PREFACE
[Thetheoryofrelativity]didnotatalloriginateinarevolutionaryactbutas anaturaldevelopmentofalinethatcanbetracedthroughcenturies.¹ (Albert Einstein,1921)
Therelativitytheoriesoftheearlytwentiethcenturydeeplyalteredourmostbasicnotions ofspace,time,andmotion.Theymadetimedependontheobserver,theyentwinedspace andtime,andtheyreducedgravitationtoinertialmotioninacurvedspace–time.Historianshavetriedtoexplaintheradicalnessofthesechangesinvariousways:throughEinstein’s peculiargenius,throughcontemporarydevelopmentsinphilosophyortechnology,and throughamajorcrisisinphysicsaround1900.Thesethreeexplanationscorrespondto threeallegedpreconditionsofradicalchange:singularminds,externalcausation,and internalcrisis.Theinternalcrisisapproachisperhapsthemostconvincing,becausephilosophicalandtechnologicalconsiderationsmayhavebeeninternalizedbeforetheyserved thefoundersofrelativity,andbecausethesingularityofthesefoundersmayboildownto theirabilitytodetectandsolvecrises.Butthisapproachcanonlybeafirstapproximation, andafullerhistoryshouldcombineinternal,external,andbiographicalelements. Whetherornottheycomeclosetothisideal,thereceivedhistoriesofrelativityare short-term,fragmentedhistories.Specialandgeneralrelativityareusuallytreatedseparately,andthemaximaltimespanisaboutacenturylong(intheinternalcrisisapproach), withafocusonthelatenineteenthandearlytwentiethcentury.Thereareexceptions: JürgenRennhascomparedtheemergenceofrelativitytheorywiththeemergenceofearly modernmechanics;JeanEisenstaedthasdevelopedanalogiesbetweeneighteenth-century Newtonianopticsandgeneralrelativity;severalphilosophers,includingRobertoTorretti andRobertDiSalle,haveexploredthevariousconceptsofspaceandtimeinthehistory ofphysicsfromAristotletorelativity;andonephysicist,JulianBarbour,hasprobedthe long-termhistoryofdynamicsfromaMachianviewpoint.Buttheirmainintentionwas nottodemonstrategenuine,long-termhistoricalconnections:itwastoinstructonthevarietyofpossibleconcepts,toreflectontheirsimilaritiesanddifferences,or(inRenn’scase) toillustrateagenerictheoryofconceptualdevelopment.²
Incontrast,thisbookpresentsalong-term,multi-approachhistoryofrelativityfrom GalileotoEinstein,inthevariouscontextsofmechanics,mathematics,philosophy,astronomy,optics,andelectrodynamics.Onemightfearthatanyattempttoconnecteventsthree
¹Einstein[1921],p.431.
²Torretti1978; Renn1993; Barbour2001; Eisenstaedt2005b; DiSalle2006.AlsoworthmentioningareMarieAntoinetteTonnelat’s Histoireduprincipederelativité (Tonnelat1971)forananalysisofthesuccessivemeanings ofrelativityfromantiquitytogeneralrelativity;andRaffaellaToncelli’sdissertationontheroleofprinciplesinthe constructionofrelativitytheory(Toncelli2010),whichalsohasaverylargescope.
centuriesapartwouldbesuperficialandartificial,andthiswasmyownopinionuntilIrealized,afterafewpartialpre-histories,thatgenuinehistoricalconnectionsexistedbetween thevarioususesofrelativityprinciplesacrossthreecenturies.Iobservedtwokindsofconnectedness.First,atleastuntiltheearlytwentiethcenturyphysicistshadamuchlonger memorythantheyhavenowadays.Theyreadandexploitedliteraturewrittencenturies earlier,whereastodayscientiststendtodisregardanythingolderthanafewyears.Second, therewereindirecthistoricalconnectionsthroughchainsofsuccessiveborrowings.Inboth casesitisessential,inaproperlyhistoricalstudy,totakeintoaccountthechangeofcontext intheborrowingprocess,evenwhentheborrowerisnotawareofthischange.Otherwise wewouldmisunderstandthesuccessivesystemsofthought,andwewouldmisrepresent thetransferofknowledgefromonesystemtoanother.³
Whendevelopedwithsufficientcare,thislong-termviewhasseveraladvantages.It showsthatquestionsaboutthenatureofspace,time,andinertiatraversedthehistoryof physicsfromtheearlymodernperiodtotherelativitytheories,althoughtheanswersgiven tothesequestionsvariedconsiderably.Itexplainshowtherelativityprincipleemerged asatrue,constructiveprincipleasearlyastheseventeenthcentury,howitcametobe namedsointhemid-nineteenthcenturytomarkitsconstructivepowerinmechanics, andhowthisnameandconceptreachedtheyoungEinstein.Itdemonstratesdeepanalogiesbetweensomeauthors’dismissalofNewton’sabsolutespaceandEinstein’slater dismissaloftheelectromagneticether.Itestablishesanindirecthistoricallinkbetween acorollaryofNewton’s Principia andEinstein’sequivalenceprinciple,throughaFrench principleofaccelerativerelativity.Itsituatesbothprinciplesinaspace–time–inertiatangle thatoriginatedinGalileo’sandNewton’stheoriesofmotionandevolvedintoEinstein’s finalreductionofgravitationtoinertia,throughintermediatestepsintheeighteenthand nineteenthcentury.Also,thereareadvantagesintreatingthehistoriesofspecialandgeneralrelativityinthesamevolume:thishighlightsthecontinuityofEinstein’sendeavors, andshowstheimportanceofMinkowski’sandLaue’sversionsofthespecialtheoryin Einstein’squestforarelativisticfieldtheoryofgravitation.
Anylong-termhistoryfacesdifficultiesinnamingbasicconceptsandprinciplesacross time.Today,weusuallydefinetherelativityprinciplethroughthecompleteequivalenceof allinertialreferenceframes.Itisimpossibleoratleastdangeroustoapplythisdefinition uniformlyfromGalileotoEinstein.Thename“referenceframe”firstappearedin1884, underJamesThomson’spen.Theconceptearlierexistedunderdifferentnames,guises, andextensions:ametonymicalboatforGalileo,a“space”forNewton,anda“system ofaxes”forLaplace.Statementsofrelativityhadavariablestatus:anempiricalfactfor Galileo,atheoremforNewton,andagenuineprincipleforHuygens,Laplace,andlater Frenchauthors.Thereweredifferenttheoremsandprinciplesofrelativity(hencetheplural inmytitle)accordingtotheimpliedclassofequivalentframes:inertialframesandacceleratedframesforNewton,Laplace,andBélanger;justinertialframesforHuygens,Poincaré, andtheearlyEinstein;andanyacceleratedframeandevenaframing“mollusk”forthe laterEinstein.Untilthelatenineteenthcentury,relativityprinciplesandtheoremswere
³BycontextIheremeanintellectualandexperimentalcontext,althoughthechangesinthisnarrowercontext usuallyimplythebroadersocio-culturalcontext.
mostfrequentlyformulatedintheactiveform,astheabsenceofeffectsofacommonlyimpressedmotionontherelativemotionsofasystemofbodiesundergivenforces,andmore rarelyinthepassiveform,asthelackofeffectsofachangeofreferenceframe.Allthese ambiguitiesshouldbekeptinmindwhen,fortheconvenienceofthereader,Iwillsometimesuse“referenceframe”insteadofthetruehistoricalterm.Thatsaid,thereisenough kinshipbetweenthevariouskindsofrelativitytojustifyaglobalhistoricalaccountoftheir meaningandpurpose.
Chapter 1 describesthenewtheoriesofmotionoftheseventeenthcenturyandtheemergenceofrelativityfacts,theorems,andprinciplesinconjunctionwithnewconceptsof mechanicalinertia.Chapter 2 recountshow,followingHuygens’spioneeringderivation ofthelawsoffreefall,anumberofFrenchauthorsusedarelativityprincipletoderive Newton’ssecondlawofmotion,howthename“principleofrelativemotions”appearedin thiscontext,howPoincarémodifiedthisname,andhowhepassedittoEinstein.Chapter 3 discussesdifficultiesinthecorrelativedefinitionsofspace,time,andinertiafromNewton tolate-nineteenth-centurycriticsofthefoundationsofmechanics,therepeatedrejections ofNewton’sabsolutespace,andStreintz’sfinalsuggestionofanintrinsicGalileanspace–time.Atthispoint,thereaderwillhaveviewedtheemergenceofamechanicalrelativity principlewithconstructiveandrepresentationalfunctions.Chapter 4 isabouttherelativitythatemergedincorpuscularopticsandsurvivedwaveopticsdespitetheexistenceofthe etherasaprivilegedframe.AsisexplainedinChapter 5,asimilartensionbetweenether andrelativityexistedintheelectrodynamicsofmovingbodies,andLorentzalleviatedit withoutsuppressingitinhiselectrontheoryofthe1890s.Chapter 6 showshowPoincaré raisedopticalandelectrodynamicrelativitytoaprincipleonparwiththemechanicalrelativityprinciple,andhowhetransformedLorentz’stheoryintoafirstversionofspecial relativity.Chapter 7 describestheelaborationofmoreradicalandmorepotentformsof specialrelativityinthehandsofEinstein,Minkowski,andLaue.Chapter 8 showshow RiemannandhisfollowersinventedthemathematicsEinsteinemployedingeneralrelativity,alsohowtheemergenceofnon-EuclideangeometryinspiredPoincaréandMinkowski. Chapter 9 isamedium-sizedaccountofthelongandtwistedgenesisofgeneralrelativity. Chapter 10 coversthedifficultiesEinstein’searlyreadersencounteredintryingtomake senseofthistheory,andthegreatclarificationbroughtbyWeyl,Eddington,andothers. Theepiloguebringsouttheinterconnectionsofthedevelopmentsdescribedintheprevious chapters.Twoofthesechapters,theeighthandtheninth,areformallyandmathematically moredemandingthantheothers.Readerswhodonotwishtoentertechnicalitiesmay contentthemselveswiththesummariesgivenintheirconcludingsections.
Thereisplentyofsecondaryliteratureformostofthetopicsaddressedinthisbook. Themostimportantstudiesareindicatedatthebeginningofeachchapter,andtherest infootnotes.Forthemoment,itwillbesufficienttosignalthesourcesIfoundmost usefulformyproject.Ontheether-theoreticalbackgroundtospecialrelativity,thereis EdmundWhittaker’soldandrich AHistoryofAetherandElectricity,TetuHirosigeon Lorentzandelectrodynamicorigins,andRussellMcCormmachontheelectromagnetic worldview.ArthurMiller’s AlbertEinstein’sSpecialRelativity remainsastandardreference ofgreatvalue.MorerecentcontributionstothehistoryandcontextsofspecialrelativityincludeMichelJanssen’sdissertationontherelationbetweenLorentz’stheoryand
specialrelativityinthelightoftheTrouton–Nobleexperiments,ScottWalter’sdissertationonMinkowskiandthemathematicalreceptionofrelativity,PeterGalison’s Einstein’s Clocks,Poincaré’sMaps,RichardStaley’s Einstein’sGeneration,Jean-PierreProvostand ChristianBracco’sstudiesofPoincaré’srelativitytheory,MarcoGiovanelli’sthorough studyofprinciples-basedvs.constructiveapproachestorelativisticdynamics,andGalina Weinstein’s Einstein’sPathway.Onthehistoryofgeneralrelativity,muchvaluecanbe foundinAbrahamPais’s SubtleIstheLord,inJohnStachel’scontemporarydiscussions oftherigid-bodyproblem,inJohnNorton’sgroundbreakingstudyofEinstein’sZürich notebook,inJeanEisenstaedt’shistoryoftheSchwarzschildsingularity,inWeinstein’s GeneralRelativityConflictandRivalries,andinthemulti-volume Genesis directedby JürgenRenn.Thislast,collectiveenterprisegroupshighlycompetentandthoroughstudiesbyJohnNorton,JohnStachel,JürgenRenn,TilmanSauer,MichelJanssen,andafew otherscholars.Usefulstudiesofpre-relativisticconsiderationsontherelativityofmotionincludeChristianeVilain’s LamécaniquedeChristiaanHuygens,GiulioMaltese’s studyofEuler’srelativityconsiderations,MariusStan’sarticlesonHuygens,Kant,and relativity,JeanEisenstaedt’s AvantEinstein,andAlbertoMartínez’s Kinematics.Lastly, afewhistorico-philosophicalstudieshelpedmeclarifyconceptualissuesencounteredin theprimarysources:LawrenceSklar’stwo Spacetime books,RobertoTorretti’s PhilosophyofGeometry,MichaelFriedman’s FoundationsofSpacetimeTheories andhis Kant’s ConstructionofNature,JohnEarman’s WorldEnough,MichelPaty’s EinsteinPhilosophe, JulianBarbour’s TheDiscoveryofDynamics,HarveyBrown’s PhysicalRelativity,Robert DiSalle’s UnderstandingSpacetime,ThomasRyckman’s TheReignofRelativity andhis recent Einstein ⁴
Thisbookistheendresultofanoldinterestofmineintheoriginsofrelativitytheory. Ifellinlovewiththistheoryinmyteenyears,whilereadingLandauandLifshitzonaCap FerretbeachonthewesternshoresofFrance.Inthe1980s,IcontributedtotheFrench EinsteinprojectunderthedirectionofFrançoiseBalibar,benefitingfromJohnStachel’s expertiseforthevolumesonrelativity.Inthe1990s,Istudiedtheelectrodynamicoriginsof thespecialtheoryasthenaturalconclusionofahistoryofelectrodynamics.Morerecently Iworkedontheopticsofmovingbodies,onRiemann’sgeometryandconsequences,onthe earlyreceptionofgeneralrelativity,andonthegenesisofthistheory.Myreturntogeneral relativityandmyideatowriteahistoryofrelativityfromGalileotoEinsteinderivedfrom myphilosophicalinterestinthecoordinationbetweentheoryandexperience,andfrom mydecision,inthelightofRyckman’s ReignofRelativity,tofocusonthemostbasicand oldestproblemofcoordination,thespatialandtemporalorderingofevents.Ryckman havingexploredthisproblemthroughthephilosophicalreceptionofgeneralrelativity, Iwantedtoretraceitsevolutionfromearlymodernmechanicstogeneralrelativity.
⁴Whittaker1951; Hirosige1966, 1969, 1976; McCormmach1970; Sklar1974, 1985; Torretti1978; Stachel1980, 1989a, 1989b; Miller1981a; Eisenstaedt1982, 2005; Pais1982; Friedman1983, 2013; Earman1989; Paty1993; Janssen1995; Vilain1996; Walter1996a; Maltese2000;Barbour2002; Galison2003; Brown2005; Ryckman2005, 2017; ProvostandBracco2006,2009; DiSalle2006; Staley2008; Martínez2009; Stan2009, 2016; Weinstein2015a, 2015b; Giovanelli2020;Rennetal. GGR.AlsoworthmentioningisChristopherRay’s Theevolutionofrelativity (Ray1987)foritsanalysisofthefoundationsofgeneralrelativityfromEinsteintoHawking,withMach’s philosophyofmotionasabackdrop.
Duringalltheseyears,Ibenefitedfromthelivelyandpleasantenvironmentofthe SPHereresearchteamatCentrenationaldelarecherchescientifique(CNRS)inParis, withunabatedsupportfromitssuccessivedirectorsMichelPaty,KarineChemla,Pascal Crozet,DavidRabouin,andSabineRommevaux.Ihadnumerous,instructiveconversationswithmyclosestcolleaguesinthisteam:NadinedeCourtenay,JanLacki,Sara Franceschelli,andMartha-CeciliaBustamante.MyresearchwasgreatlyeasedandenrichedbymyaffiliationtoUC-Berkeley’srenownedOfficeforHistoryofScienceand TechnologyunderthewingsofJohnHeilbron,CathrynCarson,andMassimoMassotti. Whileresearchingthehistoryandprehistoryofrelativitytheory,IprofitedfromenlighteningexchangeswithJohnHeilbron,JedBuchwald,JohnNorton,PeterGalison,Harvey Brown,TomRyckman,andThibaultDamour.BackontheverybeachonwhichIfirst encounteredrelativitytheory,IfondlyrememberallthesecolleaguesandfriendsandI warmlythankthemforeasing,guiding,andilluminatingmyintellectualvoyagethrough themysteriesofspaceandtimecoordination.
CONVENTIONSANDNOTATIONS
• Forthesakeofthephysicistreader,Iusemodernizedandstandardizednotation whenevertheoriginalnotationdidnotessentiallyconditionthehistoricaldevelopments.Forexample,Iusethemodernvectornotationinoldermechanicsand electrodynamicsalthoughitspropagationbeganonlyinthelatenineteenthcentury. Forgeneralrelativity,Iusethemoderntensornotation,whichdepartsfromEinstein’s originalnotationinwaysindicatedatthebeginningofChapter9.Forthehistory ofthesenotations,IinvitethereadertoreadtheclassicalstudiesbyMichaelCrowe andKarinReich.Whilethehistoricalsignificanceofachangeofnotationmaysometimesbehigh,itsconceptualandpracticalimportancehasoftenbeenexaggerated. Forinstance,inoldermechanics,thegeometricalrepresentationofvectorquantitiespermittedanintrinsicconceptionofvectorswithoutthevectornotation;in olderelectrodynamics,thepracticeofwritingonlyoneofthethreecomponentsofa Cartesian-coordinateequation(thetwoothercomponentsbeingimplicitlygivenby circularpermutation)compensatedforthelackofanintrinsicnotation.⁵
• Citationsareintheauthor–dateformatandrefertotheappendedbibliography (Iuse“cf.”intheFrenchway,torefertoasourceinthesecondaryliterature).Square bracketsaroundadateindicateamanuscriptsource.Abbreviationsarelistedon p.416beforethebibliography.
• TranslationsfromLatin,German,andFrencharemineunlessthecitedsource alreadyisatranslation.
⁵Crowe1967;Reich1994.Forthehistoricalrepresentationofmechanicalquantities,cf.Martínez2009,chap.4. Ontheuseoffour-vectorsinMinkowskispace,cf.Walter2007b.
RETHINKINGMOTIONINTHESEVENTEENTHCENTURY
Iknowthatyou,Simplicio,havegonefromPaduabyboatmanytimes,and,if youwilladmitthetruthofthematter,youhaveneverfeltwithinyourselfyour participationinthatmotionexceptwhentheboathasbeenstoppedbyrunning agroundorbystrikingsomeobstacle,whenyouandtheotherpassengers,taken bysurprise,havestumbledperilously.¹ (Galileo Galilei,1632)
TheScientificRevolutionbroughtadrasticrethinkingofmotion,nowsubjectedtolaws neverdreamtofbyAristotelianphilosophers.AfewbravesupportersoftheCopernican cosmos,KeplerandGalileoforemost,measuredtheconsequencesofchangingthereferenceofplanetarymotionfromtheearthtothesun.Byerasingtheancientdifference betweenthesublunaryandcelestialworlds,thenewsystemrequiredcommonconcepts ofmotioninthetwoworlds.Inaddition,GalileoandDescartespromotedafullymathematical,geometricunderstandingofthephysicalworld,basedonthesimplestandleast speculativeconceptofchange,thatis,locomotion.Beeckman,Galileo,andNewtonappealedtothemotionofatoms;DescartesandHuygenstotherelativemotionsofthe particlesofaplenum.Thepurposeofthischapteristointroducetherelevanttheories ofmotion,withspecialattentiontotherelativityissue.²
Themotionofabodyisalwaysdefinedinreferencetosomethingregardedasimmobile.Aswewillseeinthischapter,theconceptofthis“something”variedconsiderably inseventeenth-centurymechanicalphilosophy.ForDescartes(asforAristotle),itwasthe mediuminwhichthebodyisimmersed.ForGalileo,itwasadistinguishedmaterialsystem, forinstancethesunandfixedstars,theearth,oraboat;forNewton,itwasspaceregardedasanimmaterial,rigid,immobile,penetrablesubstratumofallthings.Toeachkind ofreferencecorrespondedadifferentconceptofinertia.Galileo,Beeckman,Descartes, Huygens,andNewtonallassumedthepersistenceofthemotionofabodylefttoitself foravarietyofreasons:responsetotheneedtoharmonizecelestialandterrestrialmotions,continuitywiththemedievalconceptofimpetus,logicalnecessityinaneo-atomist worldpicture,God’spredilectionforpermanence.ForGalileo,inertiacouldbecircular, anditcoveredboththemotionsofplanetsaroundthesunandthemotionofaballon ahorizontaltableonearth;forDescartes,inertialmotionwasrectilinearandreferredto
¹Galilei1632,p.255.
²OntheScientificRevolution,cf. Cohen2015,andfurtherreferencestherein.
RelativityPrinciplesandTheoriesfromGalileotoEinstein.OlivierDarrigol,OxfordUniversityPress. ©OlivierDarrigol(2022).DOI:10.1093/oso/9780192849533.003.0001
thesurroundingmedium;forNewton,itwasrectilinearinabsolutespace.Thesedifferent conceptsofinertiaimplieddifferentlawsofmotionanddifferenttakesontherelativityof motion.
ForGalileo,themotionofasystemofbodiesobeyedthesamelawswhetheritwas referredtotheearthortoaboatuniformlymovingonthesea.ForHuygens,thesimilarityextendedtoanyreferenceframeinwhichthelawofuniform,rectilinearinertiais valid.ForNewton,thissimilarityappliedtoanyreferenceframemovinguniformlyand rectilinearlyinabsolutespace;italsoappliedtoasystemofbodiessubjectedtoacommonacceleration,asisthecasewhenthebodiesareallsubjectedtotheattractionofa remoteheavymass.ForDescartes,thelawsofmotioncoulddependonlyontherelative positionoftheparticlesoftheworld.Inmodernparlance,DescartesanticipatedMach’s eliminationofanyimmaterialreferenceofmotion,whileGalileo,Huygens,andNewton detectedabasicinvarianceofthelawsofmotionwithinaprivilegedclassofreference frames.Newtonincludeduniformlyandrectilinearlyacceleratedframesinthisclass,thus partiallyanticipatingEinstein’sequivalenceprinciple.
AnotherimportantdistinctionconcernsthearchitectonicroleoftheinvarianceassertedintherelativitystatementsofGalileo,Newton,andHuygens.Aswewillsee, Galileowasmostlyconcernedwithalawofrelativity,directlyinferredfromobservations(onaboat)orindirectlyfromtheempiricallawsoffreefall.Incontrast,Newton’s relativitystatementsweretheoremsdeducedfromhisthreelawsofmotion(whichhebelievedhehaddeducedfromexperience).Huygensinauguratedarelativityprinciplethat comesbeforethemechanicallawsandcontributestotheirderivation.Thiswasacrucial stepinregardtothelaterroleoftheprincipleinjustifyingorconstructingtheoriesof motion.
Itwouldbeartificialtolimitthischaptertotheconsiderationofafewextractsofthe majortextinwhichrelativitystatementsappearedintheseventeenthcentury.Aswasjust indicated,suchconsiderationscannotbeseparatedfromthetheoriesofmotionandthe philosophiesofnaturetowhichtheybelong.Also,thesetheoriesandphilosophiesform partofthebackgroundinwhichlaterrecoursetorelativityprinciplesoccurred.Withthis inmind,wewillsuccessivelyconsiderthedecisivecontributionsofGalileo,Beeckman, Descartes,Newton,andHuygens.
1.1 Galileo’sscienceofmotion
Inertiaandrelativity
In1632,aftermanyyearsofself-restraint,GalileoGalileipublishedhislengthy,witty,and risky Dialogosopraiduemassimisistemidelmondo,apowerfuldefenseoftheCopernicansystemthroughadialoguebetweenthreefictionalcharacters:thebrilliantCopernican Salviati,theobtusescholiastSimplicio,andtheinquisitiveSagredo.Animportantpartof theirseconddayisdevotedtoathenfrequentobjectiontotherotationoftheearth:that itwouldimplyneverobservedalterationsofthewaybodiesmoveonearth.Inparticular, astone,whenfallingfromaverticaltower,wouldnotfallatthefootofthetowerbecause duringthefallthegroundandthetowerwouldbecarriedeastwardbytherotatingearth. Tofortifythispointbyexperiment,theobjectormightaddthatonamovingship,astone
releasedfromthetopofamastshouldfallbehindthemastasaresultoftheprogression oftheship.³
Salviatideniesboththepointandtheexperiment.TheAristotelians,heexplains,have erredinignoringthehorizontalvelocityofthestoneatthebeginningofitsfallfromthe topofatower.Thisvelocity,beingpartofthenaturalcircularmotionofallterrestrial bodies,hastoremainconstantduringthefallsothatstoneandtowerconstantlysharethe samehorizontalmotionandthestonethereforefallsalongthemast.Thepersistencyof horizontalmotionwasindeedalientoAristotle’sphysics.Inthisdoctrine,naturalmotion inthesublunaryworldisdirectedtoorfromthecenteroftheuniverse,whichcoincideswith thecenteroftheearth.Horizontalmotionis“violent”orforcedmotionofexternalorigin, anditcannotpersistwithoutasustainingforce.Inthefirstdayofthedialogue,Salviatihas purposelyredefinednaturalmotionasmotionthatdoesnotdisturbtheglobalarrangement oftheworld.Accordingly,circularmotionisequallynaturalfortherotatingearthandfor thecirculationoftheearthandtheplanetsaroundthesun,whereasforAristotlecircular motionisnaturalonlyforcelestialbodiesmovingaroundtheearth.⁴
Besidesthiscosmologicalconsideration,Salviatiassertsthatforthemovingshiptoo, thefallingstoneaccompaniestheshipinitshorizontalmotion.IfAristotle’sfollowershad trulyperformedtheexperiment,SalviatitellsSimplicio,theywouldhaveseenthattheship’s motionhasnoeffectonthefallobservedbythesailors.Salviatibaseshisconvictionon twoempiricalfacts:theconstantvelocityofaballrollingonaperfectlysmoothhorizontal table(theresistanceoftheairbeingnegligible),andtheindependencebetweenvertical andhorizontalmotionduringtherollingoftheballonaninclinedplane.Mostvividly,he arguesthatcommonexperienceconfirmstheabsenceofeffectsofaship’suniformmotion onthepassengers’internalobservations:⁵
Shutyourselfupwithsomefriendinthemaincabinbelowdecksonsomelargeship, andhavewithyoutheresomeflies,butterflies,andothersmallflyinganimals.Havea largebowlofwaterwithsomefishinit;hangupabottlethatemptiesdropbydrop intoawidevesselbeneathit.Withtheshipstandingstill,observecarefullyhowthe littleanimalsflywithequalspeedtoallsidesofthecabin.Thefishswimindifferently inalldirections;thedropsfallintothevesselbeneath;and,inthrowingsomething toyourfriend,youneedthrowitnomorestronglyinonedirectionthananother, thedistancesbeingequal;jumpingwithyourfeettogether,youpassequalspacesin everydirection.Whenyouhaveobservedallthesethingscarefully(thoughthereis nodoubtthatwhentheshipisstandingstilleverythingmusthappeninthisway), havetheshipproceedwithanyspeedyoulike,solongasthemotionisuniformand notfluctuatingthiswayandthat.Youwilldiscovernottheleastchangeinallthe effectsnamed,norcouldyoutellfromanyofthemwhethertheshipwasmovingor
³Galilei1632,p.126;alsoGalileitoIngoli1624,in Finocchiaro1989,pp.182–188.Cf. Drake1978; Vilain1996, chap. 2; Heilbron2010,pp.262–263,270–271; Swerdlow2013
⁴Galilei1632,pp.138–143(stoneandtower),31–32(perpetualcircularmotion).
⁵Galilei1632,pp.145–149,186–188(citation).AsGalileounderstoodit,theinertialmotionfromthetopofthe towerslightlyexceedsthehorizontalmotionofthelowerpartsofthetower(sincetheyareclosertotheaxisof therotatingearth),sothatthefallingbodyshouldexperienceaslightdeviationtotheeast;cf. Burstyn1965.On anticipationsofGalileo’sship-cabinbyGiordanoBrunoandothers,cf. Capecchi2014,pp.158–159.
standingstill.Injumping,youwillpassonthefloorthesamespacesasbefore,nor willyoumakelargerjumpstowardthesternthantowardtheproweventhoughthe shipismovingquiterapidly,despitethefactthatduringthetimethatyouareinthe airthefloorunderyouwillbegoinginadirectionoppositetoyourjump.Inthrowing somethingtoyourcompanion,youwillneednomoreforcetogetittohimwhether heisinthedirectionoftheboworthestern,withyourselfsituatedopposite.The dropletswillfallasbeforeintothevesselbeneathwithoutdroppingtowardthestern, althoughwhilethedropsareintheairtheshiprunsmanyspans.Thefishintheir waterwillswimtowardthefrontoftheirbowlwithnomoreeffortthantowardthe back,andwillgowithequaleasetobaitplacedanywherearoundtheedgesofthe bowl.Finally,thebutterfliesandflieswillcontinuetheirflightsindifferentlytoward everyside,norwilliteverhappenthattheyareconcentratedtowardthestern,as iftiredoutfromkeepingupwiththecourseoftheship,fromwhichtheywillhave beenseparatedduringlongintervalsbykeepingthemselvesintheair.Andifsmoke ismadebyburningsomeincense,itwillbeseengoingupintheformofalittlecloud, remainingstillandmovingnomoretowardonesidethantheother.Thecauseofall thesecorrespondencesofeffectsisthefactthattheship’smotioniscommontoall thethingscontainedinit,andtotheairalso.ThatiswhyIsaidyoushouldbebelow decks;forifthistookplaceaboveintheopenair,whichwouldnotfollowthecourseof theship,moreorlessnoticeabledifferenceswouldbeseeninsomeoftheeffectsnoted. Nodoubtthesmokewouldfallasmuchbehindastheairitself.Theflieslikewise,and thebutterflies,heldbackbytheair,wouldbeunabletofollowtheship’smotionif theywereseparatedfromitbyaperceptibledistance.Butkeepingthemselvesnear it,theywouldfollowitwithouteffortorhindrance;fortheship,beinganunbroken structure,carrieswithitapartofthenearbyair.Forasimilarreasonwesometimes, whenridinghorseback,seepersistentfliesandhorsefliesfollowingourhorses,flying nowtoonepartoftheirbodiesandnowtoanother.Butthedifferencewouldbesmall asregardsthefallingdrops,andastothejumpingandthethrowingitwouldbequite imperceptible.
ItiseasytomisunderstandthekindofrelativityGalileohadinmindhere.InSalviati’s words,“thecauseofallthesecorrespondencesofeffectsisthattheship’smotioniscommon toallthethingscontainedinit.”Thisseemstobeechoinghisearlierremarkthatthe commonreferenceforthemotionofasystemofbodiescanbearbitrarilychangedwithout alteringtherelativemotionsofthesebodies:
Itisobvious...thatmotionwhichiscommontomanymovingthingsisidleand inconsequentialtotherelationofthesemovablesamongthemselves,nothingbeing changedamongthem,andthatitisoperativeonlyintherelationthattheyhavewith otherbodieslackingthatmotion,amongwhichtheirlocationischanged.
ThecontextofthisstatementiswhatwewouldnowcallthefullobservationalequivalencebetweenCopernic’sheliocentricsystemandTycho’ssysteminwhichthesamerelative motionsaredescribedinanearth-boundframe(toputitinmodernterms).Thecommonmotionneednotbeuniforminthiscase,whereasSalviatirequireshisshiptomove uniformly.Ashelatermakesclear,nonuniformitywoulddisturbthesimilarityofthe phenomenaobservedinthemovingshipandintheanchoredship:
Iknowthatyou,Simplicio,havegonefromPaduabyboatmanytimes,and,ifyouwill admitthetruthofthematter,youhaveneverfeltwithinyourselfyourparticipationin thatmotionexceptwhentheboathasbeenstoppedbyrunningagroundorbystriking someobstacle,whenyouandtheotherpassengers,takenbysurprise,havestumbled perilously.
Thepointisthatasuddenaccelerationoftheshipisnotsharedbyitspassengers,sothat themotion(orrest)ofthepassengerswithrespecttotheshipisnotconserved.Amodern physicisthererecognizestheeffectofinertia:Galileoimplicitlyselectsthereferenceframe sothattheprincipleofinertiaholdsinit.Namely,hewantstherollingofaballonatable orthehorizontalmotionofafallingstonetobestilluniformwhentheshipismoving.⁶
TheGalileaninvarianceoftheotherphenomenaobservedinamovingship’scabinis anexperimentalfact.Inaddition,theGalileaninvarianceoffreefallderivesfromtheconstancyofhorizontalmotionandtheindependenceofverticalandhorizontalmotionsin theearth-basedframe.SothereisnodoubtthatGalileoperceivestheintimateconnection betweenthepersistenceofunimpededmotionandhisrelativityprinciple.Still,oneshould thinktwicebeforeidentifyinghisconceptofpersistingmotionwiththemodernconceptof inertia.Inthecaseofastonefallingfromatower,heregardsthehorizontalmotionasa naturalmotionsharedwiththerotatingearth.Thismotioniscircularandnotrectilinear asmoderninertiawouldhaveit.Inthecaseofastonefallingfromthetopofamastina movingship,thehorizontalmotionisnolongernaturalbutitspersistenceremainsempiricallytrue.AsGalileoextrapolatesfromaballrollingonacurvedhorizontalsurface,he regardsthepersistingmotionascircularinthiscasetoo.⁷
Yet,whenSalviatilaterdiscussestheeffectsoftheearth’srotationonthemotionof abulletshotbyacannon,heassumesthebullet’s(absolute)motiontoberectilinearand uniformaslongastheeffectsofgravityandairresistancecanbeignored.Heisherenearing themodernconceptofinertia,evenmoresowhenheespousesthemedievalconceptofan inherenttendencytomotion,the“impetus,”anddeniestheorthodoxAristotelianviewthat themediumisresponsibleforthepersistenceofmotionstartedbyhumanforce.InGalileo’s view,themediumcanonlyimpedethemotionandthebullet’smotionisconservedaslong astheeffectsofthemediumcanbeignored.⁸
SalviaticoncedestoSimpliciothatthekindofrelativityhehasinmindisnoteasily conceived,atleastforanyonewithascholasticbackground:“Youarenotthefirsttofeel agreatrepugnancetowardrecognizingthisnonoperativequalityofmotionamongthe thingswhichshareitincommon.”ThisiswhySalviatispendssomuchtimearguingthat theconservationofhorizontalvelocityandtheindependenceoftheverticalandhorizontal velocityinfreefallimplythefallofastonealongthemastofthemovingship.Sagredothen notesthatthetrajectoryofthefallingstone,asseenfromtheshore,wouldbeanalogous tothetrajectoryofabulletshotfromahorizontalcannon.Accordingly,thebulletshould takethesametimetoreachtheground(whichissupposedtobeperfectlylevel)forany
⁶Galilei1632,pp.116,255.
⁷Galilei1632,pp.142(stoneandtower),148(stoneandmast).
⁸Galilei1632,pp.178(cannon),149–151(medium).GiambattistaBenedettianticipatedrectilinearinertiain 1585:cf. DrakeandDrabkin1969,p.156.Onimpetus,cf. Drake1975; Wallace1981; Capecchi2014,pp.63–78.
valueoftheinitialvelocity.Atfirstglance,thislookslikeaproto-applicationofGalilean relativitytoderivingalawofmotion.Butthisisnotquitesobecause,inordertoprove thefallofthestonealongthemast,Salviatihasearlierreliedontheindependenceofthe verticalandhorizontalmotions,whichbyitselfimpliesSagredo’slaw.Allinall,Galileo’s relativitystatementismorealawthanaprinciple.⁹
Thelawsoffreefall
Inhisresponsetoaclassicalobjectiontoarotatingearth,Salviatiisnaturallyledtothe considerationoffreefallandherepeatedlyreliesonresultsearlierestablishedbyhisfriend “theAcademician,”thatis,Galileo.Forexample,Salviatireliesontheindependenceof theverticalandhorizontalcomponentsofmotion,ontheslowerfallofaballrollingonan inclinedplane,onthesymmetryoftheascentanddescentofaballshotverticallyupward, andevenontheproportionalitybetweenthedescentandthesquareofthetimeoffallfrom rest(whenSagredoaskshimwhatwouldbethetrajectoryofabodyfallingwithaninitial horizontalvelocity).¹⁰
AlthoughGalileoobtainedtheseresultsearlyinthecentury,hepublishedtheevidence verylate,inthe Discorsi of1638.Inthereceivedscholasticview,afallingbodyissuddenly accelerateduntilitreachesawell-definedvelocity.Thevelocityoffallisproportionaltothe weightofthebody,andinverselyproportionaltothedensityofthemedium.Galileorefutes alltheseclaimsbyamixtureofreasoningandexperimentation.Experimentswithinclined planesandpendulumshaveconvincedhimthatthevelocityandimpetusacquiredduring thefalldependonlyontheverticaldistancetraveled,andthatthisimpetuswouldbringthe fallingbodybacktoitsoriginalheightifthemotionwereredirectedupward(asoccursfor apendulumwhenthebobpassesitslowestpoint).Forslightlyinclinedplanes,thefallis muchslowerthaninfreeverticalfall,becausethetraveleddistanceismuchlonger.Galileo wasabletomeasurethetimeofthisfallwithawaterclockandfoundittobeproportional tothesquarerootofthedescent.Asheunderstood,theeffectoftheinclinedplaneisto diminishtheeffectivepropellingforceinaconstantproportion(thesineoftheinclination), andthefreeverticalfallthereforeobeysasimilarlaw.¹¹
Galileothenprovesthatthislawimpliesaconstantaccelerationofthefallingbody, thatis,avelocitylinearlyincreasingwiththetimeofdescent.Forabodylaunchedwitha horizontalvelocity,hederivestheparabolicshapeofthetrajectorybycomposingauniform horizontalmotionwithaconstantlyacceleratedfall.Forabodythrownupwardormoving againsttheslopeofaninclinedplane,heunderstandsthatgravityimpliesadecreaseofthe velocityataconstantrateuntilreversalatthepointofzerovelocity.¹²
Galileo’ssuccessinunveilingthesesimplelawscruciallydependsonhisfocusonthe idealcaseinwhichtheresistanceoftheaircanbeignoredbecausethedensityofthefalling
⁹Galilei1632,pp.171(citation),154–155(canon).
¹⁰Galilei1632,pp.30–31,145–149,163–164,221–222.
¹¹Galilei1638,pp.105–106(scholastics),205(inclinedplane),206–207(pendulum),212–213(experiment).
¹²Galilei1638,pp.208–210(accelerationlaw),268–308(parabola),200–201(upwardfreemotion),243–245 (upwardmotiononinclinedplane).
bodyissufficientlylargeandthetimeoffallissufficientlysmall.Also,herefrainsfrom speculatingonthecauseoftheaccelerationandrathercomputesandmeasuresitseffects. Heknowsonlythatsome“mutualcooperationoftheparts”orsome“concordantinstinct andnaturaltendency”isresponsiblefortherotundityoftheearth,themoon,thesun,and theplanets,fortheweightandfallofobjectsneartheirsurface,andperhapsevenforthe velocityoftheearthandplanetsaroundthesun(ifthisvelocityresultsfromafallfroma fixedsolardistance).¹³
Asfortheeffectoftheweightofbodiesontheirfall,Galileoclaimsthatsufficiently densebodiesallfallwiththesamevelocity,irrespectiveoftheirweight.Whetherornot hedroppedobjectsfromthetopoftheTowerofPisa,hecouldeasilyverifythislawfor constrainedfallonaninclinedplaneorwithapendulum.Inthe Discorsi,healsoreasons thatthecontraryassumptionofafasterfallofheavierbodieswouldleadtoalogicalcontradiction:ontheonehand,thecompoundbodyformedbyattachingaheavybodytoa lightbodyshouldfallatanintermediatespeedbecausethelightbodyslowsdowntheheavy one;ontheotherhand,astheweightofthecompoundbodyisgreaterthantheweightof itstwocomponents,itshouldfallfasterthanbothofthem.Thisreasoninghassometimes beenregardedasapieceofsophistry,becauseitwouldforinstanceimplythatallelectricchargesshouldmoveatthesamespeedinagivenuniformelectricfield.Theobjection isinvalidbecauseGalileoassumesthespeedoffalldependsonweightonly,whereasthe accelerationofanelectricchargedependsontwoparameters,chargeandmass.Thattwo differentparameters,inertialmassandgravitationalmass,mightalsoapplytofreefallwas notanoptioninGalileo’scontextofreasoning:eventhoughherejectedtheAristotelian proportionalitybetweenspeedoffallandweight,heconservedtheweakerassumptionthat thespeedoffalldependsonweightonly.Quiteoften,thesingularityofGalileo’sreasoning inoureyesdependsonhisbeingatanintermediatestagebetweenAristotleandNewton.
Thedefinitionofmotion
Whatisnowcalled“motion”trulywas motolocale or movimientolocale,inordertoavoid confusionwiththebroaderAristotelianconceptofmotion,whichincludesanychange ofstate.In IlSaggiatore (1623),Galileomadeclearthatheregardedthekindofmotion impliedinchangeofplaceasmoredefiniteandlesssubjectivethanchangesinotherperceivedpropertiesofbodies.Heevenflirtedwiththeatomistideaofreducingthelatter properties(secondaryqualities)totheshape,size,andmotionofatoms.Atanyrate,he wasmostlyconcernedwithastronomyandmechanicsinwhichthenarrowerconceptof motionisdominant.¹⁵
¹³Galilei1638,pp.275–277(idealization),p.202(causeunknown); Galilei1632,pp.33–34(universalattraction), pp.20–21,29(velocityofplanets).
¹⁴Galilei1638,pp.128–129(pendulums),107–108(compoundbody).Galileorestrictshisreasoningtobodies madeofthesamematerial,presumablybecauseotherwisetheArchimedeanpushwouldnotbeaconstantfraction oftheweight.Buthebelievestheresulttoextendtoallbodies(ibid.pp.116–117).Ontheallegedsophistry,see, forexample, Hacyan2015
¹⁵Galilei1623,pp.275–276[Draketransl.].Cf. DrakeandO’Malley1960,pp.310–311.
Inthe Dialogo,Galileomadeclearthatforhimplaceandmotionweredefinedwithrespecttoaconcretebodyorsystemofbodiesconsideredatrest:itcouldbeaship,the earth,orthesuntogetherwiththefixedstars.Forfreemotionreferredtoaspherical bodyliketheearth,heassumedconstantaccelerationtowardthecenteraswellasuniformcircularmotionaroundthecenter.Forfreemotionreferredtodistantbodies(the sunandfixedstars),heassumeduniformrectilinearmotion.Hethussubvertedthedistinctionbetweennaturalandviolentmotion,andhedeniedtheessentialroleofthemedium intheAristotelianunderstandingofthesetwokindsofmotion.Thatsaid,hisconcept offreemotionremainedtiedtothenaturaldistributionofmatterintheworld,andhe didnotgiveacompletetheoryofmotionthatcouldserveasthefoundationofanew physics.¹⁶
1.2 BeeckmanandDescartesonfreefall
Asstatedinthepreviousparagraph,forGalileotheplaceofanobjectisdefinedwithrespect toaconcretereferencebody.ForAristotle,theplace(τóπoϛ)ofanobjectisthetouching surfaceofthemediuminwhichitisimmersed;andchangeofplace,orlocomotion,depends onthepropertiesofthismedium.Forboththinkers,thereisnoconceptofemptyspaceas acontainerinwhichobjectsareplacedandcanmove.Theword“space”(χῶρoϛ, διάστημα, spazio),wheneveritoccurs,referstoroomorintervalbetweenconcretebodies.¹⁷
Themodernconceptofemptyspaceasanimmovablereceptacleofobjectshasroots inancientGreekatomismandalsoinperipateticcriticismofAristotle’sconceptofmotion(e.g.,byAristotle’sdefinition,thespinningofaglobedoesnotseemtoinvolveany motion).InthelateRenaissance,thisnewconceptofspacepenetratedthewritingsofFranciscusPatricius,GiordanoBruno,andTommasoCampanella.Intheseventeenthcentury, itprosperedintheneo-atomisthandsofIsaacBeeckmanintheNetherlandsandPierre GassendiinFrance.Foranatomistphilosopher,theworldismadeofatomstraveling throughemptiness,anditbecomesnaturaltodefinespaceastheimmovablecontainerof matter.¹⁸
Space,beingempty,isalsohomogeneousandnonsubstantial.Consequently,anatom originallyatrestmustremainatrestandanatomoriginallyinmotionmustkeepmovingin thesamedirectionwiththesamevelocityuntilacollisionwithanotheratomoccurs.Both BeeckmanandGassendiassertedthepersistenceofmotioninemptyspace.AsBeeckman putitin1614:“Themindconceivesveryeasilythatinavacuummotionneverturnsto restbecausethereisnocausetoalterthemotion:indeed,nothingcanbechangedwithout somecauseofchange.”Byasimilarargument,thedirectionofmotioninavacuumcannotchangebecausenothingcancausethischange.BeeckmanandGassendinonetheless
¹⁶Galilei1632,pp.115–117.AsEdmundHalleyassertedin1718fromanunreliablecomparisonbetweenancient GreekandTycho’sobservations,andasJacquesCassinidefinitelyprovedin1738,theso-calledfixedstarsactually move(Halley1718; Cassini1738;cf. VerbuntandvanderSluys2019).
¹⁷Cf. Jammer1954,chap. 1
¹⁸Jammer1954,chaps. 2–3; DeRisi2015
maintainedthenaturalnessofcircularmotion(aroundthesunandaroundthecenterof theearth)andthusdidnotquitereachthemodernconceptofrectilinearinertia.¹⁹
Beeckmanalsohadaninterestinfreefall,wellbeforeGalileopublishedhisviewson thistopic.Beeckmanbelieved(rectilinear)inertiaplayedafundamentalroleinthisprocess. InJune1618,heenteredthefollowingremarkinhisdiary:
Hereisthereasonwhytwomotionsarecompounded[forafallingstone]:firstly,there isthenaturalmotiondownwards;secondly,thestone,oncesetinmotion,persistsin itsmotion,andthenaturalmotionagainaddstothismotion.
ThisideaissimilartoGalileo’scompositionofhorizontalinertialmotionandverticalacceleratedmotioninthecaseofahorizontallyshotbullet.Laterinthesameyear,Beeckman askedyoungRenéDescartes,theninBredaasamercenaryfortheDutcharmy,todeterminethelawoffreefallbasedonthisidea.AportionofDescartes’sreplyisrecordedina lettertoMarinMersenneofNovember1629:
Firstly,Iassumethatthemotionimpressedonabodygoesonforeverunlessitis removedbysomeothercause.Thatistosay,once[abody]hasbeensetinmotion,it willkeepmovingforeverwithequalvelocity.NowsupposethataweightatAispulled byitsgravitytowardC.Assoonasithasbeguntomove,ifitsgravityissuppressed, itwillneverthelesspersistinthesamemotionuntilitreachesC,anditwilldescend fromAtoBatthesamespeedasfromBtoC[A,B,andCarethreeverticallyaligned points,withAB = BC].Sinceinrealitygravitykeepsactingandaddsnewdownward forcesateachsuccessiveinstant,thebodymusttravelmuchfasterinBCthaninAB becausewhilemovinginBCitretainsalltheimpetusofitsmotioninABandin additionitkeepsacquiringnewimpetusbygravity.
Descartesgoesonwithafalsedeductionofthetimeratiosforthesuccessiveintervals ABandBC.Whatmatterstousistherecoursetoaprincipleofinertiaaswellasthe superpositionofpreviouslyacquiredandnewlyimpressedmotion.NotethatforBeeckman andDescartesthesuperpositionconcernsparallel,verticalvelocities,whereasforGalileo ahorizontalinertialmotioniscombinedwithaverticalfall.²⁰
Amonthlater,DescartescommunicatedtoMersennehisproofofthelinearincreaseof velocityofafallingbody:
Inthefirstmoment,thevelocityoneisimpressedbygravity;thenagainthevelocity oneinthesecondmoment,andsoforth.Oneinthefirstmomentandoneinthesecond momentmakestwo;andoneinthethirdmomentmakesthree,sothatthevelocity increasesinarithmeticproportion.IthoughtIhadsufficientlyprovedthisfromthe factthatgravityactspermanently....Suppose,forexample,thataplumbmassfallsby
¹⁹Beeckman1939–1953,Vol.1,pp.24–25.OnBeeckman,cf. Arthur2007; vanBerkelandUltee2013.On Gassendi,cf. Pav1966; Fischer2014
²⁰Beeckman1939,Vol.1,p.174;DescartestoMersenne,November13,1629,in AdamandTannery1897, pp.72(citation,mostlyinLatinandprobablytakenfromamanuscriptof1619),75(TanneryontheBeeckman connection).Cf. Bouasse1895,pp.103–105; Jouguet1908,Vol.1,pp.81–82; Damerowetal.1992; Richard2007 AsnotedbyDamerowetal.,JeanBuridananticipatedBeeckman’sapproach.
theforceofgravityandthat,afterthefirstmomentoffall,Godwithholdsallgravity fromtheplumb.Thenthismasskeepsgoingdowninavacuumbecauseithasbeen setinmotionandbecausenoreasoncanbegivenwhyitsvelocitywouldincrease;but itcannotdecreaseeither(rememberthatIsupposethatonce[abody]hasbeensetin motion,itwillmoveforeverinavacuum;andIwillproveitinmytreatise).Ifafter sometimeGodrestitutesgravitytotheplumbforanequalmoment,andthenagain withholdsgravity,theforceofgravitywillpulltheplumbasmuchasitdidduringthe firstmoment,sothatthevelocityofthemotionisdoubled.
Again,theproofprobablydatesfrom1618,whenDescarteswasundertheinfluenceof Beeckman’satomism.Descarteswasthenassumingvacuum,inertialmotioninavacuum, andthemutualindependenceoftheimpressedandinertialmotions.²¹
In1631,DescartestoldMersennethathenolongerheldthethirdoftheseassumptions:
Inotonlyassumedavacuum,butIalsoassumedthatthemovingforce...wasacting inanalwaysequalmanner,whichopenlyconflictswiththelawsofNature:indeedall naturalpowersacttoalargerorsmallerextentaccordingastheobjectismoreorless disposedtoreceivetheiraction;anditiscertainthatastoneisnotequallydisposed toreceiveanewmotion,oravelocityincrease,whenitisalreadymovingveryrapidly andwhenitismovingveryslowly.
When,ayearlater,DescartesbecameawareofGalileo’s Dialogo,hedisagreedwiththe universalityoffreefallaswellaswithGalileo’streatmentoffallwithaninitialhorizontal velocity.Heneverpublishedhisearlyrationaldeductionofthelawofconstantacceleration, sincehehadceasedtobelieveinitstruth.Asweareabouttosee,inhismaturephilosophy theacceleratingeffectofaforceonabodyhadtodependonitsinitialvelocity.²²
1.3 Descartes’sworld
WhileDescarteswasreportingtoMersennehisoldreplytoBeeckman’squeryonfreefall, hewasalsowritinganambitioustreatise, Lemonde,propoundinganewmechanicalphilosophytosupplantAristotle’ssystem.FromBeeckmanheretainedthecorpuscles,their inertia,andtheircollisions,buthestronglyrejectedtheconceptofemptyspaceattheheart ofneo-atomism.InDescartes’world,anyspatialextensionismatter,andviceversa.The infiniteworldismadeofcontiguousrigidfiguresofvariousshape:thegrossparticlesof ordinarymatter(thirdelement),contiguousballsbetweentheformerparticles(secondelement),anddustfillingtheremaininginterstices(firstelement).Thesecorpusclesorrigid figurespossessnoqualityotherthantheirshapeandextension.Theymovewithrespect toeachotherinaperpetualrearrangementimplyingmutualcollisions.Thereisnoabsolute,emptyspace.Motion,incommonparlance,beingreferredtoalargerigidbody (mostfrequentlytheearth),isill-definedsincetheworldcontainsmanyrigidassemblies
²¹DescartestoMersenne,December18,1629,in AdamandTannery1897,pp.88–90(inLatin).
²²DescartestoMersenne,OctoberorNovember1631,in AdamandTannery1897,p.230;Novemberor December1632, ibid.p.261;14Aug.1634, ibid.pp.304–305.
ofcorpusclesthatcouldserveasreferencebodies.ThisiswhyDescartesdefinesthetrue motionofabodyasmotionwithrespecttoadjacentbodies:²³
If,insteadofsettlingonanotionfoundedonordinaryusageonly,wewishtoknow whatismotionintruth,weshallsay,forthesakeofdeterminateness,thatmotionis thetransportofaportionofmatterorofabodyfromthevicinityofthosethattouch itimmediately.
FromGod’simmutability,Descartesderivestheglobalconservationofthismotionas wellasthetendencyofindividualparticlestopreservetheirrestortheirmotion.Thisleads himtothree“lawsofnature”:²⁴
(1) Everythingremainsinitsstateofbeingaslongasnothingchangesit.
(2) Everymovingbodytendstocontinueitsmotiononastraightline.
(3) Ifamovingbodyencountersastrongerbody,itdoesnotloseanyofitsmotion;if amovingbodyencountersaweakeronethatitmaymove,theformerbodylosesas muchmotionasitgivestothelatter.
AmodernreaderofDescartesmaybetemptedtoconflatethetwofirstlawswithourprincipleofinertia,andthethirdwiththeconservationofmomentumorofkineticenergy inacollision.TheseguessesaredeeplyincompatiblewithDescartes’struemeaning.First, Descartes’sstatementofthepersistenceofrestandmotionshouldnotbeconfusedwiththe modernprincipleofinertia,becauseinhissystemtherestandmotionofabodyaredefined withrespecttotheadjacentcorpuscles,notwithrespecttoanabstractimmovablespace. Inparticular,Descartesexplainstherigidityofamacroscopicbodybypersistenceofthe mutualrestofitsparticles,andfluiditybytheconstant,erraticmotionoftheparticlesof thefluid.Second,thepropositionthatalargerparticleremainsatrestwhenimpactedbya smallerparticlecontradictsourconservationofmomentum(yetforDescartesitiscompatiblewiththecontrarybehaviorofmacroscopicbodies,becausethemediumcontributesto thisbehavior).Third,Descartes’sideathatinacollisiononeparticlelosesasmuchmotion astheothergainsdoesnotagreewithanymodernconservationlaw,becausehemeasures themotionofaparticlebyvolumetimesvelocitymodulus.
Descartes’soriginaldefinitionoftruemotionhasimportantconsequences.First,the earthandtheplanetsdonothaveanytruemotion,becausetheyareatrestwithrespectto thesubtlematterinwhichtheybathe:
Properlyspeaking,[motion]isonlythetransportofabodyfromthevicinityofthose thattouchit....Inearthandtheotherplanetsnomotioninthepropersensecan befound,becausetheyarenottransportedfromthevicinityofthepartsofthesky thattouchthem....Ifweseemtoattributesomemotiontotheearth,weshould thinkthatwedosoimproperly,inthesamewayaswesometimessaythatthosewho sleepandrestinashipnonethelesstravelfromCalaistoDover,owingtotheshipthat carrythem.
²³Descartes[1633];1644,p.140.
²⁴Descartes1644,pp.150–156.Cf. Garber1992; Vilain1996,pp.60–67.
