandwillshowthemto you;foryouwillenjoythe diversityofmethodemployedby thesetwoauthorsin reachingandprovingthe sameconclusions;youwillalsofindthat someoftheseconclusionsareexplainedindifferentways,althoughbothareinfactequallycorrect.

SAoR.I shallbe muchpleasedto seethemandwillconsiderit a greatfavorifyouwillbringthemto our regularmeeting.But in themeantime,consideringthe strengthofa solidformedfroma prismby meansofa parabolicsection,wouldit not, inviewofthe fa_ that thisresultpromisesto bebothinterestingandusefulinmanymechanicaloperations,beafinethingifyouwereto givesomequickandeasyruleby whicha mechanicianmightdrawa parabolauponaplanesurface?

SALV.Thereare manywaysof tracingthesecurves;I will omentionmerelythe twowhichare the quickestof all. Oneofthese is really remarkable;becauseby it I can trace thirtyor fortyparaboliccurveswithno lessneatnessand precision,and in a shortertime than anothermancan,by the aid of acompass,neatlydrawfouror sixcirclesof differentsizesuponpaper. I take a perfedtIyroundbrassball aboutthe sizeof awalnutandprojectit alongthesurfaceofametallicmirrorheld

*An eminent Italian mathematician, contemporarywith Galileo.[Trans.]

SECONDDAY 149in a nearlyuprightposition,sothat the ballin itsmotionwillpressslightlyuponthe mirrorandtraceout a finesharppara-bolicline;this parabolawillgrowlongerandnarroweras theangleof elevationincreases.Theaboveexperimentfurnishesclearandtangibleevidencethat the pathof a projectileis aparabola;a factfirstobservedbyourfriendanddemonstratedbyhiminhisbookonmotionwhichweshalltakeupat ournextmeeting. In the executionof this method,it is advisabletoslightlyheatandmoistentheballbyrollinginthehandinorderthatitstraceuponthemirrormaybemoredistincct.

[I86]Theothermethodofdrawingthedesiredcurveupontheface

of the prismis the following:Drivetwonailsintoa wallata convenientheightandat the samelevel;makethe distancebetweenthesenailstwicethewidthoftherecCtangleuponwhichit is desiredto tracethe semiparabola.Overthesetwonailshanga lightchainof sucha lengththat the depthof its sagis equalto thelengthofthe prism.Thischainwillassumetheformof a parabola,*sothat if this formbe markedbypointsonthe wallweshallhavedescribeda completeparabolawhichcanbe dividedintotwoequalpartsby drawinga verticallinethrougha pointmidwaybetweenthe twonails. The transferofthiscurveto the twoopposingfacesoftheprismisa matterofnodifficulty;anyordinarymechanicwillknowhowto doit.

By use of the geometricallinesdrawnuponour friend'scompass,tonemay easilylayoffthosepointswhichwilllocatethissamecurveuponthe samefaceoftheprism.

Hithertowehavedemonstratednumerousconclusionsper-tainingto the resistancewhichsolidsofferto fracture. Asa startingpointforthisscience,weassumedthat theresistanceofferedby the solidto a stralght-awaypullwasknown;fromthis baseonemightproceedto the discoveryof manyotherresultsandtheirdemonstrations;oftheseresultsthenumberto

* It isnowwellknownthat this curveisnot a parabolabut a eatenarythe equationof whichwasfirstgiven,49yearsafter Galileo'sdeath, byJamesBernoulli. [Trans.]

t Thegeometricaland militarycompassof Galileo,describedinNat.Ed.Vol.2. [Trans.l

i5o THE TWO NEW SCIENCESOF GALtT._Obe foundin natureisinfinite. But, in orderto bringourdailyconferenceto an end,I wishto discussthe strengthofhollowsolids,whichareemployedin art--andstilloftenerinnature--in a thousandoperationsfor thepurposeof greatlyincreasingstrengthwithoutaddingto weight;examplesoftheseare seenin thebonesofbirdsandin manykindsofreedswhicharelightand highlyresistantboth to bendingand braking. For if astemof strawwhichcarriesa headof wheatheavierthan theentirestalkweremadeup of the samemount ofmaterialin

[I87]solidformitwouldofferlessresistanceto bendingandbreaking.Thisis anexperiencewhichhasbeenverifiedandconfirmedinpracticewhereit isfoundthat ahollowlanceor a tubeofwoodormetalismuchstrongerthanwouldbea solidoneofthesamelengthandweight,onewhichwouldnecessarilybe t_nner;menhavediscovered,therefore,that in orderto makelancesstrongas wellas lighttheymustmakethemhollow.We shallnowshowthat:

In the caseof two cylinders,onehollowthe other solidbut havingequalvolumesandequallengths,theirresist-ances[bendingstrengths]areto eachotherin the ratiooftheirdiameters.

LetAEdenotea hollowcylinderandIN a solidoneof theA same weight and length;

(_ .-_' _-- _____2 --_ then, I say, that the resist-ance against fracCtureex-hibitedbythetubeAEbearsB

! to that of the solidcylinder-_--_-- _. ' ,-_, IN the sameratioas thedi-

ameterAB to the diameterI _L _ -¢,r n--i-iL_ne • • qIL. lh_s _sveryewdent;/or

Fig.37 sincethe tubeandthe solidcylinderIN havethesamevolumeandlength,theareaofthecir-cularbaseILwillbeequaltothat oftheannulusABwhichisthebaseofthetubeAE. (Byannulusis heremeantthe areawhichliesbetweentwo concentriccirclesof differentradii.) Hencetheirresistancestoa straight-awaypullareequal;butinproduc-

ing

SECONDDAY I5Iingfra_urebya transversepullweemploy,in thecaseof thecylinderIN, thelengthLN asoneleverarm,the pointL asafulcrum,andthediameterLI,orits half,asthe opposingleverarm:whilein thecaseofthe tube,the lengthBEwhichplaysthepartofthefirstleverarmisequalto LN,theopposingleverarmbeyondthe fulcrum,B, is thediameterAB,or its half.Manifestlythentheresistance[bendingstrength]of the tubeexceedsthatofthesolidcylinderintheproportioninwhichthediameterABexceedsthediameterIL"whichisthedesiredresult.

[ 881Thus the strengthof a hollowtube exceedsthat of a solidcylinderin the ratioof theirdiameterswheneverthe two aremadeofthesamematerialandhavethesameweightandlength.

It maybewellnextto investigatethegeneralcaseoftubesandsolidcylindersofconstantlength,butwiththeweightandthehollowportionvariable.Firstweshallshowthat:

Givena hollowtube,a solidcylindermaybedeterminedwhichwillbeequal[eguale]to it.

Themethodisverysimple. LetABdenotetheexternalandCD the internaldiameterofthe tube. In the largercirclelayoff thelineAE equalin lengthto the di- /LameterCD; join the pointsE and B. fNowsincethe angleat E inscribedin a //_semicircle,AEB,i8a right-angle,the area/ /ofthecirclewhosediameterisABisequal[jto the sumof the areasofthe twocircles|/whoserespecCtivediametersareAE and_"EB. ButAEi8thediameterofthehollow\portionof the tube. Thereforethe areaofthe circlewhosediameteris EB is thesameas the area of the annulusACBD. Fig. 38Hencea solidcylinderof circularbasehavinga diameterEBwillhavethe samevolumeas the wallsof the tube of equallength.

Byuseofthistheorem,it iseasy:Tofindtheratiobetweenthe resistance[bendingstrength]of anytubeandthatofanycylinderof equallength. Let

I52 THE TWO NEW SCIENCESOF GALII.EOLetABEdenotea tubeandRSMa cylinderofequallength:itis requiredto findthe ratiobetweentheirresistances.Usingtheprecedingproposition,determinea cylinderILNwhichshall

have the same volumeand"_% - _ ]engthas the tube. Drawa¢) lineV of sucha lengththat] :B it willbe relatedto IL and

RS (diametersof the bases1_i ofthecylindersINandR_M),

(_ ' _._ as ,ollows:V:RS--RS:IL.yrT Then,I say,theresistanceofIkl/ /

v, thetubeAEisto that oftheIk cylinderRM as the length(D )_: ofthelineABisto thelength[189]

V.For,sincethe tubeAEisFig.39 equalboth in volumeand

length,to the cylinderIN, theresistanceofthe tubewillbeartotheresistanceofthecylinderthesameratioasthelineABto IL;but the resistanceof the cylinderIN is to that ofthe cylinderRMasthe cubeofIL isto thecubeofRS,that is,asthe lengthIL is to lengthV:therefore,excequali,the resistance[bendingstrength]ofthe tubeAEbearsto the resistanceofthe cylinderRM thesameratioasthe lengthABto V. q. _. D.

END OF SECOND DAY.

THIRD DAY[ 9o]

CHANGEOF POSITION.[DeMotuLocal ].................................................Y purposeis to set fortha very new science

_ dealingwith a veryancientsubjecCt.There

is, in nature, perhaps nothing older thanmotion,concerningwhichthe bookswrittenby philosophersare neither{ewnor small;neverthelessI have discoveredby experi-ment somepropertiesof it whichare worthknowingand whichhave not hitherto been

eitherobservedor temonstrated. Somesuperficialobservationshavebeenmade,as, forinstance,that thefreemotion[naturalemmotum]of a heavy fallingbody is continuouslyaccelerated;*but to just what extentthis accelerationoccurshasnot yet beenannounced;for sofar as I know,noonehas yet pointedout thatthe distancestraversed, during equal intervals of time, by abody fallingfrom rest, stand toone anotherin the sameratio asthe oddnumbersbeginningwithunity/f

It has been observedthat missilesand projecCtilesdescribea curvedpath of somesort;howevernoone haspointedout thefagt that this path is a parabola. But this and otherfacets,notfew in number or less worth knowing,I have succeededinproving; and what I considermoreimportant,there havebeenopenedup to this vast and most excellentscience,of whichmy

• "Naturalmotion"oftheauthorhasherebeentranslatedinto"freemotion"--sincethisisthetermusedto-daytodistinguishthe"natural"fromthe"violent"motionsoftheRenaissance.[Trans.]

t Atheoremdemonstratedonp. 175below.[Trans.]

I54 THE TWO NEW SCIENCESOF GALII.F.Oworkis merelythebeginning,waysandmeansbywhichothermindsmoreacutethar_minewillexploreits remotecomers.

This discussionis dividedinto three parts; the firstpartdealswithmotiorrwhichissteadyoruniform;thesecondtreatsofmotionaswefindit acceleratedin nature;thethirddealswiththeso-calledviolentmotionsandwithprojecCtiles.

[I911UNIFORMMOTION

In dealingwith steadyor uniformmotion,weneeda singledefinitionwhichI giveasfollows:

DEFINITION

By steadyor uniformmotion,I meanonein whichthe dis-tances traversedby the movingparticleduringany equalintervalsof time_arethemselvesequal.

CAtmo_Wemustaddto the olddefinition(whichdefinedsteadymo-

tion simplyas onein whichequaldistancesare traversedinequaltimes)theword"any," meaningby this,allequalinter-valsof time; for it may happenthat the movingbodywilltraverseequaldistancesduringsomeequalintervalsof timeandyet the distancestraversedduringsomesmallportionofthesetime-intervalsmaynot be equal,eventhoughthe time_intel_ralsbeequal.

Fromtheabovedefinition,fouraxiomsfollow,namely:Axio_I

In the caseofoneandthe sameuniformmotion,the distancetraversedduringa longerintervalof time is greaterthan thedistancetraversedduringa shorterintervaloftime.

Axao_tIIIn the caseof oneand the sameu_formmotion,the time

requiredto traversea greaterdistanceis longerthanthe timerequiredforalessdistance.

6

THIRD DAY I55_oM III

In oneand thesameinte_alof time,he distancetraver6edat a greaterspeedis largerthan the distancetraversedat alessspeed.

[i921A_OM IV

The speedrequiredto traversea longerdistanceis greaterthan that requiredto traversea shorterdis_nceduringthesametime-interval.

THEOREMI, PROPOSITION!If amovingparticle,carrieduniformlyat a constantspeed,traversestwo distancesthe tim_e-intervalsrequiredaretoeachotherin the ratioofthesedistances.

Let a particlemoveuniformlywith constantspeedthroughtwodistancesAB,BC,andletthetimerequiredto traverseABberepresentedbyDE;thetimerequiredto traverseBC,byEF;

• _-DT;E TF .

i_.,, : I l 'L ,t £...... : , : TJlk"rB TC . i" ' ..... IX

Fig.4othenI saythat the distanceABis to the distanceBCas thetimeDEisto thetimeEF.

LetthedistancesandtimesbeextendedonbothsidestowardsG,H andI, K; letAGbe dividedintoanynumberwhateverofspaceseach equalto AB,and in like mannerlay off in DIexactlythe samenumberd time-intervalseachequalto DE.Againlay off in CH anynumberwhateverof distanc_eachequalto BC; and in FK exa&lythe samenumberof time-intervalseachequalto EF; thenwillthe distanceBGandthetimeEl he equaland arbitrarymultiplesof the distanceBAandthe timeED; and likewisethe distanceHB andthe timeKE are equalandarbitrarymultiplesofthe distanceCB andthe timeFE.

AndsinceDE isthe timerequiredto traverseAB,thewholetime

I56 THE TWO NEW SCW.NCESOF GALILEOtimeEI willbe requiredfor the w_oledistanceBG,andwhenthemotionisuniformtherewillbein E1asro_nytime-intervalseachequaltoDEastherearedistancesinBGeachequalto BA;andlikewiseit followsthat KE representsthe timerequiredtotraverseHB.

Since,however,the motionis uniform,it followsthat if thedistanceGBis equalto the distanceBH, thenmustalsothetimeIE beequalto thetimeEK;andifGBisgreaterthanBH,thenalsoIE willbegreaterthanEK;andif less,less.*There

[ 93]arethenfourquantities,the firstAB,the secondBC, the thirdDE, andthe fourthEF; the timeIE andthe distanceGB arearbitrarymultiplesof the first and the third, namelyof thedistanceABandthetimeDE.

But it hasbeenprovedthat bothof theselatter quantitiesareeitherequalto, greaterthan,or lessthanthe timeEK andthe spaceBH,whicharearbitrarymultiplesof the secondandthe fourth. Thereforethe first is to the second,namelythedistanceABisto thedistanceBC,asthe thirdis to thefourth,namelythetimeDEisto the timeEF. q.w.D.

Tao= II,PRoPosmoIIIf a movingparticletraversestwodistancesin equal in-tervalsof time,thesedistanceswillbeartoeachotherthesameratioas the speeds. Andconverselyif thedistancesareas the speedsthenthe timesareequal.

ReferringtoFig.4o,letABandBCrepresentthetwodistancestraversedin equaltime-intervals,thedistanceAB forinstancewith thevelocityDE, andthe distanceBCwith the velocityEF. Then,I say, thedistanceABisto the distanceBCas thevelocityDE is to the velocityEF. Forif equalmultiplesofbothdistancesandspeedsbe taken,asabove,namely,GB andIE ofABandDErespe_ively,and in likemannerIIB andKEof BC and EF, then onemay infer, in the samemannerasabove,that themultiplesGBandIE areeitherlessthan,equal

* The methodhereemployedby Galileois that of Euclid as set forthin the famous5thDefinitionof the Fifth Bookof hisElements,forwhichseeart.GeometryEncy. Brit. IxthEd. p. 683. [Trans.]

THIRD DAY 157to, or greaterthanequalmultiplesof BHandEK. Hencethetheoremisestablished.

nI,P oPosmoNnIIn the caseofunequalspeeds,the time-intervalsrequiredto traversea givenspaceareto eachotherinverselyasthespeeds.

Let the largerofthe twounequalspeedsbeindicatedbyA;the smaller,by B; andlet the motioncorrespondingto bothtraversethe givenspaceCD. ThenI saythetimerequiredtotraversethedistanceCDat speedAk:A isto thetimerequiredto trav-ersethe samedistanceat speed_ j _: _ .:B, asthe speedB isto the speed _ D'A. For let CD be to CE as Ais to B; then, fromthe preced-]_t ting, it followsthat the timere- Fig.41qi_iredtocompletethe distanceCD at speedAis the sameas

[I94]the time necessaryto completeCE at speedB; but the timeneededto traversethedistanceCEat speedB isto the timere-quiredto traversethe distanceCD at the samespeedas CEisto CD; thereforethe timein whichCD is coveredat speedA isto the timeinwhichCDis coveredat speedBas CEistoCD,that is,asspeedB isto speedA. Q.E.D.

TE_OR_MIV,Pgo_osrrioNIVIftwoparticlesarecarriedwithuniformmotion,buteachwith a differentspeed,thedistancescoveredbythemdur-ingunequalintervalsoftimebearto eachotherthe com-poundratioof the speedsandtimeintervals.

Let the twoparticleswhicharecarriedwithuniformmotionbeE andFandlettheratioofthespeedofthebodyEbeto thatofthebodyF asAisto B;butlettheratioofthetimeconsumedby the motionofE be to thetimeconsumedbythe motionofF asCistoD. Then,I say,thatthedistancecoveredbyE, withspeedA intimeC,bearsto the spacetraversedbyF with"speedB

I58 THE TWONEW SCIENCESOF GALII,_OBinthn.eDa ratiowhichistheproductoftheratioofthespeedA tothespeedBbytheratioofthetimeCtothetimeD. ForifGisthedistancetraversedbyE atspeedA duringthetime-1_.A----.----_ G_ L ; intervalC,andifG istoI asC_ "thesp.e¢¢!Aistothesp_dB;

B: : I: .... -_ andif alsothetime-intervalx.:: : C is to the time-intervalD_.;..... : asI is to L, thenit follows

Fig.4z that I is thedistancetrav-ersedby F in thesametimethatG istraversedbyE sinceGistoI in thesameratioasthespeedAto thespeedB. AndsinceI isto L in thesameratioasthetime-intervalsCandD,if I isthedistancetraversedbyFduringtheintervalC,thenLwillbethedistancetraversedbyFduringtheintervalDat thespeedB.

ButtheratioofG to L istheproductoftheratiosG to IandI to L, thatis,oftheratiosofthespeedAto thespeedBandofthetime-intervalCtothetime-intervalD. Q.g, D.

[,9s]T_Eo_a V,PgoPosmo_V

If twoparticlesaremovedata uniformrate,butwithun-equalspuds,throughunequaldistances,thentheratioofthetime-intervalsoccupiedwillbetheprodu_oftheratioofthedistancesbytheinverseratioofthespeeds.

LetthetwomovingparticlesbedenotedbyA andB, andletthe speedof A bed]_w. _ ¢. _,tothespeedofBin :._. :theratioofVtoT; *"- .:_ 'in likemannerlet_ T: _: c. .......thedistancestray- _:ersedbeintheratio Fig.43ofStoR; thenI saythattheratioofthetime-intervalduringwhichthemotionofAoccurs_tothetime-intervaloccupiedbythemotionofBistheprodu_oftheratioofthespeedT to thespeedVbytheratioofth_di_st_.nc¢Stothedistanc¢R.

LetCbethetime-intervaloccupiedbythemotionofA,andlet

THIRD DAY 159letthe time-intervalC bearto atime-intervalE the sameratio

! asthespeedT tothespeedV.'_ AndsinceC is the time-intervalduringwhichA,withspeed_ V,traversesthedistanceSandsinceT, the speedofB,isto the

speedV,as the time-intervalC is to thetime-intervalE, thenE willbe the timerequiredby the particleB to traversethedistanceS. If nowwelet the time-intervalE beto the time-intervalGas thedistanceSis to thedistanceR, thenit followsthat G isthetimerequiredbyB totraversethe spaceR. SincetheratioofC to G isthe productoftheratiosC to E andE toG (whilealsothe ratioofCto EistheinverseratioofthespeedsofAandB respeCtively,i.e.,theratioofT toV);andsincetheratioofE to G is the sameas that of the distancesS andRrespectively,thepropositionisproved.

[x961T_sotu_MVt, PRox-osrrlO_VI

If twoparticlesare carriedat a uniformrate,the ratiooftheirspeedswillbetheproductof theratioofthe distancestraversedbytheinverseratioofthetime-intervalsoccupied.

LetA andB be the twoparticleswhichmoveat a uniformrate; andlet the respectivedistancestraversedby themhayethe ratio of V.._- C ,to T, but letthetime-intervalsbe 1_asStoR. Then _I say the speedB Gof A will bear t_" "_to the speedof Fig.44B a ratiowhichisthe productoftheratioof thedistanceV tothedistanceT andthetime-intervalRto thetime-intervalS.

LetCbethespeedat whichA traversesthedistanceVduringthe time-intervalS;andletthe speedC bearthe-sameratiotoanotherspeedE asV bearsto T; thenE willbe the speedatwhichB traversesthe distanceT duringthe time-intervalS.If nowthe speedE isto anotherspeedGasthe time-intervalRis to thetime-intervalS,thenG willbethe speedat whichthe

particle

16o THE TWO NEW SCIENCESOF GALIT,EOparticleB traversesthe distanceT duringthe time-intervalR.Thuswehave the speedC at whichthe particleA coversthedistanceV duringthetimeSandalsothespeedG at whichtheparticleB traversesthe distanceT duringthe time R. TheratioofC to G is the produc°cofthe ratioC to E andE to G;theratioofC to E isby definitionthe sameasthe ratioof thedistanceVto distanceT; andthe ratioofE to G isthe sameastheratioofR to S. Hencefollowstheproposition.

SAT.V.TheprecedingiswhatourAuthorhaswrittenconcern-inguniformmotion. Wepassnowto a newandmorediscrim-inatingconsiderationof naturallyacceleratedmotion,suchasthat generallyexperiencedby heavyfallingbodies;followingisthe titleandintrodu_ion.

[197]NATURAT,I,YACCEI,ERATt_,DMOTION

The propertiesbelongingto uniformmotionhavebeendis-cussedin the precedingsection;but acceleratedmotionremainsto beconsidered.

And firstofallit seemsdesirableto findandexplaina defini-t_n bestfittingnaturalphenomena.Foranyonemayinventanarbitrarytype of motionand discussits properties;thus, forinstance,somehaveimaginedhelicesandconchoidsasdescribedbycertainmotionswhicharenotmetwithin nature,andhaveverycommendablyestablishedthepropertieswhichthesecurvespossessinvirtueoftheirdefinitions;but wehavedecidedto con-siderthe phenomenaof bodiesfallingMth an accelerationsuchas ad'tuallyoccursin nature and to makethis definitionofacceleratedmotionexhibitthe essentialfeaturesof observedacceleratedmotions.Andthis,at last,afterrepeatedeffortswetrustwehavesucceededindoing. Inthisbeliefweareconfirmedmainlyby the considerationthat experimentalresultsare seento agreewith and exa£tlycorrespondwith thosepropertieswhichhave been, one after another,demonstratedby us.Finally,in the investigationof naturallyacceleratedmotionwewereled,byhandasitwere,infollowingthehabitandcustomof

nature

THIRD DAY I6inatureherself,in all her variousother processes,to employonlythosemeanswhicharemostcommon,simpleandeasy.

For I thinknoonebelievesthat swimmingor flyingcanbei accomplishedin a mannersimpleror easierthanthat instinc-

tivelyemployedbyfishesandbirds.: When,therefore,I observea stoneinitiallyat rest falling

froman elevatedpositionand continuallyacquiringnewin-crementsofspeed,whyshouldI notbelievethat suchincreasestakeplacein a mannerwhichisexceedinglysimpleandratherobviousto everybody?If nowweexaminethemattercarefullywefindnoadditionor incrementmoresimplethanthat whichrepeatsitselfalwaysin the samemanner. This we readilyunderstandwhenweconsiderthe intimaterelationshipbetweentimeandmotion;forjustasuniformityofmotionisdefinedbyandconceivedthroughequaltimesandequalspaces(thuswecalla motionuniformwhenequaldistancesaretraversedduringequal time-intervals),so alsowemay, in a similarmanner,throughequal time-intervals,conceiveadditionsof speedastakingplacewithoutcomplication;thuswemaypic°mreto our

[I981minda motionasuniformlyandcontinuouslyacceleratedwhen,duringanyequalintervalsoftimewhatever,equalincrementsof speedare givento it. Thusif anyequalintervalsof timewhateverhaveelapsed,countingfromthe timeat whichthemovingbodyleftits positionofrestandbeganto descend,theamountof speedacquiredduringthe first two time-intervalswill be doublethat acquiredduringthe first time-intervalalone;so theamountaddedduringthreeofthesetime-intervalswillbetreble;andthat infour,quadruplethatofthefirsttime-interval. To put the mattermoreclearly,if a bodyweretocontinueits motionwiththe samespeedwhichit hadacquiredduringthe firsttime-lntervalandwereto retainthissameuni-formspeed,thenitsmotionwouldbetwiceasslowasthatwhichit wouldhaveif itsvelocityhadbeenacquiredduringtwotime-intervals.

Andthus,it seems,weshallnot be farwrongifweput theincrementof speedas proportionalto the incrementof time;

hence

,6z THE TWO NEW SCW.NCESOF GALTT,FOhencethedefinitionofmotionwhichweareaboutto discussmaybestatedasfollows:Amotionissaidtobeuniformlyaccelerated,whenstartingfromrest,it acquires,duringequaltime-intervals,equalincrementsofspeed.

SAGR.AlthoughI can offerno rationalobjecCtionto this orindeedto anyotherdefinition,devisedbyanyauthorwhomso-ever, sinceall definitionsare arbitrary,I may neverthelesswithoutoffensebe allowedto doubtwhethersuchadefinitionastheabove,establishedinanabstracCtmanner,correspondsto anddescribesthat kind of acceleratedmotionwhichwemeet innaturein thecaseoffreelyfallingbodies.AndsincetheAuthorapparentlymaintainsthat the motiondescribedin his defini-tion is that of freelyfallingbodies,I wouldliketo clearmymind of certaindifficultiesin order that I may later applymyselfmore earnestlyto the propositionsand their demon-strations.

SALV.It iswellthat youandSimplicioraisethesedifficulties.They are, I imagine,the samewhichoccurredto me whenIfirstsawthistreatise,andwhichwereremovedeitherbydiscus-sionwiththe Authorhimself,orby turningthematteroverinmyownmind.

SAGI_..WhenI thinkofaheavybodyfallingfromrest,that is,startingwithzerospeedandgainingspeedin proportionto the

['99]timefromthe beginningof themotion;suchamotionaswould,forinstance,in eightbeatsofthe pulseacquireeightdegreesofspeed;havingat the endof the fourthbeat acquiredfourde-grees;at the endofthe second,two;at the endofthefirst,one:andsincetimeisdivisiblewithoutlimit,it followsfromalltheseconsiderationsthat if the earlierspeedofa bodyislessthanitspresentspeedin a constantratio, then thereis no degreeofspeedhoweversmall(or,onemaysay, no degreeof slownesshowevergreat)withwhichwemaynotfindthisbodytravellingafterstartingfrominfiniteslowness,i. e.,fromrest. Sothat ifthat speedwhichit hadat the endofthe fourthbeatwassuchthat, ifkeptuniform,the bodywouldtraversetwomilesin anhour,and ifkeepingthe speedwhichit hadat the endof the

second

THIRD DAY I63secondbeat, it wouldtraverseonemileanhour,wemustinferthat, as the instantof startingis moreandmorenearlyap-proached,thebodymovessoslowlythat,if itkeptonmovingatthisrate,it wouldnottraverseamilein anhour,or ina day,orin ayearor ina thousandyears;indeed,it wouldnottraverseaspanin an evengreatertime;a phenomenonwhichbafflestheimagination,whileoursensesshowusthat a heavyfallingbodysuddenlyacquiresgreatspeed.

SM_v.Thisisoneofthe difficultieswhichI alsoat thebegin-ning,experienced,butwhichI shortlyafterwardsremoved;andtheremovalwaseffecCtedbythe veryexperimentwhichcreatesthedifficultyforyou. Yousaytheexperimentappearsto showthat immediatelyaftera heavybodystartsfromrest itacquiresa veryconsiderablespeed:andI saythat thesameexperimentmakesclearthefacetthat theinitialmotionsofafallingbody,nomatterhowheavy,arevery slowandgentle. Placea heavybodyupona yieldingmaterial,andleaveit therewithoutanypressureexceptthat owingto its ownweight;it is clearthat ifoneliftsthisbodya cubitor twoandallowsit to falluponthesamematerial,it will,withthisimpulse,exertanewandgreaterpressurethanthat causedbyits mereweight;andthiseffec°cisbroughtaboutbythe [weightofthe]fallingbodytogetherwiththe velocityacquiredduringthe fall,an effecCtwhichwillbegreaterand greateraccordingto the heightofthe fall,that isaccordingas the velocityof the fallingbodybecomesgreater.Fromthequalityandintensityofthe blowwearethusenabledto accuratelyestimatethespeedofafallingbody. Buttellme,gentlemen,isit not truethat if ablockbeallowedto falluponastakefroma heightoffourcubitsanddrivesit intotheearth,

[zoo]say, fourfinger-breadths,that comingfroma heightof twocubitsitwilldrivethe stakeamuchlessdistance,andfromtheheightofonecubitastilllessdistance;andfinallyiftheblockbeliftedonlyonefinger-breadthhowmuchmorewillit accomplishthan if merelylaid on top of the stake withoutpercussion?Certainlyvery little. If it be liftedonlythe thicknessof aleaf,the effectwillbealtogetherimperceptible.Andsincetheeffec°c

164 THE TWO NEW SCIENCESOF GALIT._'Oeffec_tof the blowdependsuponthe velocityof this strikingbody,cananyonedoubtthemotionisveryslowandthe speedmorethansmallwheneverthe eiTecCt[oftheblow]isimpercepti-ble? Seenowthepoweroftruth;the sameexperimentwhichatfirst glanceseemedto showone thing,whenmorecarefullyexamined,assuresusofthecontrary.

Butwithoutdependinguponthe aboveexperiment,whichisdoubtlessvery conclusive,it seemsto methat it oughtnot tobedifficultto establishsuchafa_ by reasoningalone. Imaginea heavystoneheldin theairat rest;the supportisremovedandthestonesetfree;thensinceit isheavierthantheairit beginstofall,andnot withuniformmotionbut slowlyat thebeginningandwithacontinuouslyacceleratedmotion.Nowsincevelocitycanbe increasedanddiminishedwithoutlimit,what reasonisthereto believethat sucha movingbodystartingwithinfiniteslowness,that is,fromrest,immediatelyacquiresa speedoftendegreesratherthan oneof four,or of two,or of one,or of ahalf,orofa hundredth;or,indeed,ofanyoftheinfinitenumberofsmallvalues[ofspeed]? Praylisten. I hardlythinkyouwillrefuseto grantthat the gainof speedof the stonefallingfromrestfollowsthe samesequenceasthediminutionandlossofthissamespeedwhen,by someimpellingforce,thestoneisthrowntoits formerelevation:but evenifyoudonotgrantthis,I donotseehowyoucandoubtthat the ascendingstone,diminishinginspeed,mustbeforecomingto restpass througheverypossibledegreeofslowness.

SIMP.But if the numberof degreesof greaterand greaterslownessislimitless,theywillneverbe allexhausted,thereforesuchan ascendingheavybodywillneverreachrest,but willcontinuetomovewithoutlimitalwaysat a slowerrate;but thisisnottheobservedfa_.

SA_v.This wouldhappen,Simplicio,if the movingbodyweretomaintainitsspeedforanylengthoftimeat eachdegreeof velocity;but it merelypasseseachpointwithoutdelayingmorethan an instant:and sinceeachtime-intervalhowever

[2oi]smallmay bedividedintoan infinitenumberof instants,these

will

THIRD DAY 165willalwaysbesu_cient[innumber]tocorrespondto theinfinitedegreesofdiminishedvelocity.

That suchaheavyrisingbodydoesnotremainforanylengthoftimeat anygivendegreeofvelocityis evidentfromthe fol-lowing:becauseif,sometime-intervalhavingbeenassigned,thebodymoveswith the samespeedin the lastas in thefirstin-stantofthat time-interval,it couldfromthisseconddegreeofelevationbe in like mannerraisedthroughan equalheight,justas it wastransferredfromthe firstelevationto thesecond,andby the samereasoningwouldpassfromthe secondto thethird and wouldfinallycontinuein uniformmotionforever.

Saam Fromtheseconsiderationsit appearsto me that wemay obtain a propersolutionof the problemdiscussedbyphilosophers,namely,what causesthe accelerationin thenaturalmotionofheavybodies? Since,as it seemsto me,theforce[_irt_t]impressedbytheagentprojecCtingthebodyupwardsdiminishescontinuously,thisforce,solongasit wasgreaterthanthe contraryforceof gravitation,impelledthebodyupwards;whenthe two are in equilibriumthe bodyceasesto riseandpassesthroughthestateofrestin whichthe impressedimpetus[impeto]is notdestroyed,but onlyitsexcessovertheweightofthebodyhasbeenconsumed--theexcesswhichcausedthebodyto rise. Thenasthediminutionoftheoutsideimpetus[impeto]continues,andgravitationgainstheupperhand,the fallbegins,but slowlyat firston accountof the opposingimpetus[virt_zimpressa],alargeportionofwhichstillremainsin thebody;butas thiscontinuesto diminishit alsocontinuesto bemoreandmoreovercomebygravity,hencethe continuousaccelerationofmotion.

Sn_P.Theideaisclever,yetmoresubtlethansound;forevenif the argumentwereconclusive,it wouldexplainonlythe casein whicha naturalmotionis precededby a violentmotion,inwhichtherestillremainsaCtivea portionofthe externalforce[virthesterna];butwherethereisnosuchremainingportionandthebodystartsfroman antecedentstateofrest,the cogencyofthewholeargumentfails.

SACR.I believethat youaremistakenandthat thisdistinc-tion

i66 THE TWO NEW SCIENCESOF GALILEOtionbetweencaseswhichyoumakeissuperfluousorrathernon-existent. But, tell me, cannota proje&ilereceivefromtheproje&oreithera largeor asmallforce[v/rt_]suchaswillthrowit to a heightof a hundredcubits,andeventwentyor fourorone ?

[zoz]S_w.Undoubtedly,yes.SAcg.So thereforethis impressedforce[virt_impressa]may

exceedthe resistanceof gravityso slightlyas to raiseitonlya finger-breadth;and finallythe force[v/rt_]of the projectormaybe just largeenoughto exac°dybalancethe resistanceofgravityso that the body is not liftedat all but merelysus-tained. Whenoneholdsa stoneinhishanddoeshedoanythingbut giveit a forceimpelling[v/rt_impellente]it upwardsequaltothepower[facoltd]ofgravitydrawingit downwards? Anddoyounot continuouslyimpressthis force[virtU]uponthe stoneas longas youholdit in the hand? Doesit perhapsdiminishwiththetimeduringwhichoneholdsthestone?

Andwhatdoesit matterwhetherthissupportwhichpreventsthe stonefromfallingis furnishedby one'shandor bya tableor by a ropefromwhichit hangs? Certainlynothingat all.You must conclude,therefore,Simplicio,that it makesnodifferencewhateverwhetherthefallofthestoneisprecededbyaperiodof rest whichis long,short,or instantaneousprovidedonlythe falldoesnot take placeso longas the stoneis a&eduponby a force[v/rtfi]opposedto its weightandsufficienttoholdit at rest.

SAT.v.The presentdoesnot seemto be daepropertime toinvestigatethe causeofthe accelerationof naturalmotioncon-cerningwhichvariousopinionshavebeenexpressedbyvariousphilosophers,someexplainingk by attra&ionto the center,othersto repulsionbetweenthe very smallpartsof the body,whilestillothersattributeit to acertainstressinthesurroundingmediumwhichclosesin behindthe fallingbodyand drivesitfromoneof its positionsto another. Now,all thesefantasies,andotherstoo,oughtto be examined;but it isnotreallyworthwhile. At presentit is the purposeof ourAuthormerelyto

investigate

THIRD DAY 167investigateand to demonstratesomeof the propertiesof ac-celeratedmotion(whateverthe causeof thisaccelerationmaybe)--meaningtherebyamotion,suchthatthemomentumofitsvelocity[imomentidellasuavelocitY]goeson increasingafterdeparturefromrest,in simpleproportionalitytothetime,whichis the sameas sayingthat in equaltime-intervalsthe bodyreceivesequalincrementsofvelocity;andifwefindtheproper-ties[ofacceleratedmotion]whichwillbedemonstratedlaterarerealizedinfreelyfallingandacceleratedbodies,wemayconcludethat the assumeddefinitionincludessucha motionof failingbodiesandthat theirspeed[accelerazione]goesonincreasingasthetimeandthedurationofthemotion.

[203]SAGR.So far as I seeat present,the definitionmighthave

beenput a little moreclearlyperhapswithoutchangingthefundamentalidea,namely,uniformlyacceleratedmotionissuchthat its speedincreasesin proportionto thespacetraversed;sothat, forexample,the speedacquiredby a bodyin fallingfourcubitswouldbe doublethat acquiredin fallingtwocubitsandthislatterspeedwouldbedoublethat acquiredin thefirstcubit.Becausethereis nodoubtbut that a heavybodyfallingfromthe heightof sixcubitshas, and strikeswith, a momentum[impeto]doublethat it hadatthe endofthreecubits,triplethatwhichithadattheendofone.

SALV.It isvery comfortingto me to havehadsucha com-panioninerror;andmoreoverletmetellyouthatyourproposi-tionseemssohighlyprobablethat ourAuthorhimselfadmitted,whenI advancedthisopinionto him,thathehadforsometimesharedthe samefallacy. Butwhatmostsurprisedmewastosee two propositionsso inherentlyprobablethat they com-mandedthe assentofeveryoneto whomtheywerepresented,provenin a few simplewordsto be not onlyfalse,but im-possible.

SIM1,.I am oneof thosewhoacceptthe proposition,andbelievethat afallingbodyacquiresforce[vires]initsdescent,itsvelocityincreasingin proportionto the space,and that themomentum[momento]ofthefallingbodyisdoubledwhenit falls

from

I68 THE TWO NEW SCIENCESOF GALII.EOfroma doubledheight;thesepropositions,it appearsto me,oughtto beconcededwithouthesitationorcontroversy.

SALV.Andyettheyareasfalseandimpossibleasthat motionshouldbe completedinstantaneously;andhereis a verycleardemonstrationof it. If the velocitiesarein proportionto thespacestraversed,or to be traversed,then these spacesaretraversedin equalintervalsof time;if, therefore,the velocitywithwhic_thefailingbodytraversesa spaceofeightfeetweredoublethat withwhichit coveredthefirstfourfeet(justas theone distanceis doublethe other)then the time-intervalsre-quiredforthesepassageswouldbe equal. But foroneandthesamebody to falleightfeet andfourfeetin the sametime ispossibleonlyinthecaseofinstantaneous[discontinuous]motion;

[204]but observationshowsus that the motionofa fallingbodyoc-cupiestime,andlessofit incoveringa distanceoffourfeetthanofeightfeet;thereforeit isnottruethat itsvelocityincreasesinproportiontothespace.

Thefalsityofthe otherpropositionmaybeshownwithequalclearness.Forifweconsiderasinglestrikingbodythedifferenceofmomentumin its blowscan dependonlyupondifferenceofvelocity;forif the strikingbodyfallingfroma doubleheightwereto delivera blowofdoublemomentum,it wouldbeneces-saryfor this bodyto strikewith a doubledvelocity;but withthis doubledspeedit wouldtraversea doubledspacein thesametime-interval;observationhowevershowsthat the timerequiredforfallfromthegreaterheightislonger.

SAQI_.Youpresentthese reconditematterswithtoo muchevidenceand ease;this greatfacilitymakesthemlessappre-ciatedthan theywouldbe had theybeenpresentedin a moreabstrusemanner. For, in my opinion,peopleesteemmorelightlythat knowledgewhichtheyacquirewithsolittle laborthanthatacquiredthroughlongandobscurediscussion.

S_LV.If thosewhodemonstratewithbrevityand clearnessthe fallacyofmanypopularbeliefsweretreatedwithcontemptinsteadof gratitudethe injurywouldbe quitebearable;butonthe otherhandit is veryunpleasantandannoyingto seemen,

who

THIRD DAY 169whoclaimtobepeersofanyoneina certainfieldofstudy,takefor grantedcertainconclusionswhichlater arequicklyandeasilyshownbyanotherto be false. I donot describesuchafeelingas oneof envy,whichusuallydegeneratesintohatredandangeragainstthosewhodiscoversuchfallacies;Iwouldcallit a strongdesireto maintainolderrors,ratherthan acceptnewlydiscoveredtruths. Thisdesireat timesinducesthemtouniteagainstthesetruths,althoughat heartbelievinginthem,merelyforthepurposeof loweringtheesteemin whichcertainothersareheldbytheunthinkingcrowd.Indeed,I haveheardfromourAcademicianmany suchfallaciesheld as true buteasilyrefutable;someoftheseIhaveinmind.

SAGI_.You must not withholdthemfromus, but, at thepropertime,tellusaboutthemeventhoughanextrasessionbenecessary.Butnow,continuingthethreadofourtalk,it would[205]seemthat upto thepresentwehaveestablishedthedefinitionofuniformlyacceleratedmotionwhichis expressedas follows:

A motionis said to be equallyor uniformlyacceleratedwhen,startingfromrest,itsmomentum(celeritatismomenta)receivesequalincrementsin equaltimes.

SALv.Thisdefinitionestablished,theAuthormakesa singleassumption,namely,

The speedsacquiredby oneand the samebodymovingdownplanesof differentinclinationsare equalwhentheheightsoftheseplanesareequal.

By theheightofan inclinedplanewemeantheperpendicularletfallfromthe upperendoftheplaneuponthehorizontallinedrawnthroughthe lowerend of the sameplane. Thus, toillustrate,let the lineABbe horizontal,and lettheplanesCAandCDbeinclinedtoit; thentheAuthorcallstheperpendicularCBthe "height"of theplanesCAandCD;he supposesthatthe speedsacquiredby oneand the samebody,descendingalongtheplanesCAandCDtotheterminalpointsAandD areequalsincethe heightsof theseplanesarethe same,CB;andalsoit mustbe understoodthatthis speedisthat whichwouldbeacquiredbythe samebody.fallingfromC toB. Sagr.

17o THE TWO NEW SCIENCESOF GALILEOSAog.Yourassumptionappearsto meso reasonablethat it

oughtto beconcededwithoutquestion,providedofcoursethereare no chanceor outsideresistances,andthat the planesare

_o hard and smooth,and that thefigureof the movingbodyis per-felly round,so that neitherplanenormovingbodyisrough.Allre-sistanceand oppositionhavingbeen removed,my reason tells

A me at oncethat a heavyandper-Fig.45 fec°dyroundballdescendingalong

the linesCA,CD, CBwouldreachthe terminalpointsA,D,B,withequalmomenta[impetieguah].

SALV.Yourwordsarevery plausible;but I hopeby experi-mentto increasetheprobabilityto anextentwhichshallbelittleshortofa rigiddemonstration.

[206]Imaginethis page to representa verticalwall,with a nail

drivenintoit; and fromthe naillet therebe suspendeda leadbulletof oneor twoouncesby meansof a fineverticalthread,AB,sayfromfourto sixfeetlong,onthiswalldrawahorizontallineDC,at rightanglesto theverticalthreadAB,whichhangsabouttwofinger-breadthsin frontof thewall. NowbringthethreadABwiththeattachedballintothepositionACandsetitfree;firstit willbeobservedto descendalongthe arc CBD,topassthe pointB, andto travelalongthe arcBD, till it almostreachesthehorizontalCD,aslightshortagebeingcausedbytheresistanceof the air and the string;fromthiswemayrightlyinferthat theballin its descentthroughthe arcCBacquiredamomentum[impeto]on reachingB, whichwasjust sufficienttocarryit througha similararc BDto the sameheight. Havingrepeatedthisexperimentmanytimes,letusnowdriveanailintothe wallcloseto the perpendicularAB,sayat E or F, so thatit proje_sout somefiveor sixfinger-breadthsin orderthat thethread,againcarryingthebulletthroughthearcCB,maystrikeuponthenailEwhenthebulletreachesB,andthuscompelit totraversethe arcBG,describedaboutE as center. Fromthis

we

THIRD DAY :y:wecanseewhatcanbe doneby thesamemomentum[impeto]whichpreviouslystartingat the samepointB carriedthesamebodythroughthe arcBD tothehorizontalCD. Now,gentle-men,youwillobservewithpleasurethat theballswingstothepointG in the horizontal,andyouwouldseethe samethinghappenif theobstaclewereplacedat somelowerpoint,sayatF,aboutwhichtheballwoulddescribethearcBI,theriseofthe

A/

Fig.46ballalwaysterminatingexa&lyonthe lineCD. Butwhenthenailis placedso lowthat the remainderof thethreadbelowitwillnotreachtotheheightCD(whichwouldhappenifthenailwereplacednearerB than to the interse&ionofABwiththe[207]horizontalCD)thenthe threadleapsoverthe nailandtwistsitselfaboutit.

Thisexperimentleavesnoroomfordoubtas to the truthofoursupposition;forsincethetwoarcsCBandDBareequalandsimilarlyplaced,the momentum[momento]acquiredbythe fallthroughthearcCBisthesameasthatgainedbyfallthroughthearcDB; but themomentum[momento]acquiredat B,owingtofallthroughCB,isableto liftthesamebody[mobile]throughthearc BD; therefore,the momentumacquiredin the fallBD isequalto that whichliftsthe samebodythroughthe samearcfromB to D; so,in general,everymomentumacquiredby fallthrough

i7z THE TWO NEW SCIENCESOF GALILEOthroughan arc is equalto that whichcanlift the samebodythroughthe samearc. Butall thesemomenta[momentz]whichcausea risethroughthe arcsBD,BG,andBI are equal,sincethey are producedby the samemomentum,gainedby fallthroughCB,as experimentshows.Thereforeall the momentagainedbyfallthroughthearcsDB,GB,IBareequal.

SAcR.The argumentseemsto mesoconclusiveandthe ex-perimentsowelladaptedto establishthe hypothesisthat wemay,indeed,considerit asdemonstrated.

SALV.I donotwish,Sagredo,that wetroubleourselvestoomuchaboutthismatter,sincewearegoingtoapplythisprinciplemainlyin motionswhichoccuronplanesurfaces,andnotuponcurved,alongwhichaccelerationvariesin a mannergreatlydifferentfromthat whichwehaveassumedforplanes.

So that, althoughthe aboveexperimentshowsus that thedescentofthemovingbodythroughthe arcCBconfersuponitmomentum[momento]just sufficientto carry it to the sameheightthroughanyofthearcsBE),BG,BI,wearenotable,bysimilarmeans,to showthat the eventwouldbe identicalinthe caseofa perfedtlyroundballdescendingalongplaneswhoseinclinationsare respe&ivelythe sameas the chordsof thesearcs. It seemslikely,ontheotherhand,that, sincetheseplanesformanglesat the pointB,they willpresentan obstacleto theballwhichhasdescendedalongthe chordCB,andstartsto risealongthe chordBD,BG,BI.

In strikingtheseplanessomeof its momentum[impeto]willbelostanditwillnotbeableto riseto theheightofthelineCD;but this obstacle,whichinterfereswith the experiment,onceremoved,it is clearthat the momentum[im2e[v](whichgains

[ o8]in strengthwithdescent)willbe ableto carrythe bodyto thesameheight. Letus then,for the present,take thisas a pos-tulate,the absolutetruthofwhichwillbe establishedwhenwefind that the inferencesfromit correspondto andagreeper-fe&lywithexperiment.Theauthorhavingassumedthissingleprinciplepassesnextto the propositionswhichhe clearlydem-onstrates;thefirstoftheseisasfollows:

THIRD DAY 173T_EoP,x_ I, PRo_osmoNI

The timeinwhichanyspaceis traversedby a bodystart-ingfromrestanduniformlyacceleratedisequaltothetimein whichthat samespacewouldbetraversedby thesamebodymovingat a uniformspeedwhosevalueisthe meanofthe highestspeedandthe speedjustbeforeaccelerationbegan.

Let us representby the lineABthe timein whichthe spaceCD is traversedby a bodywhichstartsfromrestat C andisuniformlyaccelerated;let the finaland highestvalueof thespeedgainedduringthe intervalABbe representedbythelineEB drawnat rightanglesto AB;drawthe lineAE, then alllinesdrawnfromequidistantpointsonABandparallelto BEwillrepresenttheincreasingvaluesofthespeed, cbeginningwiththe instantA. Let thepointF G _. -bisecCtthe lineEB; drawFG parallelto BA,andGAparallelto FB,thusformingaparallel-ogramAGFBwhichwillbeequalinareato thetriangleAEB,sincethe sideGFbisecCtsthesideAEat thepointI; forif theparallellinesinthetriangleAEBareextendedto GI, thenthesum x,ofalltheparallelscontainedinthequadrilateral/isequalto thesumofthosecontainedinthetri-angleAEB;for thosein the triangleIEF areequal to thosecontainedin the triangleGIA,whilethoseincludedinthetrapeziumAIFBare icommon.Sinceeachandeveryinstantoftime_ 1_in the time-intervalABhas its corresponding jpointon the lineAB,fromwhichpointspar- ,_allelsdrawninandlimitedbythetriangleAEB Drepresentthe increasingvaluesof thegrowing Fig.47velocity,andsinceparallelscontainedwithinthe re.angle rep-resentthevaluesofaspeedwhichisnotincreasing,butconstant,it appears,inlikemanner,thatthemomenta[momenta]assumedbythe movingbodymayalsoberepresented,inthe caseof theacceleratedmotion,by the increasingparallelsof the triangleAEB;

174 THETWONEWSCIENCESOFG.ATJTFO[2o9]

AEB,and,inthecaseoftheuniformmotion,bytheparallelsoftherec°cangleGB. For,whatthemomentamaylackinthefirstpartoftheacceleratedmotion(thedeficiencyofthemomentabeingrepresentedbytheparallelsofthetriangleAGI)ismadeupbythemomentarepresentedbytheparallelsofthetriangleIEF.

Henceit isclearthatequalspaceswillbetraversedinequaltimesbytwobodies,oneofwhich,startingfromrest,moveswith

Art a uniformacceleration,whilethe momentumof"_.theother,movingwithuniformspeed,is one-half

o D. its maximummomentumunderacceleratedmo-_ tion. Q._.D.F._ THEOREMII, PROPOSITIONII

Thespacesdescribedbyabodyfallingfromresta witha uniformlyacceleratedmotionareto each

otherasthesquaresofthe time-intervalsem-ployedintraversingthesedistances.

. Y LetthetimebeginningwithanyinstantAberep-resentedbythestraightlineABinwhicharetakenanytwotime-intervalsADandAE. LetHIrepre-sentthedistancethroughwhichthebody,startingfromrestat H, fallswithuniformacceleration.IfHLrepresentsthespacetraversedduringthetime-intervalAD,andHM that coveredduringthe in-

Bi t tervalAE,thenthespaceR_Istandsto thespace- " LHina ratiowhichisthesquareoftheratioofthe

timeAEto thetimeAD;orwemaysaysimplythatthedistancesHMandHI.arerelatedasthesquares

Fig.48 ofAEandAD.DrawthelineACmakinganyanglewhateverwiththe line

AB;andfromthepointsD andE, drawtheparallellinesDOandEP;ofthesetwolines,DOrepresentsthegreatestvelocityattainedduringthe intervalAD,whileEPrepresentsthemax-imumvelocityacquiredduringthe intervalAE. But it hasjust beenprovedthat sofar as distancestraversedarecon-

cerned

THIRD DAY _75cernedit is preciselythe samewhethera bodyfallsfromrestwith a uniformaccelerationorwhetherit fallsduringan equaltime-intervalwitha constantspeedwhichisone-halfthe max-imumspeedattainedduringthe acceleratedmotion.It followsthereforethat thedistancesH1VIandHLarethesameaswouldbetraversed,duringthe time-intervalsAEandAD,byuniformvelocitiesequalto one-halfthoserepresentedby DOand EPrespecCtively.If, therefore,one can showthat the distancesHMandHL are in the sameratioas the squaresofthe time-intervalsAEandAD,ourpropositionwillbeproven.

[2IO1Butin the fourthpropositionofthe firstbook[p.157above]

it hasbeenshownthat thespacestraversedbytwoparticlesinuniformmotionbeartooneanothera ratiowhichisequalto theproductof the ratioof the velocitiesby the ratioof thetimes.But inthiscasetheratioofthevelocitiesisthesameasthe ratioofthe time-intervals(forthe ratioofAEto ADisthe sameasthat ofx/4EPto I/4DOorofEPto DO). Hencetheratioofthespacestraversedis the sameas the squaredratioof the time-intervals. Q.v..D.

Evidentlythenthe ratioofthe distancesisthe squareof theratioofthefinalvelocities,that is,ofthelinesEPandDO,sincethesearetoeachotherasAEtoAD.

COROT.IARY I

Henceit is clearthat ifwetakeanyequalintervalsof timewhatever,countingfromthe beginningofthe motion,suchasAD,DE, EF, FG, in whichthe spacesHL,LM,MN,NI aretraversed,thesespaceswillbearto oneanotherthesameratioastheseriesofoddnumbers,I, 3,5,7;forthisistheratioofthedifferencesof the squaresof the lines[whichrepresenttime],differenceswhichexceedoneanotherby equalamounts,thisexcessbeingequalto thesmallestline[viz.theonerepresentingasingletime-interval]:orwemaysay[thatthisisthe ratio]ofthedifferencesofthe squaresofthe naturalnumbersbeginningwithunity. While,

176 THE TWO NEW SCIENCESOF GALTT,EOWhile,therefore,duringequalintervalsoftimethe velocities

increaseasthe naturalnumbers,theincrementsin thedistancestraversedduringtheseequaltime-intervalsaretooneanotherastheoddnumbersbeginningwithunity.

SA6R.Pleasesuspendthe discussionfora momentsincetherejust occursto mean ideawhichI wantto illustratebymeansof adiagramin orderthat it maybe dearerboth to you andtome.

Let thelineAI representthelapseoftimemeasuredfromtheinitialinstantA throughA drawthe .straightlineAFmaking

ajA any.anglewhatever;.,join the terminal/i "pointsI andF; dlvldethetimeAI inhalf

at C; drawCBparallelto IF. Let usconsiderCBas the maximumvalueof

] the velocitywhichincreasesfromzero"B p-c at the beginning,in simpleproportion-ality to the interceptson the triangleABCof linesdrawnparallelto BC;or

/ whatisthesamething,letussupposethe

velocityto increasein proportionto the2v g _ _t z time;thenI admitwithoutquestion,in

' viewoftheprecedingargument,that the

spacedescribedby a bodyfailingin thei aforesaidmannerwill be equalto the

spacetraversedbythesamebodyduringthesamelengthof timetravellingwithauniformspeedequalto EC, the halfof

It. il. o BC. Furtherlet us imaginethat theFig.49 [21I]

bodyhas fallenwith acceleratedmotionso that, at the in-stant C, it has the velocityBC. It is clearthat if the bodycontinuedto descendwith the samespeedBC, withoutac-celeration,it would in the next time-intervalCI traversedoublethe distancecoveredduringthe intervalAC,with theuniformspeedECwhichishalfofBC;but sincethefallingbodyacquiresequalincrementsof speedduringequalincrementsoftime, it followsthat the velocityBC, duringthe nexttime-

interval

THIRD DAY 177interval CI willbe increasedby an amountrepresentedby theparallelsofthe triangleBFGwhichis equalto the triangleABC.ILthen, oneadds to the velocityGI halfof thevelocityFG,thehighest speed acquiredby the acceleratedmotion and deter-mined by the parallelsof the triangleBFG, he will have theuniformvelocitywith which the same spacewouldhave beendescribedin the time CI; and sincethis speedIN is three timesas great as ECit followsthat the spacedescribedduringthe in-tervalCI isthree timesasgreatasthat describedduringtheinter-val AC. Let us imaginethe motionextendedoveranotherequaltime-intervalIO, and the triangleextendedto APO; it is thenevident that if the motioncontinuesduring the intervalIO, attheconstantrate IF acquiredby accelerationduringthe timeAI,the spacetraversedduringthe intervalIO willbe fourtimesthattraversedduringthe firstintervalAC, becausethe speed IF isfourtimes the speedEC. But ifwe enlargeourtrianglesoastoincludeFPQ whichis equalto ABC, stillassumingLheaccelera-tion to be constant,we shalladd to the uniformspeedan incre-mentRQ, equal toEC; then the valueofthe equivalentuniformspeedduring the time-intervalIO willbe five timesthat duringthe firsttime-intervalAC; thereforethe spacetraversedwillbequintuplethat duringthe first intervalAC. It is thus evidentby simple computationthat a movingbody starting fromrestand acquiringvelocityat a rate proportionalto the time,will,during equal intervals of time, traversedistances which arerelatedto eachother as the oddnumbersbeginningwithunity,I, 3, 5;* or consideringthe total spadetraversed,that covered

[212[in doubletimewill be quadruplethat coveredduringunit time;in triple time, the space is nine timesas greatas in unit time.

*Asillustratingthegreatereleganceandbrevityofmodernanalyticalmethods,onemayobtaintheresultof Prop.n directlyfromthe fun-damentalequation

s=I/,g(t',-t l)=g/2 +t3 tl)whereg istheaccelerationofgravityands,thespacetraversedbetweentheinstantstxandh. Ifnowt2- tl= I,sayonesecond,thens = g]_(t2+ ta)wheret_+tx,mustalwaysbean oddnumber,seeingthatit isthesumoftwoconsecutivetermsintheseriesofnaturalnumbers. [Trans.]

I78 THE TWO NEWSCIENCESOF GALILEOAndin generalthe spacestraversedarein +.heduplicateratioo_fthetimes,i.e.,inthe ratioofthesquaresofthe times.

SnwP.In truth, I findmorepleasurein thissimpleandclearargumentofSagredothanin theAuthor'sdemonstrationwhichto me appearsratherobscure;so that I am convincedthatmattersareas described,oncehavingacceptedthe definitionofuniformlyacceleratedmotion. Butasto whetherthisaccelera-tion is that whichonemeetsin nature in the caseof fallingbodies,I amstilldoubtful;andit seemsto me,notonlyformyownsakebut alsoforall thosewhothink as I do, that thiswouldbe the propermomentto introduceoneof thoseexperi-mentswandthere are many of them, I understand--whichillustratein severalwaystheconclusionsreached.

SALV.Therequestwhichyou,asa manofscience,make,is averyreasonableone;for this is the custom--andproperlyso---in thosescienceswheremathematicaldemonstrationsareappliedto naturalphenomena,as is seenin the caseof perspecCtive,astronomy,mechanics,music,andotherswherethe principles,onceestablishedbywell-chosenexperiments,becomethefounda-tionsofthe entiresuperstrucCture.I hopethereforeit willnotappearto be awasteoftimeifwediscussat considerablelengththis first and most fundamentalquestionupon whichhingenumerousconsequencesof whichwehave in thisbookonlyasmallnumber,placedthereby the Author,whohasdonesomuchto openapathwayhithertoclosedto mindsofspeculativeturn. So far as experimentsgothey havenotbeenneglecCtedbythe Author;andoften,inhiscompany,I haveattemptedinthe followingmannerto assuremyselfthat the accelerationacCtuallyexperiencedbyfallingbodiesisthatabovedescribed.

A pieceof woodenmouldingor scantling,about I2 cubitslong,half a cubit wide,and three finger-breadthsthick,wastaken;on its edgewascut a channela littlemorethan onefingerinbreadth;havingmadethisgrooveverystraight,smooth,and polished,andhaving lined it with parchment,alsoassmoothand polishedas possible,we rolledalongit a hard,smooth,and very roundbronzeball. Having placed this

[ I3] board

THIRD DAY 179boardin a slopingposition,by liftingoneendsomeoneortwocubitsabovetheother,werolledtheball,asI wasjust saying,alongthechannel,noting,in amannerpresentlyto bedescribed,the timerequiredto makethe descent.We repeatedthisex-perimentmorethanoncein orderto measurethe timewithanaccuracysuch that the deviationbetweentwo observationsneverexceededone-tenthof a pulse-beat.Havingperformedthisoperationandhavingassuredourselvesofitsreliability,wenowrolledthe ballonlyone-quarterthelengthofthe channel;andhavingmeasuredthe timeof its descent,wefoundit pre-ciselyone-halfoftheformer. Nextwetriedotherdistances,com-paringthetimeforthewholelengthwiththatforthehalf,orwiththat for two-thirds,orthree-fourths,orindeedforanyfra_ion;in suchexperiments,repeateda fullhundredtimes,wealwaysfoundthat thespacestraversedwereto eachotherasthesquaresofthe times,andthiswastrueforall inclinationsofthe plane,i.e.,ofthe channel,alongwhichwerolledthe ball. Wealsoobservedthatthetimesofdescent,forvariousinclinationsoftheplane,boreto oneanotherpreciselythat ratiowhich,asweshallseelater,theAuthorhadpredictedanddemonstratedforthem.

Forthe measurementof time,weemployeda largevesselofwater placedin an elevatedposition;to the bottomof thisvesselwassoldereda pipeofsmalldiametergivinga thinjet ofwater,whichwecolle_edinasmallglassduringthetimeofeachdescent,whetherforthewholelengthofthechannelorforapartofits length;the waterthuscollecCtedwasweighed,aftereachdescent,ona veryaccuratebalance;thedifferencesandratiosoftheseweightsgaveus thedifferencesandratiosofthetimes,andthis with suchaccuracythat althoughthe operationwasre-peatedmany,manytimes,therewasnoappreciablediscrepancyin theresults.

S_P. I wouldliketo havebeenpresentat theseexperiments;but feelingconfidencein the carewithwhichyou performedthem,and in the fidelitywith whichyou relatethem, I amsatisfiedandacceptthemastrueandvalid

SM,v. Therrwecanproceedwithoutdiscussion.[2_4]

x8o THE TWO NEW SCIENCESOF GALILEO

COROT,I,ARYIISecondly,itfollowsthat, startingfromanyinitialpoint,ifwe

takeany twodistances,traversedin anytime-intervalswhatso-"i_ ever,thesetime-intervalsbearto oneanotherthe same• ratioas oneofthe distancesto themeanproportionalof

the twodistances.ForifwetaketwodistancesSTandSYmeasuredfrom

the initialpointS,themeanproportionalofwhichisSX,T the timeoffall throughSTis to thetimeoffallthrough.:I¢:SYasSTistoSX;oronemaysaythetimeoffallthrough

SYis to thetimeoffallthroughSTasSYisto SX. Nowsinceit has beenshownthat the spacestraversedare in

Y¢"thesameratioasthesquaresofthe times;andsince,more-Fig.5Oover' the ratio of the spaceSY to the spaceST is thesquareof the ratioSYto SX,it followsthat the ratioofthetimesof fall throughSYandST is the ratioofthe respec°dvedistancesSYandSX.

SCHOLIUM

The abovecorollaryhasbeenprovenfor the caseofverticalfall;but it holdsalsoforplanesinclinedat anyangle;forit istobeassumedthat alongtheseplanesthevelocityincreasesin thesameratio,that is,inproportionto the time,or,ifyouprefer,asthe seriesofnaturalnumbers.*

SAr,v.Here,Sagredo,I shouldllke,if it be not too tedioustoSimplicio,to interruptfora momentthe presentdiscussioninorderto makesomeadditionson the basisofwhathasalreadybeenprovedandof whatmechanicalprincipleswehavealreadylearnedfromourAcademician.This additionI makefor thebetterestablishmenton logicalandexperimentalgrounds,oftheprinciplewhichwe haveaboveconsidered;andwhat is moreimportant,forthepurposeofderivingit geometrically',afterfirstdemonstratinga singlelemmawhichisfundamentalin thescienceofmotion[impet_].

*ThedialoguewhichintervenesbetweenthisScholiumandthefollow-ingtheoremwaselaboratedbyViviani,at thesuggestionofGalileo.SeeNationalEdition,viii,23. [Trans.]

THIRD DAY I8iSAGR.If the advancewhichyou proposeto make is such as

will confirmand fully establishthesesciencesof motion,I willgladly devoteto it any lengthof time. Indeed,I shallnot only

[215]be gladto haveyou proceed,but I begofyou at onceto satisfythe curiositywhichyou have awakenedin me concerningyourproposition;andIthinkthat Simplicioisofthe samemind.

SIMP.QuiterighLSALV.SincethenI haveyourpermission,let usfirstof allcon-

siderthis notable£ac°c,that the momentaor speeds[imomentiolevelocitY]ofone and the samemovingbodyvarywith the inclina-tionof the plane.

The speedreachesa maximumalonga verticaldirection,andfor other dirertionsdiminishesas the plane divergesfrom thevertical. Therefore the impetus, ability, energy, [l'impeto,iltalento,l'energia]or, one might say, the momentum[ilmomento]of descentof the movingbody is diminishedby the plane uponwhichit is supportedand alongwhichit rolls.

For the sakeofgreaterclearnessereCtthe lineABperpendicularto the horizontalAC; next drawAID,AE, AF, etc., at differentinclinationsto the horizontal. ThenI saythat all themomentumof the fallingbodyis alongthe verticaland is a maximumwhenitfallsin that dirertlon;the momentumis lessalongDA and stilllessalongEA,and evenlessyet alongthe moreinclinedplaneFA.Finallyon the horizontalplane the mo- Bmentum vanishesaltogether;the body J)findsitself in a conditionof indifferenceasto motionor rest;hasno inherenttend- Igencyto movein any dirertion,and offersnoresistanceto beingset inmotion. Forjust as a heavy bodyor systemof bodiescannotof itselfmoveupwards,or recedefrom the commoncenter [comuncentro]H(31towardwhichall heavythingstend, soit ! Ais impossibleforany bodyof its ownac- Ocord to assumeany motionother than Fig.51one which carriesit nearerto the aforesaidcommoncenter.Hence,alongthe horizontal,by whichwe understanda surface,everypointofwhichisequidistantfromthissamecommoncenter,the bodywillhavenomomentumwhatever. This

I82 THE TWO NEW SCIENCES OF GALILEO[516]

This changeof momentumbeingclear,it is herenecessaryforme to explainsomethingwhichour Academicianwrotewhen inPadua, embodyingit in a treatiseon mechanicspreparedsolelyfor the use of his students, and provingit at lengthand conclu-sivelywhen consideringthe originand nature of that marvellousmachine,the screw. What he provedis the mannerin whichthemomentum[impeto]varieswiththe inclinationofthe plane,asforinstancethat oftheplaneFA,oneendofwhichiselevatedthrougha verticaldistanceFC. Thisdire_ionFC is that alongwhichthemomentumof a heavybodybecomesa maximum;let us discoverwhatratio thismomentumbearsto that of the samebodymovingalongthe inclinedplane FA. This ratio, I say, is the inverseofthat of the aforesaidlengths. Suchis the lemmaprecedingthetheoremwhichI hopeto demonstratea littlelater.

It is clearthat the impellingforce[impeto]acCtingon a bodyindescentis equal to the resistanceor least force[resistenzaoforzaminima]sufficientto holdit at rest. In orderto measurethis forceand resistance_orzae resistenza]I proposeto use the weightofanotherbody. Let us placeupon the plane FA a body G con-necCtedto the weightH by means of a cord passingover thepoint F; then the body H will ascend or descend,along theperpendicular,the same distancewhichthe body G ascendsordescendsalongthe inclinedplane FA; but this distancewill notbe equalto theriseorfallofG alongthe verticalinwhichdirectionaloneG, asother bodies,exertsits force[resistenza].Thisis clear.For if weconsiderthe motionof the bodyG, fromA to F, in thetriangleAFC to be madeup ofa horizontalcomponentAC and averticalcomponentCF, and rememberthat this bodyexperiencesno resistanceto motionalongthe horizontal(becauseby such a

[2x71motionthe bodyneithergainsnor losesdistancefromthecommoncenterof heavythings) it followsthat resistanceis met only inconsequenceof the body risingthroughthe verticaldistanceCF.Sincethenthe bodyG inmovingfromA toF offersresistanceonlyin so far as it rises throughthe vertical distanceCF, whiletheother bodyH must fallverticallythroughthe entiredistanceFA,and sincethis ratio is maintainedwhetherthe motionbe largeorsmall,the two bodiesbeinginextensiblyconnected,we are ableto assertpositivelythat, in.caseof equilibrium(bodiesat rest) the

momenta,

THIRD DAY 183momenta,thevelocities,or theirtendencyto motion[propensionial mot@i. e., the spaceswhichwouldbe traversedby them inequaltimes,mustbe in the inverseratioto theirweights. This iswhathas beendemonstratedineverycaseof mechanicalmotion.*So that, inorderto hold the weight(3at rest,onemustgive H aweightsmallerin the sameratio asthedistanceCFis smallerthanFA. If wedothis,FA:FC=weightG:weightH; then equilibriumwilloccur,that is,the weightsH andG willhavethe sameimpell-ing forces [momentieguali],and the two bodieswill come torest.

And sincewe are agreedthat the impetus,energy,momentumor tendencytomotionof a movingb.odyis asgreatas theforceorleast resistance[forzaoresistenzamzn_ma]sufficientto stopit,andsincewe havefoundthat the weightH is capableof preventingmotionin theweightG, it followsthat the lessweightHwhoseen-tireforce[momentototale]is alongtheperpendicular,FC,willbeanexactmeasureof the componentof force[momentoparziale]whichthe largerweightG exertsalongthe planeFA. But the measureof the total force[totalmomento]onthe bodyG is its ownweight,sinceto preventits fall it is onlynecessaryto balanceit withanequal weight,providedthis secondweightbe freeto moveverti-cally;thereforethe componentof the force[momentoparziale]onG alongthe inclinedplaneFAwillbearto themaximumandtotalforceon this samebodyG alongthe perpendicularFC the sameratio as the weightH to the weightG. This ratio is, by con-struction, the samewhichthe height, FC, of the inclinedplanebearsto the lengthFA. We havehere the lemmawhichI pro-posedto demonstrateand which,as you will see,has been as-sumedby our Authorin the secondpart of the sixthpropositionofthepresenttreatise.

SAGg.From what you haveshownthus far, it appearsto methat one might infer, arguingex aequaliconla proportioneper-turbata,that the tendencies[momentz]ofoneandthe samebodytomovealongplanesdifferentlyinclined,but havingthe sameverti-cal height, as FA and FI, are to each other inverselyas thelengthsofthe planes.

[218]

SAT.V.Perfectlyright. This point established,I pass to thedemonstrationofthe followingtheorem:

*A nearapproachto the principleof virtualworkenunciatedbyJohnBernoulliin17I7. [Trans.]

I84 THE TWO NEW SCIENCES OF GALII.F.OIf a bodyfallsfreelyalongsmoothplanesinclinedat anyanglewhatsoever,but of the sameheight,the speedswithwhichitreachesthe bottomarethe same.

First we must recallthe fact that on a planeof any inclinationwhatevera body starting from rest gains speedor momentum[laquantit&dell'impao]in directproportionto the time, in agree-mentwiththe definitionofnaturallyacceleratedmotiongivenbytheAuthor. Hence,ashe hasshownin theprecedingproposition,the distancestraversedare proportionalto the squaresof thetimes and thereforeto the squaresof the speeds. The speedrelationsarehere the sameas in the motionfirst studied[i. e.,verticalmotion],sinceineachcasethe gainof speedis proportionalto the time.

LetABbe an inclinedplanewhoseheightabovethe levelBC isAC. Aswehaveseenabovethe forceimpelling[l'impeto]a body

A tofallalongtheverticalACis totheforce

whichdrivesthe samebodyalongthe in-

clinedplaneABasAB is toAC. OntheinclineAB, lay off AD a third propor-tionaltoABand AC; then the forcepro-ducingmotionalongAC is to that along

" AB (i.e.,alongAD)as thelengthACis toB B IJthe lengthAD. Andthereforethe body

Fig.52 willtraverse the spaceAD,alongthe in-clineAB,in the sametimewhichit wouldoccupyin fallingthever-ticaldistanceAC, (sincetheforces[moment,]arein thesameratioas thesedistances);alsothe speedat C is to thespeedat D asthedistanceACis to thedistanceAD. But,accordingtothedefini-tionof acceleratedmotion,the speedat B is to the speedof thesamebody at D asthe timerequiredto traverseAB is to the timerequiredforAD; and,accordingto the last corollaryofthe secondproposition,the timeof passingthroughthedistanceAB bearstothe time of passingthroughAD the same ratioas the distanceAC (a mean proportionalbetweenABand AD)to AD. Accord-inglythe two speedsat B and C eachbearto the speedat D thesameratio,namely,that of thedistancesACandAD;hencetheyareequal. This is the theoremwhichI setoutto prove.

Fromtheabovewe arebetterableto demonstratethefollowingthirdpropositionof theAuthorin whichhe employsthe followingprinciple,namely,the timerequiredto traversean inclinedplane

is

THIRD DAY i85is to that requiredto fall throughtheverticalheightofthe planeinthe sameratioasthelengthofthe planeto itsheight.

[219]For,accordingto the secondcorollaryofthesecondproposition,

if BArepresentsthe timerequiredto passover the distanceBA,the timerequiredto passthe distanceADwillbe a meanpropor-tional betweenthese two distancesand will be representedbythe lineAC; but ifACrepresentsthe timeneededto traverseADit willalsorepresentthe timerequiredto fallthroughthe distanceAC,sincethe distancesACand AD aretraversedinequaltimes;consequentlyif ABrepresentsthe timerequiredforABthen ACwillrepresentthe timerequiredforAC. Hencethetimesrequiredto traverseABand ACare toeachotheras the distancesABandAC.

In like mannerit can be shownthat the timerequiredto fallthroughAC is to the timerequiredfor any other inclineAE asthe lengthACisto thelengthAE; therefore,exaequali,thetimeoffall alongthe inclineABis to that alongAEas the distanceABisto the distanceAE,etc.*

Onemightby applicationof this sametheorem,as Sagredowillreadilysee,immediatelydemonstratethe sixthpropositionof theAuthor; but let us here end this digressionwhich Sagredohasperhapsfoundrathertedious,thoughI considerit quite importantforthe theoryofmotion.

SACR.On the contraryit has givenme greatsatisfaction,andindeed I find it necessaryfor a completegrasp of this principle.

SALv.I willnowresumethereadingof thetext.[215]TttEOltEMIII, PROPOSXTXONIII

If one and the same body, starting from rest, fallsalongan inclinedplaneand alsoalonga vertical,eachhavingthesameheight, the times of descentwill be to eachother asthe lengthsof the inclinedplane and the vertical.

Let ACbe the inclinedplaneandAB the perpendicular,eachhaving the same verticalheight above the horizontal,namely,BA;then I say, the time of descentof one and the samebody

*Putting_hisargumentina modernandevidentnotation,onehasAC= 1h gt_andAD= 1[__--gt_.If now_-_2= tLB.AD,itfoUowsatoncethat t_= t_. [Trans.] Q"_"_"

186 THE TWO NEW SCIENCESOF GALILEO[zI6l

alongthe planeACbearsa ratioto the timeoffall alongtheperpendicularAB,whichis the sameasthe ratioofthe lengthACto thelengthAB. LetDG,EI andLFbe anylinesparallel

to the horizontalCB; then it followsfrom• whathasprecededthat abodystartingfrom

A willacquirethesamespeedat the pointGa Dasat D, sincein eachcasethe verticalfallis

the same;in likemannerthe speedsat I andE willbe the same;soalsothoseat L andF.Andingeneralthe speedsat the twoextremi-tiesof anyparalleldrawnfromanypointonAB to the correspondingpointonACwillbe

© _ _ equal.Fig.53 Thus the two distancesAC and AB are

traversedat the samespeed. But it hasalreadybeenproved[zI7]

that iftwodistancesaretraversedbyabodymovingwithequalspeeds,thenthe ratioofthe timesofdescentwillbethe ratioofthe distancesthemselves;therefore,the timeof descentalongACis to that alongAB as the lengthoftheplaneAC is to theverticaldistanceAB. O.v.D.

[181SacR.It seemsto methat theabovecouldhavebeenproved

clearlyandbrieflyonthebasisofa propositionalreadydemon-strated,namely,that the distancetraversedin the caseofacceleratedmotionalongACorABis the sameasthat covered

[zr9]by auniformspeedwhosevalueisone-halfthemaximumspeed,CB;the twodistancesAC andAB havingbeentraversedat thesameuniformspeedit is evident,fromPropositionI, that thetimesofdescentwillbeto eachotherasthedistances.

COROLLARY

Hencewemay inferthat the timesof descentalongplaneshavingdifferentinclinations,but thesameverticalheightstand

to

THIRD DAY 187to oneanotherin the sameratioas thelengthsof theplanes.ForconsideranyplaneAMextendingfromA tothe horizontalCB;thenit maybe demonstratedinthe samemannerthatthetimeof descentalongAMis to the timealongAB as thedis-tanceAMistoAB; butsincethetimealongAB istothat alongACasthelengthABistothelengthAC,itfollows,excequali,thatasAMistoACsois the timealongAMto thetimealongAC.

IV,PRoPosmoIVThetimesof descentalongplanesof the samelengthbutof differentinclinationsare to eachotherin the inverseratio of the squareroots of their heights

Froma singlepointB drawthe planesBAandBC,havingthe samelengthbut differentinclinations;letAE andCD behorizontallinesdrawnto meet the perpendicularBD; and

[22olletBErepresentthe heightoftheplaneAB,andBDtheheightofBC; alsoletBIbe a meanproportionalto BDandBE; thenthe ratioof BD to BI is equalto the Bsquarerootof the ratioof BD to BE. .m,Now,I say,the ratioofthe timesofde- /,I/scentalongBAandBCistheratioofBD / /toBI;so that the timeof descentalong / /BAis relatedto the heightof the other / /planeBC,namelyBD asthe timealong / /BCis relatedto the heightBI. Nowit / / _-mustbe provedthat the timeof descent& _/alongBAisto thatalongBCasthelength _yBDisto thelengthBI.

DrawIS parallelto DC;and sinceit Fig.54hasbeenshownthat the timeoffallalongBAisto that alongthe verticalBE as BAisto BE;and alsothat the timealongBE isto that alongBDas BEis to BI; and likewisethat thetimealongBIDisto that alongBCasBDisto BC,or asBI toBS;it follows,excequali,that thetimealongBAisto that alongBC asBAto BS,or BCto BS. However,BCisto BSas BDisto BI;hencefollowsourproposition.

x88 THE 2_zO NEW SCIENCESOF GALII.EO

THEO_ V, PRoPosmo_VThe timesofdescentalongplanesof differentlength,slopeandheightbearto oneanothera ratiowhichis equaltothe productof theratioofthe lengthsbythe squarerootoftheinverseratiooftheirheights.

Draw the planesAB and AC, havingdifferentinclinations,lengths,andheights. My theoremthenis that the ratioofthe

A timeof descentalongACto that alongABis.41 equaltotheproductof theratioofACto AB

//' [ by thesquarerootoftheinverseratiooftheir/./ ! heights.

/ / I ForletADbeaperpendicularto whichare/ / ]Gdrawnthe horizontallinesBGandCD;alsoB ¢ /_ i letAL bea meanproportionalto theheights

/ ] AGandAD;fromthe pointL drawa hori-1_/ [I_zontallinemeetingAC inF; accordinglyAF/ ] willbea meanproportionalbetweenACand

/ [ AE. NowsincethetimeofdescentalongACC/ 110is to that alongAE as the lengthAF is to

Fig.55 AE; and sincethe timealongAEis to thatalongAB asAE istoAB,it isclearthat thetimealongACistothatalongAB asAF istoAB.[2i]

Thusit remainsto beshownthat theratioofAFtoABisequalto the productofthe ratioofACto ABby the ratioofAGtoAL,whichis the inverseratioof thesquarerootsoftheheightsDA andGA. Nowit is evidentthat, ifweconsiderthe lineACin connectionwithAFandAB,the ratioofAFto ACisthesameasthat orAL toAD,orAGto ALwhichisthesquarerootof the ratioof theheightsAGandAD;but theratioofACtoAB is theratioof the lengthsthemselves.Hencefollowsthetheorem.

THEOm_MVI, PROPOSITIOnVII__romthehighestorlowestpointina verticalcircletherebe drawnanyinclinedplanesmeetingthecircumferencethetimes

THIRD DAY I89timesofdescentalongthesechordsare eachequalto theother.

Onthe horizontallineGHconstrufitaverticalcircle.Fromitslowestpoint--thepointof tangencywiththehorizontal,drawthe diameterFAandfromthe highestpoint,A,drawinclinedplanesto B and (2,anypointswhateveron the circumference;thenthe timesof descentalong Btheseare equal. DrawBD andCEperpendicularto the diame-ter; makeAI a meanpropor-¢tionalbetweentheheightsoftheplanes,AEand AD; and sincethe refitanglesFA.AEand FA.ADarerespefitivelyequalto thesquaresofACandAB,whiletherectangleFAakEis to the refit- ....angleFA_ADas AE is to AD, G _it followsthat the squareofAC Fig.56is to the squareofAB as the lengthAEis to the lengthAD.Butsincethe lengthAEisto ADas the squareofAI isto thesquareofAD, it followsthat the squareson the linesACandABareto eachotherasthesquaresonthelinesAI andAD,andhencealsothe lengthAC is to the lengthABas AI is to AD.But it haspreviouslybeendemonstratedthat the ratioof thetime of descentalongAC to that alongAB is equalto theprodufitof the tworatiosACto AB andADto AI; but thislastratioisthesameasthatofABtoAC. Thereforetheratioofthe timeof descentalongACto that alongABis the productofthe tworatios,ACto ABandABto AC. Theratioofthesetimesisthereforeunity. Hencefollowsourproposition.

Byuseoftheprinciplesofmechanics[exmechanic#]onemayobtainthe sameresult,namely,that a fallingbodywillrequireequaltimesto traversethe distancesCAandDA,indicatedinthefollowingfigure.LayoffBAequaltoDA,andletfallthe

[zzzlperpendicularsBE and DF; it followsfromthe principlesofmechanics

I9o THE TWONEW SCIENCESOF GALII.FOmechanicsthat the componentof the momentum[momentumponderis]acCtingalongthe inclinedplaneABCis to the totalmomentum[i.e.,the momentumof the bodyfallingfreely]as

BEis to BA; in likemannerthe momentumalong theplaneADis to its total mo-mentum[i.e.,themomentumof the bodyfallingfreely]asDFistoDA,ortoBA.There-fore the momentumof thissameweightalongthe plane

_ DAisto that alongthe planeABC as the lengthDF is to

_._ thelengthBE;forthisreason,thissameweightwillin equalA mF timesaccordingto thesecond

Fig.57 propositionof the firstbook,traversespacesalongthe planesCAandDAwhichareto eachotherasthelengthsBE andDF. Butit canbeshownthatCAisto DAasBE isto DF. Hencethefallingbodyvcilltraversethe twopathsCAandDAin equaltimes.

Moreoverthe fa_ that CAiSto DAasBEisto DF maybedemonstratedas follows:JoinC andD; throughD, drawthelineDGLparallelto AF andcuttingthelineACin I; throughB drawthe lineBH,alsoparallelto AF. Thenthe angleADIwillbe equalto the angleDCA,sincetheysubtendequalarcsLAandDA,andsincethe angleDACis common,the sidesofthe triangles,CADandDM, aboutthecommonanglewillbeproportionalto eachother; accordinglyas CAis to DAso isDAto IA,that is as BAis to IA, or asHAis to GA,that isasBEisto DF. _. D.

The samepropositionmaybe moreeasilydemonstratedasfollows:OnthehorizontallineABdrawa circlewhosediameterDCisvertical. Fromtheupperendofthisdiameterdrawanyinclinedplane,DF,extendingtomeetthecircumference;then,Isay, a bodywilloccupythe sametime in fallingalongtheplaneDF as alongthe diameterDC. For drawFG parallel

to

THIRD DAY 191to ABandperpendicularto DC;joinFC; andsincethe timeoffall alongDC is to that alongDG as the meanproportional

[223]betweenCD andGDis to GDitself;and sincealsoDF is ameanproportionalbetweenDC andDG, the angleDFC in-scribedina semicirclebeinga right-angle,and FG beingperpendicular

to DC, it followsthat the timeof /"_ \____ 1_

fallalongDCisto that alongDGas f JthelengthFDisto GD. Butit has //,,Ialreadybeendemonstratedthat theF_"timeof descentalongDF is to that [alongDGasthelengthDFistoDG; _hencethetimesofdescentalongDF \andDCeachbearto thetimeof fall _ _ " /_,alongDG the same ratio; conse- _ \quentlytheyareequal.

In likemannerit maybe shown_ cthat ifonedrawsthechordCEfrom Fig.58the lowerendof the diameter,alsothe lineEH parallelto thehorizon,andjoinsthepointsE andD, thetimeofdescentalongEC,willbethe sameasthat alongthe diameter,DC.

COROLLARYIFromthisit followsthat thetimesofdescentalongallchords

drawnthrougheitherCorDareequaloneto another.

COROIJ,ARYIIIt alsofollowsthat, if fromanyonepointtherebe drawna

verticallineandaninclinedonealongwhichthetimeofdescent.isthe same,the inclinedlinewillbe a chordof a semicircleofwhichtheverticallineisthediameter.

COROllARYIIIMoreoverthe timesof descentalonginclinedplaneswillbe

equalwhenthe verticalheightsofequallengthsof theseplanesare

I92 THE TWO NEW SCIENCESOF GALILEOareto eachotherasthelengthsoftheplanesthemselves;thusitis clearthat the timesof descentalongCA andDA, in thefigurejust beforethe last, are equal,providedthe verticalheightof AB (ABbeingequalto AD),namely,BE, is to theverticalheightDF asCAisto DA.

SAcR.Pleaseallowmeto interruptthelecturefora momentinorderthatI mayclearupanideawhichjustoccurstome;onewhich,if it involvenofallacy,suggestsat leasta freakishand

[2241interestingcircumstance,suchas oftenoccursin natureandintherealmofnecessaryconsequences.

If, fromanypointfixedin a horizontalplane,straightlinesbe drawnextendingindefinitelyin all dire_ions,and ff weimagineapointto movealongeachoftheselineswithconstantspeed,allstartingfromthefixedpointat the sameinstantandmovingwithequalspeeds,thenit isclearthat allofthesemov-ing pointswilllie uponthe circumferenceof a circlewhichgrowslargerandlarger,alwayshavingtheaforesaidfixedpointasitscenter;thiscirclespreadsoutinpreciselythesamemanneras the littlewavesdo in the caseof a pebbleallowedto dropintoquietwater,wherethelmpac°cofthestonestartsthemotionin alldirections,whilethe pointof lmpac_cremainsthe centeroftheseever-expandingcircularwaves. But imaginea verticalplanefromthe highestpointofwhichare drawnlinesinclinedat everyangleand extendingindefinitely;imaginealsothatheavyparticlesdescendalongtheselineseachwitha naturallyacceleratedmotionandeachwith a speedappropriateto theinclinationof its line. If thesemovingparticlesare alwaysvisible,whatwillbethe locusoftheirpositionsat anyinstant?Nowtheanswerto thisquestionsurprisesme,forI amledby_eprecedingtheoremsto believethat theseparticleswillalwayslieuponthe circumferenceof a singlecircle,everincreasinginsizeasthe particlesrecedefartherandfartherfromthepointatwhichtheirmotionbegan. To be moredefinite,let A be thefixedpointfromwhicharedrawnthe linesAFandM-Iinclinedat any anglewhatsoever.On the perpendicular/_BtakeanytwopointsC _ndDaboutwhich,ascenters,circlesaredescribed

passing

THIRD DAY I93passingthrough the point A, and cutting the inclined linesat the pointsF, H, B, E, G, I. Fromthe precedingtheoremsitis clearthat, ifparticlesstart, at the sameinstant, fromA anddescendalongthese lines,when one is at E anotherwill be atG and anotherat I; at a later instant Atheywillbe foundsimultaneouslyatF, H and B; these, and indeedan

infinite numbe_22_°ther particles_ /

travellingalong an infinitenumberof differentslopeswill at successiveinstants always lie upon a single xever-expandingcircle.Thetwokindsof motion occurringin nature giverise thereforeto two infiniteseriesof circles,at once resemblingand Fig.59differingfromeachother; the onetakes its risein the centerofan infinitenumberof concentriccircles;the other has its originin the contac°c,at their highestpoints,of an infinitenumberofeccentriccircles;the formerareproducedby motionswhichareequal and uniform; the latter by motions which are neitheruniformnor equal amongthemselves,but whichvary fromoneto anotheraccordingto theslope.

Further, if fromthe two points chosenas originsof motion,we draw lines not only along horizontal and vertical planesbut inall direcCtionsthen just as in the formercases,beginningat a singlepoint ever-expandingcirclesare produced,so in thelatter casean infinitenumberof spheresare producedabout asinglepoint, or rather a singlesphere which expandsin sizewithoutlimit; and this in twoways,onewith the originat thecenter,the otheron the surfaceof the spheres.

S_v. The idea is reallybeautifuland worthy of the clevermindofSagredo.

Sn_P.Asfor me, I understandin a generalway howthe twokinds of natural motionsgive rise to the circlesand spheres;and yet as to the producCtionofcirclesby acceleratedmotionandits proof,I am notentirelyclear;but the facetthat onecantake

the

I94 THE TWONEW SCIENCESOF GALII.F.Otheoriginofmotioneitherat theinmostcenterorat theverytopofthe sphereleadsoneto think that theremaybe somegreatmysteryhiddenin thesetrueandwonderfulresults,a mysteryrelatedto the creationof the universe(whichis said to besphericalin shape),and relatedalso to the seatof the firstcause[primacausa].

SaJ.v.I haveno hesitationin agreeingwithyou. But pro-foundconsiderationsofthSskindbelongto ahighersciencethanours[api_taltedottrinechelenostre].Wemustbe satisfiedtobelongto that classof lessworthyworkmenwhoprocurefromthe quarrythe marbleout ofwhich,later,the giftedsculptorproducesthosemasterpieceswhichlayhiddenin thisroughandshapelessexterior. Now,ifyouplease,letusproceed.

[2z6]THEOREMVII, PROPOSITIONVII

I{theheightsoftwoinclinedplanesaretoeachotherinthesameratioasthe squaresof theirlengths,bodiesstartingfromrestwilltraversetheseplanesin equaltimes.

Taketwoplanesofdifferentlengthsanddifferentinclinations,AEandAB,whoseheightsareAF andAD:letAFbeto ADas

Athe squareofAE is to the squareof_A_B;then,I say,that a body,startingfromrestatA,willtraversetheplanesAEandABinequaltimes. Fromtheverticalline,drawthehorizontalpar-

G I_allellinesEF andDB, the latter cut-tingAE at G. SinceFA:DA=EA*:B--_2,andsinceFA:DA=EA:GA,it

_, followsthat EA:GA--_2 : B--_2.:FHenceBAis a meanproportionalbe-

Fig.6o tweenEA and GA. Now sincethetimeofdescentalongABbearsto thetimealongAGthe sameratiowhichABbearsto AGandsincealsothe timeof descentalongAGis to the timealongAEasAGis to a meanpropor-tionalbetweenAGandAE,that is,to AB,it follows,ex¢equali,

that

THIRD DAY 195thatthetimealongABisto the timealongAEasABistoitself.Thereforethe timesare equal. Q.E.D.

THEOm_aVIII, P_oPosmoNVIIIThetimesofdescentalongallinclinedplaneswhichintersedtoneandthe sameverticalcircle,eitherat its highestorlowestpoint,areequalto the timeoffallalongtheverticaldiameter;forthoseplaneswhichfallshortofthisdiameterthetimesareshorter;forplaneswhichcutthisdiameter,thetimesarelonger.

Let ABbe the verticaldiameterof a circlewhichtouchesthehorizontalplane. It hasalreadybeenproventhat the timesofde- _a.scent alongplanesdrawnfromeitherend,A or B, to the cir-cumferenceare equal. In orderto showthat the timeofdescentI}

[227]alongthe planeDF whichfallsshortof the diameteris shorterwemaydrawtheplaneDBwhichisboth longerandlesssteeplyin-dinedthanDF;whenceit follows .....thatthetimealongDFislessthan"' Bthat alongDB andconsequently Fig.6IalongAB. In like manner,it is shownthat the timeof de-scentalongCOwhichcutsthediameterisgreater:forit isbothlongerand lesssteeplyinclinedthan CB. Hencefollowsthetheorem.

THEOREMIX,PRoeosmoNIXIffromanypointonahorizontalllnetwoplanes,inclinedatanyangle,aredrawn,andiftheyarecutbyalinewhichmakeswiththemanglesalternatelyequaltotheanglesbe-tweentheseplanesandthehorizontal,thenthetimesre-quiredto traversethoseportionsoftheplanecutoffbythe aforesaidline are equal.

Through

*96 THE TWO NEW SCIENCESOF GALILEOThroughthe point C bn the horizontallineX, drawtwo

planesCD and CE inclinedat any anglewhatever:at anypointin the lineCD layoffthe angleCDFequalto the angleXCE;let thelineDF cut CEat F sothat the anglesCDFandCFDarealternatelyequalto XCEandLCD;then,I say,the

.... timesof descentoverCD

x and CF are equal. Now

sincethe angle CDF isequalto the angleXCEbyconstrucCtion,it is evident

D _B //'_ c that the angleCFD must

_ _¢: beequalto theangleDCL.For if the commonangleDCF be subtracCtedfromthe threeanglesof the tri-

Fig.6_ angleCDF,togetherequalto tworightangles,(towhichare alsoequalalltheangleswhichcanbedescribedaboutthepointConthe lowersideof thelineLX)there remainin the triangletwo angles,CDFand CFD,equalto thetwoanglesXCEandLCD;but,by hypothesis,theanglesCDFandXCEareequal;hencetheremainingangleCFDisequalto theremainderDCL. TakeCEequaltoCD;fromthepointsDandEdrawDAandEBperpendiculartothehorizontallineXL; andfromthe pointC drawCGperpendicularto DF.NowsincetheangleCDGis equalto the angleECBand sinceDGCandCBEarerightangles,itfollowsthatthetrianglesCDGandCBEareequiangular;consequentlyDC:CG=CE:EB. ButDCisequalto CE,andthereforeCGisequaltoEB. Sincealsothe anglesat C andatA, in the triangleDAC,areequalto theanglesat F and (3 in the triangleCGF,wehaveCD:DA=FC:CGand, permutando,DC:CF--DA:CG=DA:BE.Thusthe ratio of the heightsof the equalplanesCD and CE isthesameasthe ratioofthe lengthsDCandCF. Therefore,by

[2 8]CorollaryI ofProp.VI, thetimesofdescentalongtheseplaneswillbe equal. Q.E.D.

Analternativeproofis thefollowing:DrawFSperpendicularto

THIRD DAY 197to the horizontalline AS. Then,since the triangleCSF issimilarto the triangleDGC,wehave SF:FC--GC:CD;andsincethe triangleCFG is similarto the triangleDCA,wehaveFC:CG=CD:DA.L z. c _" x:

Hence,ex cquali, SF: _"/_ [ //_xa c,

CG=CG:DA. There-foreCG isa meanpro-portionalbetween SFandDA,whileDA:SF=D-A_:C-G_. Againsince xthetriangleACT)issim- _ [_.---"7_

ilarto thetriangleCGF,we have DA'_C=GC:CF and, I_ermutando,DA:CG= DC:CF:also Fig.63b-A2:C-G_=D--C_:_2. Butit hasbeenshownthat D-A2:C-G_=DA:SF. ThereforeD-C_.'CF*=DA:FS.Hencefromthe aboveProp.VII,sincetheheightsDAandFSof the planesCDandCFareto eachotherasthesquaresofthelengthsofthe planes,it followsthat the timesof descentalongtheseplaneswillbeequal.

THEOREM X, PROPOSITIONX

Thetimesofdescentalonginclinedplanesofthesameheight,butofdifferentslope,aretoeachotherasthelengthsof theseplanes;andthis istruewhetherthe motionstartsfrom rest or whetherit is precededby a fall from aconstantheight.

LetthepathsofdescentbealongABCandABDto thehorizon-talplaneDCsothat thefallsalongBDandBCareprecededbythefallalongAB;then,I say,that thetimeofdescentalongBDis to the timeofdescentalongBCasthe lengthBDis to BC.Drawthe horizontalline_A_FandextendDB untilit cuts this

[229]lineat F; letFE bea meanproportionalbetweenDF andFB;drawEOparallelto DC;thenAOwillbe a meanproportionalbetweenCAandAB. If nowwerepresentthetimeoffallalong

AB

198 THE TWO NEW SCIENCESOF GALII.EOABbythelengthA_B,thenthe timeofdescentalongFBwillberepresentedbythedistanceFB; soalsothetimeoffallthroughthe entiredistanceAC willbe representedby the meanpro-portionalAO:andfor the entiredistanceFD by FE. Hencethe timeoffallalongthe remainder,BC,willbe representedby

A. "_!. BO,andthat alongtheremainder,BD,

"byBE; but sinceBE:BO--BD'_C,it

follows,ifwe allowthe bodiesto fallfirstalongAB andFB,or,whatis thesamething,alongthe commonstretchAB,thatthe timesofdescentalongBDand BCwillbe to eachother as the

o lengthsBDandBC.Butwehavepreviouslyproventhat_ thetimeofdescent,fromrestatB, along

Fig.64 BDis tothe timealongBCin the ratiowhichthelengthBDbearsto BC. Hencethetimesofdescentalongdifferentplanesofconstantheightareto eachotherasthelengthsoftheseplanes,whetherthemotionstartsfromrestor isprecededbyafallfroma constantheight. Q._. 9.

THeOReMX_I,P_oPosmoNXIIf a planebedividedintoanytwopartsandifmotionalongit startsfromrest, thenthe timeofdescentalongthe firstpart is to the timeof descentalongthe remainder "-_as the lengthof thisfirstpart is to the excessofameanproportionalbetweenthisfirstpartandtheen-tire lengthoverthisfirstpart.

Let the fall take place,fromrestat A, throughtileentiredistanceABwhichis dividedat anypointC; also eletAFbea meanproportionalbetweentheentirelength pBAandthefirstpartAC;thenCFwilldenotetheexcessofthemeanproportionalFAoverthefirstpartAC. Now,I say,the timeof descentalongACwillbetothe timeof BsubsequentfallthroughCB as the lengthACisto CF.Fig.65Thisis evident,becausethetimealongACis to the timealongthe entiredistanceABasACis to themeanproportionalAF.

Therefore,

THIRD DAY I99Therefore,divldendo,the time alongAC willbe to the timealongthe remainderCBasACisto CF. If weagreeto repre-sentthe timealongACby the lengthACthenthe timealongCBwillberepresentedbyCF. Q._. D.

[ 3o]In casethemotionisnotalongthestraightlineACBbutalongthebrokenlineACDto thehorizon- A

tal lineBD,andif fromF wedraw _thehorizontallineFE, itmayinlikemannerbe provedthat the timealongACisto thetimealongthein-clinedlineCDasACisto CE. Forthe time alongAC is to the timealongCBas AC is to CF; but ithas alreadybeenshownthat the D_'timealongCB,afterthefallthrough Fig.66thedistanceAC,isto thetimealongCD,afterdescentthroughthe samedistance_AC,asCBisto CD,or,asCFisto CE;there-fore,excequali,the timealongACwillbeto thetimealongCDasthelengthACisto thelengthCE.

THEOg_XII, PRoPosmonXIIIf a verticalplaneand anyinclinedplanearelimitedbytwohorizontals,andifwetakemeanproportionalsbetweenthe lengthsof theseplanesand thoseportionsof themwhichliebetweentheirpointofintersecCtionandtheupperhorizontal,then the time of fall alongthe perpendicularbearsto thetimerequiredto traversetheupperpartoftheperpendicularplusthe timerequiredto traversethe lowerpart of the intersecCtingplanethe sameratio whichtheentirelengthof the verticalbearsto a lengthwhichisthesum of the meanproportionalon the verticalplus theexcessof the entirelengthof the inclinedplaneoveritsmeanproportional.

LetAFandCDbetwohorizontalplaneslimitingtheverticalplaneACandthe inclinedplaneDF; letthe twolast-mentlonedplanesintersecCtat B. LetARbe ameanproportionalbetween

the

2OO THE TWO NEW SCIENCESOF GALILEOthe entireverticalACandits upperpart AB;andlet FSbe ameanproportionalbetweenFD andits upperpart FB. Then,I say,the timeoffallalongtheentireverticalpathACbearstothe timeoffallalongitsupperportionABplusthe timeoffall

A_1_ alongthe lowerpart ofthe inclinedplane,namely,BD, the sameratiowhichthe lengthAC bearsto themeanproportionalon the vertical,namely,AR,plusthelengthSDwhich

is the excessof theentireplaneDF

over its meanproportionalFS.Jointhe pointsR andS givinga

horizontallineRS. NowsincetheD o c timeof fall throughthe entiredis-

Fig.67 tanceAC is to the time alongtheportionAB as CAis to themeanproportionalAR it followsthat, if we agreeto representthe timeof fallthroughACbythe distanceAC,the timeof fallthroughthe distanceABwillberepresentedbyAR;andthe timeof descentthroughthe re-mainder,BC,willberepresentedbyRC. But,if thetimealongACis takento be equalto the lengthAC, thenthe timealongFDwillbeequalto thedistanceFD;andwemaylikewiseinferthat thetimeofdescentalongBE),whenprecededbyafallalongFB orAB,is numericallyequalto the distanceDS. Therefore

thetime requiredto fallalongthepathACisequalto ARplusPC;whilethetimeofdescentalongthebrokenlineABDwillbeequaltoARplusSD. Q.E.D.

The samethingis true if, in placeof a verticalplane,onetakesanyotherplane,asfor instanceNO;the methodofproofisalsothesame.

PROBLEMI, PROPOSITIONXIIIGivena perpendicularlineof limitedlength,it is requiredto finda planehavinga verticalheightequalto the givenperpendicularandso inclinedthat a body,havingfallenfromrest alongthe perpendicular,willmakeits descent

along

THIRD DAY 2o:alongthe inclinedplanein the sametimewhichit occu-piedinfallingthroughthegivenperpendicular.

LetABdenotethe givenperpendicular:prolongthis linetoC makingBCequalto AB,anddrawthe horizontallinesCEandAG. It isrequiredto drawaplanefromBto thehorizontalline CE suchthat aftera bodystartingfromrestat A hasfallenthroughthe distanceAB, it willcompleteits pathalongthisplanein anequaltime. LayoffCDequalto BC,anddrawthelineBD. ConstructthelineBEequalto thesumofBDandDC;then, I say,BE is the requiredplane. ProlongEB tillitintersectsthe horizontalAGat G. Let GF be a meanpro-portionalbetweenGE and GB; _ @thenEF:FB=EG.K3F,and _--_2: --F--B2=E--O2:G--F2=EG:GB. But /EGistwiceGB;hencethesquareof EF istwicethe squareofFB;

soalsois the squareofDBtwice_../fif/the squareofBC. ConsequentlyEF:FB=DB:BC,andcomponendoet permutando,EB'J)B+BC= D 0BF:BC. But EB=DB+ BC; Fig.68henceBF=BC=BA. If weagreethat thelengthABshallrep-resentthetimeoffallalongthelineAB,thenGBwillrepresentthe timeof descentalongGB,andGFthetimealongtheentiredistanceGE; thereforeBF willrepresentthe timeof descentalongthedifferenceof thesepaths,namely,BE,afterfallfromGot fromA. Q._. F.

[23 ]PRosn_-mIf,PP.oPosiTIO_XIV

Given an inclinedplaneand a perpendicularpassingthroughit, to finda lengthonthe upperpart of the per-pendicularthroughwhichabodywillfallfromrest in thesametimewhichis requiredto traversethe inclinedplaneafterfallthroughtheverticaldistancejustdetermined.

LetACbe the inclinedplaneandDB theperpendicular.It isrequiredto findon McverticalAD a lengthwhichwillbe

traversed

2oz THE TWO NEW SCIENCESOF GALILEOtraversedbya body,fallingfromrest,inthe sametimewhichisneededby the samebodyto traversethe planeAC aftertheaforesaidfall. Drawthe horizontalCB; layoffAEsuchthatBA+ zAC'.AC---AC'-A_E,andlay offAR suchthat BA-_A_C=EA-AkR.FromR drawRX perpendicularto DR;then,I say,X is the point sought. For sinceBAq-2AC'.AC=AC'_A_E,itfollows,dividendo,that BA+AC-_AC=CE'akE.And sinceBA'_C=EA'A_R,we have, componendo,BA+ AC'_A_C=ER:RA. But BA+ AC'.AC=CE'AE,henceCE:EA=ER'_RA=sum of the antecedents:sum of the consequents=CR'aRE.

D ThusRE is seento bea meanpropor-tionalbetweenCRandRA. Mordoversinceit has been assumedthat BA:

XI RAC=EA'.AR,andsinceby similartri-

] / angleswe have BA'JkC=XA'a_-R,itfollowsthat EA-_AR=XA'JkR. Hence

A EA andXA areequal. But ifweagree

// that the timeof fallthroughRA shallbe representedbythe lengthRA,thenthe timeoffallalongRCwillbe repre-sentedby the lengthRE whichis a

¢ !_ meanproportionalbetweenRAandRC;Fig. 69 likewiseAEwillrepresentthe timeof

descentalongACafterdescentalongRAor alongAN. Butthe timeof fallthroughXA is representedby the lengthXA,whileRA representsthe timethroughRA. But it has beenshownthat XAandAEareequal. O.F..F.

PROBLE_III, PRoPosrrIo_"XVGivena verticalline anda planeinclinedto it, it is re-quiredto finda lengthon the verticallinebelowitspointofinterseCtionwhichwillbe traversedin thesametimeasthe inclinedplane,eachof thesemotionshavingbeenpre-cededby a fallthroughthegivenverticalline.

LetABrepresenttheverticallineandBCtheinclinedplane;itisrequiredto finda lengthontheperpendicularbelowitspointofinterseCtion,whichafterafallfromAwillbetraversedin the

same

THIRD DAY 203sametimewhichisneededforBCafteranidenticalfallfromA.DrawthehorizontalAD,intersecCtingtheprolongationofCBatD; letDE be a meanproportionalbetweenCD andDB;lay[233]offBF equalto BE;alsoletAGbe a thirdproportionalto BAandAF. Then,I say,BGis thedistancewhicha body,afterfallingthroughAB,willtraversein the Asametimewhichisneededfortheplane "-'-'7BC after the same preliminaryfall. /Forif weassumethat the timeof fallalongAB is representedbyAB, then/'thetimeforDBwillbe representedby/DB. AndsinceDEis ameanpropor-tionalbetweenBDandDC, thissameDEwillrepresentthe timeof descentalongtheentiredistanceDCwhileBEwillrepresentthetimerequiredforthedifferenceof thesepaths,namely,BC,providedin eachcasethe fall is fromrestat D or at A. In likemannerwemayinferthat BF representsthe time Fig.7°ofdescentthroughthe distanceBGafterthesamepreliminaryfall;but BF isequalto BE. Hencethe problemissolved.

THEO_SMXIII, PROPOSITIONXVIIf a limitedinclinedplaneanda limitedverticallinearedrawnfromthe samepoint,andif thetimerequiredforabody,startingfromrest,to traverseeachof theseis thesame,thenabodyfallingfromanyhigheraltitudewilltrav-ersetheinclinedplanein lesstimethanisrequiredfortheverticalline.

Let EBbe the verticalllneandCE the inclinedplane,bothstartingfromthecommonpointE, andbothtraversedin equaltimesby a bodystartingfromrest at E; extendthe verticalline upwardsto any pointA, fromwhichfallingbodiesareallowedtostart. Then,I saythat,afterthefallthroughAE,theinclinedplaneEC willbe traversedin lesstime thanthe per-

pendicular

204 THE TWO NEW SCIENCESOF GALII.ROpendicularEB. JoinCB,drawthe horizontalAD,andprolongCEbackwardsuntilit meetsthe latterinD; letDF beameanproportionalbetweenCD andDE whileAGis madea meanproportionalbetweenBAandAE. DrawFG andDG; then

[234]sincethetimesofdescentalongECandEB, startingfromrestat E, areequal,it follows,accordingto CorollaryII ofProposi-

tionVI that theangleat C isaDrightangle;but the angleatA

is alsoa rightangleand theanglesat the vertex E areequal;hencethetrianglesAEDand CEBare equiangularandthesidesabouttheequalanglesare proportional;henceBE:EC= DE:EA. Consequently

G the re&angleBE.EAis equalto the re&angleCE.ED;and

c sincethe re&angleCD.DEex-ceedsthe rectangleCE.EDbythesquareofED,andsincethere&angleBA.AEexceedstherectangleBE.EAbythesquareofEA,it followsthattheexcessof the re&angleCD.DEoverthe re&angleBA_kE,or whatisthe samething,theexcessofthe squareof FD over the

Fig.71 squareof AG,willbe equaltotheexcessofthe squareofDEoverthesquareofAE,whichex-cessis equalto the squareof AD. ThereforeF-D*=G--A_-t--_)*=G--D*.HenceDF is equalto DG,and the angleDGFis equalto the angleDFGwhilethe angleEGFislessthantheangleEFG,andtheoppositesideEF islessthan the oppositesideEG. If nowweagreeto representthe timeoffallthroughAE bythelengthAE,thenthetimealongDEwillberepresentedbyDE. AndsinceAGisa meanproportionalbetweenBAand

AE,

THIRD DAY 2o5AE,it followsthat AGwillrepresentthetimeoffallthroughthetotaldistanceA.B,andthe differenceEGwillrepresentthe timeoffall,fromrestat A, throughthedifferenceofpathEB.

In likemannerEF representsthe timeofdescentalongEC,startingfromrestatDorfallingfromrestatA. ButithasbeenshownthatEFislessthanEG;hencefollowsthetheorem.

COROLLARYFromthisandthe precedingproposition,it is clearthat the

verticaldistancecoveredby a freelyfallingbody,aftera pre-liminaryfall,andduringthe time-intervalrequiredto traversean inclinedplane,is greaterthan the lengthof the inclinedplane, but less than the distancetraversedon the inclinedplaneduringan equaltime,withoutanypreliminaryfall. Forsincewehavejust shownthat bodiesfallingfroman elevatedpointA willtraversethe planeECin Fig.7I in a shortertimethanthe verticalEB, it is evidentthat the distancealongEBwhichwillbe traversedduringa timeequalto that ofdescentalongECwillbe lessthanthewholeofEB. Butnowin orderto showthat thisverticaldistanceisgreaterthanthelengthofthe inclinedplaneEC,wereproduce A I_Fig.70 of the precedingtheoremin ' /whichtheverticallengthBGistray-

./ersedin the sametimeas BCafterapreliminaryfall throughAB. That /BG is greaterthan BC is shownas /"follows:sinceBE and FB are equal _ _ F

[23s]whileBA is lessthan BD, it follows"_"thatFBwillbeartoBAagreaterratiothan EB bearsto BD; and,compon-endo,FA willbear to BA a greaterratiothan El) to DB; but FA'.AB-- @GF:FB (sinceAF is a meanpropor- Fig.72tionalbetweenBAandAG)andin likcmannerED:BE)=CE:EB. HenceGBbearsto BF a greaterratiothanCBbearstoBE;thereforeGB isgreaterthanBC.

_6 THE TWO NEW SCIENCESOF GALILEO

PROBLEMIV, PROPOSITIONXVII

Givena verticallineandan inclinedplane,it is requiredto layoffa distancealongthegivenplanewhichwillbetrav-ersedby a body,afterfallalongthe perpendicular,in thesametime-intervalwhichis neededfor this bodyto fallfromrestthroughthegivenperpendicular.

LetABbe theverticallineandBE the inclinedplane. Theproblemis to determineon BE a distancesuchthat a body,

__ DafterfallingthroughAB,willtraverseit

B_// inatimeequaltothat requiredto traversetheperpendicularABitself,startingfrom

. rest.

l_/ DrawthehorizontalADandextendtheplaneuntil it meetsthis linein D. Lay

_/ offFB equalto BA;andchoosethepoint

E suchthat BD:FD=DF'.DE. Then,Isay,the timeof descentalongBE,afterfallthroughAB,isequaltothetimeoffall,

Fig.73 fromrestat A, throughAB. For, ifweassumethat the lengthABrepresentsthe timeof fallthroughAB,thenthetimeoffallthroughDBwillberepresentedby thetimeDB;andsinceBD:FD=DF:DE,it followsthat DF willrepresentthe timeof descentalongthe entireplaneDEwhile

* BF representsthe timethroughthe portionBE startingfromrestat D; but the timeof descentalongBEafterthe prelimi-nary descentalongDB is the sameasthat after apreliminaryfallthroughAB. HencethetimeofdescentalongBEafterABwillbeBF whichof courseisequalto the timeof fallthroughABfromrestatA/ O.E.F.

[zs6]PROBLEMV, PROPOSITIONX-VIII

Giventhedistancethroughwhichabodywill{allverticallyfromrestduringagiventlme-lnterval,andg_vena|soasmallertime-interval,itisrequiredtolocateanother[equal]

vertical

THIRD DAY 2o7verticaldistancewhichthe bodywilltraverseduringthisgivensmallertime-interval.

Let the verticallinebe drawnthroughA,andon this linelayoffthe distanceABwhichistraversedby a bodyfallingfromrestat A,duringa timewhichmay alsobe Arepresentedby AB. Draw the horizontal ./IxlineCBE,andonit layoffBCto represent _ [\the givenintervalof timewhichis shorter_ (" BI \ CthanAB. It is requiredto locate,in the \perpendicularabovementioned,a distancewhichis equalto ABandwhichwillbede-scribedinatimeequaltoBC. JointhepointsAandC;then,sinceBC<BA,it followsthat xthe angleBAC<angleBCA. ConstrucCtthe \_angleCAEequalto BCAandlet E be the xxpointwhereAEintersectsthehorizontalline; _x_drawEl) at rightanglesto AE,cuttingtheverticalatD; layoffDFequaltoBA. Then,I say,thatFD is that portionofthevertical Fig.74whicha bodystartingfromrestatAwilltraverseduringtheas-signedtime-lntervalBC. For, if in the right-angledtriangleAEDaperpendicularbedrawnfromthe right-angleatE to theoppositesideAD,thenAEwillbeameanproportionalbetweenDAandABwhileBEwillbe a meanproportionalbetweenBDand BA,or betweenFA andAB (seeingthat FA isequaltoDB);andsinceit hasbeenagreedto representthetimeof fallthroughABbythe distanceAB,itfollowsthat AE,orEC,willrepresentthetimeoffallthroughtheentiredistanceAD,whileEBwillrepresentthe timethroughAF. Consequentlythe re-mainderBCwillrepresentthe timeof fallthroughtheremain-ingdistanceFD. Q.E.F.

[_371PROm,EMVI,PRoPosrrro_XIX

Giventhedistancethroughwhicha bodyfallsina verticalllnefromrestandgivenalsothe timeoffall,it isrequiredto findthetimeinwhichthe samebodywill,later,traverse

an

zo8 THE TWO NEW SCIENCESOF GALIT._Oan equaldistancechosenanywherein the sameverticalline.

OntheverticallineAB,layoffACequaltothedistancefallenfromrest at A, alsolocateat randoman equaldistanceDB.

._ Let the timeof fallthroughACberepresentedbythe lengthAC.It is

requiredto findthe timenecessary

.Cto traverseDB afterfallfromrest• at A. Aboutthe entirelengthAB

describethe semicircleAEB;fromA C drawCE perpendicularto AB;

_ join the pointsA and E; the line

AEwillbe longerthanEC; layoffEFequalto EC. Then,I say,thedifferenceFAwillrepresentthetimerequiredfor fallthroughDB. For

BsinceAEis ameanproportionalbe-tweenBAandACandsinceACrep-resentsthetimeoffallthroughAC,

C_ it followsthat AE willrepresentthetimethroughtheentiredistanceAB. AndsinceCE is a meanpro-portionalbetweenDAandAC (see-

1_ ing that DA=BC) it followsthatFig.75 CE, that is,EF, willrepresentthe

timeoffallthroughAD. Hencethe difference.A_Fwillrepresentthe timeoffallthroughthe differenceDB. Q._. D.

COROLIARYHenceit is inferredthat if the timeof fallfromrestthrough

anygivendistanceis representedby that distanceitself,thenthetimeoffall,afterthegivendistancehasbeenincreasedby acertainamount,willbe representedby the excessof the meanproportionalbetweenthe increaseddistanceand the originaldistanceoverthe meanproportionalbetweenthe originaldis-tanceandthe increment.Thus,for instance,ifweagreethat

AB

THIRD DAY 2o9ABrepresentsthetimeof fall,fromrestat A, throughthe dis-tanceAB,andthat ASis the increment,thetimerequiredtotraverse_AB,afterfallthroughSA,willbetheexcessofthemeanproportionalbetweenSBandBAoverthe meanproportionalbetweenBAandAS.

[z:38] "<PROBLEMVII,PROPOSITION

Givenanydistancewhateverandaportionofitlaidofffromthe pointat whichmotionbegins,it isre-quiredto findanotherportionwhichliesat theotherendofthedistanceandwhichistraversedinthesametimeasthefirstgivenportion. Fig. 76

Let the givendistancebe CBandletCDbe that part ofitc whichis laid off fromthe beginningof motion.It is

requiredto findanotherpart, at the end B, whichistraversedin the same time as the assignedportion

D CD. Let BA be a mean proportionalbetweenBCand CD; also let CE be a third proportionalto BCandCA. Then,I say,EB willbe the distancewhich,after fall fromC, willbe traversedin the sametimeas CD itself. For if we agree that CB shallrepre-sentthe timethroughthe entiredistanceCB,thenBA

._ (which,of course,is a meanproportionalbetweenBCand CD)willrepresentthe time alongCD; and sinceCAis a meanproportionalbetweenBCandCE,it fol-lowsthat CA willbe the time throughCE; but thetotal lengthCBrepresentsthe time throughthe totaldistanceCB. Thereforethe differenceBAwillbe the

Fig.77timealongthe differenceof distances,EB,afterfallingfromC; but this sameBAwasthe timeof fallthroughCD.ConsequentlythedistancesCDandEBaretraversed,fromrestat A,in equaltimes, q._..F.

THEO_XIV,PRoPosmo_XXIIf, onthe pathof a bodyfallingverticallyfromrest,onelaysoffaportionwhichis traversedinanytimeyoupleaseand

.zto THE TWO NEW.SCIFNCESOF GALTT,_Oandwhoseupperterminuscoincideswiththe pointwherethe motionbegins,andif thisfallis followedby a motiondeflegtedalonganyinclinedplane,thenthespacetraversedalongtheinclinedplane,duringatime-intervalequaltothatoccupiedin the previousverticalfall,willbe greaterthantwice,andlessthanthreetimes,the lengthofthe verticalfall.

LetABbe averticallinedrawndownwardsfromthehorizon-tal lineAE,andletit representthepathofa bodyfallingfromrestat A; chooseany portionACof this path. ThroughCdrawany inclinedplane,CG,alongwhichthe motionis con-tinuedafterfallthroughAC. Then,I say,that the distance

[539]traversedalongthisplaneCG,duringthe time-intervalequalto that of the fall throughAC, is morethantwice,but less

A B thanthreetimes,thissame

c_///_ distanceAC. Letuslayoff

A _ CF equal to AC,and ex-tendthe planeGC untilitmeetsthe horizontalinE;

chooseG such that CE:

EF=EF:EG. If nowwep assumethat the time of

fallalongACisrepresentedG bythelengthAC,thenCE

willrepresentthe time of!_ descentalong CE, while

CF, or CA,willrepresent•Fig.78 the timeof descentalong

CG. It nowremainsto be shownthatthedistanceCGismorethan twice,and lessthan threetimes,the distanceCAitself.SinceCE:EF=EF_G, it followsthat CE:EF=CF:FG;butEC<EF; thereforeCF willbe lessthan FG andGC willbemorethantwiceFC,orAC. AgainsinceFE<_EC(forEC isgreaterthanCA,or CF),wehaveGFlessthantwiceFC, andalsoGClessthanthreetimesCF,or CA. Q.E.D.

Thispropositionmaybe statedina moregeneralform;sincewhat

THIRD DAY 211whathasbeenprovenforthecaseofaverticalandinclinedplaneholdsequallywellinthecaseofmotionalongaplaneofanyinclinationfollowedby motionalonganyplaneoflesssteepness,ascanbeseenfromtheadjoiningfigure.Themethodofproofisthesame.

[_o]PRo_._VIII,PRoPosmo_XXII

Giventwounequaltime-intervals,alsothedistancethroughwhicha bodywillfallalonga verticalline,fromrest, duringthe shorterofthese intervals,it is requiredtopassthroughthe highest point of this vertical line a plane so inclinedthat the timeof descentalongit willbe equalto the longerof the givenintervals.

Let A representthe longerand B the shorterof the twoun-equal time-intervals,also let CD represent the length of the

J 'ISFig.79

verticalfall,fromrest,duringthe time]3. It is requiredto passthroughthe point C a planeof such a slopethat it willbe trav-ersedin the timeA.

Draw _romthepoint C to thehorizontala lineCX of suchalength that B:A =CD:CX. It is clear that CX is the planealongwhich a body will descendin the giventimeA. For ithas beenshownthat the timeof descentalongan inclinedplanebears to the time of fall through its vertical height the sameratio whichthe lengthof the plane bearsto its verticalheight.Thereforethe time alongCX is to the time alongCD as thelengthCX is to the lengthCD, that is, as the time-intervalA is

tO

212 THE TWONEW SCIENCESOF GALII.E0to thetime-intervalB:butB isthetimerequiredto traversetheverticaldistance,CD,startingfromrest;thereforeA isthe timerequiredfordescentalongtheplaneCX.

PROBT_MIX, PROPOSmONXXlIIGiventhetimeemployedbyabodyinfallingthroughacer-tain distancealonga verticalline, it is requiredto passthroughthe lowerterminusofthisverticalfall,a planesoinclinedthat this bodywill,afterits verticalfall,traverseon thisplane,duringa time-intervalequalto thatof theverticalfall,a distanceequalto anyassigneddistance,pro-

[2411videdthisassigneddistanceismorethantwiceandlessthanthreetimes,theverticalfall.

Let AS be any verticalline,and let AC denoteboth thelengthof the verticalfall,fromrestat A, and alsothe time

_ r _ _ _. requiredfor this fall. LetIR bea distancemorethan

'-- IA _._'l_,twice and less than three[_ times,AC. It is required

_ to passa planethroughthe_' [ pointC so inclinedthat a

- " [ body,afterfallthroughAC,will,duringthe time AC,traversea distanceequalto

,1,._ IR. Lay offRN andNMFig.8o eachequaltoAC. Through

thepointC,drawaplaneCEmeetingthehorizontal,A.E,atsucha pointthat IM:MN--AC:CE.Extendthe planeto O,andlayoffCF,FG andGOequalto RN,NM,andMI respectively.Then,I say, the timealongthe inclinedplaneCO,afterfallthroughAC,isequalto thetimeoffall,fromrestat A,throughAC. For since OG:GF=FC:CE,it follows,componendo,that OF:FG_-OF:FC=FE:EC,and sincean antecedentisto its consequentasthesumof theantecedentsisto the sumofthe consequents,we haveOE'.EF=EF:EC. Thus EF is ameanproportionalbetweenOE and EC. Havingagreedto

represent

THIRD DAY 213representthetimeoffallthroughACbythelengthACit followsthat ECwillrepresentthetimealongEC,andEFthetimealongtheentiredistanceEO,whilethedifferenceCFwillrepresentthetime alongthe differenceCO; but CF=CA; thereforetheproblemissolved.ForthetimeCAisthetimeoffall,fromrestat A, throughCAwhileCF (whichis equalto CA)is thetimerequiredto traverseCOafter descentalongEC or afterfallthroughAC. Q.E.F.

It is to be remarkedalsothat the samesolutionholdsif theantecedentmotiontakesplace,notalongavertical,butalonganinclinedplane. This caseis illustratedin the followingfigurewherethe antecedentmotionis alongthe inclinedplaneAS

[242]underneaththe horizontalAE. Theproofisidenticalwiththepreceding.

SCHOLIUM

Oncarefulattention,it willbeclearthat,thenearerthegivenlineIR approachesto threetimesthe lengthAC,thenearerthe

I M l_ l_.I ' '| I i , I

/aFig.8I

inclinedplane, CO, alongwhichthe secondmotiontakesplace,approachesthe perpendicularalongwhichthe spacetraversed,duringthe timeAC,willbe threetimesthe distanceAC. Fori/IR be takennearlyequalto threetimesAC,thenIM willbe almostequalto MN;and since,by constrnd_don,IM:

2i4 THE TWONEW SCIENCESOF GALII.EOIM-.MN=AC:CE,it followsthat CE is but littlegreaterthanCA:consequentlythepointE willlienearthepointA,andthelinesCOandCS,forminga veryacuteangle,willalmostcoin-cide. But,ontheotherhand,if thegivenline,IR,beonlytheleastbit longerthantwiceAC,the lineIM willbeveryshort;fromwhichit followsthat ACwillbeverysmallin comparisonwithCEwhichis nowsolongthat it almostcoincideswiththehorizontallinedrawnthroughC. Hencewecaninferthat, if,afterdescentalongtheinclinedplaneACoftheadjoiningfigure,themotioniscontinuedalongahorizontalline,suchasCT,thedistancetraversedby a body,duringa timeequalto the timeoffallthroughAC,willbeexactlytwicethe distanceAC. Theargumenthereemployedisthe sameasthepreceding.Forit isclear,sinceOE:EF=EF'.EC,that FC measuresthe time ofdescentalongCO. But, ifthehorizontallineTCwhichistwiceas longas CA,be dividedintotwoequalpartsat V thenthislinemustbe extendedindefinitelyin the directionofX beforeit will intersectthe line AE produced;and accordinglytheratioofthe infinitelengthTX to theinfinitelengthVXisthesameas the ratioof the infinitedistanceVX to the infinitedistanceCX.

The sameresultmaybe obtainedby anothermethodof ap-proach,namely,byreturningtothesamelineofargumentwhichwasemployedin the proofof the first proposition.Let us

[z431considerthe triangleABC,which,by linesdrawnparallelto itsbase,representsforusa velocityincreasinginproportionto thetime;if theselinesareinfinitein number,justas the pointsinthe lineAC are infiniteor as the numberof instantsin anyintervaloftimeisinfinite,theywillformtheareaofthetriangle.Letusnowsupposethat themaximumvelocityattained--thatrepresentedby thelineBC--tobe continued,withoutaccelera-tionandatconstantvaluethroughanotherintervaloftimeequalto thefirst. Fromthesevelocitieswillbe builtup, in a similarmanner,the area of the parallelogramADBC,whichis twicethat of the triangleABC;accordinglythe distancetraversedwiththesevelocitiesduringanygivenintervaloftimewillbe

twice

THIRD DAY zI5twice that traversedwith the velocitiesrepresentedby thetriangleduringan equalintervalof time. But alonga horizontalplane the motion is uniformsince here it experiencesneitheraccelerationnor retardation; thereforewe con-l). Acludethat thedistanceCDtraversedduringatime-_ ;interval equal to AC is twice the distanceAC; _ .-for the latter is coveredby a motion,starting : :from rest and increasingin speed in proportion :to the parallel lines in the triangle, whilethe -:former is traversed by a motion representedby L:the parallel lines of the parallelogramwhich,beingalsoinfinitein number,yieldan area twice_ _that of the triangle. Fig.82

Furthermorewemay remarkthat anyvelocityonceimpartedto a moving body will be rigidlymaintained as longas theexternal causesof accelerationor retardation are removed,aconditionwhich is foundonly on horizontalplanes;for in thecaseof planeswhichslopedownwardsthere is alreadypresentacauseof acceleration,whileon planes slopingupward there isretardation;from this it followsthat motionalonga horizontalplane is perpetual; for, if the velocitybe uniform,it cannotbediminishedor slackened,much less destroyed. Further, al-thoughany velocitywhicha bodymay haveacquiredthroughnatural fall is permanentlymaintainedsofar as its ownnature[suaptenaturalis concerned,yet it must be rememberedthat if,after descent along a plane inclineddownwards,the body isdeflectedto a plane inclinedupward,thereis alreadyexistinginthis latter plane a causeof retardation;for in any such planethis samebody is subjecCtto a natural accelerationdownwards.Accordinglywe have here the superpositionof two differentstates, namely,the velocityacquiredduringthe precedingfallwhichif actingalonewouldcarry the bodyat a uniformrate toinfinity,and the velocitywhichresultsfroma natural accelera-tion downwardscommonto all bodies. It seemsaltogetherreasonable,therefore,ifwewish to tracethe futurehistoryofabody whichhas descendedalongsomeinclinedplane and hasbeen deflected along someplane inclinedupwards,for us to

a3$11ine

2x6 THE TWO NEW SCIENCESOF GALII.EOassumethat the maximumspeedacquiredduringdescentispermanentlymaintainedduringthe ascent. In the ascent,however,there supervenesa naturalinclinationdownwards,namely,amotionwhich,startingfromrest,is acceleratedat the

[z44]usualrate. If perhapsthis discussionis a littleobscure,thefollowingfigurewillhelpto makeitclearer.

Let us supposethat the descenthasbeenmadealongthcdownwardslopingplaneAB,fromwhichthe bodyisdefle_edsoas to continueits motionalongthe upwardslopingplaneBC;andfirstlet theseplanesbeofequallengthandplacedsoasto makeequalangleswith the horizontallineGH. Nowit iswellknownthat abody,startingfromrestatA,anddescendingalongAB,acquiresa speedwhichis proportionalto the tlme,

¢_ _, A. whichisa maximum.... at B, andwhichis........ "" maintainedby the

body solongas allcausesof fresh ac-celerationorretarda-

I4. tion are removed;Fig.83 the accelerationto

whichI refer is that to whichthe bodywouldbe subje_if its motionwere continuedalongthe planeAB extended,whilethe retardationis that whichthe bodywouldencounterif its motionweredeflectedalongthe planeBCinclinedup-wards;but, upon the horizontalplaneGH, the bodywouldmaintaina uniformvelocityequalto that whichit hadac-quiredat B afterfall fromA;moreoverthisvelocityis suchthat, duringan intervalof time equalto the time of descentthroughAB,the bodywilltraversea horizontaldistanceequalto twiceAB. Nowletus imaginethissamebodyto movewiththe sameuniformspeedalongthe planeBCso that herealsoduringa time-intervalequalto that ofdescentalongAB,it willtraversealongBCextendeda distancetwiceAB; but let ussupposethat, at theveryinstant_.]_ebodybeginsits ascentit issubje6ted,by its very--nature_m the same influenceswhich

surrounded

THIRD DAY 217surroundedit duringits descentfromA along_A_B,namely,itdescendsfromrestunderthesameaccelerationasthatwhichwaseffec°dvein AB, andit traverses,duringan equalintervaloftime,the samedistancealongthissecondplaneas it didalongAB;it isclearthat,bythussuperposinguponthebodyauniformmotionofascentandanacceleratedmotionofdescent,itwillbecarriedalongtheplaneBCasfarasthepointCwherethesetwovelocitiesbecomeequal.

If nowweassumeanytwopointsD andE, equallydistant{tomthe vertex]3,wemaythen inferthat the descentalongBDtakesplacein thesametimeastheascentalongBE. DrawDF parallelto BC;weknowthat, afterdescentalongAD,thebodywillascendalongDF; or, if,on reachingD, the bodyiscarriedalongthe horizontalDE, it willreachE withthe samemomentum[impetus]withwhichit leftD; hencefromE thebodywillascendasfar asC,provingthat the velocityat E isthesameasthat atD.

Fromthiswemaylogicallyinferthat abodywhichdescends[ 4s]

alonganyinclinedplaneandcontinuesits motionalonga planeinclinedupwardswill,on accountof the momentumacquired,ascendto an equalheightabovethe horizontal;sothat if thedescent is along D ¢_ A EABthe bodywill -x \ / /be carriedup the _ _ / /planeBCasfarasthehorizontallineACD:andthis istrue whetherthe Binclinationsofthe Fig.84planesare the sameor different,as in the caseof the planesAB andBD. But by a previouspostulate[p.184]the speedsacquiredby fall alongvariouslyinclinedplaneshavingthesameverticalheightare the same. If thereforethe planesEB and BD havethe sameslope,the descentalongEBwillbe ableto drive the bodyalongBDas far as D; and sincethis propulsioncomesfromthe speedacquiredon reaching

the

218 THE TWO NEW SCIENCESOF GALILI_Othe pointB,it followsthat thisspeedat B isthesamewhetherthebodyhasmadeitsdescentalongABorEB. Evidentlythenthe bodywillbe carriedup BDwhetherthe descenthasbeenmadealongABor alongEB. Thetimeof ascentalongBD ishowevergreaterthan that alongBC,justas the descentalongEB occupiesmoretime than that alongAB;moreoverit hasbeendemonstratedthat the ratiobetweenthelengthsofthesetimesisthesameasthat betweenthelengthsoftheplanes.Wemust next discoverwhat ratio existsbetweenthe distancestraversedin equaltimesalongplanesofdh°/erentslope,butofthe sameelevation,that is, alongplaneswhichare includedbetweenthe sameparallelhorizontallines. This is doneasfollows:

THEOREMXV, PROPOSITIONGiventwoparallelhorizontalplanesandaverticallinecon-nectingthem;givenalsoan inclinedplanepassingthroughthelowerextremityofthisverticalline;then,if abodyfallfreelyalongthe verticallineandhaveits motionreflectedalongthe inclinedplane,the distancewhichit willtraversealongthisplane,duringa timeequalto that oftheverti-calfall,isgreaterthanoncebut lessthantwicetheverticalline.

LetBCandHGbe the twohorizontalplanes,connecCtedbytheperpendicular.A.E;alsoletEB representthe inclinedplane

B A Ci , |1¢

H 1_ GFig.85

alongwhichthe motiontakesplaceafterthe bodyhas fallenalongAE andhasbeenreflecCtedfromE towardsB. Then,Isay,that,duringatimeequalto that offallalongAE,thebodywill ascendthe inclinedplanethrougha distancewhichis

greater

THIRD DAY 219greaterthanAEbut lessthantwiceAE. LayoffEl) equaltoAE and chooseF so that EB:BD=BD-S3F.Firstwe shall

[245]showthat F is the point to whichthe movingbodywillbecarriedafterrefle_ionfromE towardsB duringa timeequaltothat offallalongAE;andnextweshallshowthat thedistanceEFisgreaterthanEAbutlessthantwicethatquantity.

Letus agreeto representthe timeof fallalongAEby thelengthAE,thenthe timeofdescentalongBE,orwhatis thesamething,ascentalongEBwillberepresentedbythedistanceEB.

Now,sinceDB is ameanproportionalbetweenEB andBF,andsinceBEis thetimeofdescentfortheentiredistanceBE,it followsthat BDwillbethetimeofdescentthroughBF,whiletheremainderDEwillbethetimeofdescentalongtheremainderFE. But thetimeofdescentalongthefallfromrestat Bisthesameasthe timeofascentfromE to F afterreflectionfromEwiththe speedacquiredduringfalleitherthroughAEorBE.ThereforeDE representsthe timeoccupiedby the body inpassingfromE to F, afterfallfromAto E andafterrefie_ionalongEB. But by construgfionED is equalto AE. Thisconcludesthefirstpartofourdemonstration.

Nowsincethe wholeof EB is to the wholeof BD as theportionDB is to the portionBF,wehavethe wholeofEBisto the wholeof BD asthe remainderEl) is to the remainderDF; but EB>BD andhenceED>DF,and EF is lessthantwiceDE or AE. Q.E.D.

Thesameis truewhenthe initialmotionoccurs,notalongaperpendicular,but uponan inclinedplane:the proofisalsothesameprovidedtheupwardslopingplaneislesssteep,i.e.,longer,thanthedownwardslopingplane.

THEOREMXVI,P_OPOSmONXXVIf descentalonganyinclinedplaneis followedby motionalongahorizontalplane,the timeofdescentalongthe in-dinedplanebearsto the timerequiredto traverseanyas-signedlengthofthe horizontalplanethe sameratiowhich

twice

zzo THE TWO NEW SCIENCESOF GALILEOtwicethe lengthof the inclinedplanebearsto the givenhorizontallength.

Let CBbeanyhorizontallineandABan inclinedplane;afterdescentalongABlet themotioncontinuethroughthe assigned

A horizontaldistanceBD. Then,

I say,thetimeof descentalongABbearsto the _imespentin

' traversingBD the same ratioc, D, z . B whichtwiceAB bearsto BD.

Fig.86 For, layoff;BCequalto twiceABthenit follows,fromapreviousproposition,that thetimeofdescentalongABisequalto thetimerequiredto traverseBC;but the timealongBC is to the timealongDB as the lengthCBis to the lengthBD. Hencethe timeofdescentalongAB

[z47]isto the timealongBD astwicethe distanceABisto the dis-tanceBD. q. _. D.

PROBLEMX, PROPOSITIONXX-VI

Givena verticalheightjoiningtwohorizontalparallellines;givenalsoa distancegreaterthanonceandlessthantwicethisverticalheight,it is requiredto passthroughthefootofthegivenperpendicularaninclinedplanesuchthat,afterfall throughthe givenverticalheight,a bodywhosemo-tionis defle6tedalongthe planewilltraversetheassigneddistanceina timeequalto thetimeofverticalfall.

Let AB be the verticaldistanceseparatingtwo parallelhorizontallinesAOandBC;alsoletFEbegreaterthanonceandlessthantwiceBA. Theproblemisto passa planethroughB,extendingto the upperhorizontalline,andsuchthat a body,afterhavingfallenfromA to B, will,if its motionbedeflecCtedalongthe inclinedplane,traversea distanceequalto EF in atimeequalto that offallalongAB. Lay off;EDequalto AB;then the remainderDF willbe lessthanAB sincethe entirelengthEF islessthantwicethisquantity;alsolayoff;DI equalto DF, and choosethe point X such that EI:ID=DF:FX;fromB,drawtheplaneBOequalin lengthto EX. Then,I say,

that

THIRD DAY 221thatthe planeBOis theonealongwhich,afterfallthroughAB,a bodywilltraversetheassigneddistanceFE ina timeequaltothe timeof fallthroughAB. LayoffBRandRSequalto EDand DF respectively;then sinceEI:ID=DF:FX,we have,componendo,El) =DX-Y_=ED-J3F=EX.ZZD=BO:OR=

o A

Fig.87RO:OS.If we representthe timeof fall alongAB by thelengthAB,thenOBwillrepresentthe timeof descentalong

[248]OB,andRO willstandfor the timealongOS,whilethe re-mainderBRwillrepresentthetimerequiredforabodystartingfromrestatO to traversethe remainingdistanceSB. But thetimeofdescentalongSBstartingfromrestat O is equalto thetimeofascentfromB to S afterfallthroughAB. HenceBOisthat plane,passingthroughB, alongwhicha body,afterfallthroughAB,willtraversethedistanceBS,equalto theassigneddistanceEF, in thetime-intervalBRorB_A. Q.E.F.

XVlI,P oPosiTioXXVlIIf a bodydescendsalongtwoinclinedplanesofdifferentlengthsbut ofthesameverticalheight,thedistancewhichit willtraverse,inthelowerpartofthelongerplane,duringa time-intervalequalto that of descentoverthe shorterplane,is equalto the lengthof the shorterplaneplusaportionof it to whichthe shorterplanebearsthe sameratiowhichthe longerplanebearsto the excessof thelongerovertheshorterplane.

LetAC be the longerplane,AB,the shorter,andAD thecommonelevation;on the lowerpartofAClay offCE equal

to

222 THE TWO NEW SCIENCESOF GALII,EOto AB. ChooseF such that CA'akE=CA:CA-AB=CE'.EF.Then,I say,that FCisthat distancewhichwill,afterfallfromA,betraversedduringatime-intervalequalto that requiredfor

A descentalongAB. For since

¢_*_ CA'.AE=CE'.EF,it follows

that the remainderEA: theremainder AF = CA:AE.ThereforeAE is a meanpro-porfionalbetweenAC andAF. Accordinglyifthelength

DABis employedto measureFig.88 thetimeoffallalongAB,then

the distanceACwillmeasurethetimeofdescentthroughAC;but the timeof descentthroughAFismeasuredby the lengthAE, andthat throughFC by EC. NowEC=AB;andhencefollowstheproposition.

[249]PROBLEMXI, PROPOSITIONXXVIII

LetAGbeanyhorizontallinetouchinga circle;letABbethediameterpassingthroughthe pointof contagt;andletAEandEB representany twochords. The problemis to determinewhatratiothetime offallthrough A GAB bears to the time of descent ,._-.,. ' /overbothAEandEB. ExtendBEtill it meetsthe tangentat G,anddrawAF soas to bisegtthe angle :_'\".../BAE.Then,I say,thetimethrough 3ABisto thesumof the timesalongAEandEB asthe lengthAE istothe sumofthe lengthsAEandEF.For sincethe angleFABis equalto the angleFAE,whilethe angle .BEAGis equalto the angleABF it Fig.89followsthattheentireangleGAFisequaltothesumoftheanglesFABandABF. ButtheangleGFAisalsoequalto thesumofthesetwoangles.Hencethe lengthGF is equalto the length

GA

THIRD DAY 2z3GA;andsincethe re6tangleBG.GEis equalto the squareofGA, it willalsobe equalto the squareofGF, or BG:GF=GF._E. If nowwe agreeto representthe time of descentalpngAEby the lengthAE,thenthelengthGEwillrepresentthe timeofdescentalongGE,whileGFwillstandforthe timeof descentthroughthe entiredistanceGB; so alsoEF willdenotethe timethroughEB afterfallfromGor fromAalongAE. Consequentlythe timealongAE,orAB,is to the timealongAEandEBasthelengthAEistoAE+EF. Q.E.D.

A shortermethodis to layoffGFequaltoGA,thusmakingGF a meanproportionalbetweenBG andGE. The restoftheproofisasabove.

THEOm_MXVIII, PROPOSmONXXIXGivena limitedhorizontalline,at oneend ofwhichiserecteda limitedverticallinewhoselengthis equalto one-haKthe givenhorizontalline;thena body,fallingthroughthis givenheightandhavingits motiondeflecCtedintoahorizontaldirecCtion,willtraversethe givenhorizontaldis-tanceandverticallinein less time than it owillanyotherverti- -Acaldistanceplusthe Ngivenhorizontaldis- _/

[25ol L , oLetBCbethegivendis-_ c

tance in a horizontalplane;at the endB erecCta perpendicular,onwhich 'c. ....'D..lay offBA equalto half Fig. 9°BC. Then,I say,that the timerequiredfora body,startingfromrestat A, to traversethe twodistances,AB andBC,isthe leastof allpossibletimesin whichthissamedistanceBCtogetherwith a verticalportion,whethergreateror lessthanAB,canbetraversed.

LayoffEB greaterthanAB,as in the firstfigure,andlessthan

2z4 THE TWO NEW SCIENCESOF GALILEOthan.A_B,as in the second.It must be shownthat the timerequiredto traversethe distanceEB plusBCis greaterthanthat requiredforABplusBC. Letus agreethat the lengthABshallrepresentthe timealongAB,thenthetimeoccupiedintraversingthe horizontalportionBCwillalsobe AB, seeingthat BC=2AB;consequentlythe timerequiredforbothABandBCwillbe twiceAB. Choosethe point0 suchthat EB:B0=BO:BA,thenB0 willrepresentthe time of fall throughEB. Againlay offthe horizontaldistanceBD equalto twiceBE; whenceit is clearthat BOrepresentsthe time alongBDafterfallthroughEB. SeleCta pointN suchthat DB:BC--EB:BA=OB_N. Now sincethe horizontalmotionis uni-formandsince0B is the timeoccupiedin traversingBD,afterfallfromE, it followsthat NBwillbe the timealongBCafterfallthroughthe sameheightEB. Henceit isclearthat 0B plusBN representsthe timeof traversingEB plusBC;and, sincetwiceBAisthe timealongABplusBCoit remainsto be shownthat OB+BN>2BA.

But sinceEB'.BO=B0_A,it followsthat EB_A--0--Bh_--A_.MoreoversinceEB_A =OB:BNit followsthatOB:BN=0-B_:B--A_. But 0B:BN--(OB:BA)(BA:BN),and thereforeAB:BN=OB:BA,that is,BAis a meanproportionalbetweenB0 and BiN. Consequently0B+BN>2BA. q. _. D.

[2s ]THEOIU_MX]TK,PRoPosmoNXXX

A perpendicularis let fallfromanypointin a horizontalline;it is requiredto passthroughanyotherpointin thissamehorizontallineaplanewhichshallcuttheperpendicu-larandalongwhichabodywilldescendtotheperpendicularintheshortestpossibletime. Suchaplanewillcutfromtheperpendiculara portionequalto the distanceof the as-sumedpoint in the horizontalfromthe upperendof theperpendicular.

LetACbeanyhorizontallineandBanypointinitfromwhichis droppedthe verticallineBE). Chooseanypoint C in thehorizontalline and l_iyoff,on the Vertical,the distanceBE

equal

THIRD DAY 225equalto BC; joinC andE. Then,I say,that of all inclinedplanesthat canbepassedthroughC,cuttingtheperpendicular,CE isthat onealongwhichthe descentto the perpendicularisaccomplishedin the shortesttime. For,drawthe planeCFcuttingtheverticalaboveE, andtheplaneCGcuttingtheverticalbelowE; anddrawIK,a parallelverticalline,touchingat C acir- xcledescribedwithBCas radius.LetEKbedrawnparallelto CF,_ c_andextendedto meetthe tan-gent,aftercuttingthecircleat L. r,Nowit is clearthat the timeoffallalongLEisequalto thetimealongCE; but the time alongKE is greaterthan alongLE;thereforethe time alongKE isgreaterthanalongCE. Butthe G[time alongKE is equalto the .ItimealongCF, sincetheyhavethe samelengthand the sameslope;and,in likemanner,itfol-lowsthat the planesCGandIE,havingthe samelengthandthe Fig.9Isameslope,willbe traversedinequaltimes. Also,sinceHE<IE, the time alongHE willbe lessthan the time alongIE.ThereforealsothetimealongCE(equalto the timealongHE),willbeshorterthanthetimealongIE. Q._.D.

THFoUXX,PgoPosmoNXXXIIf a straightlineisinclinedat anyangleto thehorizontalandif,fromanyassignedpointinthehorizontal,aplaneofquickestdescentis to be drawnto the inclinedline,thatplanewillbe the onewhichbisecCtsthe anglecontained

[ 52]betweentwo linesdrawnfromthe givenpoint,oneper-

pendicular

zz6 THE TWO NEW SCIENCESOF GALILEOpendicularto the horizontalline,the otherperpendicularto theinclinedline.

Let CDbe a lineinclinedat anyangleto the horizontalAB;andfromanyassignedpointA in the horizontaldrawACper-pendicularto A_B,andAE perpendicularto CD; drawFA soasto biseCtthe angleCAE. Then,I say,that ofallthe planeswhichcanbe drawnthroughthe pointA, cuttingthe lineCD

at any pointswhatsoeverAF is the oneof quickestdescent[in quo temporeomniumbrevissimofiatde-

_ scensus].Draw FG par-allelto AE; the alternate

_ anglesGFAandFAEwill_.be equal;also the angleEAFisequalto the angleFAG. Thereforethe sidesGFandGAof thetriangle

/ FGAare equal. Accord-inglyifwedescribea circle

A li_ aboutGascenter,withGAas radius,this circlewillFig.9_ passthroughthe pointF,

andwilltouchthe horizontalat the pointA andthe inclinedlineat F; forGFCisa rightangle,sinceGFandAEareparallel.It isclearthereforethat all linesdrawnfromA to the inclinedline,with the singleexceptionof FA,willextendbeyondthecircumferenceof thecircle,thusrequiringmoretimetotraverseanyofthemthanisneededforFA. q.E.D.

LEMMAIf two circlesonelyingwithinthe otherare in contaCt,and if any straightline be drawntangentto the innercircle,cuttingthe outercircle,andif threelinesbe drawnfromthe pointat whichthe circlesarein contaCtto threepointsonthe tangentialstraightline,namely,thepointoftangencyon the innercircleandthe twopointswherethe

straight

THIRD DAY 227straightlineextendedcutsthe outercircle,then these threelineswillcontainequalanglesat the pointof contacCt.

Let the two circlestoucheachother at the point A, the centerof the smallerbeingat B, the centerof the largerat C. Draw

[2s3lthe straight lineFG touchingthe innercircleat H, and cuttingthe outer at the points F and G; alsodrawthe threelinesAF,All, and AG. Then, I say, the anglescontainedby theselines,FAIl and GAll,are equal. Pro- XlongAH to the circumferenceatI; from the centersof the circles,drawBHand CI; jointhe centers /B and C and extendthe lineuntil _ {_it reachesthe point of contacCtat r-N<,A and cuts the circles at the -\points O and N. But nowthe _ HlinesBH and CI are parallel,be-causethe anglesICN and HBOare equal, each being twice theangleIAN. AndsinceBH,drawnfrom the center to the point of Fig.93contacCtisperpendicularto FG,it followsthat CI will alsobeperpendicularto FGand that the arc FI is equalto the arc IG;consequentlythe angleFAI is equalto the angle LAG.Q.E.D.

XXI,PRoPosmoXXXlIIf in a horizontalline any two points are chosenand ifthroughoneofthesepointsa linebedrawninclinedtowardsthe other, and if from this other point a straight line isdrawnto the inclinedline in sucha direcCtionthat it cutsoff from the inclinedline a portionequal to the distancebetweenthe two chosenpointson the horizontalline,thenthe timeofdescentalongthelinesodrawnislessthan alongany other straight linedrawnfrom the samepoint to thesame inclinedline. Alongother lineswhich make equalanglesonoppositesidesof thisline,the timesofdescentarethe same.

Let

2z8 THE TWO NEW SCIENCESOF GALII.KOLetA andB be anytwopointsona horizontalline:through

B drawan inclinedstraightlineBC,andfromB layoffa dis-tanceBD equalto BA;jointhe pointsA andD. Then,I say,the timeof descentalongAD islessthan alonganyotherlinedrawnfromA to the inclinedlineBC. Fromthe pointA drawAEperpendicularto BA;andfromthe pointD drawDEper-pendicularto BD, intersectingAEat E. Sincein the isosceles

_k triangleABD,we havetheanglesBADandBDAequal,

[254]theircomplementsDAEandEDA are equal. Hence if,with E as centerandEA asradius,wedescribea circleitwillpassthroughD andwilltouchthe linesBA and BDat the pointsAandD. NowsinceAistheendoftheverti-callineAE,thedescentalongADwilloccupylesstimethan

Fig.94 alongany other line drawnfromthe extremityA to the lineBCandextendingbeyondthecircumferenceofthe circle;whichconcludesthefirstpartoftheproposition.

If however,weprolongtheperpendicularlineAE,andchooseanypointF uponit,aboutwhichascenter,wedescribea circleof radiusFA, thiscircle,AGC,willcut the tangentlinein thepointsGandC. Drawthe linesAGandACwhichwillaccord-ingto the precedinglemma,deviateby equalanglesfromthemedianlineAD. The time of descentalongeitherof theselinesisthe same,sincetheystartfromthe highestpointA,andterminateonthecircurrfferenceofthe circleAGC.

PROBT.vMXII, PRoPosrrlo_XXXIIIGivena limitedverticallineandan inclinedplaneofequalheight,havinga commonupperterminal;it is requiredtofinda pointonthe verticalline,extendedupwards,from

which

THIRD DAY 229whicha bodywillfalland,whendeflectedalongtheinclinedplane,willtraverseit in the sametime-intervalwhichisrequiredforfall,fromrest,throughthegivenverticalheight.

Let AB be the givenlimitedverticalline andAC an in-dinedplanehavingthe samealtitude. It isrequiredto findontheverticalBA,extendedaboveA,a pointfromwhicha fallingbodywilltraversethe distanceACin the sametimewhichisspentin falling,fromrestat A, throughthe givenverticallineAB. Drawthe lineDCEat rightanglesto AC,andlayoffCDequalto AB;alsojointhepointsA andD; thentheangleADCwillbe greaterthantheangleCAD,sincethesideCAisgreaterthaneitherABorCD. MaketheangleDAEequalto theangle

[z55]ADE,anddrawEF perpendicularto AE; thenEFwillcuttheinclinedplane, ex- [ /tendedbothways,at ItF. Lay offAI andAGeachequaltoCF; AthroughG drawthe xlg _ //[ho zontallineGH.K'- /IThen,Isay, H isthe / _ / " / [pointsought. _ _ / / [

For,ifweagreeto _ Cx_/ [ [Blet the length AB _ _/_ [representthetimeof V _ /fall alongthe verfi- J_. _ /calAB,thenACwill/ - Notlikewiserepresentthetimeofdescentfromxrestat A,alongAC; Fig.95andsince,intheright-angledtriangleAEF,thelineEChasbeendrawnfromtherightangleatEperpendiculartothebaseAF,itfollowsthat AEwillbeameanproportionalbetweenFAandAC,whileCEwillbe ameanproportionalbetweenACandCF,thatisbetweenCAandAI. Now,sinceACrepresentsthe timeofdescentfromA alongAC,it followsthat AEwillbe the timealongthe entiredistanceAF,andECthe timealongAI. Butsince

230 THE TWO NEW SCIENCESOF GALILEOsincein the isoscelestriangleAEDthe sideEAis equalto thesideED it followsthat EDwillrepresentthe timeoffallalongAF,whileEC isthe timeof fallaIongAI. ThereforeCD, thatis A.B,willrepresentthe timeoffall,fromrestat A, alongIF;whichisthesameassayingthatABisthetimeoffall,fromGorfromH, alongAC. E.F.

PROBLEMXIII,PROPOSITIONi_Givena limitedinclinedplaneanda verticallinehavingtheir highestpoint in common,it is requiredto findapointin the verticallineextendedsuchthat a bodywillfallfromit andthentraversetheinclinedplanein thesametimewhichisrequiredto traversethe inclinedplanealonestartingfromrestat thetopofsaidplane.

Let AC and AB be an inclinedplaneand a verticallinerespeCtively,havinga commonhighestpointat A. It is re-quiredto finda pointin theverticalline,aboveA,suchthat abody,fallingfromit andafterwardshavingits motiondireCtedalongA_B,willtraverseboth the assignedpart of the vertical

[256]lineandthe planeAB in the sametimewhichis requiredfortheplaneABalone,startingfromrestat A. DrawBCa hori-zontallineandlayoffANequalto .A_C;choosethe pointL sothat AB'.BN--AL:LC,andlayoffAI equalto AL;choosethepointE suchthat CE,laidoffonthe verticalACproduced,willbe a thirdproportionalto ACandBI. Then,I say,CEis thedistancesought;so that, if the verticallineis extendedaboveAandifaportionAXislaidoffequalto CE,thenabodyfallingfromX willtraverseboth the distances,XA andAB, in thesametime as that required,whenstartingfromA,to traverseABalone.

DrawXR parallelto BCand intersecCtingBAproducedinR; nextdrawEr) parallelto BCandmeetingBAproducedinD; onADas diameterdescribea semicircle;fromB drawBFperpendicularto AD,and prolongit till it meetsthe circum-ferenceof the circle;evidentlyFB is a meanproportionalbetweenABandBD,whileFAisa meanproportionalbetween

DA

THIRD DAY 231DAandAB. TakeBSequalto BI andFHequalto FB. NowsinceAB_D =AC:CEand sinceBF is a meanproportional

[ sT]betweenABandBD,whileBI isa meanproportionalbetweenAC and CE, it followsthat BA:AC=FB:BS,and sinceBA:AC=BA_N=FB_S we shall have, convertendo,BF-.FS=AB:BN=AL'.LC. Consequentlythe rectangleformedby FB

lg I IJ

c

Fig.96andCL is equalto the recCtanglewhosesidesare ALandSF;moreover,thisrectangle_A_L.SFis the excessof the recCtangleAL.FB,orAI.BF,overtherecCtangleAI.BS,orM.IB. ButtherecCtangleFB.LCisthe excessoftherectangleAC.BFoverthere,angleAL.BF;andmoreovertherecCtangleAC.BFisequaltothe rec°cangleAB.BIsinceBA-_A_C=FB:BI;hencethe excessofthe recCtangleAB.BIoverthe recCtangleAI.BF,orAI.FH,isequalto the excessofthe rectangleAI.FHoverthe rectangleAI.IB;thereforetwicethe recCtangleM.FH is equalto the sumof

23z THE TWO NEW SCIENCESOF GALII.F.Oof the redtanglesAB.BI and AI.IB, or 2AI.FH=2AI.IB+B-I_. Add_ to eachside,then_2AI.IB+BI2 +_ =_=zAI.FH+AI_. AgainaddB-F_to eachside,thenAB2q-BF_=_ = 2AI.FH+ _2 + _-_ = zAI.FH+ __T_+ F--H_. But_* =zAH.HF+ _" + H-_'_;and hence zAI.FH+ _2 +F--H2=zAH.I-IF+ _2 + _i_. Subtracering_ from eachsidewe have2AI.FH+_* =2AH.HF+AH_. Sincenow_FHis a fadtorcommonto both re,angles, it followsthat AH isequalto AI; for ifAHwereeithergreateror smallerthanAI,thenthetworecCtanglesAH.HFplusthesquareofHAwouldbeeitherlargeror smallerthanthe tworecCtanglesAI.FHplusthesquareof IA, a resultwhichis contraryto whatwehavejustdemonstrated.

If nowweagreeto representthe timeof descentalongABby the lengthAB, thenthe timethrough_ACwilllikewisebemeasuredbyAC;andIB,whichisa meanproportionalbetweenAC andCE,willrepresentthe timethroughCE,orX_A,fromrestat X. Now,sinceAF isa meanproportionalbetweenDAandAB,or betweenRBandAB,andsinceBF,whichisequalto FH, is a meanproportionalbetweenAB and BD, that isbetweenAB andAR, it follows,froma precedingproposition[PropositionXIX, corollary],that the differenceAHrepresentsthe timeofdescentalongABeitherfromrestat R or afterfallfromX,whilethe timeofdescentalongAB,fromrestat A, ismeasuredby the lengthAB. Butas hasjust beenshown,thetimeof fall throughXA ismeasuredby IB,whilethe timeofdescentalongA_B,afterfall,throughRAorthroughXA,isIA.Thereforethe timeofdescentthroughXA plusABismeasuredby the lengthAB,which,of course,alsomeasuresthe timeofdescent,fromrestat A,alongABalone. Q.F..F.

[zSs]PROBLEMXIV,PROPOSITIONXX.XV

Givenaninclinedplaneandalimitedverticalline,itisre-quiredtofindadistanceontheinclinedplanewhichabody,startingfromrest,willtraverseinthesametimeasthatneededtotraverseboththeverticalandtheinclinedplane.

Let

THIRD DAY 233LetABbe theverticallineandBCtheinclinedplane. It is

requiredto layoffonBCadistancewhichabody,startingfromrest,willtraversein a timeequalto that whichis occupiedbyfallthroughtheverticalABandbydescentoftheplane. DrawthehorizontallineAD,whichintersecCtsat E theprolongationofthe inclinedplaneCB;layoffBF equalto BA,andaboutE ascenter,withEF as radiusdescribethe circleFIG. ProlongFEuntilit interse_sthe circumferenceat G. Choosea pointHsuchthat GB'J3F=BH'_-I_F.Drawthe lineHI tangentto the

I" D

Fig.97circleat I. AtB drawthelineBKperpendicularto FC,cuttingthelineEILat L;alsodrawLMperpendiculartoELandcuttingBCatM. Then,I say,BMisthedistancewhichabody,start-ingfromrestat B,willtraversein the sametimewhichis re-quiredto descendfromrestat A throughbothdistances,ABand BM. Lay offEN equal to EL; then sinceGB:BF--BH'.HF,we shallhave, permutando,GB:BH=BF'MF,and,dividendo,GH:BH--BH'_-I-F.Consequentlythe recCtangleGH.HFisequalto the squareonBH;but thissamerectangleis alsoequalto the squareonHI; thereforeBHis equalto HI.Since,in the quadrilateralILBH,the sidesliB and HI are

equal

'4

z34 THE TWO NEW SCIENCESOF GALI!.EOequalandsincethe anglesat B andI arefightangles,it followsthat the sidesBLandLI arealsoequal:but EI --EF;therefore

[ s9]thetotal lengthT,E,orNE, is equalto the sumofLBandEF.If wesubtracCtthe commonpartEF, the remainderFN willbeequalto LB: but, by construc°don,FB=BA and, therefore,LB=AB4-BN. If againweagreeto representthe timeof fallthroughABby the lengthA_B,thenthe timeofdescentalongEBwillbemeasuredbyEB;moreoversinceENis ameanpro-portionalbetweenME and EB it willrepresentthe time ofdescentalongthe wholedistanceEM; thereforethe differenceofthesedistances,BM,willbetraversed,afterfallfromEB,orAB,in a timewhichisrepresentedbyBN. ButhavingalreadyassumedthedistanceABasameasureofthetimeoffallthroughAB,thetimeofdescentalongABandBMismeasuredbyAB4-BN. SinceEBmeasuresthe timeof fall,fromrestat E, alongEB, the timefromrest at B alongBM willbe the meanpro-portionalbetweenBEandBM,namely,BL. The timethere-

A 13foreforthe pathAB+BM,startingfromrestat A is AB4-BN;butthetimeforBMalone,startingfromrestat 13,isBL; andsinceit hasalready been shownthat BL= AB+ BN,thepropositionfollows.

Anotherandshorterproofis the following:

Fig.98 Let BCbetheinclinedplaneand BAthe vertical;at B draw a perpendicularto EC,extendingit both ways;lay off BH equal to the excessofBE overBA;makethe angleHEL equalto the angleBHE;prolongELuntilitcutsBKinL; atL drawLMperpendiculartoEL andextendit till it meetsBCinM; then,I say,BM istheportionof BC sought. For, sincethe angleMLE is a fightangle,BL willbe a meanproportionalbetweenMB andBE,

while

THIRD DAY 235whileLEis a meanproportionalbetweenMEandBE;layoffEN equalto I,E;thenNE=EL=LH,andHB=NE-BL.ButalsoHB=NE-(NB+BA);thereforeBN+BA=BL. If nowweassumethelengthEBasa measureofthe timeof descentalongEB,the timeofdescent,fromrestatB, alongBMwillberepresentedbyBL;but,ifthedescentalongBMisfromrestatE oratA,thenthetimeofdescentwillbemeasuredbyBN;andAB willmeasurethe timealongAB. Thereforethe timere-quiredto traverseABandBM,namely,thesumofthedistancesABandBN, is equalto the timeof descent,fromrestat B,alongBMalone. Q._. F.

[26o]LEMMA

LetDCbedrawnperpendiculartO thediameterBA;fromtheextremityB drawthe line BED at /k_random;drawthe lineFB. Then,Isay, FB is a meanproportionalbe-tweenDB and BE. Jointhe pointsE and F. ThroughB, draw thetangentBGwhichwillbeparalleltoCD. Now,sincethe angleDBGis Cequalto theangleFDB,andsincethe Salternateangleof GBD is equaltoEFB, it followsthat the trianglesFDBand FEBaresimilarandhence_i _sBD:BF=FB:BE. Fig.99 '

LEMMALetACbe a linewhichis longerthanDF,andlet the ratio

of ABto BCbe greaterthanthat ofDE to EF. Then,I say,AB is greaterthan DE. For, if AB

_ ._ bearsto BCa ratiogreaterthanthatof_ _ p DEtoEF, thenDEwillbearto some

, , , _ lengthshorterthanEF, thesameratioFig.ioo whichABbearstoBC. Callthislength

EG;thensinceAB:BC=DE=EG,it follows,componendoetcon-vertendo,

z36 THE TWO NEW SCIENCESOF GALILEOvertendo,that CA'./kB=GD'.DE.ButsinceCAis greaterthanGD,it followsthat BAisgreaterthanDE.

#- B LFAVIMA

LetACIBbethe quadrantofacircle;fromBdrawBEparalleltoAC;about anypoint in the line

$ BEdescribeacircleBOES,touch-ingABat B andintersec°dngthecircumferenceof the quadrantatI. JointhepointsC andB; drawthe line CI, prolongingit to S.Then,I say,the lineCI is alwayslessthan CO. Drawthe lineAItouchingthe circleBOE. Then,

[z6I]if the lineDI be drawn,it willbeequal to DB; but, since DBtouchesthe quadrant,DI willalsobetangentto it andwillbeatrightangle_to AI; thusAI touchesthecircleBOE at I. And sincetheangleAICisgreaterthantheangle

/k _,,. ABC,subtendingas it does a

1rtoo t t enSINisalsogreaterthanthe angleABC. Whereforethe arc IES isgreaterthan the arc BO,andtheline CS,beingnearerthe center,

¢:: is longerthanCB. Consequently

COis greaterthan CI, sinceSC:CB--OC:CI.

This result wouldbe all theFig.ioi moremarkedif, as in the second

figure,the arc RICwerelessthan a quadrant. For the per-pendicularDBwouldthencut thecircleCIB;andsoalsowould

DI

THIRD DAY 237DI whichis equalto BD;the angleDIAwouldbeobtuseandthereforethelineAINwouldcutthecircleBIE. SincetheangleABCislessthanthe angleAIC,whichisequalto SIN,andstilllessthantheanglewhichthetangentat I wouldmakewiththelineSI, it followsthat the arcSEI is fargreaterthanthe arcBO;whence,etc. Q.E.D.

Ta_.ORSMXXII,PROPOSITIONXXXVIIf fromthe lowestpoint of a verticalcircle,a chordisdrawnsubtendingan arc not greaterthan a quadrant,and if fromthe twoendsof this chordtwootherchordsbe drawnto anypointonthearc,thetimeofdescentalongthe twolatter chordswillbe shorterthanalongthe first,andshorteralso,bythe sameamount,thanalongthelowerofthesetwolatterchords.

[262]Let CBDbe an arc, notexceedinga quadrant,takenfromaverticalcirclewhoselowestpoint is C; let CD be the chord[planumelevatum]sub-m B &tendingthisarc,andletwt" ' fA" .... ..A_'there be two otherI f//j,chordsdrawn from C/ _ {/ ]/.andD to any point B/ _[ ///onthearc. Then,I say,[ /o \/_/'--thetimeofdescentalong[ / v_.the two chords[plana][/_ -DB and BC is shorterId_--1"thanalongDCalone,or[¢alongBCalone,startingfromrestatB. Throughthe point D, drawthe _" G _ l_ 1_horizontalline MDAcuttingCBextendedat Fig. lO2A:drawDN andMC at rightanglesto MD,andBNat rightanglesto BD;aboutthe right-angledtriangleDBNdescribethesemicircleDFBN,cuttingDCat F. ChoosethepointO suchthat DOwillbeameanproportionalbetweenCDandDF;inlikemanner

238 THE TWO NEW SCIENCESOF GALII.EOmannerselec°cVsothat AVisa meanproportionalbetweenCAandAB. Let thelengthPS representthetimeofdescentalongthewholedistanceDC orBC,bothof whichrequirethe sametime. LayoffPR suchthat CD'.DO=timePS.timePR. ThenPRwillrepresentthetimeinwhichabody,startingfromD,willtraversethedistanceDF,whileRSwillmeasurethetimeinwhichthe remainingdistance,FC,willbe traversed.ButsincePS isalsothe time ofdescent,fromrest at B,alongBC,and ifwechooseT suchthat BC:CD=PS:_ thenPT willmeasurethetimeofdescentfromAto C,forwehavealreadyshown[Lemma]that DC isa meanproportionalbetweenACandCB. FinallychoosethepointGsuchthatCA:AV_lYI'_G,thenPGwillbethe timeofdescentfromA toB,whileGTwillbe the residualtimeofdescentalongBCfollowingdescentfromAto B. But,sincethe diameter,DN,ofthe circleDFNisa verticalline,thechordsDF andDBwillbetraversedin equaltimes;whereforeif onecan provethat a bodywilltraverseBC, afterdescentalongDB, in a shortertimethanit willFC afterdescentalongDF he willhaveprovedthe theorem.Buta bodydescendingfromD alongDBwilltraverseBCin the sametimeasif it hadcomefromA alongAB,seeingthat thebodyacquiresthe same[2631momentumin descendingalongDB as alongAB. Henceitremainsonlyto showthat descentalongBCafterABisquickerthanalongFCafterDF. Butwehavealreadyshownthat GTrepresentsthe timealongBCafterAB;alsothat RSmeasuresthe time alongFC afterDF. Accordinglyit mustbe shownthat RS is greaterthan GT, whichmaybe doneas follows:SinceSP_R =CD'.DO,it follows,invertendoet convertendo,that RS:SP=OC:CD;alsowe haveSP.'_=DC:CA. AndsinceTP:PG=CA'_AV,it follows,invertendo,that PT:TG=AC:CV,therefore,ex wquali,RS:GT=OC:CV.But, as weshallpresentlyshow,OCis greaterthan CV;hencethe timeRSisgreaterthanthe timeGT,whichwasto be shown.Now,since[Lemma]CFis greaterthanCBandFD smallerthanBA,it followsthat CD:DF>CA-.AB.But CD'J)F=CO.£)F,seeingthat CDff)O=DO'.DF;andCA'dkB=_*:_". There-

fore

THIRD DAY z39foreCO£)F>CV:VB,and,accordingto the precedinglemma,CO>CV. Besidesthisit is clearthatthetimeofdescentalongDC is to the time alongDBCas DOCis to the sumof DOandCV.

SCHOLIUM

Fromthe precedingit is possibleto inferthat the path ofquickestdescent[lationemomniumvelocissimam]from onepointto anotheris not the shortestpath,namely,a straightline,but the arc ofa circle.*In the quadrantBAEC,havingthe sideBCvertical,dividethe arc ACinto anynumberof .equalparts,AD,DE,EF, FG,GC,andfromC drawstraightlinesto the pointsA,D,E, F, G; _ Adrawalsothe straightlinesAD,DE, EF, FG,GC. Evidentlyde-scentalongthepathADCisquicker

[264] DthanalongACaloneor alongDCfromrestat D. Butabody,start-ingfrom rest at A, willtraverseDC morequicklythan the pathADC;while,if it startsfromrestatA, itwilltraversethepathDEC 1_in a shortertimethanDC alone.CHence descentalong the three Fig. lO3chords,ADEC,willtakelesstimethan alongthe twochordsADC. Similarly,followingdescentalongADE,thetimerequiredto traverseEFCislessthanthat neededforECalone. There-foredescentis morerapidalongthefourchordsADEFCthanalongthe threeADEC.Andfinallyabody,afterdescentalongADEF,willtraversethe twochords,FGC,morequicklythanFC alone. Therefore,alongthefivechords,ADEFGC,descentwillbe morerapidthanalongthefour,ADEFC.Consequently

* It is wellknownthat the first correctsolutionfor the problemofquickestdescent,under the conditionof a constantforcewasgivenbyJohn Bernoulli0667-I748). [Trans.]

z4o THE TWO NEW SCIENCESOF GALII.EOthe nearerthe inscribedpolygonapproachesa circletheshorteris thetimerequiredfordescentfromAto C.

Whathasbeenprovenfor the'quadrantholdstrue alsoforsmallerarcs;the reasoningisthesame.

PROBLEMXV,PRoPosmosXXXVIIGivenalimitedverticallineandan inclinedplaneofequalaltitude;it is requiredto finda distanceon the inclinedplanewhichis equal to the verticalline and whichistraversedin an intervalequalto the timeoffallalongtheverticalline.

LetABbetheverticallineandACtheinclinedplane. Wemustlocate,on the inclinedplane,a distanceequalto the vertical

A_line AB and whichwill be

traversedby a bodystarting

from rest at A in the sametimeneededforfallalongthevertical line. Lay off ADequal to AB,and bisect theremainderDC at I. Choose

1_the pointE suchthat AC:CIFig.Io4 --CI:AE and lay off DG

equalto AE. ClearlyEGisequaltoAD,andalsoto AB. Andfurther,I say that EG isthat distancewhichwillbe traversedby a body,startingfromrestat A, in the sametimewhichisrequiredfor that bodyto fall throughthe distanceAB. Forsince AC:CI--CI:AE--ID:DG,we have, convertecuto,CA:AI =DI:IG. Andsincethe wholeofCAis to the wholeofAIas the portionCI is to the portionIG, it followsthat the re-

[z6S]mainderIAisto the remainderAGasthewholeofCAisto thewholeof AI. ThusAI is seento be a meanproportionalbe-tweenCAandAG,whileCIisa meanproportionalbetweenCAandAE. IfthereforethetimeoffallalongABisrepresentedbythe lengthAB,the timealongACwillbe representedbyAC,whileCI,or ID,willmeasurethetimealongAE. SinceAI isameanproportionalbetweenCA andAG,and sinceCA is a

measure

THIRD DAY 241measureofthetimealongtheentiredistanceAC,itfollowsthatAI isthetimealongAG,andthedifferenceICisthetimealongthe differenceC_; but DI wasthe time alongAE. Conse-quentlythe lengthsDI andICmeasurethetimesalongAEandCGrespe_ively.Thereforethe remainderDArepresentsthetimealongEG,whichofcourseis equalto thetimealongAB.

Q. E. F.

COROT,T.ARY

Fromthisit is clearthat thedistancesoughtis boundedateachendbyportionsof theinclinedplanewhicharetraversedinequaltimes.

PRObLeMXVI,PRoPosrrloNXXXVlIIGiventwohorizontalplanescut by a verticalline,it isrequiredto finda pointonthe upperpartof the verticallinefromwhichbodiesmayfalltothehorizontalplanesandthere, havingtheir motiondefle_edinto a horizontaldire_ion,will,duringan intervalequalto thetimeoffall,traversedistanceswhichbearto eachotherany assignedratioofasmallerquantityto alarger.

LetCD andBEbethe horizontalplanescut by the verticalACB,and let the ratioof the smallerquantityto the largerbe that of N to FG. It is requiredto findin the upperpartof theverticalline,AB,a pointfromwhicha bodyfallingtothe planeCD andtherehavingitsmotiondefle_edalongthisplane,willtraverse,duringan intervalequalto itstimeoffalladistancesuchthat ifanotherbody,fallingfromthissamepointto the planeBE, there haveits motiondefle_edalongthisplaneandcontinuedduringan intervalequaltoitstimeoffall,willtraversea distancewhichbearsto the formerdistancethe

[266]ratioofFG tON. LayoffGHequalto N,andsele_the pointLsothatFHa_IG=BC:CL.Then,I say,Listhepointsought.For,if welay offCM equalto twiceCL,anddrawthe lineL'V[cuttingthe planeBEat O,then]30willbeequalto twiceBL

..... _ - ~ " "" 7 .....

24z THE TWO NEW SCIENCESOF GALIt,F.OBL. And since FH:HG=BC:CL,we have, componendoetconvertendo,HG._F=N._F=CL.q,B=CM',BO. It is clearthat, sinceCM is doublethe di_stanceLC, the spaceCM isthat whicha bodyfallingfromL throughLCwilltraverseinthe pIaneCD; and,forthe samereason,sinceBOis twicethedistanceBL,it is clearthat BOis the distancewhicha body,

A

L

: 'N

!f L t - I

Fig.Io5afterfallthroughLB,willtraverseduringan intervalequaltothe timeof itsfallthroughLB. Q.v.F.

SAox.Indeed,I thinkwemayconcedeto ourAcademician,withoutflattery,hisclaimthat in the principle[principio,i. e.,acceleratedmotion]laiddownin thistreatisehehasestablisheda newsciencedealingwitha veryoldsubjec°c.Observingwithwhateaseanddearnesshe deducesfroma singleprincipletheproofsof somanytheorems,I wondernot a little howsucha questionescapedthe attentionof Archimedes,Apollonius,Euclidand so many other mathematiciansand illustriousphilosophers,especiallysincesomanyponderoustomeshavebeendevotedto thesubjedtofmotion.

[_671S_v. Thereisa fragmentofEudid whichtreatsofmotion,

but

THIRD DAY z43but init thereisno indicationthatheeverbeganto investigatethepropertyof accelerationandthemannerinwhichit varieswithslope. Sothatwemaysaythedoorisnowopened,forthefirsttime,to a newmethodfraughtwithnumerousandwonder-fulresultswhichinfutureyearswillcommandtheattentionofotherminds.

SACg.I reallybelievethat just as, for instance,the fewpropertiesof the circleprovenbyEuclidintheThirdBookofhisElementsleadto manyothersmorerecondite,so the prin-cipleswhicharesetforthin this littletreatisewill,whentakenupbyspeculativeminds,leadtomanyanothermoreremarkableresult;andit istobebelievedthatitwillbesoonaccountofthenobilityofthesubjec*c,whichissuperiortoanyotherinnature.

Duringthis longandlaboriousday,I haveenjoyedthesesimpletheoremsmorethantheirproofs,manyof which,fortheircompletecomprehension,wouldrequiremorethananhoureach;thisstudy,ifyouwillbegoodenoughto leavethebookinmyhands,isonewhichI meanto takeupat myleisureafterwehavereadtheremainingportionwhichdealswiththemotionof projedkiles;andthisifagreeableto youweshalltakeupto-morrow.

SALV.I shallnotfailto bewithyou.

END OF THE THIRD DAY.

[26s1

FOURTH DAY_ALVIATI. Oncemore,Simpliciois hereon;l_]time; so let uswithoutdelaytakeup the

i.__. questionofmotion.ThetextofourAuthor'_/_=_isasfollows:

THE MOTIONOFPROJECrILESIntheprecedingpageswehavediscussedthe

propertiesof uniformmotionand of motionnaturallyaccel-eratedalongplanesof all inclinations.I nowproposeto setforththosepropertieswhichbelongto a bodywhosemotioniscompoundedoftwoothermotions,namely,oneuniformandonenaturallyaccelerated;theseproperties,wellworthknowing,Iproposeto demonstratein a rigidmanner.Thisis the kindofmotionseenin amovingprojectile;itsoriginI conceiveto beasfollows:

Imagineanyparticleprojectedalonga horizontalplanewith-out friction;then weknow,fromwhat hasbeenmorefullyexplainedin the precedingpages,that this particlewillmovealongthis sameplanewith a motionwhichis uniformandperpetual,providedthe planehasnolimits. But if theplaneislimitedandelevated,thenthemovingparticle,whichweimag-ineto bea heavyone,willonpassingovertheedgeoftheplaneacquire,in additionto its previousuniformand perpetualmotion,a downwardpropensitydueto itsownweight;so thatthe resultingmotionwhichI callprojection[projectio],is com-poundedofonewhichis uniformandhorizontalandofanotherwhichisverticalandnaturallyaccelerated.Wenowproceedto

demonstrate

FOURTHDAY 245demonstratesomeof its properties,thefirstofwhichis as fol-lows:

[269]THEOREMI, PROPOSITIONI

Aprojectilewhichiscarriedbyauniformhorizontalmotioncompoundedwith a naturallyacceleratedverticalmotiondescribesa pathwhichis a seml-parabola.

SAGR.Here,Salviati,it willbe necessaryto stop a littlewhileformy sakeand,I believe,alsofor the benefitof Sim-plicio;for it sohappensthat I havenotgoneveryfar in mystudyofApolloniusandammerelyawareof the fact that hetreatsofthe parabolaandotherconicsections,withoutan un-.derstandingof whichI hardlythinkonewillbe ableto followthe proofof otherpropositionsdependinguponthem. Sinceevenin thisfirstbeautifultheoremtheauthorfindsit necessaryto provethat thepathofa projecCtileisa parabola,andsince,asI imagine,weshallhaveto dealwithonlythiskindofcurves,it willbeabsolutelynecessarytohaveathoroughacquaintance,ifnotwithallthepropertieswhichApolloniushasdemonstratedfor thesefigures,at leastwiththosewhichare neededfor thepresenttreatment.

SALV.You are quitetoo modest,pretendingignoranceoffactswhichnot longagoyou acknowledgedaswellknown--Imeanat the time whenwewerediscussingthe strengthofmaterialsand neededto usea certaintheoremofApolloniuswhichgaveyounotrouble.

SACl_.I mayhavechancedto knowit ormaypossiblyhaveassumedit, solongasneeded,forthat discussion;butnowwhenwehavetofollowallthesedemonstrationsaboutsuchcurvesweoughtnot,astheysay,toswallowit whole,andthuswastetimeandenergy.

SIMP.NoweventhoughSagredois,asIbelieve,wellequippedforallhisneeds,I donotunderstandeventheelementaryterms;foralthoughourphilosophershavetreatedthe motionof pro-jedkiles,I do not recalltheirhavingdescribedthe pathof aproje_ileexceptto statein a generalwaythat it is alwaysa

curved

246 THE TWO NEW SCIENCESOF GATJLFDcurvedline,unlessthe projecCtionbe verticallyupwards.But

[7o]if the littleEuclidwhichI havelearhedsinceourpreviousdis-cussiondoesnot enableme to understandthe demonstrationswhichareto follow,thenI shallbe obligedto acceptthe the-oremsonfaithwithoutfullycomprehendingthem.

SALv.On the contrary,I desirethat youshouldunderstandthemfromtheAuthorhimself,who,whenhe allowedmeto seethisworkof his,wasgoodenoughto proveformetwoof theprincipalpropertiesoftheparabolabecauseI didnothappentohaveat handthebooksofApollonius.Tbzeseproperties,whicharethe onlyonesweshallneed in the presentdiscussion,heprovedin sucha waythat noprerequisiteknowledgewasre-quired. Thesetheoremsare,indeed,givenby Apollonius,butaftermanyprecedingones,to followwhichwouldtakea longwhile. I wishto shortenourtaskbyderivingthefirstproperty

purelyandsimplyfromthe modeofgen-erationof the parabolaandprovingthesecondimmediatelyfromthefirst.

Beginningnowwiththe first,imaginea rightcone,ereftedupon the circularbaseibkcwithapexat 1. The see°donofthisconemadebya planedrawnparalleltothe sidelk isthecurvewhichis calleda parabola.The baseof thisparabolabc

,i .]_cutsatrightanglesthediameterik ofthecircleibkc,and theaxisad is paralleltotheside/k;nowhavingtakenanypointfin thecurvebfadrawthe straightlinefe

Fig./o6 parallelto bd; then, I say, the squareof bd is to the squareoffe in the sameratioas the axisadis to the portion02. Throughthe point epassaplaneparallelto thecircleibkc,producinginthe conea circularse6tionwhosediameteris thelinegeh. Sincebdisat rightanglesto ik in thecircleibk,thesquareofbdisequalto the recCtangleformedby k/and dk; soalso in the uppercirclewhichpassesthroughthepointsgfhthe squareoffe isequalto the re_Langleformedby

ge

FOURTH DAY z47geandeh;hencethe squareof bdis to the squareoffe asthetee°tangleid.dkisto therec°canglege.eh.Andsincethelineedisparallelto hk, the lineeh,beingparallelto dk, is equalto it;thereforethe reftangleid.dkisto the rec°mnglege.ehas/d isto

[2711ge,that is,asdaisto a¢;whencealsotherectangle/d.dkisto therectanglege.eh,that is,the squareofbdis tothe squareofre,asthe axisda isto the portionae. Q._. D.

Theotherpropositionnecessaryforthisdiscussionwedemon-strateas follows.Letusdrawa parabolawhoseaxiscaispro-longedupwardsto a pointd;fromanypointbdrawthelinebeparallelto thebaseoftheparabola;ifnowthepointd ischosenso that da= ca, then, I say, thestraight line drawn throughthe dpointsb andd willbe tangenttothe parabolaat b. Forimagine,ifpossible,thatthislinecutsthepar-abolaaboveor that its prolonga-tioncutsitbelow,andthroughanypointg in it drawthe straightlinefge. Andsincethe squareoffe is agreaterthanthe squareofge,thesquareoffewillbearagreaterratioto the squareofbcthanthesquareofgeto thatofbc;andsince,bytheprecedingproposition,the squareoffeisto that ofbcasthelineeaisto ca,it followsthat the lineeawillbear to the line caa greaterratiothan the squareofgeto thatof bc,or, than the squareofedtothatofcd(thesidesofthetrianglesdegand dcbbeingproportional). Fig.xo7Butthe lineeaisto ca,orda,inthesameratioasfourtimestherec°cangleea.adis to fourtimesthesquareofad,or,whatisthesame,the squareofcd,sincethisis fourtimesthesquareofad;hencefourtimesthe rec°mngleea.adbearsto the squareof cda

248 THE TWONEW SCIENCESOF GALn,_oa greaterratiothan the squareofed to the squareof cd;butthat wouldmakefour timesthe rectangleea.adgreaterthanthe squareof ed;whichis false,the°fac'tbeingjust the oppo-site,becausethe two portionseaandadof the lineed are notequal. Thereforethe line db touchesthe parabolawithoutcuttingit. Q.E.D.

Sire,.Yourdemonstrationproceedstoorapidlyand,it seemsto me,you keeponassumingthat allofEuclid'stheoremsare

[272]as familiarand availableto me as his firstaxioms,whichisfarfromtrue. Andnowthisfa_ whichyou springuponus,that fourtimesthe rec_ngleea.adis lessthan the squareofdebecausethe two portionsea andadof the linede arenotequalbringsmelittlecomposureofmind,but ratherleavesmein suspense.

SALv.Indeed,allrealmathematiciansassumeonthe partofthe readerperfecCtfamiliaritywith at least the elementsofEuclid;and hereit is necessaryin yourcaseonlyto recallapropositionofthe SecondBookin whichheprovesthat whenalineis cut intoequalandalsointotwounequalparts,the rec-tangleformedon the unequalparts is lessthan that formedon theequal(i.e.,lessthanthe squareonhalfthe line),by anamountwhichis the squareofthe differencebetweentheequalandunequalsegments.Fromthis it is clearthat the squareofthe wholelinewhichis equalto fourtimesthe squareof thehalf is greaterthan four timesthe recCtangleof the unequalparts. In orderto understandthe followingportionsof thistreatiseit willbe necessaryto keepin mindthe twoelementaltheoremsfromconicsecCtionswhichwehavejustdemonstrated;and thesetwo theoremsare indeedthe only oneswhichtheAuthoruses. We can nowresumethe text and seehowhedemonstrateshis firstpropositionin whichhe showsthat abodyfallingwitha motioncompoundedofa uniformhorizontaland a naturallyaccelerated[naturaledescendente]onedescribesa semi-parabola.

Let us imaginean elevatedhorizontallineor planeabalongwhicha bodymoveswithuniformspeedfroma to b. Suppose

this

FOURTHDAY 249t_s planeto endabruptlyat b;thenat thispointthebodywill,on accountof its weight,acquirealsoa naturalmotiondown-wardsalongthe perpendicular/m.Drawthe linebealongtheplaneba to representtheflow,ormeasure,oftime;dividethislineinto a numberof segments,bc,cd,de,representingequalintervalsoftime;fromthepointsb,c,d,e,letfalllineswhichareparallel to the per-pendicularbn.Onthe - • d e_ _ d_firstof these lay off _ _ 0

anydistanceci,onthe /_seconda distancefour j_,

timesas long,df; on /[273]the third, one ninetimesas long,eh;andsoon,in proportiontothesquaresof cb,db,eb,or,wemaysay,in Fig. IO8the squaredratioof thesesamelines. Accordinglyweseethatwhilethebodymovesfrombto cwithuniformspeed,it alsofallsperpendicularlythroughthe distanceci,andat the endof thetime-intervalbcfindsitselfat thepointi. In likemannerat theendofthetime-intervalbd,whichisthedoubleofbc,theverticalfallwillbefourtimesthefirstdistanceci;forithasbeenshownin a previousdiscussionthat thedistancetraversedby a freelyfallingbodyvariesasthe squareofthetime;inlikemannerthespaceehtraversedduringthe time bewillbe ninetimesci;thus it is evidentthat the distanceseh,dr, ciwillbe to oneanotheras the squaresof the linesbe,bd,bc. Nowfromthepointsi,f, h drawthe straightlinesio,fg,hlparallelto be;theselineshl,fg, ioareequalto eb,dbandcb,respeCtively;soalsoarethelinesbo,bg,blrespeCtivelyequaltod, dr,andeh. Thesquareofhlisto that offgasthelinelbisto bg;andthe squareoffg isto that ofioasgbis to bo;thereforethe points_,f, h,lieononeandthe sameparabola. In likemannerit maybe shownthat,ifwetakeequaltime-intervalsofanysizewhatever,andifweimaginetheparticletobecarriedbyasimilarcompoundmotion,the

25o THE TWONEWSCIENCESOF GALII,FOthepositionsofthisparticle,at theendsofthesetime-intervals,willlieononeandthesameparabola, q._. D.

S_v. Thisconclusionfollowsfromtheconverseofthefirstof the twopropositionsgivenabove.For,havingdrawnaparabolathroughthepointsbandh,anyothertwopoints,f andi, notfallingontheparabolamustlieeitherwithinorwithout;consequentlythelinefg iseitherlongerorshorterthanthelinewhichterminatesontheparabola.ThereforethesquareofMwillnotbeartothesquareoffgthesameratioasthelinelbtobg,butagreaterorsmaller;thefactis,however,thatthesquareofMdoesbearthissameratioto thesquareoffg. Hencethepointfdoeslieontheparabola,andsodoalltheothers.

SACX.Onecannotdenythattheargumentisnew,subtleandconclusive,restingas it doesuponthishypothesis,namely,that thehorizontalmotionremainsuniform,that theverticalmotioncontinuesto beaccelerateddownwardsinproportiontothesquareof thetime,andthatsuchmotionsandvelocitiesasthesecombinewithoutaltering,disturbing,or hinderingeachother,*sothatasthemotionproceedsthepathoftheproje_iledoesnotchangeintoa differentcurve:butthis,inmyopinion,

[z74]is impossible.Fortheaxisoftheparabolaalongwhichweimaginethenaturalmotionofafallingbodytotakeplacestandsperpendicularto a horizontalsurfaceandendsat thecenteroftheearth;andsincetheparaboladeviatesmoreandmorefromitsaxisnoproje_ilecaneverreachthecenteroftheearthor,ifit does,asseemsnecessary,thenthepathoftheproje_ilemusttransformitselfintosomeothercurveverydifferentfromtheparabola.

Srm,.Tothesedifficulties,Imayaddothers.Oneoftheseisthatwesupposethehorizontalplane,whichslopesneitherupnordown,toberepresentedbya straightlineasifeachpointonthislinewereequallydistantfromthecenter,whichisnotthecase;forasonestartsfromthemiddle[ofthe line]andgoestowardeitherend,he departsfartherand fartherfromthecenter[oftheearth]andisthereforeconstantlygoinguphill.Whenceit followsthat the motioncannotremainuniform

*AverynearapproachtoNewton'sSecondLawofMotion.[Trans.]

FOURTHDAY 25ithroughanydistancewhatever,butmustcontinuallydiminish.Besides,I donotseehowit ispossibletoavoidtheresistanceofthemediumwhichmustdestroytheuniformityofthehorizon-tal motionandchangethe lawof accelerationoffallingbodies.Thesevariousdifficultiesrenderit highlyimprobablethat aresultderivedfromsuchunreliablehypothesesshouldholdtruein pracCtice.

SALv.All thesedifficultiesandobjecCtionswhichyou urgearesowellfoundedthat it is impossibleto removethem;and,asforme,I am readyto admitthemall,whichindeedI thinkourAuthorwouldalsodo. I grantthattheseconclusionsprovedin the abstracCtwillbe differentwhenappliedin the concreteandwillbefallacioustothisextent,that neitherwillthehorizon-tal motionbe uniformnor the naturalaccelerationbe in theratioassumed,nor the path of the pro_ecCtilea parabola,etc.But,on theotherhand,I askyounottobegrudgeourAuthorthatwhichothereminentmenhaveassumedevenifnotstri_lytrue. Theauthority-ofArchimedesalonewillsatisfyeverybody.In hisMechanicsandin hisfirstquadratureofthe parabolahetakesforgrantedthat the beamofa balanceorsteelyardis astraightline, everypoint of whichis equidistantfromthecommoncenterofallheavybodies,andthatthecordsbywhichheavybodiesaresuspendedareparalleltoeachother.

Someconsiderthis assumptionpermissiblebecause,in prac-tice,ourinstrumentsandthe distancesinvolvedaresosmallincomparisonwiththe enormousdistancefromthecenteroftheearththatwemayconsideraminuteofarconagreatcircleasastraightline,andmayregardtheperpendicularsletfallfromitstwoextremitiesasparallel.Forifin accrualpracCticeonehadto[ZTS]considersuchsmallquantities,it wouldbe necessaryfirstofalltocriticisethe architecCtswhopresume,byuseofaplumbline,toere_ hightowerswithparallelsides. I mayaddthat, in alltheirdiscussions,Archimedesandthe othersconsideredthem-selvesas locatedat an infinitedistancefromthe centerof theearth,in whichcasetheirassumptionswerenot false,andthereforetheirconclusionswereabsolutelycorrecCt.When'wewish

zSz THE TWO NEW SCIENCESOF GALII.KOwishto applyourprovenconclusionsto distanceswhich,thoughfinite,areverylarge,it isnecessary_forusto infer,onthebasisofdemonstratedtruth, whatcorrectionisto bemadefor the factthat our distancefromthe centerof the earth is not reallyinfinite,but merelyvery great in comparisonwith the smalldimensionsof ourapparatus.Thelargestof thesewillbe therangeof ourproje_iles--andevenhereweneedconsideronlythe artillery--which,howevergreat,willneverexceedfourofthosemilesof whichas manythousandseparateus fromthecenterof the earth;and sihcethesepathsterminateuponthesurfaceofthe earthonlyvery slightchangescantake placeintheir parabolicfigurewhich,it is conceded,wouldbe greatlyalteredif theyterminatedat thecenteroftheearth.

As to the perturbationarisingfromthe resistanceof themediumthis ismoreconsiderableanddoesnot,onaccountofitsmanifoldforms,submitto fixedlawsand exa_ description.Thusifweconsideronlytheresistancewhichtheairoffersto themotionsstudiedby us,weshallseethat itdisturbsthemallanddisturbsthemin aninfinitevarietyofwayscorrespondingto theinfinitevarietyin the form,weight,andvelocityof the pro-jec_iles.Foras to velocity,the greaterthis is,the greaterwillbe the resistanceofferedby the air; a resistancewhichwillbegreateras the movingbodiesbecomeless dense[mengram].Sothat althoughthefallingbodyoughtto be displaced[andareaccelerandosz]in proportionto the squareof the durationofitsmotion,yet no matterhowheavythe body,if it fallsfromaveryconsiderableheight,theresistanceoftheairwillbesuchasto preventany increasein speedand willrenderthe motion

. [2761uniform;and in proportionas the movingbodyis lessdense[mengrave]this uniformitywillbe somuchthe morequicklyattainedandafterashorterfall. Evenhorizontalmotionwhich,ifnoimpedimentwereoffered,wouldbeuniformandconstantisalteredby theresistanceof the air andfinallyceases;andhereagainthe less dense[pig leggiero]the bodythe quickertheprocess.Of theseproperties[accidentz]of weight,of velocity,andalsoofform[figura],infiniteinnumber,it isnot possibleto

give

FOURTH DAY z53giveany exaCtdescription;hence,inorderto handlethis matterina scientificway, it is necessarytocut loosefromthesedifficul-ties; and havingdiscoveredand demonstratedthe theorems,inthe caseof noresistance,to use themandapplythemwith suchlimitationsas experiencewillteach. And the advantageof thismethod will not be small; for the material and shapeof theprojectilemay be chosen,as denseand round as possible,sothat it willencounterthe least resistancein the medium. Norwillthe spacesand velocitiesin generalbe sogreatbut that weshallbe easilyabletocorreCtthemwithprecision.

In the caseof thoseprojectileswhichwe use,made of dense[grave]material and round in shape,or of lightermaterial andcylindricalin shape, such as arrows,thrown from a slingorcrossbow,the deviationfrom an exacCtparabolicpath is quiteinsensible. Indeed, if youwillallowme a little greaterliberty,I canshowyou, by twoexperiments,that the dimensionsofourapparatus are so small that these externaland incidentalre-sistances,amongwhich that of the mediumis the most con-siderable,arescarcelyobservable.

I now proceedto the considerationof motionsthroughtheair, sinceit is with thesethat we are nowespeciallyconcerned;the resistanceof the air exhibits itself in two ways: first byofferinggreater impedanceto less dense than to very densebodies,and secondlyby offeringgreaterresistanceto a bodyinrapid motionthan to the samebodyin slowmotion.

Regarding the first of these, considerthe caseof two ballshaving the same dimensions,but one weighingten or twelvetimes as much as the other; one,say,of lead, the other of oak,bothallowedto fallfroman elevationof I5oor2oocubits.

Experimentshowsthat they willreach the earthwith slightdifferenceinspeed,showingus that inbothcasestheretardationcausedby the air is small;for if both ballsstart at the samemoment and at the sameelevation,and if the leadenone beslightlyretardedand thewoodenonegreatlyretarded,thentheformer ought to reach the earth a considerabledistance inadvanceof the latter, sinceit is ten times as heavy. But this

[z771does

:z54 THE TWO NEW SCIENCESOFGALtt.F.Odoesnot happen;indeed,the gainin distanceof oneovertheotherdoesnot amountto the hundredthpartofthe entirefall.Andinthe caseofaballofstoneweighingonlyathirdor halfasmuchasoneoflead,thedifferenceintheirtimesofreachingtheearthwillbe scarcelynoticeable.Nowsincethe speed[irnpeto]acquiredbya leadenballin fallingfromaheightofzoocubitsissogreatthat if themotionremaineduniformthe ballwould,inanintervaloftimeequalto thatofthe fall,traverse4oocubits,andsincethisspeedis soconsiderablein comparisonwiththosewhich,by useofbowsorothermachinesexceptfirearms,weareableto giveto ourproje_iles,it followsthat wemay,withoutsensibleerror,regardasabsolutelytruethosepropositionswhichweareaboutto provewithoutconsideringthe resistanceofthemedium.

Passingnowto the secondcase,wherewehaveto showthatthe resistanceof the air lcora rapidlymovingbodyisnot verymuchgreaterthan foronemovingslowly,ampleproofis givenby the followingexperiment.Attachto two threadsof equallength--say four or five yards--twoequal leadenballsandsuspendthemfromthe ceiling;nowpullthemasidefromtheperpendicular,the onethrough8oor moredegrees,the otherthroughnot morethanfouror fivedegrees;so that, whensetfree, the onefalls,passesthroughthe perpendicular,andde-scribeslargebut slowlydecreasingarcsof I6o,I5o, I4Odegrees,etc.;theotherswingingthroughsmallandalsoslowlydiminish-ingarcsofIO,8,6, degrees,etc.

In the firstplaceit must be remarkedthat onependulumpassesthroughits arcsofI8o0,I6O0,etc.,inthesametimethattheother swingsthroughits Io0,8°,etc.,fromwhichit followsthat the speedofthe firstball is I6 andI8 timesgreaterthanthat ofthesecond.Accordingly,if theairoffersmoreresistanceto thehighspeedthanto the low,the frequencyofvibrationinthe largearcsof I8o° or I6o0,etc.,oughttobe lessthan in thesmallarcsof zoo,8°,40,etc.,andevenlessthanin arcsofz0,orI°; but thispredictionis not verifiedby experiment;becauseiftwopersonsstart to countthevibrations,the onethe large,theother the small,they willdiscoverthat after countingtens

and

FOURTHDAY 255andevenhundredstheywillnot differbya singlevibration,notevenbyafra6tionofone.

[2781Thisobservationjustifiesthe twofollowingpropositions,

namely,thatvibrationsofverylargeandverysmallamplitudealloccupythesametimeandthat theresistanceof theairdoesnotaffectmotionsofhighspeedmorethanthoseoflowspeed,contraryto theopinionhithertogenerallyentertained.

SAoR.Onthecontrary,sincewecannotdenythat theairhindersbothofthesemotions,bothbecomingslowerandfinallyvanishing,wehaveto admitthattheretardationoccursinthesameproportionineachcase.Buthow?How,indeed,couldthe resistanceofferedto theonebodybe greaterthan thatofferedtotheotherexceptbytheimpartationofmoremomen-tumandspeed[impetoe velocitlz]to thefastbodythanto theslow?Andifthisissothespeedwithwhichabodymovesisatoncethecauseandmeasure[cagioneemisura]oftheresistancewhichit meets.Therefore,allmotions,fastor slow,arehin-deredanddiminishedinthesameproportion;a result,it seemstome,ofnosmallimportance.

S_v. Weareable,therefore,inthissecondcasetosaythattheerrors,negle_ingthosewhichareaccidental,intheresultswhichweareabouttodemonstratearesmallinthecaseofourmachineswherethevelocitiesemployedaremostlyverygreatand the distancesnegligiblein comparisonwith the semi-diameteroftheearthoroneofitsgreatcircles.

SIMP.I wouldliketo hearyourreasonforputtingthepro-jec2ilesoffirearms,i. e.,thoseusingpowder,ina differentclassfromtheprojec2ilesemployedinbows,slings,andcrossbows,onthegroundoftheirnotbeingequallysubje6tto changeandresistancefromtheair.

S_v. I amledtothisviewbytheexcessiveand,soto speak,supernaturalviolencewithwhichsuchproje6t_ilesarelaunched;for,indeed,itappearstomethatwithoutexaggerationonemightsaythatthespeedofa ballfiredeitherfroma musketorfromapieceofordnanceissupernatural.Forifsucha ballbeallowedto fallfromsomegreatelevationits speedwill,owingto the

resistance

z56 THE TWO NEW SCIENCESOF GALII,KOresistanceoftheair,notgoonincreasingindefinitely;that whichhappensto bodiesof smalldensityin fallingthroughshort.distances--Imeanthereduc°donoftheirmotionto uniformity--willalsohappento a ballof ironorleadafterit hasfallena fewthousandcubits;this terminalor finalspeed[termina_veloci_]isthe maximumwhichsuchaheavybodycannaturallyacquire

[z79]in fallingthroughthe air. ThisspeedI estimateto be muchsmallerthanthat impresseduponthe ballby the burningpow-der.

An appropriateexperimentwiUserveto demonstratethisfaCt. Froma heightof onehundredor morecubitsfirea gun[archibuso]loadedwith a lead bullet,verticallydownwardsupon a stonepavement;with the samegun shootagainstasimilarstonefroma distanceofoneortwocubits,andobservewhichofthe twoballsis the moreflattened. Nowif the ballwhichhascomefromthe greaterelevationis foundto be thelessflattenedof the two, thiswillshowthat the air hashin-deredanddiminishedthe speedinitiallyimpartedto the bulletby the powder,andthat the air willnot permita bulletto ac-quiresogreataspeed,nomatterfromwhatheightit falls;forifthe speedimpresseduponthe ball by the firedoesnot exceedthat acquiredbyit infallingfreely[naturalmente]thenits down-wardblowoughtto begreaterratherthanless.

ThisexperimentIhavenotperformed,butI amoftheopinionthat a musket-ballor cannon-shot,fallingfroma height asgreatasyouplease,willnot deliversostronga blowas it wouldiffiredintoa wallonlyafewcubitsdistant,i. e.,at sucha shortrangethat the splittingorrendingoftheairwillnotbesufficienttorobtheshotofthat excessofsupernaturalviolencegivenitbythepowder.

Theenormousmomentum[impel.o]oftheseviolentshotsmaycausesomedeformationofthetrajectory,makingthebeginningoftheparabolaflatterandlesscurvedthantheend;but,sofarasourAuthoris concerned,this is a matterof smallconsequencein praCticaloperations,themainoneofwhichisthepreparationof a tableof rangesforshotsof highelevation,givingthe dis-

tance

FOURTHDAY 257tanceattainedby the ball as a func°donof the angleof eleva-tion;andsinceshotsofthiskindarefiredfrommortars[mortart]usingsmallchargesandimpartingnosupernaturalmomentum[impetosopranaturale]theyfollowtheirprescribedpaths veryexacCtly.

But nowlet us proceedwith the discussionin whichtheAuthorinvitesus to the studyandinvestigationofthe motionofabody[impetoarelmobile]whenthatmotioniscompoundedoftwoothers;andfirstthe caseinwhichthetwoareuniform,theonehorizontal,theothervertical.

[280]THEOREMII, PROPOSITIONII

W'hen themotionofa bodyistheresultantoftwouniformmotions,onehorizontal,theotherperpendicular,thesquareof the resultantmomentumis equalto the sum of thesquaresofthetwocomponentmomenta.*

Letus imagineanybodyurgedbytwouniformmotionsandlet abrepresentthe verticaldisplacement,whilebcrepresentsthedisplacementwhich,inthesameintervalof time,takesplacein a horizontaldirec-tion. If then the distancesaband bcaretraversed,duringthe sametime-interval,_ Lwith uniformmotionsthe corresponding Fig.lO9momentawillbeto eachotherasthedistancesabandbcare toeachother;but the bodywhichisurgedby thesetwomotionsdescribesthediagonalac; its momentumisproportionalto ac.Alsothe squareof acis equalto the sumof the squaresof abandbc. Hencethe squareoftheresultantmomentumis equalto thesumofthe squaresofthetwomomentaabandbc.Q.E.D.

Shay._Atthis point thereis just oneslightdifficultywhichneedsto be clearedup; for it seemsto me that the conclusion

*Intheoriginalthistheoremreadsasfollows:"Si aliquodmobileduplicimotu_equabilimoveatur,nempeorizontaliet

perpendiculari,impetusseumomentumlationisex utroquemotucom-po.ri_eritpotentiacequalisambobusmoment#priorummotuum."

Forthejustificationofthistranslationoftheword"potentia"andoftheuseoftheadjective"resultant"seep. 266below.[Trans.]

258 THE TWO NEW SCIENCESOF GALILEOjustreachedcontradi_sa previousproposition* in whichit isclaimedthat thespeed[impel.o]ofabodycomingfroma to bisequalto that in comingfroma to c_whilenowyou concludethat thespeed[impao]atc isgreaterthanthat atb.

SALV.Both propositions,Simplicio,are true, yet therei8 agreat differencebetweenthem. Here we are speakingof abodyurgedby a singlemotionwhichi8 the resultantof twouniformmotions,whiletherewewerespeakingof twobodieseachurgedwith naturallyacceleratedmotions,one alongtheverticalab the other alongthe inclinedplaneac. Besidesthetime-intervalsweretherenot supposedto be equal,that alongthe inclineac beinggreaterthan that alongthe verticalab;but the motionsof whichwe nowspeak,thosealongab,bc,ac,areuniformandsimultaneous.

SIMP.Pardonme;I amsatisfied;praygoon.[28 ]

SALV.OurAuthornextundertakesto explainwhathappenswhenabodyisurgedby a motioncompoundedofonewhichi8horizontaland uniformand of anotherwhichis verticalbutnaturallyaccelerated;fromthesetwo componentsresultsthepath of a projecCtile,whichis a parabola. The problemis todeterminethe speed[impeto]of the projecCtileat eachpoint.Withthispurposein viewourAuthorsetsforthasfollowsthemanner,orratherthemethod,ofmeasuringsuchspeed[impeto]alongthe pathwhichis takenby a heavybodystartingfromrestandfallingwithanaturallyacceleratedmotion.

Ta o= III,PgoPosi=oIIILet the motiontake placealongthe lineab,startingfrom

restat a,andin this linechooseanypointc. Letacrepresentthe time,or the measureof the time,requiredforthe bodytofall throughthe spaceac; let ac alsorepresentthe velocity[impetusseumomentum]at c acquiredby a fall throughthedistanceac. In thelineabsele_ anyotherpointb. Theprob-lemnowis to determinethe velocityat b acquiredby a bodyin fallingthroughthe distanceab andto expressthisin termsofthe velocityatc, themeasureofwhichisthelengthac. Take

• Seep. 169 above. [Trans.]

FOURTHDAY 259as a meanproportionalbetweenac and ab. We shallprovethat the velocityat b is to that at c as the lengthas isto thelengthac. Drawthe horizontallinecd,havingtwicethe length .aof ac,and be,havingtwicethe __ .... clengthof ba. It then follows,from the precedingtheorems, sthat a bodyfallingthroughthe e [distanceac, and turned so as _-.... j , [mto movealongthe horizontalcd Fig.IiOwith a uniformspeedequalto that acquiredon reachingc

[28z1willtraversethe distancecd in the sameintervalof time asthat requiredto fallwithacceleratedmotionfroma to c. Like-wisebewillbetraversedin the sametimeas ha. But the timeof descentthroughab is as; hencethe horizontaldistancebeis alsotraversedin the timeas. Takea pointl suchthat thetimeas is to the timeac asbeis to bl;sincethe motionalongbeisuniform,the distancebl,if traversedwiththe speed[mo-mentumceleritatis]acquiredat b, willoccupythe timeac;butin thissametime-interval,ac,thedistancecdis traversedwiththe speedacquiredin c. Nowtwospeedsareto eachotherasthe distancestraversedin equalintervalsof time. Hencethespeedat c is to the speedat b as cdis to bl. Butsincedcistobeas theirhalves,namely,as cais to ha,and sincebeis to blas baisto sa;it followsthat dcisto blascaisto sa. In otherwords,the speedat c isto that at bas caisto sa,that is,asthetimeoffallthroughab.

Themethodofmeasuringthespeedofa bodyalongthedirec-tion of its fall is thus clear;the speedis assumedto increasedirecCtlyasthe time.

But beforeweproceedfurther,sincethis discussionis todealwiththe motioncompoundedof a uniformhorizontaloneandoneacceleratedverticallydownwards--thepath of a pro-je&ile,namely,a parabola--itisnecessarythat wedefinesomecommonstandardby whichwemayestimatethe velocity,ormomentum[velocitatem,impetumseumomentum]of both mo-tions

z6o THE TWO NEW SCIENCESOF GALII.V.Otions;and sincefromthe innumerableuniformvelocitiesoneonly,andthat notsele&edat random,isto becompoundedwitha velocityacquiredby naturallyacceleratedmotion,I canthinkofnosimplerwayofsele&ingandmeasuringthisthantoassumeanotherof the samekind.* For the sakeof clearness,drawthe verticalline_ to meetthe hor_ntal linebc..dc isthe heightandbcthe amplitudeof the semi-parabolaab,whichisthe resultantof the twomotions,onethat of a bodyfalling

[283]from rest at a, throughthe distanceac, wlth naturallyac-celeratedmotion,the otherauniformmotionalongthehorizon-

tal ad. The speedacquiredat c by a fall.9throughthe distanceac is determinedby

the heightac;for the speedof a bodyfall-ingfromthe sameelevationis alwaysoneandthe same;but alongthehorizontalonemaygivea bodyan infinitenumberof uni-

_; _ a_formspeeds. However,inorderthat Imay

[_ sele&oneout of thismultitudeand sepa-

rate it fromthe rest in a perfectlydefinitemanner,I willextendthe heightcaupwardsto e just as faras is necessaryandwillcallthisdistanceae the "sublimity." Imaginea bodyto fallfromrestat e;it isclearthatwemaymakeits terminalspeedat a the

_: ¢ same as that with whichthe samebodyFig.ix1 travels alongthe horizontalline ad; this

speedwillbe suchthat, in the timeofdescentalongea,it willdescribea horizontaldistancetwicethe lengthof ea. Thispreliminaryremarkseemsnecessary.

Thereaderisremindedthat aboveI havecalledthehorizontallinecbthe "amplitude"of the semi-parabolaab; the axisacofthisparabola,I havecalledits "altitude";but thelineeathefallalongwhichdeterminesthe horizontalspeedI havecalledthe "sublimity." These matters havingbeen explained,Iproceedwiththedemonstration.

*Galileohereproposestoemployasastandardofvelocitytheterminalspeedofabodyfallingfreelyfromagivenheight.[Trans.]

FOURTHDAY 26xSAGR.Allowme,please,to interruptin orderthat I may

pointoutthe beautifulagreementbetweenthis thoughtof theAuthorand the viewsof Plato concerningthe originof thevariousuniformspeedswithwhichtheheavenlybodiesrevolve.The latter chanceduponthe ideathat a bodycouldnot passfromrest to anygivenspeedandmaintainit uniformlyexceptbypassingthroughallthedegreesofspeedintermediatebet_veenthegivenspeedandrest. Platothoughtthat God,afterhavingcreatedthe heavenlybodies,assignedthem the properanduniformspeedswithwhichtheywereforeverto revolve;andthat Hemadethemstartfromrestandmoveoverdefinitedis-tances undera natural and recCtilinearaccelerationsuch asgovernsthe motionof terrestrialbodies. He addedthat oncethesebodieshadgainedtheirproperandpermanentspeed,theirredtilinearmotionwasconvertedintoa circularone,theonly

[2841motioncapableofmaintaininguniformity,a motionin whichthe bodyrevolveswithouteitherrecedingfromor approachingits desiredgoal. ThisconceptionistrulyworthyofPlato;andit isto be all themorehighlyprizedsinceitsunderlyingprinci-plesremainedhiddenuntil discoveredby ourAuthorwhore-movedfromthemthemaskandpoeticaldressandsetforththeideain corre_historicalperspe&ive.In viewofthe fa_ thatastronomicalsciencefurnishesus suchcompleteinformationconcerningthe sizeof the planetaryorbits,the distancesofthesebodiesfromtheircentersofrevolution,andtheirvelocities,I cannothelpthinkingthat ourAuthor(towhomthis ideaofPlatowasnotunknown)hadsomecuriosityto discoverwhetheror nota definite"sublimity"mightbe assignedto eachplanet,suchthat, if it wereto startfromrestat thisparticularheightandtofallwithnaturally'acceleratedmotionalongastraightline,andwerelaterto changethe speedthusacquiredintouniformmotion,the sizeofitsorbitanditsperiodofrevolutionwouldbethosea_uallyobserved.

SALv.I thinkI rememberhishavingtoldmethatheoncemadethe computationandfounda satisfactorycorrespondencewithobservation.But he did notwishto speakof it, lestin

view

z6z THE TWO NEW SCIENCESOF C_T,IT,I_Oviewofthe odiumwhichhismanynewdiscoverieshadalreadybroughtuponhim,this mightbe addingfuelto the fire. Butifanyonedesiressuchinformationhecanobtainit forhimselffromthetheorysetforthinthepresen_treatment.

Wenowproceedwiththematterin hand,whichis to prove:PROBLEMI, PROPOSITIONIV

TodeterminethemomentumofaprojecCtileateachparticularpointin its givenparabolicpath.

Let beebe the semi-parabolawhoseamplitudeis cd andwhoseheightisdb,whichlatterextendedupwardscutsthe tan-gentof the parabolaca in a. Throughthe vertexdrawthehorizontallinebi parallelto cd. Nowif the amplitudecd isequalto the entireheightda, then bi willbe equalto baandalsoto bd;and if wetake abas the measureof the time re-quiredforfallthroughthe distanceabandalsoof themomen-tumacquiredat b in consequenceofits fallfromrestat a, thenif.weturn intoa horizontaldirectionthe momentumacquiredby fallthroughab[impetumab]thespacetraversedin the sameintervaloftimewillberepresentedbydcwhichistwicebi. Buta bodywhichfallsfromrestat b alongthe lineM willduringthe sametime-intervalfallthroughthe heightof the parabola

[z851bd. Hencea bodyfallingfromrestata,turnedintoa horizontaldirectionwith the speedabwilltraversea spaceequalto dc.Nowifonesuperposesuponthismotiona fallalongbd,travers-ingthe heightbdwhilethe parabolabcis described,thenthemomentumofthe bodyat the terminalpointc is the resultantof a uniformhorizontalmomentum,whosevalueis representedby ab,andofanothermomentumacquiredby fallfromb to theterminalpointd or c;thesetwomomentaare equal. If, there-fore,wetake ab to be the measureof oneof thesemomenta,say,the uniformhorizontalone, thenbf,whichis equalto bd,willrepresentthe momentumacquiredat d or c; and ia willrepresentthe resultantofthesetwomomenta,that is,the totalmomentumwithwhichthe proje_ile,travellingalongthe pa-rabola,strikesat c.

With

FOURTHDAY . z63Withthis inmindletus take anypointontheparabola,say

e, and determinethe momentumwith whichthe projectilepassesthat point. Drawthe horizontalefandtakebga meanproportionalbetweenbdand bf. Nowsinceab,or bd, is as-sumedtObe the measureof the ¢_time andof the momentum[too- AIrnentumvelocitatis]acquiredbyfall- J/[ingfromrestat b throughthedis- // [tance bd, it followsthat bgwill , / / Imeasurethe time and also the _o__.._.._gmomentum[impetus]acquiredat f J_-- ].by fallfromb. If thereforewelay _ [7offbo,equaltobg,the diagonalline _" - ]qjoininga ando willrepresentthe / q'• C"momentumat thepointe;because / [dthe lengthabhasbeenassumedtorepresent the momentumat b Fig.ii2which,afterdiversionintoa horizontaldirection,remainscon-stant; andbecausebomeasuresthe momentumat f or e, ac-quiredby fall,fromrestat b,throughthe heightbf. But thesquareof aoequalsthe sumofthe squaresofabandbo. Hencethe theoremsought.

SAcg.The mannerin whichyou compoundthesedifferentmomentato obtaintheir resultantstrikesme as sonovelthatmy mind is left in no smallconfusion.I do not referto thecompositionof twouniformmotions,evenwhenunequal,andwhenone takes placealonga horizontal,the other alongaverticaldire_ion;becausein this caseI am thoroughlycon-vincedthat theresultantisamotionwhosesquareisequalto thesum of the squaresof the two components.The confusionariseswhenoneundertakesto compounda uniformhorizontalmotionwith a verticalonewhichis naturallyaccelerated.Itrust, therefore,Wemaypursuethisdiscussionmoreat length.

[286]Sn_P.AndI needthisevenmorethanyousinceI amnotyet

as clearin my mindasI oughtto be concerningthosefunda-mentalpropositionsuponwhichthe othersrest. Evenin the

case

264 THE TWO NEWSCIENCES OF GALTT.F.Ocaseof the two uniformmotions,one horizontal,the otherperpendicular,I wishto understandbetterthe mannerinwhichyouobtaintheresultantfromthecomponents.Now,Salviati,youunderstandwhatweneedandwhatwedesire.

SALv.Your requestis altogetherreasonableand I willseewhethermylongconsiderationofthesematterswillenablemeto makethemclearto you. Butyoumustexcuseme if in theexplanationI repeatmanythingsalreadysaidby theAuthor.

Concerningmotionsand theirvelocitiesor momenta[mov/-mentie lorvelocit_o imperilwhetheruniformor naturallyac-celerated,onecannotspeakdefinitelyuntilhe hasestablisheda measureforsuchvelocitiesandalsofortime. Asfortimewehavethealreadywidelyadoptedhours,firstminutesandsecondminutes. Soforvelocities,just asforintervalsoftime,thereisneedof a commonstandardwhichshallbe understoodandacceptedby everyone,andwhichshallbethe sameforall. Ashasalreadybeenstated,theAuthorconsidersthevelocityofafreelyfallingbodyadaptedto thispurpose,sincethisvelocityincreasesaccordingto the samelawin allparts of the world;thusforinstancethe speedacquiredbya leadenballofa poundweightstartingfrom rest and fallingverticallythroughtheheightof,say,a spear'slengthis the samein allplaces;it isthereforeexcellentlyadaptedforrepresentingthe momentum[impeto]acquiredin thecaseofnaturalfall.

It still remainsfor us to discovera methodof measuringmomentumin thecaseofuniformmotionin suchawaythat allwhodiscussthe subje&willformthe sameconceptionof itssizeand velocity[grander._e veloci@.This willpreventonepersonfromimaginingit larger,anothersmaller,than it reallyis; so that in the compositionof a givenuniformmotionwithonewhichis accelerateddifferentmenmaynotobtaindifferentvaluesfor the resultant. In orderto determineand representsuchamomentumandparticularspeed[imt_etoevelocithpartico-late]ourAuthorhas foundno bettermethodthan to use themomentumacquiredby abodyinnaturallyacceleratedmotion.

[2871The speedof a bodywhichhas in this manneracquiredany

momentum

FOURTHDAY z65momentumwhateverwill,whenconvertedintouniformmotion,retainpreciselysucha speedas,duringa time-intervalequaltothat ofthe fall,willcarrythebodythrougha distanceequaltotwicethat of the fall. But sincethismatter is onewhichisfundamentalin our discussionit is wellthat wemakeit per-fec°dyclearbymeansofsomeparticularexample.

Letusconsiderthespeedandmomentumacquiredby abodyfallkngthroughthe height,say,of a spear[picca]asa standardwhichwemay use in the measurementof other speedsandmomentaas occasiondemands;assumefor instancethat thetimeofsuchafallisfourseconds[minutisecondid'ora];nowinorderto measurethe speedacquiredfroma fallthroughanyotherheight,whethergreateror less,onemust not concludethat thesespeedsbearto oneanotherthe sameratioas theheightsoffall;forinstance,itisnottruethat a fallthroughfourtimesa givenheightconfersa speedfourtimesasgreatasthatacquiredby descentthroughthe givenheight;becausethespeedof a naturallyacceleratedmotiondoesnotvary in pro-portionto the time. Ashasbeenshownabove,theratioofthespacesisequalto thesquareoftheratioofthetimes.

ILthen,as is often donefor the sakeof brevity,we takethe samelimitedstraightlineas themeasureofthe speed,andof the time,andalsoofthe spacetraversedduringthat .atime,it followsthat the durationof fall andthe speedacquiredby the samebodyin passingoveranyotherdistance,isnotrepresentedbythisseconddistance,but •f'by a meanproportionalbetweenthe two distances.This I canbetterillustrateby an example.In thever-ticallineac,lay offtheportionabto representthe dis-tancetraversedby a bodyfallingfreelywith acceler-atedmotion:thetimeof fallmayberepresentedbyanylimitedstraightline,but forthesakeofbrevity,weshall ¢representit bythe samelengthab;thislengthmayalsobeemployedas a measureofthemomentumandspeedFig.II 3acquiredduringthe motion;in short,let abbe a measureofthevariousphysicalquantitieswhichenterthisdiscussion.

Havingagreedarbitrarilyuponab as a measureof thesethree

z66 THE TWO NEW SCIENCESOF GALILFOthreedifferentquantities,namely,space,time,andmomentum,ournext task is to findthe time requiredforfall througha[z88]givenverticaldistanceac,alsothe momentumacquiredat theterminalpointc,bothofwhichare to be expressedin termsofthetimeandmomentumrepresentedbyab. Thesetworequiredquantitiesare obtainedby layingoffad,a meanproportionalbetweenabandac;inotherwords,the timeoffallfroma to c isrepresentedby adon the samescaleon whichweagreedthatthe timeoffallfroma to bshouldbe representedbyab. In likemannerwe may say that the momentum[impetoo gradodiveloci_]acquiredat c isrelatedto that acquiredatb,inthesamemannerthat thelineadis relatedto ab,sincethevelocityvariesdirec°dyas the time,a conclusion,whichalthoughemployedas a postulate in PropositionIII, is hereamplifiedby theAuthor.

This point beingclearandwell-establishedwe passto theconsiderationof the momentum[impeto]in the caseof twocompoundmotions,oneof whichis compoundedof a uniformhorizontaland a uniformverticalmotion,whilethe other iscompoundedofauniformhorizontalanda naturallyacceleratedverticalmotion. If both componentsare uniform,andoneatrightanglesto theother,wehavealreadyseenthat the squareofthe resultantis obtainedby addingthe squaresof the compo-nents[19.257]aswillbeclearfromthe followingillustration.

Letus imaginea bodyto movealongthe verticalabwith auniformmomentum[impeto]of 3, andon reachingb to move

a toward c with a momentum[veloc#Red

irnpeto]of 4, 8othat duringthe sametime-

intervalit willtraverse3 cubitsalongtheverticaland4 alongthe horizontal. Buta

c [_particlewhichmoveswiththe resultantve-Fig.II 4 locity[velocitY]will,in the sametime,trav-

ersethediagonalac,whoselengthis not 7 cubits--thesumofab(3)andbc (4)wbut5,whichis in_otenzaequalto the sumof3 and4, that is,the squaresof3 and4 whenaddedmakezS,whichisthe squareofac,andisequalto thesumofthe squaresof

FOURTHDAY 267ofabandbc. Henceacis representedby thesidc orwemaysaytheroot--ofasquarewhoseareais25,namelyS-

Asa fixedandcertainruleforobtainingthemomentumwhich[2891

results.fromtwo uniformmomenta,one vertical,the otherhorizontal,wehavethereforethe following:takethe squareofeach,addthesetogether,andextra(tthesquarerootofthesum,whichwillbethe momentumresultingfromthe two. Thus,inthe aboveexample,the bodywhichin virtueof its verticalmotionwouldstrikethe horizontalplanewith a momentum[for_]of3,wouldowingto itshorizontalmotionalonestrikeatcwithamomentumof4;but if thebodystrikeswitha momen-tumwhichisthe resukantofthesetwo,itsblowwillbethatofabodymovingwithamomentum[velocitaeforza]of5;andsuchablowwillbe the sameat allpointsofthe diagonalac,sinceitscomponentsarealwaysthe sameandneverincreaseordiminish.

Letus nowpassto the considerationofa uniformhorizontalmotioncompoundedwiththe verticalmotionofa freelyfallingbodystartingfromrest. It is at onceclearthat the diagonalwhichrepresentsthe motioncompoundedofthesetwois notastraightline,but, as hasbeendemonstrated,a semi-parabola,in whichthe momentum[impeto]is alwaysincreasingbecausethe speed[velocita]of theverticalcomponentis alwaysincreas-ing. Wherefore,to determinethe momentum[impeto]at anygivenpoint in the parabolicdiagonal,it is necessaryfirst tofixuponthe uniformhorizontalmomentum[impeto]andthen,treatingthe body as one fallingfreely,to find the verticalmomentumat the givenpoint;this latter can be determinedonlybytakingintoaccountthe durationoffall,a considerationwhichdoesnotenterintothe compositionoftwouniformmo-tionswherethe velocitiesandmomentaarealwaysthe same;but herewhereoneof the componentmotionshasan initialvalueofzeroandincreasesitsspeed[velodta]indirecCtproportionto the time,it followsthat the timemustdeterminethe speed[veloci_]at the assignedpoint. It onlyremainsto obtainthemomentumresultingfromthesetwocomponents(asinthe caseofuniformmotions)byplacingthe squareoftheresultantequal

to

268 THE TWO NEW SCIENCESOF GALII.F.Oto the sumof the squaresof the two components.But hereagainit isbetterto illustratebymeansofanexample.

On the verticalaclayoff:anypoetionabwhichweshallem-ployasameasureofthe spacetraversedbya bodyfallingfreelyalongthe perpendicular,likewiseas ameasureof the timeandalsoof the speed_gradodi velocitY]or,wemay say,ofthe mo-menta[impet,].It is at onceclearthat if themomentumof a

[29o]bodyat b, afterhavingfallenfromrest at a, be divertedalongthehorizontaldirec'tionbd,withuniformmotion,its speedwillbe suchthat, duringthe time-intervalab, it willtraverseadistancewhichisrepresentedbythelineMandwhichistwiceas

great as ab. Now chooseapoint c, suchthat bcshallbe

J equal to ab, and throughc

draw the line ce equal andparallel to bd; throughthepointsb ande drawthe pa-d rabolabei. Andsince,during

.1_--./_ the time-intervalab,the hori-zontaldistanceMorce,doublethelengthab,istraversedwithe othe momentumab,and since

_ ' duringan equaltime-intervaltheverticaldistancebcistrav-ersed,thebodyacquiringat c

'./_ . - ¢_a momentumrepresentedbyFig.II5 the samehorizontal,bd,k fol-

lowsthat duringthe timeabthebodywillpassfrombto ealongthe parabolabe,andwillreachewitha momentumcompoundedof twomomentaeachequalto ab. Andsinceoneof theseishorizontalandtheothervertical,thesquareoftheresultantmo-mentumisequalto the sumofthe squaresofthesetwo compo-nents,i. e.,equalto twiceeitheroneof them.

Therefore,ifwelayoff:the distancebf,equalto ha,anddrawthe diagonalaf, it followsthat the momentum[impetoe per-cossa]atewillexceedthat ofabodyat bafterhavingfallenfrom

a,

FOURTHDAY z69a, orwhatisthe samething,willexceedthe horizontalmomen-tum [percossadell'impeto]alongbd,in the ratioofaf to ab.

Supposenowwe choosefor the heightof fall a distancebowhichis notequalto but greaterthanab,andsupposethat bgrepresentsa meanproportionalbetweenbaandbo;then,stillre-tainingbaas ameasureofthedistancefallenthrough,fromrestat a, to b,alsoasa measureofthe timeandofthemomentumwhichthe fallingbodyacquiresat b,it followsthat bgwillbethe measureof the timeandalsoof themomentumwhichthebodyacquiresinfallingfrombtoo. Likewisejustasthemomen-tumabduringthe timeabcarriedthe bodyadistancealongthehorizontalequalto twiceab,sonow,duringthe time-intervalbg,the bodywillbe carriedin a horizontaldirecCtionthroughadistancewhichis greaterin the ratioof bgto ba. Lay offlbequalto bganddrawthe diagonala/, fromwhichwehaveaquantitycompoundedoftwovelocities[imperilonehorizontal.,theothervertical;thesedeterminetheparabola.Thehorizontalanduniformvelocityisthat acquiredat b infallingfroma; theotheristhat acquiredat o,or,wemaysay,at i,byabodyfallingthroughthe distancebo,duringa timemeasuredby thelinebg,

[29z]whichllnebgalsorepresentsthe momentumofthe body. Andin likemannerwemay,by takingameanproportionalbetweenthe two heights,determinethe momentum[bnpeto]at theextremeendof the parabolawherethe heightis lessthanthesublimityab;thismeanproportionalis to be drawnalongthehorizontalin placeof bf,andalsoanotherdiagonalin placeofaf,whichdiagonalwillrepresentthemomentumat the extremeendoftheparabola.

To what has hithertobeen saidconcerningthe momenta,blowsor shocksof projecCtiles,wemustadd anothervery im-portantconsideration;to determinetheforceandenergyoftheshock[for_ edenergiadellapercossa]it isnot sufficientto con-sideronlythespeedoftheprojecCtiles,butwemustalsotakeintoaccountthe nature and conditionof the targetwhich,in nosmalldegree,determinestheefficiencyofthe blow. Firstof allit is wellknownthat the targetsuffersviolencefromthe speed

[veloci_]

z7o THE TWO NEW SCIENCESOF _ALIT,EO[veloci_]of the projectilein proportionas it partlyor entirelystopsthemotion;becauseifthe blowfallsuponanobjectwhichyieldsto theimpulse[velocit_delperc_ziente]withoutresistancesucha blowwillbe ofno effect;likewisewhenoneattackshisenemywitha spearandovertakeshimat aninstantwhenheisfleeingwith equalspeedtherewillbe no blowbut merelyaharmlesstouch. But if the shockfallsuponan objectwhichyieldsonlyin part thenthe blowwillnot haveks fulleffe&,but the damagewillbe in proportionto the excessofthe speedoftheprojectileoverthat oftherecedingbody;thus,forexam-ple,if the shotreachesthe targetwitha speedof IOwhilethelatter recedeswith a speedof 4, the momentumand shock[impetoepercossa]willbe representedby 6. Finallythe blowwillbeamaximum,in sofarastheproje&ileisconcerned,whenthe targetdoesnot recedeat allbut ifpossiblecompletelyre-sistsandstopsthe motionofthe proje&ile. I havesaidin sofar as theproje_ileis concernedbecauseif the target shouldapproachthe projecetilethe shockofcollision[colpoel'incontro]wouldbe greaterin proportionasthe sumof the twospeedsisgreaterthanthatoftheprojectilealone.

Moreoverit isto beobservedthat the amountofyieldinginthe targetdependsnot onlyuponthe qualityof the material,as regardshardness,whetherit be ofiron,lead,wool,etc.,but

[z9z]alsouponits position. If the positionis suchthat the shotstrikesit at rightangles,themomentumimpartedby theblow[impetodelcolpo]willbe a maximum;but if the motionbeoblique,that is to say slanting,the blowwillbe weaker;andmoreandmoresoin proportionto theobliquity;for,nomatterhowhard the materialof the targetthus situated,the entiremomentum[impetoe moto]of the shotwillnot be spent andstopped;the projecCtilewillslideby and will,to someextent,continueits motionalongthe surfaceof the opposingbody.

Allthathasbeensaidaboveconcerningtheamountofmomen-tum in the projecCtileat the extremityof the parabolamustbeunderstoodtoreferto ablowreceivedonalineat rightanglestothisparabolaoralongthe tangentto theparabolaat thegiven

point

FOURTHDAY z7,point;for, eventhoughthe motionhas two components, onehorizontal,the othervertical,neitherwillthemomentumalongthe horizontalnor that upon a planeperpendicularto thehorizontalbe a maximum,sinceeachof thesewillbe receivedobliquely.

SAcR.Yourhavingmentionedtheseblowsandshocksrecallsto myminda problem,or rathera question,in mechanicsofwhichno authorhasgivena solutionor saidanythingwhichdiminishesmy astonishmentor evenpartlyrelievesmy mind.

My difficultyand surpriseconsistin not beingableto seewhenceanduponwhatprincipleis derivedtheenergyandim-menseforce[energiaeforzaimmensa]whichmakesitsappearancein a blow;for instancewe seethe simpleblowof a hammer,weighingnot morethan 8 or IOlbs.,overcomingresistanceswhich,withouta blow,wouldnotyieldto theweightofabodyproducingimpetusby pressurealone,eventhoughthat bodyweighedmanyhundredsofpounds. I wouldliketo discoveramethodofmeasuringtheforce[forza]ofsuchapercussion.I canhardlythinkit infinite,butinclineratherto theviewthat it hasits limitand canbe counterbalancedand measuredby otherforces,suchasweights,orbyleversorscrewsorothermechanicalinstrumentswhichareusedtomultiplyforcesin amannerwhichI satisfacCtorilyunderstand.

SAnv.Youarenot alonein yoursurpriseat thiseffecCtor inobscurityasto thecauseofthisremarkableproperty. I studiedthismattermyselffora whilein vain;butmyconfusionmerelyincreaseduntilfinallymeetingourAcademicianI receivedfrom

[293]himgreatconsolation.Firsthetoldmethat he alsohadforalongtimebeengropinginthe dark;butlaterhesaidthat,afterhavingspentsomethousandsof hoursin speculatingandcon-templatingthereon,he hadarrivedat somenotionswhicharefar removedfromourearlierideasandwhichare remarkablefortheirnovelty.AndsincenowI knowthat youwouldgladlyhearwhatthesenovelideasareI shallnotwaitforyouto askbut promisethat, as soonas our discussionof projectilesiscompleted,I willexplainall thesefantasies,or if you please,

vagaries

z7z THE TWO NEW SCIENCESOF GALILEOvagaries,as far as I can recallthem fromthe wordsof ourAcademician.In the meantimeweproceedwiththe proposi-tionsoftheauthor.

PROPOSITIONV, PROBLEM

Havinggivenaparabola,findthepoint,initsaxisextendedupwards,fromwhichaparticlemustfallinordertodescribethissameparabola.

Letabbe the givenparabola,hb its amplitude,andheits axisextended.Theproblemisto findthepointefromwhichabodymustfallinorderthat,afterthemomentumwhichit acquiresata hasbeendivertedintoa horizontaldire&ion,it willdescribethe parabolaab. Drawthe horizontalag,parallelto bh, and

e havinglaidoffaf equal to ah,I drawthe straightlinebfwhich] willbe a tangentto the parab-J it"ola at b, andwillinterse&the

/[_" horizontalagatg:chooseesuch[ that agwillbe a meanpropor-

eq/ ! tionalbetweena/and ae. Now__.._..._ a I saythat e is the pointabove

] sought.That is,if a bodyfalls/__'_- It fromrestat thispoint e,andif

"" the momentumacquiredat theFig.If6 pointa be divertedintoa hori-

zontal dire&ion,and compoundedwith the momentumac-quiredat h in fallingfromrestat a, thenthe bodywilldescribethe parabolaab. Forifweunderstandeato be the measureofthe timeoffallfrome to a, andalsoofthemomentumacquiredat a,thenag(whichisa meanproportionalbetweeneaandaf)willrepresentthe time andmomentumof fall fromf to a or,whatisthe samething,froma toh;andsinceabodyfallingfrome,duringthe timeea,will,owingto the momentumacquiredata, traverse at uniformspeeda horizontaldistancewhichistwaceea,it followsthat, the bodywillif impelledby the samemomentum,duringthe time-intervalag traversea distanceequalto twiceagwhichis the halfofbh. This istrue because,

in

FOURTHDAY 273in the caseof uniformmotion,the spacestraversedvary di-recCtlyas thetimes. Andlikewiseif themotionbeverticalandstart fromrest, the bodywilldescribethe distanceah in the

[2941timeag. Hencethe amplitudebhandthe altitudeaharetrav-ersedby a bodyin the sametime. Thereforethe parabolaabwillbe describedby a bodyfallingfromthe sublimityof e.

Q. E. F.

COROLLARYHenceit followsthat halfthe base,oramplitude,ofthesemi-

parabola(whichisone-quarteroftheentireamplitude)isameanproportionalbetweenits altitudeandthe sublimityfromwhicha fallingbodywilldescribethissameparabola.

PRoPosrrioNVI, PROBLEMGiventhe sublimityandthealtitudeof aparabola,to finditsamplitude.

Let the lineac,in whichlie the givenaltitudecbandsub-limity ab, be perpendicularto Fathehorizontallinecd. Theprob-lem is to find the amplitude,alongthe horizontalcd,of thesemi-parabolawhichis describedwith the sublimitybaand alti-tude bc. Lay off cd equal to btwicethe meanproportionalbe-tweencband ba. Then cdwillbe the amplitudesought,as isevidentfromthe precedingprop-d -: c_osition. Fig. xx7

THEOREM. PROPOSITIONVII

I_ projec°dlesdescribesemi-parabolasof the sameampli-tude,themomentumrequiredto describethatonewhoseamplitudeis doubleits altitudeis lessthan that requiredforanyother. Let

_74 THE TWO NEW SCIENCESOF GALILFOLet bdbe a semi-parabolawhoseamplitudecd is doubleitsaltitudecb;onits axisextendedupwardslayoffbaequalto ksahitudebc. Drawthe linead whichwillbe a tangentto theparabolaat d andwillcut the horizontallinebeat the pointe,makingbeequalto bcandalsoto ha. It isevidentthat thisparabolawillbedescribedbyaprojedtilewhoseuniformhorizon-talmomentumisthat whichit wouldacquireat binfallingfromrestat a andwhosenaturallyacceleratedverticalmomentumisthat ofthebodyfallingto c,fromrestat b. Fromthisit follows

f

C , C

Fig. I:8

that the momentumat the terminalpoint d, compoundedofthesetwo,is representedby the diagonalae,whosesquareisequalto the sumof the squaresofthe two components.Nowletgdbe anyotherparabolawhateverhavingthe sameampli-tudecd,butwhosealtitudecgis eithergreateror lessthanthealtitudebc. Let /u/ be the tangent cutting the horizontal[29S1throughg at k. Sele_a pointl suchthat hg:gk=gk:gl.Thenfroma precedingproposition[V],k followsthat glwillbe the

height

FOURTHDAY 275heightfromwhicha bodymust fallin orderto describetheparabolagd.

Let gmbe a meanproportionalbetweenabandgl;thengmwill[Prop.IV]representthe timeandmomentumacquiredatg.bya fallfroml; forabhasbeenassumedasa measureofbothume and momentum.Againlet gnbe a meanproportionalbetweenbcandcg;it willthenrepresentthe timeandmomen-tumwhichthe bodyacquiresat c in fallingfromg. If nowwejoinm andn, this linemnwillrepresentthemomentumat d ofthe projec°dletraversingthe paraboladg;whichmomentumis,I say,greaterthan that of the projec°ciletravellingalongtheparabolabdwhosemeasurewasgivenbyae. Forsincegnhasbeentakenasameanproportionalbetweenbcandgc;andsincebcis equalto beandalsoto kg (eachof thembeingthehalfofdc__)it follows__that cg:gn=gn:gk,and ascgor (hg)is to gk so__isng_ to gk2:but by constru_ionhg:gk=gk:gl.Henceng2:gk_=gk:gl. But gk:gl=-gk_:gm_,since gmis a meanpropor-tionalbetweenkg andgl. Thereforethe threesquaresng,kg,mgforma continuedproportion,g-n_:_ =gk2:gin._ Andthesumof the twoextremeswhichis equalto the squareof mnisgreaterthan twicethe squareof gk;but the squareof aeisdoublethesquareofgk. Hencethesquareofmnisgreaterthanthesquareofaeandthelengthmnisgreaterthanthelengthae.

Q.E.D.

[z96]COROIJARY

Converselyit is evidentthat lessmomentumwillbe requiredto sendaproje6tilefromtheterminalpointd alongtheparabolabdthanalonganyotherparabolahavingan elevationgreaterorlessthan that of the parabolabd,forwhichthe tangentat dmakesan angleof 45° with the horizontal.Fromwhichitfollowsthat ifprojec°dlesarefiredfromthe terminalpointd, allhavingthesamespeed,buteachhavingadifferentelevation,themaximumrange,i. e.,amplitudeofthe semi-parabolaor oftheentireparabola,willbe obtainedwhentheelevationis45°:the

other

Z76 THE TWO NEW SCIENCESOF GALII,V.Ot

othershots,firedat anglesgreateror lesswillhavea shorterrange.

SACR.Theforceof rigiddemonstrataonssuchas occuronlyin mathematicsfillsme with wonderand delight. Fromac-countsgivenbygunners,I wasalreadyawareofthe facetthatin the useof cannonandmortars,the mnx_rnumrange,that isthe onein whichthe shotgoesfarthest,is obtainedwhentheelevationis45° or, asthey say,at the sixthpointof the quad-rant; but to understandwhythis happensfar outweighsthemereinformationobtainedby the testimonyof othersor evenbyrepeatedexperiment.

S_u,v.Whatyousayisverytrue. Theknowledgeof asinglefa_ acquiredthrougha discoveryof its causespreparesthemindto understandandascertainotherfac%swithoutneedof recourseto experiment,preciselyas in the presentcase,wherebyargumentationalonetheAuthorproveswithcertaintythatthemaximumrangeoccurswhentheelevationis45°. Hethusdemonstrateswhathasperhapsneverbeenobservedinexperience,namely,that of othershotsthosewhichexceedorfallshortof 45° by equalamountshaveequalranges;so thatif theballshavebeenfiredoneat anelevationof 7 points,theotherat 5, theywillstrikethe levelat the samedistance:thesameistrueif theshotsarefiredat 8 andat4 points,at9 andat3,etc. Nowletushearthedemonstrationofthis.

[2971TaEOr.E_.PxoFosrnoNVIII

Theamplitudesof two parabolasdescribedby proje_ilesfiredwiththe samespeed,butat anglesofelevationwhichexceedandfall shortof 45° by equalamounts,areequalto eachother.

In the trianglemcblet the horizontalsidebcandthe verticalcm,whichforma rightangleat c,beequalto eachother;thenthe anglembcwillbea semi-rightangle;let thelinecmbepro-longedto d, sucha pointthat the twoanglesatb, namelyrobeandmbd,oneaboveandtheotherbelowthediagonalrob,shallbeequal.It isnowtobeprovedthatinthecaseoftwoparabolas

described

FOURTHDAY 277describedby twoprojecCtilesfiredfromb withthe samespeed,oneat the angleofebc,the otherat the angleofdbc,theiram-plitudeswillbe equal. Nowsincethe externalanglebmcisequalto thesumofthe internalanglesrndbanddbmwemayalsoequateto themthe anglembc;but ifwere-placethe angledbmby robe,thenthissame danglembcis equalto the tworobeandbdc: /and ifwe subtraCtfrom each sideof thisequationtheanglerobe,wehavetheremain- [derbdcequalto the remainderebc. Hence mthe two trianglesdcbandbceare similar. _BiseCtthe straightlinesdc and ec in the /_/epointsh andf: anddrawthe lineshi andfg /JJparallelto the horizontalcb,and chooseI /Jj,_ fsuchthat dh:hl=ih:hl. Thenthe triangle_'hll _ "willbesimilarto i/M,andalsotothe triangleor - ,"_ andsinceih andgfareequal,eachbeing Fig.ii9

ofbc,itfollowsthat h/is equalto fe andalsoto fc; and ifweaddto eachofthesethecommonpartfh,it willbe seenthatchisequaltoil.

Letus nowimaginea paraboladescribedthroughthe pointshandbwhosealtitudeishcandsublimityhL Its amplitudewillbecbwhichisdoublethelengthhisincehiisameanproportionalbetweendh(orc/z)andh/. Thelinedbistangentto theparabolaat b, sincechis equalto hd. If againweimaginea paraboladescribedthroughthe pointsf andb, with a sublimityf/andaltitudefc, of whichthemeanproportionalisfg, orone-halfofcb,then,as before,willcbbe the amplitudeandthe lineebatangentat b;forefandfcareequal.

[z98]But the twoanglesdbcandebc,the anglesof elevation,differbyequalamountsfroma 45° angle. Hencefollowsthe proposi-tion.

TIIEORE_.PROPOSrrIONIX

Theamplitudesoftwoparabolasareequalwhentheiralti-tudesandsublimitiesareinverselyproportional.

Let

278 THE TWO NEW SCW.NCESOF GALII,I_.O.Let the altitudegf of the parabolafh bearto the altitudecboftheparabolaMthesameratiowhichthesublimitybabearstothe sublimityre; thenI say the amplitudehg is equalto theamplitudea_. Forsincethefirstofthesequantities,gf,bearsto

_'the secondcbthesamee ratiowhichthe third,

ha,bearsto thefourthre, it followsthat thearea of the re&angle

___ _.feisequalto that of

the re&_nglecb.ba;therefore squareswhich are equal to

g these re&anglesareFig.i2o equal to eachother.

But [byPropositionVIIthe squareof halfofghisequalto there&anglegf.fe;andthe squareof halfof cdis equalto the rec-tanglecb.ba.Thereforethesesquaresand theirsidesandthedoublesoftheirsidesareequal. But theselastare the ampli-tudes ghandcd. Hencefollowsthe proposition.

L_M__AFOgT_EVm.TOWmOPRovosrrm_Ifa straightlinebecutat anypointwhateverandmeanpro-portionalsbetweenthis lineandeachofits partsbe taken,the sumofthe squaresof thesemeanproportionalsis equalto thesquareoftheentireline.

Letthe lineabbe cutat c. ThenI saythat thesquareofthemean proportionalbetweenaband acplusthe squareofthemeanproportionalbetweenabandcbisequalto the squareofthewholelineab. This is evidentas soonas wede-scribea semicircleuponthe entirelineab,a tere&a perpendicularcdat c,and drawda Fig.i2ianddb. Forda isa meanproportionalbetweenabandacwhile

[299]db is a meanproportionalbetweenaband bc:and sincetheangleadb,inscribedin a semicircle,is a rightanglethe_sumofthe

FOURTHDAY 279thesquaresofthe linesdaanddbis equaltothe squareoftheentirelineab. Hencefollowstheproposition.

TaEOREM.PROPOSmONXThe momentum[impetusseu momentum]acquiredby aparticleat theterminalpointofany semi-parabolaisequaltothat whichit wouldacquireinfallingthroughaverticaldistanceequaltothesumofthesublimityandthealtitudeofthesemi-parabola.*

Letabbe a semi-parabolahavinga sublimitydaandan altitudeac,thesumofwhichistheperpendiculardc. NowI saythemomentumoftheparticleatbisthesame dasthatwhichitwouldacquireinfallingfreelyfrom ed to c. Letus take the lengthof dc itselfasa 2 ameasureof timeandmomentum,and layoffcf

/

equalto the meanproportionalbetweencd andda;alsolay offcea meanproportionalbetweencdandca. Nowcf isthemeasureofthetimeandof the momentumacquiredbyfall,fromrestatd, throughthe distanceda;whileceis the time jandmomentumof fall,fromrest at a, through_the distanceca;alsothe diagonalefwillrepre-_'(_* csenta momentumwhichisthe resultantof these Fig.122two,andis thereforethe momentumat the terminalpointoftheparabola,b.

And sincedchasbeencut at somepointa and sincecfandcearemeanproportionalsbetweenthewholeofcdanditsparts,daandac,it follows,fromthe precedinglemma,that the sumofthe squaresofthesemeanproportionalsisequalto thesquareof the whole:but the squareof ef is alsoequalto the sumofthesesamesquares;whenceit followsthat the lineef isequaltodc.

Accordinglythe momentumacquiredat c by a particleinfailingfromd is the sameas that acquiredat b by a particletraversingthe parabolaab. Q.E.V.

* In modemmechanicsthiswell-knowntheoremassumesthe followingform: Thespeedofa projectileatanypointis thatproducedbyafall frorathedirectrix. [Trans.]

280 THE TWO NEW SCIENCESOF GALTI.EO

COROTJ.ARY

Henceit followsthat, in the caseof allparabolaswherethesumofthe sublimityandaltitudeis a constant,themomentumat theterminalpointisa constant.

PROBLEM.PROPOSITIONXI

Giventheamplitudeandthespeed[impetus]attheter-minalpointofasemi-parabola,tofinditsaltitude.

Letthegivenspeedberepresentedbytheverticalllneab,andthe amplitudeby the horizontallinebe;it is requiredto findthe sublimityof the semi-parabolawhoseterminalspeedis aband amplitudebe. Fromwhat precedes[Cor.Prop.V] it isclearthat halfthe amplitudebcisa meanproportionalbetween[3oolthealtitudeandsublimityoftheparabolaofwhichtheterminalspeedisequal,inaccordancewiththeprecedingproposition,to

_t_the speedacquiredby a bodyinfalling

//_ fromrestat a throughthe distanceab.CThereforethe lineba mustbe cutat aJ_/f I".p°intsuchthat thereCtangleformedbyA \ I its twopartswillbe equalto the square

/ \ _ [ ofhalfbc,namelybd. Necessarily,there-/ _N_] _ fore,bdmustnot exceedthe halfof ba;

/_ forof all the rectanglesformedby parts

ofastraightlinetheoneof greatestareaisobtainedwhenthe lineisdividedintotwo equalparts. Let e be the middle

_ point of the line ab;and nowifbdbe,¢_ equalto betheproblemissolved;for bewillbe the altitudeandeathe sublimity

Fig.I_3 of the parabola. (Incidentallywe mayobservea consequencealreadydemonstrated,namely:of allparabolasdescribedwith any giventerminal speedthat forwhichthe elevationis 45° willhave the maximumamplitude.)

But supposethat bd is lessthan half of bawhichis to bedivided

FOURTH DAY 281dividedin sucha waythat theredtang|euponitspartsmaybeequalto the squareofbd. Uponeaasdiameterdescribeasemi-circleefa,in whichdrawthe chordaf,equalto bd:joinfe andlay offthe distanceegequalto re. Thenthe rectanglebg.gaplusthe squareofegwillbeequalto thesquareofea,andhencealsoto the sumofthe squaresofaf andre. If nowwesubtracCtthe equalsquaresoffe andgethereremainsthe recCtanglebg.gaequalto the squareofaf, that is,of bd,a linewhichis a meanproportionalbetweenbgandga;fromwhichit isevidentthatthe semi-parabolawhoseamplitudeis bc andwhoseterminalspeed[impetus]is representedby bahasan altitudebgand asublimityga.

If howeverwelay offbiequalto ga,thenbiwillbe the alti-tudeofthesemi-parabolaic,andiawillbeitssublimity.Fromthe precedingdemonstrationweareableto solvethe followingproblem.

PROBLEM. PROPOSITIONXII

To computeand tabulate the amplitudesof all semi-parabolaswhichare describedbyproje6tilesfiredwiththesameinitialspeed[impetus].

Fromthe foregoingit followsthat, wheneverthe sumofthe altitudeand sublimityis a constantverticalheightforany set of parabolas,theseparabolasare describedby pro-je_ileshavingthe sameinitialspeed;allverticalheightsthus

[3o ]obtainedare thereforeincludedbetweentwoparallelhorizontallines. Letcbrepresenta horizontallineandaba verticallineofequallength;drawthe diagonalac;the angleacbwillbeoneof45°;letd bethemiddlepointoftheverticallineab. Thenthesemi-paraboladc is the onewhichis determinedby the sub-limityad andthe altitudedb,whileits terminalspeedat c isthat whichwouldbeacquiredat bbyaparticlefallingfromrestat a. If nowagbedrawnparallelto bc,thesumofthe altitudeand sublimityfor any other semi-parabolahavingthe sameterminalspeedwill,in the mannerexplained,be equalto thedistancebetweenthe parallellinesagandbc. Moreover,sinceit

z8z THE TWO NEW SCIENCESOF GALILEOit has alreadybeenshownthat the amplitudesof two semi-parabolasarethesamewhentheiranglesofelevationdifferfrom45°bylikeamounts,it followsthat the samecomputationwhichisemployedforthelargerelevationwillservealsoforthesmaller.

Let us also assumeIOOOOas the• greatestamplitudefor a parabola

whoseangleofelevationis45°; thisthenwillbe the lengthof the line

o baand the amplitudeof the semi-parabolabc. This number,IOOOO,

a'is sele_edbecausein thesecalcula-tionsweemploya tableoftangents

_in whichthis is the valueof thetangentof 45°. And now,comingdownto business,drawthe straightline cemakingan acute angleecbgreaterthan acb:the problemnowis to draw the semi-parabolatowhichthe lineec is a tangentand

tr _ forwhichthe sumofthe sublimityFig.x24 and the altitudeis the distanceba.

Take the lengthof the tangent* be fromthe table of tan-gents,usingthe anglebceas an argument:letf bethe middlepointof be;nextfind a third proportionalto bf and bi (thehalf of bc),whichis of necessitygreaterthan fa.t Callthisfo. Wehavenowdiscoveredthat, for the parabolainscribed

[3ozlin the triangleecbhavingthe tangentce and the amplitudecb,the altitudeis bfandthe sublimityfo. But thetotallengthof bo exceedsthe distancebetweenthe parallelsag and cb,whileour problemwas to keepit equal to this distance:forboth the parabolasoughtand the paraboladc are described

• Thereaderwillobservethattheword"tangent"ishereusedinasensesomewhatdifferentfromthat of the precedingsentence.The"tangentec"isa linewhichtouchestheparabolaat c;butthe"tan-genteb"isthesideoftheright-angledtrianglewhichliesoppositethean - - -gleecb,a linewhoselengthisproportionaltothenumericalvalueofthetangentofthisangle.[Trans.]

t Thisfactisdemonstratedinthethirdparagraphbelow.[Tra_.l

FOURTHDAY 283by proje_ilesfiredfromcwiththesamespeed.Nowsinceaninfinitenumberof greaterand smallerparabolas,similartoeachother,maybedescribedwithinthe anglebcewemustfindanotherparabolawhichlikecdhasforthesumofitsaltitudeandsublimitythe heightha,equaltobc.

Thereforelayoffcrsothat, ob:ba=bc:cr;thencrwillbe theamplitudeofa semi-parabolaforwhichbceisthe angleofeleva-tionandforwhichthesumofthe altitudeandsublimityis thedistancebetweentheparallelsgaandcb,asdesired.Theprocessis thereforeas follows:One drawsthe tangentof the givenanglebce;takeshalfofthistangent,andaddstoit thequantity,fo,whichis a thirdproportionalto the halfofthis tangentandthe halfofbc;thedesiredamplitudecris thenfoundfromthefollowingproportionob:ba=bc:cr.For examplelet the angleecbbeoneof5o°;itstangentis11918,halfofwhich,namelybf,is 5959;halfofbcis5o00;thethirdproportionalofthesehalvesis4195,whichaddedto bfgivesthevalue10154forbo. Further,asobisto ab,that is,as 10154isto IOO_,soisbc,or IOOOO(eachbeingthe tangentof45°) to cr,whichis the amplitudesoughtandwhichhas the value9848, the maximumamplitudebeingbc,orIOOOO.Theamplitudesoftheentireparabolasaredoublethese,namely,19696and2cx:x:x).Thisisalsotheamplitudeofaparabolawhoseangleofelevationis4°0, sinceit deviatesby anequalamountfromoneof45°.

[303]SAca.In orderto thoroughlyunderstandthisdemonstration

I needto beshownhowthethirdproportionalofbfandbiis,astheAuthorindicates,necessarilygreaterthanfa.

SAT.v.Thisresultcan,I think,beobtainedas follows.Thesquareofthemeanproportionalbetweentwolinesisequalto therec°cangleformedby thesetwolines. Thereforethe squareof/z/(orof bdwhichisequaltob0mustbeequaltothe re(tangleformedby fb andthe desiredthirdproportional.This thirdproportionalisnecessarilygreaterthanfa becausethe rec°cangleformedby bfandfa is lessthanthesquareofbdbyanamountequalto thesquareofdr,asshowninEuclid,II.1. Besidesit istobeobservedthatthepointf, whichisthemiddlepointof the

tangent

284 THE TWO NEW SCIENCES OF GALII,V.Otangent eb,fallsin generalabovea and onlyonceat a; in whichcasesit is self-evidentthat the third proportionalto the halfofthe tangent and to the sublimityb/lies whollyabovea. Butthe Author has taken a case where it is not evident that thethird proportionalis alwaysgreater than fa, so that when laidoffabovethe pointf it extendsbeyondthe parallelag.

Now let usproceed. It willbeworth while,by the use ofthistable, to compute another givingthe altitudes of these semi-parabolasdescribedby projectileshavingthe sameinitialspeed.The construfftionis asfollows:

[3o41Amplitudesofseml-parabolas Altitudesof semi-parabolasde-

describedwiththe samein- scribedwiththe sameinitialitialspeed, speed.

Angleof Angleof Angleof AngleofElevation Elevation Elevation Elevation

45° 1oooo 1° 3 460 517346 9994 44° 2 I3 47 534647 9976 43 3 28 48 552348 9945 42 4 50 49 569849 99oz 41 5 76 50 586850 9848 40 6 lO8 51 6o3851 9782 39 7 15o 5z 62o75z 9704 38 8 194 53 637953 9612 37 9 245 54 654654 9511 36 Io 3oz 55 67IO55 9396 35 11 365 5_; 687356 927z 34 12 432 57 703357 9136 33 13 5o6 58 719°

58 8989 32 14 585 75o259 8829 31 15 670 _ 734860 8659 30 16 76o 61 764961 8481 29 17 855 62 7796

62 829o 28 18 I_ 63 793963 8o9o 27 29 64 807864 7880 26 2o H7o 65 821465 766o 25 zz 2285 66 834666 7431 24 zz I4oz 67 847467 7191 23 23 I527 68 859768 6944 22 24 I685 69 871569 6692 21 25 1786 70 8830

FOURTH DAY 285Amplitudesof semi-parabolas Altitudes of semi-parabolasde-

describedwith the same in- scribedwith the same initialitial speed, speed.

Angleof Angleof Angleof AngleofElevation Elevation Elevation Elevation

7°0 6428 2o° 26° I922 7l° 894071 6X57 I9 27 2o6I 72 904572 5878 x8 28 2204 73 914473 5592 x7 29 2351 74 924074 5300 I6 3° 2499 75 933075 5°00 15 31 2653 76 941576 4694 14 32 281o 77 949377 4383 13 33 2967 78 956778 4067 12 34 3128 79 963679 3746 xl 35 3289 80 969880 342o IO 36 3456 81 975581 309° 9 37 362x 82 980682 2756 8 38 3793 83 985x83 2419 7 39 3962 84 989084 2079 6 40 4132 85 992485 1736 5 41 43o2 86 995186 x39I 4 42 4477 87 997287 lO44 3 43 4654 88 998788 698 2 44 4827 89 999889 349 I 45 5oo0 9o xoooo

[305]PROBLEM. PROPOSITIONXIII

Fromthe amplitudesof semi-parabolasgivenin the pre-cedingtableto findthe altitudesofeachof theparabolasdescribedwiththesameinitialspeed.

Let bcdenotethe givenamplitude;andlet ob,the sumofthealtitudeandsublknity,bethemeasureoftheinitialspeedwhich

understoodto remainconstant. Next we must findanddeterminethealtitude,whichweshallaccomplishbysodividing

that the recCtanglecontainedbyitspartsshallbe equalto thesquareof halfthe amplitude,bc. Letf denotethis pointofdivisionandd and{be the middlepointsof obandbcrespec-tively. The squareofib isequalto the rectanglebf.fo;but thesquareof do is equalto the sumof the rectanglebf.foandthe

square

z86 THE TWO NEW SCIV.NCESOF GALILEOsquareoffd. If, therefore,fromthe squareof dowesubtractthe squareof biwhichisequalto the recCtanglebf.fo,therewillremainthe squareoffd. The altitudein question,bf, is nowobtainedby addingto this length,fd, thelinebe{.Theprocess

isthenas follows:Fromthe squareof halfof boi_whichis known,subtractthe squareof bi which• is alsoknown;take the squarerootof the re-mainder,andaddto it theknownlengthdb;then

youhavetherequiredaltitude,bf.Example.Tofindthealtitudeofa semi-paraboladescribedwithanangleofelevationof55°. Fromthe precedingtable the amplitudeis seento be

c. i _ 9396,of whichthe half is 4698,and the square_ 22o7Izo4.'When this is subtractedfrom theFig. Iz5 square of the half of bo, which is always

25,ooo,ooo,the remainderis z928796,of whichthe squareroot[3o6]

is approximatelyI7IO. Addingthis to the halfof bo,namely5oo%wehave67Iofortheakitudeofbf.

It willbeworthwhileto adda thirdtablegivingthe altitudesandsublimitiesforparabolasin whichthe amplitudeis a con-stant.

SACg.I shallbe very gladto seethis; for fromit I shalllearnthe differenceof speedandforce[degl'irnpetiedelleforze]requiredto fireprojectilesover the samerangewithwhat wecallmortarshots. This differencewill,I believe,vary greatlywiththeelevationsothat if,forexample,onewishedto employanelevationof 3° or 4°, or 87° or 88° andyet givethe ballthesamerangewhichit hadwithanelevationof45° (wherewehaveshownthe initialspeedto be a minimum)the excessof forcerequiredwill,I think,beverygreat.

SALV.Youarequkefight,sir;andyouwillfindthat in orderto performthisoperationcompletely,at allanglesofelevation,youwillhaveto makegreatstridestowardan infinitespeed.Wepassnowto the considerationofthetable.

FOURTHDAY 287[307]

Tablegivingthe altitudesandsublimitiesof parabolasofconstant amplitude,namelyIoooo,computedfor

eachdegreeofelevation.Angleof AngleofElevation Altitude Sublimity Elevation Altitude Sublimity

I ° 87 286533 460 5177 48282 175 14245o 47 5363 46623 z6z 95802 48 5553 45°24 349 71531 49 5752 43455 437 5714z 5° 5959 41966 525 47573 51 6174 40487 614 40716 5z 6399 39°68 7o2 35587 53 6635 37659 792 3H65 54 6882 3632

IO 88I 28367 55 7141 35°0II 972 25720 56 7413 3372IZ XO63 23518 57 7699 324713 II54 21701 58 8002 3123I4 I246 20056 59 8332 3°°415 1339 I8663 60 8600 2887I6 I434 I7405 6I 9020 277II7 I529 I6355 62 9403 265818 I624 15389 63 98X3 254719 1722 14522 64 XO25I 2438ZO X8ZO 13736 65 10722 233I2X 1919 13O24 66 II230 222622 2ozo I2376 67 II779 212223 2123 ii778 68 i2375 202024 2226 II230 69 I3025 I91925 2332 IO722 7° I3237 181926 2439 10253 71 I452I I72I27 2547 9814 72 I5388 162428 2658 9404 73 I6354 I52829 2772 9°20 74 I7437 I4333O 2887 8659 75 X8660 H393I 3°o8 8336 76 20054 I24632 3124 8OOX 77 21657 X_5433 3247 7699 78 23523 lO6234 3373 7413 79 _5723 97235 35oI 7141 8o 28356 88136 3633 6882 81 31569 79z37 3768 6635 82 35577 7o238 39o6 6395 83 40222 613

288 THE TWONEW SCIENCESOF GALILEOAngleof AngleofElevation Altitude Sublimity Elevation Altitude Sublimity

39° 4049 6x74 " 84° 4757z 52540 4196 5959 85 5715o 4374x 4346 575z 86 7x5o3 3494z 45oz 5553 87 95405 z6z43 4662 5362 88 14318I I7444 4828 5x77 89 286499 8745 5°oo 5°oo 9o infinita

[308]PROPOSITIONXIV

Tofindforeachdegreeofelevationthealtitudesandsub-limitiesofparabolasofconstantamplitude.

Theproblemis easilysolved.Forif weassumea constantamplitudeof IOOOO,thenhalfthe tangentat anyangleofelevationwillbethealtitude.Thus,to illustrate,a parabolahavinganangleofelevationof3o°andanamplitudeof xoooo,willhaveanaltitudeofz887,whichisapproximatelyone-halfthe tangent.Andnowthe altitudehavingbeenfound,thesublimityisderivedasfollows.Sinceit hasbeenprovedthathalftheamplitudeofa semi-parabolaisthemeanproportionalbetweenthealtitudeandsublimity,andsincethealtitudehasalreadybeenfound,andsincethesemi-amplitudeisaconstant,namely5oo%it followsthatifwedividethesquareofthesemi-amplitudebythealtitudeweshallobtainthesublimitysought.Thusinourexamplethealtitudewasfoundto bez887:thesquareof5ooois25,ooo,ooo,whichdividedby_887givestheapproximatevalueofthesublimity,namely8659.

SALV.Herewesee,firstof all,howverytrueisthestate-mentmadeabove,that,fordifferentanglesofelevation,thegreaterthedeviationfromthemean,whetheraboveorbelow,the greaterthe initialspeed[impetoe violenza]requiredtocarrytheproje_ileoverthesamerange.Forsincethespeedisthe resultantof two motions,namely,onehorizontalanduniform,the otherverticalandnaturallyaccelerated;andsincethe sumof the altitudeandsublimityrepresentsthisspeed,it isseenfromtheprecedingtablethatthissumis a

minimum

FOURTH DAY _89minimumforan elevationof 45° wherethe altitudeandsub-]imit_are equal,namely,each 5ooo;and their sum ioooo.Butifwechooseagreaterelevation,say5o°,weshallfindthealti-tude5959,andthesublimity4196,givingasumo{1o155;in likemannerweshallfindthat thisispreciselythevalueofthespeedat 4°0elevation,both anglesdeviatingequallyfromthemean.

Secondlyit is to be notedthat, whileequalspeedsare re-quiredforeachof twoelevationsthat areequidistantfromthemean,thereisthiscuriousalternation,namely,thatthealtitudeand sublimityat the greaterelevationcorrespondinverselytothe sublimityandaltitudeat the lowerelevation.Thusin the

bog]precedingexamplean elevationof 5o° givesan altitudeof5959anda sublimityof 4196;whilean elevationof 4°ocorrespondsto an altitudeof4196anda sublimityof5959-Andthisholdstrue in general;but it is to be rememberedthat, in ordertoescapetediouscalculations,no accounthas been taken offra_ionswhichare of littlemomentin comparisonwith suchlargenumbers.

SAc_.I note alsoin regardto the two componentsof theinkialspeed[irnpeto]that the higherthe shotthe lessis thehorizontalandthegreaterthe verticalcomponent;ontheotherhand,at lowerelevationswherethe shotreachesonlya smallheightthe horizontalcomponentof the initialspeedmust begreat. In the caseof a projecCtilefiredat an elevationof9o°,I quiteunderstandthat all the force[forza]in theworldwouldnotbesufficienttomakeitdeviateasinglefinger'sbreadth{romthe perpendicularandthat it wouldnecessarilyfall backintoits initialposition;but in the caseofzeroelevation,whentheshotis firedhorizontally,I am notsocertainthat someforce,lessthaninfinite,wouldnotcarrytheproje_ilesomedistance;thUSnot evena cannoncanfirea shotin a pedec°dyhorizontaldired_don,oraswesay,pointblank,that is,withnoelevationatall. HereI admitthereissomeroomfordoubt. ThefacetI donot denyoutright,becauseof anotherphenomenonapparentlyno lessremarkable,but yet onefor whichI haveconclusiveevidence.Thisphenomenonis the impossibilityof stretching

a

290 THE TWO NEW SCIENCESOF GALILEOa ropeinsuchawaythat it shallbeat oncestraightandparallelto the horizon;the fac°cisthat the cordalwayssagsandbendsandthat no forceis sufficientta stretchit peffecCtlystraight.

SALV.In this caseof the ropethen, Sagredo,you ceasetowonderat thephenomenonbecauseyouhaveitsdemonstration;but ifweconsiderit withmorecarewemaypossiblydiscoversomecorrespondencebetweenthecaseofthegunandthatofthestring. The curvatureof thepathoftheshotfiredhorizontallyappearstoresultfromtwoforces,one(thatoftheweapon)drivesit horizontallyandtheother(itsownweight)drawsitverticallydownward.Soin stretchingtheropeyouhavethe forcewhichpullsit horizontallyanditsownweightwhicha&sdownwards.Thecircumstancesin thesetwocasesare,therefore,verysimilar.If then you attribute to the weightof the ropea powerand

[310]energy[possanzaedenergia]sufficientto opposeandovercomeany stretchingforce,no matter howgreat, why deny thispowerto the bullet?

BesidesI musttell you somethingwhichwillboth surpriseand pleaseyou, namely,that a cord stretchedmoreor lesstightlyassumesacurvewhichcloselyapproximatestheparabola.Thissimilarityisclearlyseenifyoudrawa paraboliccurveonaverticalplaneandtheninvertit sothat theapexwilllieat thebottom and the base remainhorizontal;for, on hangingachainbelowthe base,oneendattachedto eachextremityofthebase,youwillobservethat,onslackeningthechainmoreorless,it bendsandfits itselfto the parabola;and the coincidenceismoreexac°cin proportionas the parabolais drawnwith lesscurvatureor,so to speak,morestretched;so that usingparab-olasdescribedwith elevationslessthan 45° the chainfits itsparabolaalmostperfe&ly.

SAQR.Then witha finechainonewouldbe ableto quicklydrawmanyparaboliclinesuponaplanesurface.

SXLV.Certainlyandwithnosmalladvantageas I shallshowyoulater.

SIMv.Butbeforegoing,further,I amanxiousto be convincedat leastof that propositionof whichyou say that there is a

rigid

FOURTHDAY 29Irigiddemonstration;I referto thestatementthat it isimpossibleby anyforcewhateverto stretcha cord8othat it willlieper-fecCtlystraightandhorizontal.

SAcR.I willseeifI canrecallthedemonstration;but inorderto understandit, Simplicio,it willbenecessaryforyouto takeforgrantedconcerningmachineswhatisevidentnotalonefromexperimentbut alsofromtheoreticalconsiderations,namely,that the velocityof a movingbody[velocithdelmovente],evenwhenits force[for_] is small,canovercomea verygreatre-sistanceexertedbya slowlymovingbody,wheneverthevelocityofthemovingbodybearsto that ofthe resistingbodyagreaterratio than the resistance[resisten_a]of the resistingbodytotheforce[forza]ofthemovingbody.

SIMP.ThisI knowverywellforit hasbeendemonstratedbyAristotlein his _uestionsin Mechanics;it is alsoclearlyseenin the leverand the steelyardwherea counterpoiseweighingnotmorethan4 poundswillliftaweightof4ooprovidedthatthe distanceof the counterpoisefromthe axisaboutwhichthe steelyardrotatesbe morethanonehundredtimesasgreatas the distancebetweenthisaxisandthe pointof supportfor

[3II]the largeweight. This is truebecausethe counterpoisein itsdescenttraversesa spacemorethanonehundredtimesasgreatas that movedoverby the largeweightin the sametime;inotherwordsthe smallcounterpoisemoveswithavelocitywhichis morethan onehundredtimesas greatas that of the largeweight.

SAcg.Youarequiteright;youdonothesitateto admitthathoweversmallthe force[forza]ofthemovingbodyit willover-comeany resistance,howevergreat,providedit gainsmorein velocitythan it losesin forceandweight[_igoree gravi_].Nowletusreturnto the caseofthecord. In theaccompanyingfigureabrepresentsalinepassingthroughtwofixedpointsa andb; at the extremitiesof this linehang,as you see,two largeweightsc andd, whichstretchit withgreatforceandkeepittruly straight,seeingthat it is merelya linewithoutweight.NowI wishto remarkthat iffromthe middlepointofthisline,which

z9z THE TWO NEW SCT_.NCESOF GALTt._Dwhichwemaycalle,you suspend:anysmallweight,sayh,thelineabwillyieldtowardthe pointf andonaccountofitselonga-tionwillcompelthe twoheavyweightscandd to rise. This Ishalldemonstrateasfollows:withthepointsa andb ascentersdescribethetwoquadrants,eigandelm;nowsincethetwosemi-diametersaiandblareequalto aeandeb,theremaindersfi andfl aretheexcessesofthelinesaf andfboveraeandeb;theythere-

q .... t'

Fig.xe6foredeterminetheriseoftheweightscandd,assumingofcoursethat the weighth has takenthe positionf. But the weighth

[zIz]willtake the positionf, wheneverthe lineefwhichrepresentsthe descentof h bearsto the linefi--that is,to the riseof theweightsc andd--a ratiowhichis greaterthan the ratioof theweightofthetwolargebodiesto thatofthebodyh. Evenwhentheweightsofcandd areverygreatandthat ofhverysmallthiswillhappen;fortheexcessoftheweightscanddovertheweightofh canneverbe sogreatbut that the excessofthe tangentefoverthe segmentfi maybe proportionallygreater. This maybe

FOURTHDAY 293be provedas follows:Drawa Circleof diametergai;drawthelinebosuchthat the ratioof its lengthto anotherlengthc,c>d,isthe sameasthe ratiooftheweightscanddto theweighth. Sincec>d,theratioofboto d isgreaterthanthat ofboto c.Takebea thirdproportionalto obandd;prolongthe diametergi to a pointf suchthat gi:if=oe:eb;andfromthe pointfdrawthe tangentfn; then sincewe alreadyhaveoe:eb=gi:if, weShallObtain,by compoundingratios,ob:eb=if:if. But d is ameanproportionalbetweenobandbe;whilenf is a meanpro-portionalbetweenif andfi. Hencenf bearsto fi the sameratioasthat ofcbtod,whichisgreaterthanthat oftheWeightscandd to the weighth. Sincethenthe descent,or velocity,ofthe Weighth bearsto the rise,or velocity,of the weightscandd agreaterratiothantheweightofthebodiescandd bearsto the weightof h, it is clearthat the weighth willdescendandthe lineabwillceaseto be straightandhorizontal.

Andnowthiswhichhappensin the caseof aweightlesscordabwhenanysmallweight/_isattachedat thepointe,happensals6whenthe Cordis madeof ponderablematterbut withoutanyattachedweight;becausein thiscasethematerialofwhichthe cordis composedfunctionsasa suspendedweight.

S_a,. I am fullysatisfied.So nowSalviaticanexplain,ashe pi_mised,the advantageof sucha chainand, afterwards,presentthe speculationsof ourAcademicianon the subjectofimpulsiveforces[forr_della_ercossa].

SAa_v.Let the precedingdiscussionssufficefor to-day;thehourisalreadylateandthetimeremainingwillnotpermitustoclearupthe subjec°csproposed;wemaythereforepostponeourmeetinguntilanotheraridmoreopp6rmneoccasion.

SAoR.I concurin youropinion,becauseafter variouscon-versationswithintimatefriendsofourAcademicianI havecon-cludedthat thisquestionofimpulsiveforcesisveryobscure,andI thinkthat, upto thepresent,noneofthosewhohavetreated[313]thissubjecthavebeenableto clearup.its darkcornerswhichliealmostbeyondthe reachof humannnagination;amongthevariousviewswhichI haveheardexpressedone,strangelyfan-tastic

294 THE TWONEWSCIENCESOF GALILEOtastic,remainsinmymemory,namely,"thatimpulsiveforcesareindeterminate,ifnotinfinite.Letus,therefore,awaitthecon-venienceof Salviati.Meanwhiletellmewhatis thiswhichfollowsthediscussionofproje_iles.

S_v. Thesearesometheoremspertainingto thecentersofgravityofsolids,discoveredbyourAcademicianinhisyouth,andundertakenbyhimbecauseheconsideredthetreatmentofFederigoComandinotobesomewhatincomplete.Theproposi-tionswhichyouhavebeforeyouwould,he thought,meetthedeficienciesof Comandino'sbook. The investigationwasundertakenat the instanceof the IllustriousMarquisGuid'UbaldoDalMonte,a verydistinguishedmathematicianofhisday,asisevidencedbyhisvariouspublications.Tothisgentle-manourAcademiciangavea copyofthiswork,hopingto ex-tendtheinvestigationtoothersolidsnottreatedbyComandino.Butalittlelatertherechancedtofallintohishandsthebookofthegreatgeometrician,LucaValerio,wherehefoundthesub-je&treatedsocompletelythatheleftoff:hisowninvestigations,althoughthemethodswhichheemployedwerequitedifferentfromthoseofValerio.

SAcg.Pleasebegoodenoughto leavethisvolumewithmeuntilournextmeetingsothatI maybeableto readandstudythesepropositionsintheorderinwhichtheyarewritten.

SALv.It isapleasureto complywithyourrequestandI onlyhopethatthepropositionswillbeofdeepinteresttoyou.

END OF FOURTHDAY.

APPENDIX

Containing some theorems, and their proofs, dealing withcenters of gravity of solid bodies, written by the same Author atan earlier date.*

* Followingtheexampleof theNationalEdition,thisAppendixwhichcoversI8 pagesof the LeydenEditionof I638is hereomittedas beingof minorinterest. [Trans.]

[FINIS]

INDEXAccelerandosl= displace,252 Beamcarryinga constantloadAcceleratedmotion,x6o whichmovesfromoneend to

reduced to theother,I4Ouniformmotion,x73 Beamcarryinga constantload

Acceleration,definitionof uni- in a fixedposition,14oet $q.form,I69 Beams,similar,relativestrength

Accelerationof gravitymeasured, of, I24I78 _, strengthof,II5

Accelerazione,used in the sense , variationofstrengthwithofspeed,I67 diameter,II9

Achillesand the Tortoise,prob- , variationofstrengthwithlemof,x64 length,I23

Acoustics,95 Bendingstrength,II9Adhesionof plates,xI, IZ Bernoulli,James,I49Aggiunti,x , John,I83Air, resistanceof, to projectiles,Bones,proportionof, I3X

252,256 , ofbirds,I5o,239Altitudeof parabola,determina-Brachistochrone,225,239

tion of,285Amplitudeof paraboladefined, Canne= fathom

Capillarity,effectseeninshorten-26o,273 tablesof,284 ingofrope,2o

Apollonius,242,245,246 as seenin dropsontheoremof,38,44 cabbageleaves,7o

Aproino,Paolo,ix Catenary,I49,29oArcetri,xi,xviii Cause,speedconsideredas a, 255Archimedes,v, 4I, IiO, I44,I47, Causes,searchafter,futile,I66

242,25x Cavalieri,4IArchimedes'principle,8I Chordsofcircles,timesofdescentAriosto,quotationfrom,I3I along,I88etsq.Aristotle,I2, I3,2o,26,49,64,76, Cicero,xix

77,80,95,IIO,I25,I35, 29I Cohesionof solids,II etsg.,I8Aristotle'sideaoffallingbodies,6I Comandino,Federigo,294Arrighetti,x Compass,Galileo'smilitary,I49Atmosphericpressure,I7 Compositionofmotions,244,257,AugmentedFourth,Io4 267Axiomsof uniformmotion,I54 Continuity,mathematical,24etsq.

298 INDEXContinuousquantities,3I, 34-36 Gra_ty,centersof,294,295Contraction,theoryof,51 .,experimentaldetermina-Cylinders,relative strength of tionof,178

similar,Iz4 Guevara(di)Giovani,zo,I26

DefinitionofuniformlyacceleratedHammerblow,problemof,27imotion,I61 Heath,T. L., v, I45

Definitionofuniformmotion,x54 Huygens,vDefinitions,mathematical,28 Impenetrabilityof matter,49,5I,Density,seeSpecificGravity 6IDiapason,99Diapente,99 Impulsiveforces,293DiminishedFifth,Io4 Inclinedplane,principleof, I84Diodati,Elia,ix,xii etsq.

--, usedto "dilute'"Energia,269,27I,290 gravity,178Elzevir,Louis,xi Inclinedplanesof equalheight,Elzevirs,The,xviii speedacquiredon,I69,184Errors in parabolictrajectories, Indivisibility,30,36

25Ietsq. Infinity,26,30,3I, 37,39_Euclid,242,243,246,248,283 Isochronismof the simplepen-Expansion,theoryof,51 dulum,97

of pendulumsof dif-Fallingbodies,Aristotle'snotion ferentmaterials,85

of,6I,64-68 Isoperimetricproblems,58--., lawsof,265--; sphereas locusof, Jealousyof investigators,83

I9Z,I93Featherandcoinresultpredicted, Lawoflever,II2

72 Lawofmotion,Newton'sfirst,244Fifth,a musicalinterval,ioo Newton'ssecond,Fire escape,devisedby kinsman 25o

ofGalileo,Io Lawofsimplependulum,96Force,synonymsfor,1I4,267,269, Lawsofairresistance,253

27I, 286,29I, 293 Lawsof fallingbodies,I74et sq.,Forza,in the senseof mechanical 265

advantage,124 Lever,lawof, II2.,inthesenseofmomentum,Levityofair,77,78

267,269,271,286 Lift pump,theoryof,I6Frequenza= frequencyof pen- Light,speedof,42

dulum,97 Limitingspeedof bodiesin vacuo,7z

GeometrycomparedwithLogic, Lined,Academyofthe,xxI37 Loadedbeams,I4o

Gilding,thicknessof,53etsq. Logicandgeometrycompared,I37

INDEX 299Marsili,x Padua, ix,x, xii, 182Mathematicaldefinitions,28 Parabola,descriptionof, 148,246,Mathematics,r61eof in physics, 258

276 - , quadratureof, 147Maximumrangeofaprojectile,275 --., to find the sublimityofMedia,effectof,onspeedoffalling a given,272

bodies,76 Parallelogramof velocities,257Medici,Antoniode', x Pendulum motion, law of air

, PrinceMattia de', xi resistanceto, 254Mersenne,M, v. Pendulumsof leadand cork,84Micanzio,xi Pendulums,time of descent,95Minimum momentum theorem, Pendulum with string striking

273 againstnail,17oMirrors,spherical,41 Periodo= periodof pendulum,97Mole= volume,8o, 82 Peripatetics,48,61Moment,used in sense of impor- Peso= specificgravity,69,72

tance,65 Pieroni,G.,xlMomento-- force,183 Planetaryorbits,261Momentoin the senseof magni- Plato, 9o, 137,261

tude, 124 Point blankfiring,289Momentum, 258, 259, z63, z66; Potenza,meaningof"in potenza,"

z69,270 257,266, measureof, 264 Principleof virtualwork,183

Monochord,99 Problem of Achilles and theMotion, axiomsof uniform,154 Tortoise,164

, natural,153 of stretchedrope,290of fallingbodies,16o Projectile,speedof, 279of projectiles,244 Projectiles,motionof, 244supernatural,255 Pythagoreanprobleminmusic,95,uniform,lawsof, I54 lO3

Monte, G. del, ix,294Motte, v Quadratureof parabola,147,251Music,95 Quickestdescent,225,239

Naturally acceleratedmotion,16o Range,maximum,275"Natural" Motion, 153 Range,minimummomentumre-Newton, v quiredfor a given,273Newton's First Law of Motion, Rangetables, 2.56,284

215,216,244 Resistanceof mr proportionaltoSecondLaw of Motion, speed,74

of air to projectiles,250,257,263Noailles,Countof, xii, xvii 252

Resonanceof a bell,98Octave,ratio involvedin, 99 of pendulum,97Orbitsof planets, 261 of strings,99

3oo INDEX

Resultant,257,258, 267 Strings,lawsof vibrating,IooRipples, produced hy sounding "gul_limlty" de_ned,z60

a goblet,99 of any parabola, toRope-making,theoryof, 8 et sq. find,272Ropes,shorteningdueto moisture, Subsidenceofsmallparticles,87

20_, stretched,problemof, z9o Tangent, used in two senses,28z

Terminalspeed, 9z, 256, 26oSack,properdesignof, 56 Theoremof Apollonius,38, 44Sacrobosco(John Holywood)57 Tritono,lO4Salusbury,Thomas,v Tubes,bendingstrengthof, I5OSarpi,Fra Paolo,ix, xSecondminute, as a unit of time, Ubaldi,ix

z64, 265 Uniform acceleration, definitionSiena,x of, 169Space,describedin uniformlyac- Uniformmotion,lawsof, 154

celeratedmotion,173et sq.Spearas unit oflength, 265 Vahrio,Luca, 30, 148,z94Specificgravity, halls adjusted to Velocities,parallelogramof, 257

equal that of water, 69 Velocity,standardof, 26oSpecificgravity of air, 78 tt sq. , uniform, definitionof,

of aquatic 154animals,13z , virtual, z9I

Speedof projectile,z79 Venice,xi, standard of, 260, 265 Figore= force,z91, terminal,9I Vinta, Belisario,x•, uniform,definitionof, 154 Virtual work, principleof, z9z

Statics, fundamentalprincipleof, Viviani,Vincenzio,ix, I8oIIO

Steelyard,29I .Weighingin vacuo,81-83Strengthof materials,lO9 Weston,Thomas,vi