SlideShare a Scribd company logo

Las dos nuevas ciencias de Galileo.pdf

•
•
G
gian1232

Libro recopilatorio de Galileo donde describe sus observaciones y conclusiones sobre la naturaleza del movimiento y otros fenómenos.

Science
1 of 328
Download to read offline
TWO NEW SCIENCESBY GALILEO
°_o° THE MACMILLAN COMPANY _W YORJK • BOSTON • CItlCAGO • DALLAS ATLANTA • SANyP.ANCISCO MACMILLAN & CO., Lmrr_ LONDON • BOMBAY • CALCUTTA _T,_ OUP._NE T_L_; MACMILLAN CO. OF CANADA, LTD. TORONTO
GALILEO GALILEI. Subterman's portrait, paintedaboutI64O; nowin the Galler_a dr' Pittiat Florence.
DIALOGUES CONCERNING TWO NEW SCIENCES BY GALILEOGALILEI Translatedfromthe Italianand Latin intoEnglishby HENRY 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 the learning of the ancients." Benjamin Franklin, Phil. Trans. 64,445.(I774.) N_ _ork THE MACMILLAN COMPANY I914 All rightsreserved
COPX_IGBT. X914 B_ THEMACMILLAN COMPANY Set upande[ectrotyped.Published May_I914.
"La Dynamique estlascience desforcesaccN6ra- tricesor retardatrices, et des mouvemens varies qu'ellesdoiventproduire.Cette science est due enti_rement auxmodernes, et Galilee est cduiqui en a jet_lespremiers fondemens." Lagrange Mec. dad. I. zzl, TRANSLATORS' PREFACE OR morethana centuryEnglishspeaking students havebeenplaced intheanomalous position of hearingGalileo constantlyre- ferred to asthefounder ofmodern physical science, without havinganychance toread, intheirownlanguage, whatGalileo himself has to say. Archimedes has beenmade available byHeath; Huygens' Light hasbeen turnedinto Englishby Thompson, whileMotte has put the Principia of Newtonbackintothe language in whichit was conceived.To renderthePhysics of Galileo alsoaccessible to English andAmerican studentsis thepurpose ofthefollowing translation. The lastof the greatcreators of theRenaissance wasnot a prophetwithouthonorin hisowntime;for it wasonlyone groupofhiscountry-men thatfailedto appreciate him. Even during hislifetime,hisMechanics hadbeenrendered intoFrench byoneoftheleading physicists oftheworld, Mersenne. Withintwenty-five yearsafterthedeathofGalileo, hisDia- logues onAstronomy, andthoseon TwoNewSciences, hadbeen doneinto Englishby ThomasSalusbury and wereworthily printed in two handsome quarto volumes. The TwoNew Sciences, which contains practically allthatGalileo hastosayon thesubje&ofphysics, issuedfromthe English pressin I665.
vi TRANSLATORS' PREFACE It issupposed thatmostofthecopies were destroyed inthegreat London firewhichoccurred inihe yearfollowing. Wearenot aware ofanycopyinAmerica: eventhatbelonging totheBritish Museum isanimperfecCt one. Againin 173 ° the TwoNewSciences wasdoneintoEnglish by Thomas Weston;but thisbook,nownearlytwocenturies old,is scarceand expensive.Moreover, the literalness with whichthistranslation wasmaderenders manypassages either ambiguous orunintelligible to themodernreader.Otherthan thesetwo,noEnglish version hasbeenmade. Quiterecentlyan eminentItalian scholar,after spending thirtyofthebestyearsofhislifeuponthesul_jecCt, hasbrought to completion the greatNationalEditionof the Worksof Galileo.Werefertothetwentysuperb volumes inwhichPro- fessor AntonioFavaro ofPaduahasgivena definitive presenta- tionofthelaborsofthemanwhocreatedthemodern science of physics. The following rendition includes neitherLe Mechaniche of Galileo norhispaperDeMotuAccelerato, sincethe formerof thesecontainslittlebut the Staticswhichwascurrentbefore thetimeofGalileo, andthelatterisessentially included inthe Dialogue oftheThirdDay. Dynamics wastheonesubjecCt to whichundervariousforms,suchas Ballistics, Acoustics, As- tronomy,he consistently and persistently devotedhis whole life. Into theone volumeheretranslatedhe seemsto have gathered,duringhislast years,pracCtically allthat is of value either to the engineeror the physicist. The historian,the philosopher, and the astronomer willfindthe othervolumes replete withinteresting material. tt ishardlynecessary toaddthatwehavestric°dy followed the textoftheNationalEdition---essentially theElzevireditionof 1638.Allcomments andannotations havebeenomittedsave hereandtherea foot-note intendedto economize the reader's time. To eachofthesefootnotes hasbeenattachedthesigna- ture[Trans.] inordertopreserve theoriginal asnearly intacCt as possible. Muchofthevalueofanyhistorical document liesinthelan- guage employed, andthisis doubly truewhenoneattemptsto
TRANSLATORS' PREFACE vii trace the rise and growthof any set of conceptssuch as those employedin modem physics. We have thereforemade this translationasliteral as isconsistentwithclearnessandmodem- ity. In caseswherethere isany importantdeviationfromthis rule, and in the caseof many technicaltermswherethere isno deviationfrom it, we have giventhe originalItalian or Latin phrase in italics enclosedin square brackets. The intention here isto illustratethe great varietyof termsemployedby the earlyphysiciststo describea singledefiniteidea,and conversely, to illustratethe numeroussensesinwhich,then as now,a single wordisused. For the fewexplanatoryEnglishwordswhichare placedin squarebrackets withoutitalics,the translatorsalone are responsible.Thepagingofthe NationalEditionisindicated in squarebrackets insertedalongthe medianlineof the page. The imperfecCtions of the followingpageswouldhave been manymorebut forthe aidof threeof our colleagues.Professor D. R. Curtiss was kind enoughto assistin the translationof thosepageswhichdiscussthe natureofInfinity:ProfessorO.H. Basquingavevaluablehelp in the renditionof the chapteron Strengthof Materials;and ProfessorO.F. Longclearedup the meaningofa numberof Latinphrases. To ProfessorA. Favaroof the UniversityofPaduathe trans- lators share,with every reader, a feelingof sincereobligation forhisIntroducCtion. H.C. A. DE S. EVANSTON) ILLINOIS) x5February, I9x4.
INTRODUCTION .................. '............................... RITINGtohisfaithful friend EliaDiodati, _ Galileo speaks ofthe"NewSciences" which hehadinmindtoprintas being"superior to everythingelseof minehithertopub- lished";elsewhere he says"they contain results-which I consider themostimportant ofallmy studies";andthisopinion which heexpressed concerning hisownworkhas beenconfirmed by posterity: the"NewSciences" are,indeed, themasterpiece ofGalileo whoat thetimewhenhemadethe above remarks hadspentuponthemmorethanthirtylaborious years. Onewhowishes to tracethehistory ofthisremarkable work willfindthat thegreatphilosopher laiditsfoundations during the eighteen best yearsof his lifc thosewhichhe spentat Padua. As welearnfromhislast scholar, Vincenzio Viviani, thenumerous resultsatwhichGalileo hadarrivedwhile inthis city,awakened intenseadmiration inthefriends whohadwit- nessed variousexperiments by meansof whichhewasaccus- tomed toinvestigate interesting questions inphysics.FraPaolo Sarpiexclaimed: To giveus the Science ofMotion,Godand NaturehavejoinedhandsandcreatedtheintellecCt ofGalileo. Andwhenthe"NewSciences" camefromthepressoneofhis foremost pupils, Paolo Aproino, wrotethatthevolume contained muchwhichhehad "alreadyheardfromhisownlips"during student daysat Padua. Limitingourselves to onlythe moreimportant documents which mightbecitedinsupport ofourstatement, it willsuffice tomention the letter,writtentoGuidobaldo delMonteon the 29thofNovember, I6O2, concerning thedescent ofheavy bodies
x INTRODUCTION alongthearcsofcircles andthechords subtended bythem;that to Sarpi,datedI6thofOctober, I6o4,dealing withthefreefall of heavybodies; the letterto Afltonio de'Medicion the IIth ofFebruary, I6o9,inwhich hestatesthathehas"completed all the theorems anddemonstrations pertaining to forcesandre- sistances of beamsof various lengths,thicknesses and shapes, proving thattheyareweaker at themiddle thanneartheends, thattheycancarrya greaterloadwhenthat loadisdistributed throughout thelengthof thebeamthanwhenconcentrated at onepoint,demonstrating alsowhatshapeshould begivento a beaminorderthat it mayhavethe samebendingstrengthat everypoint,"andthat hewasnowengaged "uponsomeques- tionsdealing withthemotionofprojeCtiles"; andfinallyinthe letterto Belisario Vinta,dated7th of May, x6IO, concerning hisreturnfromPaduatoFlorence, heenumerates various pieces ofworkwhichwerestillto becompleted, mentioning explicitly threebooks onanentirelynewscience dealing withthetheory ofmotion. Although at varioustimesafterthe returnto his nativestatehedevoted considerable thoughttotheworkwhich, evenat that date,hehadinmindas isshown by certainfrag- mentswhichclearlybelongto different periods of hislifeand whichhave,forthefirsttime,beenpublished in theNational Edition;andalthough thesestudieswerealwaysuppermost in histhoughtit doesnotappear that hegavehimself seriously to themuntilafterthe publication of theDialogue andthe com- pletion of thattrialwhich wasrightly described as thedisgrace ofthecentury. InfaCtaslateasOctober,x63 o,hebarely men- tionstoAggiuntihisdiscoveries in thetheoryof motion,and onlytwoyearslater,inalettertoMarsiliconcerning themotion ofprojeCtiles, hehintsat abooknearlyreadyforpublication in whichhewilltreat alsoofthis subject;andonlya yearafter thishewritestoArrighetti thathehasinhandatreatiseonthe resistance ofsolids. But theworkwasgivendefinite formby Galileo duringhis enforced residence at Siena: in thesefivemonthsspentquietly withtheArchbishop he himself writesthat hehascompleted "a treatiseona newbranchofmechanics fullofinteresting and usefulideas";sothat a fewmonthslaterhewasableto send
INTRODUCTION xi wordtoMicanzio that the "workwasready";as soonashis friends learned ofthis,theyurgeditspublication.It was,how- ever,no easymatterto printtheworkofa manalreadycon- demned bytheHolyOffice: andsince Galileo could nothopeto printit eitherinFlorence or inRome,heturnedtothefaithful Micanzio asking himtofindoutwhether thiswould bepossible inVenice, fromwhence hehadreceived offers toprinttheDia- logue onthePrincipal Systems, as soonasthenewshadreached therethathewasencountering difficulties. At firsteverything wentsmoothly; sothatGalileo commenced sending toMicanzio someofthemanuscript whichwasreceived by thelatterwith anenthusiasm inwhichhewassecond tononeof thewarmest admirers of the greatphilosopher.But whenMicanzio con- sultedthe Inquisitor, he received the answerthat therewas an express orderprohibiting theprintingor reprinting of any workof Galileo,eitherinVeniceor in anyotherplace,hullo excepto. As soonasGalileo received thisdiscouraging newshebegan tolookwithmore favoruponoffers which hadcome tohimfrom Germany wherehisfriend,andperhapsalsohisscholar, Gio- vanniBattistaPieroni,wasinthe service of theEmperor, as militaryengineer; consequently Galileo gaveto PrinceMattia de'Medici whowasjust leaving forGermany thefirsttwoDia- logues tobehandedtoPieroniwhowasundecided whether to publish thematVienna orPrague or atsome placeinMoravia; inthemeantime, however, hehadobtained permission toprint bothatVienna andatOlmtitz.But Galileo recognized danger at everypointwithinreachof the longarmof the Courtof Rome; hence, availing himself oftheopportunity offered bythe arrivalof LouisElzevirinItalyin 1636, alsoofthe friendship betweenthe latter and Micanzio, not to mentiona visit at Arcetri,hedecided to abandon allotherplansandentrustto theDutchpublishertheprintingof hisnewworkthemanu- scriptofwhich,although notcomplete, Elzevirtookwithhim onhisreturnhome. In thecourse oftheyear1637, thepriming wasfinished, and at thebeginning ofthe following yeartherewaslacking only the index,the title-page and the dedication.This last had,
xii INTRODUCTION through thegoodoffices ofDiodati, beenoffered totheCountof Noailles, a formerscholar ofGalileo at Padua,andsince1634 ambassador ofFranceatRome,amanwhodidmuch toalleviate the distressing consequences of the celebrated trial; and the offerwasgratefully accepted.The phrasing of thededication deserves briefcomment.Since Galileo wasaware,ontheone hand,ofthe prohibition againsttheprintingofhisworksand since, on theotherhand,hedidnotwishto irritatetheCourt ofRome fromwhose handshewasalways hoping forcomplete freedom, hepretends inthededicatory letter(where, probably through excess of caution, hegives onlymainoutlines) thathe hadnothingtodowiththeprintingofhisbook,asserting that hewillneveragainpublishanyof his researches, andwillat mostdistributehereandthere a manuscript copy. He even expresses greatsurprise thathisnewDialogues havefalleninto thehandsof the Elzevirsandweresoonto bepublished; so that, having beenaskedtowritea dedication, hecould thinkof no manmoreworthywhocouldalsoon this occasion defend himagainsthisenemies. As to the title whichreads:Discourses andMathematical Demonstrations concerning TwoNewSciences pertaining toMe- chanics andLocal Motions, thisonlyisknown, namely, that the titleisnot theonewhichGalileo haddevised andsuggested; in fac_heprotestedagainstthe publishers takingthe libertyof changing it andsubstituting "a lowandcommon title forthe noble anddignified onecarried uponthetitle-page." In reprinting thisworkintheNational Edition,I havefol- lowed theLeydentextof 1638faithfully but notslavishly, be- causeI wishedto utilizethelargeamountofmanuscript ma- terialwhich hascome downtous,forthepurpose of colTeccting a considerable numberof errorsin this firstedition,and also forthesakeofinserting certainadditions desired bytheauthor himself.IntheLeyden Edition, thefourDialogues arefollowed by an"Appendix containing some theorems andtheir proofs, deal- zngwithcenters of gravity of solidbodies, writtenbythesame Author atanearlier date,"whichhasnoimmediate connecCtion withthesubjec°cs treatedintheDialogues; thesetheorems were foundby Galileo, as hetellsus,"at theageoftwenty-two and
INTRODUCTION _5ii aftertwoyearsstudyofgeometry" andwerehereinserted only tosavethemfromoblivion. But it wasnot the intentionof Galileo that theDialogues ontheNewSciences should contain onlythefourDaysandthe above-mentioned appendix whichconstitute theLeydenEdi- tion;while,on theonehand,theElzevirs werehastening the printing andstriving tocomplete it attheearliest possible date, Galileo, on theotherhand,kepton speaking of anotherDay, besides thefour,thusembarrassing andperplexing theprinters. Fromthe correspondence whichwenton between authorand publisher, it appears that thisFifthDaywasto havetreated "of theforceof percussion andtheuseofthecatenary";, but as thetypographical workapproached completion, thepnnter became anxious forthebookto issuefromthepresswithout furtherdelay;and thus it cameto passthat the Discorsi e Dimostrazioni appeared containing onlythefourDaysandthe Appendix, in spiteof thefactthat inApril,I638,Galileo had plunged moredeeply thanever"intotheprofound question of percussion" and"hadalmost reached a complete solution." The "NewSciences" nowappearinanedition following the textwhichI, afterthemostcareful anddevoted study,deter- mineduponfortheNational Edition. It appears alsointhat language inwhich, above allothers, I havedesired to seeit. In thistranslation, thelastandripestwork ofthegreatphilosopher makesits firstappearance in the NewWorld:if towardthis important resultI mayhopetohavecontributed insome meas- ureI shallfeelamplyrewarded forhavinggiven to thisfieldof research thebestyearsofmylife. AwroNm FAv_a_o. UNIVERSITY OF PADUA_ 2_ o] October, I9x3.
DISCOR, SI E DIMOSTR.AZIONI MI.A.TEMA TI C H E, intorno _duenuoue fiiene_e Atxenenti alla M_-CANICA_ i Mo VIMENTI LOCALt; delSignor GALILEO GALILEI LINCEO, Filofofo eMatematico primario delSereniilimo Grand Duca diTofcana. c°o_ z_a_ppendice del ceutro digrauit_ d'klc-_ni ._olidi._ IN LEIDA, Apprdro gliEl/_virii. _.D. c,xxxv_.
' [431 TO THE MOST ILLUSTRIOUSLORD COUNT OF NOAILLES Counsellor ofhisMostChristian Majesty, KnightoftheOrder of theHolyGhost, FieldMarshal andCommander, Seneschal andGovernor ofRouergue, andHis Majesty's Lieutenant inAuvergne, my LordandFForshipful Patron OSTILLUSTRIOUS LORD:- In the pleasure whichyou derivefrom the possession of thisworkof mineI rec- i]_/i[ ll_r_-_[_ ognize yourLordship's magnanimity. The i[lV/[]_-4_}disappointment anddiscouragement I have _'_[,_ feltovertheill-fortune whichhasfollowed __ myotherbooksarealready known to you. Indeed,I had decided not to publish any moreof my work. Andyet in order to saveit fromcom- pleteoblivion,it seemedto mewiseto leavea manuscript copyinsome place whereit would beavailable atleasttothose who followintelligently the sub jet% whichI havetreated. Accordingly I chosefirstto placemyworkinyourLordship's hands,askingno moreworthydepository, andbelieving that, onaccount ofyouraffecCtion forme,youwould haveat heartthe preservation of mystudiesand labors.Therefore, whenyou werereturning homefromyourmission toRome, I came topay myrespecCts inpersonasI hadalready donemanytimesbefore byletter. At thismeeting I presented toyourLordship acopy ofthesetwoworks whichat thattimeIhappened tohaveready. Inthegracious reception which yougavetheseI found assurance of
xviii TO THE COUNT OF NOAILLES oftheirpreservation. Thefacet ofyourcarrying themtoFrance andshowing themto friendsofyourswhoareskilled inthese sciences gaveevidence thatmysilence wasnottobeinterpreted ascomplete idleness.Alittlelater,justasI wasonthepointof [44] sending othercopies toGermany, Flanders, England, Spainand possibly to some placesinItaly,I wasnotified by theElzevirs thattheyhadtheseworksofmineinpressandthatI oughtto decide upona dedication andsendthema replyat once. This sudden andunexpecCted newsledmetothinkthat theeagerness of your Lordship to reviveand spreadmy nameby passing theseworkson to variousfriendswasthe realcauseof their falling intothehandsofprinterswho,because theyhadalready published otherworks ofmine,nowwishedtohonormewitha beautiful andornateedition ofthiswork.Butthesewritings of minemusthavereceived additional valuefromthecriticism of so excellent a judgeas yourLordship,whoby the unionof manyvirtueshaswonthe admiration of all. Yourdesireto enlarge therenown ofmyworkshows yourunparalleled generos- ity and your zealforthe publicwelfarewhichyouthought wouldthus be promoted. Underthese circumstances it is eminently fittingthat I should,in unmistakable terms,grate- fullyacknowledge thisgenerosity on thepartofyourLordship, whohasgiven tomyfame wings thathavecarried it intoregions moredistantthanI haddaredtohope. It is,therefore, proper that I dedicateto yourLordship thischildof my brain. To thiscourseI amconstrained notonlybytheweightof obliga- tion underwhichyouhaveplacedme,but also,if I may so speak,by the interestwhichI havein securing yourLordship as thedefender ofmyreputation againstadversaries whomay attackit while I remain underyourprotecCtion. Andnow,advancing underyourbanner,I paymy respecCts toyoubywishing thatyoumayberewarded forthesekindnesses bytheachievement ofthehighest happiness andgreatness. I amyourLordship's Most devotedServant, GALILEO GALILEn _lrcetri, 6March, I638.
THE PUBLISHERTO THE READER INCEsociety isheldtogether bythemutual services whichmenrenderoneto another, andsinceto thisendtheartsandsciences havelargelycontributed, investigations in thesefields havealways beenheldingreat esteem andhave beenhighly regarded byour wiseforefathers.Thelarger theutilityand excellence oftheinventions, thegreater has beenthehonorandpraise bestowed upontheinventors.Indeed, menhaveevendeified themandhaveunitedintheattemptto perpetuate thememory of theirbenefafftors bythebestowal of thissupreme honor. Praiseand admiration are likewise dueto thosecleverin- tellecCts who, confining their attentionto the known,have discovered and corre&edfallaciesand errorsin manyand many a propositionenunciated by men of distincCtion and accepted forages asfacet.Although these menhaveonlypointed outfalsehood andhavenotreplaced it bytruth,theyarenever- thelessworthyof commendation whenweconsider the well- knowndifficulty of discovering facet, a difficulty whichledthe princeof oratorsto exclaim: Utinara tamfacilepossem vera reperire, quamfalsa convincere.* And indeed,theselatest centuries meritthis praisebecause it is duringthemthat the artsandsciences, discovered bytheancients, havebeenreduced to so great and constantly increasing perle&ion throughthe investigations and experiments of clear-seeing minds. This development is particularly evidentin thecaseof themathe- maticalsciences.Here,without mentioning various menwho haveachieved success, wemustwithout hesitation andwiththe *Cicero. deNatura Deorum, I,9I. [Trans.]
xx THE PUBLISHER TO THE READER unanimousapprovalof scholarsassignthe first placeto Galileo Galilei,Memberof theAcademyoftheLincei. Thishedeserves not only becausehe has effectivelydemonstratedfallaciesin many of our current conclusions,as is amply shownby his publishedworks, but also becauseby meansof the telescope (inventedin this countrybut greatlyperfectedby him) he has discoveredthe four satellitesof Jupiter, has shownus the true charaCterof the MilkyWay, and has madeus acquaintedwith spotson the Sun, with the rough and cloudyportionsof the lunar surface, with the threefold nature of Saturn, with the phasesof Venus and with the physical charaCterof comets. Thesematterswereentirelyunknownto the ancientastronomers andphilosophers;sothat wemay truly say that he has restored to the worldthe scienceof astronomyand has presentedit in a newlight. Remembering that the wisdomand powerand goodness ofthe Creator are nowhereexhibitedso well as in the heavensand celestialbodies,we can easilyrecognizethe great merit of him who has brought these bodies to our knowledgeand has, in spite of their almost infinite distance, rendered them easily visible. For, accordingto the commonsaying,sight can teach moreandwith greatercertaintyina singleday than canprecept even though repeateda thousand times; or, as another says, intuitiveknowledge keepspacewithaccuratedefinition. But the divine and natural gifts of this man are shownto best advantagein the present'workwhere he is seen to have discovered,though not w_6hout many labors and long vigils, twoentirelynewsciencesand to have demonstratedthem in a rigid, that is, geometric,manner: and what is even more re- markablein this workis the facetthat one of the two sciences dealswith a subjeCtof never-endinginterest,perhapsthe most importantin nature, onewhichhas engagedthe mindsof allthe great philosophersand one concerningwhichan extraordinary numberof bookshave been written. I refer to motion [moto locale], a phenomenonexhibitingvery many wonderfulproper- ties,noneofwhichhas hithertobeendiscovered or demonstrated by any one. Theotherscience whichhehas alsodevelopedfrom its
THE PUBLISHER TO THE READER xxi its veryfoundationsdealswiththe resistancewhichsolidbodies offer to fracture by external forces_er violenza], a subjectof great utility, especiallyin the sciencesand mechanicalarts, and onealsoaboundinginpropertiesand theoremsnothitherto observed. In this volume one finds the first treatment of these two sciences,full of propositionsto which,as time goes on, able thinkers willadd many more;alsoby meansof a largenumber of clear demonstrationsthe author points the way to many other theoremsas willbe readilyseenandunderstoodby allin- telligentreaders.
TABLE OF CONTENTS I Page Firstnewscience, treating oftheresistance which solid bodies offer tofracture.FirstDay.......................... I II Concerning thecause ofcohesion. Second Day ............ lO 9 III Second newscience, treating of motion[movimenti locah]. ThirdDay........................................ 153 Uniform motion ...................................... 154 Naturally accelerated motion ........................... 16o IV Violent motions.ProjeFtiles. Fourth Day............... 244 V Appendix; theorems anddemonstrations concerning thecenters ofgravity ofsolids .................................. 295
TWO NEW SCIENCESBYGALILEO
FIRST DAY INTERLOCUTORS: SALVIATI, SA- GREDO AND SIMPLICIO :' .............. :'.............................. ALV. Theconstant activity which youVene- _ tiansdisplay inyourfamous arsenal suggests tothestudious minda large fieldforinvesti- gation,especially that part of the work which involves mechanics; forinthisdepart- mentalltypesofinstruments andmachines areconstantly beingconstrue'ted by many artisans, amongwhom theremustbesome who,partlybyinherited experience andpartlyby theirownob- servations, havebecome highly expert andclever inexplanation. SAc_.Youarequitefight. Indeed,I myself, beingcurious by nature,frequently visitthis placeforthemerepleasure of observing theworkofthosewho,onaccount oftheirsuperiority overotherartisans, wecall"firstrankmen." Conference with themhasoftenhelped meintheinvestigation ofcertaineffec2s including notonlythosewhich arestriking, butalsothose which arerecondite andalmost incredible.AttimesalsoI havebeen put toconfusion anddriventodespair ofeverexplaining some- thingforwhichI couldnotaccount, butwhich mysenses told metobetrue. Andnotwithstanding thefadtthatwhattheold man told us a little whileago is proverbial and commonly accepted, yetit seemed tomealtogether false, likemanyanother sayingwhichis currentamongtheignorant; forI thinkthey introduce theseexpressions inordertogivetheappearance of knowing something about matters which theydonotunderstand. Salv.
2 THE TWO NEW SCIENCESOF GALILEO [3o] SAJ_v. Yourefer, perhaps,tothatlastremarkofhiswhenwe askedthe reasonwhy they employed stocks,scaffolding and bracing oflargerdimensions forlaunching abigvessel thanthey dofora small one;andheanswered thattheydidthisinorderto avoid thedanger oftheshippartingunderitsownheavyweight [vasta mole], a danger towhichsmall boatsarenotsubject? SAcR. Yes,that iswhatI mean;andI referespecially to his lastassertion whichI havealways regarded asa false,though current,opinion; namely,that in speaking of theseandother similar machines onecannotarguefromthesmallto thelarge, because many devices whichsucceed on a smallscaledo not workona largescale.Now,since mechanics hasitsfoundation ingeometry, where meresizecutsnofigure, I donotseethatthe properties of circles, triangles, cylinders, conesandothersolid figures willchange withtheirsize. If,therefore, alargemachine beconstrucCted insuchawaythatitspartsbeartooneanother thesameratioasinasmaller one,andifthesmaller issufficiently strongforthepurpose forwhichit wasdesigned, I donot see whythelargeralsoshould notbeabletowithstandanysevere anddestrucCtive teststowhich it maybesubjected. SAT.v. Thecommon opinion ishereabsolutely wrong.Indeed, it is sofar wrongthat precisely theopposite is true, namely, thatmanymachines canbeconstrucCted evenmoreperfecCtly ona largescale thanonasmall; thus,forinstance, aclock which indi- catesandstrikes thehourcanbemademoreaccurate ona large scalethan on a small. Therearesomeintelligent people who maintainthis sameopinion, but on morereasonable grounds, whenthey cut loose fromgeometry and arguethat thebetter performance ofthelargemachine isowing totheimperfecCtions andvariations ofthematerial.HereI trustyouwillnotcharge Is1] mewitharrogance if I say that imperfections in thematerial, eventhosewhicharegreat enough to invalidate the clearest mathematical proof,arenotsufficient to explain thedeviations observed between machines intheconcrete andintheabstra_. Yet I shallsayit andwillaffirm that,eveniftheimperfecCtions did
FIRST DAY 3 didnotexistandmatterwereabsolutelyperfecCt, unalterable and freefrom all accidentalvariations,stillthe merefact that it is matter makesthe larger machine,built of the samematerial and in the same proportionas the smaller,correspondwith exacCtness to the smallerineveryrespecCt exceptthat itwillnot be so strong or so resistant against violent treatment; the larger the machine,the greaterits weakness. SinceI assume matter to be unchangeableand alwaysthe same,it isclearthat weare nolessabletotreat this constantandinvariableproperty in a rigidmannerthan ifit belongedto simpleand puremathe- matics. Therefore,Sagredo,you woulddo wellto changethe opinionwhich you, and perhapsalso many other studentsof mechanics,haveentertainedconcerning the abilityof machines and structures to resist external disturbances,thinkingthat whenthey are builtof the samematerialand maintainthesame ratio betweenparts, they are able equally,or rather propor- tionally, to resist or yield to such external disturbancesand blows. For we can demonstrateby geometrythat the large machineisnotproportionately strongerthan thesmall. Finally, we may say that, for every machineand strucCture, whether artificialor natural, there isset a necessarylimitbeyondwhich neitherart nor nature canpass; it ishereunderstood,of course, that the material is the sameand the proportionpreserved. SAGI_. Mybrainalreadyreels. Mymind,likea cloudmomen- tarily illuminatedby a lightning-flash, is for an instantfilled withan unusuallight,whichnowbeckonsto meand whichnow suddenly minglesand obscuresstrange, crude ideas. From what you have said it appearsto me impossible to buildtwo similarstrucCtures ofthe samematerial,but ofdifferentsizesand have themproportionatelystrong;and if this wereso, it would [52] not be possibleto tindtwosinglepolesmadeof the same-wood which shall be alike in strength and resistancebut unlikein size. SALv. Soit is,Sagredo. _And to makesurethat weunderstand each other, I say that if we take a woodenrod of a certain length and size, fitted, say,into a wall at right angles,i. e., parallel
4 THE TWO NEW SCIENCES OF GALILEO parallelto the horizon,it may be reducedto sucha lengththat it willjust supportitself;sothat if a ha_r'sbreadthbe addedto its lengthit willbreakunderits ownweightandwillbe the only rodof the kindinthe world.* Thus if,forinstance,its lengthbe a hundredtimesits breadth,youwillnotbe abletofindanother rodwhoselengthisalsoa hundredtimesits breadthand which, like the former,is just ableto sustainits own weight and no more:allthe largeroneswillbreakwhileallthe shorteroneswill be strong enough to support somethingmorethan their own weight. And thiswhichI havesaidabout the abilityto support itselfmust beunderstoodto applyalsotoothertests;sothat if a pieceof scantling[corrente] willcarrythe weightoften similarto itself,a beam [trave] having the sameproportionswillnot be ableto supportten similarbeams. Please observe, gentlemen,how faCtswhich at first seem improbablewill, even on scant explanation,drop the cloak which has hidden them and stand forth in naked and simple beauty. Who doesnot knowthat a horsefallingfrom a height of three or four cubitswillbreak hisbones,whilea dog falling fromthe sameheightor a catfroma heightof eightor ten cubits willsufferno injury? Equally harmlesswouldbe the fall of a grasshopper from a toweror the fallof an ant fromthe distance of the moon. Do not childrenfallwith impunityfromheights whichwouldcosttheir eldersabrokenlegor perhapsa fraCtured skull? And just assmalleranimalsare proportionatelystronger and morerobustthan the larger,soalsosmallerplantsare able to stand upbetter than larger. I amcertainyoubothknowthat an oak two hundred cubits [braccia] highwouldnot be ableto sustainits ownbranchesif they weredistributedas in a tree of ordinarysize;and that nature cannotproducea horseas largeas twenty ordinary horsesor a giant ten times taller than an ' [53] ordinary man unless by miracle or by greatly altering the proportionsofhislimbsand especiallyofhisbones,whichwould have to be considerablyenlargedover the ordinary. Likewise the currentbeliefthat, inthe caseof artificialmachinesthevery *The authorhereapparently meansthat the solutionis unique. [Trans.]
FIRST DAY 5 largeand the smallare equallyfeasibleand lastingisa man_fest error. Thus, for example,a smallobeliskor columnor other solidfigurecancertainlybe laiddownor setup withoutdanger ofbreaking,whilethe verylargeoneswillgotopiecesunderthe slightestprovocation,and that purelyon accountof their own weight. AndhereI must relatea circumstance whichisworthy ofyourattention asindeedare alleventswhichhappencontrary to expecCtation, especiallywhen a precautionarymeasureturns out to be a causeof disaster. A largemarblecolumnwaslaid out so that its two ends rested each upon a pieceof beam; a little laterit occurredto a mechanicthat, inorderto be doubly sureof its notbreakinginthe middleby its ownweight,itwould be wise to lay a third support midway;this seemedto all an excellentidea;but the sequelshowedthat it wasquitethe oppo- site, fornot many monthspassedbeforethe columnwasfound crackedand brokenexadtlyabovethe newmiddlesupport. Sn_P.A very remarkableand thoroughlyunexpectedacci- dent, especiallyif causedby placingthat newsupport in the middle. SALV. Surely this is the explanation,and the moment the cause is knownour surprisevanishes;forwhenthe two pieces of the columnwereplacedon levelgroundit wasobservedthat one of the end beamshad, after a longwhile,becomedecayed and sunken,but that the middleone remainedhard and strong, thus causingone halfof the columnto projecCt inthe airwithout any support. Under these circumstancesthe body therefore behaveddifferentlyfrom what it wouldhavedoneif supported only upon the first beams; becauseno matter howmuch they might have sunken the columnwouldhave gonewith them. Thisisan accidentwhichcouldnotpossiblyhavehappenedto a smallcolumn,eventhoughmadeofthe samestoneand havinga length corresponding to its thickness,i. e., preservingthe ratio betweenthicknessand lengthfoundinthe largepillar. [541 SAc_.I am quite convinced ofthe fa_s of the case,but I do not understandwhy the strength and resistanceare not multi- pliedinthe sameproportionas the material;and I am the more puzzled
6 THE TWO NEW SCIENCES OF GALILEO puzzledbecause,on the contrary,I havenoticedin othercases that the strength and resistanceagainstbreakingincreasein a largerratio than the amountof material. Thus,forinstance,if two nails be driveninto a wall,the one which is twiceas big as the otherwillsupportnot only twiceas muchweightas the other,but threeor fourtimesasmuch. SALv.Indeedyouwillnotbe far wrongifyousay eighttimes as much; nor doesthis phenomenoncontradicCt the othereven thoughinappearancethey seemsodifferent. SACR. Will you not then, Salviati,removethese difficulties and clear awaytheseobscuritiesif possible:for I imaginethat this problemofresistanceopensup a fieldofbeautifuland useful ideas;and if youare pleasedtomakethisthe subjecCt of to-day's discourseyou willplaceSimplicioand me undermany obliga- tions. SALV. I am at your serviceif onlyI cancall to mindwhat I learned from our Academician* who had thoughtmuch upon this subjecCt and accordingto his custom had demonstrated everything by geometricalmethods so that one might fairly call this a new science. For, althoughsomeof his conclusions had been reachedby others,first of all by Aristotle,these are not the most beautifuland, what is moreimportant,they had not beenprovenina rigidmannerfromfundamentalprinciples. Now,sinceI wishto convinceyou by demonstrativereasoning rather than to persuadeyou by mereprobabilities,I shallsup- posethat youare familiarwithpresent-daymechanicssofar as it is needed in our discussion. First of all it is necessary,to considerwhat happenswhena pieceofwoodor any other solid . which coheresfirmly is broken; for this is the fundamental facet, involvingthe firstand simpleprinciplewhichwemusttake forgrantedas wellknown. To graspthis moreclearly,imaginea cylinderor prism,AB, made of wood or other solid coherent material. Fasten the upper end, A, so that the cylinderhangs vertically. To the lowerend, B, attach the weight C. It is clear that however great they may be, the tenacity and coherence[tenacit_e • I. e. Galileo: Theauthorfrequently refersto himself underthis name. [Tran_r.]
FIRST DAY 7 [55] eoeren_] between thepartsofthissolid, solongastheyarenot infinite, canbeovercome by thepulloftheweight C, aweight which canbeincreased indefinitely untilfinally thesolid breaks likea rope.Andas inthecaseoftheropewhose strength we knowto be derived froma multitudeof hempthreadswhich compose it, sointhecaseofthewood, weobserve itsfibres and filaments runlengthwise and render it muchstronger than a hempropeof thesamethickness.But in the caseof a stoneormetallic cylinder wherethe' coherence seemsto be stillgreater the cement whichholdsthe partstogether mustbe some- thingotherthan filaments and fibres;and yet eventhiscanbebroken bya strong pull. Srme.If thismatterbeasyousayI canwell understand thatthefibres ofthewood,beingas longas thepieceofwooditself,renderit strong and resistantagainstlargeforcestendingto breakit. But howcan onemakea ropeone hundredcubitslongoutofhempen fibres which arenotmorethantwoor threecubitslong,and stillgiveit somuchstrength ? Besides, I should begladtohearyouropinion as tothemanner in whichthepartsof metal,stone,andotherma- terialsnot showing a filamentous strucCture are Fig.i put together;for,if I mistakenot,theyexhibitevengreater tenacity. SALV. To solve theproblems which youraiseit willbeneces- sarytomakea digression intosubjecCts which havelittlebearing uponourpresentpurpose. SAcg.But if,by digressions, wecanreachnewtruth,what harmis there in makingonenow,so thatwe maynot lose thisknowledge, remembering that suchan opportunity, once omitted, maynotreturn;remembering alsothatwearenottied down toa fixed andbriefmethod butthatwemeetsolely forour ownentertainment? Indeed,whoknows butthatwemaythus [S6] frequently
8 THE TWO NEW SCIENCESOF GALILEO frequently discover something moreinteresting andbeautiful thanthesolution originally sought._ I begofyou,therefore, to grantthe request of Simplicio, whichis alsomine;forI amno lesscurious anddesirous thanheto learnwhatis thebinding materialwhichholdstogetherthepartsof solidssothat they canscarcely beseparated.Thisinformation is alsoneededto understand the coherence of thepartsof fibresthemselves of Which some solids arebuiltup. SAJ_V. I am at yourservice,sinceyoudesireit. The first question is,Howarefibres, eachnot morethantwoor three cubitsinlength, sotightlyboundtogetherinthecaseof arope onehundredcubitslongthatgreatforce[violent] isrequired to breakit? Nowtellme,Simplicio, canyounotholda hempen fibreso tightlybetween yourfingers that I, pullingby theotherend, wouldbreakit beforedrawing it awayfromyou? Certainly youcan. Andnowwhenthefibres ofhempareheldnotonlyat theends,but aregrasped bythesurrounding medium through- outtheirentirelengthisit notmanifestly moredii_cult totear themloose fromwhatholdsthemthantobreakthem? Butin thecaseoftheropetheveryacCt oftwisting causes thethreads tobindoneanother insuchawaythatwhen theropeisstretched witha greatforcethe fibresbreakratherthan separatefrom eachother. At thepointwherea ropepartsthefibresare,as everyone knows, veryshort,nothing likea cubitlong, astheywoaldbeif the partingof the ropeoccurred, not by thebreaking of the filaments, but bytheirslipping oneovertheother. SAGR. In confirmation ofthisit mayberemarked that ropes sometimes breaknot by a lengthwise pull but by excessive twisting.This,it seems tome,isaconclusive argument because the threadsbindoneanothersotightlythat thecompressing fibres donotpermitthose whicharecompressed to lengthen the spiralseventhat littlebitbywhichit is necessary forthemto lengthen inorder to surround therope which, ontwisting, grows shorterandthicker. SALv. Youarequiteright. Nowseehowonefa_ suggests another
FIRST DAY 9 another. Thethreadheldbetween thefingers doesnot yield [ST] toonewhowishes to draw it awayevenwhenpulledwithcon- siderable force, but resistsbecause it is heldbackby a double compression, seeingthat the upperfinger presses againstthe lower as hamasthelower againsttheupper.Now,ifwecould retainonlyoneof thesepressures thereis no doubtthatonly half the original resistance wouldremain;but sincewe are _ not able,by lifting,say,the upperfinger, to removeoneof thesepressures withoutalsoremoving the other,it becomes necessary to preserve oneof themby meansof a newdevice whichcausesthe threadto pressitselfagainstthe finger or againstsome othersolid bodyuponwhichit rests; andthusit is brought aboutthattheveryforce which pulls it inorderto snatchit awaycompresses it moreand moreas the pullincreases. This is accomplished by wrappingthe thread aroundthe solidinthe mannerof a spiral; _I_ andwillbebetterunderstood bymeansofa figure.LetABandCDbetwocylinders be- tween whichis stretched thethreadEF:and _ O forthesakeof greaterclearness wewillim- agineit to be a smallcord. If thesetwo cylinders be pressedstronglytogether,the cordEF,whendrawnbytheendF,willun- doubtedly standa considerable pullbeforeit slipsbetweenthe two compressing solids. But ifweremove oneofthesecylinders the cord,thoughremaining in contacCt withthe other,willnot therebybe prevented from slipping freely.Onthe otherhand,if one holdsthecordloosely againstthetopof the Fig. 2 cylinderA, windsit in the spiralformAFLOTR, and then pullsit by the endR, it is evident that thecordwillbeginto bindthe cylinder; the greaterthenumber of spiralsthemore tightlywillthe cordbe pressedagainstthe cylinderby any given pull. Thusasthenumber of turnsincreases, thelineof contacCt
Io THE TWO NEW SCIENCESOF GALILEO contactbecomes longerandin consequence moreresistant;so thatthecordslipsandyieldsto thetractiveforce withincreas- ingdifficulty. [58] Isitnotclearthatthisisprecisely thekindofresistance which onemeetsinthecaseofa thickhempropewhere thefibres form thousands andthousands of similar spirals?And,indeed,the qbinding effecCt of theseturnsis sogreatthata fewshortrushes woventogetherintoa fewinterlacing spiralsformoneof the strongestof ropes whichI believetheycallpackrope[susta]. SAoR. Whatyousayhascleared up twopointswhichI did notpreviously understand.Onefactis howtwo,or at most three,turnsofa ropearoundtheaxleofa windlass cannot only holdit fast,but canalsopreventit fromslipping whenpulled by the immense forceof the weight[forzadelpeso] whichit sustains; andmoreover how,byturningthewindlass, thissame axle,by merefricCtion of theropearoundit, canwindup and lifthugestoneswhilea mereboyi'sableto handle theslack of therope. TheotherfaCt hastodowith asimple butclever device, invented byayoung kins- manof mine,forthe purpose of descending froma window by meansof a ropewithoutlacerating the palmsofhishands,ashadhappened tohimshortly before andgreatlytohisdiscomfort.Asmall sketch willmakethis clear. He tooka woodencylinder, AB,aboutasthickasa walking stickandaboutone spanlong:on thishecut a spiralchannel of about oneturnanda half,andlargeenough tojust receive theropewhich hewished touse. Havingintroduced theropeat theendAandledit outagainat theend BB, heenclosed boththe cylinderandthe ropeina caseofwoodor tin,hingedalongthe81de sothat it Fig. 3 couldbe easilyopenedand closed. After hehad Iastenedtheropeto afirmsupportabove,he could, on grasp- ingandsqueezing thecasewithbothhands,hangbyhisarms. The pressure on therope,lyingbetweenthe caseand thecyl- inder,wassuchthat he could,at will,eithergraspthe case more
FIRST DAY II moretightlyand holdhimselffromslipping, or slackenhis hold anddescend asslowly ashewished. IS9] SALV. A truly ingenious device! I feel,however, that for a complete explanation otherconsiderations mightwellenter; yetI mustnotnowdigress uponthisparticular topicsinceyou arewaitingtohearwhatI thinkaboutthebreaking strength of othermaterials which,unlikeropesandmostwoods, do not showa filamentous structure. The coherence of thesebodies is,in myestimation, produced by othercauses whichmaybe grouped undertwoheads. Oneis that much-talked-of repug- nance whichnatureexhibits towards avacuum; butthishorror of a vacuumnot beingsufficient, it is necessary to introduce anothercause intheformof agluey or viscous substance which bindsfirmly togetherthecomponent partsofthebody. FirstI shallspeakofthevacuum, demonstrating bydefinite experiment thequalityandquantityofitsforce[o/rt_].If you taketwohighlypolished andsmooth platesofmarble, metal,or glassandplacethemfacetoface,onewillslideovertheother withthegreatestease,showing conclusively that thereisnoth- ingof aviscous naturebetween them. Butwhenyouattempt to separatethemandkeepthemat a constantdistanceapart, youfindtheplatesexhibit sucha repugnance toseparation that theupperonewillcarrythelower onewithit andkeepit lifted indefinitely, evenwhenthelatterisbigandheavy. This experiment showsthe aversionof naturefor empty space,evenduring thebriefmomentrequired fortheoutside air to rushinandfillup theregion between thetwoplates. It is alsoobserved that if twoplatesarenot thoroughly polished, theircontactisimperfect sothatwhenyouattempttoseparate them slowly the onlyresistance offeredis that of weight;if, however, the pullbe sudden, then the lowerplaterises,but quickly fallsback,havingfollowed theupper plateonlyforthat veryshortintervalof timerequired forthe expansion of the smallamountof air remaining betweenthe plates,in conse- quence oftheirnotfitting, andfortheentrance ofthesurround- ingair. Thisresistance whichis exhibited between the two plates
Iz THE TWO NEW SCIENCESOF GALILEO platesisdoubtless likewise present between thepartsofa solid, and enters,at leastin par[,as a concomitant causeof their coherence. [6o] SAGR. Allow meto interruptyoufora moment, please; for I wantto speakof something which justoccurs tome,namely, whenI seehowthelower platefollows theupperoneandhow rapidlyit is lifted,I feelsurethat, contrary to theopinion of manyphilosophers, including perhapsevenAristotlehimself, motionina vacuumis notinstantaneous.If thisweresothe twoplatesmentioned abovewouldseparatewithoutany re- sistance whatever, seeingthat thesameinstantof timewould suffice fortheirseparation andforthesurrounding medium to rushinandfillthevacuum between them. The fa&that the lowerplatefollows theupperoneallows us to infer,not only that motionin a vacuumis not instantaneous, but alsothat, betweenthetwoplates,a vacuum reallyexists,at leastfora veryshorttime,sufficient to allow thesurrounding mediumto rushinandfillthevacuum; foriftherewerenovacuumthere would benoneedofanymotion inthemedium.Onemustadmit thenthat a vacuumis sometimes produced by violentmotion [violenza] or contraryto the lawsof nature,(although in my opinion nothing occurs contrary tonatureexcept theimpossible, andthat never occurs). But hereanotherdifficulty arises. Whileexperiment con- vincesmeofthecorrecCtness of thisconclusion, mymindis not entirelysatisfied as to thecauseto whichthis effe&is to be attributed. For the separationof the platesprecedesthe formation of thevacuumwhichis produced as a consequence ofthisseparation; andsince it appears tomethat,intheorderof nature,the causemustprecedetheeffe&, eventhoughit ap- pearsto follow inpointoftime,andsinceevery positive effecCt musthavea positive cause,I do notseehowtheadhesion of twoplatesandtheirresistance to separation--acCrual fa_s---can bereferredto a vacuumas cause whenthisvacuumis yet to follow.According to theinfallible maximof thePhilosopher, thenon-existent canproduce noeffe&. Simp.
FIRST DAY 13 Sire,. Seeingthat youacceptthis axiomofAristotle,I hardly thinkyouwillreje_ anotherexcellent and reliable maximof his, namely,Nature undertakesonly that which happenswithout resistance;and inthis saying,it appearsto me,youwillfindthe solutionof your difficulty. Sincenature abhorsa vacuum,she preventsthat fromwhicha vacuumwouldfollowas a necessary consequence.Thus it happensthat naturepreventsthe separa- tionofthe twoplates. [6i] SACR. Nowadmittingthat what Simplicio saysisan adequate solutionof my difficulty,it seemsto me, ifI may be allowedto resume my former argument, that this very resistanceto a vacuumought to be sufficientto holdtogetherthe parts either of stoneor of metalor the parts of any othersolidwhichisknit togethermorestronglyandwhichismoreresistanttoseparation. If for one effe_ there be onlyone cause,or if,morebeingas- signed,they canbe reducedto one,thenwhyisnotthis vacuum whichreallyexistsa sufficientcausefor allkindsof resistance ? SALV. I do not wishjust nowto enter this discussionas to whether the vacuum alone is sufficientto hold together the separateparts of a solidbody;but I assureyouthat the vacuum whichacCts as a sufficient causeinthe caseofthetwoplatesisnot alonesufficientto bind togetherthe partsof a solidcylinderof marble or metal which, whenpulled violently,separatesand divides. Andnow if I finda methodof distinguishing thiswell known resistance,dependingupon the vacuum, from every other kind which might increasethe coherence,and if I show you that the aforesaidresistancealoneis not nearlysufficient for such an effect, willyou not grant that we are bound to introduceanother cause. _ Help him, Simplicio,sincehe does not knowwhat replytomake. SIMP. Surely,Sagredo'shesitationmust be owingto another reason,fortherecanbe nodoubtconcerning a conclusion which isat oncesoclearandlogical. SACra. Youhaveguessedrightly,Simplicio. I waswondering whether, if a millionof gold each year from Spain were not sufficientto pay the army, it might not be necessary to make
I4 THE TWO NEW SCIENCES OF GALILEO make provisionother than small coin for the pay of the soldiers.* But go ahead,Salviati;assumethat I admityour conclusion and showusyourmethodof separatingtheacCtion of thevacuum from other causes;and by measuringit showus how it is not sufficient to producethe effectin question. SALV. Your good angel assistyou. I willtell you how to separatethe forceof the vacuumfrom the others, and a{ter- wards how to measure it. For this purposelet us considera continuoussubstancewhoseparts lackall resistanceto separa- tionexceptthat derivedfroma vacuum,suchas isthe casewith water,a fact fullydemonstratedbyourAcademician inoneof his treatises. Whenever a cylinderofwaterissubjectedto apulland [62] offersa resistanceto the separation of itsparts this canbe attrib- uted tonoothercausethantheresistance of the /k_j vacuum. In orderto try suchan experiment I have invented a devicewhichI can better explainby meansof a sketchthan by mere words. Let CABDrepresentthe crosssection of a cylindereither of metal or, preferably, of glass,hollowinsideand accuratelyturned. G I-I Into this is introduced a perfec°dyfitting i C . Dcylinderof wood,representedin crosssection by EGHF, and capableof up-and-downmo- tion. Through the middleof this cylinderis boreda holeto receivean ironwire,carrying a hook at the end K, while the upper end of the wire, I, is providedwith a conical head. The woodencylinderis countersunk Fig.4 at the top so as to receive,witha perfect fit, the conicalhead I of the wire,IK,whenpulleddown by theendK. NowinsertthewoodencylinderEH inthe hollow cyllnderAD, soas not to touchtheupperend of thelatterbut to leavefreea spaceof two or threefinger-breadths; this spaceis to be filled *Thebearing ofthisremark becomes clear onreading whatSalviati says onp. 18below.[Trans.]
FIRST DAY 15 withwaterbyholding thevessel withthemouthCD upwards, pushing down onthestopper EH,andat thesametimekeeping theconical headofthewire,I, away fromthehollow portion of thewooden cylinder.Theairisthusallowed toescape alongside theironwire(which doesnotmakea close fit)assoonas one pressesdownon the woodenstopper. The air havingbeen allowed to escape andtheironwirehaving beendrawn backso that it fits snugly againstthe conical depression inthewood, invert thevessel, bringing itmouthdownwards, andhangonthe hookK a vesselwhichcanbe filled withsandor anyheavy materialin quantitysufficient to finallyseparatethe upper surface ofthestopper, EF, fromthelowersurface ofthewater towhichit wasattached onlyby theresistance ofthevacuum. Nextweighthe stopperandwiretogetherwiththeattached vesseland its contents; we shallthenhavethe forceof the vacuum [forza ddvacuo].Ifoneattaches toacylinder ofmarble [63] or glassa weight which, together withtheweight ofthemarble or glassitself,is just equalto the sumof theweights before mentioned, andifbreaking occurs weshallthenbejustified in sayingthatthevacuum aloneholdsthepartsofthemarble and glasstogether; but ifthisweight doesnotsuffice andifbreaking occursonlyafteradding, say,fourtimesthisweight, weshall thenbe compelled to saythat thevacuumfurnishes onlyone fifthofthetotalresistance [resf._ema]. SLurP. Noonecandoubtthecleverness ofthedevice; yetit presentsmanydifficulties which makemedoubtitsreliability. Forwhowillassure usthattheairdoesnotcreepinbetween the glassandstopperevenif it is wellpackedwithtowor other yielding material._ I question alsowhether oiling withwaxor turpentine willsuffice tomakethecone, I,fitsnugly onitsseat. Besides, maynot the partsof the waterexpandand dilate? Whymaynottheairorexhalations orsome othermoresubtile substances penetrate theporesofthewood,orevenoftheglass itself? SAT.v. Withgreatskillindeed hasSimplicio laidbefore usthe difficulties; andhehasevenpartlysuggested howtopreventthe air
x6 THE TWO NEW SCIENCESOF GALILEO airfrompenetrating thewood or passing between thewood and theglass. Butnowletmepoifitoutthat, asourexperience in- creases, weshalllearnwhetheror notthesealleged difficulties reallyexist. For if,as is thecasewithair,wateris bynature expansible, although onlyunderseveretreatment, weshallsee thestopperdescend; and ifweput a smallexcavation in the upperpart oftheglassvessel, suchas indicated byV,thenthe air or anyothertenuousandgaseous substance, whichmight penetratethe poresof glassor wood,wouldpassthroughthe waterandcolle& inthisreceptacle V. Butifthesethingsdonot happenwemayrestassured thatourexperknent hasbeenper- formedwithpropercaution;andweshalldiscover that water doesnot dilateand that glassdoesnot allowany material, however tenuous, topenetrate it. SAGm Thanks tothisdiscussion, Ihavelearned thecause ofa certaineffe& whichI havelongwondered at anddespaired of understanding.I oncesawa cistern whichhadbeenprovided witha pumpunderthe mistakenimpression that the water mightthusbedrawnwithlesseffort oringreater quantitythan bymeansoftheordinary bucket. Thestockofthepumpcar- [64] rieditssucker andvalveintheupperpartsothatthewaterwas liftedby attra&ion andnotbya pushasisthecase withpumps inwhichthesuckerisplacedlowerdown.Thispumpworked peffedtly solong asthewaterinthecistern stoodabove acertain level;but belowthis levelthepumpfailedto work. WhenI firstnoticed thisphenomenon I thoughtthemachine wasoutof order;but theworkman whomI called in to repairit toldme the defecCt wasnot in the pumpbut in thewaterwhichhad fallentoolowtoberaisedthrough sucha height;andheadded that it wasnot possible, eitherby a pumpor by any other machine working on theprinciple of attra&ion, to liftwatera hair'sbreadthaboveeighteencubits;whetherthe pump be largeor smallthisis theextreme limitof thelift. Upto this timeI hadbeensothoughtless that,although I knewa rope,or rodofwood,orof iron,if sufficiently long,wouldbreakby its ownweight whenheldbytheupperend,it never occurred tome " that
FIRST DAY 17 thatthesamethingwould happen, onlymuchmoreeasily, toa columnof water. And reallyis not that thingwhichis at- tra_ed inthepumpa column ofwaterattachedat theupper endandstretched moreandmoreuntilfinally apointisreached whereit breaks,likea rope,onaccount ofitsexcessive weight ? SALV. That isprecisely thewayit works; thisfixed elevation ofeighteen cubits istrueforanyquantityofwaterwhatever, be thepumplargeor smallor evenasfineas a straw. Wemay therefore saythat, onweighing thewatercontained in a tube eighteen cubitslong,no matterwhatthe diameter, we shall obtainthevalueoftheresistance ofthevacuum ina cylinder of anysolidmaterialhavinga boreof thissamediameter.And havinggonesofar, let us seehoweasyit is to findto what lengthcylinders ofmetal,stone,wood, glass,etc.,ofanydiam- etercanbe elongated withoutbreaking by theirownweight. [6S] Takeforinstance a copper wireofanylengthandthickness; fixthe upperend andto theotherend attacha greaterand greaterloaduntilfinally thewirebreaks;letthemaximum load be, say,fiftypounds.Thenit is clearthat if fiftypoundsof copper,inadditionto theweightofthewireitselfwhichmay be, say,z/sounce,is drawnout intowireof thissamesizewe shallhavethegreatest length ofthiskindofwirewhich cansus- tainitsownweight. Suppose thewirewhichbreaks to beone cubitin lengthandI/sounceinweight;thensinceit supports 5olbs.inadditionto itsownweight,i.e.,48ooeighths-of-an- ounce,it follows that allcopper wires, independent ofsize,can sustainthemselves up to a lengthof48Olcubitsandnomore. Sincethen a copperrod can sustainitsownweightup to a lengthof48Olcubitsit follows that thatpart ofthebreaking strength [resistenza] which depends uponthevacuum, comparing itwiththeremaining facetors ofresistance, isequaltotheweight ofa rodofwater,eighteen cubits longandasthickasthecopper rod. If,forexample, copper isninetimesasheavyaswater,the breaking strength[resistenza allostrappars.z] of anycopperrod, insofarasit depends uponthevacuum, asequalto theweight of twocubitsof thissamerod. By a similar methodonecan find
I8 THE TWO NEW SCIENCESOF GALILEO findthemaximum lengthofwireorrodofanymaterial which willjust sustainitsownweight,andcanat the sametimedis- cover thepartwhichthevacuum playsinitsbreaking strength. SACR. It stillremains foryouto tellus uponwhatdepends theresistance tobreaking, otherthanthatofthevacuum; what is the glueyor viscous substance whichcementstogetherthe partsof the solid? For I cannotimagine a gluethat willnot burnup ina highlyheatedfurnace intwoor threemonths,or certainly withintenor a hundred.Forifgold,silverandglass arekeptfora longwhileinthemoltenstateandareremoved fromthe furnace,theirparts,on cooling, immediately reunite and bind themselves togetheras before. Not only so,but whatever difficulty arises withrespe_tothecementation ofthe partsof theglass arisesalsowithregard tothepartsoftheglue; inotherwords, whatisthatwhichholdsthesepartstogetherso firmly? [661 SALv. A littlewhile ago,I expressed thehopethatyourgood angelmightassistyou. I nowfindmyself inthesamestraits. Experiment leavesno doubtthat the reasonwhytwoplates cannotbeseparated, exceptwithviolent effort,isthat theyare heldtogetherby theresistance ofthe vacuum; andthe same canbesaidof twolargepiecesof amarble or bronzecolumn. Thisbeingso,I donotseewhythissame cause maynotexplain thecoherence ofsmaller partsandindeed of theverysmallest particles of thesematerials.Now,since eacheffe_musthave onetrueandsufficient cause andsince Ifindnoothercement, am I notjustified intryingtodiscover whether thevacuum isnot a sufficient cause? S_. But seeingthat youhavealready provedthat there- sistancewhichthe largevacuumoffersto the separation of twolargepartsofasolid isreally verysmall incomparison with thatcohesive force which bindstogether themostminute parts, whydo youhesitateto regardthis latter as something very different fromtheformer ? S_v. Sagredo hasalready [p.I3 above] answered thisques- tionwhenhe remarked that eachindividual soldier wasbeing paid
FIRST DAY 19 paidfromcoincoiled-ted by a general taxofpennies andfarth- ings,whileevena million ofgoldwould not suffice to paythe entirearmy. Andwhoknowsbut that theremay be other extremely minutevacuawhichaffecCt thesmallest particles so thatthatwhich bindstogether thecontiguous partsisthrough- outofthesamemintage ? Letmetellyousomething which has justoccurred tomeandwhich I donotoffer asanabsolute facet, but ratheras a passing thought, stillimmature andcalling for morecareful consideration. Youmaytakeofit whatyoulike; andjudgetherestasyouseefit. Sometimes whenI haveob- servedhowfirewindsits way in betweenthe mostminute particles ofthisorthatmetaland,eventhough thesearesolidly cemented together,tearsthemapartandseparates them,and whenI haveobserved that,onremoving thefire,theseparticles reunitewiththesametenacityas at first,withoutany lossof quantityin thecaseofgoldandwithlittlelossinthecaseof othermetals, eventhoughthesepartshavebeenseparated fora longwhile,I havethoughtthattheexplanation mightlieinthe factthat the extremely fineparticles of fire,penetrating the slenderporesof themetal(toosmallto admiteventhefinest particles of air or of manyotherfluids), wouldfillthe small intervening vacuaandwould setfreethesesmallparticles from the attracCtion whichthesesamevacuaexertuponthemand which prevents theirseparation.Thustheparticles areableto [671 movefreely sothatthemass[rnassa] becomes fluidandremains soaslong astheparticles offireremain inside; butiftheydepart andleavetheformer vacuathentheoriginal attraction [attraz- zione] returnsandthepartsareagaincemented together. In replytothequestion raised bySimplicio, onemaysaythat althougheachparticularvacuumis exceedingly minuteand therefore easily overcome, yettheirnumber issoextraordinarily greatthat theircombined resistance is,soto speak,multipled almostwithoutlimit. The natureand the amountof force [forza] whichresults[risulta] fromadding togetheranimmense numberof smallforces[debolissimi rnornent_] is clearlyillus- tratedbythefa_ thataweight ofmillions ofpounds, suspended by
20 THE TWO NEW SCIENCES OF GALILEO by great cables,is overcomeand lifted,whenthe south wind carries innumerableatoms of water, suspendedin thin mist, whichmovingthroughthe airpenetratebetweenthefibresof the tense ropes in spite of the tremendousforceof the hanging weight. When these particles enter the narrow pores they swell the ropes, thereby shorten them, and perforcelift the heavymass[mole]. SAcR. There canbe no doubtthat any resistance,solongas it is not infinite,may be overcomeby a multitudeof minute forces. Thus a vast numberof ants might carryashorea ship laden with grain. And sinceexperienceshowsus daily that one ant caneasilycarry onegrain,it isclearthat the numberof rains in the shipis not infinite,but fallsbelowa certainlimit. you take anothernumberfouror six timesas great, and if you set to worka corresponding numberof ants theywillcarry the grainashoreand the boat also. It istrue that thiswillcall fora prodigiousnumberof ants,but in my opinionthis ispre- ciselythe casewith the vacua which bind togetherthe least particlesofa metal. SALV. But even if this demandedan infinitenumberwould you stillthink 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 seeif wecannotprovethat withina finiteex- tent it ispossibletodiscoveran infinitenumberofvacua. Atthe sametimeweshallat least reacha solutionof the mostremark- ableof all that list of problemswhichAristotle himselfcalls wonderful;I referto hisQuestions in Mechanics.This solution may be no lessclearand conclusive than that whichhe himself givesand quitedifferentalsofromthat socleverly expoundedby themostlearnedMonsignordiGuevara.* First it is necessaryto considera proposition, not treated by others,but uponwhichdependsthe solutionofthe problemand from which, if I mistake not, we shallderiveother new and remarkable facts. For the sake of clearnesslet us draw an *Bishop ofTeano; b.x56x ,d.I64I. [Trans.]
FIRST DAY 2I accurate figure. AboutG as a centerdescribe anequiangular andequilateral polygon ofanynumber ofsides, saythehexagon ABCDEF. Similarto this and concentric with it, describe anothersmaller onewhich weshallcallHIIZT.MN. Prolong the F , ..... ff"i ............ 4 , , . T -- ,,I ! t , I ( . ": iim] B Fig. 5 sideAB,of thelargerhexagon, indefinitely towardS; in like mannerprolong thecorresponding sideHI ofthe smaller hex- agon,inthe samedirecCtion, sothat thelineHT isparallelto AS;andthroughthe centerdrawthelineGVparallel to the othertwo. Thisdone,imagine thelargerpolygon torollupon [69] thelineAS,carrying withit thesmaller polygon.It isevident that,ifthepointB,theendofthesideAB,remains fixed at the beginning oftherotation,thepointAwillriseandthepointC willfalldescribing thearcCQuntilthesideBCcoincides with thelineBQ,equaltoBC. Butduring thisrotation thepointI, onthesmaller polygon, willriseabove thelineITbecause IBis oblique toAS;andit willnotagainreturn tothelineITuntilthe pointC shallhavereachedtheposition Q. ThepointI, having described thearcIOabove thelineHT,willreachtheposition Oat
2z THE TWO NEW SCIENCES OF GALILEO 0 at the sametimethe sideIK assumesthe position0P; but in the meantimethe centerG has traverseda path aboveGVand doesnot return to it until it has _ompletedthe arc GC. This stephavingbeentaken,the largerpolygonhas beenbroughtto rest withits sideBC coinciding withthe lineBQwhilethe side IK of the smallerpolygonhas beenmadeto coincidewith the lineOP,havingpassedoverthe portionI0 withouttouchingit; alsothe centerG willhavereachedthe positionC after having traversedallits courseabovethe parallellineGV. Andfinally the entire figurewillassumea positionsimilarto the first,so that ifwecontinuethe rotationand cometo the next step,the sideDC of the largerpolygonwillcoincidewiththe portionQX , and the sideKL of the smallerpolygon, havingfirstskippedthe arc PY,willfallon YZ, whilethe centerstillkeepingabovethe lineGV willreturn to it at R after havingjumpedthe interval CR. At the endofonecompleterotationthe largerpolygonwill havetraced upon the lineAS,withoutbreak,sixlinestogether equal to its perimeter; the lesserpolygonwill likewisehave imprintedsixlinesequalto its perimeter,but separatedby the interpositionof five arcs, whose chords represent the parts of HT not touchedby the polygon:the centerG neverreaches the lineGV exceptat sixpoints. From this it isclearthat the spacetraversedby the smallerpolygonis almostequalto that traversedby the larger,that is, the lineHT approximatesthe lineAS,differingfrom it onlyby the lengthof one chordof one ofthesearcs,provided weunderstand the lineI-ITto include the fiveskippedarcs. Now this exposition whichI have givenin the caseof these hexagonsmust be understoodto be applicableto all other polygons, whatever the numberof sides,providedonlytheyare [70] similar, concentric,and rigidlyconnecCted, so that when the greateronerotatesthe lesserwillalsoturn howeversmallit may be. Youmust alsounderstandthat the linesdescribedby these two are nearlyequalprovidedwe includein the spacetraversed by the smallerone the intervalswhichare not touchedby any part ofthe perimeterofthis smallerpolygon. Let
FIRST DAY z3 Let a largepolygonof, say, one thousandsides makeone completerotationand thus layoffa lineequalto its perimeter; at the sametimethe smallonewillpassoveran approximately equal distance, made up of a thousand smallportions each equalto oneof its sides,but interruptedby a thousandspaces which,in contrastwiththe portionsthat coincide withthe sides of the polygon,we may call empty. Sofar the matter isfree fromdifficulty or doubt. But nowsupposethat about any center,say A,we describe two concentricand rigidlyconneCtedcircles;and supposethat from the points C and B, on their radii, there are drawn the tangentsCEand BF and that throughthe centerAthe lineAD is drawnparallelto them, then if the large circlemakesone completerotation alongthe lineBF, equalnotonly to its cir- cumference but alsoto the othertwo linesCE andAD, tellme what the smallercirclewilldoand alsowhat the centerwilldo. Asto the center it willcertainlytraverseand touch the entire lineAD whilethe circumference of the smallercirclewillhave measuredoffby its pointsof contaCtthe entirelineCE,just as wasdoneby theabovementionedpolygons.Theonlydifference is that the lineI-ITwasnot at everypointin contactwiththe perimeterof the smallerpolygon,but therewereleftuntouched as manyvacant spacesas therewerespacescoinciding withthe sides. But hereinthe caseofthe circles the circumference ofthe smalleroneneverleavesthe lineCE, sothat nopartof the latter isleftuntouched,noristhereevera timewhensomepointonthe circleisnot in contaCt withthe straightline. Hownowcanthe smallercircletraversea lengthgreater than its circumference unlessit goby jumps? 8AGmIt seemsto methat onemaysaythat just as thecenter ofthe circle,by itself,carriedalongthe lineAD isconstantlyin contac2 withit, althoughit isonlya singlepoint,sothepointson the circumference of the smaller circle,carried alongby the motionof the largercircle,wouldslideoversomesmallparts of the lineCE. : [7I] : SALV. There are two reasonswhy this cannothappen. First because ?
24 THE TWO NEW SCIENCESOF GALILEO because thereis nogroundforthinking that onepointof con- taCt,suchas that at C, ratherthan another, shouldslipover certain portionsofthelineCE. Butifsuchslidings alongCE didoccurtheywould beinfiniteinnumber sincethepointsof contaCt (being merepoints)areinfinite innumber:aninfinite number offiniteslipswillhowever makeaninfinitely longline, while asamatteroffaCtthelineCEisfinite. Theotherreason isthat asthegreatercircle,initsrotation, changes itspointof contactcontinuously thelesser circle mustdothesame because Bistheonlypointfrom whichastraightlinecanbedrawntoA andpassthrough C. Accordingly thesmallcircle mustchange itspointofcontactwhenever thelargeonechanges: nopointof thesmallcircletouches thestraightlineCE inmorethanone point. Notonlyso,but evenin therotationof thepolygons therewasnopointon theperimeter ofthesmaller whichcoin- cidedwithmorethanonepointonthe linetraversed by that perimeter; this is at onceclearwhenyouremember that the lineIKisparallel toBCandthattherefore IKwillremain above IPuntilBCcoincides withBQ,andthatIKwillnotlieuponIP except attheveryinstant whenBCoccupies theposition BQ;at thisinstanttheentireline IKcoincides withOPandimmediately afterwards risesabove it. SAOl_. Thisisaveryintricate matter. I seenosolution.Pray explain it tous. SALV. Let usreturnto theconsideration ofthe above men- tionedpolygons whosebehavior wealready understand.Now inthecaseofpolygons withIOOOOO sides, thelinetraversed by the perimeterof thegreater,i. e.,the linelaiddownby its IOOCXX) sides oneafteranother, isequaltothelinetracedoutby theIOCX:_ sidesofthesmaller, provided weinclude theIO(Xx_ vacantspaces interspersed.Soin thecaseofthecircles, poly- gonshavingan infinitude of sides,the linetraversed by the continuously distributed[continuamente dispostz] infinitude of sidesisinthegreatercircle equalto thelinelaiddownby the infinitude of sidesinthe smaller circle but withtheexception that theselatteralternatewithemptyspaces;and sincethe sidesarenotfiniteinnumber, butinfinite, soalsoaretheinter- vening
FIRST DAY 25 veningempty spacesnotfinitebut infinite. The linetraversed by the largercircleconsiststhen of an infinitenumberof points whichcompletely fillit; whilethat whichistracedbythe smaller circleconsistsof an infinitenumberof pointswhichleaveempty spacesand only partly fill the line. And here I wishyou to observethat after dividingand resolvinga line into a finite numberof parts, that is,intoa numberwhichcanbecounted,it [72] isnotpossibleto arrangethem againintoa greaterlengththan that whichthey occupiedwhenthey formeda continuum[con- tinuate]and were conne_ed without the interpositionof as many empty spaces. But if weconsiderthe lineresolvedinto an infinitenumberof infinitelysmalland indivisible parts, we shallbe ableto conceivethe lineextendedindefinitely by the interposition,not of a finite,but of an infinitenumberof in- finitelysmallindivisible emptyspaces. Nowthiswhichhasbeensaidconcerning simplelinesmust be understoodto holdalsointhe caseof surfacesand solidbodies, it being assumedthat they are made up of an infinite,not a finite,number of atoms. Such a body once divided into a finitenumberofparts itisimpossible toreassemble themsoas to occupymore space than beforeunless we interposea finite numberof empty spaces,that is to say, spacesfree from the substanceof whichthe solidis made. But if we imaginethe body, by someextreme and final analysis,resolvedinto its primaryelements,infinitein number,then weshallbe ableto think of them as indefinitelyextended in space,not by the interpositionof a finite,but of an infinitenumberof empty spaces. Thus one can easilyimaginea smallball of goldex- panded into a very largespacewithout the introducCtion of a finite number of empty spaces,alwaysprovidedthe gold is madeupof aninfinitenumberof indivisible parts. SIM1,. It seemsto me that you are travellingalongtoward thosevacuaadvocatedby a certainancientphilosopher. SAzv.But youhavefailedto add,"whodeniedDivineProvi- dence,"an inapt remarkmadeon a similaroccasionby a cer- tain antagonistofour Academician. Simp.
26 THE TWO NEW SCIENCESOF GALILEO Sr_P.I noticed,andnotwithoutindignation, the rancorof thisill-natured opponent;furtherreferences to theseaffairsI omit,not onlyas a matterof goodform,but alsobecause I know howunpleasant theyaretothegoodtemperedandwell orderedmindof oneso religious and pious,soorthodox and God-fearing asyou. Butto returnto oursubject,yourprevious discourse leaves withmemanydifficulties whichI amunableto solve. First among theseis that,ifthecircumferences ofthetwocircles are equalto the twostraightlines,CE and BF, the latter con- sideredasa continuum, theformeras interrupted withan in- finityofemptypoints,I donotseehowit ispossible to saythat thelineADdescribed bythecenter,andmadeupofaninfinity ofpoints, isequaltothiscenter which isa single point. Besides, thisbuilding upof linesoutofpoints, divisibles outofindivisi- bles,andfinites outofinfinites, offers meanobstacle difficult to avoid;andthe necessity of introducing a vacuum,soconclu- sively refutedbyAristotle, presents thesame difficulty. [73] SAr.V. Thesedifficulties arereal;andtheyarenot theonly ones. Butlet usremember thatwearedealing withinfinities and indivisibles, both of whichtranscendour finiteunder- standing, theformer on account oftheirmagnitude, thelatter because oftheirsmallness.In spiteofthis,mencannotrefrain fromdiscussing them,eventhough it mustbedoneina round- aboutway. Therefore I alsoshould liketotaketheliberty topresent some of my ideaswhich,thoughnot necessarily convincing, would, onaccount of theirnovelty, at least,provesomewhat startling. But sucha diversion mightperhaps carryustoofarawayfrom thesubjectunderdiscussion andmighttherefore appear toyou inopportune andnotverypleasing. SACR. Prayletusenjoytheadvantages andprivileges which comefromconversation between friends, especially uponsub- jects freelychosenand not forceduponus, a matter vastly different fromdealing withdeadbooks whichgiverisetomany doubtsbutremove none. Share withus,therefore, thethoughts which
FIRST DAY z7 which ourdiscussion hassuggested toyou;forsince wearefree fromurgentbusiness therewillbeabundant timetopursue the topics alreadymentioned; and in particularthe obje£tions raised bySimplicio oughtnotinanywisetobenegle&ed. S_J_v. Granted,sinceyousodesire.Thefirstquestion was, Howcana single pointbeequaltoa line? Since I cannotdo moreat presentI shallattempttoremove, orat leastdiminish, oneimprobability by introducing a similaror a greaterone, justas sometimes awonder isdiminished byamiracle.* AndthisI shalldoby showing youtwoequalsurfaces, to- getherwithtwoequalsolidslocated uponthesesamesurfaces asbases, allfourofwhichdiminish continuously anduniformly in sucha waythat theirremainders always preserve equality among themselves, andfinally boththesurfaces andthesolids terminate theirprevious constant equality by degenerating, the onesolidandtheonesurfaceintoa verylongline,theother solidand the other surface into a singlepoint;that is, the latterto onepoint,theformer toaninfinite number ofpoints. [74] SACR. Thisproposition appearsto me wonderfial, indeed; butletusheartheexplanation anddemonstration. SALV. Sincethe proofis purelygeometrical we shallneed a figure. Let_FB be a semicircle withcenterat C;aboutit describe the re&angle ADEBand fromthe centerdrawthe straightlinesCDandCEto thepointsD andE. Imagine the radius CFtobedrawn perpendicular toeitherofthelines ABor DE,andtheentirefigure to rotateaboutthisradiusasanaxis. It isclearthatthere&angle ADEBwillthusdescribe acylinder, thesemicircle AFBahemisphere, andthetriangle CDE,a cone. Nextletus remove thehemisphere but leave theconeandthe restofthecylinder, which, onaccount ofitsshape, wewillcalla "bowl." Firstweshallprovethat thebowlandthe coneare equal;thenweshall show thataplane drawn parallel tothecircle which forms thebaseofthebowlandwhich hasthelineDEfor diameter andF foracenterwaplane whose traceisGN---cuts thebowlinthepoints G,I,O,N,andtheconeinthepoints I-I,L, sothatthepartofthecone indicated byCHLisalways equal to *Cf.p.3obelow. [Trans.]
z8 THE TWO NEW SCIENCESOF GALILEO thepartofthebowlwhose profile isrepresented bythetriangles GAIandBON. Besides thisweshallprovethatthebaseofthe cone,i.e.,thecircle whose diameter isHL,isequaltothecircular A C 5 surface whichforms thebaseof ___ thisportionof thebowl,or as onemightsay,equaltoa ribbon G N whosewidthis OI. (Noteby the waythenatureof mathe- maticaldefinitions whichcon- . sistmerely intheimposition of D F _ names or,ifyouprefer, abbrevi- Fig.6 ations ofspeech established and introduced in orderto avoidthetediousdrudgerywhich you and I now experience simplybecause we have not agreed to call this surfacea "circularband" and that sharpsolid portionof the bowla "round razor.") Nowcallthem by [75] whatnameyouplease, itsuffices tounderstand that theplane, drawnat any heightwhatever,so longas it is parallelto thebase,i.e.,to thecircle whose diameter isDE,alwayscuts thetwosolids sothattheportion CHLofthecone isequal tothe upperportionofthebowl;likewise thetwoareaswhich arethe basesofthesesolids, namely thebandandthecircle I-IL, arealso equal. Herewehavethemiracle mentioned above; asthecut- tingplaneapproaches thelineABtheportions ofthesolids cut offarealways equal,soalsotheareasoftheirbases.Andasthe cuttingplanecomes nearthetop,thetwosolids (always equal) aswellastheirbases(areas which arealsoequal) finally vanish, onepairofthemdegenerating intothecircumference ofa circle, theotherintoasingle point,namely, theupperedge ofthebowl andtheapexof thecone. Now,sinceas thesesolids diminish equality ismaintained between themuptotheverylast,weare justified in sayingthat, at the extreme andfinalendof this diminution, theyarestillequaland that oneis not infinitely greaterthan the other. It appearsthereforethat we may equatethecircumference ofa largecircle to asingle point. And thiswhichistrueofthesolids istruealsoofthesurfaces which form
FIRST DAY 29 formtheirbases;forthesealsopreserveequalitybetweenthem- selvesthroughouttheir diminutionand in the end vanish,the one into the circumference of a circle,the other into a single point. Shallwenotthen callthemequalseeing thattheyarethe last tracesand remnantsof equalmagnitudes ? Note alsothat, even if these vesselswere large enoughto contain immense celestial hemispheres, both their upperedgesand the apexesof the conestherein containedwould alwaysremainequal and wouldvanish,the formerinto circleshavingthe dimensions of the largestcelestialorbits, the latterinto singlepoints. Hence in conformitywith the precedingwe may say that all circum- ferencesof circles,howeverdifferent,are equalto each other, andareeachequalto a singlepoint. SAtin.This presentationstrikesme as so clever and novel that, even if I were able, I wouldnot be willingto opposeit; forto defacesobeautifula stru_ure by a bluntpedanticattack wouldbe nothingshortofsinful. But forour completesatisfac- [76] tion pray give us this geometricalproof that there is always equality between thesesolidsand between their bases; for it cannot,I think, fail to be very ingenious, seeinghow subtleis the philosophical argumentbaseduponthis result. SAJ_v. The demonstrationisboth short and easy. Referring to the precedingfigure,sinceIPC isa rightanglethe squareof the radiusIC isequalto the sumofthe squareson thetwo sides IP, PC; but the radiusIC isequalto ACand alsoto GP, while CP isequalto PH. Hencethe squareof the lineGP isequalto the sumof the squaresof IP andPH, ormuklplyingthroughby 4,wehavethesquareof the diameterGN equaltothe sumofthe squareson IO and HL. And, sincethe areasof circlesare to eachotheras the squaresof their diameters,it followsthat the areaofthe circle whosediameterisGN isequaltothe sumofthe areasof circles havingdiametersIOandI-i-L, sothat ifweremove the commonarea of the circlehavingIO for diameterthe re- mainingareaof the circleGN willbe equalto the areaof the circlewhosediameterisHL. Somuchforthefirstpart. Asfor the otherpart, weleaveitsdemonstration forthe present,partly because
30 THE TWO NEW SCIENCES OF GALILEO because thosewhowishto follow it willfindit in thetwelfth proposition ofthesecond bookof Decentro gravitatis solidorum bytheArchimedes ofourage,LucaValerio,* whomadeuseofit fora different objec°c, andpartlybecause, forourpurpose, it suffices to have seenthat the above-mentioned surfacesare always equalandthat, as theykeepon diminishing uniformly, theydegenerate, theoneintoa single point,theotherintothe circumference ofa circle largerthananyassignable; inthisfa& liesourmiracle.t SACR. The demonstration is ingenious and the inferences drawnfromit areremarkable. Andnowletushearsomething concerning theotherdifficulty raised by Simplicio, ifyouhave anythingspecialto say,which,however, seemsto mehardly possible, sincethematterhasalready beensothoroughly dis- cussed. S_mv. ButI dohavesomething special to say,andwillfirst of all repeatwhatI saida littlewhileago,namely,that in- finityandindivisibility areintheirverynatureincomprehensi- bleto us;imagine thenwhattheyarewhencombined.Yetif [77] wewishto buildup a lineout of indivisible points,wemust take aninfinite numberof them,andare,therefore, boundto understand boththe infiniteand the indivisible at the same time. Manyideas havepassed through mymindconcerning this subjecCt, some ofwhich, possibly themoreimportant, I maynot beableto recallonthe spurof themoment;but inthecourse ofourdiscussion it mayhappen thatI shallawaken inyou,and especially in Simplicio, objecCtions and difficulties whichin turn willbringto memory thatwhich, withoutsuchstimulus, wouldhavelaindormantinmymind. Allow metherefore the customary libertyofintroducing some ofourhumanfancies, for indeedwemayso callthemin comparison withsupernatural truth whichfurnishes theonetrue andsaferecourse fordeci- sionin ourdiscussions andwhichis aninfallible guideinthe darkanddubious pathsofthought. *Distinguished Italian mathematician; bornatFerrara about I5S2; admitted totheAccademia dei Lincel I612; died I618.[Trans.] Jf Cf.p.27above.[Trans.]
FIRST DAY 3I Oneof the mainobjec°cions urgedagainst this building up of continuous quantities out of indivisible quantities[continuo d' Cndivisibih] is that the additionof one indivisible to an- othercannotproduce a divisible, forifthiswereso it would rendertheindivisible divisible.Thusif twoindivisibles, say twopoints,canbe unitedto forma quantity,saya divisible line,thenanevenmoredivisible linemightbeformed by the unionofthree,five,seven, or anyotheroddnumber ofpoints. Since however theselinescanbecut intotwoequalparts,it becomes possible to cuttheindivisible whichliesexac°dy inthe middle oftheline. In answer tothisandotherobjec°dons ofthe sametypewereplythat a divisible magnitude cannot becon- stru(tedoutoftwoortenorahundred orathousand indivisibleS, butrequires aninfinite number ofthem. Sire,.Herea difficulty presentsitselfwhichappears to me insoluble.Sinceit is clearthat wemayhaveonelinegreater than another,eachcontaining an infinitenumberof points, we areforcedto admitthat, withinoneand the sameclass, wemayhavesomething greaterthaninfinity, because thein- finityof pointsin thelonglineis greaterthan theinfinity of pointsin theshortline. Thisassigning to aninfinite quantity avaluegreaterthaninfinity isquitebeyond mycomprehension. SALv. This is oneof the difficulties whicharisewhenwe attempt,withourfiniteminds, todiscuss theinfinite, assigning toitthoseproperties which wegivefothefinite andlimited; but [78] thisI thinkiswrong, forwecannot speak ofinfinite quantities as beingtheonegreater orlessthanorequalto another.To provethisI haveinmindanargument which, forthesakeof clearness, I shallputintheform ofquestions to Simplicio who raised thisdifficulty. I takeit forgrantedthatyouknow which ofthenumbers are squares andwhich arenot. Sn_P. Iamquiteaware thatasquared number isonewhich re- sultsfromthemultiplication of another number byitself;thus 4,9,etc.,aresquared numbers which come from multiplying 2,3, etc.,bythemselves. Salv.
37 THE TWO NEW SCIENCES OF GALILEO SALV. Very well;and youalsoknowthat just as the products are calledsquaresso the favors are calledsidesor roots;while on the otherhand those numberswhichdo not consistof two equal facCtors are not squares. Thereforeif I assert that all numbers, includingboth squares and non-squares,are more than the squaresalone, I shall speakthe truth, shall I not? Snvn,. Most certainly. SALV. If t shouldask furtherhowmanysquaresthereareone might replytruly that there are as many as file corresponding numberof roots,sinceeverysquarehas its own rootand every root its own square,whileno squarehas more than one root and norootmorethan onesquare. SIMP.Preciselyso. SALV. But if I -inquirehowmany rootsthere are, it cannot be deniedthat there are as manyas there are numbersbecause every numberis a root of some square. This beinggranted wemust say that there are as many squaresas there are num- bersbecausethey are just as numerousas their roots,and all the numbers are roots. Yet at the outset we said there are many morenumbersthan squares, sincethe largerportion of them are not squares. Not only so, but the proportionate number of squares diminishesas wepass to larger numbers. Thusup to IoowehaveIOsquares,that is,the squaresconstitute I/IO part of all the numbers;up to IOOOO, we findonly I/IO0 [79] part to be squares;and up to a milliononly I/IOOO part; on the otherhand in an infinitenumber,ifone couldconceive of sucha thing, he wouldbe forcedto admit that there are as many squaresas therearenumbersalltakentogether. SAGR. What then must one concludeunder these circum- stances? SALV. Sofar as I seewe can only inferthat the totality of all numbersis infinite,that the numberof squares is infinite, and that the numberof their roots is infinite;neither is the numberof squaresless than the totality of all numbers,nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicableto infinite, but
FIRST DAY 33 but only to finite,quantkies. When thereforeSimpllc[oin- troducesseverallinesof differentlengthsand asksme how it is possiblethat the longerones do not contain more points than the shorter,I answerhimthat one linedoesnot contain moreor lessor just as many points as another,but that each line containsan infinitenumber. Or if I had repliedto him that the pointsinone llnewereequalin numberto the squares; inanother,greaterthan thetotality ofnumbers;andinthe little one,asmany asthe numberof cubes,mightI not, indeed,have satisfiedhim by thus placingmorepoints in one line than in another and yet maintainingan infinitenumberin each? So muchforthe firstdifficulty. SAGg. Pray stop a momentand let me add to what has al- readybeen said an ideawhichjust occursto me. If the pre- cedingbe true, it seemsto me impossible to say eitherthat one infinitenumberisgreaterthan anotheror eventhat it isgreater than a finitenumber,becauseifthe infinitenumberweregreater than, say, a millionit wouldfollowthat on passingfrom the millionto higherand highernumberswe wouldbe approach- ing the infinite;but this is not so;on the contrary,the lar- ger the numberto which we pass, the more we recedefrom [thispropertyof]infinity,becausethe greaterthe numbersthe fewer [relatively] are the squarescontainedin them; but the squaresin infinity cannotbe lessthan the totality of all the numbers,as wehavejust agreed;hencethe approachto greater and greaternumbersmeansa departurefrominfinity.* SAT.v. And thus fromyour ingeniousargumentweareled to [8o] concludethat the attributes "larger," "smaller,"and "equal" have no placeeitherin comparinginfinitequantitieswith each otheror in comparinginfinitewithfinitequantities. I pass now to another consideration. Sincelines and all continuousquantitiesare divisibleintoparts whichare them- selvesdivisiblewithout end, I do not see how it is possible *Acertain confusion ofthought appears tobeintroduced herethrough a failure todistinguish between thenumber n andtheclass ofthefirstn numbers; andlikewise froma failure todistinguish infinity asa number from infinity astheclass ofallnumbers.[Trans.]
34 THE TWO NEW SCIENCESOF GALILEO to avoidthe conclusion that theselinesarebuiltup of an in- finitenumberofindivisible quantities because a division anda subdivision whichcan be carriedon indefinitely presupposes that thepartsareinfinite innumber, otherwise thesubdivision wouldreachanend;andifthepartsareinfinite innumber, we mustconclude that theyarenot finitein size,because an in- finitenumberoffinitequantities would giveaninfinite magni- tude. Andthuswehavea continuous quantitybuiltupof an infinite number of indivisibles. Shay.But if wecan carryon indefinitely thedivision into finitepartswhatnecessity is therethen forthe introduction ofnon-finlte parts? SALV. The veryfacet that oneis ableto continue, without end,thedivision intofiniteparts[inpattiquante] makesit nec- essaryto regardthequantityas composed of aninfinite num- ber of immeasurably smallelements[di infinitinonquanta]. Nowin orderto settlethismatterI shallaskyouto tellme whether,in youropinion, a continuum is madeup of a finite orofaninfinite number offiniteparts[parti quante]. SIMI,. My answeris that their numberis bothinfiniteand finite;potentiallyinfinitebut afftually finite[infinite, in po.. tenza;efinite,inatto]; that is to say,potentially infinite before division andactually finiteafterdivision; because partscannot besaidto existina bodywhichis notyet divided or at least marked out;ifthisisnotdone wesaythattheyexistpotentially. SALV. Sothat a linewhichis, for instance,twentyspans longis notsaidtocontain afftually twentylineseachonespan inlengthexceptafterdivision intotwentyequalparts;before division it is saidto containthemonlypotentially.Suppose thefacets areasyousay;tellmethenwhether, whenthedivision is oncemade,the sizeof the original quantityis therebyin- creased, diminished, orunaffecCted. SIMV. It neither increases nordiminishes. SALV. That is my opinionalso. Therefore the finiteparts [pattiquante] in a continuum, whethera&uallyor potentially present,donotmakethequantityeitherlargeror smaller; but it is perfecCtly clearthat, if thenumberoffinitepartsaCtually contained
FIRST DAY 35 contained inthewhole isinfinite innumber, theywillmakethe magnitude infinite.Hencethenumber offinite parts,although existing onlypotentially, cannot beinfinite unless themagnitude containing thembeinfinite; andconversely ifthemagnitude is finiteit cannotcontain aninfinite number offinitepartseither actually or potentially. SAGe. Howthenisit possible todivide a continuum without limitintopartswhicharethemselves always capable ofsubdivi- sion? SAT.V. ThisdistinCtion ofyours between actualandpotential appears torendereasybyonemethod whatwould beimpossible by another. But I shallendeavor to reconcile thesematters in anotherway;and as to thequerywhether thefiniteparts of a limitedcontinuum [continuo terminato] arefiniteor in- finitein numberI will,contraryto the opinion of Simplicio, answer thattheyareneither finite norinfinite. SIMP. Thisanswerwould neverhaveoccurred tomesinceI didnotthinkthatthereexisted anyintermediate stepbetween thefiniteandtheinfinite, sothat theclassification or distinc- tionwhich assumes thata thingmustbeeitherfinite or infinite isfaulty anddefective. SALv. Soit seems tome. Andifweconsider discrete quanti- ties I thinkthereis, between finiteandinfinite quantities, a third intermediate termwhichcorresponds to everyassigned number;sothat if asked,as in thepresentcase,whether the finitepartsof a continuum arefiniteor infinite innumber the bestreplyisthat theyareneither finitenorinfinite but corre- spondto everyassigned number. In orderthat thismaybe possible, it isnecessary that those partsshould notbeincluded withina limited number, forinthatcasetheywould notcorre- spondtoa number whichisgreater; norcantheybeinfinite in numbersincenoassigned number is infinite; andthus at the pleasure ofthequestioner wemay,to anygivenline,assign a hundred finite parts,athousand, ahundred thousand, or indeed anynumber wemayplease solongasit benotinfinite.I grant, therefore, to the philosophers_ that thecontinuum contains as many
36 THE TWO NEW SCIENCESOF GALILEO manyfinitepartsas theypleaseand I concede alsothat it con- tainsthem,eitheraCtually or potentially, astheymaylike;but I mustaddthatjustasalinetenfathoms Jeanne] inlengthcon- tainstenlineseachof onefathomandfortylineseachof one cubit[braccia] andeightylineseachof halfa cubit,etc.,soit containsaninfinitenumber of points;callthemaCtual or po- tential,asyoulike, foras tothisdetail,Simplicio, Idefer toyour opinion andtoyourjudgment. [821 SL_P. I cannothelp admiring your discussion; but I fear that this parallelism betweenthe pointsand the finiteparts contained ina linewillnotprovesatisfaCtory, andthatyouwill notfindit soeasyto divide a givenlineintoaninfinite num- berofpointsasthephilosophers dotocutit intotenfathoms or fortycubits;notonlyso,butsucha division is quiteimpossible to realizein praCtice, sothat thiswillbeoneof thosepoten- tialities which cannot bereduced toactuality. SALV. Thefactthatsomething canbedoneonlywitheffort ordiligence orwithgreatexpenditure oftimedoesnotrenderit impossible; forI thinkthatyouyourself could noteasilydivide a lineinto a thousand parts,andmuchlessif thenumberof partswere937or any otherlargeprimenumber. But if I wereto accomplish thisdivision whichyoudeemimpossible as readily as anotherperson woulddivide thelineintofortyparts would youthenbemore willing, inourdiscussion, toconcede the possibility ofsuchadivision ? Snvn,. In general I enjoygreatlyyourmethod;andreplying to yourquery,I answer that it wouldbemorethansufficient ifit provenotmoredifficult to resolve a lineintopointsthanto divide it intoa thousand parts. SALv. I willnowsaysomething which mayperhaps astonish you;it refersto thepossibility of dividing a lineinto its in- finitely smallelements by following thesameorderwhichone employs individing thesamelineintoforty,sixty,ora hundred parts,thatis,bydividing it intotwo,four,etc. Hewhothinks that, byfollowing thismethod, hecanreachaninfinite number ofpointsisgreatly mistaken; forifthisprocess werefollowed to etemiw / 7
FIRST DAY 37 eternity there wouldstill remainfiniteparts whichwere un- divided. Indeedby such a methodone is very far from reachingthe goal of indivisibility;on the contrary he recedesfrom it and whilehe thinksthat, by continuingthis division and by multi- plyingthe multitude of parts, he willapproachinfinity,he is, inmy opinion,gettingfarther and fartherawayfromit. My reasonis this. In the precedingdiscussion we concluded that, inan infinitenumber,it isnecessarythat the squaresand cubes shouldbe as numerousas the totality of the natural numbers [tuttii numerz],becauseboth of theseare as numerousas their roots which constitute the totality of the natural numbers. Nextwesawthat thelargerthe numberstakenthemoresparsely distributedwerethe squares,and stillmoresparselythe cubes; thereforeit isclearthat the largerthe numbersto whichwepass the fartherwerecedefromthe infinitenumber;henceit follows [8S] that, sincethisprocesscarriesus fartherandfartherfromthe endsought,if on turningback weshallfindthat any number can be said to be infinite,it mustbe unity. Hereindeedare satisfiedall those conditions whichare requisite foran infinite number; I meanthat unity contains in itselfasmanysquares as therearecubesandnaturalnumbers[tuttiinumen]. SIMP.I donotquitegraspthemeaningofthis. SALV. There isno difficulty in the matterbecauseunityisat once a square,a cube, a squareof a squareand all the other powers[dignity]; noristhereany essentialpeculiarityinsquares or cubeswhichdoesnot belongto unity; as, forexample,the propertyof twosquarenumbersthat they havebetweenthema meanproportional;take any squarenumberyou pleaseas the first term and unity forthe other,then youwillalwaysfind a numberwhichisa meanproportional. Consider the twosquare numbers,9 and 4; then 3 is the mean proportionalbetween 9 and I ;while2isameanproportional between4and I; between 9 and 4 wehave6 as a meanproportional.A propertyof cubes is that they must have betweenthem two meanproportional numbers; take 8 and 27; betweenthem lie IZ and 18;while between
38 THE TWO NEW SCIENCESOF-GALILEO between Iand8wehave2and4intervening; andbetween I and 27therelie3and9. Therefore weconclude thatunity is the onlyinfinite number.Thesearesome ofthemarvels which our imagination cannotgraspandwhichshould warnusagainst the serious errorof thosewhoattemptto discussthe infiniteby assigning to it the sameproperties whichweemployfor the finite, thenatures ofthetwohaving nothing incommon. Withregard to thissubjecCt I musttellyouof a remarkable property whichjustnowoccurs to meandwhich willexplain thevastalteration andchange ofcharacCter which afinitequan- tity wouldundergo in passing to infinity.Let us drawthe straight lineABofarbitrary length andletthepointC divide it intotwounequal parts; thenI saythat,ifpairsoflinesbe drawn, onefrom eachof theterminal pointsA andB, andif theratiobetween thelengths oftheselinesisthesame asthat between ACandCB,theirpoints ofinterse&ion willalllieupon the circumference of oneandthe samecircle. Thus,for ex- [84] ample, ALandBLdrawn from AandB,meeting atthepointL, beating to oneanother the sameratioasACto BC,andthe pair AK and BK meetingat K also beatingto one an- other thesame ratio, and likewise thepairs A 6 _c B-ii"-"""_ EM, BI,AH,BH,AG, BG, AF, BF, AE, BE,havetheirpoints ofintersec°don L,K, I,H, G,F,E,all ly- Fig.7 inguponthecircum- ference ofoneandthesamecircle.Accordingly ifweimagine thepointCtomovecontinuously insuch amanner thatthelines drawn fromittothefixed terminal points, Aand B,always main- tainthesame ratiobetween theirlengths asexistsbetween the original parts, ACandCB,thenthepoint C will, asI shall pres- entlyprove, describe acircle.Andthecircle thusdescribed will _crcase r
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf
Las dos nuevas ciencias de Galileo.pdf

Recommended

Fundamentals of Crystallography Giacovazzo 2000
Fundamentals of Crystallography Giacovazzo 2000Fundamentals of Crystallography Giacovazzo 2000
Fundamentals of Crystallography Giacovazzo 2000PaReJaiiZz
 
Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...
Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...
Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...PaReJaiiZz
 
A text book of practical entomology ( pdf drive )
A text book of practical entomology ( pdf drive )A text book of practical entomology ( pdf drive )
A text book of practical entomology ( pdf drive )Kudzai Mafuwe
 
Types of paragraph
Types of paragraphTypes of paragraph
Types of paragraphpmartine1247
 
Types of paragraph (3)
Types of paragraph (3)Types of paragraph (3)
Types of paragraph (3)SOFIMATI01
 
257341652 reactor-shielding-for-engineers-pdf (1)
257341652 reactor-shielding-for-engineers-pdf (1)257341652 reactor-shielding-for-engineers-pdf (1)
257341652 reactor-shielding-for-engineers-pdf (1)manojg1990
 
Cut spelling-handbook
Cut spelling-handbookCut spelling-handbook
Cut spelling-handbookH4llud4l
 
Essay About Globalisation.pdf
Essay About Globalisation.pdfEssay About Globalisation.pdf
Essay About Globalisation.pdfElizabeth Brown
 

Recommended

Fundamentals of Crystallography Giacovazzo 2000
Fundamentals of Crystallography Giacovazzo 2000Fundamentals of Crystallography Giacovazzo 2000
Fundamentals of Crystallography Giacovazzo 2000PaReJaiiZz
 
Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...
Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...
Carmelo giacovazzo, h. l. monaco, d. viterbo, f. scordari, g. gilli, g. zanot...PaReJaiiZz
 
A text book of practical entomology ( pdf drive )
A text book of practical entomology ( pdf drive )A text book of practical entomology ( pdf drive )
A text book of practical entomology ( pdf drive )Kudzai Mafuwe
 
Types of paragraph
Types of paragraphTypes of paragraph
Types of paragraphpmartine1247
 
Types of paragraph (3)
Types of paragraph (3)Types of paragraph (3)
Types of paragraph (3)SOFIMATI01
 
257341652 reactor-shielding-for-engineers-pdf (1)
257341652 reactor-shielding-for-engineers-pdf (1)257341652 reactor-shielding-for-engineers-pdf (1)
257341652 reactor-shielding-for-engineers-pdf (1)manojg1990
 
Cut spelling-handbook
Cut spelling-handbookCut spelling-handbook
Cut spelling-handbookH4llud4l
 
Essay About Globalisation.pdf
Essay About Globalisation.pdfEssay About Globalisation.pdf
Essay About Globalisation.pdfElizabeth Brown
 
Group work_27 Translations techniques
Group work_27 Translations techniquesGroup work_27 Translations techniques
Group work_27 Translations techniquesPaula Motais de N.
 
The renaissance, 1500–1650. By Manuel Gracia Baez
The renaissance, 1500–1650. By Manuel Gracia BaezThe renaissance, 1500–1650. By Manuel Gracia Baez
The renaissance, 1500–1650. By Manuel Gracia BaezManuell GB
 
Diccionarioparaingenieros
DiccionarioparaingenierosDiccionarioparaingenieros
DiccionarioparaingenierosLucy Hart
 
Diccionario para ingenieros Español-Ingles
Diccionario para ingenieros Español-InglesDiccionario para ingenieros Español-Ingles
Diccionario para ingenieros Español-InglesPablo Saúl
 
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docxcomparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docxrobert345678
 
2001052491
20010524912001052491
2001052491KhanhHoa Tran
 
Arthur Danto And The End Of Art
Arthur Danto And The End Of ArtArthur Danto And The End Of Art
Arthur Danto And The End Of ArtLeslie Schulte
 
A collection of college words and customs
A collection of college words and customsA collection of college words and customs
A collection of college words and customsAslam Malik
 
The paragraph
The paragraphThe paragraph
The paragraphAMANDA LEON LEAL
 
American Dictionary of The English Language- 1828 Version
American Dictionary of The English Language- 1828 VersionAmerican Dictionary of The English Language- 1828 Version
American Dictionary of The English Language- 1828 VersionChuck Thompson
 
A translation of the chapter: what is conservation biology?
A translation of the chapter: what is conservation biology? A translation of the chapter: what is conservation biology?
A translation of the chapter: what is conservation biology? Christian Espinoza
 
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docxwww.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docxjeffevans62972
 
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...Becky Gulson
 
Oxford dictionary earth science
Oxford dictionary earth scienceOxford dictionary earth science
Oxford dictionary earth scienceWael Sharba
 
Running head SOURCES LIST1SOURCES LIST2Sources Li.docx
Running head SOURCES LIST1SOURCES LIST2Sources Li.docxRunning head SOURCES LIST1SOURCES LIST2Sources Li.docx
Running head SOURCES LIST1SOURCES LIST2Sources Li.docxtodd521
 
Household Cyclopedia
Household CyclopediaHousehold Cyclopedia
Household CyclopediaEdward Dobson
 
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)Erin Taylor
 
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...reynaldogonzales13
 
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-PickingAllusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-PickingAddison Coleman
 
Language bloomfield, leonard, 1887-1949
Language bloomfield, leonard, 1887-1949Language bloomfield, leonard, 1887-1949
Language bloomfield, leonard, 1887-1949Mona khosravii
 
Lubrication System in forced feed system
Lubrication System in forced feed systemLubrication System in forced feed system
Lubrication System in forced feed systemADB online India
 
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...Ansari Aashif Raza Mohd Imtiyaz
 

More Related Content

Similar to Las dos nuevas ciencias de Galileo.pdf

Group work_27 Translations techniques
Group work_27 Translations techniquesGroup work_27 Translations techniques
Group work_27 Translations techniquesPaula Motais de N.
 
The renaissance, 1500–1650. By Manuel Gracia Baez
The renaissance, 1500–1650. By Manuel Gracia BaezThe renaissance, 1500–1650. By Manuel Gracia Baez
The renaissance, 1500–1650. By Manuel Gracia BaezManuell GB
 
Diccionarioparaingenieros
DiccionarioparaingenierosDiccionarioparaingenieros
DiccionarioparaingenierosLucy Hart
 
Diccionario para ingenieros Español-Ingles
Diccionario para ingenieros Español-InglesDiccionario para ingenieros Español-Ingles
Diccionario para ingenieros Español-InglesPablo Saúl
 
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docxcomparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docxrobert345678
 
Arthur Danto And The End Of Art
Arthur Danto And The End Of ArtArthur Danto And The End Of Art
Arthur Danto And The End Of ArtLeslie Schulte
 
A collection of college words and customs
A collection of college words and customsA collection of college words and customs
A collection of college words and customsAslam Malik
 
American Dictionary of The English Language- 1828 Version
American Dictionary of The English Language- 1828 VersionAmerican Dictionary of The English Language- 1828 Version
American Dictionary of The English Language- 1828 VersionChuck Thompson
 
A translation of the chapter: what is conservation biology?
A translation of the chapter: what is conservation biology? A translation of the chapter: what is conservation biology?
A translation of the chapter: what is conservation biology? Christian Espinoza
 
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docxwww.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docxjeffevans62972
 
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...Becky Gulson
 
Oxford dictionary earth science
Oxford dictionary earth scienceOxford dictionary earth science
Oxford dictionary earth scienceWael Sharba
 
Running head SOURCES LIST1SOURCES LIST2Sources Li.docx
Running head SOURCES LIST1SOURCES LIST2Sources Li.docxRunning head SOURCES LIST1SOURCES LIST2Sources Li.docx
Running head SOURCES LIST1SOURCES LIST2Sources Li.docxtodd521
 
Household Cyclopedia
Household CyclopediaHousehold Cyclopedia
Household CyclopediaEdward Dobson
 
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)Erin Taylor
 
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...reynaldogonzales13
 
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-PickingAllusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-PickingAddison Coleman
 
Language bloomfield, leonard, 1887-1949
Language bloomfield, leonard, 1887-1949Language bloomfield, leonard, 1887-1949
Language bloomfield, leonard, 1887-1949Mona khosravii
 

Similar to Las dos nuevas ciencias de Galileo.pdf (20)

Group work_27 Translations techniques
Group work_27 Translations techniquesGroup work_27 Translations techniques
Group work_27 Translations techniques
 
The renaissance, 1500–1650. By Manuel Gracia Baez
The renaissance, 1500–1650. By Manuel Gracia BaezThe renaissance, 1500–1650. By Manuel Gracia Baez
The renaissance, 1500–1650. By Manuel Gracia Baez
 
Diccionarioparaingenieros
DiccionarioparaingenierosDiccionarioparaingenieros
Diccionarioparaingenieros
 
Diccionario para ingenieros Español-Ingles
Diccionario para ingenieros Español-InglesDiccionario para ingenieros Español-Ingles
Diccionario para ingenieros Español-Ingles
 
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docxcomparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
comparative literature studies, vol. 48, no. 4, 2011.Copyrig.docx
 
2001052491
20010524912001052491
2001052491
 
Arthur Danto And The End Of Art
Arthur Danto And The End Of ArtArthur Danto And The End Of Art
Arthur Danto And The End Of Art
 
A collection of college words and customs
A collection of college words and customsA collection of college words and customs
A collection of college words and customs
 
The paragraph
The paragraphThe paragraph
The paragraph
 
American Dictionary of The English Language- 1828 Version
American Dictionary of The English Language- 1828 VersionAmerican Dictionary of The English Language- 1828 Version
American Dictionary of The English Language- 1828 Version
 
A translation of the chapter: what is conservation biology?
A translation of the chapter: what is conservation biology? A translation of the chapter: what is conservation biology?
A translation of the chapter: what is conservation biology?
 
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docxwww.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
www.cambridge.org9781107008519CAMBRIDGE TEXTS IN THEH.docx
 
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
Memo Essay. 316 Micro Memo - Grade: A - Memorandum 26 September 2018 From:, B...
 
Oxford dictionary earth science
Oxford dictionary earth scienceOxford dictionary earth science
Oxford dictionary earth science
 
Running head SOURCES LIST1SOURCES LIST2Sources Li.docx
Running head SOURCES LIST1SOURCES LIST2Sources Li.docxRunning head SOURCES LIST1SOURCES LIST2Sources Li.docx
Running head SOURCES LIST1SOURCES LIST2Sources Li.docx
 
Household Cyclopedia
Household CyclopediaHousehold Cyclopedia
Household Cyclopedia
 
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
A Bird S-Eye View On Pattern Research (Paper For Feedback Discussion)
 
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
vsip.info_diccionario-para-ingenierios-espanol-ingles-ingles-espanol-2da-ed-p...
 
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-PickingAllusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
Allusion And Ambiguity In Seamus Heaney S Quot Blackberry-Picking
 
Language bloomfield, leonard, 1887-1949
Language bloomfield, leonard, 1887-1949Language bloomfield, leonard, 1887-1949
Language bloomfield, leonard, 1887-1949
 

Recently uploaded

Lubrication System in forced feed system
Lubrication System in forced feed systemLubrication System in forced feed system
Lubrication System in forced feed systemADB online India
 
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...Ansari Aashif Raza Mohd Imtiyaz
 
RACEMIzATION AND ISOMERISATION completed.pptx
RACEMIzATION AND ISOMERISATION completed.pptxRACEMIzATION AND ISOMERISATION completed.pptx
RACEMIzATION AND ISOMERISATION completed.pptxArunLakshmiMeenakshi
 
X-rays from a Central “Exhaust Vent” of the Galactic Center Chimney
X-rays from a Central “Exhaust Vent” of the Galactic Center ChimneyX-rays from a Central “Exhaust Vent” of the Galactic Center Chimney
X-rays from a Central “Exhaust Vent” of the Galactic Center ChimneySérgio Sacani
 
EU START PROJECT. START-Newsletter_Issue_4.pdf
EU START PROJECT. START-Newsletter_Issue_4.pdfEU START PROJECT. START-Newsletter_Issue_4.pdf
EU START PROJECT. START-Newsletter_Issue_4.pdfStart Project
 
In-pond Race way systems for Aquaculture (IPRS).pptx
In-pond Race way systems for Aquaculture (IPRS).pptxIn-pond Race way systems for Aquaculture (IPRS).pptx
In-pond Race way systems for Aquaculture (IPRS).pptxMAGOTI ERNEST
 
Film Coated Tablet and Film Coating raw materials.pdf
Film Coated Tablet and Film Coating raw materials.pdfFilm Coated Tablet and Film Coating raw materials.pdf
Film Coated Tablet and Film Coating raw materials.pdfPharmatech-rx
 
Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...
Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...
Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...kevin8smith
 
MSC IV_Forensic medicine - Mechanical injuries.pdf
MSC IV_Forensic medicine - Mechanical injuries.pdfMSC IV_Forensic medicine - Mechanical injuries.pdf
MSC IV_Forensic medicine - Mechanical injuries.pdfSuchita Rawat
 
Technical english Technical english.pptx
Technical english Technical english.pptxTechnical english Technical english.pptx
Technical english Technical english.pptxyoussefboujtat3
 
Information science research with large language models: between science and ...
Information science research with large language models: between science and ...Information science research with large language models: between science and ...
Information science research with large language models: between science and ...Fabiano Dalpiaz
 
GBSN - Microbiology (Unit 5) Concept of isolation
GBSN - Microbiology (Unit 5) Concept of isolationGBSN - Microbiology (Unit 5) Concept of isolation
GBSN - Microbiology (Unit 5) Concept of isolationAreesha Ahmad
 
MODERN PHYSICS_REPORTING_QUANTA_.....pdf
MODERN PHYSICS_REPORTING_QUANTA_.....pdfMODERN PHYSICS_REPORTING_QUANTA_.....pdf
MODERN PHYSICS_REPORTING_QUANTA_.....pdfRevenJadePalma
 
NUMERICAL Proof Of TIme Electron Theory.
NUMERICAL Proof Of TIme Electron Theory.NUMERICAL Proof Of TIme Electron Theory.
NUMERICAL Proof Of TIme Electron Theory.syedmuneemqadri
 
Adaptive Restore algorithm & importance Monte Carlo
Adaptive Restore algorithm & importance Monte CarloAdaptive Restore algorithm & importance Monte Carlo
Adaptive Restore algorithm & importance Monte CarloChristian Robert
 
PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)
PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)
PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)kushbuR
 
Electricity and Circuits for Grade 9 students
Electricity and Circuits for Grade 9 studentsElectricity and Circuits for Grade 9 students
Electricity and Circuits for Grade 9 studentslevieagacer
 
NuGOweek 2024 programme final FLYER short.pdf
NuGOweek 2024 programme final FLYER short.pdfNuGOweek 2024 programme final FLYER short.pdf
NuGOweek 2024 programme final FLYER short.pdfpablovgd
 

Recently uploaded (20)

Lubrication System in forced feed system
Lubrication System in forced feed systemLubrication System in forced feed system
Lubrication System in forced feed system
 
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
Molecular and Cellular Mechanism of Action of Hormones such as Growth Hormone...
 
RACEMIzATION AND ISOMERISATION completed.pptx
RACEMIzATION AND ISOMERISATION completed.pptxRACEMIzATION AND ISOMERISATION completed.pptx
RACEMIzATION AND ISOMERISATION completed.pptx
 
X-rays from a Central “Exhaust Vent” of the Galactic Center Chimney
X-rays from a Central “Exhaust Vent” of the Galactic Center ChimneyX-rays from a Central “Exhaust Vent” of the Galactic Center Chimney
X-rays from a Central “Exhaust Vent” of the Galactic Center Chimney
 
EU START PROJECT. START-Newsletter_Issue_4.pdf
EU START PROJECT. START-Newsletter_Issue_4.pdfEU START PROJECT. START-Newsletter_Issue_4.pdf
EU START PROJECT. START-Newsletter_Issue_4.pdf
 
In-pond Race way systems for Aquaculture (IPRS).pptx
In-pond Race way systems for Aquaculture (IPRS).pptxIn-pond Race way systems for Aquaculture (IPRS).pptx
In-pond Race way systems for Aquaculture (IPRS).pptx
 
Film Coated Tablet and Film Coating raw materials.pdf
Film Coated Tablet and Film Coating raw materials.pdfFilm Coated Tablet and Film Coating raw materials.pdf
Film Coated Tablet and Film Coating raw materials.pdf
 
Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...
Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...
Harry Coumnas Thinks That Human Teleportation is Possible in Quantum Mechanic...
 
Chemistry Data Delivery from the US-EPA Center for Computational Toxicology a...
Chemistry Data Delivery from the US-EPA Center for Computational Toxicology a...Chemistry Data Delivery from the US-EPA Center for Computational Toxicology a...
Chemistry Data Delivery from the US-EPA Center for Computational Toxicology a...
 
MSC IV_Forensic medicine - Mechanical injuries.pdf
MSC IV_Forensic medicine - Mechanical injuries.pdfMSC IV_Forensic medicine - Mechanical injuries.pdf
MSC IV_Forensic medicine - Mechanical injuries.pdf
 
Technical english Technical english.pptx
Technical english Technical english.pptxTechnical english Technical english.pptx
Technical english Technical english.pptx
 
Information science research with large language models: between science and ...
Information science research with large language models: between science and ...Information science research with large language models: between science and ...
Information science research with large language models: between science and ...
 
GBSN - Microbiology (Unit 5) Concept of isolation
GBSN - Microbiology (Unit 5) Concept of isolationGBSN - Microbiology (Unit 5) Concept of isolation
GBSN - Microbiology (Unit 5) Concept of isolation
 
MODERN PHYSICS_REPORTING_QUANTA_.....pdf
MODERN PHYSICS_REPORTING_QUANTA_.....pdfMODERN PHYSICS_REPORTING_QUANTA_.....pdf
MODERN PHYSICS_REPORTING_QUANTA_.....pdf
 
NUMERICAL Proof Of TIme Electron Theory.
NUMERICAL Proof Of TIme Electron Theory.NUMERICAL Proof Of TIme Electron Theory.
NUMERICAL Proof Of TIme Electron Theory.
 
Adaptive Restore algorithm & importance Monte Carlo
Adaptive Restore algorithm & importance Monte CarloAdaptive Restore algorithm & importance Monte Carlo
Adaptive Restore algorithm & importance Monte Carlo
 
ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY // USES OF ANTIOBIOTICS TYPES OF ANTIB...
ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY // USES OF ANTIOBIOTICS TYPES OF ANTIB...ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY // USES OF ANTIOBIOTICS TYPES OF ANTIB...
ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY // USES OF ANTIOBIOTICS TYPES OF ANTIB...
 
PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)
PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)
PHOTOSYNTHETIC BACTERIA (OXYGENIC AND ANOXYGENIC)
 
Electricity and Circuits for Grade 9 students
Electricity and Circuits for Grade 9 studentsElectricity and Circuits for Grade 9 students
Electricity and Circuits for Grade 9 students
 
NuGOweek 2024 programme final FLYER short.pdf
NuGOweek 2024 programme final FLYER short.pdfNuGOweek 2024 programme final FLYER short.pdf
NuGOweek 2024 programme final FLYER short.pdf
 

Las dos nuevas ciencias de Galileo.pdf

  • 2. °_o° THE MACMILLAN COMPANY _W YORJK • BOSTON • CItlCAGO • DALLAS ATLANTA • SANyP.ANCISCO MACMILLAN & CO., Lmrr_ LONDON • BOMBAY • CALCUTTA _T,_ OUP._NE T_L_; MACMILLAN CO. OF CANADA, LTD. TORONTO
  • 3.
  • 5. DIALOGUES CONCERNING TWO NEW SCIENCES BY GALILEOGALILEI Translatedfromthe Italianand Latin intoEnglishby HENRY 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 the learning of the ancients." Benjamin Franklin, Phil. Trans. 64,445.(I774.) N_ _ork THE MACMILLAN COMPANY I914 All rightsreserved
  • 7. "La Dynamique estlascience desforcesaccN6ra- tricesor retardatrices, et des mouvemens varies qu'ellesdoiventproduire.Cette science est due enti_rement auxmodernes, et Galilee est cduiqui en a jet_lespremiers fondemens." Lagrange Mec. dad. I. zzl, TRANSLATORS' PREFACE OR morethana centuryEnglishspeaking students havebeenplaced intheanomalous position of hearingGalileo constantlyre- ferred to asthefounder ofmodern physical science, without havinganychance toread, intheirownlanguage, whatGalileo himself has to say. Archimedes has beenmade available byHeath; Huygens' Light hasbeen turnedinto Englishby Thompson, whileMotte has put the Principia of Newtonbackintothe language in whichit was conceived.To renderthePhysics of Galileo alsoaccessible to English andAmerican studentsis thepurpose ofthefollowing translation. The lastof the greatcreators of theRenaissance wasnot a prophetwithouthonorin hisowntime;for it wasonlyone groupofhiscountry-men thatfailedto appreciate him. Even during hislifetime,hisMechanics hadbeenrendered intoFrench byoneoftheleading physicists oftheworld, Mersenne. Withintwenty-five yearsafterthedeathofGalileo, hisDia- logues onAstronomy, andthoseon TwoNewSciences, hadbeen doneinto Englishby ThomasSalusbury and wereworthily printed in two handsome quarto volumes. The TwoNew Sciences, which contains practically allthatGalileo hastosayon thesubje&ofphysics, issuedfromthe English pressin I665.
  • 8. vi TRANSLATORS' PREFACE It issupposed thatmostofthecopies were destroyed inthegreat London firewhichoccurred inihe yearfollowing. Wearenot aware ofanycopyinAmerica: eventhatbelonging totheBritish Museum isanimperfecCt one. Againin 173 ° the TwoNewSciences wasdoneintoEnglish by Thomas Weston;but thisbook,nownearlytwocenturies old,is scarceand expensive.Moreover, the literalness with whichthistranslation wasmaderenders manypassages either ambiguous orunintelligible to themodernreader.Otherthan thesetwo,noEnglish version hasbeenmade. Quiterecentlyan eminentItalian scholar,after spending thirtyofthebestyearsofhislifeuponthesul_jecCt, hasbrought to completion the greatNationalEditionof the Worksof Galileo.Werefertothetwentysuperb volumes inwhichPro- fessor AntonioFavaro ofPaduahasgivena definitive presenta- tionofthelaborsofthemanwhocreatedthemodern science of physics. The following rendition includes neitherLe Mechaniche of Galileo norhispaperDeMotuAccelerato, sincethe formerof thesecontainslittlebut the Staticswhichwascurrentbefore thetimeofGalileo, andthelatterisessentially included inthe Dialogue oftheThirdDay. Dynamics wastheonesubjecCt to whichundervariousforms,suchas Ballistics, Acoustics, As- tronomy,he consistently and persistently devotedhis whole life. Into theone volumeheretranslatedhe seemsto have gathered,duringhislast years,pracCtically allthat is of value either to the engineeror the physicist. The historian,the philosopher, and the astronomer willfindthe othervolumes replete withinteresting material. tt ishardlynecessary toaddthatwehavestric°dy followed the textoftheNationalEdition---essentially theElzevireditionof 1638.Allcomments andannotations havebeenomittedsave hereandtherea foot-note intendedto economize the reader's time. To eachofthesefootnotes hasbeenattachedthesigna- ture[Trans.] inordertopreserve theoriginal asnearly intacCt as possible. Muchofthevalueofanyhistorical document liesinthelan- guage employed, andthisis doubly truewhenoneattemptsto
  • 9. TRANSLATORS' PREFACE vii trace the rise and growthof any set of conceptssuch as those employedin modem physics. We have thereforemade this translationasliteral as isconsistentwithclearnessandmodem- ity. In caseswherethere isany importantdeviationfromthis rule, and in the caseof many technicaltermswherethere isno deviationfrom it, we have giventhe originalItalian or Latin phrase in italics enclosedin square brackets. The intention here isto illustratethe great varietyof termsemployedby the earlyphysiciststo describea singledefiniteidea,and conversely, to illustratethe numeroussensesinwhich,then as now,a single wordisused. For the fewexplanatoryEnglishwordswhichare placedin squarebrackets withoutitalics,the translatorsalone are responsible.Thepagingofthe NationalEditionisindicated in squarebrackets insertedalongthe medianlineof the page. The imperfecCtions of the followingpageswouldhave been manymorebut forthe aidof threeof our colleagues.Professor D. R. Curtiss was kind enoughto assistin the translationof thosepageswhichdiscussthe natureofInfinity:ProfessorO.H. Basquingavevaluablehelp in the renditionof the chapteron Strengthof Materials;and ProfessorO.F. Longclearedup the meaningofa numberof Latinphrases. To ProfessorA. Favaroof the UniversityofPaduathe trans- lators share,with every reader, a feelingof sincereobligation forhisIntroducCtion. H.C. A. DE S. EVANSTON) ILLINOIS) x5February, I9x4.
  • 10.
  • 11. INTRODUCTION .................. '............................... RITINGtohisfaithful friend EliaDiodati, _ Galileo speaks ofthe"NewSciences" which hehadinmindtoprintas being"superior to everythingelseof minehithertopub- lished";elsewhere he says"they contain results-which I consider themostimportant ofallmy studies";andthisopinion which heexpressed concerning hisownworkhas beenconfirmed by posterity: the"NewSciences" are,indeed, themasterpiece ofGalileo whoat thetimewhenhemadethe above remarks hadspentuponthemmorethanthirtylaborious years. Onewhowishes to tracethehistory ofthisremarkable work willfindthat thegreatphilosopher laiditsfoundations during the eighteen best yearsof his lifc thosewhichhe spentat Padua. As welearnfromhislast scholar, Vincenzio Viviani, thenumerous resultsatwhichGalileo hadarrivedwhile inthis city,awakened intenseadmiration inthefriends whohadwit- nessed variousexperiments by meansof whichhewasaccus- tomed toinvestigate interesting questions inphysics.FraPaolo Sarpiexclaimed: To giveus the Science ofMotion,Godand NaturehavejoinedhandsandcreatedtheintellecCt ofGalileo. Andwhenthe"NewSciences" camefromthepressoneofhis foremost pupils, Paolo Aproino, wrotethatthevolume contained muchwhichhehad "alreadyheardfromhisownlips"during student daysat Padua. Limitingourselves to onlythe moreimportant documents which mightbecitedinsupport ofourstatement, it willsuffice tomention the letter,writtentoGuidobaldo delMonteon the 29thofNovember, I6O2, concerning thedescent ofheavy bodies
  • 12. x INTRODUCTION alongthearcsofcircles andthechords subtended bythem;that to Sarpi,datedI6thofOctober, I6o4,dealing withthefreefall of heavybodies; the letterto Afltonio de'Medicion the IIth ofFebruary, I6o9,inwhich hestatesthathehas"completed all the theorems anddemonstrations pertaining to forcesandre- sistances of beamsof various lengths,thicknesses and shapes, proving thattheyareweaker at themiddle thanneartheends, thattheycancarrya greaterloadwhenthat loadisdistributed throughout thelengthof thebeamthanwhenconcentrated at onepoint,demonstrating alsowhatshapeshould begivento a beaminorderthat it mayhavethe samebendingstrengthat everypoint,"andthat hewasnowengaged "uponsomeques- tionsdealing withthemotionofprojeCtiles"; andfinallyinthe letterto Belisario Vinta,dated7th of May, x6IO, concerning hisreturnfromPaduatoFlorence, heenumerates various pieces ofworkwhichwerestillto becompleted, mentioning explicitly threebooks onanentirelynewscience dealing withthetheory ofmotion. Although at varioustimesafterthe returnto his nativestatehedevoted considerable thoughttotheworkwhich, evenat that date,hehadinmindas isshown by certainfrag- mentswhichclearlybelongto different periods of hislifeand whichhave,forthefirsttime,beenpublished in theNational Edition;andalthough thesestudieswerealwaysuppermost in histhoughtit doesnotappear that hegavehimself seriously to themuntilafterthe publication of theDialogue andthe com- pletion of thattrialwhich wasrightly described as thedisgrace ofthecentury. InfaCtaslateasOctober,x63 o,hebarely men- tionstoAggiuntihisdiscoveries in thetheoryof motion,and onlytwoyearslater,inalettertoMarsiliconcerning themotion ofprojeCtiles, hehintsat abooknearlyreadyforpublication in whichhewilltreat alsoofthis subject;andonlya yearafter thishewritestoArrighetti thathehasinhandatreatiseonthe resistance ofsolids. But theworkwasgivendefinite formby Galileo duringhis enforced residence at Siena: in thesefivemonthsspentquietly withtheArchbishop he himself writesthat hehascompleted "a treatiseona newbranchofmechanics fullofinteresting and usefulideas";sothat a fewmonthslaterhewasableto send
  • 13. INTRODUCTION xi wordtoMicanzio that the "workwasready";as soonashis friends learned ofthis,theyurgeditspublication.It was,how- ever,no easymatterto printtheworkofa manalreadycon- demned bytheHolyOffice: andsince Galileo could nothopeto printit eitherinFlorence or inRome,heturnedtothefaithful Micanzio asking himtofindoutwhether thiswould bepossible inVenice, fromwhence hehadreceived offers toprinttheDia- logue onthePrincipal Systems, as soonasthenewshadreached therethathewasencountering difficulties. At firsteverything wentsmoothly; sothatGalileo commenced sending toMicanzio someofthemanuscript whichwasreceived by thelatterwith anenthusiasm inwhichhewassecond tononeof thewarmest admirers of the greatphilosopher.But whenMicanzio con- sultedthe Inquisitor, he received the answerthat therewas an express orderprohibiting theprintingor reprinting of any workof Galileo,eitherinVeniceor in anyotherplace,hullo excepto. As soonasGalileo received thisdiscouraging newshebegan tolookwithmore favoruponoffers which hadcome tohimfrom Germany wherehisfriend,andperhapsalsohisscholar, Gio- vanniBattistaPieroni,wasinthe service of theEmperor, as militaryengineer; consequently Galileo gaveto PrinceMattia de'Medici whowasjust leaving forGermany thefirsttwoDia- logues tobehandedtoPieroniwhowasundecided whether to publish thematVienna orPrague or atsome placeinMoravia; inthemeantime, however, hehadobtained permission toprint bothatVienna andatOlmtitz.But Galileo recognized danger at everypointwithinreachof the longarmof the Courtof Rome; hence, availing himself oftheopportunity offered bythe arrivalof LouisElzevirinItalyin 1636, alsoofthe friendship betweenthe latter and Micanzio, not to mentiona visit at Arcetri,hedecided to abandon allotherplansandentrustto theDutchpublishertheprintingof hisnewworkthemanu- scriptofwhich,although notcomplete, Elzevirtookwithhim onhisreturnhome. In thecourse oftheyear1637, thepriming wasfinished, and at thebeginning ofthe following yeartherewaslacking only the index,the title-page and the dedication.This last had,
  • 14. xii INTRODUCTION through thegoodoffices ofDiodati, beenoffered totheCountof Noailles, a formerscholar ofGalileo at Padua,andsince1634 ambassador ofFranceatRome,amanwhodidmuch toalleviate the distressing consequences of the celebrated trial; and the offerwasgratefully accepted.The phrasing of thededication deserves briefcomment.Since Galileo wasaware,ontheone hand,ofthe prohibition againsttheprintingofhisworksand since, on theotherhand,hedidnotwishto irritatetheCourt ofRome fromwhose handshewasalways hoping forcomplete freedom, hepretends inthededicatory letter(where, probably through excess of caution, hegives onlymainoutlines) thathe hadnothingtodowiththeprintingofhisbook,asserting that hewillneveragainpublishanyof his researches, andwillat mostdistributehereandthere a manuscript copy. He even expresses greatsurprise thathisnewDialogues havefalleninto thehandsof the Elzevirsandweresoonto bepublished; so that, having beenaskedtowritea dedication, hecould thinkof no manmoreworthywhocouldalsoon this occasion defend himagainsthisenemies. As to the title whichreads:Discourses andMathematical Demonstrations concerning TwoNewSciences pertaining toMe- chanics andLocal Motions, thisonlyisknown, namely, that the titleisnot theonewhichGalileo haddevised andsuggested; in fac_heprotestedagainstthe publishers takingthe libertyof changing it andsubstituting "a lowandcommon title forthe noble anddignified onecarried uponthetitle-page." In reprinting thisworkintheNational Edition,I havefol- lowed theLeydentextof 1638faithfully but notslavishly, be- causeI wishedto utilizethelargeamountofmanuscript ma- terialwhich hascome downtous,forthepurpose of colTeccting a considerable numberof errorsin this firstedition,and also forthesakeofinserting certainadditions desired bytheauthor himself.IntheLeyden Edition, thefourDialogues arefollowed by an"Appendix containing some theorems andtheir proofs, deal- zngwithcenters of gravity of solidbodies, writtenbythesame Author atanearlier date,"whichhasnoimmediate connecCtion withthesubjec°cs treatedintheDialogues; thesetheorems were foundby Galileo, as hetellsus,"at theageoftwenty-two and
  • 15. INTRODUCTION _5ii aftertwoyearsstudyofgeometry" andwerehereinserted only tosavethemfromoblivion. But it wasnot the intentionof Galileo that theDialogues ontheNewSciences should contain onlythefourDaysandthe above-mentioned appendix whichconstitute theLeydenEdi- tion;while,on theonehand,theElzevirs werehastening the printing andstriving tocomplete it attheearliest possible date, Galileo, on theotherhand,kepton speaking of anotherDay, besides thefour,thusembarrassing andperplexing theprinters. Fromthe correspondence whichwenton between authorand publisher, it appears that thisFifthDaywasto havetreated "of theforceof percussion andtheuseofthecatenary";, but as thetypographical workapproached completion, thepnnter became anxious forthebookto issuefromthepresswithout furtherdelay;and thus it cameto passthat the Discorsi e Dimostrazioni appeared containing onlythefourDaysandthe Appendix, in spiteof thefactthat inApril,I638,Galileo had plunged moredeeply thanever"intotheprofound question of percussion" and"hadalmost reached a complete solution." The "NewSciences" nowappearinanedition following the textwhichI, afterthemostcareful anddevoted study,deter- mineduponfortheNational Edition. It appears alsointhat language inwhich, above allothers, I havedesired to seeit. In thistranslation, thelastandripestwork ofthegreatphilosopher makesits firstappearance in the NewWorld:if towardthis important resultI mayhopetohavecontributed insome meas- ureI shallfeelamplyrewarded forhavinggiven to thisfieldof research thebestyearsofmylife. AwroNm FAv_a_o. UNIVERSITY OF PADUA_ 2_ o] October, I9x3.
  • 16.
  • 17. DISCOR, SI E DIMOSTR.AZIONI MI.A.TEMA TI C H E, intorno _duenuoue fiiene_e Atxenenti alla M_-CANICA_ i Mo VIMENTI LOCALt; delSignor GALILEO GALILEI LINCEO, Filofofo eMatematico primario delSereniilimo Grand Duca diTofcana. c°o_ z_a_ppendice del ceutro digrauit_ d'klc-_ni ._olidi._ IN LEIDA, Apprdro gliEl/_virii. _.D. c,xxxv_.
  • 18.
  • 19. ' [431 TO THE MOST ILLUSTRIOUSLORD COUNT OF NOAILLES Counsellor ofhisMostChristian Majesty, KnightoftheOrder of theHolyGhost, FieldMarshal andCommander, Seneschal andGovernor ofRouergue, andHis Majesty's Lieutenant inAuvergne, my LordandFForshipful Patron OSTILLUSTRIOUS LORD:- In the pleasure whichyou derivefrom the possession of thisworkof mineI rec- i]_/i[ ll_r_-_[_ ognize yourLordship's magnanimity. The i[lV/[]_-4_}disappointment anddiscouragement I have _'_[,_ feltovertheill-fortune whichhasfollowed __ myotherbooksarealready known to you. Indeed,I had decided not to publish any moreof my work. Andyet in order to saveit fromcom- pleteoblivion,it seemedto mewiseto leavea manuscript copyinsome place whereit would beavailable atleasttothose who followintelligently the sub jet% whichI havetreated. Accordingly I chosefirstto placemyworkinyourLordship's hands,askingno moreworthydepository, andbelieving that, onaccount ofyouraffecCtion forme,youwould haveat heartthe preservation of mystudiesand labors.Therefore, whenyou werereturning homefromyourmission toRome, I came topay myrespecCts inpersonasI hadalready donemanytimesbefore byletter. At thismeeting I presented toyourLordship acopy ofthesetwoworks whichat thattimeIhappened tohaveready. Inthegracious reception which yougavetheseI found assurance of
  • 20. xviii TO THE COUNT OF NOAILLES oftheirpreservation. Thefacet ofyourcarrying themtoFrance andshowing themto friendsofyourswhoareskilled inthese sciences gaveevidence thatmysilence wasnottobeinterpreted ascomplete idleness.Alittlelater,justasI wasonthepointof [44] sending othercopies toGermany, Flanders, England, Spainand possibly to some placesinItaly,I wasnotified by theElzevirs thattheyhadtheseworksofmineinpressandthatI oughtto decide upona dedication andsendthema replyat once. This sudden andunexpecCted newsledmetothinkthat theeagerness of your Lordship to reviveand spreadmy nameby passing theseworkson to variousfriendswasthe realcauseof their falling intothehandsofprinterswho,because theyhadalready published otherworks ofmine,nowwishedtohonormewitha beautiful andornateedition ofthiswork.Butthesewritings of minemusthavereceived additional valuefromthecriticism of so excellent a judgeas yourLordship,whoby the unionof manyvirtueshaswonthe admiration of all. Yourdesireto enlarge therenown ofmyworkshows yourunparalleled generos- ity and your zealforthe publicwelfarewhichyouthought wouldthus be promoted. Underthese circumstances it is eminently fittingthat I should,in unmistakable terms,grate- fullyacknowledge thisgenerosity on thepartofyourLordship, whohasgiven tomyfame wings thathavecarried it intoregions moredistantthanI haddaredtohope. It is,therefore, proper that I dedicateto yourLordship thischildof my brain. To thiscourseI amconstrained notonlybytheweightof obliga- tion underwhichyouhaveplacedme,but also,if I may so speak,by the interestwhichI havein securing yourLordship as thedefender ofmyreputation againstadversaries whomay attackit while I remain underyourprotecCtion. Andnow,advancing underyourbanner,I paymy respecCts toyoubywishing thatyoumayberewarded forthesekindnesses bytheachievement ofthehighest happiness andgreatness. I amyourLordship's Most devotedServant, GALILEO GALILEn _lrcetri, 6March, I638.
  • 21. THE PUBLISHERTO THE READER INCEsociety isheldtogether bythemutual services whichmenrenderoneto another, andsinceto thisendtheartsandsciences havelargelycontributed, investigations in thesefields havealways beenheldingreat esteem andhave beenhighly regarded byour wiseforefathers.Thelarger theutilityand excellence oftheinventions, thegreater has beenthehonorandpraise bestowed upontheinventors.Indeed, menhaveevendeified themandhaveunitedintheattemptto perpetuate thememory of theirbenefafftors bythebestowal of thissupreme honor. Praiseand admiration are likewise dueto thosecleverin- tellecCts who, confining their attentionto the known,have discovered and corre&edfallaciesand errorsin manyand many a propositionenunciated by men of distincCtion and accepted forages asfacet.Although these menhaveonlypointed outfalsehood andhavenotreplaced it bytruth,theyarenever- thelessworthyof commendation whenweconsider the well- knowndifficulty of discovering facet, a difficulty whichledthe princeof oratorsto exclaim: Utinara tamfacilepossem vera reperire, quamfalsa convincere.* And indeed,theselatest centuries meritthis praisebecause it is duringthemthat the artsandsciences, discovered bytheancients, havebeenreduced to so great and constantly increasing perle&ion throughthe investigations and experiments of clear-seeing minds. This development is particularly evidentin thecaseof themathe- maticalsciences.Here,without mentioning various menwho haveachieved success, wemustwithout hesitation andwiththe *Cicero. deNatura Deorum, I,9I. [Trans.]
  • 22. xx THE PUBLISHER TO THE READER unanimousapprovalof scholarsassignthe first placeto Galileo Galilei,Memberof theAcademyoftheLincei. Thishedeserves not only becausehe has effectivelydemonstratedfallaciesin many of our current conclusions,as is amply shownby his publishedworks, but also becauseby meansof the telescope (inventedin this countrybut greatlyperfectedby him) he has discoveredthe four satellitesof Jupiter, has shownus the true charaCterof the MilkyWay, and has madeus acquaintedwith spotson the Sun, with the rough and cloudyportionsof the lunar surface, with the threefold nature of Saturn, with the phasesof Venus and with the physical charaCterof comets. Thesematterswereentirelyunknownto the ancientastronomers andphilosophers;sothat wemay truly say that he has restored to the worldthe scienceof astronomyand has presentedit in a newlight. Remembering that the wisdomand powerand goodness ofthe Creator are nowhereexhibitedso well as in the heavensand celestialbodies,we can easilyrecognizethe great merit of him who has brought these bodies to our knowledgeand has, in spite of their almost infinite distance, rendered them easily visible. For, accordingto the commonsaying,sight can teach moreandwith greatercertaintyina singleday than canprecept even though repeateda thousand times; or, as another says, intuitiveknowledge keepspacewithaccuratedefinition. But the divine and natural gifts of this man are shownto best advantagein the present'workwhere he is seen to have discovered,though not w_6hout many labors and long vigils, twoentirelynewsciencesand to have demonstratedthem in a rigid, that is, geometric,manner: and what is even more re- markablein this workis the facetthat one of the two sciences dealswith a subjeCtof never-endinginterest,perhapsthe most importantin nature, onewhichhas engagedthe mindsof allthe great philosophersand one concerningwhichan extraordinary numberof bookshave been written. I refer to motion [moto locale], a phenomenonexhibitingvery many wonderfulproper- ties,noneofwhichhas hithertobeendiscovered or demonstrated by any one. Theotherscience whichhehas alsodevelopedfrom its
  • 23. THE PUBLISHER TO THE READER xxi its veryfoundationsdealswiththe resistancewhichsolidbodies offer to fracture by external forces_er violenza], a subjectof great utility, especiallyin the sciencesand mechanicalarts, and onealsoaboundinginpropertiesand theoremsnothitherto observed. In this volume one finds the first treatment of these two sciences,full of propositionsto which,as time goes on, able thinkers willadd many more;alsoby meansof a largenumber of clear demonstrationsthe author points the way to many other theoremsas willbe readilyseenandunderstoodby allin- telligentreaders.
  • 24.
  • 25. TABLE OF CONTENTS I Page Firstnewscience, treating oftheresistance which solid bodies offer tofracture.FirstDay.......................... I II Concerning thecause ofcohesion. Second Day ............ lO 9 III Second newscience, treating of motion[movimenti locah]. ThirdDay........................................ 153 Uniform motion ...................................... 154 Naturally accelerated motion ........................... 16o IV Violent motions.ProjeFtiles. Fourth Day............... 244 V Appendix; theorems anddemonstrations concerning thecenters ofgravity ofsolids .................................. 295
  • 26.
  • 28.
  • 29. FIRST DAY INTERLOCUTORS: SALVIATI, SA- GREDO AND SIMPLICIO :' .............. :'.............................. ALV. Theconstant activity which youVene- _ tiansdisplay inyourfamous arsenal suggests tothestudious minda large fieldforinvesti- gation,especially that part of the work which involves mechanics; forinthisdepart- mentalltypesofinstruments andmachines areconstantly beingconstrue'ted by many artisans, amongwhom theremustbesome who,partlybyinherited experience andpartlyby theirownob- servations, havebecome highly expert andclever inexplanation. SAc_.Youarequitefight. Indeed,I myself, beingcurious by nature,frequently visitthis placeforthemerepleasure of observing theworkofthosewho,onaccount oftheirsuperiority overotherartisans, wecall"firstrankmen." Conference with themhasoftenhelped meintheinvestigation ofcertaineffec2s including notonlythosewhich arestriking, butalsothose which arerecondite andalmost incredible.AttimesalsoI havebeen put toconfusion anddriventodespair ofeverexplaining some- thingforwhichI couldnotaccount, butwhich mysenses told metobetrue. Andnotwithstanding thefadtthatwhattheold man told us a little whileago is proverbial and commonly accepted, yetit seemed tomealtogether false, likemanyanother sayingwhichis currentamongtheignorant; forI thinkthey introduce theseexpressions inordertogivetheappearance of knowing something about matters which theydonotunderstand. Salv.
  • 30. 2 THE TWO NEW SCIENCESOF GALILEO [3o] SAJ_v. Yourefer, perhaps,tothatlastremarkofhiswhenwe askedthe reasonwhy they employed stocks,scaffolding and bracing oflargerdimensions forlaunching abigvessel thanthey dofora small one;andheanswered thattheydidthisinorderto avoid thedanger oftheshippartingunderitsownheavyweight [vasta mole], a danger towhichsmall boatsarenotsubject? SAcR. Yes,that iswhatI mean;andI referespecially to his lastassertion whichI havealways regarded asa false,though current,opinion; namely,that in speaking of theseandother similar machines onecannotarguefromthesmallto thelarge, because many devices whichsucceed on a smallscaledo not workona largescale.Now,since mechanics hasitsfoundation ingeometry, where meresizecutsnofigure, I donotseethatthe properties of circles, triangles, cylinders, conesandothersolid figures willchange withtheirsize. If,therefore, alargemachine beconstrucCted insuchawaythatitspartsbeartooneanother thesameratioasinasmaller one,andifthesmaller issufficiently strongforthepurpose forwhichit wasdesigned, I donot see whythelargeralsoshould notbeabletowithstandanysevere anddestrucCtive teststowhich it maybesubjected. SAT.v. Thecommon opinion ishereabsolutely wrong.Indeed, it is sofar wrongthat precisely theopposite is true, namely, thatmanymachines canbeconstrucCted evenmoreperfecCtly ona largescale thanonasmall; thus,forinstance, aclock which indi- catesandstrikes thehourcanbemademoreaccurate ona large scalethan on a small. Therearesomeintelligent people who maintainthis sameopinion, but on morereasonable grounds, whenthey cut loose fromgeometry and arguethat thebetter performance ofthelargemachine isowing totheimperfecCtions andvariations ofthematerial.HereI trustyouwillnotcharge Is1] mewitharrogance if I say that imperfections in thematerial, eventhosewhicharegreat enough to invalidate the clearest mathematical proof,arenotsufficient to explain thedeviations observed between machines intheconcrete andintheabstra_. Yet I shallsayit andwillaffirm that,eveniftheimperfecCtions did
  • 31. FIRST DAY 3 didnotexistandmatterwereabsolutelyperfecCt, unalterable and freefrom all accidentalvariations,stillthe merefact that it is matter makesthe larger machine,built of the samematerial and in the same proportionas the smaller,correspondwith exacCtness to the smallerineveryrespecCt exceptthat itwillnot be so strong or so resistant against violent treatment; the larger the machine,the greaterits weakness. SinceI assume matter to be unchangeableand alwaysthe same,it isclearthat weare nolessabletotreat this constantandinvariableproperty in a rigidmannerthan ifit belongedto simpleand puremathe- matics. Therefore,Sagredo,you woulddo wellto changethe opinionwhich you, and perhapsalso many other studentsof mechanics,haveentertainedconcerning the abilityof machines and structures to resist external disturbances,thinkingthat whenthey are builtof the samematerialand maintainthesame ratio betweenparts, they are able equally,or rather propor- tionally, to resist or yield to such external disturbancesand blows. For we can demonstrateby geometrythat the large machineisnotproportionately strongerthan thesmall. Finally, we may say that, for every machineand strucCture, whether artificialor natural, there isset a necessarylimitbeyondwhich neitherart nor nature canpass; it ishereunderstood,of course, that the material is the sameand the proportionpreserved. SAGI_. Mybrainalreadyreels. Mymind,likea cloudmomen- tarily illuminatedby a lightning-flash, is for an instantfilled withan unusuallight,whichnowbeckonsto meand whichnow suddenly minglesand obscuresstrange, crude ideas. From what you have said it appearsto me impossible to buildtwo similarstrucCtures ofthe samematerial,but ofdifferentsizesand have themproportionatelystrong;and if this wereso, it would [52] not be possibleto tindtwosinglepolesmadeof the same-wood which shall be alike in strength and resistancebut unlikein size. SALv. Soit is,Sagredo. _And to makesurethat weunderstand each other, I say that if we take a woodenrod of a certain length and size, fitted, say,into a wall at right angles,i. e., parallel
  • 32. 4 THE TWO NEW SCIENCES OF GALILEO parallelto the horizon,it may be reducedto sucha lengththat it willjust supportitself;sothat if a ha_r'sbreadthbe addedto its lengthit willbreakunderits ownweightandwillbe the only rodof the kindinthe world.* Thus if,forinstance,its lengthbe a hundredtimesits breadth,youwillnotbe abletofindanother rodwhoselengthisalsoa hundredtimesits breadthand which, like the former,is just ableto sustainits own weight and no more:allthe largeroneswillbreakwhileallthe shorteroneswill be strong enough to support somethingmorethan their own weight. And thiswhichI havesaidabout the abilityto support itselfmust beunderstoodto applyalsotoothertests;sothat if a pieceof scantling[corrente] willcarrythe weightoften similarto itself,a beam [trave] having the sameproportionswillnot be ableto supportten similarbeams. Please observe, gentlemen,how faCtswhich at first seem improbablewill, even on scant explanation,drop the cloak which has hidden them and stand forth in naked and simple beauty. Who doesnot knowthat a horsefallingfrom a height of three or four cubitswillbreak hisbones,whilea dog falling fromthe sameheightor a catfroma heightof eightor ten cubits willsufferno injury? Equally harmlesswouldbe the fall of a grasshopper from a toweror the fallof an ant fromthe distance of the moon. Do not childrenfallwith impunityfromheights whichwouldcosttheir eldersabrokenlegor perhapsa fraCtured skull? And just assmalleranimalsare proportionatelystronger and morerobustthan the larger,soalsosmallerplantsare able to stand upbetter than larger. I amcertainyoubothknowthat an oak two hundred cubits [braccia] highwouldnot be ableto sustainits ownbranchesif they weredistributedas in a tree of ordinarysize;and that nature cannotproducea horseas largeas twenty ordinary horsesor a giant ten times taller than an ' [53] ordinary man unless by miracle or by greatly altering the proportionsofhislimbsand especiallyofhisbones,whichwould have to be considerablyenlargedover the ordinary. Likewise the currentbeliefthat, inthe caseof artificialmachinesthevery *The authorhereapparently meansthat the solutionis unique. [Trans.]
  • 33. FIRST DAY 5 largeand the smallare equallyfeasibleand lastingisa man_fest error. Thus, for example,a smallobeliskor columnor other solidfigurecancertainlybe laiddownor setup withoutdanger ofbreaking,whilethe verylargeoneswillgotopiecesunderthe slightestprovocation,and that purelyon accountof their own weight. AndhereI must relatea circumstance whichisworthy ofyourattention asindeedare alleventswhichhappencontrary to expecCtation, especiallywhen a precautionarymeasureturns out to be a causeof disaster. A largemarblecolumnwaslaid out so that its two ends rested each upon a pieceof beam; a little laterit occurredto a mechanicthat, inorderto be doubly sureof its notbreakinginthe middleby its ownweight,itwould be wise to lay a third support midway;this seemedto all an excellentidea;but the sequelshowedthat it wasquitethe oppo- site, fornot many monthspassedbeforethe columnwasfound crackedand brokenexadtlyabovethe newmiddlesupport. Sn_P.A very remarkableand thoroughlyunexpectedacci- dent, especiallyif causedby placingthat newsupport in the middle. SALV. Surely this is the explanation,and the moment the cause is knownour surprisevanishes;forwhenthe two pieces of the columnwereplacedon levelgroundit wasobservedthat one of the end beamshad, after a longwhile,becomedecayed and sunken,but that the middleone remainedhard and strong, thus causingone halfof the columnto projecCt inthe airwithout any support. Under these circumstancesthe body therefore behaveddifferentlyfrom what it wouldhavedoneif supported only upon the first beams; becauseno matter howmuch they might have sunken the columnwouldhave gonewith them. Thisisan accidentwhichcouldnotpossiblyhavehappenedto a smallcolumn,eventhoughmadeofthe samestoneand havinga length corresponding to its thickness,i. e., preservingthe ratio betweenthicknessand lengthfoundinthe largepillar. [541 SAc_.I am quite convinced ofthe fa_s of the case,but I do not understandwhy the strength and resistanceare not multi- pliedinthe sameproportionas the material;and I am the more puzzled
  • 34. 6 THE TWO NEW SCIENCES OF GALILEO puzzledbecause,on the contrary,I havenoticedin othercases that the strength and resistanceagainstbreakingincreasein a largerratio than the amountof material. Thus,forinstance,if two nails be driveninto a wall,the one which is twiceas big as the otherwillsupportnot only twiceas muchweightas the other,but threeor fourtimesasmuch. SALv.Indeedyouwillnotbe far wrongifyousay eighttimes as much; nor doesthis phenomenoncontradicCt the othereven thoughinappearancethey seemsodifferent. SACR. Will you not then, Salviati,removethese difficulties and clear awaytheseobscuritiesif possible:for I imaginethat this problemofresistanceopensup a fieldofbeautifuland useful ideas;and if youare pleasedtomakethisthe subjecCt of to-day's discourseyou willplaceSimplicioand me undermany obliga- tions. SALV. I am at your serviceif onlyI cancall to mindwhat I learned from our Academician* who had thoughtmuch upon this subjecCt and accordingto his custom had demonstrated everything by geometricalmethods so that one might fairly call this a new science. For, althoughsomeof his conclusions had been reachedby others,first of all by Aristotle,these are not the most beautifuland, what is moreimportant,they had not beenprovenina rigidmannerfromfundamentalprinciples. Now,sinceI wishto convinceyou by demonstrativereasoning rather than to persuadeyou by mereprobabilities,I shallsup- posethat youare familiarwithpresent-daymechanicssofar as it is needed in our discussion. First of all it is necessary,to considerwhat happenswhena pieceofwoodor any other solid . which coheresfirmly is broken; for this is the fundamental facet, involvingthe firstand simpleprinciplewhichwemusttake forgrantedas wellknown. To graspthis moreclearly,imaginea cylinderor prism,AB, made of wood or other solid coherent material. Fasten the upper end, A, so that the cylinderhangs vertically. To the lowerend, B, attach the weight C. It is clear that however great they may be, the tenacity and coherence[tenacit_e • I. e. Galileo: Theauthorfrequently refersto himself underthis name. [Tran_r.]
  • 35. FIRST DAY 7 [55] eoeren_] between thepartsofthissolid, solongastheyarenot infinite, canbeovercome by thepulloftheweight C, aweight which canbeincreased indefinitely untilfinally thesolid breaks likea rope.Andas inthecaseoftheropewhose strength we knowto be derived froma multitudeof hempthreadswhich compose it, sointhecaseofthewood, weobserve itsfibres and filaments runlengthwise and render it muchstronger than a hempropeof thesamethickness.But in the caseof a stoneormetallic cylinder wherethe' coherence seemsto be stillgreater the cement whichholdsthe partstogether mustbe some- thingotherthan filaments and fibres;and yet eventhiscanbebroken bya strong pull. Srme.If thismatterbeasyousayI canwell understand thatthefibres ofthewood,beingas longas thepieceofwooditself,renderit strong and resistantagainstlargeforcestendingto breakit. But howcan onemakea ropeone hundredcubitslongoutofhempen fibres which arenotmorethantwoor threecubitslong,and stillgiveit somuchstrength ? Besides, I should begladtohearyouropinion as tothemanner in whichthepartsof metal,stone,andotherma- terialsnot showing a filamentous strucCture are Fig.i put together;for,if I mistakenot,theyexhibitevengreater tenacity. SALV. To solve theproblems which youraiseit willbeneces- sarytomakea digression intosubjecCts which havelittlebearing uponourpresentpurpose. SAcg.But if,by digressions, wecanreachnewtruth,what harmis there in makingonenow,so thatwe maynot lose thisknowledge, remembering that suchan opportunity, once omitted, maynotreturn;remembering alsothatwearenottied down toa fixed andbriefmethod butthatwemeetsolely forour ownentertainment? Indeed,whoknows butthatwemaythus [S6] frequently
  • 36. 8 THE TWO NEW SCIENCESOF GALILEO frequently discover something moreinteresting andbeautiful thanthesolution originally sought._ I begofyou,therefore, to grantthe request of Simplicio, whichis alsomine;forI amno lesscurious anddesirous thanheto learnwhatis thebinding materialwhichholdstogetherthepartsof solidssothat they canscarcely beseparated.Thisinformation is alsoneededto understand the coherence of thepartsof fibresthemselves of Which some solids arebuiltup. SAJ_V. I am at yourservice,sinceyoudesireit. The first question is,Howarefibres, eachnot morethantwoor three cubitsinlength, sotightlyboundtogetherinthecaseof arope onehundredcubitslongthatgreatforce[violent] isrequired to breakit? Nowtellme,Simplicio, canyounotholda hempen fibreso tightlybetween yourfingers that I, pullingby theotherend, wouldbreakit beforedrawing it awayfromyou? Certainly youcan. Andnowwhenthefibres ofhempareheldnotonlyat theends,but aregrasped bythesurrounding medium through- outtheirentirelengthisit notmanifestly moredii_cult totear themloose fromwhatholdsthemthantobreakthem? Butin thecaseoftheropetheveryacCt oftwisting causes thethreads tobindoneanother insuchawaythatwhen theropeisstretched witha greatforcethe fibresbreakratherthan separatefrom eachother. At thepointwherea ropepartsthefibresare,as everyone knows, veryshort,nothing likea cubitlong, astheywoaldbeif the partingof the ropeoccurred, not by thebreaking of the filaments, but bytheirslipping oneovertheother. SAGR. In confirmation ofthisit mayberemarked that ropes sometimes breaknot by a lengthwise pull but by excessive twisting.This,it seems tome,isaconclusive argument because the threadsbindoneanothersotightlythat thecompressing fibres donotpermitthose whicharecompressed to lengthen the spiralseventhat littlebitbywhichit is necessary forthemto lengthen inorder to surround therope which, ontwisting, grows shorterandthicker. SALv. Youarequiteright. Nowseehowonefa_ suggests another
  • 37. FIRST DAY 9 another. Thethreadheldbetween thefingers doesnot yield [ST] toonewhowishes to draw it awayevenwhenpulledwithcon- siderable force, but resistsbecause it is heldbackby a double compression, seeingthat the upperfinger presses againstthe lower as hamasthelower againsttheupper.Now,ifwecould retainonlyoneof thesepressures thereis no doubtthatonly half the original resistance wouldremain;but sincewe are _ not able,by lifting,say,the upperfinger, to removeoneof thesepressures withoutalsoremoving the other,it becomes necessary to preserve oneof themby meansof a newdevice whichcausesthe threadto pressitselfagainstthe finger or againstsome othersolid bodyuponwhichit rests; andthusit is brought aboutthattheveryforce which pulls it inorderto snatchit awaycompresses it moreand moreas the pullincreases. This is accomplished by wrappingthe thread aroundthe solidinthe mannerof a spiral; _I_ andwillbebetterunderstood bymeansofa figure.LetABandCDbetwocylinders be- tween whichis stretched thethreadEF:and _ O forthesakeof greaterclearness wewillim- agineit to be a smallcord. If thesetwo cylinders be pressedstronglytogether,the cordEF,whendrawnbytheendF,willun- doubtedly standa considerable pullbeforeit slipsbetweenthe two compressing solids. But ifweremove oneofthesecylinders the cord,thoughremaining in contacCt withthe other,willnot therebybe prevented from slipping freely.Onthe otherhand,if one holdsthecordloosely againstthetopof the Fig. 2 cylinderA, windsit in the spiralformAFLOTR, and then pullsit by the endR, it is evident that thecordwillbeginto bindthe cylinder; the greaterthenumber of spiralsthemore tightlywillthe cordbe pressedagainstthe cylinderby any given pull. Thusasthenumber of turnsincreases, thelineof contacCt
  • 38. Io THE TWO NEW SCIENCESOF GALILEO contactbecomes longerandin consequence moreresistant;so thatthecordslipsandyieldsto thetractiveforce withincreas- ingdifficulty. [58] Isitnotclearthatthisisprecisely thekindofresistance which onemeetsinthecaseofa thickhempropewhere thefibres form thousands andthousands of similar spirals?And,indeed,the qbinding effecCt of theseturnsis sogreatthata fewshortrushes woventogetherintoa fewinterlacing spiralsformoneof the strongestof ropes whichI believetheycallpackrope[susta]. SAoR. Whatyousayhascleared up twopointswhichI did notpreviously understand.Onefactis howtwo,or at most three,turnsofa ropearoundtheaxleofa windlass cannot only holdit fast,but canalsopreventit fromslipping whenpulled by the immense forceof the weight[forzadelpeso] whichit sustains; andmoreover how,byturningthewindlass, thissame axle,by merefricCtion of theropearoundit, canwindup and lifthugestoneswhilea mereboyi'sableto handle theslack of therope. TheotherfaCt hastodowith asimple butclever device, invented byayoung kins- manof mine,forthe purpose of descending froma window by meansof a ropewithoutlacerating the palmsofhishands,ashadhappened tohimshortly before andgreatlytohisdiscomfort.Asmall sketch willmakethis clear. He tooka woodencylinder, AB,aboutasthickasa walking stickandaboutone spanlong:on thishecut a spiralchannel of about oneturnanda half,andlargeenough tojust receive theropewhich hewished touse. Havingintroduced theropeat theendAandledit outagainat theend BB, heenclosed boththe cylinderandthe ropeina caseofwoodor tin,hingedalongthe81de sothat it Fig. 3 couldbe easilyopenedand closed. After hehad Iastenedtheropeto afirmsupportabove,he could, on grasp- ingandsqueezing thecasewithbothhands,hangbyhisarms. The pressure on therope,lyingbetweenthe caseand thecyl- inder,wassuchthat he could,at will,eithergraspthe case more
  • 39. FIRST DAY II moretightlyand holdhimselffromslipping, or slackenhis hold anddescend asslowly ashewished. IS9] SALV. A truly ingenious device! I feel,however, that for a complete explanation otherconsiderations mightwellenter; yetI mustnotnowdigress uponthisparticular topicsinceyou arewaitingtohearwhatI thinkaboutthebreaking strength of othermaterials which,unlikeropesandmostwoods, do not showa filamentous structure. The coherence of thesebodies is,in myestimation, produced by othercauses whichmaybe grouped undertwoheads. Oneis that much-talked-of repug- nance whichnatureexhibits towards avacuum; butthishorror of a vacuumnot beingsufficient, it is necessary to introduce anothercause intheformof agluey or viscous substance which bindsfirmly togetherthecomponent partsofthebody. FirstI shallspeakofthevacuum, demonstrating bydefinite experiment thequalityandquantityofitsforce[o/rt_].If you taketwohighlypolished andsmooth platesofmarble, metal,or glassandplacethemfacetoface,onewillslideovertheother withthegreatestease,showing conclusively that thereisnoth- ingof aviscous naturebetween them. Butwhenyouattempt to separatethemandkeepthemat a constantdistanceapart, youfindtheplatesexhibit sucha repugnance toseparation that theupperonewillcarrythelower onewithit andkeepit lifted indefinitely, evenwhenthelatterisbigandheavy. This experiment showsthe aversionof naturefor empty space,evenduring thebriefmomentrequired fortheoutside air to rushinandfillup theregion between thetwoplates. It is alsoobserved that if twoplatesarenot thoroughly polished, theircontactisimperfect sothatwhenyouattempttoseparate them slowly the onlyresistance offeredis that of weight;if, however, the pullbe sudden, then the lowerplaterises,but quickly fallsback,havingfollowed theupper plateonlyforthat veryshortintervalof timerequired forthe expansion of the smallamountof air remaining betweenthe plates,in conse- quence oftheirnotfitting, andfortheentrance ofthesurround- ingair. Thisresistance whichis exhibited between the two plates
  • 40. Iz THE TWO NEW SCIENCESOF GALILEO platesisdoubtless likewise present between thepartsofa solid, and enters,at leastin par[,as a concomitant causeof their coherence. [6o] SAGR. Allow meto interruptyoufora moment, please; for I wantto speakof something which justoccurs tome,namely, whenI seehowthelower platefollows theupperoneandhow rapidlyit is lifted,I feelsurethat, contrary to theopinion of manyphilosophers, including perhapsevenAristotlehimself, motionina vacuumis notinstantaneous.If thisweresothe twoplatesmentioned abovewouldseparatewithoutany re- sistance whatever, seeingthat thesameinstantof timewould suffice fortheirseparation andforthesurrounding medium to rushinandfillthevacuum between them. The fa&that the lowerplatefollows theupperoneallows us to infer,not only that motionin a vacuumis not instantaneous, but alsothat, betweenthetwoplates,a vacuum reallyexists,at leastfora veryshorttime,sufficient to allow thesurrounding mediumto rushinandfillthevacuum; foriftherewerenovacuumthere would benoneedofanymotion inthemedium.Onemustadmit thenthat a vacuumis sometimes produced by violentmotion [violenza] or contraryto the lawsof nature,(although in my opinion nothing occurs contrary tonatureexcept theimpossible, andthat never occurs). But hereanotherdifficulty arises. Whileexperiment con- vincesmeofthecorrecCtness of thisconclusion, mymindis not entirelysatisfied as to thecauseto whichthis effe&is to be attributed. For the separationof the platesprecedesthe formation of thevacuumwhichis produced as a consequence ofthisseparation; andsince it appears tomethat,intheorderof nature,the causemustprecedetheeffe&, eventhoughit ap- pearsto follow inpointoftime,andsinceevery positive effecCt musthavea positive cause,I do notseehowtheadhesion of twoplatesandtheirresistance to separation--acCrual fa_s---can bereferredto a vacuumas cause whenthisvacuumis yet to follow.According to theinfallible maximof thePhilosopher, thenon-existent canproduce noeffe&. Simp.
  • 41. FIRST DAY 13 Sire,. Seeingthat youacceptthis axiomofAristotle,I hardly thinkyouwillreje_ anotherexcellent and reliable maximof his, namely,Nature undertakesonly that which happenswithout resistance;and inthis saying,it appearsto me,youwillfindthe solutionof your difficulty. Sincenature abhorsa vacuum,she preventsthat fromwhicha vacuumwouldfollowas a necessary consequence.Thus it happensthat naturepreventsthe separa- tionofthe twoplates. [6i] SACR. Nowadmittingthat what Simplicio saysisan adequate solutionof my difficulty,it seemsto me, ifI may be allowedto resume my former argument, that this very resistanceto a vacuumought to be sufficientto holdtogetherthe parts either of stoneor of metalor the parts of any othersolidwhichisknit togethermorestronglyandwhichismoreresistanttoseparation. If for one effe_ there be onlyone cause,or if,morebeingas- signed,they canbe reducedto one,thenwhyisnotthis vacuum whichreallyexistsa sufficientcausefor allkindsof resistance ? SALV. I do not wishjust nowto enter this discussionas to whether the vacuum alone is sufficientto hold together the separateparts of a solidbody;but I assureyouthat the vacuum whichacCts as a sufficient causeinthe caseofthetwoplatesisnot alonesufficientto bind togetherthe partsof a solidcylinderof marble or metal which, whenpulled violently,separatesand divides. Andnow if I finda methodof distinguishing thiswell known resistance,dependingupon the vacuum, from every other kind which might increasethe coherence,and if I show you that the aforesaidresistancealoneis not nearlysufficient for such an effect, willyou not grant that we are bound to introduceanother cause. _ Help him, Simplicio,sincehe does not knowwhat replytomake. SIMP. Surely,Sagredo'shesitationmust be owingto another reason,fortherecanbe nodoubtconcerning a conclusion which isat oncesoclearandlogical. SACra. Youhaveguessedrightly,Simplicio. I waswondering whether, if a millionof gold each year from Spain were not sufficientto pay the army, it might not be necessary to make
  • 42. I4 THE TWO NEW SCIENCES OF GALILEO make provisionother than small coin for the pay of the soldiers.* But go ahead,Salviati;assumethat I admityour conclusion and showusyourmethodof separatingtheacCtion of thevacuum from other causes;and by measuringit showus how it is not sufficient to producethe effectin question. SALV. Your good angel assistyou. I willtell you how to separatethe forceof the vacuumfrom the others, and a{ter- wards how to measure it. For this purposelet us considera continuoussubstancewhoseparts lackall resistanceto separa- tionexceptthat derivedfroma vacuum,suchas isthe casewith water,a fact fullydemonstratedbyourAcademician inoneof his treatises. Whenever a cylinderofwaterissubjectedto apulland [62] offersa resistanceto the separation of itsparts this canbe attrib- uted tonoothercausethantheresistance of the /k_j vacuum. In orderto try suchan experiment I have invented a devicewhichI can better explainby meansof a sketchthan by mere words. Let CABDrepresentthe crosssection of a cylindereither of metal or, preferably, of glass,hollowinsideand accuratelyturned. G I-I Into this is introduced a perfec°dyfitting i C . Dcylinderof wood,representedin crosssection by EGHF, and capableof up-and-downmo- tion. Through the middleof this cylinderis boreda holeto receivean ironwire,carrying a hook at the end K, while the upper end of the wire, I, is providedwith a conical head. The woodencylinderis countersunk Fig.4 at the top so as to receive,witha perfect fit, the conicalhead I of the wire,IK,whenpulleddown by theendK. NowinsertthewoodencylinderEH inthe hollow cyllnderAD, soas not to touchtheupperend of thelatterbut to leavefreea spaceof two or threefinger-breadths; this spaceis to be filled *Thebearing ofthisremark becomes clear onreading whatSalviati says onp. 18below.[Trans.]
  • 43. FIRST DAY 15 withwaterbyholding thevessel withthemouthCD upwards, pushing down onthestopper EH,andat thesametimekeeping theconical headofthewire,I, away fromthehollow portion of thewooden cylinder.Theairisthusallowed toescape alongside theironwire(which doesnotmakea close fit)assoonas one pressesdownon the woodenstopper. The air havingbeen allowed to escape andtheironwirehaving beendrawn backso that it fits snugly againstthe conical depression inthewood, invert thevessel, bringing itmouthdownwards, andhangonthe hookK a vesselwhichcanbe filled withsandor anyheavy materialin quantitysufficient to finallyseparatethe upper surface ofthestopper, EF, fromthelowersurface ofthewater towhichit wasattached onlyby theresistance ofthevacuum. Nextweighthe stopperandwiretogetherwiththeattached vesseland its contents; we shallthenhavethe forceof the vacuum [forza ddvacuo].Ifoneattaches toacylinder ofmarble [63] or glassa weight which, together withtheweight ofthemarble or glassitself,is just equalto the sumof theweights before mentioned, andifbreaking occurs weshallthenbejustified in sayingthatthevacuum aloneholdsthepartsofthemarble and glasstogether; but ifthisweight doesnotsuffice andifbreaking occursonlyafteradding, say,fourtimesthisweight, weshall thenbe compelled to saythat thevacuumfurnishes onlyone fifthofthetotalresistance [resf._ema]. SLurP. Noonecandoubtthecleverness ofthedevice; yetit presentsmanydifficulties which makemedoubtitsreliability. Forwhowillassure usthattheairdoesnotcreepinbetween the glassandstopperevenif it is wellpackedwithtowor other yielding material._ I question alsowhether oiling withwaxor turpentine willsuffice tomakethecone, I,fitsnugly onitsseat. Besides, maynot the partsof the waterexpandand dilate? Whymaynottheairorexhalations orsome othermoresubtile substances penetrate theporesofthewood,orevenoftheglass itself? SAT.v. Withgreatskillindeed hasSimplicio laidbefore usthe difficulties; andhehasevenpartlysuggested howtopreventthe air
  • 44. x6 THE TWO NEW SCIENCESOF GALILEO airfrompenetrating thewood or passing between thewood and theglass. Butnowletmepoifitoutthat, asourexperience in- creases, weshalllearnwhetheror notthesealleged difficulties reallyexist. For if,as is thecasewithair,wateris bynature expansible, although onlyunderseveretreatment, weshallsee thestopperdescend; and ifweput a smallexcavation in the upperpart oftheglassvessel, suchas indicated byV,thenthe air or anyothertenuousandgaseous substance, whichmight penetratethe poresof glassor wood,wouldpassthroughthe waterandcolle& inthisreceptacle V. Butifthesethingsdonot happenwemayrestassured thatourexperknent hasbeenper- formedwithpropercaution;andweshalldiscover that water doesnot dilateand that glassdoesnot allowany material, however tenuous, topenetrate it. SAGm Thanks tothisdiscussion, Ihavelearned thecause ofa certaineffe& whichI havelongwondered at anddespaired of understanding.I oncesawa cistern whichhadbeenprovided witha pumpunderthe mistakenimpression that the water mightthusbedrawnwithlesseffort oringreater quantitythan bymeansoftheordinary bucket. Thestockofthepumpcar- [64] rieditssucker andvalveintheupperpartsothatthewaterwas liftedby attra&ion andnotbya pushasisthecase withpumps inwhichthesuckerisplacedlowerdown.Thispumpworked peffedtly solong asthewaterinthecistern stoodabove acertain level;but belowthis levelthepumpfailedto work. WhenI firstnoticed thisphenomenon I thoughtthemachine wasoutof order;but theworkman whomI called in to repairit toldme the defecCt wasnot in the pumpbut in thewaterwhichhad fallentoolowtoberaisedthrough sucha height;andheadded that it wasnot possible, eitherby a pumpor by any other machine working on theprinciple of attra&ion, to liftwatera hair'sbreadthaboveeighteencubits;whetherthe pump be largeor smallthisis theextreme limitof thelift. Upto this timeI hadbeensothoughtless that,although I knewa rope,or rodofwood,orof iron,if sufficiently long,wouldbreakby its ownweight whenheldbytheupperend,it never occurred tome " that
  • 45. FIRST DAY 17 thatthesamethingwould happen, onlymuchmoreeasily, toa columnof water. And reallyis not that thingwhichis at- tra_ed inthepumpa column ofwaterattachedat theupper endandstretched moreandmoreuntilfinally apointisreached whereit breaks,likea rope,onaccount ofitsexcessive weight ? SALV. That isprecisely thewayit works; thisfixed elevation ofeighteen cubits istrueforanyquantityofwaterwhatever, be thepumplargeor smallor evenasfineas a straw. Wemay therefore saythat, onweighing thewatercontained in a tube eighteen cubitslong,no matterwhatthe diameter, we shall obtainthevalueoftheresistance ofthevacuum ina cylinder of anysolidmaterialhavinga boreof thissamediameter.And havinggonesofar, let us seehoweasyit is to findto what lengthcylinders ofmetal,stone,wood, glass,etc.,ofanydiam- etercanbe elongated withoutbreaking by theirownweight. [6S] Takeforinstance a copper wireofanylengthandthickness; fixthe upperend andto theotherend attacha greaterand greaterloaduntilfinally thewirebreaks;letthemaximum load be, say,fiftypounds.Thenit is clearthat if fiftypoundsof copper,inadditionto theweightofthewireitselfwhichmay be, say,z/sounce,is drawnout intowireof thissamesizewe shallhavethegreatest length ofthiskindofwirewhich cansus- tainitsownweight. Suppose thewirewhichbreaks to beone cubitin lengthandI/sounceinweight;thensinceit supports 5olbs.inadditionto itsownweight,i.e.,48ooeighths-of-an- ounce,it follows that allcopper wires, independent ofsize,can sustainthemselves up to a lengthof48Olcubitsandnomore. Sincethen a copperrod can sustainitsownweightup to a lengthof48Olcubitsit follows that thatpart ofthebreaking strength [resistenza] which depends uponthevacuum, comparing itwiththeremaining facetors ofresistance, isequaltotheweight ofa rodofwater,eighteen cubits longandasthickasthecopper rod. If,forexample, copper isninetimesasheavyaswater,the breaking strength[resistenza allostrappars.z] of anycopperrod, insofarasit depends uponthevacuum, asequalto theweight of twocubitsof thissamerod. By a similar methodonecan find
  • 46. I8 THE TWO NEW SCIENCESOF GALILEO findthemaximum lengthofwireorrodofanymaterial which willjust sustainitsownweight,andcanat the sametimedis- cover thepartwhichthevacuum playsinitsbreaking strength. SACR. It stillremains foryouto tellus uponwhatdepends theresistance tobreaking, otherthanthatofthevacuum; what is the glueyor viscous substance whichcementstogetherthe partsof the solid? For I cannotimagine a gluethat willnot burnup ina highlyheatedfurnace intwoor threemonths,or certainly withintenor a hundred.Forifgold,silverandglass arekeptfora longwhileinthemoltenstateandareremoved fromthe furnace,theirparts,on cooling, immediately reunite and bind themselves togetheras before. Not only so,but whatever difficulty arises withrespe_tothecementation ofthe partsof theglass arisesalsowithregard tothepartsoftheglue; inotherwords, whatisthatwhichholdsthesepartstogetherso firmly? [661 SALv. A littlewhile ago,I expressed thehopethatyourgood angelmightassistyou. I nowfindmyself inthesamestraits. Experiment leavesno doubtthat the reasonwhytwoplates cannotbeseparated, exceptwithviolent effort,isthat theyare heldtogetherby theresistance ofthe vacuum; andthe same canbesaidof twolargepiecesof amarble or bronzecolumn. Thisbeingso,I donotseewhythissame cause maynotexplain thecoherence ofsmaller partsandindeed of theverysmallest particles of thesematerials.Now,since eacheffe_musthave onetrueandsufficient cause andsince Ifindnoothercement, am I notjustified intryingtodiscover whether thevacuum isnot a sufficient cause? S_. But seeingthat youhavealready provedthat there- sistancewhichthe largevacuumoffersto the separation of twolargepartsofasolid isreally verysmall incomparison with thatcohesive force which bindstogether themostminute parts, whydo youhesitateto regardthis latter as something very different fromtheformer ? S_v. Sagredo hasalready [p.I3 above] answered thisques- tionwhenhe remarked that eachindividual soldier wasbeing paid
  • 47. FIRST DAY 19 paidfromcoincoiled-ted by a general taxofpennies andfarth- ings,whileevena million ofgoldwould not suffice to paythe entirearmy. Andwhoknowsbut that theremay be other extremely minutevacuawhichaffecCt thesmallest particles so thatthatwhich bindstogether thecontiguous partsisthrough- outofthesamemintage ? Letmetellyousomething which has justoccurred tomeandwhich I donotoffer asanabsolute facet, but ratheras a passing thought, stillimmature andcalling for morecareful consideration. Youmaytakeofit whatyoulike; andjudgetherestasyouseefit. Sometimes whenI haveob- servedhowfirewindsits way in betweenthe mostminute particles ofthisorthatmetaland,eventhough thesearesolidly cemented together,tearsthemapartandseparates them,and whenI haveobserved that,onremoving thefire,theseparticles reunitewiththesametenacityas at first,withoutany lossof quantityin thecaseofgoldandwithlittlelossinthecaseof othermetals, eventhoughthesepartshavebeenseparated fora longwhile,I havethoughtthattheexplanation mightlieinthe factthat the extremely fineparticles of fire,penetrating the slenderporesof themetal(toosmallto admiteventhefinest particles of air or of manyotherfluids), wouldfillthe small intervening vacuaandwould setfreethesesmallparticles from the attracCtion whichthesesamevacuaexertuponthemand which prevents theirseparation.Thustheparticles areableto [671 movefreely sothatthemass[rnassa] becomes fluidandremains soaslong astheparticles offireremain inside; butiftheydepart andleavetheformer vacuathentheoriginal attraction [attraz- zione] returnsandthepartsareagaincemented together. In replytothequestion raised bySimplicio, onemaysaythat althougheachparticularvacuumis exceedingly minuteand therefore easily overcome, yettheirnumber issoextraordinarily greatthat theircombined resistance is,soto speak,multipled almostwithoutlimit. The natureand the amountof force [forza] whichresults[risulta] fromadding togetheranimmense numberof smallforces[debolissimi rnornent_] is clearlyillus- tratedbythefa_ thataweight ofmillions ofpounds, suspended by
  • 48. 20 THE TWO NEW SCIENCES OF GALILEO by great cables,is overcomeand lifted,whenthe south wind carries innumerableatoms of water, suspendedin thin mist, whichmovingthroughthe airpenetratebetweenthefibresof the tense ropes in spite of the tremendousforceof the hanging weight. When these particles enter the narrow pores they swell the ropes, thereby shorten them, and perforcelift the heavymass[mole]. SAcR. There canbe no doubtthat any resistance,solongas it is not infinite,may be overcomeby a multitudeof minute forces. Thus a vast numberof ants might carryashorea ship laden with grain. And sinceexperienceshowsus daily that one ant caneasilycarry onegrain,it isclearthat the numberof rains in the shipis not infinite,but fallsbelowa certainlimit. you take anothernumberfouror six timesas great, and if you set to worka corresponding numberof ants theywillcarry the grainashoreand the boat also. It istrue that thiswillcall fora prodigiousnumberof ants,but in my opinionthis ispre- ciselythe casewith the vacua which bind togetherthe least particlesofa metal. SALV. But even if this demandedan infinitenumberwould you stillthink 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 seeif wecannotprovethat withina finiteex- tent it ispossibletodiscoveran infinitenumberofvacua. Atthe sametimeweshallat least reacha solutionof the mostremark- ableof all that list of problemswhichAristotle himselfcalls wonderful;I referto hisQuestions in Mechanics.This solution may be no lessclearand conclusive than that whichhe himself givesand quitedifferentalsofromthat socleverly expoundedby themostlearnedMonsignordiGuevara.* First it is necessaryto considera proposition, not treated by others,but uponwhichdependsthe solutionofthe problemand from which, if I mistake not, we shallderiveother new and remarkable facts. For the sake of clearnesslet us draw an *Bishop ofTeano; b.x56x ,d.I64I. [Trans.]
  • 49. FIRST DAY 2I accurate figure. AboutG as a centerdescribe anequiangular andequilateral polygon ofanynumber ofsides, saythehexagon ABCDEF. Similarto this and concentric with it, describe anothersmaller onewhich weshallcallHIIZT.MN. Prolong the F , ..... ff"i ............ 4 , , . T -- ,,I ! t , I ( . ": iim] B Fig. 5 sideAB,of thelargerhexagon, indefinitely towardS; in like mannerprolong thecorresponding sideHI ofthe smaller hex- agon,inthe samedirecCtion, sothat thelineHT isparallelto AS;andthroughthe centerdrawthelineGVparallel to the othertwo. Thisdone,imagine thelargerpolygon torollupon [69] thelineAS,carrying withit thesmaller polygon.It isevident that,ifthepointB,theendofthesideAB,remains fixed at the beginning oftherotation,thepointAwillriseandthepointC willfalldescribing thearcCQuntilthesideBCcoincides with thelineBQ,equaltoBC. Butduring thisrotation thepointI, onthesmaller polygon, willriseabove thelineITbecause IBis oblique toAS;andit willnotagainreturn tothelineITuntilthe pointC shallhavereachedtheposition Q. ThepointI, having described thearcIOabove thelineHT,willreachtheposition Oat
  • 50. 2z THE TWO NEW SCIENCES OF GALILEO 0 at the sametimethe sideIK assumesthe position0P; but in the meantimethe centerG has traverseda path aboveGVand doesnot return to it until it has _ompletedthe arc GC. This stephavingbeentaken,the largerpolygonhas beenbroughtto rest withits sideBC coinciding withthe lineBQwhilethe side IK of the smallerpolygonhas beenmadeto coincidewith the lineOP,havingpassedoverthe portionI0 withouttouchingit; alsothe centerG willhavereachedthe positionC after having traversedallits courseabovethe parallellineGV. Andfinally the entire figurewillassumea positionsimilarto the first,so that ifwecontinuethe rotationand cometo the next step,the sideDC of the largerpolygonwillcoincidewiththe portionQX , and the sideKL of the smallerpolygon, havingfirstskippedthe arc PY,willfallon YZ, whilethe centerstillkeepingabovethe lineGV willreturn to it at R after havingjumpedthe interval CR. At the endofonecompleterotationthe largerpolygonwill havetraced upon the lineAS,withoutbreak,sixlinestogether equal to its perimeter; the lesserpolygonwill likewisehave imprintedsixlinesequalto its perimeter,but separatedby the interpositionof five arcs, whose chords represent the parts of HT not touchedby the polygon:the centerG neverreaches the lineGV exceptat sixpoints. From this it isclearthat the spacetraversedby the smallerpolygonis almostequalto that traversedby the larger,that is, the lineHT approximatesthe lineAS,differingfrom it onlyby the lengthof one chordof one ofthesearcs,provided weunderstand the lineI-ITto include the fiveskippedarcs. Now this exposition whichI have givenin the caseof these hexagonsmust be understoodto be applicableto all other polygons, whatever the numberof sides,providedonlytheyare [70] similar, concentric,and rigidlyconnecCted, so that when the greateronerotatesthe lesserwillalsoturn howeversmallit may be. Youmust alsounderstandthat the linesdescribedby these two are nearlyequalprovidedwe includein the spacetraversed by the smallerone the intervalswhichare not touchedby any part ofthe perimeterofthis smallerpolygon. Let
  • 51. FIRST DAY z3 Let a largepolygonof, say, one thousandsides makeone completerotationand thus layoffa lineequalto its perimeter; at the sametimethe smallonewillpassoveran approximately equal distance, made up of a thousand smallportions each equalto oneof its sides,but interruptedby a thousandspaces which,in contrastwiththe portionsthat coincide withthe sides of the polygon,we may call empty. Sofar the matter isfree fromdifficulty or doubt. But nowsupposethat about any center,say A,we describe two concentricand rigidlyconneCtedcircles;and supposethat from the points C and B, on their radii, there are drawn the tangentsCEand BF and that throughthe centerAthe lineAD is drawnparallelto them, then if the large circlemakesone completerotation alongthe lineBF, equalnotonly to its cir- cumference but alsoto the othertwo linesCE andAD, tellme what the smallercirclewilldoand alsowhat the centerwilldo. Asto the center it willcertainlytraverseand touch the entire lineAD whilethe circumference of the smallercirclewillhave measuredoffby its pointsof contaCtthe entirelineCE,just as wasdoneby theabovementionedpolygons.Theonlydifference is that the lineI-ITwasnot at everypointin contactwiththe perimeterof the smallerpolygon,but therewereleftuntouched as manyvacant spacesas therewerespacescoinciding withthe sides. But hereinthe caseofthe circles the circumference ofthe smalleroneneverleavesthe lineCE, sothat nopartof the latter isleftuntouched,noristhereevera timewhensomepointonthe circleisnot in contaCt withthe straightline. Hownowcanthe smallercircletraversea lengthgreater than its circumference unlessit goby jumps? 8AGmIt seemsto methat onemaysaythat just as thecenter ofthe circle,by itself,carriedalongthe lineAD isconstantlyin contac2 withit, althoughit isonlya singlepoint,sothepointson the circumference of the smaller circle,carried alongby the motionof the largercircle,wouldslideoversomesmallparts of the lineCE. : [7I] : SALV. There are two reasonswhy this cannothappen. First because ?
  • 52. 24 THE TWO NEW SCIENCESOF GALILEO because thereis nogroundforthinking that onepointof con- taCt,suchas that at C, ratherthan another, shouldslipover certain portionsofthelineCE. Butifsuchslidings alongCE didoccurtheywould beinfiniteinnumber sincethepointsof contaCt (being merepoints)areinfinite innumber:aninfinite number offiniteslipswillhowever makeaninfinitely longline, while asamatteroffaCtthelineCEisfinite. Theotherreason isthat asthegreatercircle,initsrotation, changes itspointof contactcontinuously thelesser circle mustdothesame because Bistheonlypointfrom whichastraightlinecanbedrawntoA andpassthrough C. Accordingly thesmallcircle mustchange itspointofcontactwhenever thelargeonechanges: nopointof thesmallcircletouches thestraightlineCE inmorethanone point. Notonlyso,but evenin therotationof thepolygons therewasnopointon theperimeter ofthesmaller whichcoin- cidedwithmorethanonepointonthe linetraversed by that perimeter; this is at onceclearwhenyouremember that the lineIKisparallel toBCandthattherefore IKwillremain above IPuntilBCcoincides withBQ,andthatIKwillnotlieuponIP except attheveryinstant whenBCoccupies theposition BQ;at thisinstanttheentireline IKcoincides withOPandimmediately afterwards risesabove it. SAOl_. Thisisaveryintricate matter. I seenosolution.Pray explain it tous. SALV. Let usreturnto theconsideration ofthe above men- tionedpolygons whosebehavior wealready understand.Now inthecaseofpolygons withIOOOOO sides, thelinetraversed by the perimeterof thegreater,i. e.,the linelaiddownby its IOOCXX) sides oneafteranother, isequaltothelinetracedoutby theIOCX:_ sidesofthesmaller, provided weinclude theIO(Xx_ vacantspaces interspersed.Soin thecaseofthecircles, poly- gonshavingan infinitude of sides,the linetraversed by the continuously distributed[continuamente dispostz] infinitude of sidesisinthegreatercircle equalto thelinelaiddownby the infinitude of sidesinthe smaller circle but withtheexception that theselatteralternatewithemptyspaces;and sincethe sidesarenotfiniteinnumber, butinfinite, soalsoaretheinter- vening
  • 53. FIRST DAY 25 veningempty spacesnotfinitebut infinite. The linetraversed by the largercircleconsiststhen of an infinitenumberof points whichcompletely fillit; whilethat whichistracedbythe smaller circleconsistsof an infinitenumberof pointswhichleaveempty spacesand only partly fill the line. And here I wishyou to observethat after dividingand resolvinga line into a finite numberof parts, that is,intoa numberwhichcanbecounted,it [72] isnotpossibleto arrangethem againintoa greaterlengththan that whichthey occupiedwhenthey formeda continuum[con- tinuate]and were conne_ed without the interpositionof as many empty spaces. But if weconsiderthe lineresolvedinto an infinitenumberof infinitelysmalland indivisible parts, we shallbe ableto conceivethe lineextendedindefinitely by the interposition,not of a finite,but of an infinitenumberof in- finitelysmallindivisible emptyspaces. Nowthiswhichhasbeensaidconcerning simplelinesmust be understoodto holdalsointhe caseof surfacesand solidbodies, it being assumedthat they are made up of an infinite,not a finite,number of atoms. Such a body once divided into a finitenumberofparts itisimpossible toreassemble themsoas to occupymore space than beforeunless we interposea finite numberof empty spaces,that is to say, spacesfree from the substanceof whichthe solidis made. But if we imaginethe body, by someextreme and final analysis,resolvedinto its primaryelements,infinitein number,then weshallbe ableto think of them as indefinitelyextended in space,not by the interpositionof a finite,but of an infinitenumberof empty spaces. Thus one can easilyimaginea smallball of goldex- panded into a very largespacewithout the introducCtion of a finite number of empty spaces,alwaysprovidedthe gold is madeupof aninfinitenumberof indivisible parts. SIM1,. It seemsto me that you are travellingalongtoward thosevacuaadvocatedby a certainancientphilosopher. SAzv.But youhavefailedto add,"whodeniedDivineProvi- dence,"an inapt remarkmadeon a similaroccasionby a cer- tain antagonistofour Academician. Simp.
  • 54. 26 THE TWO NEW SCIENCESOF GALILEO Sr_P.I noticed,andnotwithoutindignation, the rancorof thisill-natured opponent;furtherreferences to theseaffairsI omit,not onlyas a matterof goodform,but alsobecause I know howunpleasant theyaretothegoodtemperedandwell orderedmindof oneso religious and pious,soorthodox and God-fearing asyou. Butto returnto oursubject,yourprevious discourse leaves withmemanydifficulties whichI amunableto solve. First among theseis that,ifthecircumferences ofthetwocircles are equalto the twostraightlines,CE and BF, the latter con- sideredasa continuum, theformeras interrupted withan in- finityofemptypoints,I donotseehowit ispossible to saythat thelineADdescribed bythecenter,andmadeupofaninfinity ofpoints, isequaltothiscenter which isa single point. Besides, thisbuilding upof linesoutofpoints, divisibles outofindivisi- bles,andfinites outofinfinites, offers meanobstacle difficult to avoid;andthe necessity of introducing a vacuum,soconclu- sively refutedbyAristotle, presents thesame difficulty. [73] SAr.V. Thesedifficulties arereal;andtheyarenot theonly ones. Butlet usremember thatwearedealing withinfinities and indivisibles, both of whichtranscendour finiteunder- standing, theformer on account oftheirmagnitude, thelatter because oftheirsmallness.In spiteofthis,mencannotrefrain fromdiscussing them,eventhough it mustbedoneina round- aboutway. Therefore I alsoshould liketotaketheliberty topresent some of my ideaswhich,thoughnot necessarily convincing, would, onaccount of theirnovelty, at least,provesomewhat startling. But sucha diversion mightperhaps carryustoofarawayfrom thesubjectunderdiscussion andmighttherefore appear toyou inopportune andnotverypleasing. SACR. Prayletusenjoytheadvantages andprivileges which comefromconversation between friends, especially uponsub- jects freelychosenand not forceduponus, a matter vastly different fromdealing withdeadbooks whichgiverisetomany doubtsbutremove none. Share withus,therefore, thethoughts which
  • 55. FIRST DAY z7 which ourdiscussion hassuggested toyou;forsince wearefree fromurgentbusiness therewillbeabundant timetopursue the topics alreadymentioned; and in particularthe obje£tions raised bySimplicio oughtnotinanywisetobenegle&ed. S_J_v. Granted,sinceyousodesire.Thefirstquestion was, Howcana single pointbeequaltoa line? Since I cannotdo moreat presentI shallattempttoremove, orat leastdiminish, oneimprobability by introducing a similaror a greaterone, justas sometimes awonder isdiminished byamiracle.* AndthisI shalldoby showing youtwoequalsurfaces, to- getherwithtwoequalsolidslocated uponthesesamesurfaces asbases, allfourofwhichdiminish continuously anduniformly in sucha waythat theirremainders always preserve equality among themselves, andfinally boththesurfaces andthesolids terminate theirprevious constant equality by degenerating, the onesolidandtheonesurfaceintoa verylongline,theother solidand the other surface into a singlepoint;that is, the latterto onepoint,theformer toaninfinite number ofpoints. [74] SACR. Thisproposition appearsto me wonderfial, indeed; butletusheartheexplanation anddemonstration. SALV. Sincethe proofis purelygeometrical we shallneed a figure. Let_FB be a semicircle withcenterat C;aboutit describe the re&angle ADEBand fromthe centerdrawthe straightlinesCDandCEto thepointsD andE. Imagine the radius CFtobedrawn perpendicular toeitherofthelines ABor DE,andtheentirefigure to rotateaboutthisradiusasanaxis. It isclearthatthere&angle ADEBwillthusdescribe acylinder, thesemicircle AFBahemisphere, andthetriangle CDE,a cone. Nextletus remove thehemisphere but leave theconeandthe restofthecylinder, which, onaccount ofitsshape, wewillcalla "bowl." Firstweshallprovethat thebowlandthe coneare equal;thenweshall show thataplane drawn parallel tothecircle which forms thebaseofthebowlandwhich hasthelineDEfor diameter andF foracenterwaplane whose traceisGN---cuts thebowlinthepoints G,I,O,N,andtheconeinthepoints I-I,L, sothatthepartofthecone indicated byCHLisalways equal to *Cf.p.3obelow. [Trans.]
  • 56. z8 THE TWO NEW SCIENCESOF GALILEO thepartofthebowlwhose profile isrepresented bythetriangles GAIandBON. Besides thisweshallprovethatthebaseofthe cone,i.e.,thecircle whose diameter isHL,isequaltothecircular A C 5 surface whichforms thebaseof ___ thisportionof thebowl,or as onemightsay,equaltoa ribbon G N whosewidthis OI. (Noteby the waythenatureof mathe- maticaldefinitions whichcon- . sistmerely intheimposition of D F _ names or,ifyouprefer, abbrevi- Fig.6 ations ofspeech established and introduced in orderto avoidthetediousdrudgerywhich you and I now experience simplybecause we have not agreed to call this surfacea "circularband" and that sharpsolid portionof the bowla "round razor.") Nowcallthem by [75] whatnameyouplease, itsuffices tounderstand that theplane, drawnat any heightwhatever,so longas it is parallelto thebase,i.e.,to thecircle whose diameter isDE,alwayscuts thetwosolids sothattheportion CHLofthecone isequal tothe upperportionofthebowl;likewise thetwoareaswhich arethe basesofthesesolids, namely thebandandthecircle I-IL, arealso equal. Herewehavethemiracle mentioned above; asthecut- tingplaneapproaches thelineABtheportions ofthesolids cut offarealways equal,soalsotheareasoftheirbases.Andasthe cuttingplanecomes nearthetop,thetwosolids (always equal) aswellastheirbases(areas which arealsoequal) finally vanish, onepairofthemdegenerating intothecircumference ofa circle, theotherintoasingle point,namely, theupperedge ofthebowl andtheapexof thecone. Now,sinceas thesesolids diminish equality ismaintained between themuptotheverylast,weare justified in sayingthat, at the extreme andfinalendof this diminution, theyarestillequaland that oneis not infinitely greaterthan the other. It appearsthereforethat we may equatethecircumference ofa largecircle to asingle point. And thiswhichistrueofthesolids istruealsoofthesurfaces which form
  • 57. FIRST DAY 29 formtheirbases;forthesealsopreserveequalitybetweenthem- selvesthroughouttheir diminutionand in the end vanish,the one into the circumference of a circle,the other into a single point. Shallwenotthen callthemequalseeing thattheyarethe last tracesand remnantsof equalmagnitudes ? Note alsothat, even if these vesselswere large enoughto contain immense celestial hemispheres, both their upperedgesand the apexesof the conestherein containedwould alwaysremainequal and wouldvanish,the formerinto circleshavingthe dimensions of the largestcelestialorbits, the latterinto singlepoints. Hence in conformitywith the precedingwe may say that all circum- ferencesof circles,howeverdifferent,are equalto each other, andareeachequalto a singlepoint. SAtin.This presentationstrikesme as so clever and novel that, even if I were able, I wouldnot be willingto opposeit; forto defacesobeautifula stru_ure by a bluntpedanticattack wouldbe nothingshortofsinful. But forour completesatisfac- [76] tion pray give us this geometricalproof that there is always equality between thesesolidsand between their bases; for it cannot,I think, fail to be very ingenious, seeinghow subtleis the philosophical argumentbaseduponthis result. SAJ_v. The demonstrationisboth short and easy. Referring to the precedingfigure,sinceIPC isa rightanglethe squareof the radiusIC isequalto the sumofthe squareson thetwo sides IP, PC; but the radiusIC isequalto ACand alsoto GP, while CP isequalto PH. Hencethe squareof the lineGP isequalto the sumof the squaresof IP andPH, ormuklplyingthroughby 4,wehavethesquareof the diameterGN equaltothe sumofthe squareson IO and HL. And, sincethe areasof circlesare to eachotheras the squaresof their diameters,it followsthat the areaofthe circle whosediameterisGN isequaltothe sumofthe areasof circles havingdiametersIOandI-i-L, sothat ifweremove the commonarea of the circlehavingIO for diameterthe re- mainingareaof the circleGN willbe equalto the areaof the circlewhosediameterisHL. Somuchforthefirstpart. Asfor the otherpart, weleaveitsdemonstration forthe present,partly because
  • 58. 30 THE TWO NEW SCIENCES OF GALILEO because thosewhowishto follow it willfindit in thetwelfth proposition ofthesecond bookof Decentro gravitatis solidorum bytheArchimedes ofourage,LucaValerio,* whomadeuseofit fora different objec°c, andpartlybecause, forourpurpose, it suffices to have seenthat the above-mentioned surfacesare always equalandthat, as theykeepon diminishing uniformly, theydegenerate, theoneintoa single point,theotherintothe circumference ofa circle largerthananyassignable; inthisfa& liesourmiracle.t SACR. The demonstration is ingenious and the inferences drawnfromit areremarkable. Andnowletushearsomething concerning theotherdifficulty raised by Simplicio, ifyouhave anythingspecialto say,which,however, seemsto mehardly possible, sincethematterhasalready beensothoroughly dis- cussed. S_mv. ButI dohavesomething special to say,andwillfirst of all repeatwhatI saida littlewhileago,namely,that in- finityandindivisibility areintheirverynatureincomprehensi- bleto us;imagine thenwhattheyarewhencombined.Yetif [77] wewishto buildup a lineout of indivisible points,wemust take aninfinite numberof them,andare,therefore, boundto understand boththe infiniteand the indivisible at the same time. Manyideas havepassed through mymindconcerning this subjecCt, some ofwhich, possibly themoreimportant, I maynot beableto recallonthe spurof themoment;but inthecourse ofourdiscussion it mayhappen thatI shallawaken inyou,and especially in Simplicio, objecCtions and difficulties whichin turn willbringto memory thatwhich, withoutsuchstimulus, wouldhavelaindormantinmymind. Allow metherefore the customary libertyofintroducing some ofourhumanfancies, for indeedwemayso callthemin comparison withsupernatural truth whichfurnishes theonetrue andsaferecourse fordeci- sionin ourdiscussions andwhichis aninfallible guideinthe darkanddubious pathsofthought. *Distinguished Italian mathematician; bornatFerrara about I5S2; admitted totheAccademia dei Lincel I612; died I618.[Trans.] Jf Cf.p.27above.[Trans.]
  • 59. FIRST DAY 3I Oneof the mainobjec°cions urgedagainst this building up of continuous quantities out of indivisible quantities[continuo d' Cndivisibih] is that the additionof one indivisible to an- othercannotproduce a divisible, forifthiswereso it would rendertheindivisible divisible.Thusif twoindivisibles, say twopoints,canbe unitedto forma quantity,saya divisible line,thenanevenmoredivisible linemightbeformed by the unionofthree,five,seven, or anyotheroddnumber ofpoints. Since however theselinescanbecut intotwoequalparts,it becomes possible to cuttheindivisible whichliesexac°dy inthe middle oftheline. In answer tothisandotherobjec°dons ofthe sametypewereplythat a divisible magnitude cannot becon- stru(tedoutoftwoortenorahundred orathousand indivisibleS, butrequires aninfinite number ofthem. Sire,.Herea difficulty presentsitselfwhichappears to me insoluble.Sinceit is clearthat wemayhaveonelinegreater than another,eachcontaining an infinitenumberof points, we areforcedto admitthat, withinoneand the sameclass, wemayhavesomething greaterthaninfinity, because thein- finityof pointsin thelonglineis greaterthan theinfinity of pointsin theshortline. Thisassigning to aninfinite quantity avaluegreaterthaninfinity isquitebeyond mycomprehension. SALv. This is oneof the difficulties whicharisewhenwe attempt,withourfiniteminds, todiscuss theinfinite, assigning toitthoseproperties which wegivefothefinite andlimited; but [78] thisI thinkiswrong, forwecannot speak ofinfinite quantities as beingtheonegreater orlessthanorequalto another.To provethisI haveinmindanargument which, forthesakeof clearness, I shallputintheform ofquestions to Simplicio who raised thisdifficulty. I takeit forgrantedthatyouknow which ofthenumbers are squares andwhich arenot. Sn_P. Iamquiteaware thatasquared number isonewhich re- sultsfromthemultiplication of another number byitself;thus 4,9,etc.,aresquared numbers which come from multiplying 2,3, etc.,bythemselves. Salv.
  • 60. 37 THE TWO NEW SCIENCES OF GALILEO SALV. Very well;and youalsoknowthat just as the products are calledsquaresso the favors are calledsidesor roots;while on the otherhand those numberswhichdo not consistof two equal facCtors are not squares. Thereforeif I assert that all numbers, includingboth squares and non-squares,are more than the squaresalone, I shall speakthe truth, shall I not? Snvn,. Most certainly. SALV. If t shouldask furtherhowmanysquaresthereareone might replytruly that there are as many as file corresponding numberof roots,sinceeverysquarehas its own rootand every root its own square,whileno squarehas more than one root and norootmorethan onesquare. SIMP.Preciselyso. SALV. But if I -inquirehowmany rootsthere are, it cannot be deniedthat there are as manyas there are numbersbecause every numberis a root of some square. This beinggranted wemust say that there are as many squaresas there are num- bersbecausethey are just as numerousas their roots,and all the numbers are roots. Yet at the outset we said there are many morenumbersthan squares, sincethe largerportion of them are not squares. Not only so, but the proportionate number of squares diminishesas wepass to larger numbers. Thusup to IoowehaveIOsquares,that is,the squaresconstitute I/IO part of all the numbers;up to IOOOO, we findonly I/IO0 [79] part to be squares;and up to a milliononly I/IOOO part; on the otherhand in an infinitenumber,ifone couldconceive of sucha thing, he wouldbe forcedto admit that there are as many squaresas therearenumbersalltakentogether. SAGR. What then must one concludeunder these circum- stances? SALV. Sofar as I seewe can only inferthat the totality of all numbersis infinite,that the numberof squares is infinite, and that the numberof their roots is infinite;neither is the numberof squaresless than the totality of all numbers,nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicableto infinite, but
  • 61. FIRST DAY 33 but only to finite,quantkies. When thereforeSimpllc[oin- troducesseverallinesof differentlengthsand asksme how it is possiblethat the longerones do not contain more points than the shorter,I answerhimthat one linedoesnot contain moreor lessor just as many points as another,but that each line containsan infinitenumber. Or if I had repliedto him that the pointsinone llnewereequalin numberto the squares; inanother,greaterthan thetotality ofnumbers;andinthe little one,asmany asthe numberof cubes,mightI not, indeed,have satisfiedhim by thus placingmorepoints in one line than in another and yet maintainingan infinitenumberin each? So muchforthe firstdifficulty. SAGg. Pray stop a momentand let me add to what has al- readybeen said an ideawhichjust occursto me. If the pre- cedingbe true, it seemsto me impossible to say eitherthat one infinitenumberisgreaterthan anotheror eventhat it isgreater than a finitenumber,becauseifthe infinitenumberweregreater than, say, a millionit wouldfollowthat on passingfrom the millionto higherand highernumberswe wouldbe approach- ing the infinite;but this is not so;on the contrary,the lar- ger the numberto which we pass, the more we recedefrom [thispropertyof]infinity,becausethe greaterthe numbersthe fewer [relatively] are the squarescontainedin them; but the squaresin infinity cannotbe lessthan the totality of all the numbers,as wehavejust agreed;hencethe approachto greater and greaternumbersmeansa departurefrominfinity.* SAT.v. And thus fromyour ingeniousargumentweareled to [8o] concludethat the attributes "larger," "smaller,"and "equal" have no placeeitherin comparinginfinitequantitieswith each otheror in comparinginfinitewithfinitequantities. I pass now to another consideration. Sincelines and all continuousquantitiesare divisibleintoparts whichare them- selvesdivisiblewithout end, I do not see how it is possible *Acertain confusion ofthought appears tobeintroduced herethrough a failure todistinguish between thenumber n andtheclass ofthefirstn numbers; andlikewise froma failure todistinguish infinity asa number from infinity astheclass ofallnumbers.[Trans.]
  • 62. 34 THE TWO NEW SCIENCESOF GALILEO to avoidthe conclusion that theselinesarebuiltup of an in- finitenumberofindivisible quantities because a division anda subdivision whichcan be carriedon indefinitely presupposes that thepartsareinfinite innumber, otherwise thesubdivision wouldreachanend;andifthepartsareinfinite innumber, we mustconclude that theyarenot finitein size,because an in- finitenumberoffinitequantities would giveaninfinite magni- tude. Andthuswehavea continuous quantitybuiltupof an infinite number of indivisibles. Shay.But if wecan carryon indefinitely thedivision into finitepartswhatnecessity is therethen forthe introduction ofnon-finlte parts? SALV. The veryfacet that oneis ableto continue, without end,thedivision intofiniteparts[inpattiquante] makesit nec- essaryto regardthequantityas composed of aninfinite num- ber of immeasurably smallelements[di infinitinonquanta]. Nowin orderto settlethismatterI shallaskyouto tellme whether,in youropinion, a continuum is madeup of a finite orofaninfinite number offiniteparts[parti quante]. SIMI,. My answeris that their numberis bothinfiniteand finite;potentiallyinfinitebut afftually finite[infinite, in po.. tenza;efinite,inatto]; that is to say,potentially infinite before division andactually finiteafterdivision; because partscannot besaidto existina bodywhichis notyet divided or at least marked out;ifthisisnotdone wesaythattheyexistpotentially. SALV. Sothat a linewhichis, for instance,twentyspans longis notsaidtocontain afftually twentylineseachonespan inlengthexceptafterdivision intotwentyequalparts;before division it is saidto containthemonlypotentially.Suppose thefacets areasyousay;tellmethenwhether, whenthedivision is oncemade,the sizeof the original quantityis therebyin- creased, diminished, orunaffecCted. SIMV. It neither increases nordiminishes. SALV. That is my opinionalso. Therefore the finiteparts [pattiquante] in a continuum, whethera&uallyor potentially present,donotmakethequantityeitherlargeror smaller; but it is perfecCtly clearthat, if thenumberoffinitepartsaCtually contained
  • 63. FIRST DAY 35 contained inthewhole isinfinite innumber, theywillmakethe magnitude infinite.Hencethenumber offinite parts,although existing onlypotentially, cannot beinfinite unless themagnitude containing thembeinfinite; andconversely ifthemagnitude is finiteit cannotcontain aninfinite number offinitepartseither actually or potentially. SAGe. Howthenisit possible todivide a continuum without limitintopartswhicharethemselves always capable ofsubdivi- sion? SAT.V. ThisdistinCtion ofyours between actualandpotential appears torendereasybyonemethod whatwould beimpossible by another. But I shallendeavor to reconcile thesematters in anotherway;and as to thequerywhether thefiniteparts of a limitedcontinuum [continuo terminato] arefiniteor in- finitein numberI will,contraryto the opinion of Simplicio, answer thattheyareneither finite norinfinite. SIMP. Thisanswerwould neverhaveoccurred tomesinceI didnotthinkthatthereexisted anyintermediate stepbetween thefiniteandtheinfinite, sothat theclassification or distinc- tionwhich assumes thata thingmustbeeitherfinite or infinite isfaulty anddefective. SALv. Soit seems tome. Andifweconsider discrete quanti- ties I thinkthereis, between finiteandinfinite quantities, a third intermediate termwhichcorresponds to everyassigned number;sothat if asked,as in thepresentcase,whether the finitepartsof a continuum arefiniteor infinite innumber the bestreplyisthat theyareneither finitenorinfinite but corre- spondto everyassigned number. In orderthat thismaybe possible, it isnecessary that those partsshould notbeincluded withina limited number, forinthatcasetheywould notcorre- spondtoa number whichisgreater; norcantheybeinfinite in numbersincenoassigned number is infinite; andthus at the pleasure ofthequestioner wemay,to anygivenline,assign a hundred finite parts,athousand, ahundred thousand, or indeed anynumber wemayplease solongasit benotinfinite.I grant, therefore, to the philosophers_ that thecontinuum contains as many
  • 64. 36 THE TWO NEW SCIENCESOF GALILEO manyfinitepartsas theypleaseand I concede alsothat it con- tainsthem,eitheraCtually or potentially, astheymaylike;but I mustaddthatjustasalinetenfathoms Jeanne] inlengthcon- tainstenlineseachof onefathomandfortylineseachof one cubit[braccia] andeightylineseachof halfa cubit,etc.,soit containsaninfinitenumber of points;callthemaCtual or po- tential,asyoulike, foras tothisdetail,Simplicio, Idefer toyour opinion andtoyourjudgment. [821 SL_P. I cannothelp admiring your discussion; but I fear that this parallelism betweenthe pointsand the finiteparts contained ina linewillnotprovesatisfaCtory, andthatyouwill notfindit soeasyto divide a givenlineintoaninfinite num- berofpointsasthephilosophers dotocutit intotenfathoms or fortycubits;notonlyso,butsucha division is quiteimpossible to realizein praCtice, sothat thiswillbeoneof thosepoten- tialities which cannot bereduced toactuality. SALV. Thefactthatsomething canbedoneonlywitheffort ordiligence orwithgreatexpenditure oftimedoesnotrenderit impossible; forI thinkthatyouyourself could noteasilydivide a lineinto a thousand parts,andmuchlessif thenumberof partswere937or any otherlargeprimenumber. But if I wereto accomplish thisdivision whichyoudeemimpossible as readily as anotherperson woulddivide thelineintofortyparts would youthenbemore willing, inourdiscussion, toconcede the possibility ofsuchadivision ? Snvn,. In general I enjoygreatlyyourmethod;andreplying to yourquery,I answer that it wouldbemorethansufficient ifit provenotmoredifficult to resolve a lineintopointsthanto divide it intoa thousand parts. SALv. I willnowsaysomething which mayperhaps astonish you;it refersto thepossibility of dividing a lineinto its in- finitely smallelements by following thesameorderwhichone employs individing thesamelineintoforty,sixty,ora hundred parts,thatis,bydividing it intotwo,four,etc. Hewhothinks that, byfollowing thismethod, hecanreachaninfinite number ofpointsisgreatly mistaken; forifthisprocess werefollowed to etemiw / 7
  • 65. FIRST DAY 37 eternity there wouldstill remainfiniteparts whichwere un- divided. Indeedby such a methodone is very far from reachingthe goal of indivisibility;on the contrary he recedesfrom it and whilehe thinksthat, by continuingthis division and by multi- plyingthe multitude of parts, he willapproachinfinity,he is, inmy opinion,gettingfarther and fartherawayfromit. My reasonis this. In the precedingdiscussion we concluded that, inan infinitenumber,it isnecessarythat the squaresand cubes shouldbe as numerousas the totality of the natural numbers [tuttii numerz],becauseboth of theseare as numerousas their roots which constitute the totality of the natural numbers. Nextwesawthat thelargerthe numberstakenthemoresparsely distributedwerethe squares,and stillmoresparselythe cubes; thereforeit isclearthat the largerthe numbersto whichwepass the fartherwerecedefromthe infinitenumber;henceit follows [8S] that, sincethisprocesscarriesus fartherandfartherfromthe endsought,if on turningback weshallfindthat any number can be said to be infinite,it mustbe unity. Hereindeedare satisfiedall those conditions whichare requisite foran infinite number; I meanthat unity contains in itselfasmanysquares as therearecubesandnaturalnumbers[tuttiinumen]. SIMP.I donotquitegraspthemeaningofthis. SALV. There isno difficulty in the matterbecauseunityisat once a square,a cube, a squareof a squareand all the other powers[dignity]; noristhereany essentialpeculiarityinsquares or cubeswhichdoesnot belongto unity; as, forexample,the propertyof twosquarenumbersthat they havebetweenthema meanproportional;take any squarenumberyou pleaseas the first term and unity forthe other,then youwillalwaysfind a numberwhichisa meanproportional. Consider the twosquare numbers,9 and 4; then 3 is the mean proportionalbetween 9 and I ;while2isameanproportional between4and I; between 9 and 4 wehave6 as a meanproportional.A propertyof cubes is that they must have betweenthem two meanproportional numbers; take 8 and 27; betweenthem lie IZ and 18;while between
  • 66. 38 THE TWO NEW SCIENCESOF-GALILEO between Iand8wehave2and4intervening; andbetween I and 27therelie3and9. Therefore weconclude thatunity is the onlyinfinite number.Thesearesome ofthemarvels which our imagination cannotgraspandwhichshould warnusagainst the serious errorof thosewhoattemptto discussthe infiniteby assigning to it the sameproperties whichweemployfor the finite, thenatures ofthetwohaving nothing incommon. Withregard to thissubjecCt I musttellyouof a remarkable property whichjustnowoccurs to meandwhich willexplain thevastalteration andchange ofcharacCter which afinitequan- tity wouldundergo in passing to infinity.Let us drawthe straight lineABofarbitrary length andletthepointC divide it intotwounequal parts; thenI saythat,ifpairsoflinesbe drawn, onefrom eachof theterminal pointsA andB, andif theratiobetween thelengths oftheselinesisthesame asthat between ACandCB,theirpoints ofinterse&ion willalllieupon the circumference of oneandthe samecircle. Thus,for ex- [84] ample, ALandBLdrawn from AandB,meeting atthepointL, beating to oneanother the sameratioasACto BC,andthe pair AK and BK meetingat K also beatingto one an- other thesame ratio, and likewise thepairs A 6 _c B-ii"-"""_ EM, BI,AH,BH,AG, BG, AF, BF, AE, BE,havetheirpoints ofintersec°don L,K, I,H, G,F,E,all ly- Fig.7 inguponthecircum- ference ofoneandthesamecircle.Accordingly ifweimagine thepointCtomovecontinuously insuch amanner thatthelines drawn fromittothefixed terminal points, Aand B,always main- tainthesame ratiobetween theirlengths asexistsbetween the original parts, ACandCB,thenthepoint C will, asI shall pres- entlyprove, describe acircle.Andthecircle thusdescribed will _crcase r