ToExplaintheWorld:TheDiscoveryofModernScienceStevenWeinbergHarperCollinsPublishers(2015)

Rating:★★★☆☆

In this rich, irreverent, compelling history, Nobel Prize-winning physicist StevenWeinberg takes usfrom ancientMiletus to medieval Baghdad and Oxford, from Plato's Academy and theMuseum ofAlexandria to the cathedral school ofChartres and theRoyal Society ofLondon.He shows that thescientists of ancient andmedieval times not only did not understand what we now know about theworld,theydidnotunderstandwhatthereistobeunderstood,orhowtolearnit.Yetoverthecenturies,throughthestruggletosolvesuchmysteriesasthecuriousapparentbackwardmovementoftheplanetsor the rise and fall of the tides, science eventually emerged as amodern discipline.Along theway,Weinbergexamineshistoricclashesandcollaborationsbetweenscienceand thecompetingspheresofreligion,technology,poetry,mathematics,andphilosophy.

An illuminating exploration of howwe have come to consider and analyze theworld around us,ToExplaintheWorldisasweeping,ambitiousaccountofhowdifficultitwastodiscoverthegoalsandmethodsofmodernscience,andtheimpactofthisdiscoveryonhumanunderstandinganddevelopment.

Dedication

ToLouise,Elizabeth,andGabrielle

Epigraph

Thesethreehoursthatwehavespent,Walkinghere,twoshadowswentAlongwithus,whichweourselvesproduced;But,nowthesunisjustaboveourhead,Wedothoseshadowstread;Andtobraveclearnessallthingsarereduced.

JohnDonne,“ALectureupontheShadow”

Contents

DedicationEpigraphPreface

PARTI:GREEKPHYSICS

1.MatterandPoetry2.MusicandMathematics3.MotionandPhilosophy4.HellenisticPhysicsandTechnology5.AncientScienceandReligion

PARTII:GREEKASTRONOMY

6.TheUsesofAstronomy7.MeasuringtheSun,Moon,andEarth8.TheProblemofthePlanets

PARTIII:THEMIDDLEAGES

9.TheArabs10.MedievalEurope

PARTIV:THESCIENTIFICREVOLUTION

11.TheSolarSystemSolved12.ExperimentsBegun13.MethodReconsidered14.TheNewtonianSynthesis15.Epilogue:TheGrandReductionAcknowledgmentsTechnicalNotesEndnotesBibliographyIndex

AbouttheAuthorAlsobyStevenWeinbergCreditsCopyrightAboutthePublisher

Preface

Iamaphysicist,notahistorian,butovertheyearsIhavebecomeincreasinglyfascinatedbythehistoryofscience.Itisanextraordinarystory,oneofthemostinterestinginhumanhistory.Itisalsoastoryinwhichscientistslikemyselfhaveapersonalstake.Today’sresearchcanbeaidedandilluminatedbyaknowledgeof its past, and for some scientists knowledgeof thehistoryof sciencehelps tomotivatepresent work. We hope that our research may turn out to be a part, however small, of the grandhistoricaltraditionofnaturalscience.Wheremyownpastwritinghastouchedonhistory,ithasbeenmostlythemodernhistoryofphysics

andastronomy, roughly fromthe latenineteenthcentury to thepresent.Although in thiserawehavelearnedmanynewthings, thegoalsandstandardsofphysicalsciencehavenotmateriallychanged.Ifphysicists of 1900 were somehow taught today’s Standard Model of cosmology or of elementaryparticlephysics, theywouldhavefoundmuchtoamazethem,but theideaofseekingmathematicallyformulated and experimentally validated impersonal principles that explain a wide variety ofphenomenawouldhaveseemedquitefamiliar.AwhileagoIdecidedthatIneededtodigdeeper,tolearnmoreaboutanearliererainthehistoryof

science,whenthegoalsandstandardsofsciencehadnotyettakentheirpresentshape.Asisnaturalforanacademic,whenIwanttolearnaboutsomething,Ivolunteertoteachacourseonthesubject.OverthepastdecadeattheUniversityofTexas,Ihavefromtimetotimetaughtundergraduatecoursesonthehistoryofphysicsandastronomytostudentswhohadnospecialbackgroundinscience,mathematics,orhistory.Thisbookgrewoutofthelecturenotesforthosecourses.Butasthebookhasdeveloped,perhapsIhavebeenabletooffersomethingthatgoesalittlebeyonda

simplenarrative:it is theperspectiveofamodernworkingscientistonthescienceofthepast.Ihavetakenthisopportunitytoexplainmyviewsaboutthenatureofphysicalscience,andaboutitscontinuedtangledrelationswithreligion,technology,philosophy,mathematics,andaesthetics.Before history there was science, of a sort. At any moment nature presents us with a variety of

puzzlingphenomena:fire,thunderstorms,plagues,planetarymotion,light,tides,andsoon.Observationoftheworldledtousefulgeneralizations:firesarehot;thunderpresagesrain;tidesarehighestwhentheMoon is full or new, and soon.These becamepart of the common sense ofmankind.But here andthere,somepeoplewantedmorethanjustacollectionoffacts.Theywantedtoexplaintheworld.Itwasnoteasy.Itisnotonlythatourpredecessorsdidnotknowwhatweknowabouttheworld—

moreimportant,theydidnothaveanythinglikeourideasofwhattherewastoknowabouttheworld,andhowtolearnit.AgainandagaininpreparingthelecturesformycourseIhavebeenimpressedwithhowdifferenttheworkofscienceinpastcenturieswasfromthescienceofmyowntimes.AsthemuchquotedlinesofanovelofL.P.Hartleyputit,“Thepastisaforeigncountry;theydothingsdifferentlythere.”IhopethatinthisbookIhavebeenabletogivethereadernotonlyanideaofwhathappenedinthehistoryoftheexactsciences,butalsoasenseofhowhardithasallbeen.So this book is not solely about how we came to learn various things about the world. That is

naturallyaconcernofanyhistoryofscience.Myfocusinthisbookisalittledifferent—itishowwecametolearnhowtolearnabouttheworld.Iamnotunawarethattheword“explain”inthetitleofthisbookraisesproblemsforphilosophersof

science.Theyhavepointedoutthedifficultyindrawingaprecisedistinctionbetweenexplanationanddescription.(IwillhavealittletosayaboutthisinChapter8.)Butthisisaworkonthehistoryratherthanthephilosophyofscience.ByexplanationImeansomethingadmittedlyimprecise,thesameasismeant in ordinary life whenwe try to explainwhy a horse haswon a race or why an airplane hascrashed.

Theword “discovery” in the subtitle is also problematic. I had thought of usingThe InventionofModernScienceasasubtitle.Afterall,sciencecouldhardlyexistwithouthumanbeingstopracticeit.Ichose“Discovery”insteadof“Invention”tosuggestthatscienceisthewayitisnotsomuchbecauseofvarious adventitious historic acts of invention, but because of the way nature is. With all itsimperfections,modernscienceisatechniquethatissufficientlywelltunedtonaturesothatitworks—itisapracticethatallowsustolearnreliablethingsabouttheworld.Inthissense,itisatechniquethatwaswaitingforpeopletodiscoverit.Thus one can talk about the discovery of science in the way that a historian can talk about the

discoveryofagriculture.Withall itsvarietyandimperfections,agricultureis thewayit isbecauseitspracticesare sufficientlywell tuned to the realitiesofbiology so that itworks—it allowsus togrowfood.Ialsowantedwiththissubtitletodistancemyselffromthefewremainingsocialconstructivists:those

sociologists,philosophers,andhistorianswhotrytoexplainnotonlytheprocessbuteventheresultsofscienceasproductsofaparticularculturalmilieu.Amongthebranchesofscience,thisbookwillemphasizephysicsandastronomy.Itwasinphysics,

especiallyasappliedtoastronomy,thatsciencefirsttookamodernform.Ofcoursetherearelimitstotheextenttowhichscienceslikebiology,whoseprinciplesdependsomuchonhistoricalaccidents,canorshouldbemodeledonphysics.Nevertheless,thereisasenseinwhichthedevelopmentofscientificbiology as well as chemistry in the nineteenth and twentieth centuries followed the model of therevolutioninphysicsoftheseventeenthcentury.Science is now international, perhaps the most international aspect of our civilization, but the

discoveryofmodernsciencehappenedinwhatmaylooselybecalledtheWest.ModernsciencelearneditsmethodsfromresearchdoneinEuropeduringthescientificrevolution,whichinturnevolvedfromworkdoneinEuropeandinArabcountriesduringtheMiddleAges,andultimatelyfromtheprecociousscienceoftheGreeks.TheWestborrowedmuchscientificknowledgefromelsewhere—geometryfromEgypt, astronomical data from Babylon, the techniques of arithmetic from Babylon and India, themagneticcompassfromChina,andsoon—butasfarasIknow,itdidnotimportthemethodsofmodernscience. So this book will emphasize theWest (includingmedieval Islam) in just the way that wasdeploredbyOswaldSpenglerandArnoldToynbee: Iwillhave little tosayabout scienceoutside theWest, and nothing at all to say about the interesting but entirely isolated progress made in pre-ColumbianAmerica.Intellingthisstory,Iwillbecomingclosetothedangerousgroundthatismostcarefullyavoidedby

contemporary historians, of judging the past by the standards of the present. This is an irreverenthistory;Iamnotunwillingtocriticizethemethodsandtheoriesofthepastfromamodernviewpoint.IhaveeventakensomepleasureinuncoveringafewerrorsmadebyscientificheroesthatIhavenotseenmentionedbyhistorians.A historian who devotes years to study the works of some great man of the past may come to

exaggeratewhathisherohasaccomplished.IhaveseenthisinparticularinworksonPlato,Aristotle,Avicenna, Grosseteste, and Descartes. But it is not my purpose here to accuse some past naturalphilosophersofstupidity.Rather,byshowinghowfartheseveryintelligentindividualswerefromourpresentconceptionofscience,Iwanttoshowhowdifficultwasthediscoveryofmodernscience,howfarfromobviousareitspracticesandstandards.Thisalsoservesasawarning,thatsciencemaynotyetbeinitsfinalform.AtseveralpointsinthisbookIsuggestthat,asgreatasistheprogressthathasbeenmadeinthemethodsofscience,wemaytodayberepeatingsomeoftheerrorsofthepast.Some historians of science make a shibboleth of not referring to present scientific knowledge in

studyingthescienceofthepast.Iwillinsteadmakeapointofusingpresentknowledgetoclarifypastscience.Forinstance,thoughitmightbeaninterestingintellectualexercisetotrytounderstandhowthe

HellenisticastronomersApolloniusandHipparchusdevelopedthetheorythattheplanetsgoaroundtheEarth on looping epicyclic orbits by using only the data that had been available to them, this isimpossible,formuchofthedatatheyusedislost.ButwedoknowthatinancienttimestheEarthandplanetswentaroundtheSunonnearlycircularorbits,justastheydotoday,andbyusingthisknowledgewewillbeable tounderstandhow thedataavailable toancientastronomerscouldhavesuggested tothem their theoryofepicycles. Inanycase,howcananyone today, readingaboutancient astronomy,forgetourpresentknowledgeofwhatactuallygoesaroundwhatinthesolarsystem?Forreaderswhowanttounderstandingreaterdetailhowtheworkofpastscientistsfitsinwithwhat

actuallyexistsinnature,thereare“technicalnotes”atthebackofthebook.Itisnotnecessarytoreadthesenotestofollowthebook’smaintext,butsomereadersmaylearnafewoddbitsofphysicsandastronomyfromthem,asIdidinpreparingthem.Science is not now what it was at its start. Its results are impersonal. Inspiration and aesthetic

judgmentareimportantinthedevelopmentofscientifictheories,buttheverificationofthesetheoriesrelies finally on impartial experimental tests of their predictions. Thoughmathematics is used in theformulation of physical theories and in working out their consequences, science is not a branch ofmathematics,andscientifictheoriescannotbededucedbypurelymathematicalreasoning.Scienceandtechnology benefit each other, but at its most fundamental level science is not undertaken for anypracticalreason.ThoughsciencehasnothingtosayonewayortheotherabouttheexistenceofGodoranafterlife,itsgoalistofindexplanationsofnaturalphenomenathatarepurelynaturalistic.Scienceiscumulative; each new theory incorporates successful earlier theories as approximations, and evenexplainswhytheseapproximationswork,whentheydowork.NoneofthiswasobvioustothescientistsoftheancientworldortheMiddleAges,andallofitwas

learnedonlywithgreatdifficultyinthescientificrevolutionofthesixteenthandseventeenthcenturies.Nothing likemodern sciencewas a goal from the beginning. How then didwe get to the scientificrevolution,andbeyond it towherewearenow?That iswhatwemust try to learnasweexplore thediscoveryofmodernscience.

PARTI

GREEKPHYSICS

During or before the flowering of Greek science, significant contributions to technology,mathematics,andastronomywerebeingmadebytheBabylonians,Chinese,Egyptians,Indians,and other peoples. Nevertheless, it was from Greece that Europe drew its model and itsinspiration,anditwasinEuropethatmodernsciencebegan,sotheGreeksplayedaspecialroleinthediscoveryofscience.Onecanargueendlesslyaboutwhy itwas theGreekswhoaccomplishedsomuch. Itmaybe

significantthatGreeksciencebeganwhenGreekslivedinsmallindependentcity-states,manyofthem democracies. But as we shall see, the Greeks made their most impressive scientificachievements after these small states had been absorbed into great powers: the Hellenistickingdoms, and then the Roman Empire. The Greeks in Hellenistic and Roman times madecontributionstoscienceandmathematicsthatwerenotsignificantlysurpasseduntilthescientificrevolutionofthesixteenthandseventeenthcenturiesinEurope.Thispart ofmyaccountofGreek sciencedealswithphysics, leavingGreekastronomy tobe

discussedinPartII.IhavedividedPartIintofivechapters,dealinginmoreorlesschronologicalorder with five modes of thought with which science has had to come to terms: poetry,mathematics, philosophy, technology, and religion. The theme of the relationship of science tothesefiveintellectualneighborswillrecurthroughoutthisbook.

1

MatterandPoetry

First,tosetthescene.BythesixthcenturyBCthewesterncoastofwhatisnowTurkeyhadforsometimebeensettledbyGreeks,chieflyspeakingtheIoniandialect.TherichestandmostpowerfuloftheIonian cities wasMiletus, founded at a natural harbor near where the riverMeander flows into theAegeanSea.InMiletus,overacenturybeforethetimeofSocrates,Greeksbegantospeculateaboutthefundamentalsubstanceofwhichtheworldismade.IfirstlearnedabouttheMilesiansasanundergraduateatCornell,takingcoursesonthehistoryand

philosophyofscience.InlecturesIheardtheMilesianscalled“physicists.”Atthesametime,Iwasalsoattendingclassesonphysics,includingthemodernatomictheoryofmatter.Thereseemedtometobevery little in commonbetweenMilesian andmodern physics. Itwas not somuch that theMilesianswerewrong about the nature ofmatter, but rather that I could not understand how they could havereached theirconclusions.Thehistorical recordconcerningGreek thoughtbefore the timeofPlato isfragmentary,butIwasprettysurethatduringtheArchaicandClassicaleras(roughlyfrom600to450BCandfrom450to300BC,respectively)neithertheMilesiansnoranyoftheotherGreekstudentsofnaturewerereasoninginanythinglikethewayscientistsreasontoday.ThefirstMilesianofwhomanythingisknownwasThales,wholivedabouttwocenturiesbeforethe

timeofPlato.Hewassupposedtohavepredictedasolareclipse,onethatweknowdidoccurin585BCandwas visible fromMiletus. Evenwith the benefit ofBabylonian eclipse records it’s unlikely thatThales could have made this prediction, because any solar eclipse is visible from only a limitedgeographic region, but the fact that Thaleswas creditedwith this prediction shows that he probablyflourishedintheearly500sBC.Wedon’tknowifThalesputanyofhisideasintowriting.Inanycase,nothingwrittenbyThaleshassurvived,evenasaquotationbylaterauthors.Heisalegendaryfigure,one of those (like his contemporary Solon, who was supposed to have founded the Athenianconstitution) who were conventionally listed in Plato’s time as the “seven sages” of Greece. Forinstance,ThaleswasreputedtohaveprovedorbroughtfromEgyptafamoustheoremofgeometry(seeTechnicalNote1).Whatmatters toushere is thatThaleswassaid tohold theviewthatallmatter iscomposed of a single fundamental substance. According to Aristotle’s Metaphysics, “Of the firstphilosophers,mostthoughttheprincipleswhichwereofthenatureofmatterweretheonlyprinciplesofallthings....Thales,thefounderofthisschoolofphilosophy,saystheprincipleiswater.”1Muchlater,DiogenesLaertius(fl.AD230),abiographeroftheGreekphilosophers,wrote,“Hisdoctrinewasthatwateristheuniversalprimarysubstance,andthattheworldisanimateandfullofdivinities.”2By“universalprimarysubstance”didThalesmean thatallmatter iscomposedofwater? If so,we

havenowayoftellinghowhecametothisconclusion,butifsomeoneisconvincedthatallmatteriscomposedofasinglecommonsubstance,thenwaterisnotabadcandidate.Waternotonlyoccursasaliquidbutcanbeeasilyconvertedintoasolidbyfreezingor intoavaporbyboiling.Waterevidentlyalsoisessentialtolife.Butwedon’tknowifThalesthoughtthatrocks,forexample,arereallyformedfrom ordinarywater, or only that there is something profound that rock and all other solids have incommonwithfrozenwater.Thaleshadapupilorassociate,Anaximander,whocame toadifferentconclusion.He too thought

that there is a single fundamental substance, but he did not associate it with any commonmaterial.

Rather,heidentifieditasamysterioussubstancehecalledtheunlimited,orinfinite.Onthis,wehaveadescription of his views by Simplicius, a Neoplatonist who lived about a thousand years later.SimpliciusincludeswhatseemstobeadirectquotationfromAnaximander,indicatedhereinitalics:

Of thosewho say that [the principle] is one and inmotion andunlimited,Anaximander, sonofPraxiades,aMilesianwhobecamesuccessorandpupil toThales, said that theunlimited isbothprincipleandelementofthethingsthatexist.Hesaysthatitisneitherwaternoranyotheroftheso-called elements, but someother unlimited nature, fromwhich the heavens and theworlds inthemcomeabout;andthethingsfromwhichisthecomingintobeingforthethingsthatexistarealso those intowhich theirdestructioncomesabout, inaccordancewithwhatmustbe.For theygive justiceandreparation tooneanother for theiroffence inaccordancewith theordinanceoftime—speaking of them thus in rather poetical terms. And it is clear that, having observed thechange of the four elements into one another, he did not think fit tomake any one of these anunderlyingstuff,butsomethingelseapartfromthese.3

AlittlelateranotherMilesian,Anaximenes,returnedtotheideathateverythingismadeofsomeonecommonsubstance,butforAnaximenesitwasnotwaterbutair.Hewroteonebook,ofwhichjustonewholesentencehassurvived:“Thesoul,beingourair,controlsus,andbreathandairencompass thewholeworld.”4WithAnaximenes the contributionsof theMilesians came to an end.Miletus and theother Ionian

citiesofAsiaMinorbecamesubjecttothegrowingPersianEmpireinabout550BC.Miletusstartedarevoltin499BCandwasdevastatedbythePersians.ItrevivedlaterasanimportantGreekcity,butitneveragainbecameacenterofGreekscience.ConcernwiththenatureofmattercontinuedoutsideMiletusamongtheIonianGreeks.Thereisahint

thatearthwasnominatedasthefundamentalsubstancebyXenophanes,whowasbornaround570BCatColophoninIoniaandmigratedtosouthernItaly.Inoneofhispoems,thereistheline“Forallthingscome from earth, and in earth all things end.”5 But perhaps thiswas just his version of the familiarfunerary sentiment, “Ashes to ashes, dust to dust.” We will meet Xenophanes again in anotherconnection,whenwecometoreligioninChapter5.AtEphesus,notfarfromMiletus,around500BCHeraclitustaughtthatthefundamentalsubstanceis

fire.Hewroteabook,ofwhichonlyfragmentssurvive.Oneofthesefragmentstellsus,“Thisorderedkosmos,*whichisthesameforall,wasnotcreatedbyanyoneofthegodsorofmankind,butitwasever and is and shall be ever-livingFire, kindled inmeasure and quenched inmeasure.”6 Heraclituselsewhereemphasizedtheendlesschangesinnature,soforhimitwasmorenaturaltotakeflickeringfire,anagentofchange,asthefundamentalelementthanthemorestableearth,air,orwater.Theclassicviewthatallmatteriscomposednotofonebutoffourelements—water,air,earth,and

fire—isprobablyduetoEmpedocles.HelivedinAcragas,inSicily(themodernAgrigento),inthemid-400sBC,andhe is the firstandnearly theonlyGreek in thisearlypartof thestory tohavebeenofDorian rather than of Ionian stock.Hewrote two hexameter poems, ofwhichmany fragments havesurvived. InOnNature,we find“howfrom themixtureofWater,Earth,Aether,andSun [fire] therecame intobeing the formsandcoloursofmortal things”7 andalso“fire andwater andearth and theendlessheightofair,andcursedStrifeapartfromthem,balancedineveryway,andLoveamongthem,equalinheightandbreadth.”8ItispossiblethatEmpedoclesandAnaximanderusedtermslike“love”and“strife”or“justice”and

“injustice”onlyasmetaphors fororderanddisorder, insomething like thewayEinsteinoccasionallyused “God” as ametaphor for the unknown fundamental laws of nature. Butwe should not force amoderninterpretationontothepre-Socratics’words.AsIseeit, theintrusionofhumanemotionslike

Empedocles’ loveandstrife,orofvalues likeAnaximander’s justiceandreparation, intospeculationsabout thenatureofmatter ismore likely tobea signof thegreatdistanceof the thoughtof thepre-Socraticsfromthespiritofmodernphysics.Thesepre-Socratics, fromThales toEmpedocles, seem tohave thoughtof the elements as smooth

undifferentiatedsubstances.Adifferentviewthat iscloser tomodernunderstandingwas introducedalittlelateratAbdera,atownontheseacoastofThracefoundedbyrefugeesfromtherevoltoftheIoniancitiesagainstPersiastartedin499BC.ThefirstknownAbderitephilosopherisLeucippus,fromwhomjust one sentence survives, suggesting a deterministic worldview: “No thing happens in vain, buteverythingforareasonandbynecessity.”9MuchmoreisknownofLeucippus’successorDemocritus.HewasbornatMiletus,andhadtraveledinBabylon,Egypt,andAthensbeforesettlinginAbderainthelate 400sBC.Democrituswrote books on ethics, natural science,mathematics, andmusic, ofwhichmany fragments survive. One of these fragments expresses the view that all matter consists of tinyindivisible particles called atoms (from theGreek for “uncuttable”),moving in empty space: “Sweetexistsbyconvention,bitterbyconvention;atomsandVoid[alone]existinreality.”10Likemodernscientists,theseearlyGreekswerewillingtolookbeneaththesurfaceappearanceofthe

world,pursuingknowledgeaboutadeeperlevelofreality.Thematteroftheworlddoesnotappearatfirstglanceasifitisallmadeofwater,orair,orearth,orfire,orallfourtogether,orevenofatoms.Acceptance of the esotericwas taken to an extreme by Parmenides of Elea (themodernVelia) in

southern Italy, who was greatly admired by Plato. In the early 400s BC Parmenides taught, contraHeraclitus,thattheapparentchangeandvarietyinnatureareanillusion.HisideasweredefendedbyhispupilZenoofElea(nottobeconfusedwithotherZenos,suchasZenotheStoic).InhisbookAttacks,Zenoofferedanumberofparadoxestoshowtheimpossibilityofmotion.Forinstance,totraversethewholecourseofaracetrack,itisnecessaryfirsttocoverhalfthedistance,andthenhalftheremainingdistance,andsoonindefinitely,so that it is impossibleever to traversethewhole track.Bythesamereasoning,asfaraswecantellfromsurvivingfragments,itappearedtoZenotobeimpossibleevertotravelanygivendistance,sothatallmotionisimpossible.Ofcourse,Zeno’sreasoningwaswrong.AspointedoutlaterbyAristotle,11 thereisnoreasonwhy

wecannotaccomplishaninfinitenumberofstepsinafinitetime,aslongasthetimeneededforeachsuccessivestepdecreasessufficientlyrapidly.Itistruethataninfiniteserieslike½+⅓+¼+...hasaninfinitesum,buttheinfiniteseries½+¼+⅛+...hasafinitesum,inthiscaseequalto1.What ismost striking is not somuch that Parmenides and Zenowerewrong as that they did not

bothertoexplainwhy,ifmotionisimpossible,thingsappeartomove.Indeed,noneoftheearlyGreeksfrom Thales to Plato, in eitherMiletus or Abdera or Elea or Athens, ever took it on themselves toexplainindetailhowtheirtheoriesaboutultimaterealityaccountedfortheappearancesofthings.Thiswasnot just intellectual laziness.Therewasa strainof intellectual snobberyamong theearly

Greeks that led themtoregardanunderstandingofappearancesasnotworthhaving.This is justoneexampleof anattitude thathasblightedmuchof thehistoryof science.Atvarious times ithasbeenthoughtthatcircularorbitsaremoreperfectthanellipticalorbits,thatgoldismorenoblethanlead,andthatmanisahigherbeingthanhisfellowsimians.Arewe nowmaking similarmistakes, passing up opportunities for scientific progress becausewe

ignorephenomenathatseemunworthyofourattention?Onecan’tbesure,butIdoubtit.Ofcourse,wecannot explore everything, butwe choose problems thatwe think, rightly orwrongly, offer the bestprospectforscientificunderstanding.Biologistswhoareinterestedinchromosomesornervecellsstudyanimals like fruit flies and squid, not noble eagles and lions. Elementary particle physicists aresometimesaccusedofasnobbishandexpensivepreoccupationwithphenomenaatthehighestattainableenergies,butitisonlyathighenergiesthatwecancreateandstudyhypotheticalparticlesofhighmass,likethedarkmatterparticlesthatastronomerstellusmakeupfive-sixthsofthematteroftheuniverse.

In any case, we give plenty of attention to phenomena at low energies, like the intriguingmass ofneutrinos,aboutamillionththemassoftheelectron.Incommentingontheprejudicesofthepre-Socratics,Idon’tmeantosaythatapriorireasoninghas

no place in science. Today, for instance, we expect to find that our deepest physical laws satisfyprinciplesofsymmetry,whichstatethatphysicallawsdonotchangewhenwechangeourpointofviewin certain definiteways. Just like Parmenides’ principle of changelessness, some of these symmetryprinciples are not immediately apparent in physical phenomena—they are said to be spontaneouslybroken. That is, the equations of our theories have certain simplicities, for instance treating certainspecies of particles in the same way, but these simplicities are not shared by the solutions of theequations, which govern actual phenomena. Nevertheless, unlike the commitment of Parmenides tochangelessness,theaprioripresumptioninfavorofprinciplesofsymmetryarosefrommanyyearsofexperience in searching for physical principles that describe the real world, and broken as well asunbroken symmetries are validated by experiments that confirm their consequences. They do notinvolvevaluejudgmentsofthesortweapplytohumanaffairs.WithSocrates,inthelatefifthcenturyBC,andPlato,somefortyyearslater,thecenterofthestage

for Greek intellectual life moved to Athens, one of the few cities of Ionian Greeks on the Greekmainland.AlmostallofwhatweknowaboutSocratescomesfromhisappearanceinthedialoguesofPlato,andasacomiccharacterinAristophanes’playTheClouds.Socratesdoesnotseemtohaveputanyofhisideasintowriting,butasfaraswecantellhewasnotveryinterestedinnaturalscience.InPlato’s dialoguePhaedo Socrates recalls howhewasdisappointed in reading a bookbyAnaxagoras(aboutwhommore in Chapter 7) becauseAnaxagoras described the Earth, Sun,Moon, and stars inpurelyphysicalterms,withoutregardtowhatisbest.12Plato,unlikehisheroSocrates,wasanAthenianaristocrat.HewasthefirstGreekphilosopherfrom

whommanywritingshavesurvivedprettymuchintact.Plato,likeSocrates,wasmoreconcernedwithhumanaffairsthanwiththenatureofmatter.Hehopedforapoliticalcareerthatwouldallowhimtoputhis utopian and antidemocratic ideas into practice. In 367 BC Plato accepted an invitation fromDionysiusIItocometoSyracuseandhelpreformitsgovernment,but,fortunatelyforSyracuse,nothingcameofthereformproject.In one of his dialogues, the Timaeus, Plato brought together the idea of four elements with the

Abderitenotionofatoms.Platosupposed that thefourelementsofEmpedoclesconsistedofparticlesshaped like four of the five solid bodies known inmathematics as regular polyhedrons: bodieswithfacesthatareallidenticalpolygons,withalledgesidentical,comingtogetheratidenticalvertices.(SeeTechnicalNote2.)Forinstance,oneoftheregularpolyhedronsisthecube,whosefacesareallidenticalsquares,threesquaresmeetingateachvertex.Platotookatomsofearthtohavetheshapeofcubes.Theother regular polyhedrons are the tetrahedron (a pyramidwith four triangular faces), the eight-sidedoctahedron,thetwenty-sidedicosahedron,andthetwelve-sideddodecahedron.Platosupposedthattheatoms of fire, air, and water have the shapes respectively of the tetrahedron, octahedron, andicosahedron.Thisleftthedodecahedronunaccountedfor.Platoregardeditasrepresentingthekosmos.LaterAristotleintroducedafifthelement,theetherorquintessence,whichhesupposedfilledthespaceabovetheorbitoftheMoon.It has been common in writing about these early speculations regarding the nature of matter to

emphasizehowtheyprefigurefeaturesofmodernscience.Democritus isparticularlyadmired;oneoftheleadinguniversitiesinmodernGreeceisnamedDemocritusUniversity.Indeed,theefforttoidentifythefundamentalconstituentsofmattercontinuedformillennia,thoughwithchangesfromtimetotimein themenu of elements. By earlymodern times alchemists had identified three supposed elements:mercury, salt, and sulfur. Themodern idea of chemical elements dates from the chemical revolutioninstigated by Priestley, Lavoisier, Dalton, and others at the end of the eighteenth century, and now

incorporates92naturallyoccurringelements,fromhydrogentouranium(includingmercuryandsulfurbut not salt) plus a growing list of artificially created elements heavier than uranium.Under normalconditions, a pure chemical element consists of atoms all of the same type, and the elements aredistinguishedfromoneanotherbythetypeofatomofwhichtheyarecomposed.Todaywelookbeyondthechemicalelementstotheelementaryparticlesofwhichatomsarecomposed,butonewayoranotherwecontinuethesearch,begunatMiletus,forthefundamentalconstituentsofnature.Nevertheless,IthinkoneshouldnotoveremphasizethemodernaspectsofArchaicorClassicalGreek

science.There is an important featureofmodern science that is almostcompletelymissing inall thethinkers I have mentioned, from Thales to Plato: none of them attempted to verify or even (asideperhaps fromZeno) seriously to justify their speculations. In reading their writings, one continuallywantstoask,“Howdoyouknow?”ThisisjustastrueofDemocritusasoftheothers.Nowhereinthefragments of his books that survive dowe see any effort to show thatmatter really is composed ofatoms.Plato’s ideas about the five elements give a good example of his insouciant attitude toward

justification.InTimaeus,hestartsnotwithregularpolyhedronsbutwithtriangles,whichheproposestojoin together to form the faces of the polyhedrons.What sort of triangles? Plato proposes that theseshouldbetheisoscelesrighttriangle,withangles45°,45°,and90°;andtherighttrianglewithangles30°,60°,and90°.Thesquarefacesofthecubicatomsofearthcanbeformedfromtwoisoscelesrighttriangles,andthetriangularfacesofthetetrahedral,octahedral,andicosahedralatomsoffire,air,andwater (respectively) can each be formed from two of the other right triangles. (The dodecahedron,whichmysteriouslyrepresents thecosmos,cannotbeconstructedinthisway.)Toexplainthischoice,PlatoinTimaeussays,“Ifanyonecantellusofabetterchoiceoftrianglefortheconstructionofthefourbodies,hiscriticismwillbewelcome;butforourpartweproposetopassoveralltherest....Itwouldbe too long a story to give the reason, but if anyone can produce a proof that it is not so we willwelcomehisachievement.”13 I can imagine the reaction today if I supported anewconjecture aboutmatter in a physics article by saying that it would take too long to explain my reasoning, andchallengingmycolleaguestoprovetheconjectureisnottrue.Aristotle called the earlier Greek philosophers physiologi, and this is sometimes translated as

“physicists,”14 but that ismisleading.Thewordphysiologi simplymeans students of nature (physis),and the early Greeks had very little in commonwith today’s physicists. Their theories had no bite.Empedoclescouldspeculateabouttheelements,andDemocritusaboutatoms,buttheirspeculationsledto no new information about nature—and certainly to nothing that would allow their theories to betested.ItseemstomethattounderstandtheseearlyGreeks,itisbettertothinkofthemnotasphysicistsor

scientistsorevenphilosophers,butaspoets.IshouldbeclearaboutwhatImeanbythis.Thereisanarrowsenseofpoetry,aslanguagethatuses

verbaldevices likemeter, rhyme,oralliteration.Evenin thisnarrowsense,Xenophanes,Parmenides,andEmpedocles allwrote in poetry.After theDorian invasions and the breakup of theBronzeAgeMycenaean civilization in the twelfth centuryBC, theGreeks had become largely illiterate.Withoutwriting,poetryisalmosttheonlywaythatpeoplecancommunicatetolatergenerations,becausepoetrycanberemembered inawaythatprosecannot.LiteracyrevivedamongtheGreekssometimearound700BC,butthenewalphabetborrowedfromthePhoenicianswasfirstusedbyHomerandHesiodtowritepoetry,someofitthelong-rememberedpoetryoftheGreekdarkages.Prosecamelater.Even the early Greek philosophers who wrote in prose, like Anaximander, Heraclitus, and

Democritus,adoptedapoeticstyle.CicerosaidofDemocritusthathewasmorepoeticthanmanypoets.Platowhenyounghadwantedtobeapoet,andthoughhewroteproseandwashostiletopoetryintheRepublic,hisliterarystylehasalwaysbeenwidelyadmired.

Ihaveinmindherepoetryinabroadersense:languagechosenforaestheticeffect,ratherthaninanattempttosayclearlywhatoneactuallybelievestobetrue.WhenDylanThomaswrites,“Theforcethatthrough the green fuse drives the flower drives my green age,” we do not regard this as a seriousstatementabouttheunificationoftheforcesofbotanyandzoology,andwedonotseekverification;we(oratleastI)takeitratherasanexpressionofsadnessaboutageanddeath.AttimesitseemsclearthatPlatodidnotintendtobetakenliterally.Oneexamplementionedaboveis

hisextraordinarilyweakargumentforthechoicehemadeoftwotrianglesasthebasisofallmatter.Asan even clearer example, in the Timaeus Plato introduced the story of Atlantis, which supposedlyflourishedthousandsofyearsbeforehisowntime.Platocouldnotpossiblyhaveseriouslythoughtthathereallyknewanythingaboutwhathadhappenedthousandsofyearsearlier.Idon’tatallmeantosaythattheearlyGreeksdecidedtowritepoeticallyinordertoavoidtheneed

tovalidatetheirtheories.Theyfeltnosuchneed.Todaywetestourspeculationsaboutnaturebyusingproposedtheoriestodrawmoreorlesspreciseconclusionsthatcanbetestedbyobservation.ThisdidnotoccurtotheearlyGreeks,ortomanyoftheirsuccessors,foraverysimplereason:theyhadneverseenitdone.Therearesignshereandtherethatevenwhentheydidwanttobetakenseriously,theearlyGreeks

had doubts about their own theories, that they felt reliable knowledge was unattainable. I used oneexample in my 1972 treatise on general relativity. At the head of a chapter about cosmologicalspeculation,IquotedsomelinesofXenophanes:“Andasforcertaintruth,nomanhasseenit,norwillthereeverbeamanwhoknowsaboutthegodsandaboutthethingsImention.Forifhesucceedstothefullinsayingwhatiscompletelytrue,hehimselfisneverthelessunawareofit,andopinionisfixedbyfateuponallthings.”15Inthesamevein,inOntheForms,Democritusremarked,“Weinrealityknownothing firmly” and“That in realitywedonotknowhoweach thing is or is nothasbeen shown inmanyways.”16Thereremainsapoeticelementinmodernphysics.Wedonotwriteinpoetry;muchofthewritingof

physicists barely reaches the level of prose. But we seek beauty in our theories, and use aestheticjudgmentsasaguideinourresearch.Someofusthinkthatthisworksbecausewehavebeentrainedbycenturiesofsuccessandfailureinphysicsresearchtoanticipatecertainaspectsof thelawsofnature,and through thisexperiencewehavecome to feel that these featuresofnature’s lawsarebeautiful.17Butwedonottakethebeautyofatheoryasconvincingevidenceofitstruth.Forexample, string theory,whichdescribes thedifferentspeciesofelementaryparticlesasvarious

modes of vibration of tiny strings, is very beautiful. It appears to be just barely consistentmathematically, so that its structure is not arbitrary, but largely fixed by the requirement ofmathematical consistency. Thus it has the beauty of a rigid art form—a sonnet or a sonata.Unfortunately,stringtheoryhasnotyetledtoanypredictionsthatcanbetestedexperimentally,andasaresulttheorists(atleastmostofus)arekeepinganopenmindastowhetherthetheoryactuallyappliesto the realworld. It is this insistence on verification thatwemostmiss in all the poetic students ofnature,fromThalestoPlato.

2

MusicandMathematics

Even if Thales and his successors had understood that from their theories ofmatter they needed toderive consequences that could be compared with observation, they would have found the taskprohibitivelydifficult, inpartbecauseof the limitationsofGreekmathematics.TheBabylonianshadachievedgreatcompetenceinarithmetic,usinganumbersystembasedon60ratherthan10.Theyhadalsodevelopedsomesimple techniquesofalgebra,suchas rules (though thesewerenotexpressed insymbols) for solving various quadratic equations.But for the earlyGreeks,mathematicswas largelygeometric.Aswe have seen,mathematicians byPlato’s time had already discovered theorems abouttrianglesandpolyhedrons.MuchofthegeometryfoundinEuclid’sElementswasalreadywellknownbefore the time of Euclid, around 300 BC. But even by then the Greeks had only a limitedunderstandingofarithmetic,letalonealgebra,trigonometry,orcalculus.Thephenomenon thatwasstudiedearliestusingmethodsofarithmeticmayhavebeenmusic.This

was the work of the followers of Pythagoras. A native of the Ionian island of Samos, PythagorasemigratedtosouthernItalyaround530BC.There,intheGreekcityofCroton,hefoundedacultthatlasteduntilthe300sBC.The word “cult” seems appropriate. The early Pythagoreans left no writings of their own, but

accordingtothestoriestoldbyotherwriters1thePythagoreansbelievedinthetransmigrationofsouls.Theyaresupposedtohavewornwhiterobesandforbiddentheeatingofbeans,becausethatvegetableresembled the human fetus. They organized a kind of theocracy, and under their rule the people ofCrotondestroyedtheneighboringcityofSybarisin510BC.What is relevant to the history of science is that the Pythagoreans also developed a passion for

mathematics. According to Aristotle’sMetaphysics,2 “the Pythagoreans, as they are called, devotedthemselvestomathematics:theywerethefirsttoadvancethisstudy,andhavingbeenbroughtupinit,theythoughtitsprinciplesweretheprinciplesofallthings.”Their emphasis onmathematicsmayhave stemmed from an observation aboutmusic.They noted

that in playing a stringed instrument, if two strings of equal thickness, composition, and tension arepluckedatthesametime,thesoundispleasantifthelengthsofthestringsareinaratioofsmallwholenumbers.Inthesimplestcase,onestringis justhalf thelengthof theother.Inmodernterms,wesaythatthesoundsofthesetwostringsareanoctaveapart,andwelabelthesoundstheyproducewiththesameletterofthealphabet.Ifonestringistwo-thirdsthelengthoftheother,thetwonotesproducedaresaidtoforma“fifth,”aparticularlypleasingchord.Ifonestringisthree-fourthsthelengthoftheother,theyproduceapleasantchordcalleda“fourth.”Bycontrast,ifthelengthsofthetwostringsarenotinaratioofsmallwholenumbers(forinstanceifthelengthofonestringis,say,100,000/314,159timesthelengthoftheother),ornotinaratioofwholenumbersatall,thenthesoundisjarringandunpleasant.We now know that there are two reasons for this, having to do with the periodicity of the soundproducedbythetwostringsplayedtogether,andthematchingoftheovertonesproducedbyeachstring(seeTechnicalNote3).None of thiswas understood by thePythagoreans, or indeed by anyone elseuntil the work of the French priest Marin Mersenne in the seventeenth century. Instead, thePythagoreansaccordingtoAristotlejudged“thewholeheaventobeamusicalscale.”3Thisideahadalong afterlife. For instance,Cicero, inOn theRepublic, tells a story inwhich the ghost of the great

RomangeneralScipioAfricanusintroduceshisgrandsontothemusicofthespheres.ItwasinpuremathematicsratherthaninphysicsthatthePythagoreansmadethegreatestprogress.

EveryonehasheardofthePythagoreantheorem,thattheareaofasquarewhoseedgeisthehypotenuseofarighttriangleequalsthesumoftheareasofthetwosquareswhoseedgesaretheothertwosidesofthetriangle.NooneknowswhichifanyofthePythagoreansprovedthistheorem,orhow.Itispossibletogiveasimpleproofbasedona theoryofproportions,a theorydueto thePythagoreanArchytasofTarentum,acontemporaryofPlato.(SeeTechnicalNote4.TheproofgivenasProposition46ofBookIofEuclid’sElementsismorecomplicated.)Archytasalsosolvedafamousoutstandingproblem:givenacube,usepurelygeometricmethodstoconstructanothercubeofpreciselytwicethevolume.ThePythagoreantheoremleddirectlytoanothergreatdiscovery:geometricconstructionscaninvolve

lengthsthatcannotbeexpressedasratiosofwholenumbers.Ifthetwosidesofarighttriangleadjacenttotherightangleeachhavealength(insomeunitsofmeasurement)equalto1,thenthetotalareaofthetwosquareswiththeseedgesis12+12=2,soaccordingtothePythagoreantheoremthelengthofthehypotenusemustbeanumberwhosesquareis2.Butitiseasytoshowthatanumberwhosesquareis2cannotbeexpressedasaratioofwholenumbers.(SeeTechnicalNote5.)TheproofisgiveninBookXof Euclid’sElements, andmentioned earlier byAristotle in hisPrior Analytics4 as an example of areductioadimpossibile,butwithoutgivingtheoriginalsource.ThereisalegendthatthisdiscoveryisduetothePythagoreanHippasus,possiblyofMetapontuminsouthernItaly,andthathewasexiledormurderedbythePythagoreansforrevealingit.Wemighttodaydescribethisasthediscoverythatnumberslikethesquarerootof2areirrational—

theycannotbeexpressedasratiosofwholenumbers.AccordingtoPlato,5itwasshownbyTheodorusofCyrenethatthesquarerootsof3,5,6, . . . ,15,17,etc.(thatis, thoughPlatodoesnotsayso,thesquarerootsofall thewholenumbersother thanthenumbers1,4,9,16,etc., thatare thesquaresofwholenumbers)areirrationalinthesamesense.ButtheearlyGreekswouldnothaveexpresseditthisway.Rather,asthetranslationofPlatohasit,thesidesofsquareswhoseareasare2,3,5,etc.,squarefeetare“incommensurate”withasinglefoot.TheearlyGreekshadnoconceptionofanybutrationalnumbers,soforthemquantitieslikethesquarerootof2couldbegivenonlyageometricsignificance,andthisconstraintfurtherimpededthedevelopmentofarithmetic.ThetraditionofconcernwithpuremathematicswascontinuedinPlato’sAcademy.Supposedlythere

was a sign over its entrance, saying that no one should enter who was ignorant of geometry. Platohimselfwas nomathematician, but hewas enthusiastic aboutmathematics, perhaps in part because,during the journey toSicily to tutorDionysius theYoungerofSyracuse,hehadmet thePythagoreanArchytas.Oneof themathematicians at theAcademywhohadagreat influenceonPlatowasTheaetetusof

Athens,whowasthetitlecharacterofoneofPlato’sdialoguesandthesubjectofanother.Theaetetusiscreditedwiththediscoveryofthefiveregularsolidsthat,aswehaveseen,providedabasisforPlato’stheoryoftheelements.Theproof*offeredinEuclid’sElementsthatthesearetheonlypossibleconvexregularsolidsmaybeduetoTheaetetus,andTheaetetusalsocontributedtothetheoryofwhataretodaycalledirrationalnumbers.ThegreatestHellenicmathematicianof the fourthcenturyBCwasprobablyEudoxusofCnidus, a

pupilofArchytasandacontemporaryofPlato.ThoughresidentmuchofhislifeinthecityofCnidusonthecoastofAsiaMinor,EudoxuswasastudentatPlato’sAcademy,andreturnedlatertoteachthere.NowritingsofEudoxussurvive,butheiscreditedwithsolvingagreatnumberofdifficultmathematicalproblems,suchasshowingthatthevolumeofaconeisone-thirdthevolumeofthecylinderwiththesamebase andheight. (I haveno ideahowEudoxus couldhavedone thiswithout calculus.)Buthisgreatest contribution tomathematicswas the introduction of a rigorous style, inwhich theorems arededucedfromclearlystatedaxioms.ItisthisstylethatwefindlaterinthewritingsofEuclid.Indeed,

manyofthedetailsinEuclid’sElementshavebeenattributedtoEudoxus.Thoughagreat intellectualachievement in itself, thedevelopmentofmathematicsbyEudoxusand

the Pythagoreans was a mixed blessing for natural science. For one thing, the deductive style ofmathematical writing, enshrined in Euclid’sElements, was endlessly imitated by workers in naturalscience,whereit isnotsoappropriate.Aswewillsee,Aristotle’swritingonnaturalscienceinvolveslittlemathematics,butattimesitsoundslikeaparodyofmathematicalreasoning,asinhisdiscussionofmotioninPhysics:“A,then,willmovethroughBinatimeC,andthroughD,whichisthinner,intimeE(ifthelengthofBisequaltoD),inproportiontothedensityofthehinderingbody.ForletBbewaterandDbeair.”6PerhapsthegreatestworkofGreekphysicsisOnFloatingBodiesbyArchimedes,tobediscussed in Chapter 4. This book is written like a mathematics text, with unquestioned postulatesfollowedby deduced propositions.Archimedeswas smart enough to choose the right postulates, butscientificresearchismorehonestlyreportedasatangleofdeduction,induction,andguesswork.More important than the question of style, though related to it, is a false goal inspired by

mathematics: to reach certain truth by the unaided intellect. In his discussion of the education ofphilosopherkingsintheRepublic,PlatohasSocratesarguethatastronomyshouldbedoneinthesamewayasgeometry.AccordingtoSocrates,lookingattheskymaybehelpfulasaspurtotheintellect,inthesamewaythatlookingatageometricdiagrammaybehelpfulinmathematics,butinbothcasesrealknowledge comes solely through thought. Socrates explains in theRepublic that “we should use theheavenlybodiesmerelyasillustrationstohelpusstudytheotherrealm,aswewouldifwewerefacedwithexceptionalgeometricfigures.”7Mathematicsis themeansbywhichwededucetheconsequencesofphysicalprinciples.Morethan

that,itistheindispensablelanguageinwhichtheprinciplesofphysicalscienceareexpressed.Itofteninspiresnewideasaboutthenaturalsciences,andinturntheneedsofscienceoftendrivedevelopmentsinmathematics.Theworkofatheoreticalphysicist,EdwardWitten,hasprovidedsomuchinsightintomathematicsthatin1990hewasawardedoneofthehighestawardsinmathematics,theFieldsMedal.But mathematics is not a natural science.Mathematics in itself, without observation, cannot tell usanything about the world. And mathematical theorems can be neither verified nor refuted byobservationoftheworld.Thiswasnotclear in theancientworld,nor indeedeveninearlymoderntimes.Wehaveseenthat

Plato and the Pythagoreans consideredmathematical objects such as numbers or triangles to be thefundamental constituents of nature, and we shall see that some philosophers regarded mathematicalastronomyasabranchofmathematics,notofnaturalscience.Thedistinctionbetweenmathematicsandscience isprettywellsettled. It remainsmysterious tous

whymathematics that is invented for reasonshavingnothing todowithnatureoften turnsout tobeuseful in physical theories. In a famous article,8 the physicist Eugene Wigner has written of “theunreasonable effectiveness of mathematics.” But we generally have no trouble in distinguishing theideasofmathematicsfromprinciplesofscience,principlesthatareultimatelyjustifiedbyobservationoftheworld.Whereconflictsnowsometimesarisebetweenmathematiciansandscientists,itisgenerallyoverthe

issueofmathematical rigor.Since theearlynineteenthcentury, researchers inpuremathematicshaveregarded rigor as essential; definitions and assumptionsmust beprecise, anddeductionsmust followwith absolute certainty. Physicists are more opportunistic, demanding only enough precision andcertaintytogivethemagoodchanceofavoidingseriousmistakes.Intheprefaceofmyowntreatiseonthequantumtheoryoffields,Iadmitthat“therearepartsofthisbookthatwillbringtearstotheeyesofthemathematicallyinclinedreader.”This leads to problems in communication. Mathematicians have told me that they often find the

literatureofphysicsinfuriatinglyvague.Physicistslikemyselfwhoneedadvancedmathematicaltools

oftenfindthatthemathematicians’searchforrigormakestheirwritingscomplicatedinwaysthatareoflittlephysicalinterest.Therehasbeenanobleeffortbymathematicallyinclinedphysiciststoputtheformalismofmodern

elementary particle physics—the quantum theory of fields—on amathematically rigorous basis, andsomeinterestingprogresshasbeenmade.ButnothinginthedevelopmentoverthepasthalfcenturyoftheStandardModelofelementaryparticleshasdependedon reachingahigher levelofmathematicalrigor.Greek mathematics continued to thrive after Euclid. In Chapter 4 we will come to the great

achievementsofthelaterHellenisticmathematiciansArchimedesandApollonius.

3

MotionandPhilosophy

AfterPlato, theGreeks’speculationsaboutnature tooka turn towardastyle thatwas lesspoeticandmoreargumentative.ThischangeappearsaboveallintheworkofAristotle.NeitheranativeAtheniannoranIonian,Aristotlewasbornin384BCatStagirainMacedon.HemovedtoAthensin367BCtostudyat theschoolfoundedbyPlato, theAcademy.After thedeathofPlatoin347BC,Aristotle leftAthensandlivedforawhileontheAegeanislandofLesbosandatthecoastaltownofAssos.In343BCAristotlewascalledbacktoMacedonbyPhilipII to tutorhissonAlexander, laterAlexander theGreat.MacedoncametodominatetheGreekworldafterPhilip’sarmydefeatedAthensandThebesat the

battleofChaeroneain338BC.AfterPhilip’sdeathin336BCAristotlereturnedtoAthens,wherehefoundedhisownschool,theLyceum.ThiswasoneofthefourgreatschoolsofAthens,theothersbeingPlato’s Academy, the Garden of Epicurus, and the Colonnade (or Stoa) of the Stoics. The Lyceumcontinued for centuries, probably until itwas closed in the sackofAthens byRoman soldiers underSullain86BC.Itwasoutlasted,though,byPlato’sAcademy,whichcontinuedinoneformoranotheruntilAD529,enduringlongerthananyEuropeanuniversityhaslastedsofar.TheworksofAristotle thatsurviveappear tobechieflynotesforhis lecturesat theLyceum.They

treatanamazingvarietyofsubjects:astronomy,zoology,dreams,metaphysics, logic,ethics, rhetoric,politics, aesthetics, andwhat is usually translated as “physics.”According toonemodern translator,1Aristotle’sGreekis“terse,compact,abrupt,hisargumentscondensed,histhoughtdense,”veryunlikethepoeticstyleofPlato.IconfessthatIfindAristotlefrequentlytedious,inawaythatPlatoisnot,butalthoughoftenwrongAristotleisnotsilly,inthewaythatPlatosometimesis.PlatoandAristotlewerebothrealists,butinquitedifferentsenses.Platowasarealistinthemedieval

senseoftheword:hebelievedintherealityofabstractideas,inparticularofidealformsofthings.Itistheidealformofapinetreethatisreal,nottheindividualpinetreesthatonlyimperfectlyrealizethisform.Itistheformsthatarechangeless,inthewaydemandedbyParmenidesandZeno.Aristotlewasarealistinacommonmodernsense:forhim,thoughcategoriesweredeeplyinteresting,itwasindividualthings,likeindividualpinetrees,thatwerereal,notPlato’sforms.Aristotlewas careful to use reason rather than inspiration to justifyhis conclusions.Wecan agree

withtheclassicalscholarR.J.Hankinsonthat“wemustnotlosesightofthefactthatAristotlewasaman of his time—and for that time he was extraordinarily perspicacious, acute, and advanced.”2Nevertheless,therewereprinciplesrunningallthroughAristotle’sthoughtthathadtobeunlearnedinthediscoveryofmodernscience.Foronething,Aristotle’sworkwassuffusedwithteleology:thingsarewhattheyarebecauseofthe

purposetheyserve.InPhysics,3weread,“Butthenatureistheendorthatforthesakeofwhich.Forifathingundergoesacontinuouschange towardsomeend, that last stage isactually that for thesakeofwhich.”Thisemphasison teleologywasnatural forsomeone likeAristotle,whowasmuchconcernedwith

biology.AtAssos andLesbosAristotlehad studiedmarinebiology, andhis father,Nicomachus, hadbeenaphysicianatthecourtofMacedon.FriendswhoknowmoreaboutbiologythanIdotellmethatAristotle’swritingonanimalsisadmirable.Teleologyisnaturalforanyonewho,likeAristotleinParts

of Animals, studies the heart or stomach of an animal—he can hardly help asking what purpose itserves.Indeed,notuntil theworkofDarwinandWallace in thenineteenthcenturydidnaturalistscameto

understand that although bodily organs serve various purposes, there is no purpose underlying theirevolution.Theyarewhattheyarebecausetheyhavebeennaturallyselectedovermillionsofyearsofundirected inheritablevariations.Andof course, longbeforeDarwin, physicists had learned to studymatterandforcewithoutaskingaboutthepurposetheyserve.Aristotle’searlyconcernwithzoologymayalsohaveinspiredhisstrongemphasisontaxonomy,on

sortingthingsoutincategories.Westillusesomeofthis,forinstancetheAristotelianclassificationofgovernments intomonarchies, aristocracies, andnotdemocraciesbut constitutionalgovernments.Butmuchofitseemspointless.IcanimaginehowAristotlemighthaveclassifiedfruits:Allfruitscomeinthreevarieties—thereareapples,andoranges,andfruitsthatareneitherapplesnororanges.OneofAristotle’sclassificationswaspervasiveinhiswork,andbecameanobstacleforthefutureof

science. He insisted on the distinction between the natural and the artificial. He begins Book II ofPhysics4with “Of things that exist, some exist by nature, some from other causes.” Itwas only thenatural thatwasworthy of his attention. Perhaps it was this distinction between the natural and theartificial that keptAristotle and his followers from being interested in experimentation.What is thegoodofcreatinganartificialsituationwhenwhatarereallyinterestingarenaturalphenomena?It is not that Aristotle neglected the observation of natural phenomena. From the delay between

seeinglightningandhearingthunder,orseeingoarsonadistanttriremestrikingthewaterandhearingthesoundtheymake,heconcludedthatsoundtravelsatafinitespeed.5Wewillseethathealsomadegooduseofobservation in reachingconclusionsabout the shapeof theEarthandabout thecauseofrainbows. But this was all casual observation of natural phenomena, not the creation of artificialcircumstancesforthepurposeofexperimentation.Thedistinctionbetween thenatural andartificial playeda large role inAristotle’s thought about a

problemofgreat importance in thehistoryof science—themotionof fallingbodies.Aristotle taughtthat solid bodies fall down because the natural place of the element earth is downward, toward thecenter of the cosmos, and sparks fly upwardbecause thenatural placeof fire is in theheavens.TheEarth is nearly a sphere,with its center at the center of the cosmos, because this allows thegreatestproportionofearth toapproach thatcenter.Also,allowedtofallnaturally,a fallingbodyhasaspeedproportional to itsweight.Aswe read inOn theHeavens,6 according toAristotle, “A givenweightmovesagivendistanceinagiventime;aweightwhichisasgreatandmoremovesthesamedistanceina less time, the timesbeing in inverseproportion to theweights.For instance, ifoneweight is twiceanother,itwilltakehalfaslongoveragivenmovement.”Aristotlecan’tbeaccusedofentirelyignoringtheobservationoffallingbodies.Thoughhedidnot

knowthereason,theresistanceofairoranyothermediumsurroundingafallingbodyhastheeffectthatthespeedeventuallyapproachesaconstantvalue, the terminalvelocity,whichdoes increasewith thefallingbody’sweight. (SeeTechnicalNote6.) Probablymore important toAristotle, the observationthatthespeedofafallingbodyincreaseswithitsweightfittedinwellwithhisnotionthatthebodyfallsbecausethenaturalplaceofitsmaterialistowardthecenteroftheworld.ForAristotle, thepresenceofairorsomeothermediumwasessential inunderstandingmotion.He

thoughtthatwithoutanyresistance,bodieswouldmoveatinfinitespeed,anabsurditythatledhimtodenythepossibilityofemptyspace.InPhysics,heargues,“Letusexplainthatthereisnovoidexistingseparately, as somemaintain.”7 But in fact it is only the terminal velocity of a falling body that isinverselyproportionaltotheresistance.Theterminalvelocitywouldindeedbeinfiniteintheabsenceofallresistance,butinthatcaseafallingbodywouldneverreachterminalvelocity.

In the same chapterAristotle gives amore sophisticated argument, that in a void therewould benothingtowhichmotioncouldberelative:“inthevoidthingsmustbeatrest;forthereisnoplacetowhich things canmovemoreor less than to another; since thevoid in so far as it is void admitsnodifference.”8But this isanargumentagainstonlyan infinitevoid;otherwisemotion inavoidcanberelativetowhateverisoutsidethevoid.BecauseAristotlewasacquaintedwithmotiononlyinthepresenceofresistance,hebelievedthatall

motionhasacause.*(Aristotledistinguishedfourkindsofcause:material,formal,efficient,andfinal,of which the final cause is teleological—it is the purpose of the change.) That causemust itself becaused by something else, and so on, but the sequence of causes cannot go on forever.We read inPhysics,9“Sinceeverythingthatisinmotionmustbemovedbysomething,letustakethecaseinwhichathingisinlocomotionandismovedbysomethingthatisitselfinmotion,andthatagainismovedbysomething else that is inmotion, and that by something else, and so on continually; then the seriescannot go on to infinity, but there must be some first mover.” The doctrine of a first mover laterprovidedChristianityandIslamwithanargumentfortheexistenceofGod.Butaswewillsee,intheMiddleAgestheconclusionthatGodcouldnotmakeavoidraisedtroublesforfollowersofAristotleinbothIslamandChristianity.Aristotlewasnotbotheredbythefactthatbodiesdonotalwaysmovetowardtheirnaturalplace.A

stoneheldinthehanddoesnotfall,butforAristotlethisjustshowedtheeffectofartificialinterferencewiththenaturalorder.Buthewasseriouslyworriedoverthefactthatastonethrownupwardcontinuesfor a while to rise, away from the Earth, even after it has left the hand. His explanation, really noexplanation,wasthatthestonecontinuesupwardforawhilebecauseofthemotiongiventoitbytheair.InBookIIIofOntheHeavens,heexplainsthat“theforcetransmitsthemovementtothebodybyfirst,asitwere,tyingitupintheair.Thatiswhyabodymovedbyconstraintcontinuestomoveevenwhenthatwhichgave it the impulseceases toaccompany it.”10Aswewill see, thisnotionwas frequentlydiscussedandrejectedinancientandmedievaltimes.Aristotle’s writing on falling bodies is typical at least of his physics—elaborate though non-

mathematicalreasoningbasedonassumedfirstprinciples,whicharethemselvesbasedononlythemostcasualobservationofnature,withnoefforttotestthem.I don’t mean to say that Aristotle’s philosophy was seen by his followers and successors as an

alternative to science. There was no conception in the ancient or medieval world of science assomethingdistinct fromphilosophy.Thinkingabout thenaturalworldwas philosophy.As late as thenineteenthcentury,whenGermanuniversities institutedadoctoraldegree forscholarsof theartsandsciencestogivethemequalstatuswithdoctorsoftheology,law,andmedicine,theyinventedthetitle“doctorofphilosophy.”Whenphilosophyhadearlierbeencomparedwithsomeotherwayofthinkingaboutnature,itwascontrastednotwithscience,butwithmathematics.NooneinthehistoryofphilosophyhasbeenasinfluentialasAristotle.AswewillseeinChapter9,

hewasgreatlyadmiredbysomeArabphilosophers,evenslavishlysobyAverroes.Chapter10tellshowAristotle became influential inChristianEurope in the1200s,whenhis thoughtwas reconciledwithChristianity by Thomas Aquinas. In the high Middle Ages Aristotle was known simply as “ThePhilosopher,” andAverroes as “TheCommentator.”AfterAquinas the studyofAristotle became thecenterofuniversityeducation.IntheProloguetoChaucer’sCanterburyTales,weareintroducedtoanOxfordscholar:

AClerktherewasofOxenfordalso...Forhewouldratherhaveathisbed’sheadTwentybooks,cladinblackorred,OfAristotle,andhisphilosophy,

Thanrobesrich,orfiddle,orgaypsaltery.

Ofcourse, thingsaredifferentnow.Itwasessential in thediscoveryofscience toseparatesciencefromwhatisnowcalledphilosophy.Thereisactiveandinterestingworkonthephilosophyofscience,butithasverylittleeffectonscientificresearch.TheprecociousscientificrevolutionthatbeganinthefourteenthcenturyandisdescribedinChapter

10 was largely a revolt against Aristotelianism. In recent years students of Aristotle have mountedsomethingofacounterrevolution.Thevery influentialhistorianThomasKuhndescribedhowhewasconvertedfromdisparagementtoadmirationofAristotle:11

Aboutmotion,inparticular,hiswritingsseemedtomefullofegregiouserrors,bothoflogicandofobservation. These conclusions were, I felt, unlikely. Aristotle, after all, had been the much-admired codifier of ancient logic.For almost twomillennia after his death, hisworkplayed thesameroleinlogicthatEuclid’splayedingeometry. . . .Howcouldhischaracteristictalenthavedeserted him so systematicallywhen he turned to the study ofmotion andmechanics?Equally,whyhadhiswritingsinphysicsbeentakensoseriouslyforsomanycenturiesafterhisdeath?...Suddenlythefragmentsinmyheadsortedthemselvesoutinanewway,andfellinplacetogether.Myjawdroppedwithsurprise,forallatonceAristotleseemedaverygoodphysicistindeed,butofasortI’dneverdreamedpossible....IhadsuddenlyfoundthewaytoreadAristoteliantexts.

I heard Kuhnmake these remarks whenwe both received honorary degrees from the University ofPadua, and later asked him to explain. He replied, “What was altered by my own first reading of[Aristotle’swritingsonphysics]wasmyunderstanding,notmyevaluation,ofwhat theyachieved.” Ididn’tunderstandthis:“averygoodphysicistindeed”seemedtomelikeanevaluation.Regarding Aristotle’s lack of interest in experiment: the historian David Lindberg12 remarked,

“Aristotle’sscientificpracticeisnottobeexplained,therefore,asaresultofstupidityordeficiencyonhispart—failuretoperceiveanobviousproceduralimprovement—butasamethodcompatiblewiththeworldasheperceiveditandwellsuitedtothequestionsthatinteresthim.”OnthelargerissueofhowtojudgeAristotle’ssuccess,Lindbergadded,“ItwouldbeunfairandpointlesstojudgeAristotle’ssuccessbythedegreetowhichheanticipatedmodernscience(asthoughhisgoalwastoanswerourquestions,rather than his own).” And in a second edition of the same work:13 “The proper measure of aphilosophicalsystemorascientifictheoryisnotthedegreetowhichitanticipatedmodernthought,butitsdegreeofsuccessintreatingthephilosophicalandscientificproblemsofitsownday.”Idon’tbuyit.Whatisimportantinscience(Ileavephilosophytoothers)isnotthesolutionofsome

popularscientificproblemsofone’sownday,butunderstandingtheworld.Inthecourseofthiswork,one finds out what sort of explanations are possible, and what sort of problems can lead to thoseexplanations.Theprogressofsciencehasbeenlargelyamatterofdiscoveringwhatquestionsshouldbeasked.Ofcourse,onehastotrytounderstandthehistoricalcontextofscientificdiscoveries.Beyondthat,

thetaskofahistoriandependsonwhatheorsheistryingtoaccomplish.Ifthehistorian’saimisonlytore-create the past, to understand “how it actually was,” then it may not be helpful to judge a pastscientist’ssuccessbymodernstandards.Butthissortofjudgmentisindispensableifwhatonewantsistounderstandhowscienceprogressedfromitspasttoitspresent.Thisprogresshasbeensomethingobjective,notjustanevolutionoffashion.Isitpossibletodoubt

thatNewtonunderstoodmoreaboutmotionthanAristotle,orthatweunderstandmorethanNewton?Itnever was fruitful to ask what motions are natural, or what is the purpose of this or that physicalphenomenon.

IagreewithLindbergthatitwouldbeunfairtoconcludethatAristotlewasstupid.Mypurposehereinjudgingthepastbythestandardsofthepresentistocometoanunderstandingofhowdifficultitwasfor evenvery intelligentpersons likeAristotle to learnhow to learn aboutnature.Nothingabout thepracticeofmodernscienceisobvioustosomeonewhohasneverseenitdone.Aristotle leftAthensat thedeathofAlexander in323BC, anddied shortly afterward, in322BC.

AccordingtoMichaelMatthews,14 thiswas“adeath thatsignaled the twilightofoneof thebrightestintellectualperiodsinhumanhistory.”ItwasindeedtheendoftheClassicalera,butasweshallsee,itwasalsothedawnofanagefarbrighterscientifically:theeraoftheHellenistic.

4

HellenisticPhysicsandTechnology

FollowingAlexander’sdeathhisempiresplitintoseveralsuccessorstates.Ofthese,themostimportantfor the history of sciencewasEgypt.Egyptwas ruled by a succession ofGreekkings, startingwithPtolemy I, who had been one of Alexander’s generals, and ending with Ptolemy XV, the son ofCleopatraand(perhaps)JuliusCaesar.ThislastPtolemywasmurderedsoonafterthedefeatofAntonyandCleopatraatActiumin31BC,whenEgyptwasabsorbedintotheRomanEmpire.This age, from Alexander to Actium,1 is commonly known as the Hellenistic period, a term (in

German,Hellenismus)coinedinthe1830sbyJohannGustavDroysen.Idon’tknowifthiswasintendedby Droysen, but to my ear there is something pejorative about the English suffix “istic.” Just as“archaistic,”forinstance,isusedtodescribeanimitationofthearchaic,thesuffixseemstoimplythatHellenistic culture was not fully Hellenic, that it was a mere imitation of the achievements of theClassicalageof the fifthandfourthcenturiesBC.Thoseachievementswereverygreat,especially ingeometry,drama,historiography,architecture,andsculpture,andperhapsinotherartswhoseClassicalproductions have not survived, such as music and painting. But in the Hellenistic age science wasbrought toheights thatnotonlydwarfed thescientificaccomplishmentsof theClassicalerabutwerenotmatcheduntilthescientificrevolutionofthesixteenthandseventeenthcenturies.ThevitalcenterofHellenisticsciencewasAlexandria,thecapitalcityofthePtolemies,laidoutby

AlexanderatonemouthoftheNile.AlexandriabecamethegreatestcityintheGreekworld;andlater,intheRomanEmpire,itwassecondonlytoRomeinsizeandwealth.Around300BCPtolemy I founded theMuseumofAlexandria, aspart ofhis royal palace. Itwas

originallyintendedasacenterofliteraryandphilologicalstudies,dedicatedtothenineMuses.ButaftertheaccessionofPtolemyIIin285BCtheMuseumalsobecameacenterofscientificresearch.LiterarystudiescontinuedattheMuseumandLibraryofAlexandria,butnowattheMuseumtheeightartisticMuseswereoutshonebytheironescientificsister—Urania,theMuseofastronomy.TheMuseumandGreek science outlasted the kingdom of the Ptolemies, and, as we shall see, some of the greatestachievements of ancient science occurred in the Greek half of the Roman Empire, and largely inAlexandria.The intellectual relations between Egypt and the Greek homeland in Hellenistic times were

something like theconnectionsbetweenAmericaandEurope in the twentiethcentury.2TherichesofEgyptandthegeneroussupportofatleastthefirstthreePtolemiesbroughttoAlexandriascholarswhohadmade their names inAthens, just as European scholars flocked toAmerica from the 1930s on.Starting around 300 BC, a formermember of the Lyceum,Demetrius of Phaleron, became the firstdirectoroftheMuseum,bringinghislibrarywithhimfromAthens.AtaroundthesametimeStratoofLampsacus,anothermemberoftheLyceum,wascalledtoAlexandriabyPtolemyItoserveastutortohis son, and may have been responsible for the turn of theMuseum toward science when that sonsucceededtothethroneofEgypt.The sailing time between Athens and Alexandria during the Hellenistic and Roman periods was

similar to the time it took for a steamship to go between Liverpool andNewYork in the twentiethcentury,andtherewasagreatdealofcomingandgoingbetweenEgyptandGreece.Forinstance,StratodidnotstayinEgypt;hereturnedtoAthenstobecomethethirddirectoroftheLyceum.

Stratowasaperceptiveobserver.Hewasabletoconcludethatfallingbodiesacceleratedownward,by observing howdrops ofwater falling froma roof become farther apart as they fall, a continuousstreamofwaterbreakingup intoseparatingdrops.This isbecause thedrops thathave fallen farthesthavealsobeenfallinglongest,andsincetheyareacceleratingthismeansthattheyaretravelingfasterthandropsfollowingthem,whichhavebeenfallingforashorter time.(SeeTechnicalNote7.)Stratonotedalsothatwhenabodyfallsaveryshortdistancetheimpactonthegroundisnegligible,butifitfallsfromagreatheightitmakesapowerfulimpact,showingthatitsspeedincreasesasitfalls.3It is probably no coincidence that centers ofGreek natural philosophy likeAlexandria aswell as

Miletus and Athens were also centers of commerce. A lively market brings together people fromdifferent cultures, and relieves the monotony of agriculture. The commerce of Alexandria was far-ranging:seabornecargoesbeingtakenfromIndiatotheMediterraneanworldwouldcrosstheArabianSea,gouptheRedSea,thengooverlandtotheNileanddowntheNiletoAlexandria.ButthereweregreatdifferencesintheintellectualclimatesofAlexandriaandAthens.Foronething,

the scholars of the Museum generally did not pursue the kind of all-embracing theories that hadpreoccupied theGreeks fromThales toAristotle.AsFlorisCohenhas remarked,4 “Athenian thoughtwascomprehensive,Alexandrianpiecemeal.”TheAlexandriansconcentratedonunderstandingspecificphenomena, where real progress could be made. These topics included optics and hydrostatics, andaboveallastronomy,thesubjectofPartII.ItwasnofailingoftheHellenisticGreeksthattheyretreatedfromtheefforttoformulateageneral

theory of everything. Again and again, it has been an essential feature of scientific progress tounderstandwhichproblemsareripeforstudyandwhicharenot.Forinstance,leadingphysicistsattheturn of the twentieth century, including Hendrik Lorentz andMaxAbraham, devoted themselves tounderstanding the structure of the recently discovered electron. It was hopeless; no one could havemade progress in understanding the nature of the electron before the advent of quantummechanicssome two decades later. The development of the special theory of relativity byAlbert EinsteinwasmadepossiblebyEinstein’s refusal toworryaboutwhatelectronsare. Insteadheworriedabouthowobservations of anything (including electrons) depend on themotion of the observer. Then Einsteinhimselfinhislateryearsaddressedtheproblemoftheunificationoftheforcesofnature,andmadenoprogressbecausenooneatthetimeknewenoughabouttheseforces.Another important differencebetweenHellenistic scientists and theirClassical predecessors is that

theHellenisticerawaslessafflictedbyasnobbishdistinctionbetweenknowledgeforitsownsakeandknowledge for use—inGreek, episteme versus techne (or in Latin, scientia versus ars). Throughouthistory,many philosophers have viewed inventors inmuch the sameway that the court chamberlainPhilostrateinAMidsummerNight’sDreamdescribedPeterQuinceandhisactors:“Hard-handedmen,whoworknowinAthens,andneveryetlabor’dwiththeirminds.”Asaphysicistwhoseresearchisonsubjects like elementary particles and cosmology that have no immediate practical application, I amcertainlynotgoingtosayanythingagainstknowledgeforitsownsake,butdoingscientificresearchtofillhumanneedshasawonderfulwayofforcingthescientisttostopversifyingandtoconfrontreality.5Of course, people have been interested in technological improvement since early humans learned

howtousefiretocookfoodandhowtomakesimpletoolsbybangingonestoneonanother.ButthepersistentintellectualsnobberyoftheClassicalintelligentsiakeptphilosopherslikePlatoandAristotlefromdirectingtheirtheoriestowardtechnologicalapplications.Though this prejudice did not disappear in Hellenistic times, it became less influential. Indeed,

people,eventhoseofordinarybirth,couldbecomefamousasinventors.AgoodexampleisCtesibiusofAlexandria,abarber’sson,whoaround250BCinventedsuctionandforcepumpsandawaterclockthat kept timemore accurately than earlierwater clocks by keeping a constant level ofwater in thevessel fromwhich thewater flowed.Ctesibiuswas famous enough to be remembered two centuries

laterbytheRomanVitruviusinhistreatiseOnArchitecture.ItisimportantthatsometechnologyintheHellenisticagewasdevelopedbyscholarswhowerealso

concernedwithsystematicscientificinquiries,inquiriesthatweresometimesthemselvesusedinaidoftechnology. For instance, Philo of Byzantium,who spent time inAlexandria around 250BC,was amilitaryengineerwhoinMechanicesyntaxismwroteaboutharbors,fortifications,sieges,andcatapults(workbasedinpartonthatofCtesibius).ButinPneumatics,PhiloalsogaveexperimentalargumentssupportingtheviewofAnaximenes,Aristotle,andStratothatairisreal.Forinstance,ifanemptybottleis submerged inwaterwith itsmouthopenbut facingdownward,nowaterwill flow into it,becausethereisnowherefortheairinthebottletogo;butifaholeisopenedsothatairisallowedtoleavethebottle,thenwaterwillflowinandfillthebottle.6TherewasonescientificsubjectofpracticalimportancetowhichGreekscientistsreturnedagainand

again, even into theRomanperiod: thebehaviorof light.This concerndates to thebeginningof theHellenisticera,withtheworkofEuclid.LittleisknownofthelifeofEuclid.HeisbelievedtohavelivedinthetimeofPtolemyI,andmay

have founded the study of mathematics at theMuseum in Alexandria. His best-known work is theElements,7whichbeginswithanumberofgeometricdefinitions,axioms,andpostulates,andmovesontomoreorlessrigorousproofsofincreasinglysophisticatedtheorems.ButEuclidalsowrotetheOptics,whichdealswithperspective,andhisnameisassociatedwiththeCatoptrics,whichstudiesreflectionbymirrors,thoughmodernhistoriansdonotbelievethathewasitsauthor.Whenonethinksofit,thereissomethingpeculiaraboutreflection.Whenyoulookatthereflectionof

asmallobjectinaflatmirror,youseetheimageatadefinitespot,notspreadoutoverthemirror.Yettherearemanypathsonecandrawfromtheobjecttovariousspotsonthemirrorandthentotheeye.*Apparentlythereisjustonepaththatisactuallytaken,sothattheimageappearsattheonepointwherethis path strikes the mirror. But what determines the location of that point on the mirror? In theCatoptricsthereappearsafundamentalprinciplethatanswersthisquestion:theanglesthatalightraymakeswithaflatmirrorwhenitstrikesthemirrorandwhenitisreflectedareequal.Onlyonelightpathcansatisfythiscondition.Wedon’tknowwho in theHellenisticeraactuallydiscovered thisprinciple.Wedoknow, though,

thatsometimearoundAD60HeroofAlexandriainhisownCatoptricsgaveamathematicalproofoftheequal-anglesrule,basedontheassumptionthatthepathtakenbyalightrayingoingfromtheobjecttothemirrorandthentotheeyeoftheobserveristhepathofshortestlength.(SeeTechnicalNote8.)Bywayofjustificationforthisprinciple,Herowascontenttosayonly,“ItisagreedthatNaturedoesnothinginvain,norexertsherselfneedlessly.”8PerhapshewasmotivatedbytheteleologyofAristotle—everything happens for a purpose. But Hero was right; as we will see in Chapter 14, in theseventeenth centuryHuygenswas able to deduce the principle of shortest distance (actually shortesttime)fromthewavenatureoflight.ThesameHerowhoexploredthefundamentalsofopticsusedthatknowledgetoinventaninstrumentofpracticalsurveying,thetheodolite,andalsoexplainedtheactionofsiphonsanddesignedmilitarycatapultsandaprimitivesteamengine.ThestudyofopticswascarriedfurtheraboutAD150inAlexandriabythegreatastronomerClaudius

Ptolemy(nokinofthekings).HisbookOpticssurvivesinaLatintranslationofalostArabicversionofthe lost Greek original (or perhaps of a lost Syriac intermediary). In this book Ptolemy describedmeasurements that verified the equal-angles rule of Euclid and Hero. He also applied this rule toreflectionbycurvedmirrors,ofthesortonefindstodayinamusementparks.Hecorrectlyunderstoodthatreflectionsinacurvedmirrorarejustthesameasifthemirrorwereaplane,tangenttotheactualmirroratthepointofreflection.InthefinalbookofOpticsPtolemyalsostudiedrefraction,thebendingoflightrayswhentheypass

fromonetransparentmediumsuchasairtoanothertransparentmediumsuchaswater.Hesuspendeda

disk, marked with measures of angle around its edge, halfway in a vessel of water. By sighting asubmergedobjectalongatubemountedonthedisk,hecouldmeasuretheanglesthattheincidentandrefractedraysmakewiththenormaltothesurface,alineperpendiculartothesurface,withanaccuracyranging froma fractionofadegree toa fewdegrees.9Aswewill see inChapter13, thecorrect lawrelatingtheseangleswasworkedoutbyFermatintheseventeenthcenturybyasimpleextensionoftheprinciplethatHerohadappliedtoreflection:inrefraction,thepathtakenbyarayoflightthatgoesfromtheobject to theeye isnot theshortest,but theone that takes the least time.Thedistinctionbetweenshortest distance and least time is irrelevant for reflection, where the reflected and incident ray arepassing through thesamemedium,anddistance issimplyproportional to time;but itdoesmatter forrefraction,wherethespeedoflightchangesastheraypassesfromonemediumtoanother.ThiswasnotunderstoodbyPtolemy; thecorrect lawof refraction,knownasSnell’s law (or inFrance,Descartes’law),wasnotdiscoveredexperimentallyuntiltheearly1600sAD.Themost impressive of the scientist-technologists of theHellenistic era (or perhaps any era)was

Archimedes.Archimedeslivedinthe200sBCintheGreekcityofSyracuseinSicily,butisbelievedtohavemadeatleastonevisittoAlexandria.Heiscreditedwithinventingvarietiesofpulleysandscrews,andvariousinstrumentsofwar,suchasa“claw,”basedonhisunderstandingofthelever,withwhichships lying at anchor near shore couldbe seized and capsized.One inventionused in agriculture forcenturieswasalargescrew,bywhichwatercouldbeliftedfromstreamstoirrigatefields.ThestorythatArchimedesusedcurvedmirrorsthatconcentratedsunlighttodefendSyracusebysettingRomanshipsonfireisalmostcertainlyafable,butitillustrateshisreputationfortechnologicalwizardry.InOntheEquilibriumofBodies,Archimedesworkedouttherulethatgovernsbalances:abarwith

weightsatbothendsisinequilibriumifthedistancesfromthefulcrumonwhichthebarreststobothendsareinverselyproportionaltotheweights.Forinstance,abarwithfivepoundsatoneendandonepoundattheotherendisinequilibriumifthedistancefromthefulcrumtotheone-poundweightisfivetimeslargerthanthedistancefromthefulcrumtothefive-poundweight.ThegreatestachievementofArchimedesinphysicsiscontainedinhisbookOnFloatingBodies.10

Archimedes reasoned that if some part of a fluidwas pressed down harder than another part by theweightoffluidorfloatingorsubmergedbodiesaboveit,thenthefluidwouldmoveuntilallpartswerepresseddownbythesameweight.Asheputit,

Let it be supposed that a fluid is of such a character that, the parts lying evenly and beingcontinuous,thatpartwhichisthrustthelessisdrivenalongbythatwhichisthrustthemore;andthateachofitspartsisthrustbythefluidwhichisaboveitinaperpendiculardirectionifthefluidbesunkinanythingandcompressedbyanythingelse.

FromthisArchimedesdeduced thata floatingbodywouldsink toa levelsuch that theweightof thewater displaced would equal its own weight. (This is why the weight of a ship is called its“displacement.”)Also,asolidbodythatistooheavytofloatandisimmersedinthefluid,suspendedbya cord from the arm of a balance, “will be lighter than its true weight by the weight of the fluiddisplaced.”(SeeTechnicalNote9.)Theratioofthetrueweightofabodyandthedecreaseinitsweightwhen it is suspended inwater thus gives the body’s “specific gravity,” the ratio of itsweight to theweightof thesamevolumeofwater.Eachmaterialhasacharacteristicspecificgravity: forgold it is19.32, for lead 11.34, and so on. Thismethod, deduced from a systematic theoretical study of fluidstatics,allowedArchimedestotellwhetheracrownwasmadeofpuregoldorgoldalloyedwithcheapermetals.ItisnotclearthatArchimedeseverputthismethodintopractice,butitwasusedforcenturiestojudgethecompositionofobjects.Even more impressive were Archimedes’ achievements in mathematics. By a technique that

anticipatedtheintegralcalculus,hewasabletocalculatetheareasandvolumesofvariousplanefiguresandsolidbodies.For instance, theareaofacircle isone-half thecircumferencetimes theradius(seeTechnicalNote10).Usinggeometricmethods,hewasabletoshowthatwhatwecallpi(Archimedesdid not use this term), the ratio of the circumference of a circle to its diameter, is between 31/7and310/71.CicerosaidthathehadseenonthetombstoneofArchimedesacylindercircumscribedaboutasphere,thesurfaceofthespheretouchingthesidesandbothbasesofthecylinder,likeasingletennisballjustfittingintoatincan.ApparentlyArchimedeswasmostproudofhavingprovedthatinthiscasethevolumeofthesphereistwo-thirdsthevolumeofthecylinder.There is an anecdote about the death of Archimedes, related by the Roman historian Livy.

Archimedes died in 212BCduring the sack of Syracuse byRoman soldiers underMarcusClaudiusMarcellus.(Syracusehadbeentakenoverbyapro-CarthaginianfactionduringtheSecondPunicWar.)As Roman soldiers swarmed over Syracuse, Archimedes was supposedly found by the soldier whokilledhim,whilehewasworkingoutaproblemingeometry.Aside from the incomparableArchimedes, the greatestHellenisticmathematicianwas his younger

contemporaryApollonius.Apolloniuswasbornaround262BCinPerga,acityonthesoutheastcoastofAsiaMinor,thenunderthecontroloftherisingkingdomofPergamon,buthevisitedAlexandriainthetimesofbothPtolemyIIIandPtolemyIV,whobetweenthemruledfrom247to203BC.Hisgreatworkwasonconicsections:theellipse,parabola,andhyperbola.Thesearecurvesthatcanbeformedbyaplaneslicing throughaconeatvariousangles.Much later, the theoryofconicsectionswascruciallyimportanttoKeplerandNewton,butitfoundnophysicalapplicationsintheancientworld.Brilliant work, but with its emphasis on geometry, there were techniques missing from Greek

mathematics that are essential in modern physical science. The Greeks never learned to write andmanipulatealgebraicformulas.FormulaslikeE=mc2andF=maareattheheartofmodernphysics.(FormulaswereusedinpurelymathematicalworkbyDiophantus,whoflourishedinAlexandriaaroundAD250, but the symbols inhis equationswere restricted to standing forwholeor rational numbers,quiteunlike the symbols in the formulasofphysics.)Evenwheregeometry is important, themodernphysicist tends to derive what is needed by expressing geometric facts algebraically, using thetechniquesofanalyticgeometryinventedintheseventeenthcenturybyRenéDescartesandothers,anddescribedinChapter13.PerhapsbecauseofthedeservedprestigeofGreekmathematics,thegeometricstylepersisteduntilwellintothescientificrevolutionoftheseventeenthcentury.WhenGalileoinhis1623bookTheAssayerwantedtosingthepraisesofmathematics,hespokeofgeometry:*“Philosophyiswritteninthisall-encompassingbookthatisconstantlyopentooureyes,thatistheuniverse;butitcannot be understood unless one first learns to understand the language and knows the characters inwhichitiswritten.Itiswritteninmathematicallanguage,anditscharactersaretriangles,circles,andothergeometrical figures;without these it ishumanly impossible tounderstandawordof it,andonewanders inadark labyrinth.”Galileowas somewhatbehind the times inemphasizinggeometryoveralgebra. His own writing uses some algebra, but is more geometric than that of some of hiscontemporaries,andfarmoregeometricthanwhatonefindstodayinphysicsjournals.Inmodern timesaplacehasbeenmadeforpurescience,sciencepursuedfor itsownsakewithout

regardtopracticalapplications.Intheancientworld,beforescientistslearnedthenecessityofverifyingtheirtheories,thetechnologicalapplicationsofsciencehadaspecialimportance,forwhenoneisgoingtousea scientific theory rather than just talkabout it, there isa largepremiumongetting it right. IfArchimedesbyhismeasurementsofspecificgravityhadidentifiedagildedleadcrownasbeingmadeofsolidgold,hewouldhavebecomeunpopularinSyracuse.Idon’twanttoexaggeratetheextenttowhichscience-basedtechnologywasimportantinHellenistic

orRomantimes.ManyofthedevicesofCtesibiusandHeroseemtohavebeennomorethantoys,or

theatricalprops.Historianshavespeculatedthatinaneconomybasedonslaverytherewasnodemandforlaborsavingdevices,suchasmighthavebeendevelopedfromHero’stoysteamengine.Militaryandcivilengineeringwereimportantintheancientworld,andthekingsinAlexandriasupportedthestudyof catapults and other artillery, perhaps at theMuseum, but thiswork does not seem to have gainedmuchfromthescienceofthetime.Theone area ofGreek science that did havegreat practical valuewas also the one thatwasmost

highlydeveloped.Itwasastronomy,towhichwewillturninPartII.

There is a large exception to the remark above that the existenceofpractical applicationsof scienceprovidedastrongincentivetogetthescienceright.It isthepracticeofmedicine.Untilmoderntimesthemosthighlyregardedphysicianspersistedinpractices, likebleeding,whosevaluehadneverbeenestablishedexperimentally,andthat infactdidmoreharmthangood.Wheninthenineteenthcenturythe reallyuseful techniqueof antisepsiswas introduced, a technique forwhich therewas a scientificbasis,itwasatfirstactivelyresistedbymostphysicians.Notuntilwellintothetwentiethcenturywereclinical trials required beforemedicines could be approved for use. Physicians did learn early on torecognizevariousdiseases, and for some theyhadeffective remedies, suchasPeruvianbark—whichcontains quinine—for malaria. They knew how to prepare analgesics, opiates, emetics, laxatives,soporifics, and poisons. But it is often remarked that until sometime around the beginning of thetwentiethcenturytheaveragesickpersonwoulddobetteravoidingthecareofphysicians.Itisnotthattherewasnotheorybehindthepracticeofmedicine.Therewashumorism,thetheoryof

thefourhumors—blood,phlegm,blackbile,andyellowbile,which(respectively)makeussanguine,phlegmatic,melancholy,orcholeric.HumorismwasintroducedinclassicalGreektimesbyHippocrates,or by colleagues of his whose writings were ascribed to him. As briefly statedmuch later by JohnDonnein“TheGoodMorrow,”thetheoryheldthat“whateverdieswasnotmixedequally.”ThetheoryofhumorismwasadoptedinRomantimesbyGalenofPergamon,whosewritingsbecameenormouslyinfluentialamongtheArabsandtheninEuropeafteraboutAD1000.Iamnotawareofanyeffortwhilehumorism was generally accepted ever to test its effectiveness experimentally. (Humorism survivestodayinAyurveda,traditionalIndianmedicine,butwithjustthreehumors:phlegm,bile,andwind.)In addition to humorism, physicians in Europe until modern times were expected to understand

anothertheorywithsupposedmedicalapplications:astrology.Ironically,theopportunityforphysiciansto study these theories atuniversitiesgavemedicaldoctorsmuchhigherprestige than surgeons,whoknewhowtodoreallyusefulthingslikesettingbrokenbonesbutuntilmoderntimeswerenotusuallytrainedinuniversities.Sowhydidthedoctrinesandpracticesofmedicinecontinuesolongwithoutcorrectionbyempirical

science?Ofcourse,progressisharderinbiologythaninastronomy.AswewilldiscussinChapter8,theapparentmotionsof theSun,Moon,andplanetsare so regular that itwasnotdifficult to see thatanearlytheorywasnotworkingverywell;andthisperceptionled,afterafewcenturies,toabettertheory.But if a patient dies despite the best efforts of a learned physician,who can saywhat is the cause?Perhapsthepatientwaitedtoolongtoseethedoctor.Perhapshedidnotfollowthedoctor’sorderswithsufficientcare.Atleasthumorismandastrologyhadanairofbeingscientific.Whatwasthealternative?Goingback

tosacrificinganimalstoAesculapius?Anotherfactormayhavebeentheextremeimportancetopatientsofrecoveryfromillness.Thisgave

physicians authorityover them, an authority that physicianshad tomaintain inorder to impose theirsupposedremedies.Itisnotonlyinmedicinethatpersonsinauthoritywillresistanyinvestigationthatmightreducetheirauthority.

5

AncientScienceandReligion

Thepre-SocraticGreekstookagreatsteptowardmodernsciencewhentheybegantoseekexplanationsofnaturalphenomenawithoutreferencetoreligion.Thisbreakwiththepastwasatbesttentativeandincomplete.AswesawinChapter1,DiogenesLaertiusdescribedthedoctrineofThalesasnotonlythat“water is theuniversalprimarysubstance,”butalso that“theworld isanimateandfullofdivinities.”Still, ifonly in the teachingsofLeucippusandDemocritus,abeginninghadbeenmade.Nowhere intheirsurvivingwritingsonthenatureofmatteristhereanymentionofthegods.Itwasessentialforthediscoveryofsciencethatreligiousideasbedivorcedfromthestudyofnature.

Thisdivorce tookmanycenturies,notbeing largelycomplete inphysical scienceuntil theeighteenthcentury,norinbiologyeventhen.It is not that the modern scientist makes a decision from the start that there are no supernatural

persons.Thathappenstobemyview,buttherearegoodscientistswhoareseriouslyreligious.Rather,theideaistoseehowfaronecangowithoutsupposingsupernaturalintervention.Onlyinthiswaycanwe do science, because once one invokes the supernatural, anything can be explained, and noexplanationcanbeverified.Thisiswhythe“intelligentdesign”ideologybeingpromotedtodayisnotscience—itisrathertheabdicationofscience.Plato’s speculations were suffused with religion. In Timaeus he described how a god placed the

planets in theirorbits, andhemayhave thought that theplanetsweredeities themselves.EvenwhenHellenicphilosophersdispensedwiththegods,someofthemdescribednatureintermsofhumanvaluesand emotions,which generally interested themmore than the inanimateworld.Aswe have seen, indiscussingchanges inmatter,Anaximander spokeof justice, andEmpedoclesof strife.Plato thoughtthattheelementsandotheraspectsofnaturewereworthstudyingnotfortheirownsake,butbecauseforhimtheyexemplifiedakindofgoodness,presentinthenaturalworldaswellasinhumanaffairs.His religionwas informedby thissense,asshownbyapassage fromtheTimaeus:“ForGoddesiredthat,sofaraspossible,all thingsshouldbegoodandnothingevil;wherefore,whenHetookoverallthat was visible, seeing that it was not in a state of rest but in a state of discordant and disorderlymotion,Hebrought it intoorderoutofdisorder,deeming that theformerstatewas inallwaysbetterthanthelatter.”1Today,wecontinuetoseekorderinnature,butwedonotthinkitisanorderrootedinhumanvalues.

Not everyone has been happy about this. The great twentieth-century physicist Erwin Schrödingerargued for a return to the exampleof antiquity,2with its fusionof science andhumanvalues. In thesamespirit,thehistorianAlexandreKoyréconsideredthepresentdivorceofscienceandwhatwenowcall philosophy “disastrous.”3Myownview is that this yearning for a holistic approach to nature ispreciselywhatscientistshavehadtooutgrow.Wesimplydonotfindanythinginthelawsofnaturethatinanywaycorrespondstoideasofgoodness,justice,love,orstrife,andwecannotrelyonphilosophyasareliableguidetoscientificexplanation.It is not easy to understand in justwhat sense the pagans actually believed in their own religion.

ThoseGreekswhohadtraveledorreadwidelyknewthatagreatvarietyofgodsandgoddesseswereworshippedinthecountriesofEurope,Asia,andAfrica.SomeoftheGreekstriedtoseetheseasthesamedeitiesunderdifferentnames.For instance, thepioushistorianHerodotus reported,not that the

nativeEgyptiansworshipped a goddess namedBubastuswho resembled theGreekgoddessArtemis,butratherthattheyworshippedArtemisunderthenameofBubastus.Otherssupposedthatthesedeitieswere all different and all real, and even included foreign gods in their own worship. Some of theOlympiangods,suchasDionysusandAphrodite,wereimportsfromAsia.AmongotherGreeks,however,themultiplicityofgodsandgoddessespromoteddisbelief.Thepre-

Socratic Xenophanes famously commented, “Ethiopians have gods with snub noses and black hair,Thraciansgodswithgray eyes and redhair,” and remarked, “But if oxen (andhorses) and lionshadhandsorcoulddrawwithhandsandcreateworksofart like thosemadebymen,horseswoulddrawpicturesofgodslikehorses,andoxenofgodslikeoxen,andtheywouldmakethebodies[oftheirgods]inaccordancewiththeformthateachspeciesitselfpossesses.”4IncontrasttoHerodotus,thehistorianThucydides showed no signs of religious belief. He criticized the Athenian general Nicias for adisastrousdecisiontosuspendanevacuationofhistroopsfromthecampaignagainstSyracusebecauseofalunareclipse.ThucydidesexplainedthatNiciaswas“over-inclinedtodivinationandsuchthings.”5SkepticismbecameespeciallycommonamongGreekswhoconcernedthemselveswithunderstanding

nature.Aswe have seen, the speculations ofDemocritus about atomswere entirely naturalistic.TheideasofDemocrituswereadoptedasanantidotetoreligion,firstbyEpicurusofSamos,whosettledinAthensandatthebeginningoftheHellenisticerafoundedtheAthenianschoolknownastheGarden.EpicurusinturninspiredtheRomanpoetLucretius.Lucretius’poemOntheNatureofThingsmolderedinmonastic librariesuntil its rediscoveryin1417,afterwhichithada large influenceinRenaissanceEurope. StephenGreenblatt6 has traced the impact of Lucretius onMachiavelli,More, Shakespeare,Montaigne,Gassendi,*Newton,andJefferson.Evenwherepaganismwasnotabandoned,therewasagrowingtendencyamongtheGreekstotakeitallegorically,asacluetohiddentruths.AsGibbonsaid,“TheextravaganceoftheGrecianmythologyproclaimed,withaclearandaudiblevoice,thatthepiousinquirer, insteadof being scandalizedor satisfiedwith the literal sense, shoulddiligently explore theoccultwisdom,whichhadbeendisguised,bytheprudenceofantiquity,underthemaskoffollyandoffable.”7Thesearch forhiddenwisdomled inRoman times to theemergenceof theschoolknown tomoderns as Neoplatonism, founded in the third century AD by Plotinus and his student Porphyry.Though not scientifically creative, the Neoplatonists retained Plato’s regard for mathematics; forinstance, Porphyrywrote a life of Pythagoras and a commentary on Euclid’sElements. Looking forhiddenmeaningsbeneathsurfaceappearancesisalargepartofthetaskofscience,soitisnotsurprisingthattheNeoplatonistsmaintainedatleastaninterestinscientificmatters.Paganswerenotmuchconcernedtopoliceeachother’sprivatebeliefs.Therewerenoauthoritative

writtensourcesofpaganreligiousdoctrineanalogoustotheBibleortheKoran.TheIliadandOdysseyandHesiod’sTheogonywereunderstoodasliterature,nottheology.Paganismhadplentyofpoetsandpriests,butithadnotheologians.Still,openexpressionsofatheismweredangerous.AtleastinAthensanaccusationofatheismwasoccasionallyusedasaweaponinpoliticaldebate,andphilosopherswhoexpresseddisbeliefinthepaganpantheoncouldfeelthewrathofthestate.Thepre-SocraticphilosopherAnaxagoraswasforcedtofleeAthensforteachingthattheSunisnotagodbutahotstone,largerthanthePeloponnesus.Plato in particularwas anxious to preserve the role of religion in the study of nature.Hewas so

appalledbythenontheisticteachingofDemocritusthathedecreedinBook10oftheLawsthatinhisidealsocietyanyonewhodeniedthatthegodswererealandthattheyintervenedinhumanaffairswouldbecondemnedtofiveyearsofsolitaryconfinement,withdeathtofollowiftheprisonerdidnotrepent.Inthisasinmuchelse,thespiritofAlexandriawasdifferentfromthatofAthens.Idonotknowof

anyHellenisticscientistswhosewritingsexpressedanyinterestinreligion,nordoIknowofanywhosufferedfortheirdisbelief.

ReligiouspersecutionwasnotunknownundertheRomanEmpire.Notthattherewasanyobjectiontoforeigngods.ThepantheonofthelaterRomanEmpireexpandedtoincludethePhrygianCybele,theEgyptianIsis,andthePersianMithras.Butwhateverelseonebelieved,itwasnecessaryasapledgeofloyalty to the state also to publicly honor the official Roman religion. According to Gibbon, thereligionsoftheRomanEmpire“wereallconsideredbythepeople,asequallytrue,bythephilosopher,asequallyfalse,andbythemagistrate,asequallyuseful.”8Christianswerepersecutednotbecausetheybelieved in Jehovah or Jesus, but because they publicly denied the Roman religion; they wouldgenerallybeexoneratediftheyputapinchofincenseonthealtaroftheRomangods.NoneofthisledtointerferencewiththeworkofGreekscientistsundertheempire.Hipparchusand

Ptolemywereneverpersecutedfor theirnontheistic theoriesof theplanets.ThepiouspaganemperorJuliancriticizedthefollowersofEpicurus,butdidnothingtopersecutethem.Though illegalbecauseof its rejectionof the state religion,Christianity spreadwidely through the

empireinthesecondandthirdcenturies.Itwasmadelegalintheyear313byConstantineI,andwasmade the sole legal religion of the empire by Theodosius I in 380. During those years, the greatachievementsofGreeksciencewerecomingtoanend.ThishasnaturallyledhistorianstoaskwhethertheriseofChristianityhadsomethingtodowiththedeclineoforiginalworkinscience.In the past attention centered on possible conflicts between the teachings of religion and the

discoveriesof science.For instance,CopernicusdedicatedhismasterpieceOn theRevolutionsof theHeavenlyBodiestoPopePaulIII,andinthededicationwarnedagainstusingpassagesofScripturetocontradict theworkof science.Hecitedas ahorrible example theviewsofLactantius, theChristiantutorofConstantine’seldestson:

Butifperchancetherearecertain“idletalkers”whotakeitonthemselvestopronouncejudgment,thoughwhollyignorantofmathematics,andifbyshamelesslydistortingthesenseofsomepassageinHolyWrittosuittheirpurpose,theydaretoreprehendandtoattackmywork;theyworrymesolittle that I shall evenscorn their judgmentsas foolhardy.For it isnotunknown thatLactantius,otherwiseadistinguishedwriterbuthardlyamathematician,speaksinanutterlychildishfashionconcerningtheshapeoftheEarth,whenhelaughsatthosewhosaidthattheEarthhastheformofaglobe.9

Thiswasnotquite fair.Lactantiusdidsay that itwas impossible for sky tobeunder theEarth.10Heargued that if theworldwere a sphere then therewouldhave tobepeople and animals living at theantipodes.Thisisabsurd;thereisnoreasonwhypeopleandanimalswouldhavetoinhabiteverypartofa spherical Earth. And what would be wrong if there were people and animals at the antipodes?Lactantiussuggeststhattheywouldtumbleinto“thebottompartofthesky.”HethenacknowledgesthecontraryviewofAristotle (notquotinghimbyname) that“it is thenatureof things forweight tobedrawntothecenter,”onlytoaccusethosewhoholdthisviewof“defendingnonsensewithnonsense.”Of course it is Lactantius who was guilty of nonsense, but contrary to what Copernicus suggested,LactantiuswasrelyingnotonScripture,butonlyonsomeextremelyshallowreasoningaboutnaturalphenomena.Allinall,Idon’tthinkthatthedirectconflictbetweenScriptureandscientificknowledgewasanimportantsourceoftensionbetweenChristianityandscience.Muchmore important, it seems to me, was the widespread view among the early Christians that

paganscienceisadistractionfromthethingsofthespiritthatoughttoconcernus.ThisgoesbacktotheverybeginningsofChristianity, toSaintPaul,whowarned:“Beware lest anymanspoilyou throughphilosophyandvaindeceit,after the traditionofmen,after the rudimentsof theworld,andnotafterChrist.”11 The most famous statement along these lines is due to the church father Tertullian, whoaround theyear200asked, “WhatdoesAthenshave todowith Jerusalem,or theAcademywith the

Church?”(TertullianchoseAthensandtheAcademytosymbolizeHellenicphilosophy,withwhichhepresumably was more familiar than he was with the science of Alexandria.) We find a sense ofdisillusion with pagan learning in the most important of the church fathers, Augustine of Hippo.AugustinestudiedGreekphilosophywhenyoung(thoughonlyinLatintranslations)andboastedofhisgraspofAristotle,buthelaterasked,“AndwhatdiditprofitmethatIcouldreadandunderstandallthebooksIcouldgetintheso-called‘liberalarts,’whenIwasactuallyaslaveofwickedlust?”12AugustinewasalsoconcernedwithconflictsbetweenChristianityandpaganphilosophy.Toward theendofhislife, in426,he lookedbackathispastwriting,andcommented,“Ihavebeenrightlydispleased, too,withthepraisewithwhichIextolledPlatoorthePlatonistsortheAcademicphilosophersbeyondwhatwasproperforsuchirreligiousmen,especiallythoseagainstwhosegreaterrorsChristianteachingmustbedefended.”13Anotherfactor:Christianityofferedopportunitiesforadvancementinthechurchtointelligentyoung

men,someofwhommightotherwisehavebecomemathematiciansorscientists.Bishopsandpresbytersweregenerallyexempt from the jurisdictionof theordinarycivilcourts,and from taxation.Abishopsuch asCyril ofAlexandria orAmbrose ofMilan could exercise considerable political power,muchmorethanascholarattheMuseuminAlexandriaortheAcademyinAthens.Thiswassomethingnew.Underpaganismreligiousofficeshadgonetomenofwealthorpoliticalpower,ratherthanwealthandpower going to men of religion. For instance, Julius Caesar and his successors won the office ofsupremepontiff,notasarecognitionofpietyorlearning,butasaconsequenceoftheirpoliticalpower.Greeksciencesurvivedforawhileafter theadoptionofChristianity, thoughmostly in theformof

commentariesonearlierwork.ThephilosopherProclus,workinginthefifthcenturyattheNeoplatonicsuccessortoPlato’sAcademyinAthens,wroteacommentaryonEuclid’sElements,withsomeoriginalcontributions.InChapter8IwillhaveoccasiontoquotealatermemberoftheAcademy,Simplicius,forhisremarks,inacommentaryonAristotle,aboutPlato’sviewsonplanetaryorbits.Inthelate300stherewas Theon of Alexandria, who wrote a commentary on Ptolemy’s great work of astronomy, theAlmagest,andpreparedan improvededitionofEuclid.HisfamousdaughterHypatiabecameheadofthe city’sNeoplatonic school.A century later inAlexandria theChristian John of PhiloponuswrotecommentariesonAristotle, inwhichhetookissuewithAristotle’sdoctrinesconcerningmotion.Johnarguedthatthereasonbodiesthrownupwarddonotimmediatelyfalldownisnotthattheyarecarriedbytheair,asAristotlehadthought,butratherthatwhentheyarethrownbodiesaregivensomequalitythatkeepsthemmoving,ananticipationoflaterideasofimpetusormomentum.Buttherewerenomorecreative scientists or mathematicians of the caliber of Eudoxus, Aristarchus, Hipparchus, Euclid,Eratosthenes,Archimedes,Apollonius,Hero,orPtolemy.WhetherornotbecauseoftheriseofChristianity,sooneventhecommentatorsdisappeared.Hypatia

was killed in 415 by a mob, egged on by Bishop Cyril of Alexandria, though it is difficult to saywhetherthiswasforreligiousorpoliticalreasons.In529theemperorJustinian(whopresidedoverthereconquestofItalyandAfrica,thecodificationofRomanlaw,andthebuildingofthegreatchurchofSantaSophia inConstantinople)ordered theclosingof theNeoplatonicAcademyofAthens.On thisevent,thoughGibbonispredisposedagainstChristianity,heistooeloquentnottobequoted:

TheGothicarmswerelessfataltotheschoolsofAthensthantheestablishmentofanewreligion,whoseministerssupersededtheexerciseofreason,resolvedeveryquestionbyanarticleoffaith,andcondemnedtheinfidelorskeptictoeternalflames.Inmanyavolumeoflaboriouscontroversytheyespousedtheweaknessoftheunderstandingandthecorruptionoftheheart,insultedhumannatureinthesagesofantiquity,andproscribedthespiritofphilosophicalinquiry,sorepugnanttothedoctrine,oratleasttothetemper,ofahumblebeliever.14

TheGreekhalfof theRomanEmpiresurviveduntilAD1453,butasweshallseeinChapter9, longbeforethenthevitalcenterofscientificresearchhadmovedeast,toBaghdad.

PARTII

GREEKASTRONOMY

Thesciencethatintheancientworldsawthegreatestprogresswasastronomy.OnereasonisthatastronomicalphenomenaaresimplerthanthoseontheEarth’ssurface.Thoughtheancientsdidnotknowit,thenasnowtheEarthandtheotherplanetsmovedaroundtheSunonnearlycircularorbits,atnearlyconstantvelocities,undertheinfluenceofasingleforce—gravitation—andtheyspun on their axes at essentially constant rates. The same applied to the Moon in its motionaroundtheEarth.InconsequencetheSun,Moon,andplanetsappearedfromEarthtomoveinaregularandpredictablewaythatcouldbeandwasstudiedwithconsiderableprecision.The other special feature of ancient astronomy is that it was useful, in a way that ancient

physicsgenerallywasnot.TheusesofastronomyarediscussedinChapter6.Chapter7discusseswhat,flawedasitwas,canbeconsideredatriumphofHellenisticscience:

themeasurement of the sizes of the Sun,Moon, and Earth, and the distances to the Sun andMoon.Chapter8treatstheproblemposedbytheapparentmotionoftheplanets,aproblemthatcontinuedtoconcernastronomersthroughtheMiddleAges,andthateventuallyledtothebirthofmodernscience.

6

TheUsesofAstronomy1

Evenbeforethestartofhistory,theskymusthavebeencommonlyusedasacompass,aclock,andacalendar.ItcouldnothavebeendifficulttonoticethattheSunriseseverymorninginmoreorlessthesamedirection,thatduringthedayonecantellhowmuchtimethereisbeforenightfromtheheightoftheSuninthesky,andthathotweatherwillfollowthetimeofyearwhenthedaylastslongest.Weknow that the starswereused for similar purposesvery early inhistory.Around3000BC the

Egyptiansknew that thecrucialevent in theiragriculture, the floodingof theNile inJune,coincidedwiththeheliacalrisingofthestarSirius.(ThisisthedayintheyearwhenSiriusfirstbecomesvisiblejustbeforedawn;earlier intheyearit isnotvisibleatnight,andlater it isvisiblewellbeforedawn.)Homer,writing before 700BC, comparesAchilles to Sirius, which is high in the sky at the end ofsummer:“thatstar,whichcomesonintheautumnandwhoseconspicuousbrightnessfaroutshinesthestarsthatarenumberedinthenight’sdarkening,thestartheygivethenameofOrion’sDog,whichisbrightest among the stars, and yet is wrought as a sign of evil and brings on the great fever forunfortunatemortals.”2Alittlelater,thepoetHesiodinWorksandDaystoldfarmersthatgrapesarebestcut at theheliacal risingofArcturus, and thatplowing shouldbedoneat the cosmical settingof thePleiades constellation. (This is the day in the year when these stars first are seen to set just beforesunrise;earlierintheyeartheydonotsetatallbeforetheSuncomesup,andlatertheysetwellbeforedawn.) FollowingHesiod, calendars known asparamegmata, which gave the risings and settings ofconspicuousstars foreachday,becamewidelyusedbyGreekswhosecity-stateshadnoothersharedwayofidentifyingdates.Bywatchingthestarsatnight,notobscuredbythe lightofmoderncities,observers inmanyearly

civilizations could see clearly that,with a few exceptions (aboutwhichmore later), the stars alwaysremaininthesameplacesrelativetooneanother.Thisiswhyconstellationsdonotchangefromnighttonightorfromyeartoyear.Butthewholefirmamentofthese“fixed”starsseemstorevolveeachnightfromeasttowestaroundapointintheskythatisalwaysduenorth,andhenceisknownasthenorthcelestial pole. Inmodern terms, this is the point towardwhich the axis of the Earth extends if it iscontinuedfromtheEarth’snorthpoleoutintothesky.Thisobservationmadeitpossibleforstarstobeusedveryearlybymarinersforfindingdirectionsat

night.HomertellshowOdysseusonhiswayhometoIthacaistrappedbythenymphCalypsoonherislandinthewesternMediterranean,untilZeusordersCalypsotosendOdysseusonhisway.ShetellsOdysseustokeepthe“GreatBear,thatsomehavecalledtheWain...onhislefthandashecrossedthemain.”3TheBear,ofcourse,isUrsaMajor,theconstellationalsoknownastheWagon(orWain),andinmoderntimesastheBigDipper.UrsaMajorisnearthenorthcelestialpole.ThusinthelatitudeoftheMediterraneanUrsaMajornever sets (“wouldneverbathe inordip in theOcean stream,”asHomerputsit)andisalwaysmoreorlessinthenorth.WiththeBearonhisleft,Odysseuswouldkeepsailingeast,towardIthaca.SomeGreekslearnedtodobetterwithotherconstellations.AccordingtothebiographyofAlexander

theGreatbyArrian,althoughmostsailorsinhistimeusedUrsaMajortotellnorth,thePhoenicians,theace sailors of the ancientworld, usedUrsaMinor, a constellation that is not as conspicuous asUrsaMajorbutisclosertothenorthcelestialpole.ThepoetCallimachus,asquotedbyDiogenesLaertius,4

claimedthattheuseofUrsaMinorgoesbacktoThales.TheSunalsoseemsduringthedaytorevolvefromeasttowestaroundthenorthcelestialpole.Of

course,wecannotusuallyseestarsduringtheday,butHeraclitus5andperhapsothersbeforehimseemtohaverealizedthatthestarsarealwaysthere,thoughwiththeirlightblottedoutduringthedaybythelightoftheSun.Somestarscanbeseenjustbeforedawnorjustaftersunset,whenthepositionoftheSun in the sky is known, and from this it became clear that the Sun does not keep a fixed positionrelativetothestars.Rather,aswaswellknownveryearlyinBabylonandIndia,inadditiontoseemingtorevolvefromeasttowesteverydayalongwiththestars,theSunalsomoveseachyeararoundtheskyfromwesttoeastthroughapathknownasthezodiac,markedinorderbythetraditionalconstellationsAries,Taurus,Gemini,Cancer,Leo,Virgo,Libra,Scorpio,Sagittarius,Capricorn,Aquarius,andPisces.Aswewillsee,theMoonandplanetsalsotravelthroughthezodiac,thoughnotonpreciselythesamepaths.TheparticularpaththroughtheseconstellationsfollowedbytheSunisknownasthe“ecliptic.”Oncethezodiacwasunderstood,itwaseasytolocatetheSuninthebackgroundofstars.Justnotice

whatconstellationofthezodiacishighestintheskyatmidnight;theSunisintheconstellationofthezodiacthatisdirectlyopposite.Thalesissupposedtohavegiven365daysasthetimeittakesfortheSuntomakeonecompletecircuitofthezodiac.OnecanthinkofthefirmamentofstarsasarevolvingspheresurroundingtheEarth,withthenorth

celestialpoleabovetheEarth’snorthpole.Butthezodiacisnottheequatorofthissphere.Rather,asAnaximanderissupposedtohavediscovered,thezodiacistiltedby23½°withrespecttothecelestialequator, with Cancer and Gemini closest to the north celestial pole, and Capricorn and Sagittariusfarthestfromit.Inmodernterms,thistilt,whichisresponsiblefortheseasons,isduetothefactthattheaxisoftheEarth’srotationisnotperpendiculartotheplaneofitsorbit,whichisprettyclosetotheplaneinwhichalmostallobjectsinthesolarsystemmove,butistiltedfromtheperpendicularbyanangleof23½°;inthenorthernsummerorwintertheSunisrespectivelyinthedirectiontowardwhichorawayfromwhichtheEarth’snorthpoleistilted.Astronomy began to be a precise science with the introduction of a device known as a gnomon,

which allowed accurate measurements of the Sun’s apparent motions. The gnomon, credited by thefourth-centurybishopEusebiusofCaesarea toAnaximanderbutbyHerodotus to theBabylonians, issimplyaverticalpole,placedinalevelpatchofgroundopentotheSun’srays.Withthegnomon,onecanaccuratelytellwhenitisnoon;itisthemomentinthedaywhenSunishighest,sothattheshadowofthegnomonisshortest.AtnoonanywherenorthofthetropicstheSunisduesouth,andtheshadowofthegnomonthereforepointsduenorth,soonecanpermanentlymarkoutonthegroundthepointsofthe compass. The gnomon also provides a calendar. During the spring and summer the Sun risessomewhat northof east,whereasduring the autumnandwinter it comesup southof east.When theshadowofthegnomonatdawnpointsduewest,theSunisrisingdueeast,andthedatemustbeeitherthevernalequinox,whenwintergiveswaytospring;ortheautumnalequinox,whensummerendsandautumn begins. The summer and winter solstices are the days in the year when the shadow of thegnomonatnoon is respectivelyshortestor longest. (Asundial isdifferent fromagnomon; itspole isparalleltotheEarth’saxisratherthantotheverticaldirection,sothatitsshadowatagivenhourisinthesamedirectioneveryday.Thismakesasundialmoreusefulasaclock,butuselessasacalendar.)Thegnomonprovidesaniceexampleofanimportantlinkbetweenscienceandtechnology:anitem

of technology invented for practical purposes can open the way to scientific discoveries. With thegnomon,itwaspossibletomakeaprecisecountofthedaysineachseason,suchastheperiodfromoneequinoxtothenextsolstice,orfromthentothefollowingequinox.Inthisway,Euctemon,anAtheniancontemporaryofSocrates,discoveredthatthelengthsoftheseasonsarenotpreciselyequal.ThiswasnotwhatonewouldexpectiftheSungoesaroundtheEarth(ortheEartharoundtheSun)inacircleatconstantspeed,withtheEarth(ortheSun)atthecenter,inwhichcasetheseasonswouldbeofequal

length. Astronomers tried for centuries to understand the inequality of the seasons, but the correctexplanation of this and other anomalieswas not found until the seventeenth century,when JohannesKepler realized that theEarthmovesaround theSunonanorbit that iselliptical rather thancircular,withtheSunnotatthecenteroftheorbitbutofftoonesideatapointcalledafocus,andmovesataspeedthatincreasesanddecreasesastheEarthapproachesclosertoandrecedesfartherfromtheSun.TheMoonalsoseemstorevolvelikethestarseachnightfromeasttowestaroundthenorthcelestial

pole;andoverlongertimesitmoves, liketheSun,throughthezodiacfromwesttoeast,buttakingalittlemorethan27daysinsteadofayeartomakeafullcircleagainstthebackgroundofstars.BecausetheSunappearstomovethroughthezodiacinthesamedirection,thoughmoreslowly,theMoontakesabout 29½ days to return to the same position relative to the Sun. (Actually 29 days, 12 hours, 44minutes,and3seconds.)SincethephasesoftheMoondependontherelativepositionofitandtheSun,this intervalofabout29½days is the lunarmonth,* the timefromonenewmoon to thenext. ItwasnoticedearlythateclipsesoftheMoonoccuratfullmoonaboutevery18years,whentheMoon’spathagainstthebackgroundofstarscrossesthatoftheSun.*InsomerespectstheMoonprovidesamoreconvenientcalendarthantheSun.Observingthephaseof

theMoononanygivennight,onecaneasilytellapproximatelyhowmanydayshavepassedsincethelastnewmoon—muchmoreeasilythanonecanjudgethetimeofyearjustbylookingattheSun.Solunarcalendarswerecommonintheancientworld,andstillsurvive,forexampleforreligiouspurposesinIslam.Butofcourse,forpurposesofagricultureorsailingorwar,oneneedstoanticipatethechangesof seasons, and these are governed by the Sun.Unfortunately, there is not awhole number of lunarmonthsintheyear—theyearisapproximately11dayslongerthan12lunarmonths—sothedateofanysolsticeorequinoxwouldnotremainfixedinacalendarbasedonthephasesoftheMoon.Anotherfamiliarcomplication is that theyear itself isnotawholenumberofdays.This led to the

introduction, in the time of Julius Caesar, of a leap year every fourth year. But this created furtherproblems,becausetheyearisnotprecisely365¼days,but11minuteslonger.Countless efforts, far too many to go into here, have been made throughout history to construct

calendars that takeaccountof thesecomplications.Afundamentalcontributionwasmadearound432BCbyMetonofAthens,possiblyapartnerofEuctemon.Perhapsby theuseofBabylonian records,Metonnoticedthat19yearsisalmostprecisely235lunarmonths.Theydifferbyonly2hours.Soonecanmake a calendar covering19years rather thanoneyear, inwhichboth the timeof year and thephase of the Moon are correctly identified for each day. The calendar then repeats itself for everysuccessive19-yearperiod.Butthough19yearsisnearlyexactly235lunarmonths,itisaboutathirdofadaylessthan6,940days,soMetonhadtoprescribethataftereveryfew19-yearcyclesadaywouldbedroppedfromthecalendar.The effort of astronomers to reconcile calendars based on the Sun andMoon is illustrated by the

definitionofEaster.TheCouncilofNicaeainAD325decreedthatEastershouldbecelebratedonthefirstSundayfollowingthefirstfullmoonfollowingthevernalequinox.InthereignofTheodosiusIitwasdeclaredacapitalcrimetocelebrateEasteronthewrongday.UnfortunatelytheprecisedatewhenthevernalequinoxisactuallyobservedvariesfromplacetoplaceonthesurfaceoftheEarth.*ToavoidthehorrorofEasterbeingcelebratedondifferentdaysindifferentplaces,itwasnecessarytoprescribeadefinitedateforthevernalequinox,andalsoforthefirstfullmoonfollowingit.TheRomanchurchinlate antiquity adopted theMetonic cycle for this purpose, but the monastic communities of Irelandadopted an older Jewish 84-year cycle. The struggle in the seventh century between RomanmissionariesandIrishmonksforcontrolovertheEnglishchurchwaslargelyaconflictoverthedateofEaster.Until modern times the construction of calendars has been a major occupation of astronomers,

leadinguptotheadoptionofourmoderncalendarin1582undertheauspicesofPopeGregoryXIII.For

purposesofcalculatingthedateofEaster,thedateofthevernalequinoxisnowfixedtobeMarch21,but it isMarch21 as givenby theGregorian calendar in theWest andby the Julian calendar in theOrthodoxchurchesof theEast.SoEaster is still celebratedondifferentdays indifferentpartsof theworld.ThoughscientificastronomyfoundusefulapplicationsintheHellenicera,thisdidnotimpressPlato.

ThereisarevealingexchangeintheRepublicbetweenSocratesandhisfoilGlaucon.6Socratessuggeststhatastronomyshouldbeincludedintheeducationofphilosopherkings,andGlauconreadilyagrees:“Imean,it’snotonlyfarmersandsailorswhoneedtobesensitivetotheseasons,months,andphasesoftheyear;it’sjustasimportantformilitarypurposesaswell.”Socratescallsthisnaive.Forhim,thepointofastronomyisthat“studyingthiskindofsubjectcleansandre-ignitesaparticularmentalorgan. . .andthisorganisathousandtimesmoreworthpreservingthananyeye,sinceitistheonlyorganwhichcanseetruth.”ThisintellectualsnobberywaslesscommoninAlexandriathaninAthens,butitappearsforinstanceinthefirstcenturyAD,inthewritingofthephilosopherPhiloofAlexandria,whoremarksthat “thatwhich is appreciable by the intellect is at all times superior to thatwhich is visible to theoutwardsenses.”7Fortunately,perhapsunderthepressureofpracticalneeds,astronomerslearnednottorelyonintellectalone.

7

MeasuringtheSun,Moon,andEarth

OneofthemostremarkableachievementsofGreekastronomywasthemeasurementofthesizesoftheEarth,Sun,andMoon,andthedistancesoftheSunandMoonfromtheEarth.Itisnotthattheresultsobtainedwerenumericallyaccurate.Theobservationsonwhichthesecalculationswerebasedweretoocrudetoyieldaccuratesizesanddistances.Butforthefirsttimemathematicswasbeingcorrectlyusedtodrawquantitativeconclusionsaboutthenatureoftheworld.Inthiswork,itwasessentialfirsttounderstandthenatureofeclipsesoftheSunandMoon,andto

discover that theEarth is a sphere.Both theChristianmartyrHippolytus andAëtius, amuch-quotedphilosopher of uncertain date, credit the earliest understanding of eclipses toAnaxagoras, an IonianGreekbornaround500BCatClazomenae(nearSmyrna),whotaughtinAthens.1Perhapsrelyingonthe observation of Parmenides that the bright side of the Moon always faces the Sun, Anaxagorasconcluded,“ItistheSunthatendowstheMoonwithitsbrilliance.”2Fromthis,itwasnaturaltoinferthateclipsesoftheMoonoccurwhentheMoonpassesthroughtheEarth’sshadow.HeisalsosupposedtohaveunderstoodthateclipsesoftheSunoccurwhentheMoon’sshadowfallsontheEarth.On the shape of the Earth, the combination of reason and observation servedAristotle verywell.

Diogenes Laertius and the Greek geographer Strabo credit Parmenides with knowing, long beforeAristotle, that the Earth is a sphere, but we have no idea how (if at all) Parmenides reached thisconclusion.InOntheHeavensAristotlegaveboththeoreticalandempiricalargumentsforthesphericalshapeoftheEarth.AswesawinChapter3,accordingtoAristotle’saprioritheoryofmattertheheavyelementsearthand(lessso)waterseektoapproachthecenterofthecosmos,whileairand(moreso)firetendtorecedefromit.TheEarthisasphere,whosecentercoincideswiththecenterofthecosmos,becausethisallowsthegreatestamountof theelementearthtoapproachthiscenter.Aristotledidnotrestonthistheoreticalargument,butaddedempiricalevidenceforthesphericalshapeoftheEarth.TheEarth’sshadowontheMoonduringalunareclipseiscurved,*andthepositionofstarsintheskyseemstochangeaswetravelnorthorsouth:

Ineclipsestheoutlineisalwayscurved,and,sinceitistheinterpositionoftheEarththatmakestheeclipse,theformofthelinewillbecausedbytheformoftheEarth’ssurface,whichisthereforespherical.Again,ourobservationofthestarsmake[s]itevident,notonlythattheEarthiscircular,butalsothatitisacircleofnogreatsize.Forquiteasmallchangeofpositiononourparttosouthornorth causes amanifest alterationof thehorizon.There ismuch change, Imean, in the starswhichareoverhead,andthestarsseenaredifferent,asonemovesnorthwardorsouthward.Indeedthere are some stars seen in Egypt and in the neighborhood of Cyprus that are not seen in thenortherlyregions;andstars,whichinthenorthareneverbeyondtherangeofobservation,inthoseregionsriseandset.3

It is characteristic of Aristotle’s attitude toward mathematics that he made no attempt to use theseobservationsofstars togiveaquantitativeestimateof thesizeof theEarth.Apart fromthis, I find itpuzzling thatAristotle did not also cite a phenomenon thatmust have been familiar to every sailor.Whenashipatseaisfirstseenonacleardayatagreatdistanceitis“hulldownonthehorizon”—thecurveof theEarthhidesallbut the topsof itsmasts—but then, as it approaches, the restof the ship

becomesvisible.*Aristotle’sunderstandingofthesphericalshapeoftheEarthwasnosmallachievement.Anaximander

hadthoughtthattheEarthisacylinder,onwhoseflatfacewelive.AccordingtoAnaximenes,theEarthisflat,whiletheSun,Moon,andstarsfloatontheair,beinghiddenfromuswhentheygobehindhighpartsoftheEarth.Xenophaneshadwritten,“ThisistheupperlimitoftheEarththatweseeatourfeet;butthepartbeneathgoesdowntoinfinity.”4Later,bothDemocritusandAnaxagorashadthoughtlikeAnaximenesthattheEarthisflat.I suspect that the persistent belief in the flatness of the Earth may have been due to an obvious

problemwitha sphericalEarth: if theEarth isa sphere, thenwhydo travelersnot falloff?ThiswasnicelyansweredbyAristotle’stheoryofmatter.Aristotleunderstoodthatthereisnouniversaldirection“down,”alongwhichobjectsplacedanywheretendtofall.Rather,everywhereonEarththingsmadeofthe heavy elements earth and water tend to fall toward the center of the world, in agreement withobservation.Inthisrespect,Aristotle’stheorythatthenaturalplaceoftheheavierelementsisinthecenterofthe

cosmos worked much like the modern theory of gravitation, with the important difference that forAristotletherewasjustonecenterofthecosmos,whiletodayweunderstandthatanylargemasswilltendtocontracttoasphereundertheinfluenceofitsowngravitation,andthenwillattractotherbodiestowarditsowncenter.Aristotle’stheorydidnotexplainwhyanybodyotherthantheEarthshouldbeasphere,andyetheknew thatat least theMoon isa sphere, reasoning from thegradualchangeof itsphases,fromfulltonewandbackagain.5AfterAristotle,theoverwhelmingconsensusamongastronomersandphilosophers(asidefromafew

likeLactantius)wasthattheEarthisasphere.Withthemind’seye,ArchimedesevensawthesphericalshapeoftheEarthinaglassofwater;inProposition2ofOnFloatingBodies,hedemonstrates,“ThesurfaceofanyfluidatrestisthesurfaceofaspherewhosecenteristheEarth.”6 (Thiswouldbetrueonlyintheabsenceofsurfacetension,whichArchimedesneglected.)Now I come to what in some respects is the most impressive example of the application of

mathematicstonaturalscienceintheancientworld:theworkofAristarchusofSamos.Aristarchuswasbornaround310BContheIonianislandofSamos;studiedasapupilofStratoofLampsacus,thethirdhead of the Lyceum in Athens; and then worked at Alexandria until his death around 230 BC.Fortunately, his masterworkOn the Sizes and Distances of the Sun andMoon has survived.7 In it,Aristarchustakesfourastronomicalobservationsaspostulates:

1. “At the timeofHalfMoon, theMoon’sdistance from theSun is less thanaquadrantbyone-thirtiethofaquadrant.”(Thatis,whentheMoonisjusthalffull,theanglebetweenthelinesofsighttotheMoonandtotheSunislessthan90°by3°,soitis87°.)

2.TheMoonjustcoversthevisiblediskoftheSunduringasolareclipse.3.“ThebreadthoftheEarth’sshadowisthatoftwoMoons.”(Thesimplestinterpretationisthatat

thepositionoftheMoon,aspherewithtwicethediameteroftheMoonwouldjustfilltheEarth’sshadowduringalunareclipse.ThiswaspresumablyfoundbymeasuringthetimefromwhenoneedgeoftheMoonbegantobeobscuredbytheEarth’sshadowtowhenitbecameentirelyobscured,the time during which it was entirely obscured, and the time from then until the eclipse wascompletelyover.)

4.“TheMoonsubtendsonefifteenthpartofthezodiac.”(Thecompletezodiacisafull360°circle,butAristarchushereevidentlymeansonesignofthezodiac;thezodiacconsistsof12constellations,soonesignoccupiesanangleof360°/12=30°,andonefifteenthpartofthatis2°.)

Fromtheseassumptions,Aristarchusdeducedinturnthat:

1.ThedistancefromtheEarthtotheSunisbetween19and20timeslargerthanthedistanceoftheEarthtotheMoon.

2.ThediameteroftheSunisbetween19and20timeslargerthanthediameteroftheMoon.3.ThediameteroftheEarthisbetween108/43and60/19timeslargerthanthediameteroftheMoon.4.ThedistancefromtheEarthtotheMoonisbetween30and45/2timeslargerthanthediameterof

theMoon.

At the time of his work, trigonometry was not known, so Aristarchus had to go through elaborategeometricconstructionstoget theseupperandlowerlimits.Today,usingtrigonometry,wewouldgetmorepreciseresults;forinstance,wewouldconcludefrompoint1thatthedistancefromtheEarthtothe Sun is larger than the distance from the Earth to theMoon by the secant (the reciprocal of thecosine) of 87°, or 19.1, which is indeed between 19 and 20. (This and the other conclusions ofAristarchusarere-derivedinmoderntermsinTechnicalNote11.)From these conclusions, Aristarchus could work out the sizes of the Sun and Moon and their

distancesfromtheEarth,allintermsofthediameteroftheEarth.Inparticular,bycombiningpoints2and3,AristarchuscouldconcludethatthediameteroftheSunisbetween361/60and215/27timeslargerthanthediameteroftheEarth.ThereasoningofAristarchuswasmathematicallyimpeccable,buthisresultswerequantitativelyway

off,becausepoints1and4inthedataheusedasastartingpointwerebadlyinerror.WhentheMoonishalffull,theactualanglebetweenthelinesofsighttotheSunandtotheMoonisnot87°but89.853°,whichmakestheSun390timesfartherawayfromtheEarththantheMoon,andhencemuchlargerthanAristarchus thought.Thismeasurement couldnotpossiblyhavebeenmadebynaked-eye astronomy,thoughAristarchuscouldhavecorrectlyreportedthatwhentheMoonishalffulltheanglebetweenthelinesofsighttotheSunandMoonisnotlessthan87°.Also,thevisiblediskoftheMoonsubtendsanangleof0.519°,not2°,whichmakes thedistanceof theEarth to theMoonmore like111 times thediameter the Moon. Aristarchus certainly could have done better than this, and there is a hint inArchimedes’TheSandReckonerthatinlaterworkhedidso.*ItisnottheerrorsinhisobservationsthatmarkthedistancebetweenthescienceofAristarchusand

our own. Occasional serious errors continue to plague observational astronomy and experimentalphysics.Forinstance,inthe1930stherateatwhichtheuniverseisexpandingwasthoughttobeaboutseventimesfasterthanwenowknowitactuallyis.TherealdifferencebetweenAristarchusandtoday’sastronomers andphysicists is not that his observational datawere in error, but that he never tried tojudgetheuncertaintyinthem,orevenacknowledgedthattheymightbeimperfect.Physicists and astronomers todayare trained to take experimental uncertaintyvery seriously.Even

though as an undergraduate I knew that I wanted to be a theoretical physicist whowould never doexperiments,IwasrequiredalongwithallotherphysicsstudentsatCornelltotakealaboratorycourse.Mostofourtimeinthecoursewasspentestimatingtheuncertaintyinthemeasurementswemade.Buthistorically,thisattentiontouncertaintywasalongtimeincoming.AsfarasIknow,nooneinancientormedievaltimesevertriedseriouslytoestimatetheuncertaintyinameasurement,andaswewillseeinChapter14,evenNewtoncouldbecavalieraboutexperimentaluncertainties.WeseeinAristarchusaperniciouseffectoftheprestigeofmathematics.HisbookreadslikeEuclid’s

Elements:thedatainpoints1through4aretakenaspostulates,fromwhichhisresultsarededucedwithmathematicalrigor.Theobservationalerrorinhisresultswasverymuchgreaterthanthenarrowrangesthatherigorouslydemonstratedforthevarioussizesanddistances.PerhapsAristarchusdidnotmeantosaythattheanglebetweenthelinesofsighttotheSunandtheMoonwhenhalffullisreally87°,butonly took that as an example, to illustratewhat could be deduced.Not for nothingwasAristarchus

knowntohiscontemporariesas“theMathematician,”incontrasttohisteacherStrato,whowasknownas“thePhysicist.”ButAristarchus did get one important point qualitatively correct: theSun ismuchbigger than the

Earth.Toemphasizethepoint,AristarchusnotedthatthevolumeoftheSunisatleast(361/60)3(about218)timeslargerthanthevolumeoftheEarth.Ofcourse,wenowknowthatitismuchbiggerthanthat.There are tantalizing statements by bothArchimedes and Plutarch thatAristarchus had concluded

fromthegreatsizeoftheSunthatitisnottheSunthatgoesaroundtheEarth,buttheEarththatgoesaroundtheSun.AccordingtoArchimedesinTheSandReckoner,8AristarchushadconcludednotonlythattheEarthgoesaroundtheSun,butalsothattheEarth’sorbitistinycomparedwiththedistancetothe fixed stars. It is likely thatAristarchuswas dealingwith a problem raised by any theory of theEarth’smotion.Justasobjectsonthegroundseemtobemovingbackandforthwhenviewedfromacarousel, so the stars ought to seem tomove back and forth during the yearwhen viewed from themovingEarth.Aristotlehadseemedtorealizethis,whenhecommented9thatiftheEarthmoved,then“therewouldhave tobepassingsand turningsof the fixed stars.Yetno such thing isobserved.ThesamestarsalwaysriseandsetinthesamepartsoftheEarth.”Tobespecific,iftheEarthgoesaroundtheSun,theneachstarshouldseemtotraceoutintheskyaclosedcurve,whosesizewoulddependontheratioofthediameteroftheEarth’sorbitaroundtheSuntothedistancetothestar.SoiftheEarthgoesaroundtheSun,whydidn’tancientastronomersseethisapparentannualmotion

ofthestars,knownasannualparallax?Tomaketheparallaxsmallenoughtohaveescapedobservation,itwasnecessarytoassumethatthestarsareatleastacertaindistanceaway.Unfortunately,ArchimedesinTheSandReckonermadenoexplicitmentionofparallax,andwedon’tknowifanyoneintheancientworldusedthisargumenttoputalowerboundonthedistancetothestars.Aristotle had given other arguments against a moving Earth. Some were based on his theory of

naturalmotiontowardthecenteroftheuniverse,mentionedinChapter3,butoneotherargumentwasbased on observation.Aristotle reasoned that if the Earthweremoving, then bodies thrown straightupwardwouldbeleftbehindbythemovingEarth,andhencewouldfalltoaplacedifferentfromwheretheywerethrown.Instead,asheremarks,10“heavybodiesforciblythrownquitestraightupwardreturntothepointfromwhichtheystarted,eveniftheyarethrowntoanunlimiteddistance.”Thisargumentwasrepeatedmanytimes,forinstancebyClaudiusPtolemy(whomwemetinChapter4)aroundAD150,andbyJeanBuridan in theMiddleAges,until (aswewill see inChapter10)ananswer to thisargumentwasgivenbyNicoleOresme.ItmightbepossibletojudgehowfartheideaofamovingEarthspreadintheancientworldifwehad

a good description of an ancient orrery, amechanicalmodel of the solar system.*Cicero inOn theRepublic tellsofaconversationaboutanorrery in129BC, twenty-threeyearsbeforehehimselfwasborn.Inthisconversation,oneLuciusFuriusPhiluswassupposedtohavetoldaboutanorrerymadebyArchimedesthathadbeentakenafterthefallofSyracusebyitsconquerorMarcellus,andthatwaslaterseen in the house ofMarcellus’ grandson. It is not easy to tell from this thirdhand account how theorreryworked(andsomepagesfromthispartofDeRePublicaaremissing),butatonepoint in thestoryCicero quotes Philus as saying that on this orrery “were delineated themotion of the Sun andMoon and of those five stars that are called wanderers [planets],” which certainly suggests that theorreryhadamovingSun,ratherthanamovingEarth.11Aswewill see inChapter8, long beforeAristarchus the Pythagoreans had the idea that both the

Earth and the Sun move around a central fire. For this they had no evidence, but somehow theirspeculations were remembered, while that of Aristarchus was almost forgotten. Just one ancientastronomer is known to have adopted the heliocentric ideas ofAristarchus: the obscure Seleucus ofSeleucia,whoflourishedaround150BC.InthetimeofCopernicusandGalileo,whenastronomersand

churchmen wanted to refer to the idea that the Earth moves, they called it Pythagorean, notAristarchean.WhenIvisitedtheislandofSamosin2005,IfoundplentyofbarsandrestaurantsnamedforPythagoras,butnoneforAristarchusofSamos.ItiseasytoseewhytheideaoftheEarth’smotiondidnottakeholdintheancientworld.Wedonot

feel thismotion,andnoonebefore thefourteenthcenturyunderstood that there isnoreasonwhyweshould feel it. Also, neither Archimedes nor anyone else gave any indication that Aristarchus hadworkedouthowthemotionoftheplanetswouldappearfromamovingEarth.Themeasurementof thedistance from theEarth to theMoonwasmuch improvedbyHipparchus,

generally regarded as the greatest astronomical observer of the ancient world.12 Hipparchus madeastronomical observations inAlexandria from161BC to 146BC, and then continued until 127BC,perhaps on the island of Rhodes. Almost all his writings have been lost; we know about hisastronomicalwork chiefly from the testimony ofClaudius Ptolemy, three centuries later.One of hiscalculationswasbasedon theobservationof aneclipseof theSun,nowknown tohaveoccurredonMarch14,189BC.InthiseclipsethediskoftheSunwastotallyhiddenatAlexandria,butonlyfour-fifthshiddenontheHellespont(themodernDardanelles,betweenAsiaandEurope).SincetheapparentdiametersoftheMoonandSunareverynearlyequal,andweremeasuredbyHipparchustobeabout33'(minutesofarc)or0.55°,Hipparchuscouldconcludethat thedirectionto theMoonasseenfromtheHellespontandfromAlexandriadifferedbyone-fifthof0.55°,or0.11°.FromobservationsoftheSunHipparchusknewthelatitudesoftheHellespontandAlexandria,andheknewthelocationoftheMoonintheskyatthetimeoftheeclipse,sohewasabletoworkoutthedistancetotheMoonasamultipleoftheradiusoftheEarth.ConsideringthechangesduringalunarmonthoftheapparentsizeoftheMoon,Hipparchusconcluded that thedistance fromtheEarth to theMoonvaries from71 to83Earth radii.Theaveragedistanceisactuallyabout60Earthradii.IshouldpausetosaysomethingaboutanothergreatachievementofHipparchus,eventhoughitisnot

directlyrelevanttothemeasurementofsizesanddistances.Hipparchuspreparedastarcatalog,alistofabout 800 stars,with the celestial position given for each star. It is fitting that our bestmodern starcatalog,whichgivesthepositionsof118,000stars,wasmadebyobservationsfromanartificialsatellitenamedinhonorofHipparchus.The measurements of star positions by Hipparchus led him to the discovery of a remarkable

phenomenon, which was not understood until the work of Newton. To explain this discovery, it isnecessarytosaysomethingabouthowcelestialpositionsaredescribed.ThecatalogofHipparchushasnot survived, and we don’t know just how he described these positions. There are two possibilitiescommonlyusedfromRomantimeson.Onemethod,usedlaterinthestarcatalogofPtolemy,13picturesthefixedstarsaspointsonasphere,whoseequatoristheecliptic,thepaththroughthestarsapparentlytracedinayearbytheSun.Celestiallatitudeandlongitudelocatestarsonthissphereinthesamewaythatordinary latitudeand longitudegive the locationofpointson theEarth’s surface.* Inadifferentmethod,whichmayhavebeenusedbyHipparchus,14thestarsareagaintakenaspointsonasphere,butthissphereisorientedwiththeEarth’saxisratherthantheecliptic;thenorthpoleofthissphereisthenorthcelestialpole,aboutwhichthestarsseemtorevolveeverynight.Insteadoflatitudeandlongitude,thecoordinatesonthissphereareknownasdeclinationandrightascension.AccordingtoPtolemy,15themeasurementsofHipparchusweresufficientlyaccurateforhimtonotice

thatthecelestiallongitude(orrightascension)ofthestarSpicahadchangedby2°fromwhathadbeenobservedlongbeforeatAlexandriabytheastronomerTimocharis.ItwasnotthatSpicahadchangeditspositionrelativetotheotherstars;rather,thelocationoftheSunonthecelestialsphereattheautumnalequinox,thepointfromwhichcelestiallongitudewasthenmeasured,hadchanged.It isdifficult tobepreciseabouthowlong thischange took.Timochariswasbornaround320BC,

about 130years beforeHipparchus; but it is believed that he diedyoung around280BC, about 160yearsbeforeHipparchus. Ifweguess thatabout150yearsseparated theirobservationsofSpica, thentheseobservations indicate that thepositionof theSunat theautumnalequinoxchangesbyabout1°every75years.*Atthatrate,thisequinoctalpointwouldprecessthroughthewhole360°circleofthezodiacin360times75years,or27,000years.TodayweunderstandthattheprecessionoftheequinoxesiscausedbyawobbleoftheEarth’saxis

(likethewobbleoftheaxisofaspinningtop)aroundadirectionperpendiculartotheplaneofitsorbit,with the angle between this direction and the Earth’s axis remaining nearly fixed at 23.5°. TheequinoxesarethedateswhenthelineseparatingtheEarthandtheSunisperpendiculartotheEarth’saxis,soawobbleoftheEarth’saxiscausestheequinoxestoprecess.WewillseeinChapter14thatthiswobblewasfirstexplainedbyIsaacNewton,asaneffectofthegravitationalattractionoftheSunandMoonfortheequatorialbulgeoftheEarth.Itactuallytakes25,727yearsfortheEarth’saxistowobblebyafull360°.ItisremarkablehowaccuratelytheworkofHipparchuspredictedthisgreatspanoftime.(Bytheway,itistheprecessionoftheequinoxesthatexplainswhyancientnavigatorshadtojudgethedirectionofnorthfromthepositionintheskyofconstellationsnearthenorthcelestialpole,ratherfromthepositionoftheNorthStar,Polaris.Polarishasnotmovedrelativetotheotherstars,butinancienttimestheEarth’saxisdidnotpointatPolarisasitdoesnow,andinthefuturePolariswillagainnotbeatthenorthcelestialpole.)Returning now to celestial measurement, all of the estimates by Aristarchus and Hipparchus

expressedthesizeanddistancesoftheMoonandSunasmultiplesofthesizeoftheEarth.ThesizeoftheEarthwasmeasuredafewdecadesaftertheworkofAristarchusbyEratosthenes.Eratostheneswasbornin273BCatCyrene,aGreekcityontheMediterraneancoastoftoday’sLibya,foundedaround630BC,thathadbecomepartofthekingdomofthePtolemies.HewaseducatedinAthens,partlyattheLyceum,andthenaround245BCwascalledbyPtolemyIIItoAlexandria,wherehebecameafellowoftheMuseumandtutortothefuturePtolemyIV.HewasmadethefifthheadoftheLibraryaround234BC.Hismainworks—OntheMeasurementoftheEarth,GeographicMemoirs,andHermes—haveallunfortunatelydisappeared,butwerewidelyquotedinantiquity.Themeasurement of the sizeof theEarthbyEratostheneswasdescribedby theStoicphilosopher

CleomedesinOntheHeavens,16sometimeafter50BC.EratosthenesstartedwiththeobservationsthatatnoonatthesummersolsticetheSunisdirectlyoverheadatSyene,anEgyptiancitythatEratosthenessupposedtobeduesouthofAlexandria,whilemeasurementswithagnomonatAlexandriashowedthenoonSunatthesolsticetobeone-fiftiethofafullcircle,or7.2°,awayfromthevertical.FromthishecouldconcludethattheEarth’scircumferenceis50timesthedistancefromAlexandriatoSyene.(SeeTechnicalNote12.)ThedistancefromAlexandriatoSyenehadbeenmeasured(probablybywalkers,trainedtomakeeachstepthesamelength)as5,000stadia,sothecircumferenceoftheEarthmustbe250,000stadia.Howgoodwasthisestimate?Wedon’tknowthelengthofthestadionasusedbyEratosthenes,and

Cleomedesprobablydidn’tknowiteither,since(unlikeourmileorkilometer)ithadneverbeengivenastandard definition. But without knowing the length of the stadion, we can judge the accuracy ofEratosthenes’ use of astronomy. The Earth’s circumference is actually 47.9 times the distance fromAlexandriatoSyene(modernAswan),sotheconclusionofEratosthenesthattheEarth’scircumferenceis50timesthedistancefromAlexandriatoSyenewasactuallyquiteaccurate,whateverthelengthofthestadion.*Inhisuseofastronomy,ifnotofgeography,Eratostheneshaddonequitewell.

8

TheProblemofthePlanets

TheSunandMoonarenotaloneinmovingfromwesttoeastthroughthezodiacwhiletheysharethequickerdailyrevolutionofthestarsfromeasttowestaroundthenorthcelestialpole.Inseveralancientcivilizationsitwasnoticedthatovermanydaysfive“stars”travelfromwesttoeastthroughthefixedstarsalongprettymuchthesamepathastheSunandMoon.TheGreekscalledthemwanderingstars,orplanets,andgavethemthenamesofgods:Hermes,Aphrodite,Ares,Zeus,andCronos, translatedbytheRomansintoMercury,Venus,Mars,Jupiter,andSaturn.FollowingtheleadoftheBabylonians,theyalsoincludedtheSunandMoonasplanets,*makingseveninall,andonthisbasedtheweekofsevendays.*Theplanetsmovethroughtheskyatdifferentspeeds:MercuryandVenustake1yeartocompleteone

circuitof the zodiac;Mars takes1year and322days; Jupiter11years and315days; andSaturn29yearsand166days.All theseareaverageperiods,becausetheplanetsdonotmoveatconstantspeedthrough the zodiac—they even occasionally reverse the direction of theirmotion for awhile, beforeresumingtheireastwardmotion.Muchofthestoryoftheemergenceofmodernsciencedealswiththeeffort,extendingovertwomillennia,toexplainthepeculiarmotionsoftheplanets.AnearlyattemptatatheoryoftheplanetsandSunandMoonwasmadebythePythagoreans.They

imaginedthatthefiveplanets,togetherwiththeSunandMoonandalsotheEarth,allrevolvearoundacentralfire.ToexplainwhyweonEarthdonotseethecentralfire,thePythagoreanssupposedthatweliveonthesideoftheEarththatfacesoutward,awayfromthefire.(Likealmostallthepre-Socratics,thePythagoreansbelievedtheEarthtobeflat;theythoughtofitasadiskalwayspresentingthesamesidetothecentralfire,withusontheotherside.ThedailymotionoftheEartharoundthecentralfirewassupposedtoexplaintheapparentdailymotionofthemoreslowlymovingSun,Moon,planets,andstars around the Earth.)1 According to Aristotle and Aëtius, the Pythagorean Philolaus of the fifthcenturyBC invented a counter-Earth, orbitingwhere on our side of theEarthwe can’t see it, eitherbetweentheEarthandthecentralfireorontheothersideofthecentralfirefromtheEarth.Aristotleexplainedtheintroductionofthecounter-EarthasaresultofthePythagoreans’obsessionwithnumbers.TheEarth,Sun,Moon,andfiveplanets togetherwith thesphereof the fixedstarsmadenineobjectsabout the central fire, but thePythagoreans supposed that thenumberof theseobjectsmustbe10, aperfectnumberinthesensethat10=1+2+3+4.AsdescribedsomewhatscornfullybyAristotle,2thePythagoreans

supposedtheelementsofnumberstobetheelementsofallthings,andthewholeheaventobeamusicalscaleandanumber.Andallthepropertiesofnumbersandscaleswhichtheycouldshowtoagreewiththeattributesandpartsandthewholearrangementoftheheavens,theycollectedandfitted into their scheme, and if therewas a gap anywhere, they readilymade additions so as tomaketheirwholetheorycoherent.Forexample,asthenumber10isthoughttobeperfectandtocomprisethewholenatureofnumbers,theysaythatthebodieswhichmovethroughtheheavensare ten, but as the visible bodies are only nine, tomeet this they invent a tenth—the “counter-Earth.”

Apparently the Pythagoreans never tried to show that their theory explained in detail the apparent

motionsintheskyoftheSun,Moon,andplanetsagainstthebackgroundoffixedstars.Theexplanationoftheseapparentmotionswasataskforthefollowingcenturies,notcompleteduntilthetimeofKepler.Thisworkwasaidedbytheintroductionofdeviceslikethegnomon,forstudyingthemotionsofthe

Sun,andotherinstrumentsthatallowedthemeasurementofanglesbetweenthelinesofsighttovariousstarsandplanets,orbetweensuchastronomicalobjectsandthehorizon.Ofcourse,allthiswasnaked-eyeastronomy.ItisironicthatClaudiusPtolemy,whohaddeeplystudiedthephenomenaofrefractionandreflection(includingtheeffectsofrefractionintheatmosphereontheapparentpositionsofstars)andwhoaswewillseeplayedacrucialroleinthehistoryofastronomy,neverrealizedthatlensesandcurvedmirrors could be used tomagnify the images of astronomical bodies, as in Galileo Galilei’srefractingtelescopeandthereflectingtelescopeinventedbyIsaacNewton.Itwasnotjustphysicalinstrumentsthatfurtheredthegreatadvancesofscientificastronomyamong

theGreeks.Theseadvancesweremadepossiblealsobyimprovementsinthedisciplineofmathematics.Asmattersworkedout,thegreatdebateinancientandmedievalastronomywasnotbetweenthosewhothoughtthattheEarthortheSunwasinmotion,butbetweentwodifferentconceptionsofhowtheSunandMoonandplanetsrevolvearoundastationaryEarth.Aswewillsee,muchofthisdebatehadtodowithdifferentconceptionsoftheroleofmathematicsinthenaturalsciences.ThisstorybeginswithwhatIliketocallPlato’shomeworkproblem.AccordingtotheNeoplatonist

Simplicius,writingaroundAD530inhiscommentaryonAristotle’sOntheHeavens,

Platolaysdowntheprinciplethattheheavenlybodies’motioniscircular,uniform,andconstantlyregular. Therefore he sets the mathematicians the following problem: What circular motions,uniformandperfectlyregular,aretobeadmittedashypothesessothatitmightbepossibletosavetheappearancespresentedbytheplanets?3

“Save(orpreserve)theappearances”isthetraditionaltranslation;Platoisaskingwhatcombinationsofmotionoftheplanets(hereincludingtheSunandMoon)incirclesatconstantspeed,alwaysinthesamedirection,wouldpresentanappearancejustlikewhatweactuallyobserve.ThisquestionwasfirstaddressedbyPlato’scontemporary, themathematicianEudoxusofCnidus.4

Heconstructedamathematicalmodel,describedinalostbook,OnSpeeds,whosecontentsareknownto us from descriptions byAristotle5 andSimplicius.6 According to thismodel, the stars are carriedaroundtheEarthonaspherethatrevolvesonceadayfromeasttowest,whiletheSunandMoonandplanets are carried around the Earth on spheres that are themselves carried by other spheres. ThesimplestmodelwouldhavetwospheresfortheSun.TheoutersphererevolvesaroundtheEarthonceadayfromeasttowest,withthesameaxisandspeedofrotationasthesphereofthestars;buttheSunisontheequatorofaninnersphere,whichsharestherotationoftheoutersphereasifitwereattachedtoit,butthatalsorevolvesarounditsownaxisfromwesttoeastonceayear.Theaxisoftheinnersphereistiltedby23½°totheaxisoftheoutersphere.ThiswouldaccountbothfortheSun’sdailyapparentmotion,andforitsannualapparentmotionthroughthezodiac.LikewisetheMooncouldbesupposedtobe carried around theEarth by two other counter-rotating spheres,with the difference that the innersphereonwhichtheMoonridesmakesafullrotationfromwesttoeastonceamonth,ratherthanonceayear.Forreasonsthatarenotclear,EudoxusissupposedtohaveaddedathirdsphereeachfortheSunandMoon.Suchtheoriesarecalled“homocentric,”becausethespheresassociatedwiththeplanetsaswellastheSunandtheMoonallhavethesamecenter,thecenteroftheEarth.Theirregularmotionsoftheplanetsposedamoredifficultproblem.Eudoxusgaveeachplanetfour

spheres:theoutersphererotatingonceadayaroundtheEarthfromeasttowest,withthesameaxisofrotationasthesphereofthefixedstarsandtheouterspheresoftheSunandMoon;thenextspherelikethe inner spheres of the Sun andMoon revolvingmore slowly at various speeds fromwest to east

around an axis tilted by about 23½° to the axis of the outer sphere; and the two innermost spheresrotating,atexactlythesamerates,inoppositedirectionsaroundtwonearlyparallelaxestiltedatlargeangles to the axesof the twoouter spheres.Theplanet is attached to the innermost sphere.The twoouterspheresgiveeachplanetitsdailyrevolutionfollowingthestarsaroundtheEarthanditsaveragemotionoverlongerperiodsthroughthezodiac.Theeffectsofthetwooppositelyrotatinginnersphereswouldcanceliftheiraxeswerepreciselyparallel,butbecausetheseaxesaresupposedtobenotquiteparallel, they superimpose a figure eight motion on the average motion of each planet through thezodiac,accountingfortheoccasionalreversalsofdirectionoftheplanet.TheGreekscalledthispathahippopedebecauseitresembledthetethersusedtokeephorsesfromstraying.Themodel of Eudoxus did not quite agreewith observations of the Sun,Moon, and planets. For

instance,itspictureoftheSun’smotiondidnotaccountforthedifferencesinthelengthsoftheseasonsthat,aswesawinChapter6,hadbeenfoundwiththeuseofthegnomonbyEuctemon.ItquitefailedforMercury,anddidnotdowellforVenusorMars.Toimprovethings,anewmodelwasproposedbyCallippusofCyzicus.HeaddedtwomorespherestotheSunandMoon,andonemoreeachtoMercury,Venus, andMars. The model of Callippus generally worked better than that of Eudoxus, though itintroducedsomenewfictitiouspeculiaritiestotheapparentmotionsoftheplanets.InthehomocentricmodelsofEudoxusandCallippus,theSun,Moon,andplanetswereeachgivena

separate suite of spheres, all with outer spheres rotating in perfect unison with a separate spherecarrying the fixed stars. This is an early example of whatmodern physicists call “fine-tuning.”Wecriticize a proposed theory as fine-tuned when its features are adjusted tomake some things equal,withoutanyunderstandingofwhytheyshouldbeequal.Theappearanceoffine-tuninginascientifictheoryislikeacryofdistressfromnature,complainingthatsomethingneedstobebetterexplained.Adistasteforfine-tuningledmodernphysiciststomakeadiscoveryoffundamentalimportance.In

the late 1950s two types of unstable particle called tau and theta had been identified that decay indifferentways—thethetaintotwolighterparticlescalledpions,andthetauintothreepions.Notonlydid the tauand thetaparticleshave thesamemass—theyhad thesameaverage lifetime,even thoughtheirdecaymodeswereentirelydifferent!Physicistsassumedthatthetauandthethetacouldnotbethesameparticle,because forcomplicated reasons thesymmetryofnaturebetweenrightand left (whichdictatesthatthelawsofnaturemustappearthesamewhentheworldisviewedinamirroraswhenitisvieweddirectly)wouldforbidthesameparticlefromdecayingsometimesintotwopionsandsometimesintothree.Withwhatweknewat thetime, itwouldhavebeenpossibletoadjust theconstants inourtheoriestomakethemassesandlifetimesofthetauandthetaequal,butonecouldhardlystomachsuchatheory—itseemedhopelesslyfine-tuned.Intheend,itwasfoundthatnofine-tuningwasnecessary,because the twoparticles are in fact the sameparticle.The symmetrybetween right and left, thoughobeyedbytheforcesthatholdatomsandtheirnuclei together, issimplynotobeyedinvariousdecayprocesses, including the decay of the tau and theta.7 The physicists who realized this were right todistrusttheideathatthetauandthethetaparticlesjusthappenedtohavethesamemassandlifetime—thatwouldtaketoomuchfine-tuning.Todaywefaceanevenmoredistressingsortoffine-tuning.In1998astronomersdiscoveredthatthe

expansionoftheuniverseisnotslowingdown,aswouldbeexpectedfromthegravitationalattractionofgalaxiesforeachother,butisinsteadspeedingup.Thisaccelerationisattributedtoanenergyassociatedwithspaceitself,knownasdarkenergy.Theoryindicatesthatthereareseveraldifferentcontributionstodarkenergy.Somecontributionswecancalculate,andotherswecan’t.Thecontributionstodarkenergythatwecancalculateturnouttobelargerthanthevalueofthedarkenergyobservedbyastronomersbyabout 56 orders ofmagnitude—that is, 1 followed by 56 zeroes. It’s not a paradox, becausewe cansupposethatthesecalculablecontributionstodarkenergyarenearlycanceledbycontributionswecan’tcalculate,butthecancellationwouldhavetobepreciseto56decimalplaces.Thisleveloffine-tuningis

intolerable,andtheoristshavebeenworkinghardtofindabetterwaytoexplainwhytheamountofdarkenergy is so much smaller than that suggested by our calculations. One possible explanation ismentionedinChapter11.At the same time, it must be acknowledged that some apparent examples of fine-tuning are just

accidents.For instance, thedistancesof theSunandMoonfromtheEarthare in justabout thesameratio as their diameters, so that seen fromEarth, the Sun andMoon appear about the same size, asshownbythefactthattheMoonjustcoverstheSunduringatotalsolareclipse.Thereisnoreasontosupposethatthisisanythingbutacoincidence.Aristotle took a step to reduce the fine-tuning of the models of Eudoxus and Callippus. In

Metaphysics8heproposedtotieallthespherestogetherinasingleconnectedsystem.Insteadofgivingtheoutermostplanet,Saturn,foursphereslikeEudoxusandCallippus,hegaveitonlytheirthreeinnerspheres;thedailymotionofSaturnfromeasttowestwasexplainedbytyingthesethreespherestothesphereofthefixedstars.Aristotlealsoadded,insidethethreeofSaturn,threeextraspheresthatrotatedinopposite directions, so as to cancel the effect of themotion of the three spheres of Saturn on thespheresofthenextplanet,Jupiter,whoseouterspherewasattachedtotheinnermostofthethreeextraspheresbetweenJupiterandSaturn.Atthecostofaddingthesethreeextracounter-rotatingspheres,bytyingtheoutersphereofSaturnto

the sphere of the fixed stars Aristotle had accomplished something rather nice. It was no longernecessarytowonderwhythedailymotionofSaturnshouldpreciselyfollowthatofthestars—Saturnwasphysicallytiedtothesphereofthestars.ButthenAristotlespoileditall:hegaveJupiterallfourspheresthathadbeengiventoitbyEudoxusandCallippus.ThetroublewiththiswasthatJupiterwouldthengetadailymotionfromthatofSaturnandalsofromtheoutermostofitsownfourspheres,sothatitwouldgoaroundtheEarthtwiceaday.Didheforgetthatthethreecounter-rotatingspheresinsidethespheresofSaturnwouldcancelonlythespecialmotionsofSaturn,notitsdailyrevolutionaroundtheEarth?Worseyet,Aristotleaddedonly threecounter-rotatingspheres inside thefourspheresofJupiter, to

cancelitsownspecialmotionsbutnotitsdailymotion,andthengaveMars,thenextplanet,thefullfivespheresgiventoitbyCallippus,sothatMarswouldgoaroundtheEarththreetimesaday.Continuinginthisway,inAristotle’sschemeVenus,Mercury,theSun,andtheMoonwouldinadayrespectivelygoaroundtheEarthfour,five,six,andseventimes.IwasstruckbythisapparentfailurewhenIreadAristotle’sMetaphysics,and thenI learned that it

hadalreadybeennoticedbyseveralauthors,includingJ.L.E.Dreyer,ThomasHeath,andW.D.Ross.9Someofthemblameditonacorrupttext.ButifAristotlereallydidpresenttheschemedescribedinthestandardversionofMetaphysics, thenthiscannotbeexplainedasamatterofhisthinkingindifferenttermsfromours,orbeinginterestedindifferentproblemsfromours.Wewouldhavetoconcludethatonhisownterms,inworkingonaproblemthatinterestedhim,hewasbeingcarelessorstupid.EvenifAristotlehadputintherightnumberofcounter-rotatingspheres,sothateachplanetwould

follow the stars around theEarth just once each day, his scheme still relied on a great deal of fine-tuning. The counter-rotating spheres introduced inside the spheres of Saturn to cancel the effect ofSaturn’sspecialmotionsonthemotionsofJupiterwouldhavetorevolveatpreciselythesamespeedasthethreespheresofSaturnforthecancellationtowork,andlikewisefortheplanetsclosertotheEarth.And,justasforEudoxusandCallippus,inAristotle’sschemethesecondspheresofMercuryandVenuswouldhavetorevolveatpreciselythesamespeedasthesecondsphereoftheSun,inordertoaccountforthefactthatMercury,Venus,andtheSunmovetogetherthroughthezodiac,sothattheinnerplanetsareneverseenfarintheskyfromtheSun.Venus,forinstance,isalwaysthemorningstarortheeveningstar,neverseenhighintheskyatmidnight.Atleastoneancientastronomerseemstohavetakentheproblemoffine-tuningveryseriously.This

wasHeraclidesofPontus.HewasastudentatPlato’sAcademyinthefourthcenturyBC,andmayhavebeen left inchargeof itwhenPlatowent toSicily.BothSimplicius10 andAëtius say thatHeraclidestaught that the Earth rotates on its axis,* eliminating at one blow the supposed simultaneous dailyrevolution of the stars, planets, Sun, andMoon around the Earth. This proposal of Heraclides wasoccasionallymentionedbywritersinlateantiquityandtheMiddleAges,butitdidnotbecamepopularuntilthetimeofCopernicus,againpresumablybecausewedonotfeeltheEarth’srotation.ThereisnoindicationthatAristarchus,writingacenturyafterHeraclides,suspectedthattheEarthnotonlymovesaroundtheSunbutalsorotatesonitsownaxis.AccordingtoChalcidius(orCalcidius),aChristianwhotranslatedtheTimaeusfromGreektoLatin

inthefourthcentury,HeraclidesalsoproposedthatsinceMercuryandVenusareneverseenfarintheskyfromtheSun,theyrevolveabouttheSunratherthanabouttheEarth,thusremovinganotherbitoffine-tuning from the schemes of Eudoxus, Callippus, andAristotle: the artificial coordination of therevolutionsofthesecondspheresoftheSunandinnerplanets.ButtheSunandMoonandthreeouterplanetswerestillsupposedtorevolveaboutastationary,thoughrotating,Earth.Thistheoryworksverywell for the innerplanets,because itgives themprecisely the sameapparentmotionsas the simplestversionoftheCopernicantheory, inwhichMercury,Venus,andtheEarthallgoatconstantspeedoncirclesaroundtheSun.Asfarastheinnerplanetsareconcerned,theonlydifferencebetweenHeraclidesandCopernicusispointofview—eitherbasedontheEarthorbasedontheSun.Besides the fine-tuning inherent in the schemes of Eudoxus, Callippus, and Aristotle, there was

anotherproblem:thesehomocentricschemesdidnotagreeverywellwithobservation.Itwasbelievedthen that the planets shine by their own light, and since in these schemes the spheres onwhich theplanetsridealwaysremainatthesamedistancefromtheEarth’ssurface,theplanets’brightnessshouldnever change. It was obvious however that their brightness changed very much. As quoted bySimplicius,11aroundAD200thephilosopherSosigenesthePeripatetichadcommented:

Howeverthe[hypotheses]of theassociatesofEudoxusdonotpreservethephenomena,andjustthosewhichhadbeenknownpreviouslyandwereacceptedbythemselves.Andwhatnecessityistheretospeakaboutotherthings,someofwhichCallippusofCyzicusalsotriedtopreservewhenEudoxushadnotbeenabletodoso,whetherornotCallippusdidpreservethem?...WhatImeanisthattherearemanytimeswhentheplanetsappearnear,andtherearetimeswhentheyappeartohavemovedawayfromus.Andinthecaseofsome[planets]thisisapparenttosight.ForthestarwhichiscalledVenusandalsotheonewhichiscalledMarsappearmanytimeslargerwhentheyareinthemiddleoftheirretrogressionssothatinmoonlessnightsVenuscausesshadowstobecastbybodies.

Where Simplicius or Sosigenes refers to the size of planets,we presumably should understand theirapparentluminosity;withthenakedeyewecan’tactuallyseethediskofanyplanet,butthebrighterapointoflightis,thelargeritseemstobe.Actually,thisargumentisnotasconclusiveasSimpliciusthought.Theplanets(liketheMoon)shine

bythereflectedlightoftheSun,sotheirbrightnesswouldchangeevenintheschemesofEudoxusetal.as theygo throughdifferentphases (like thephasesof theMoon).Thiswasnotunderstooduntil thework of Galileo. But even if the phases of the planets had been taken into account, the changes inbrightnessthatwouldbeexpectedinhomocentrictheorieswouldnothaveagreedwithwhatisactuallyseen.Forprofessionalastronomers(ifnotforphilosophers)thehomocentrictheoryofEudoxus,Callippus,

andAristotlewas supplanted in theHellenistic andRoman eras by a theory that didmuch better ataccountingfortheapparentmotionsoftheSunandplanets.Thistheoryisbasedonthreemathematical

devices—the epicycle, the eccentric, and the equant—to be described below.We do not knowwhoinvented the epicycle andeccentric, but theyweredefinitelyknown to theHellenisticmathematicianApolloniusofPergaandtotheastronomerHipparchusofNicaea,whomwemetinChapters6and7.12WeknowaboutthetheoryofepicyclesandeccentricsthroughthewritingsofClaudiusPtolemy,whoinventedtheequant,andwithwhosenamethetheoryhaseverafterbeenassociated.PtolemyflourishedaroundAD150,intheageoftheAntonineemperorsattheheightoftheRoman

Empire.HeworkedattheMuseumofAlexandria,anddiedsometimeafterAD161.Wehavealreadydiscussed his study of reflection and refraction in Chapter 4. His astronomical work is described inMegale Syntaxis, a title transformed by theArabs toAlmagest, bywhich name it became generallyknown inEurope.TheAlmagestwas so successful that scribes stopped copying theworks of earlierastronomers likeHipparchus; as a result, it is difficult now to distinguishPtolemy’s ownwork fromtheirs.TheAlmagest improvedon the star catalogofHipparchus, listing1,028 stars, hundredsmore than

Hipparchus, and giving some indication of their brightness as well as their position in the sky.*Ptolemy’s theory of the planets and the Sun andMoonwasmuchmore important for the future ofscience. In one respect theworkon this theorydescribed in theAlmagest is strikinglymodern in itsmethods.Mathematicalmodels areproposed forplanetarymotions containingvarious freenumericalparameters, which are then found by constraining the predictions of the models to agree withobservation.Wewillseeanexampleofthisbelow,inconnectionwiththeeccentricandequant.In its simplest version, the Ptolemaic theory has each planet revolving in a circle known as an

“epicycle,”notabouttheEarth,butaboutamovingpointthatgoesaroundtheEarthonanothercircleknownasa“deferent.”The innerplanets,MercuryandVenus,goaround theepicycle in88and225days,respectively,whilethemodelisfine-tunedsothatthecenteroftheepicyclegoesaroundtheEarthonthedeferentinpreciselyoneyear,alwaysremainingonthelinebetweentheEarthandtheSun.Wecanseewhythisworks.Nothingintheapparentmotionofplanetstellsushowfarawaytheyare.

Hencein the theoryofPtolemy, theapparentmotionofanyplanet in theskydoesnotdependontheabsolutesizesoftheepicycleanddeferent; itdependsonlyontheratiooftheirsizes.IfPtolemyhadwanted tohecouldhaveexpanded the sizesofboth the epicycle and thedeferentofVenus,keepingtheirratiofixed,andlikewiseofMercury,sothatbothplanetswouldhavethesamedeferent,namely,theorbit of theSun.TheSunwould thenbe thepointon thedeferent aboutwhich the innerplanetstravel on their epicycles.This is not the theoryproposedbyHipparchus orPtolemy, but it gives themotionoftheinnerplanetsthesameappearance,becauseitdiffersonlyintheoverallscaleoftheorbits,whichdoesnotaffectapparentmotions.ThisspecialcaseoftheepicycletheoryisjustthesameasthetheoryattributedtoHeraclidesdiscussedabove,inwhichMercuryandVenusgoaroundtheSunwhilethe Sun goes around the Earth. As already mentioned, Heraclides’ theory works well because it isequivalenttooneinwhichtheEarthandinnerplanetsgoaroundtheSun,thetwotheoriesdifferingonlyinthepointofviewoftheastronomer.SoitisnoaccidentthattheepicycletheoryofPtolemy,whichgivesMercuryandVenusthesameapparentmotionsasthetheoryofHeraclides,alsoworksprettywellincomparisonwithobservation.Ptolemycouldhaveappliedthesametheoryofepicyclesanddeferentstotheouterplanets—Mars,

Jupiter,andSaturn—but tomake the theorywork itwouldhavebeennecessary tomake theplanets’motionaroundtheepicyclesmuchslowerthanthemotionoftheepicycles’centersaroundthedeferents.Idon’tknowwhatwouldhavebeenwrongwith this,but forone reasonoranotherPtolemychoseadifferentpath. In the simplestversionofhis scheme,eachouterplanetgoeson its epicyclearoundapointonthedeferentonceayear,andthatpointonthedeferentgoesaroundtheEarthinalongertime:1.88yearsforMars,11.9yearsforJupiter,and29.5yearsforSaturn.Herethereisadifferentsortoffine-tuning—thelinefromthecenteroftheepicycletotheplanetisalwaysparalleltothelinefromthe

EarthtotheSun.Thisschemeagreesfairlywellwiththeobservedapparentmotionsoftheouterplanetsbecausehere,as for the innerplanets, thedifferent specialcasesof this theory thatdifferonly in thescaleof theepicycleanddeferent (keeping their ratio fixed)allgive the sameapparentmotions, andthere is one special value of this scale that makes this model the same as the simplest Copernicantheory,differingonlyinpointofview:EarthorSun.Fortheouterplanets,thisspecialchoiceofscaleisthe one for which the radius of the epicycle equals the distance of the Sun from the Earth. (SeeTechnicalNote13.)Ptolemy’s theory nicely accounted for the apparent reversal in direction of planetarymotions. For

instance,Mars seems to go backward in its motion through the zodiac when it is on a point in itsepicycleclosest totheEarth,becausethenitssupposedmotionaroundtheepicycleis inthedirectionopposite to the supposed motion of the epicycle around the deferent, and faster. This is just atranscriptionintoaframeofreferencebasedontheEarthofthemodernstatementthatMarsseemstogobackwardinthezodiacwhentheEarthispassingitastheybothgoaroundtheSun.Thisisalsothetimewhen it isbrightest (asnoted in theabovequotation fromSimplicius),becauseat this time it isclosesttotheEarth,andthesideofMarsthatweseefacestheSun.ThetheorydevelopedbyHipparchus,Apollonius,andPtolemywasnotafantasythat,bygoodluck,

justhappenedtoagreefairlywellwithobservationbuthadnorelationtoreality.Asfarastheapparentmotionsof theSunandplanetsareconcerned, in itssimplestversion,with justoneepicycleforeachplanet and no other complications, this theory gives precisely the same predictions as the simplestversionof the theoryofCopernicus—that is, a theory inwhich theEarthand theotherplanetsgo incirclesatconstantspeedwiththeSunat thecenter.AsalreadyexplainedinconnectionwithMercuryandVenus (and further explained inTechnicalNote 13), this is because thePtolemaic theory is in aclassoftheoriesthatallgivethesameapparentmotionsoftheSunandplanets,andonememberofthatclass(thoughnottheoneadoptedbyPtolemy)givespreciselythesameactualmotionsoftheSunandplanetsrelativetooneanotherasgivenbythesimplestversionoftheCopernicantheory.ItwouldbenicetoendthestoryofGreekastronomyhere.Unfortunately,asCopernicushimselfwell

understood,thepredictionsofthesimplestversionoftheCopernicantheoryfortheapparentmotionsoftheplanetsdonotquiteagreewithobservation,andsoneitherdothepredictionsofthesimplestversionofthePtolemaictheory,whichareidentical.WehaveknownsincethetimeofKeplerandNewtonthattheorbitsoftheEarthandtheotherplanetsarenotexactlycircular,theSunisnotexactlyatthecenteroftheseorbits,andtheEarthandtheotherplanetsdonottravelaroundtheirorbitsatexactlyconstantspeed.Ofcourse,noneofthatwasunderstoodinmoderntermsbytheGreekastronomers.MuchofthehistoryofastronomyuntilKeplerwastakenupwithtryingtoaccommodatethesmall inaccuraciesinthesimplestversionsofboththePtolemaicandtheCopernicantheories.Platohadcalledforcirclesanduniformmotion,andasfarasisknownnooneinantiquityconceived

thatastronomicalbodiescouldhaveanymotionotherthanonecompoundedofcircularmotions,thoughPtolemywaswilling tocompromiseon the issueofuniformmotion.Workingunder the limitation toorbitscomposedofcircles,Ptolemyandhisforerunners inventedvariouscomplications tomaketheirtheoriesagreemoreaccuratelywithobservation,fortheSunandMoonaswellasfortheplanets.*One complicationwas just to addmore epicycles. The only planet for which Ptolemy found this

necessarywasMercury,whoseorbitdiffersfromacirclemore thanthatofanyotherplanet.Anothercomplicationwasthe“eccentric”;theEarthwastakentobe,notatthecenterofthedeferentforeachplanet, but at some distance from it. For instance, in Ptolemy’s theory the center of the deferent ofVenuswasdisplacedfromtheEarthby2percentoftheradiusofthedeferent.*The eccentric could be combined with another mathematical device introduced by Ptolemy, the

“equant.”Thisisaprescriptionforgivingaplanetavaryingspeedinitsorbit,apartfromthevariationduetotheplanet’sepicycle.Onemightimaginethat,sittingontheEarth,weshouldseeeachplanet,or

morepreciselythecenterofeachplanet’sepicycle,goingaroundusataconstantrate(say,indegreesofarcperday),butPtolemyknewthatthisdidnotquiteagreewithactualobservation.Onceaneccentricwasintroduced,onemightinsteadimaginethatweshouldseethecentersoftheplanets’epicyclesgoataconstantrate,notaroundtheEarth,butaroundthecentersoftheplanets’deferents.Alas,thatdidn’tworkeither.Instead,foreachplanetPtolemyintroducedwhatcametobecalledanequant,*apointontheoppositesideofthecenterofthedeferentfromtheEarth,butatanequaldistancefromthiscenter;andhesupposedthatthecentersoftheplanets’epicyclesgoataconstantangularrateabouttheequant.ThefactthattheEarthandtheequantareatanequaldistancefromthecenterofthedeferentwasnotassumed on the basis of philosophical preconceptions, but found by leaving these distances as freeparameters,andfindingthevaluesofthedistancesforwhichthepredictionsofthetheorywouldagreewithobservation.There were still sizable discrepancies between Ptolemy’s model and observation. As we will see

whenwecometoKepler inChapter11, if consistentlyused thecombinationofa singleepicycle foreachplanetandaneccentricandequantfortheSunandforeachplanetcandoagoodjobofimitatingtheactualmotionofplanetsincludingtheEarthinellipticalorbits—goodenoughtoagreewithalmostanyobservationthatcouldbemadewithouttelescopes.ButPtolemywasnotconsistent.HedidnotusetheequantindescribingthesupposedmotionoftheSunaroundtheEarth;andthisomission—sincethelocationsofplanetsarereferredtothepositionoftheSun—alsomessedupthepredictionsofplanetarymotions.AsGeorgeSmithhasemphasized,13 it isasignof thedistancebetweenancientormedievalastronomy andmodern science that no one after Ptolemy appears to have taken these discrepanciesseriouslyasaguidetoabettertheory.TheMoonpresentedspecialdifficulties: thesortof theorythatworkedprettywell for theapparent

motionsoftheSunandplanetsdidnotworkwellfortheMoon.ItwasnotunderstooduntiltheworkofIsaacNewtonthatthisisbecausetheMoon’smotionissignificantlyaffectedbythegravitationoftwobodies—the Sun aswell as the Earth—while the planets’motion is almost entirely governed by thegravitationofasinglebody:theSun.HipparchushadproposedatheoryoftheMoon’smotionwithasingleepicycle,whichwasadjustedtoaccountforthelengthoftimebetweeneclipses;butasPtolemyrecognized, thismodeldidnotdowell inpredicting the locationof theMoonon thezodiacbetweeneclipses.Ptolemywasabletofixthisflawwithamorecomplicatedmodel,buthistheoryhaditsownproblems: the distance between theMoon and theEarthwould vary a good deal, leading to amuchlargerchangeintheapparentsizeoftheMoonthanisobserved.Asalreadymentioned,inthesystemofPtolemyandhispredecessorsthereisnowaythatobservation

of theplanetscouldhave indicated the sizesof theirdeferentsandepicycles;observationcouldhavefixedonlytheratioofthesesizesforeachplanet.*PtolemyfilledthisgapinPlanetaryHypotheses,afollow-uptotheAlmagest.Inthisworkheinvokedanaprioriprinciple,perhapstakenfromAristotle,thatthereshouldbenogapsinthesystemoftheworld.EachplanetaswellastheSunandMoonwassupposedtooccupyashell,extendingfromtheminimumtothemaximumdistanceoftheplanetorSunorMoonfromtheEarth,andtheseshellsweresupposedtofittogetherwithnogaps.InthisschemetherelativesizesoftheorbitsoftheplanetsandSunandMoonwereallfixed,onceonedecidedontheirordergoingoutwardfromtheEarth.Also, theMoon iscloseenough to theEarthso that itsabsolutedistance(inunitsoftheradiusoftheEarth)couldbeestimatedinvariousways,includingthemethodofHipparchusdiscussedinChapter7.Ptolemyhimselfdevelopedthemethodofparallax:theratioofthedistancetotheMoonandtheradiusoftheEarthcanbecalculatedfromtheobservedanglebetweenthezenithandthedirectionto theMoonandthecalculatedvaluethat thisanglewouldhaveif theMoonwereobservedfromthecenteroftheEarth.14(SeeTechnicalNote14.)Hence,accordingtoPtolemy’sassumptions,tofindthedistancesoftheSunandplanetsallthatwasnecessarywastoknowtheorderoftheirorbitsaroundtheEarth.

Theinnermostorbitwasalways taken tobe thatof theMoon,because theSunand theplanetsareeachoccasionally eclipsedby theMoon.Also, itwasnatural to suppose that the farthest planets arethosethatappeartotakethelongesttogoaroundtheEarth,soMars,Jupiter,andSaturnweregenerallytakeninthatordergoingawayfromtheEarth.ButtheSun,Venus,andMercuryallonaverageappeartotakeayeartogoaroundtheEarth,sotheirorderremainedasubjectofcontroversy.PtolemyguessedthattheordergoingoutfromtheEarthistheMoon,Mercury,Venus,theSun,andthenMars,Jupiter,and Saturn. Ptolemy’s results for the distances of the Sun, Moon, and planets as multiples of thediameterof theEarthweremuchsmaller than their actualvalues, and for theSunandMoonsimilar(perhapsnotcoincidentally)totheresultsofAristarchusdiscussedinChapter7.Thecomplicationsofepicycles,equants,andeccentricshavegivenPtolemaicastronomyabadname.

But it shouldnotbe thought thatPtolemywasstubbornly introducing thesecomplications inorder tomake up for the mistake of taking the Earth as the unmoving center of the solar system. Thecomplications,beyondjustasingleepicycleforeachplanet(andnonefortheSun),hadnothingtodowithwhethertheEarthgoesaroundtheSunortheSunaroundtheEarth.Theyweremadenecessarybythefact,notunderstooduntilKepler’stime,thattheorbitsarenotcircles,theSunisnotatthecenteroftheorbits,andthevelocitiesarenotconstant.ThesamecomplicationsalsoaffectedtheoriginaltheoryofCopernicus,whoassumed that theorbitsofplanetsand theEarthhad tobecirclesand thespeedsconstant. Fortunately, this is a pretty good approximation, and the simplest version of the epicycletheory, with just one epicycle for each planet and none for the Sun, worked far better than thehomocentric spheres of Eudoxus, Callippus, andAristotle. If Ptolemy had included an equant alongwith an eccentric for the Sun as well as for each planet, the discrepancies between theory andobservationwouldhavebeentoosmalltobedetectedwiththemethodsthenavailable.ButthisdidnotsettletheissuebetweenthePtolemaicandAristoteliantheoriesofplanetarymotions.

The Ptolemaic theory agreed better with observation, but it did violence to the assumption ofAristotelianphysicsthatallcelestialmotionsmustbecomposedofcircleswhosecenteristhecenteroftheEarth. Indeed, thequeer loopingmotionofplanetsmovingonepicycleswouldhavebeenhard toswallowevenforsomeonewhohadnostakeinanyothertheory.For fifteen hundred years the debate continued between the defenders of Aristotle, often called

physicists or philosophers, and the supporters of Ptolemy, generally referred to as astronomers ormathematicians.TheAristoteliansoftenacknowledgedthatthemodelofPtolemyfittedthedatabetter,but they regarded this as just the sort of thing that might interest mathematicians, not relevant forunderstanding reality. Their attitude was expressed in a statement by Geminus of Rhodes, whoflourishedaround70BC,quotedaboutthreecenturieslaterbyAlexanderofAphrodisias,whointurnwasquotedbySimplicius,15inacommentaryonAristotle’sPhysics.Thisstatementlaysoutthegreatdebatebetweennaturalscientists(sometimestranslated“physicists”)andastronomers:

Itistheconcernofphysicalinquirytoenquireintothesubstanceoftheheavensandtheheavenlybodies,theirpowersandthenatureoftheircoming-to-beandpassingaway;byZeus,itcanrevealthetruthabouttheirsize,shape,andpositioning.Astronomydoesnotattempttopronounceonanyofthesequestions,butrevealstheorderednatureofthephenomenaintheheavens,showingthatthe heavens are indeed an ordered cosmos, and it also discusses the shapes, sizes, and relativedistancesoftheEarth,Sun,andMoon,aswellaseclipses,theconjunctionsoftheheavenlybodies,andqualitiesandquantities inherent in theirpaths.Sinceastronomy toucheson the studyof thequantity,magnitude,andqualityof theirshapes, itunderstandablyhasrecoursetoarithmeticandgeometryinthisrespect.Andaboutthesequestions,whicharetheonlyonesitpromisedtogiveanaccount of, it has the power to reach results through the use of arithmetic and geometry. Theastronomer and the natural scientist will accordingly onmany occasions set out to achieve the

sameobjective,forexample,thattheSunisasizeablebody,thattheEarthisspherical,buttheydonot use the samemethodology. For the natural scientist will prove each of his points from thesubstanceoftheheavenlybodies,eitherfromtheirpowers,orfromthefactthattheyarebetterastheyare,orfromtheircoming-to-beandchange,whiletheastronomerarguesfromthepropertiesoftheirshapesandsizes,orfromquantityofmovementandthetimethatcorrespondstoit....Ingeneralitisnottheconcernoftheastronomertoknowwhatbynatureisatrestandwhatbynatureis in motion; he must rather make assumptions about what stays at rest and what moves, andconsiderwithwhichassumptions theappearances in theheavensareconsistent.Hemustgethisfirst basic principles from the natural scientist, namely that the dance of the heavenly bodies issimple,regular,andordered;fromtheseprincipleshewillbeabletoshowthatthemovementofalltheheavenlybodiesiscircular,boththosethatrevolveinparallelcoursesandthosethatwindalongobliquecircles.

The“naturalscientists”ofGeminussharesomecharacteristicsof today’s theoreticalphysicists,butwith huge differences. Following Aristotle, Geminus sees the natural scientists as relying on firstprinciples,includingteleologicalprinciples:thenaturalscientistsupposesthattheheavenlybodies“arebetterastheyare.”ForGeminusitisonlytheastronomerwhousesmathematics,asanadjuncttohisobservations.WhatGeminusdoesnotimagineisthegive-and-takethathasdevelopedbetweentheoryandobservation.Themoderntheoreticalphysicistdoesmakedeductionsfrombasicprinciples,butheuses mathematics in this work, and the principles themselves are expressedmathematically and arelearnedfromobservation,certainlynotbyconsideringwhatis“better.”InthereferencebyGeminustothemotionsofplanets“thatrevolveinparallelcoursesandthosethat

wind alongoblique circles”one can recognize thehomocentric spheres rotatingon tilted axesof theschemesofEudoxus,Callippus,andAristotle,towhichGeminusasagoodAristotelianwouldnaturallybeloyal.Ontheotherhand,AdrastusofAphrodisias,whoaroundAD100wroteacommentaryontheTimaeus,andagenerationlaterthemathematicianTheonofSmyrnaweresufficientlyconvincedbythetheoryofApolloniusandHipparchusthattheytriedtomakeitrespectable,byinterpretingtheepicyclesand deferents as solid transparent spheres, like the homocentric spheres of Aristotle, but now nothomocentric.Somewriters,facingtheconflictbetweentherivaltheoriesoftheplanets,threwuptheirhands,and

declaredthathumanswerenotmeanttounderstandcelestialphenomena.Thus,inthemid–fifthcenturyAD,inhiscommentaryontheTimaeus,theNeoplatonistpaganProclusproclaimed:16

When we are dealing with sublunary things, we are content, because of the instability of thematerialwhich goes to constitute them, to graspwhat happens inmost instances.Butwhenwewant to know heavenly things, we use sensibility and call upon all sorts of contrivances quiteremovedfromlikelihood....Thatthisisthewaythingsstandisplainlyshownbythediscoveriesmade about these heavenly things—from different hypotheses we draw the same conclusionsrelative to the same objects. Among these hypotheses are somewhich save the phenomena bymeans of epicycles, others which do so by means of eccentrics, still others which save thephenomenabymeansofcounterturningspheresdevoidofplanets.Surelythegod’sjudgementismorecertain.Butasforus,wemustbesatisfiedto“comeclose”tothosethings,forwearemen,whospeakaccordingtowhatislikely,andwhoselecturesresemblefables.

Proclus was wrong on three counts. He missed the point that the Ptolemaic theories that usedepicycles and eccentrics did a far better job of “saving the phenomena” than theAristotelian theoryusingthehypothesisofhomocentric“counterturningspheres.”Thereisalsoaminortechnicalpoint:in

referring to hypotheses “which save the phenomena by means of epicycles, others which do so bymeansofeccentrics”Proclusseemsnottorealizethatinthecasewhereanepicyclecanplaytheroleofan eccentric (discussed in footnote *), these are not different hypotheses but different ways ofdescribingwhatismathematicallythesamehypothesis.Aboveall,ProcluswaswronginsupposingthatitishardertounderstandheavenlymotionsthanthosehereonEarth,belowtheorbitoftheMoon.Justthereverseistrue.Weknowhowtocalculatethemotionsofbodiesinthesolarsystemwithexquisiteprecision,butwestillcan’tpredictearthquakesorhurricanes.ButProcluswasnotalone.Wewillseehisunwarrantedpessimismregardingthepossibilityofunderstandingthemotionoftheplanetsrepeatedcenturieslater,byMosesMaimonides.Writinginthefirstdecadeofthetwentiethcentury,thephysicistturnedphilosopherPierreDuhem17

tookthesideofthePtolemaicsbecausetheirmodelfittedthedatabetter,buthedisapprovedofTheonandAdrastusfortryingtolendrealitytothemodel.Perhapsbecausehewasdeeplyreligious,Duhemsoughttorestricttheroleofsciencemerelytotheconstructionofmathematicaltheoriesthatagreewithobservation, rather thanencompassingefforts toexplainanything. I amnot sympathetic to thisview,becausetheworkofmygenerationofphysicistscertainlyfeelslikeexplanationasweordinarilyusetheword, not like mere description.18 True, it is not so easy to draw a precise distinction betweendescription and explanation. I would say that we explain some generalization about the world byshowing how it follows from some more fundamental generalization, but what do we mean byfundamental?Still,IthinkweknowwhatwemeanwhenwesaythatNewton’slawsofgravitationandmotion are more fundamental than Kepler’s three laws of planetary motion. The great success ofNewton was in explaining the motions of the planets, not merely describing them. Newton did notexplaingravitation,andheknew thathehadnot,but that is theway it always iswithexplanation—somethingisalwaysleftforfutureexplanation.

Becauseoftheiroddmotions,theplanetswereuselessasclocksorcalendarsorcompasses.TheywereputtoadifferentsortofuseinHellenistictimesandafterwardforpurposesofastrology,afalsesciencelearned from theBabylonians.*The sharpmodern distinction between astronomy and astrologywaslessclearintheancientandmedievalworlds,becausethelessonhadnotyetbeenlearnedthathumanconcernswereirrelevanttothelawsgoverningthestarsandplanets.GovernmentsfromthePtolemiesonsupportedthestudyofastronomylargelyinthehopethatitwouldrevealthefuture,andsonaturallyastronomersspentmuchoftheirtimeonastrology.Indeed,ClaudiusPtolemywastheauthornotonlyofthe greatest astronomical work of antiquity, the Almagest, but also of a textbook of astrology, theTetrabiblos.ButIcan’tleaveGreekastronomyonthissournote.ForahappierendingtoPartIIofthisbook,I’ll

quotePtolemyonhispleasureinastronomy:19

I know that I ammortal and the creature of a day; butwhen I search out themassedwheelingcirclesofthestars,myfeetnolongertouchtheEarth,but,sidebysidewithZeushimself,Itakemyfillofambrosia,thefoodofthegods.

PARTIII

THEMIDDLEAGES

Sciencereachedheights intheGreekpartof theancientworldthatwerenotregaineduntil thescientific revolution of the sixteenth and seventeenth centuries. The Greeks made the greatdiscoverythatsomeaspectsofnature,especiallyinopticsandastronomy,couldbedescribedbyprecisemathematicalnaturalistic theories thatagreewithobservation.Whatwas learnedaboutlightandtheheavenswasimportant,butevenmoreimportantwaswhatwaslearnedaboutthesortofthingthatcouldbelearned,andhowtolearnit.NothingduringtheMiddleAges,eitherintheIslamicworldorinChristianEurope,compares

withthis.ButthemillenniuminterveningbetweenthefallofRomeandthescientificrevolutionwas not an intellectual desert.The achievements ofGreek sciencewere preserved and in somecasesimprovedintheinstitutionsofIslamandthenintheuniversitiesofEurope.Inthisway,thegroundwaspreparedforthescientificrevolution.ItwasnotonlytheachievementsofGreeksciencethatwerepreservedintheMiddleAges.We

willseeinmedievalIslamandChristendomacontinuationoftheancientdebatesovertheroleinscienceofphilosophy,ofmathematics,andofreligion.

9

TheArabs

After thecollapse in the fifthcenturyof thewesternRomanEmpire, theGreek-speakingeasternhalfcontinued as theByzantineEmpire, and even increased in extent. TheByzantineEmpire achieved aclimacticmilitary success during the reign of the emperorHeraclius,whose army inAD 627 in thebattleofNinevehdestroyedthearmyofthePersianEmpire,theancientenemyofRome.ButwithinadecadetheByzantineshadtoconfrontamoreformidableadversary.TheArabswereknowninantiquityasabarbarianpeople,livingintheborderlandoftheRomanand

Persianempires,“thatjustdividesthedesertandthesown.”Theywerepagans,withareligioncenteredatthecityofMecca,inthesettledportionofwesternArabiaknownastheHejaz.Startingattheendofthe 500s,Muhammad, an inhabitant ofMecca, set out to convert his fellow citizens tomonotheism.Meetingopposition,heandhisacolytesfledin622toMedina,whichtheythenusedasamilitarybasefortheconquestofMeccaandofmostoftheArabianPeninsula.AfterMuhammad’sdeath in632, amajorityofMuslims followed the authorityof four successive

leadersheadquarteredatfirstatMedina:hiscompanionsandrelativesAbuBakr,Omar,Othman,andAli.Theyare recognized todaybySunniMuslimsas the“four rightlyguidedcaliphs.”TheMuslimsconqueredtheByzantineprovinceofSyriain636,justnineyearsafterthebattleofNineveh,andthenwentontoseizePersia,Mesopotamia,andEgypt.TheirconquestsintroducedtheArabstoamorecosmopolitanworld.Forinstance,theArabgeneral

Amrou,whoconqueredAlexandria, reported to thecaliphOmar,“Ihave takenacity,ofwhichIcanonly say that it contains 6000 palaces, 4000 baths, 400 theatres, 12,000 greengrocers, and 40,000Jews.”1Aminority, the forerunnersof today’sShiites, acceptedonly theauthorityofAli, the fourthcaliph

andthehusbandofMuhammad’sdaughterFatima.Thesplit intheworldofIslambecamepermanentafterarevoltagainstAli,inwhichAliaswellashissonHusseinwaskilled.Anewdynasty,theSunniUmmayadcaliphate,wasestablishedatDamascusin661.Under theUmmayadsArab conquests expanded to include the territories ofmodernAfghanistan,

Pakistan,Libya,Tunisia,Algeria,andMorocco,mostofSpain,andmuchofcentralAsiabeyond theOxusRiver.FromtheformerlyByzantinelandsthattheynowruledtheybegantoabsorbGreekscience.SomeGreek learning also came from Persia,whose rulers hadwelcomedGreek scholars (includingSimplicius) before the rise of Islam, when the Neoplatonic Academy was closed by the emperorJustinian.Christendom’slossbecameIslam’sgain.ItwasinthetimeofthenextSunnidynasty,thecaliphateoftheAbbasids,thatArabscienceentered

itsgoldenage.Baghdad,thecapitalcityoftheAbbasids,wasbuiltonbothsidesoftheTigrisRiverinMesopotamiabyal-Mansur,caliphfrom754to775.Itbecamethelargestcityintheworld,oratleastthe largestoutsideChina. Itsbest-knownrulerwasHarunal-Rashid,caliph from786 to809, famousfromAThousandandOneNights. Itwasunderal-Rashidandhissonal-Mamun,caliphfrom813 to833,thattranslationfromGreece,Persia,andIndiareacheditsgreatestscope.Al-Mamunsentamissionto Constantinople that brought back manuscripts in Greek. The delegation probably included thephysicianHunayn ibn Ishaq, the greatest of the ninth-century translators,who founded a dynasty oftranslators, traininghis son andnephew to carryon thework.Hunayn translatedworksofPlato andAristotle, as well as medical texts of Dioscorides, Galen, and Hippocrates. Mathematical works of

Euclid, Ptolemy, and others were also translated into Arabic at Baghdad, some through Syriacintermediaries. The historian Philip Hitti has nicely contrasted the state of learning at this time atBaghdadwiththeilliteracyofEuropeintheearlyMiddleAges:“ForwhileintheEastal-Rashidandal-Mamun were delving into Greek and Persian philosophy, their contemporaries in the West,Charlemagneandhislords,weredabblingintheartofwritingtheirnames.”2It is sometimes said that the greatest contribution to science of the Abbasid caliphs was the

foundation of an institute for translation and original research, the Bayt al-Hikmah, or House ofWisdom.ThisinstituteissupposedtohaveservedfortheArabssomewhatthesamefunctionthattheMuseumandLibraryofAlexandriaservedfortheGreeks.ThisviewhasbeenchallengedbyascholarofArabiclanguageandliterature,DimitriGutas.3HepointsoutthatBaytal-HikmahisatranslationofaPersiantermthathadlongbeenusedinpre-IslamicPersiaforstorehousesofbooks,mostlyofPersianhistoryandpoetryrather thanofGreekscience.Thereareonlya fewknownexamplesofworks thatweretranslatedattheBaytal-Hikmahinthetimeofal-Mamun,andthosearefromPersianratherthanGreek.Someastronomicalresearch,asweshallsee,wasgoingonat theBaytal-Hikmah,butlittleisknownof its scope.What isnot indispute is that,whetherornot at theBayt al-Hikmah, the cityofBaghdaditselfinthetimeofal-Mamunandal-Rashidwasagreatcenteroftranslationandresearch.ArabsciencewasnotlimitedtoBaghdad,butspreadwesttoEgypt,Spain,andMorocco,andeastto

Persia and centralAsia. Participating in thisworkwere not onlyArabs but also Persians, Jews, andTurks. They were verymuch a part of Arab civilization and wrote in Arabic (or at least in Arabicscript).ArabicthenhadsomethinglikethestatusinsciencethatEnglishhastoday.Insomecasesitisdifficulttodecideontheethnicbackgroundofthesefigures.Iwillconsiderthemalltogether,undertheheading“Arabs.”Asa roughapproximation,wecan identify twodifferent scientific traditions thatdivided theArab

savants.Ononehand,therewererealmathematiciansandastronomerswhowerenotmuchconcernedwith what today we would call philosophy. Then there were philosophers and physicians, not veryactive inmathematics, and strongly influenced byAristotle. Their interest in astronomywas chieflyastrological.Wheretheywereconcernedatallwiththetheoryoftheplanets,thephilosopher/physiciansfavoredtheAristoteliantheoryofspherescenteredontheEarth,whiletheastronomer/mathematiciansgenerallyfollowedthePtolemaictheoryofepicyclesanddeferentsdiscussedinChapter8.Thiswasanintellectualschismthat,asweshallsee,wouldpersistinEuropeuntilthetimeofCopernicus.TheachievementsofArabscienceweretheworkofmanyindividuals,noneofthemclearlystanding

outfromtherestas,say,GalileoandNewtonstandoutinthescientificrevolution.WhatfollowsisabriefgalleryofmedievalArabscientiststhatIhopemaygivesomeideaoftheiraccomplishmentsandvariety.Thefirstoftheimportantastronomer/mathematiciansatBaghdadwasal-Khwarizmi,*aPersianborn

around 780 in what is nowUzbekistan. Al-Khwarizmi worked at the Bayt al-Hikmah and preparedwidelyusedastronomicaltablesbasedinpartonHinduobservations.HisfamousbookonmathematicswasHisah-al-Jabrw-al-Muqabalah,dedicatedtothecaliphal-Mamun(whowashalfPersianhimself).From its titlewe derive theword “algebra.”But thiswas not really a book onwhat is today calledalgebra.Formulasliketheoneforthesolutionofquadraticequationsweregiveninwords,not inthesymbolsthatareanessentialelementofalgebra.(Inthisrespect,al-Khwarizmi’smathematicswaslessadvanced than that ofDiophantus.) From al-Khwarizmiwe also get our name for a rule for solvingproblems, “algorithm.” The text ofHisah al-Jabr w-al-Muqabalah contains a confusing mixture ofRomannumerals;Babyloniannumbersbasedonpowersof60;andanewsystemofnumberslearnedfrom India, based on powers of 10. Perhaps the most important mathematical contribution of al-KhwarizmiwashisexplanationtotheArabsoftheseHindunumbers,whichinturnbecameknowninEuropeasArabicnumbers.

Inadditiontotheseniorfigureofal-Khwarizmi,therewerecollectedinBaghdadaproductivegroupofotherninth-centuryastronomers,includingal-Farghani(Alfraganus),*whowroteapopularsummaryofPtolemy’sAlmagestanddevelopedhisownversionoftheplanetaryschemedescribedinPtolemy’sPlanetaryHypotheses.Itwasamajoroccupationof thisBaghdadgroup to improveonEratosthenes’measurementof the

size of the Earth. Al-Farghani in particular reported a smaller circumference, which centuries laterencouragedColumbus (asmentioned in an earlier footnote) to think that he could survive an oceanvoyagewestwardfromSpaintoJapan,perhapstheluckiestmiscalculationinhistory.TheArabwhowasmostinfluentialamongEuropeanastronomerswasal-Battani(Albatenius),born

around 858BC in northernMesopotamia.He used and corrected Ptolemy’sAlmagest,makingmoreaccuratemeasurementsofthe~23½°anglebetweentheSun’spaththroughthezodiacandthecelestialequator, of the lengths of the year and the seasons, of the precession of the equinoxes, and of thepositionsofstars.Heintroducedatrigonometricquantity,thesine,fromIndia,inplaceofthecloselyrelated chord used and calculated byHipparchus. (SeeTechnicalNote15.)Hisworkwas frequentlyquotedbyCopernicusandTychoBrahe.ThePersianastronomeral-Sufi(Azophi)madeadiscoverywhosecosmologicalsignificancewasnot

recognized until the twentieth century. In 964, in hisBook of the Fixed Stars, he described a “littlecloud”alwayspresentintheconstellationAndromeda.Thiswastheearliestknownobservationofwhatare now called galaxies, in this case the large spiral galaxy M31. Working at Isfahan, al-Sufi alsoparticipatedintranslatingworksofGreekastronomyintoArabic.PerhapsthemostimpressiveastronomeroftheAbbasiderawasal-Biruni.Hisworkwasunknownin

medievalEurope,so there isno latinizedversionofhisname.Al-Biruni lived incentralAsia,and in1017visitedIndia,wherehelecturedonGreekphilosophy.HeconsideredthepossibilitythattheEarthrotates, gave accurate values for the latitude and longitude of various cities, prepared a table of thetrigonometric quantity known as the tangent, and measured specific gravities of various solids andliquids. He scoffed at the pretensions of astrology. In India, al-Biruni invented a new method formeasuringthecircumferenceoftheEarth.Ashedescribedit:4

WhenIhappenedtobelivinginthefortofNandanainthelandofIndia,Iobservedfromahighmountainstandingtothewestofthefort,alargeplainlyingsouthofthemountain.ItoccurredtomethatIshouldexaminethismethod[amethoddescribedpreviously]there.So,fromthetopofthemountain,ImadeanempiricalmeasurementofthecontactbetweentheEarthandthebluesky.I found that the line of sight [to the horizon] had dipped below the reference line [that is, thehorizontaldirection]bytheamount34minutesofarc.ThenImeasuredtheperpendicularof themountain [that is, itsheight] and found it tobe652.055cubits,where thecubit is a standardoflengthusedinthatregionformeasuringcloth.*

From these data, al-Biruni concluded that the radius of the Earth is 12,803,337.0358 cubits.Somethingwentwrongwith his calculation; from the data he quoted, he should have calculated theradius as about 13.3million cubits. (SeeTechnicalNote16.)Of course, he could not possibly haveknown the height of the mountain to the accuracy he stated, so there was no practical differencebetween12.8millioncubitsand13.3millioncubits.IngivingtheradiusoftheEarthto12significantfigures,al-Biruniwasguiltyofmisplacedprecision,thesameerrorthatwesawinAristarchus:carryingout calculations and quoting results to a much greater degree of precision than is warranted by theaccuracyofthemeasurementsonwhichthecalculationisbased.I once got into trouble in this way. I had a summer job long ago, calculating the path of atoms

throughaseriesofmagnetsinanatomicbeamapparatus.Thiswasbeforedesktopcomputersorpocket

electronic calculators, but I had an electromechanical calculating machine that could add, subtract,multiply,anddividetoeightsignificantfigures.Outoflaziness,inmyreportIgavetheresultsofthecalculations to eight significant figures just as they came from the calculating machine, withoutbotheringtoroundthemofftoarealisticprecision.Mybosscomplainedtomethatthemagneticfieldmeasurementsonwhichmycalculationwasbasedwereaccurate toonlya fewpercent, and that anyprecisionbeyondthiswasmeaningless.Inanycase,wecan’tnowjudgetheaccuracyofal-Biruni’sresultthattheEarth’sradiusisabout13

millioncubits,becausenoonenowknowsthelengthofhiscubit.Al-Birunisaidthereare4,000cubitsinamile,butwhatdidhemeanbyamile?OmarKhayyam, thepoetandastronomer,wasborn in1048 inNishapur, inPersia, anddied there

around1131.HeheadedtheobservatoryatIsfahan,wherehecompiledastronomicaltablesandplannedcalendarreform.InSamarkandincentralAsiahewroteabouttopicsinalgebra,suchasthesolutionofcubic equations. He is best known to English-speaking readers as a poet, through the magnificentnineteenth-centurytranslationbyEdwardFitzgeraldof75outofalargernumberofquatrainswrittenbyOmarKhayyaminPersian,andknownasTheRubaiyat.Unsurprisinglyforthehardheadedrealistwhowrotetheseverses,hestronglyopposedastrology.ThegreatestArabcontributionstophysicsweremadeinoptics,firstattheendofthetenthcenturyby

IbnSahl,whomayhaveworkedouttherulegivingthedirectionofrefractedraysoflight(aboutwhichmore in Chapter 13), and then by the great al-Haitam (Alhazen). Al-Haitam was born in Basra, insouthernMesopotamia,around965,butworkedinCairo.HisextantbooksincludeOptics,TheLightoftheMoon,TheHaloandtheRainbow,OnParaboloidalBurningMirrors,TheFormationofShadows,The Light of the Stars, Discourse on Light, The Burning Sphere, andThe Shape of the Eclipse. Hecorrectlyattributedthebendingoflightinrefractiontothechangeinthespeedoflightwhenitpassesfromonemediumtoanother,andfoundexperimentallythattheangleofrefractionisproportionaltotheangleofincidenceonlyforsmallangles.Buthedidnotgivethecorrectgeneralformula.Inastronomy,he followed Adrastus and Theon in attempting to give a physical explanation to the epicycles anddeferentsofPtolemy.Anearly chemist, Jabir ibnHayyan, is nowbelieved tohave flourished in the late eighthor early

ninthcentury.Hislifeisobscure,anditisnotclearwhetherthemanyArabicworksattributedtohimarereallytheworkofasingleperson.ThereisalsoalargebodyofLatinworksthatappearedinEuropeinthethirteenthandfourteenthcenturiesattributedtoa“Geber,”butitisnowthoughtthattheauthoroftheseworks is not the same as the author of theArabicworks attributed to Jabir ibnHayyan. Jabirdevelopedtechniquesofevaporation,sublimation,melting,andcrystallization.Hewasconcernedwithtransmutingbasemetals intogold,andhenceisoftencalledanalchemist,but thedistinctionbetweenchemistryandalchemyaspracticedinhistimeisartificial,fortherewasthennofundamentalscientifictheorytotellanyonethatsuchtransmutationsareimpossible.Tomymind,adistinctionmoreimportantforthefutureofscienceisbetweenthosechemistsoralchemistswhofollowedDemocritusinviewingtheworkings ofmatter in a purely naturalisticway,whether their theorieswere right orwrong, andthoselikePlato(and,unless theywerespeakingmetaphorically,AnaximanderandEmpedocles),whobroughthumanorreligiousvaluesintothestudyofmatter.Jabirprobablybelongstothelatterclass.Forinstance, hemademuch of the chemical significance of 28, the number of letters in the alphabet ofArabic,thelanguageoftheKoran.Somehowitwasimportantthat28istheproductof7,supposedtobethenumberofmetals,and4,thenumberofqualities:cold,warm,wet,anddry.Theearliestmajor figure in theArabmedical/philosophical traditionwasal-Kindi (Alkindus),who

wasborninBasraofanoblefamilybutworkedinBaghdadintheninthcentury.HewasafollowerofAristotle,andtriedtoreconcileAristotle’sdoctrineswiththoseofPlatoandofIslam.Al-Kindiwasapolymath,veryinterestedinmathematics,butlikeJabirhefollowedthePythagoreansinusingitasa

sortofnumbermagic.Hewroteaboutopticsandmedicine,andattackedalchemy,thoughhedefendedastrology.Al-KindialsosupervisedsomeoftheworkoftranslationfromGreektoArabic.More impressivewasal-Razi (Rhazes), anArabic-speakingPersianof thegeneration followingal-

Kindi.HisworksincludeATreatiseon theSmallPoxandMeasles. InDoubtsConcerningGalen,hechallenged the authority of the influential Roman physician and disputed the theory, going back toHippocrates, that health is a matter of balance of the four humors (described in Chapter 4). Heexplained,“Medicine isaphilosophy,and this isnotcompatiblewithrenouncementofcriticismwithregard to the leading authors.” In an exception to the typical views ofArab physicians, al-Razi alsochallengedAristotle’steaching,suchasthedoctrinethatspacemustbefinite.The most famous of the Islamic physicians was Ibn Sina (Avicenna), another Arabic-speaking

Persian. Hewas born in 980 near Bokhara in central Asia, became court physician to the sultan ofBokhara, andwasappointedgovernorofaprovince. IbnSinawasanAristotelian,who likeal-KinditriedtoreconcilewithIslam.HisAlQanumwasthemostinfluentialmedicaltextoftheMiddleAges.Atthesametime,medicinebegantoflourishinIslamicSpain.Al-Zahrawi(Abulcasis)wasbornin

936nearCórdoba,themetropolisofAndalusia,andworkedthereuntilhisdeathin1013.Hewasthegreatest surgeon of the Middle Ages, and highly influential in Christian Europe. Perhaps becausesurgerywaslessburdenedthanotherbranchesofmedicinebyill-foundedtheory,al-Zahrawisoughttokeepmedicineseparatefromphilosophyandtheology.The divorce ofmedicine from philosophy did not last. In the following century the physician Ibn

Bajjah(Avempace)wasborninSaragossa,andworkedthereandinFez,Seville,andGranada.Hewasan Aristotelian who criticized Ptolemy and rejected Ptolemaic astronomy, but he took exception toAristotle’stheoryofmotion.Ibn Bajjah was succeeded by his student Ibn Tufayl (Abubacer), also born inMuslim Spain. He

practicedmedicineinGranada,Ceuta,andTangier,andhebecamevizierandphysiciantothesultanoftheAlmohaddynasty.HearguedthatthereisnocontradictionbetweenAristotleandIslam,andlikehisteacherrejectedtheepicyclesandeccentricsofPtolemaicastronomy.In turn, IbnTufayl had a distinguished student, al-Bitruji.Hewas an astronomer but inherited his

teacher’s commitment to Aristotle and rejection of Ptolemy. Al-Bitruji unsuccessfully attempted toreinterpretthemotionofplanetsonepicyclesintermsofhomocentricspheres.OnephysicianofMuslimSpainbecamemore famousasaphilosopher. IbnRushd (Averroes)was

bornin1126atCórdoba,thegrandsonofthecity’simam.Hebecamecadi (judge)ofSeville in1169andofCórdobain1171,andthenontherecommendationofIbnTufaylbecamecourtphysicianin1182.Asamedicalscientist,IbnRushdisbestknownforrecognizingthefunctionoftheretinaoftheeye,buthis fame rests chiefly on his work as a commentator on Aristotle. His praise of Aristotle is almostembarrassingtoread:

[Aristotle] founded and completed logic, physics, andmetaphysics. I say that he founded thembecausetheworkswrittenbeforehimonthesesciencesarenotworthtalkingaboutandarequiteeclipsedbyhisownwritings.And I say thathe completed thembecausenoonewhohas comeafter him up to our own time, that is, for nearly fifteen hundred years, has been able to addanythingtohiswritingsortofindanyerrorofanyimportanceinthem.5

The father of the modern author Salman Rushdie chose the surname Rushdie to honor the secularrationalismofIbnRushd.Naturally Ibn Rushd rejected Ptolemaic astronomy, as contrary to physics, meaning Aristotle’s

physics.HewasawarethatAristotle’shomocentricspheresdidnot“savetheappearances,”andhetriedtoreconcileAristotlewithobservationbutconcludedthatthiswasataskforthefuture:

In my youth I hoped it would be possible for me to bring this research [in astronomy] to asuccessfulconclusion.Now,inmyoldage,Ihavelosthope,forseveralobstacleshavestoodinmyway. But what I say about it will perhaps attract the attention of future researchers. Theastronomical science of our days surely offers nothing from which one can derive an existingreality. The model that has been developed in the times in which we live accords with thecomputations,notwithexistence.6

Ofcourse,IbnRushd’shopesforfutureresearcherswereunfulfilled;nooneeverwasabletomaketheAristoteliantheoryoftheplanetswork.There was also serious astronomy done inMuslim Spain. In Toledo al-Zarqali (Arzachel) in the

eleventh centurywas the first tomeasure the precession of the apparent orbit of theSun around theEarth(actuallyofcoursetheprecessionoftheorbitoftheEartharoundtheSun),whichisnowknowntobemostlyduetothegravitationalattractionbetweentheEarthandotherplanets.Hegaveavalueforthisprecessionof12.9"(secondsofarc)peryear, infairagreementwith themodernvalue,11.6"peryear.7Agroupofastronomersincludingal-Zarqaliusedtheearlierworkofal-Khwarizmiandal-BattanitoconstructtheTablesofToledo,asuccessortotheHandyTablesofPtolemy.TheseastronomicaltablesandtheirsuccessorsdescribedindetailtheapparentmotionsoftheSun,Moon,andplanetsthroughthezodiacandwerelandmarksinthehistoryofastronomy.Under the Ummayad caliphate and its successor, the Berber Almoravid dynasty, Spain was a

cosmopolitan center of learning, hospitable to Jews as well as Muslims. Moses ben Maimon(Maimonides),aJew,wasbornin1135atCórdobaduringthishappytime.JewsandChristianswerenevermorethansecond-classcitizensunderIslam,butduringtheMiddleAgestheconditionofJewswas generally far better under the Arabs than in Christian Europe. Unfortunately for benMaimon,duringhisyouthSpaincameundertheruleofthefanaticalIslamistAlmohadcaliphate,andhehadtoflee,tryingtofindrefugeinAlmeira,Marrakesh,Caesarea,andCairo,finallycomingtorestinFustat,asuburbofCairo.Thereuntilhisdeathin1204heworkedbothasarabbi,withinfluencethroughouttheworld ofmedieval Jewry, and as a highly prized physician to bothArabs and Jews.His best-knownworkistheGuidetothePerplexed,whichtakestheformofletterstoaperplexedyoungman.InitheexpressedhisrejectionofPtolemaicastronomyascontrarytoAristotle:8

You know of Astronomy as much as you have studied with me, and learned from the bookAlmagest;wehadnotsufficienttimetogobeyondthis.Thetheorythatthespheresmoveregularly,and that the assumedcoursesof the stars are inharmonywithobservation, depends, asyou areaware,ontwohypotheses:wemustassumeeitherepicycles,oreccentricspheres,oracombinationofboth.NowIwillshowthateachofthesetwohypothesesisirregular,andtotallycontrarytotheresultsofNaturalScience.

HethenwentontoacknowledgethatPtolemy’sschemeagreeswithobservation,whileAristotle’sdoesnot, and as Proclus did before him, ben Maimon despaired at the difficulty of understanding theheavens:

Butofthethingsintheheavensmanknowsnothingexceptafewmathematicalcalculations,andyouseehowfarthesego.Isayinthewordsofthepoet9“TheheavensaretheLord’s,buttheEarthhehasgiventothesonsofman”;thatistosay,Godalonehasaperfectandtrueknowledgeoftheheavens, theirnature, their essence, their form, theirmotion, and their causes;butHegavemanpowertoknowthethingswhichareundertheheavens.

Justtheoppositeturnedouttobetrue;itwasthemotionofheavenlybodiesthatwasfirstunderstoodintheearlydaysofmodernscience.ThereistestimonytotheinfluenceofArabscienceonEuropeinalonglistofwordsderivedfrom

Arabic originals: not only algebra and algorithm, but also names of stars like Aldebaran, Algol,Alphecca,Altair,Betelgeuse,Mizar,Rigel,Vega, and soon, andchemical terms likealkali, alembic,alcohol,alizarin,andofcoursealchemy.Thisbrief survey leavesuswithaquestion:whywas it specifically thosewhopracticedmedicine,

suchasIbnBajjah,IbnTufayl,IbnRushd,andbenMaimon,whoheldonsofirmlytotheteachingsofAristotle?Icanthinkofthreepossiblereasons.First,physicianswouldnaturallybemostinterestedinAristotle’s writings on biology, and in these Aristotle was at his best. Also, Arab physicians werepowerfully influencedby thewritingsofGalen,whogreatlyadmiredAristotle.Finally,medicine isafieldinwhichthepreciseconfrontationoftheoryandobservationwasverydifficult(andstillis),sothatthe failings of Aristotelian physics and astronomy to agree in detail with observationmay not haveseemedsoimportanttophysicians.Incontrast,theworkofastronomerswasusedforpurposeswherecorrect precise results are essential, such as constructing calendars; measuring distances on Earth;telling the correct times for daily prayers; and determining theqibla, the direction toMecca,whichshouldbefacedduringprayer.EvenastronomerswhoappliedtheirsciencetoastrologyhadtobeabletotellpreciselyinwhatsignofthezodiactheSunandplanetswereonanygivendate;andtheywerenotlikelytotolerateatheorylikeAristotle’sthatgavethewronganswers.The Abbasid caliphate came to an end in 1258, when the Mongols under Hulegu Khan sacked

Baghdadandkilled the caliph.Abbasid rulehaddisintegratedwellbefore that.Political andmilitarypower had passed from the caliphs to Turkish sultans, and even the caliph’s religious authoritywasweakened by the founding of independent Islamic governments: a translated Ummayad caliphate inSpain,theFatimidcaliphateinEgypt,theAlmoraviddynastyinMoroccoandSpain,succeededbytheAlmohad caliphate in North Africa and Spain. Parts of Syria and Palestine were temporarilyreconqueredbyChristians,firstbyByzantinesandthenbyFrankishcrusaders.ArabsciencehadalreadybeguntodeclinebeforetheendoftheAbbasidcaliphate,perhapsbeginning

aboutAD1100.After that, therewerenomorescientistswith thestatureofal-Battani,al-Biruni, IbnSina,andal-Haitam.Thisisacontroversialpoint,andthebitternessofthecontroversyisheightenedbytoday’spolitics.Somescholarsdenythattherewasanydecline.10It is certainly true that some science continued even after the end of the Abbasid era, under the

MongolsinPersiaandtheninIndia,andlaterundertheOttomanTurks.Forinstance,thebuildingoftheMaragha observatory in Persia was ordered byHulegu in 1259, just a year after his sacking ofBaghdad, ingratitudefor thehelpthathethought thatastrologershadgivenhiminhisconquests.Itsfounding director, the astronomer al-Tusi, wrote about spherical geometry (the geometry obeyed bygreatcirclesonasphericalsurface,likethenotionalsphereofthefixedstars),compiledastronomicaltables, and suggestedmodifications to Ptolemy’s epicycles.Al-Tusi founded a scientific dynasty: hisstudent al-Shirazi was an astronomer and mathematician, and al-Shirazi’s student al-Farisi didgroundbreakingworkonoptics,explainingtherainbowanditscolorsastheresultoftherefractionofsunlightinraindrops.More impressive, it seems to me, is Ibn al-Shatir, a fourteenth-century astronomer of Damascus.

Following earlier work of theMaragha astronomers, he developed a theory of planetarymotions inwhichPtolemy’sequantwasreplacedwithapairofepicycles,thussatisfyingPlato’sdemandthatthemotionofplanetsmustbecompoundedofmotionsatconstantspeedaroundcircles.Ibnal-Shatiralsogave a theory of the Moon’s motion based on epicycles: it avoided the excessive variation in thedistance of the Moon from the Earth that had afflicted Ptolemy’s lunar theory. The early work ofCopernicusreportedinhisCommentarioluspresentsa lunar theory that is identical to Ibnal-Shatir’s,

and a planetary theory that gives the same apparent motions as the theory of al-Shatir.11 It is nowthoughtthatCopernicuslearnedoftheseresults(ifnotoftheirsource)asayoungstudentinItaly.Someauthorshavemademuchof the fact thatageometricconstruction, the“Tusicouple,”which

hadbeeninventedbyal-Tusiinhisworkonplanetarymotion,waslaterusedbyCopernicus.(Thiswasawayofmathematicallyconvertingrotarymotionoftwotouchingspheresintooscillationinastraightline.) It is a matter of some controversy whether Copernicus learned of the Tusi couple fromArabsources,orinventedithimself.12HewasnotunwillingtogivecredittoArabs,andquotedfiveofthem,includingal-Battani,al-Bitruji,andIbnRushd,butmadenomentionofal-Tusi.Itisrevealingthat,whatevertheinfluenceofal-TusiandIbnal-ShatironCopernicus,theirworkwas

notfollowedupamongIslamicastronomers.Inanycase,theTusicoupleandtheplanetaryepicyclesofIbnal-Shatirweremeansofdealingwiththecomplicationsthat(thoughneitheral-Tusinoral-ShatirnorCopernicusknewit)areactuallyduetotheellipticalorbitsofplanetsandtheoff-centerlocationoftheSun.Thesearecomplicationsthat(asdiscussedinChapters8and11)equallyaffectedPtolemaicandCopernican theories,andhadnothing todowithwhether theSungoesaround theEarthor theEartharoundtheSun.NoArabastronomerbeforemoderntimesseriouslyproposedaheliocentrictheory.ObservatoriescontinuedtobebuiltinIslamiccountries.Thegreatestmayhavebeenanobservatory

inSamarkand,builtinthe1420sbytherulerUlughBegoftheTimuriddynastyfoundedbyTimurLenk(Tamburlaine).Theremoreaccuratevalueswerecalculatedforthesiderealyear(365days,5hours,49minutes, and15seconds)and theprecessionof theequinoxes (70 rather than75yearsperdegreeofprecession,ascomparedwiththemodernvalueof71.46yearsperdegree).AnimportantadvanceinmedicinewasmadejustaftertheendoftheAbbasidperiod.Thiswasthe

discoverybytheArabphysicianIbnal-Nafisofthepulmonarycirculation,thecirculationofbloodfromtherightsideoftheheartthroughthelungs,whereitmixeswithair,andthenflowsbacktotheheart’sleftside.Ibnal-NafisworkedathospitalsinDamascusandCairo,andalsowroteonophthalmology.Theseexamplesnotwithstanding,itishardtoavoidtheimpressionthatscienceintheIslamicworld

begantolosemomentumtowardtheendoftheAbbasidera,andthencontinuedtodecline.Whenthescientificrevolutioncame,ittookplaceonlyinEurope,notinthelandsofIslam,anditwasnotjoinedbyIslamicscientists.Evenafter telescopesbecameavailable in theseventeenthcentury,astronomicalobservatories inIslamiccountriescontinuedtobelimitedtonaked-eyeastronomy13 (thoughaidedbyelaborateinstruments),undertakenlargelyforcalendricalandreligiousratherthanscientificpurposes.Thispictureofdeclineinevitablyraisesthesamequestionthatwasraisedbythedeclineofscience

toward the end of the Roman Empire—do these declines have anything to do with the advance ofreligion? For Islam, as for Christianity, the case for a conflict between science and religion iscomplicated,andIwon’tattemptadefiniteanswer.Thereareat least twoquestionshere.First,whatwasthegeneralattitudeofIslamicscientiststowardreligion?Thatis,wasitonlythosewhosetasidetheinfluenceoftheirreligionwhowerecreativescientists?Andsecond,whatwastheattitudetowardscienceofMuslimsociety?Religiousskepticismwaswidespreadamongscientistsof theAbbasidera.Theclearestexample is

providedbytheastronomerOmarKhayyam,generallyregardedasanatheist.HerevealshisskepticisminseveralversesoftheRubaiyat:14

SomefortheGloriesoftheWorld,andsomeSighfortheProphet’sParadisetocome;Ah,taketheCash,andlettheCreditgo,NorheedtherumbleofadistantDrum!

WhyalltheSaintsandSageswhodiscuss’d

OfthetwoWorldssolearnedly,arethrustLikefoolishProphetsforth;theirWordstoScornArescatter’d,andtheirmouthsarestoptwithDust.

MyselfwhenyoungdideagerlyfrequentDoctorandSaint,andheardgreatargumentAboutit,andabout:butevermoreCameoutbythesamedoorasinIwent.

(TheliteraltranslationintoEnglishisofcourselesspoetic,butexpressesessentiallythesameattitude.)NotfornothingwasKhayyamafterhisdeathcalled“astingingserpenttotheShari’ah.”TodayinIran,government censorship requires that published versions of the poetry ofKhayyammust be edited toremoveorrevisehisatheisticsentiments.TheAristotelianIbnRushdwasbanishedaround1195onsuspicionofheresy.Anotherphysician,al-

Razi,wasanoutspokenskeptic.InhisTricksoftheProphetshearguedthatmiraclesaremere tricks,thatpeopledonotneedreligiousleaders,andthatEuclidandHippocratesaremoreusefultohumanitythan religious teachers. His contemporary, the astronomer al-Biruni, was sufficiently sympathetic totheseviewstowriteanadmiringbiographyofal-Razi.Ontheotherhand,thephysicianIbnSinahadanastycorrespondencewithal-Biruni,andsaidthatal-

Razishouldhavestucktothingsheunderstood,likeboilsandexcrement.Theastronomeral-TusiwasadevoutShiite,andwroteabouttheology.Thenameoftheastronomeral-SufisuggeststhathewasaSufimystic.It is hard to balance these individual examples.Most Arab scientists have left no record of their

religiousleanings.Myownguessisthatsilenceismorelikelyanindicationofskepticismandperhapsfearthanofdevotion.ThenthereisthequestionoftheattitudeofMuslimsingeneraltowardscience.Thecaliphal-Mamun

who founded theHouse ofWisdomwas certainly an important supporter of science, and itmay besignificant that he belonged to a Muslim sect, the Mutazalites, which sought a more rationalinterpretation of the Koran, and later came under attack for this. But theMutazalites should not beregardedasreligiousskeptics.TheyhadnodoubtthattheKoranisthewordofGod;theyarguedonlythatitwascreatedbyGod,andhadnotalwaysexisted.Norshouldtheybeconfusedwithmoderncivillibertarians;theypersecutedMuslimswhothoughtthattherewasnoneedforGodtohavecreatedtheeternalKoran.Bytheeleventhcentury,thereweresignsinIslamofoutrighthostilitytoscience.Theastronomeral-

BirunicomplainedaboutantiscientificattitudesamongIslamicextremists:15

Theextremist among themwould stamp the sciences as atheistic, andwouldproclaim that theyleadpeopleastrayinordertomakeignoramuses,likehim,hatethesciences.Forthiswillhelphimtoconcealhisignorance,andtoopenthedoortothecompletedestructionofscienceandscientists.

There is awell-known anecdote, according towhich al-Biruniwas criticized by a religious legalist,because the astronomer was using an instrument that listed the months according to their names inGreek,thelanguageoftheChristianByzantines.Al-Birunireplied,“TheByzantinesalsoeatfood.”The key figure in the growth of tension between science and Islam is often said to be al-Ghazali

(Algazel).Bornin1058inPersia,hemovedtoSyriaandthentoBaghdad.Healsomovedaboutagooddeal intellectually,fromorthodoxIslamtoskepticismandthenbacktoorthodoxy,butcombinedwithSufi mysticism. After absorbing the works of Aristotle, and summarizing them in Inventions of the

Philosophers, he later attacked rationalism in his best-known work, The Incoherence of thePhilosophers.16 (Ibn Rushd, the partisan of Aristotle, wrote a riposte, The Incoherence of theIncoherence.)Hereishowal-GhazaliexpressedhisviewofGreekphilosophy:

Thehereticsinourtimeshaveheardtheawe-inspiringnamesofpeoplelikeSocrates,Hippocrates,Plato,Aristotle,etc.Theyhavebeendeceivedbytheexaggerationsmadebythefollowersofthesephilosophers—exaggerations to the effect that the ancient masters possessed extraordinaryintellectualpowers;thatthemathematical, logical,physicalandmetaphysicalsciencesdevelopedby them are the most profound; that their excellent intelligence justifies their bold attempts todiscover the Hidden Things by deductive methods; and that with all the subtlety of theirintelligenceandtheoriginalityoftheiraccomplishmentstheyrepudiatedtheauthorityofreligiouslaws:denied thevalidityof thepositivecontentsofhistorical religions,andbelieve thatallsuchthingsareonlysanctimoniousliesandtrivialities.

Al-Ghazali’sattackonsciencetooktheformof“occasionalism”—thedoctrinethatwhateverhappensisasingularoccasion,governednotbyanylawsofnaturebutdirectlybythewillofGod.(Thisdoctrinewas not new in Islam—it had been advanced a century earlier by al-Ashari, an opponent of theMutazalites.) In al-Ghazali’s Problem XVII, “Refutation of Their Belief in the Impossibility of aDeparturefromtheNaturalCourseofEvents,”onereads:

In our view, the connection between what are believed to be the cause and the effect [is] notnecessary. . . . [God]has thepower to create the satisfactionofhungerwithout eating,ordeathwithouttheseveranceofthehead,oreventhesurvivaloflifewhentheheadhasbeencutoff,orany other thing from among the connected things (independently ofwhat is supposed to be itscause). The philosophers deny this possibility; indeed, they assert its impossibility. Since theinquiry concerning these things (which are innumerable)may go to an indefinite length, let usconsideronlyoneexample—viz., theburningofapieceofcottonat the timeof itscontactwithfire.Weadmitthepossibilityofacontactbetweenthetwowhichwillnotresultinburning,asalsoweadmitthepossibilityofatransformationofcottonintoasheswithoutcomingintocontactwithfire.Andtheyrejectthispossibility....WesaythatitisGodwho—throughtheintermediacyofangels,ordirectly—istheagentofthecreationofblacknessincotton;orofthedisintegrationofitsparts,andtheirtransformationintoasmoulderingheaporashes.Fire,whichisaninanimatething,hasnoaction.

Otherreligions,suchasChristianityandJudaism,alsoadmitthepossibilityofmiracles,departuresfromthe natural order, but here we see that al-Ghazali denied the significance of any natural orderwhatsoever.Thisishardtounderstand,becausewecertainlyobservesomeregularitiesinnature.Idoubtthatal-

Ghazaliwasunawarethatitwasnotsafetoputone’shandintofire.HecouldhavesavedaplaceforscienceintheworldofIslam,asastudyofwhatGodusuallywills tohappen,apositiontakenintheseventeenthcenturybyNicolasMalebranche.Butal-Ghazalididnottakethispath.Hisreasonisspelledout in another work, The Beginning of Sciences,17 in which he compared science to wine. Winestrengthens the body, but is nevertheless forbidden to Muslims. In the same way, astronomy andmathematicsstrengthenthemind,but“weneverthelessfearthatonemightbeattractedthroughthemtodoctrinesthataredangerous.”Itisnotonlythewritingsofal-GhazalithatbearwitnesstoagrowingIslamichostilitytosciencein

theMiddleAges.In1194inAlmohadCórdoba,attheotherendoftheIslamicworldfromBaghdad,the

Ulama (the local religious scholars) burned all medical and scientific books. And in 1449 religiousfanaticsdestroyedUlughBeg’sobservatoryinSamarkand.WeseeinIslamtodaysignsofthesameconcernsthattroubledal-Ghazali.MyfriendthelateAbdus

Salam,aPakistaniphysicistwhowonthefirstNobelPrizeinscienceawardedtoaMuslim(forworkdoneinEnglandandItaly),oncetoldmethathehadtriedtopersuadetherulersoftheoil-richPersianGulf states to invest in scientific research. He found that they were enthusiastic about supportingtechnology, but they feared that pure science would be culturally corrosive. (Salam was himself adevoutMuslim.HewasloyaltoaMuslimsect,theAhmadiyya,whichhasbeenregardedashereticalinPakistan,andforyearshecouldnotreturntohishomecountry.)It is ironic that in the twentiethcenturySayyidQutb, aguiding spiritofmodern radical Islamism,

called for the replacement of Christianity, Judaism, and the Islam of his own day with a universalpurifiedIslam,inpartbecausehehopedinthiswaytocreateanIslamicsciencethatwouldclosethegap between science and religion. But Arab scientists in their golden age were not doing Islamicscience.Theyweredoingscience.

10

MedievalEurope

AstheRomanEmpiredecayedintheWest,EuropeoutsidetherealmofByzantiumbecamepoor,rural,andlargelyilliterate.Wheresomeliteracydidsurvive,itwasconcentratedinthechurch,andthereonlyinLatin.InWesternEuropeintheearlyMiddleAgesvirtuallynoonecouldreadGreek.SomefragmentsofGreeklearninghadsurvivedinmonasterylibrariesasLatintranslations,including

partsofPlato’sTimaeusandtranslationsaroundAD500bytheRomanaristocratBoethiusofAristotle’swork on logic and of a textbook of arithmetic. Therewere alsoworkswritten in Latin byRomans,describingGreekscience.Mostnotablewasafifth-centuryencyclopediaoddlytitledTheMarriageofMercury and Philology by Martianus Capella, which treated (as handmaidens of philology) sevenliberalarts:grammar,logic,rhetoric,geography,arithmetic,astronomy,andmusic.InhisdiscussionofastronomyMartianusdescribedtheoldtheoryofHeraclidesthatMercuryandVenusgoaroundtheSunwhiletheSungoesaroundtheEarth,adescriptionpraisedamillenniumlaterbyCopernicus.Butevenwiththeseshredsofancientlearning,EuropeansintheearlyMiddleAgesknewalmostnothingofthegreat scientific achievementsof theGreeks.Batteredby repeated invasionsofGoths,Vandals,Huns,Avars,Arabs,Magyars,andNorthmen,thepeopleofWesternEuropehadotherconcerns.Europebegantoreviveinthetenthandeleventhcenturies.Theinvasionswerewindingdown,and

newtechniquesimprovedtheproductivityofagriculture.1Itwasnotuntilthelatethirteenthcenturythatsignificantscientificworkwouldbeginagain,andnotmuchwouldbeaccomplisheduntilthesixteenthcentury,butintheintervalaninstitutionalandintellectualfoundationwaslaidfortherebirthofscience.In the tenthandeleventhcenturies—a religiousage—muchof thenewwealthofEuropenaturally

wentnottothepeasantrybuttothechurch.AswonderfullydescribedaroundAD1030bytheFrenchchroniclerRaoul (orRadulfus)Glaber, “Itwas as if theworld, shaking itself andputtingoff the oldthings,wereputtingonthewhiterobeofchurches.”Forthefutureoflearning,mostimportantweretheschoolsattachedtocathedrals,suchasthoseatOrléans,Reims,Laon,Cologne,Utrecht,Sens,Toledo,Chartres,andParis.Theseschoolstrainedtheclergynotonlyinreligionbutalsoinasecularliberalartscurriculumleft

over from Roman times, based in part on the writings of Boethius and Martianus: the trivium ofgrammar, logic, and rhetoric; and, especially at Chartres, the quadrivium of arithmetic, geometry,astronomy,andmusic.SomeoftheseschoolswentbacktothetimeofCharlemagne,butintheeleventhcenturytheybegantoattractschoolmastersofintellectualdistinction,andatsomeschoolstherewasarenewedinterest inreconcilingChristianitywithknowledgeof thenaturalworld.AsremarkedbythehistorianPeterDear,2“LearningaboutGodbylearningwhatHehadmade,andunderstandingthewhysandwhereforesofitsfabric,wasseenbymanyasaneminentlypiousenterprise.”Forinstance,ThierryofChartres,whotaughtatParisandChartresandbecamechancelloroftheschoolatChartresin1142,explainedtheoriginoftheworldasdescribedinGenesisintermsofthetheoryofthefourelementshelearnedfromtheTimaeus.Anotherdevelopmentwasevenmoreimportant thanthefloweringofthecathedralschools, though

notunrelatedtoit.Thiswasanewwaveoftranslationsoftheworksofearlierscientists.Translationswereat firstnot somuchdirectly fromGreekas fromArabic:either theworksofArabscientists,orworksthathadearlierbeentranslatedfromGreektoArabicorGreektoSyriactoArabic.

The enterprise of translation began early, in the middle of the tenth century, for instance at themonastery of SantaMaria deRipoli in the Pyrenees, near the border betweenChristian Europe andUmmayadSpain.ForanillustrationofhowthisnewknowledgecouldspreadinmedievalEurope,andits influence on the cathedral schools, consider the career of Gerbert d’Aurillac. Born in 945 inAquitaine of obscure parents, he learned someArabmathematics and astronomy inCatalonia; spenttimeinRome;wenttoReims,wherehelecturedonArabicnumbersandtheabacusandreorganizedthecathedralschool;becameabbotandthenarchbishopofReims;assistedinthecoronationofthefounderof a new dynasty of French kings,HughCapet; followed theGerman emperorOtto III to Italy andMagdeburg;becamearchbishopofRavenna;andin999waselectedpope,asSylvesterII.HisstudentFulbert ofChartres studied at the cathedral school ofReims and then became bishop ofChartres in1006,presidingovertherebuildingofitsmagnificentcathedral.The pace of translation accelerated in the twelfth century. At the century’s start, an Englishman,

Adelard of Bath, traveled extensively in Arab countries; translated works of al-Khwarizmi; and, inNaturalQuestions,reportedonArablearning.SomehowThierryofChartreslearnedoftheuseofzeroin Arab mathematics, and introduced it into Europe. Probably the most important twelfth-centurytranslatorwasGerardofCremona.HeworkedinToledo,whichhadbeenthecapitalofChristianSpainbeforetheArabconquests,andthoughreconqueredbyCastiliansin1085remainedacenterofArabandJewish culture. His Latin translation from Arabic of Ptolemy’s Almagest made Greek astronomyavailabletomedievalEurope.GerardalsotranslatedEuclid’sElementsandworksbyArchimedes,al-Razi,al-Ferghani,Galen,IbnSina,andal-Khwarizmi.AfterArabSicilyfell to theNormans in1091,translationswerealsomadedirectlyfromGreektoLatin,withnorelianceonArabicintermediaries.Thetranslationsthathadthegreatest immediate impactwereofAristotle.ItwasinToledothat the

bulkofAristotle’sworkwastranslatedfromArabicsources;forinstance,thereGerardtranslatedOntheHeavens,Physics,andMeteorology.Aristotle’sworkswerenotuniversallywelcomedin thechurch.MedievalChristianityhadbeenfar

more influenced by Platonism and Neoplatonism, partly through the example of Saint Augustine.Aristotle’swritingswerenaturalisticinawaythatPlato’swerenot,andhisvisionofacosmosgovernedbylaws,evenlawsasill-developedashiswere,presentedanimageofGod’shandsinchains,thesameimagethathadsodisturbedal-Ghazali.TheconflictoverAristotlewasatleastinpartaconflictbetweentwonewmendicantorders:theFranciscans,orgrayfriars,foundedin1209,whoopposedtheteachingof Aristotle; and the Dominicans, or black friars, founded around 1216, who embraced “ThePhilosopher.”ThisconflictwaschieflycarriedoutinnewEuropeaninstitutionsofhigherlearning,theuniversities.

Oneof thecathedral schools, atParis, receiveda royal charter as auniversity in1200. (Therewasaslightly older university at Bologna, but it specialized in law and medicine, and did not play animportantroleinmedievalphysicalscience.)Almostimmediately,in1210,scholarsattheUniversityofPariswereforbidden to teach thebooksofAristotleonnaturalphilosophy.PopeGregoryIXin1231calledforAristotle’sworkstobeexpurgated,sothattheusefulpartscouldbesafelytaught.ThebanonAristotlewasnotuniversal.HisworksweretaughtattheUniversityofToulousefromits

foundingin1229.AtParisthetotalbanonAristotlewasliftedin1234,andinsubsequentdecadesthestudyofAristotlebecame thecenterofeducation there.Thiswas largely theworkof two thirteenth-century clerics:AlbertusMagnus andThomasAquinas. In the fashionof the times, theywere givengranddoctoraltitles:Albertuswasthe“UniversalDoctor,”andThomasthe“AngelicDoctor.”AlbertusMagnus studied in Padua and Cologne, became aDominican friar, and in 1241went to

Paris,wherefrom1245to1248heoccupiedaprofessorialchairforforeignsavants.LaterhemovedtoCologne, where he founded its university. Albertus was a moderate Aristotelian who favored thePtolemaic system over Aristotle’s homocentric spheres but was concerned about its conflict with

Aristotle’sphysics.HespeculatedthattheMilkyWayconsistsofmanystarsand(contrarytoAristotle)that themarkings on theMoon are intrinsic imperfections.The example ofAlbertuswas followed alittlelaterbyanotherGermanDominican,DietrichofFreiburg,whoindependentlyduplicatedsomeofal-Farisi’sworkontherainbow.In1941theVaticandeclaredAlbertusthepatronsaintofallscientists.ThomasAquinaswasbornamemberoftheminornobilityofsouthernItaly.Afterhiseducationat

themonasteryofMonteCassinoandtheUniversityofNaples,hedisappointedhisfamily’shopesthathewouldbecometheabbotofarichmonastery;instead,likeAlbertusMagnus,hebecameaDominicanfriar.ThomaswenttoParisandCologne,wherehestudiedunderAlbertus.HethenreturnedtoParis,andservedasprofessorattheuniversityin1256–1259and1269–1272.The great work of Aquinas was the Summa Theologica, a comprehensive fusion of Aristotelian

philosophyandChristiantheology.Init,hetookamiddlegroundbetweenextremeAristotelians,knownas Averroists after Ibn Rushd; and the extreme anti-Aristotelians, such as members of the newlyfounded Augustinian order of friars. Aquinas strenuously opposed a doctrine that was widely (butprobably unjustly) attributed to thirteenth-century Averroists like Siger of Brabant and Boethius ofDacia.Accordingtothisdoctrine,itispossibletoholdopinionstrueinphilosophy,suchastheeternityofmatterortheimpossibilityoftheresurrectionofthedead,whileacknowledgingthattheyarefalseinreligion.ForAquinas, therecouldbeonlyonetruth.Inastronomy,Aquinas leanedtowardAristotle’shomocentric theory of the planets, arguing that this theory was founded on reason while Ptolemaictheorymerely agreedwith observation, and another hypothesismight also fit the data.On the otherhand,AquinasdisagreedwithAristotleonthetheoryofmotion;hearguedthateveninavacuumanymotionwouldtakeafinitetime.ItisthoughtthatAquinasencouragedtheLatintranslationofAristotle,Archimedes, and others directly from Greek sources by his contemporary, the English DominicanWilliamofMoerbeke.By1255studentsatPariswerebeingexaminedontheirknowledgeoftheworksofAquinas.ButAristotle’stroubleswerenotover.Startinginthe1250s,theoppositiontoAristotleatPariswas

forcefullyledbytheFranciscanSaintBonaventure.Aristotle’sworkswerebannedatToulousein1245by Pope Innocent IV. In 1270 the bishop of Paris, Étienne Tempier, banned the teaching of 13Aristotelianpropositions.PopeJohnXXIorderedTempiertolookfurtherintothematter,andin1277Tempier condemned 219 doctrines of Aristotle or Aquinas.3 The condemnation was extended toEnglandbyRobertKilwardy,thearchbishopofCanterbury,andrenewedin1284byhissuccessor,JohnPecham.Thepropositionscondemnedin1277canbedividedaccordingtothereasonsfortheircondemnation.

Somepresentedadirectconflictwithscripture—forinstance,propositionsthatstatetheeternityoftheworld:

9.Thattherewasnofirstman,norwilltherebealast;onthecontrary,therealwayswasandalwayswillbethegenerationofmanfromman.

87.Thattheworldiseternalastoallthespeciescontainedinit;andthattimeiseternal,asaremotion,matter,agent,andrecipient.

Some of the condemned doctrines described methods of learning truth that challenged religiousauthority,forinstance:

38.Thatnothingshouldbebelievedunlessitisself-evidentorcouldbeassertedfromthingsthatareself-evident.

150.Thatonanyquestion,amanoughtnottobesatisfiedwithcertitudebaseduponauthority.153.Thatnothingisknownbetterbecauseofknowingtheology.

Finally,someofthecondemnedpropositionshadraisedthesameissuethathadconcernedal-Ghazali,thatphilosophicalandscientificreasoningseemstolimitthefreedomofGod,forexample:

34.Thatthefirstcausecouldnotmakeseveralworlds. 49.ThatGod could notmove the heavenswith rectilinearmotion, and the reason is that a vacuum

wouldremain.141.ThatGodcannotmakeanaccidentexistwithoutasubjectnormakemore[thanthree]dimensions

existsimultaneously.

ThecondemnationofpropositionsofAristotleandAquinasdidnotlast.UndertheauthorityofanewpopewhohadbeeneducatedbyDominicans,JohnXXII,ThomasAquinaswascanonizedin1323.In1325 the condemnation was rescinded by the bishop of Paris, who decreed: “We wholly annul theaforementionedcondemnationofarticlesandjudgmentsofexcommunicationastheytouch,oraresaidtotouch,theteachingofblessedThomas,mentionedabove,andbecauseofthisweneitherapprovenordisapproveofthesearticles,butleavethemforfreescholasticdiscussion.”4In1341mastersofartsatthe University of Paris were required to swear they would teach “the system of Aristotle and hiscommentator Averroes, and of the other ancient commentators and expositors of the said Aristotle,exceptinthosecasesthatarecontrarytothefaith.”5Historiansdisagreeabout the importance for the futureof scienceof thisepisodeofcondemnation

and rehabilitation. There are two questions here:Whatwould have been the effect on science if thecondemnationhadnotbeen rescinded?Andwhatwouldhavebeen theeffecton science if therehadneverbeenanycondemnationoftheteachingsofAristotleandAquinas?It seems to me that the effect on science of the condemnation if not rescinded would have been

disastrous.ThisisnotbecauseoftheimportanceofAristotle’sconclusionsaboutnature.Mostofthemwerewrong,anyway.ContrarytoAristotle,therewasatimebeforetherewereanymen;therecertainlyaremanyplanetarysystems,andtheremaybemanybigbangs;thingsintheheavenscanandoftendomoveinstraightlines;thereisnothingimpossibleaboutavacuum;andinmodernstringtheoriestherearemorethanthreedimensions,withtheextradimensionsunobservedbecausetheyaretightlycurledup. The danger in the condemnation came from the reasonswhy propositionswere condemned, notfromthedenialofthepropositionsthemselves.EventhoughAristotlewaswrongaboutthelawsofnature,itwasimportanttobelievethatthereare

lawsofnature.Ifthecondemnationofgeneralizationsaboutnaturelikepropositions34,49,and141,onthe ground thatGod can do anything, had been allowed to stand, thenChristianEuropemight havelapsedintothesortofoccasionalismurgedonIslambyal-Ghazali.Also, thecondemnationofarticlesthatquestionedreligiousauthority(suchasarticles38,150,and

153 quoted above) was in part an episode in the conflict between the faculties of liberal arts andtheologyinmedievaluniversities.Theologyhadadistinctlyhigherstatus; itsstudyledtoadegreeofdoctor of theology, while liberal arts faculties could confer no degree higher than master of arts.(Academicprocessionswereheadedbydoctorsoftheology,law,andmedicineinthatorder,followedbythemastersofarts.)Liftingthecondemnationdidnotgivetheliberalartsequalstatuswiththeology,butithelpedtofreetheliberalartsfacultiesfromintellectualcontrolbytheirtheologicalcolleagues.Itishardertojudgewhatwouldhavebeentheeffectifthecondemnationshadneveroccurred.Aswe

willsee,theauthorityofAristotleonmattersofphysicsandastronomywasincreasinglychallengedatParisandOxfordinthefourteenthcentury,thoughsometimesnewideashadtobecamouflagedasbeingmerelysecundumimaginationem—thatis,somethingimagined,ratherthanasserted.WouldchallengestoAristotle have been possible if his authority had not beenweakened by the condemnations of thethirteenthcentury?DavidLindberg6citestheexampleofNicoleOresme(aboutwhommorelater),who

in1377arguedthatitispermissibletoimaginethattheEarthmovesinastraightlinethroughinfinitespace,because“TosaythecontraryistomaintainanarticlecondemnedinParis.”7Perhapsthecourseofevents in the thirteenthcenturycanbesummarizedbysayingthat thecondemnationsavedsciencefrom dogmatic Aristotelianism, while the lifting of the condemnation saved science from dogmaticChristianity.After theeraof translationand theconflictover the receptionofAristotle,creativescientificwork

beganat last inEurope in the fourteenthcentury.The leading figurewasJeanBuridan,aFrenchmanbornin1296nearArras,whospentmuchofhislifeinParis.Buridanwasacleric,butsecular—thatis,notamemberofanyreligiousorder.Inphilosophyhewasanominalist,whobelievedintherealityofindividual things, not of classes of things. Twice Buridan was honored by election as rector of theUniversityofParis,in1328and1340.Buridan was an empiricist, who rejected the logical necessity of scientific principles: “These

principlesarenot immediatelyevident; indeed,wemaybeindoubtconcerningthemfora longtime.But they are called principles because they are indemonstrable, and cannot be deduced from otherpremises nor be proved by any formal procedure, but they are accepted because they have beenobservedtobetrueinmanyinstancesandtobefalseinnone.”8Understanding this was essential for the future of science, and not so easy. The old impossible

Platonicgoalofapurelydeductivenaturalsciencestoodinthewayofprogressthatcouldbebasedonlyoncarefulanalysisofcarefulobservation.Eventodayonesometimesencountersconfusionaboutthis.Forinstance,thepsychologistJeanPiaget9 thoughthehaddetectedsignsthatchildrenhaveaninnateunderstanding of relativity, which they lose later in life, as if relativity were somehow logically orphilosophicallynecessary,ratherthanaconclusionultimatelybasedonobservationsofthingsthattravelatornearthespeedoflight.Thoughanempiricist,Buridanwasnotanexperimentalist.LikeAristotle’s,hisreasoningwasbased

oneverydayobservation,buthewasmorecautious thanAristotle in reachingbroadconclusions.Forinstance, Buridan confronted an old problem of Aristotle: why a projectile thrown horizontally orupward does not immediately start what was supposed to be its naturalmotion, straight downward,whenitleavesthehand.Onseveralgrounds,BuridanrejectedAristotle’sexplanationthattheprojectilecontinuesforawhiletobecarriedbytheair.First,theairmustresistratherthanassistmotion,sinceitmustbedividedapartforasolidbodytopenetrateit.Further,whydoestheairmove,whenthehandthat threwtheprojectilestopsmoving?Also,a lancethat ispointedinbackmovesthroughtheairaswellasorbetterthanonethathasabroadrearonwhichtheaircanpush.Rather than supposing that airkeepsprojectilesmoving,Buridan supposed that this is aneffectof

somethingcalled“impetus,”whichthehandgivestheprojectile.Aswehaveseen,asomewhatsimilarideahadbeenproposedbyJohnofPhiloponus,andBuridan’simpetuswasinturnaforeshadowingofwhatNewtonwastocall“quantityofmotion,”orinmoderntermsmomentum,thoughitisnotpreciselythe same.Buridan sharedwithAristotle theassumption that somethinghas tokeepmoving things inmotion,andheconceivedofimpetusasplayingthisrole,ratherthanasbeingonlyapropertyofmotion,likemomentum.Heneveridentifiedtheimpetuscarriedbyabodyasitsmasstimesitsvelocity,whichishowmomentumisdefinedinNewtonianphysics.Nevertheless,hewasontosomething.Theamountofforcethatisrequiredtostopamovingbodyinagiventimeisproportionaltoitsmomentum,andinthissensemomentumplaysthesameroleasBuridan’simpetus.Buridanextendedtheideaofimpetustocircularmotion,supposingthatplanetsarekeptmovingby

their impetus, an impetus given to them by God. In this way, Buridan was seeking a compromisebetweenscienceandreligionofasort thatbecamepopularcenturieslater:Godsetsthemachineryofthecosmos inmotion,afterwhichwhathappens isgovernedby the lawsofnature.Butalthough theconservationofmomentumdoeskeeptheplanetsmoving,byitselfitcouldnotkeepthemmovingon

curved orbits as Buridan thoughtwas done by impetus; that requires an additional force, eventuallyrecognizedastheforceofgravitation.BuridanalsotoyedwithanideadueoriginallytoHeraclides,thattheEarthrotatesonceadayfrom

westtoeast.HerecognizedthatthiswouldgivethesameappearanceasiftheheavensrotatedaroundastationaryEarthonceadayfromeasttowest.Healsoacknowledgedthatthisisamorenaturaltheory,sincetheEarthissomuchsmallerthatthefirmamentofSun,Moon,planets,andstars.ButherejectedtherotationoftheEarth,reasoningthatiftheEarthrotated,thenanarrowshotstraightupwardwouldfalltothewestofthearcher,sincetheEarthwouldhavemovedunderthearrowwhileitwasinflight.ItisironicthatBuridanmighthavebeensavedfromthiserrorifhehadrealizedthattheEarth’srotationwouldgivethearrowanimpetusthatwouldcarryittotheeastalongwiththerotatingEarth.Instead,hewasmisledbythenotionofimpetus;heconsideredonlytheverticalimpetusgiventothearrowbythebow,notthehorizontalimpetusittakesfromtherotationoftheEarth.Buridan’snotionofimpetusremainedinfluentialforcenturies.ItwasbeingtaughtattheUniversity

of Padua when Copernicus studiedmedicine there in the early 1500s. Later in that century GalileolearnedaboutitasastudentattheUniversityofPisa.BuridansidedwithAristotleonanotherissue,theimpossibilityofavacuum.Buthecharacteristically

basedhisconclusiononobservations:whenairissuckedoutofadrinkingstraw,avacuumispreventedbyliquidbeingpulledupintothestraw;andwhenthehandlesofabellowsarepulledapart,avacuumispreventedbyairrushingintothebellows.Itwasnaturaltoconcludethatnatureabhorsavacuum.Aswewill see inChapter12, thecorrect explanation for thesephenomena in termsof airpressurewasnotunderstooduntilthe1600s.Buridan’sworkwas carried further by two of his students:Albert of Saxony andNicoleOresme.

Albert’swritings on philosophy becamewidely circulated, but itwasOresmewhomade the greatercontributiontoscience.Oresmewasborn in1325inNormandy,andcametoParis tostudywithBuridanin the1340s.He

wasavigorousopponentoflookingintothefuturebymeansof“astrology,geomancy,necromancy,oranysucharts,iftheycanbecalledarts.”In1377OresmewasappointedbishopofthecityofLisieux,inNormandy,wherehediedin1382.Oresme’sbookOntheHeavensandtheEarth10(writteninthevernacularfortheconvenienceofthe

kingofFrance)hastheformofanextendedcommentaryonAristotle,inwhichagainandagainhetakesissuewithThePhilosopher. In thisbookOresmereconsidered the idea that theheavensdonot rotateabout the Earth from east to west but, rather, the Earth rotates on its axis from west to east. BothBuridan and Oresme recognized that we observe only relative motion, so seeing the heavens moveleaves open the possibility that it is instead the Earth that ismoving.Oresmewent through variousobjections to the idea, and picked them apart. Ptolemy in theAlmagest had argued that if theEarthrotated,thencloudsandthrownobjectswouldbeleftbehind;andaswehaveseen,BuridanhadarguedagainsttheEarth’srotationbyreasoningthatiftheEarthrotatedfromwesttoeast,thenanarrowshotstraightupwardwouldbeleftbehindbytheEarth’srotation,contrarytotheobservationthatthearrowseems to fall straightdown to thesamespoton theEarth’ssurface fromwhich itwasshotverticallyupward.OresmerepliedthattheEarth’srotationcarriesthearrowwithit,alongwiththearcherandtheairandeverythingelseontheEarth’ssurface,thusapplyingBuridan’stheoryofimpetusinawaythatitsauthorhadnotunderstood.OresmeansweredanotherobjectiontotherotationoftheEarth—anobjectionofaverydifferentsort,

that therearepassages inHolyScripture (suchas in theBookof Joshua) that refer to theSungoingdailyaroundtheEarth.Oresmerepliedthatthiswasjustaconcessiontothecustomsofpopularspeech,aswhere it iswritten thatGodbecameangryor repented—things thatcouldnotbe taken literally. Inthis,OresmewasfollowingtheleadofThomasAquinas,whohadwrestledwiththepassageinGenesis

whereGodissupposedtoproclaim,“Let therebeafirmamentabovethewaters,andlet itdividethewatersfromthewaters.”AquinashadexplainedthatMoseswasadjustinghisspeechtothecapacityofhisaudience,andshouldnotbetakenliterally.Biblicalliteralismcouldhavebeenadragontheprogressof science, if therehadnotbeenmany inside thechurch likeAquinasandOresmewho tookamoreenlightenedview.Despiteallhisarguments,OresmefinallysurrenderedtothecommonideaofastationaryEarth,as

follows:

Afterward,itwasdemonstratedhowitcannotbeprovedconclusivelybyargumentthattheheavensmove....However,everyonemaintains,andIthinkmyself,thattheheavensdomoveandnottheEarth:ForGodhasestablished theworldwhichshallnotbemoved, inspiteofcontraryreasonsbecausetheyareclearlynotconclusivepersuasions.However,afterconsideringall thathasbeensaid,onecouldthenbelievethattheEarthmovesandnottheheavens,fortheoppositeisnotself-evident.However,atfirstsight,thisseemsasmuchagainstnaturalreasonthanallormanyofthearticles of our faith.What I have said by way of diversion or intellectual exercise can in thismannerserveasavaluablemeansofrefutingandcheckingthosewhowouldliketoimpugnourfaithbyargument.11

WedonotknowifOresmereallywasunwilling to take thefinalstep towardacknowledging that theEarthrotates,orwhetherhewasmerelypayinglipservicetoreligiousorthodoxy.OresmealsoanticipatedoneaspectofNewton’stheoryofgravitation.Hearguedthatheavythingsdo

notnecessarilytendtofalltowardthecenterofourEarth,iftheyarenearsomeotherworld.Theideathat theremaybeotherworlds,moreor less like theEarth,was theologicallydaring.DidGodcreatehumanson thoseotherworlds?DidChristcome to thoseotherworlds to redeemthosehumans?Thequestionsareendless,andsubversive.Unlike Buridan, Oresme was a mathematician. His major mathematical contribution led to an

improvementonworkdoneearlieratOxford,sowemustnowshiftourscenefromFrancetoEngland,andbackalittleintime,thoughwewillreturnsoontoOresme.By the twelfth centuryOxfordhadbecomeaprosperousmarket townon theupper reachesof the

Thames,andbegantoattractstudentsandteachers.TheinformalclusterofschoolsatOxfordbecamerecognizedasauniversityintheearly1200s.Oxfordconventionallylistsitslineofchancellorsstartingin1224withRobertGrosseteste, laterbishopofLincoln,whobegantheconcernofmedievalOxfordwithnaturalphilosophy.GrossetestereadAristotleinGreek,andhewroteonopticsandthecalendaraswellasonAristotle.HewasfrequentlycitedbythescholarswhosucceededhimatOxford.InRobertGrossetesteandtheOriginsofExperimentalScience,12A.C.Crombiewentfurther,giving

Grossetesteapivotalroleinthedevelopmentofexperimentalmethodsleadingtotheadventofmodernphysics. This seems rather an exaggeration ofGrosseteste’s importance.As is clear fromCrombie’saccount,“experiment”forGrossetestewasthepassiveobservationofnature,notverydifferentfromthemethod ofAristotle.NeitherGrosseteste nor any of hismedieval successors sought to learn generalprinciples by experiment in the modern sense, the aggressive manipulation of natural phenomena.Grosseteste’stheorizinghasalsobeenpraised,13butthereisnothinginhisworkthatbearscomparisonwiththedevelopmentofquantitativelysuccessfultheoriesoflightbyHero,Ptolemy,andal-Haitam,orofplanetarymotionbyHipparchus,Ptolemy,andal-Biruni,amongothers.Grosseteste had a great influence on Roger Bacon, who in his intellectual energy and scientific

innocencewasatruerepresentativeofthespiritofhistimes.AfterstudyingatOxford,Baconlecturedon Aristotle in Paris in the 1240s, went back and forth between Paris and Oxford, and became aFranciscanfriararound1257.LikePlato,hewasenthusiasticaboutmathematicsbutmadelittleuseof

it.Hewroteextensivelyonopticsandgeography,butaddednothing important to theearlierworkofGreeks andArabs.Toan extent thatwas remarkable for the time,Baconwas also anoptimist abouttechnology:

Alsocarscanbemadesothatwithoutanimalstheywillmovewithunbelievablerapidity....Alsoflyingmachinescanbeconstructedsothatamansitsinthemidstofthemachinerevolvingsomeenginebywhichartificialwingsaremadetobeattheairlikeaflyingbird.14

Appropriately,Baconbecameknownas“DoctorMirabilis.”In 1264 the first residential collegewas founded atOxford byWalter deMerton, at one time the

chancellorofEnglandandlaterbishopofRochester.ItwasatMertonCollegethatseriousmathematicalwork at Oxford began in the fourteenth century. The key figures were four fellows of the college:ThomasBradwardine (c. 1295–1349),WilliamHeytesbury (fl. 1335),RichardSwineshead (fl. 1340–1355),andJohnofDumbleton(fl.1338–1348).Theirmostnotableachievement,knownastheMertonCollegemeanspeedtheorem,forthefirsttimegivesamathematicaldescriptionofnonuniformmotion—thatis,motionataspeedthatdoesnotremainconstant.The earliest surviving statement of this theorem is by William of Heytesbury (chancellor of the

UniversityofOxfordin1371),inRegulaesolvendisophismata.Hedefinedthevelocityatanyinstantinnonuniformmotion as the ratio of the distance traveled to the time that would have elapsed if themotionhadbeenuniformatthatvelocity.Asitstands,thisdefinitioniscircular,andhenceuseless.Amoremoderndefinition,possiblywhatHeytesburymeant tosay, is that thevelocityatany instant innonuniformmotionistheratioofthedistancetraveledtothetimeelapsedifthevelocitywerethesameasitisinaveryshortintervaloftimearoundthatinstant,soshortthatduringthisintervalthechangeinvelocityisnegligible.Heytesburythendefineduniformaccelerationasnonuniformmotioninwhichthevelocityincreasesbythesameincrementineachequaltime.Hethenwentontostatethetheorem:15

Whenanymobilebody isuniformlyaccelerated fromrest to somegivendegree [ofvelocity], itwillinthattimetraverseone-halfthedistancethatitwouldtraverseif,inthatsametime,itweremoveduniformlyatthedegreeofvelocityterminatingthatincrementofvelocity.Forthatmotion,asawhole,willcorrespond to themeandegreeof that incrementofvelocity,which ispreciselyone-halfthatdegreeofvelocitywhichisitsterminalvelocity.

That is, the distance traveled during an interval of timewhen a body is uniformly accelerated is thedistanceitwouldhavetraveledinuniformmotionifitsvelocityinthatintervalequaledtheaverageofthe actual velocity. If something is uniformly accelerated from rest to some final velocity, then itsaveragevelocityduringthat interval ishalf thefinalvelocity,sothedistancetraveledishalf thefinalvelocitytimesthetimeelapsed.Various proofs of this theorem were offered by Heytesbury, by John of Dumbleton, and then by

Nicole Oresme. Oresme’s proof is the most interesting, because he introduced a technique ofrepresentingalgebraicrelationsbygraphs.Inthisway,hewasabletoreducetheproblemofcalculatingthe distance traveled when a body is uniformly accelerated from rest to some final velocity to theproblemofcalculatingtheareaofarighttriangle,whosesidesthatmeetattherightanglehavelengthsequal respectively to the time elapsed and to the final velocity. (See TechnicalNote 17.) The meanspeedtheoremthenfollowsimmediatelyfromanelementaryfactofgeometry,thattheareaofarighttriangleishalftheproductofthetwosidesthatmeetattherightangle.Neither any don of Merton College nor Nicole Oresme seems to have applied the mean speed

theoremtothemostimportantcasewhereitisrelevant,themotionoffreelyfallingbodies.Forthedons

andOresme the theoremwas an intellectual exercise, undertaken to show that theywere capable ofdealing mathematically with nonuniform motion. If the mean speed theorem is evidence of anincreasingabilitytousemathematics,italsoshowshowuneasythefitbetweenmathematicsandnaturalsciencestillwas.Itmustbeacknowledgedthatalthoughitisobvious(asStratohaddemonstrated)thatfallingbodies

accelerate, it is not obvious that the speed of a falling body increases in proportion to the time, thecharacteristic of uniform acceleration, rather than to thedistance fallen. If the rate of change of thedistancefallen(thatis,thespeed)wereproportionaltothedistancefallen,thenthedistancefallenoncethe body starts to fallwould increase exponentiallywith time,* just as a bank account that receivesinterest proportional to the amount in the account increases exponentially with time (though if theinterest rate is low it takesa long time to see this).The firstperson toguess that the increase in thespeed of a falling body is proportional to the time elapsed seems to have been the sixteenth-centuryDominicanfriarDomingodeSoto,16abouttwocenturiesafterOresme.From the mid-fourteenth to the mid-fifteenth century Europe was harried by catastrophe. The

HundredYears’WarbetweenEnglandandFrancedrainedEnglandanddevastatedFrance.Thechurchunderwentaschism,withapopeinRomeandanotherinAvignon.TheBlackDeathdestroyedalargefractionofthepopulationeverywhere.PerhapsasaresultoftheHundredYears’War,thecenterofscientificworkshiftedeastwardinthis

period,fromFranceandEnglandtoGermanyandItaly.ThetworegionswerespannedinthecareerofNicholasofCusa.Bornaround1401inthetownofKuesontheMoselleinGermany,hediedin1464intheUmbrianprovinceofItaly.NicholaswaseducatedatbothHeidelbergandPadua,becomingacanonlawyer,adiplomat,andafter1448acardinal.Hiswritingshowsthecontinuingmedievaldifficultyofseparatingnaturalsciencefromtheologyandphilosophy.NicholaswroteinvaguetermsaboutamovingEarthandaworldwithoutlimits,butwithnouseofmathematics.ThoughhewaslatercitedbyKeplerandDescartes,itishardtoseehowtheycouldhavelearnedanythingfromhim.ThelateMiddleAgesalsoshowacontinuationoftheArabseparationofprofessionalmathematical

astronomers, who used the Ptolemaic system, and physician-philosophers, followers of Aristotle.Amongthefifteenth-centuryastronomers,mostlyinGermany,wereGeorgvonPeurbachandhispupilJohannMüller ofKönigsberg (Regiomontanus),who together continued and extended the Ptolemaictheory of epicycles.* Copernicus later made much use of the Epitome of the Almagest ofRegiomontanus.ThephysiciansincludedAlessandroAchillini(1463–1512)ofBolognaandGirolamoFracastoroofVerona(1478–1553),botheducatedatPadua,atthetimeastrongholdofAristotelianism.Fracastorogaveaninterestinglybiasedaccountoftheconflict:17

Youarewellawarethatthosewhomakeprofessionofastronomyhavealwaysfounditextremelydifficult to account for the appearances presented by the planets. For there are two ways ofaccountingforthem:theoneproceedsbymeansofthosespherescalledhomocentric,theotherbymeansofso-calledeccentricspheres[epicycles].Eachof thesemethodshas itsdangers,each itsstumbling blocks. Those who employ homocentric spheres never manage to arrive at anexplanation of phenomena. Thosewho use eccentric spheres do, it is true, seem to explain thephenomenamoreadequately,but theirconceptionof thesedivinebodies iserroneous,onemightalmostsayimpious,fortheyascribepositionsandshapestothemthatarenotfitfortheheavens.We know that, among the ancients, Eudoxus and Callippus were misled many times by thesedifficulties.Hipparchuswasamongthefirstwhochoserathertoadmiteccentricspheresthantobefound wanting by the phenomena. Ptolemy followed him, and soon practically all astronomerswerewonoverbyPtolemy.But against these astronomers, or at least, against thehypothesisofeccentrics,thewholeofphilosophyhasraisedcontinuingprotest.WhatamIsaying?Philosophy?

Natureandthecelestialspheresthemselvesprotestunceasingly.Untilnow,nophilosopherhaseverbeen found who would allow that these monstrous spheres exist among the divine and perfectbodies.

Tobefair,observationswerenotallonthesideofPtolemyagainstAristotle.OneofthefailingsoftheAristoteliansystemofhomocentricspheres,whichaswehaveseenhadbeennotedaroundAD200bySosigenes,isthatitputstheplanetsalwaysatthesamedistancefromtheEarth,incontradictionwiththefactthatthebrightnessofplanetsincreasesanddecreasesastheyappeartogoaroundtheEarth.ButPtolemy’s theory seemed to go too far in the other direction. For instance, in Ptolemy’s theory themaximumdistanceofVenusfromtheEarthis6.5timesitsminimumdistance,soifVenusshinesbyitsown light, then (since apparent brightness goes as the inverse square of the distance) its maximumbrightness shouldbe6.52=42 timesgreater than itsminimumbrightness,which is certainlynot thecase.Ptolemy’stheoryhadbeencriticizedonthisgroundattheUniversityofViennabyHenryofHesse(1325–1397).Theresolutionoftheproblemisofcoursethatplanetsshinenotbytheirownlight,butbythereflectedlightoftheSun,sotheirapparentbrightnessdependsnotonlyontheirdistancefromtheEarthbutalso,liketheMoon’sbrightness,ontheirphase.WhenVenusisfarthestfromtheEarthitisonthesideoftheSunawayfromtheEarth,soitsfaceisfullyilluminated;whenitisclosesttotheEarthitismoreorlessbetweentheEarthandtheSunandwemostlyseeitsdarkside.ForVenustheeffectsofphase and distance therefore partly cancel,moderating its variations in brightness.None of thiswasunderstooduntilGalileodiscoveredthephasesofVenus.Soon the controversy between Ptolemaic andAristotelian astronomywas swept away in a deeper

conflict,betweenthosewhofollowedeitherPtolemyorAristotle,allofthemacceptingthattheheavensrevolvearoundastationaryEarth;andanewrevivaloftheideaofAristarchus,thatitistheSunthatisatrest.

PARTIV

THESCIENTIFICREVOLUTION

Historians used to take it for granted that physics and astronomy underwent revolutionarychangesinthesixteenthandseventeenthcenturies,afterwhichthesesciencestooksomethingliketheir modern form, providing a paradigm for the future development of all science. Theimportanceof thisrevolutionseemedobvious.Thus thehistorianHerbertButterfield*declaredthatthescientificrevolution“outshineseverythingsincetheriseofChristianityandreducestheRenaissanceandReformationtotherankofmereepisodes,mereinternaldisplacements,withinthesystemofmedievalChristendom.”1Thereissomethingaboutthissortofconsensusthatalwaysattractstheskepticalattentionof

thenextgenerationofhistorians.Inthepastfewdecadessomehistorianshaveexpresseddoubtabout the importance or even the existence of the scientific revolution.2 For instance, StevenShapin famously began a book with the sentence “There was no such thing as the scientificrevolution,andthisisabookaboutit.”3Criticismsofthenotionofascientificrevolutiontaketwoopposingforms.Ononehand,some

historiansarguethatthediscoveriesofthesixteenthandseventeenthcenturieswerenomorethananaturalcontinuationofscientificprogressthathadalreadybeenmadeinEuropeorinthelandsof Islam (or both) during theMiddleAges.This in particularwas the view of PierreDuhem.4Otherhistorianspoint to thevestigesofprescientific thinking that continued into the supposedscientific revolution—for instance, thatCopernicus andKepler in places sound like Plato, thatGalileocasthoroscopesevenwhennoonewaspayingforthem,andthatNewtontreatedboththesolarsystemandtheBibleascluestothemindofGod.Thereareelementsoftruthinbothcriticisms.Nevertheless,Iamconvincedthatthescientific

revolutionmarkedarealdiscontinuityinintellectualhistory.Ijudgethisfromtheperspectiveofa contemporary working scientist. With a few bright Greek exceptions, science before thesixteenthcenturyseemstomeverydifferentfromwhatIexperienceinmyownwork,orwhatIseeintheworkofmycolleagues.Beforethescientificrevolutionsciencewassuffusedwithreligionandwhatwe now call philosophy, and had not yet worked out its relation tomathematics. InphysicsandastronomyaftertheseventeenthcenturyI feelathome.Irecognizesomethingverylikethescienceofmyowntimes:thesearchformathematicallyexpressedimpersonal lawsthatallowprecisepredictionsofawiderangeofphenomena,lawsvalidatedbythecomparisonofthesepredictionswithobservationandexperiment.Therewasascientificrevolution,andtherestofthisbookisaboutit.

11

TheSolarSystemSolved

Whateverthescientificrevolutionwasorwasnot,itbeganwithCopernicus.NicolausCopernicuswasborn in1473inPolandofafamily that inanearliergenerationhademigratedfromSilesia.Nicolauslosthisfatherattheageoften,butwasfortunatetobesupportedbyhisuncle,whohadbecomerichinthe service of the church and a fewyears later becamebishopofVarmia (orErmeland) in northeastPoland.After an education at theUniversity ofCracow,probably including courses in astronomy, in1496 Copernicus enrolled as a student of canon law at the University of Bologna and beganastronomicalobservationsasanassistanttotheastronomerDomenicoMariaNovara,whohadbeenastudent of Regiomontanus. While at Bologna Copernicus learned that, with the help of his uncle’spatronage, he had been confirmed as one of the 16 canons of the cathedral chapter ofFrombork (orFrauenburg),inVarmia,fromwhichfortherestofhislifehederivedagoodincomewithlittleinthewayofecclesiasticalduties.Copernicusneverbecameapriest.Afterstudyingmedicinebrieflyat theUniversityofPadua,in1503hepickedupadegreeofdoctoroflawsattheUniversityofFerraraandsoon afterward returned to Poland. In 1510 he settled in Frombork, where he constructed a smallobservatory,andwhereheremaineduntilhisdeathin1543.Soon after he came to Frombork, Copernicus wrote a short anonymous work, later titled De

hypothesibus motuum coelestium a se constitutis commentariolus, and generally known as theCommentariolus, orLittle Commentary.1 TheCommentariolus was not published until long after itsauthor’sdeath,andsowasnotasinfluentialashislaterwritings,butitgivesagoodaccountoftheideasthatguidedhiswork.After a brief criticism of earlier theories of the planets, Copernicus in theCommentariolus states

sevenprinciplesofhisnewtheory.Hereisaparaphrase,withsomeaddedcomments:

1.Thereisnoonecenteroftheorbitsofthecelestialbodies.(Thereisdisagreementamonghistorianswhether Copernicus thought that these bodies are carried on material spheres,2 as supposed byAristotle.)

2.ThecenteroftheEarthisnotthecenteroftheuniverse,butonlythecenteroftheMoon’sorbit,andthecenterofgravitytowardwhichbodiesonEarthareattracted.

3.AlltheheavenlybodiesexcepttheMoonrevolveabouttheSun,whichisthereforethecenteroftheuniverse.(Butasdiscussedbelow,CopernicustookthecenteroftheorbitsoftheEarthandotherplanetstobe,nottheSun,butratherapointneartheSun.)

4.ThedistancebetweentheEarthandtheSunisnegligiblecomparedwiththedistanceofthefixedstars.(PresumablyCopernicusmadethisassumptiontoexplainwhywedonotseeannualparallax,the apparent annual motion of the stars caused by the Earth’s motion around the Sun. But theproblemofparallaxisnowherementionedintheCommentariolus.)

5.TheapparentdailymotionofthestarsaroundtheEartharisesentirelyfromtheEarth’srotationonitsaxis.

6. Theapparentmotionof theSunarises jointlyfromtherotationof theEarthon itsaxisandtheEarth’srevolution(likethatoftheotherplanets)aroundtheSun.

7.TheapparentretrogrademotionoftheplanetsarisesfromtheEarth’smotion,occurringwhentheEarthpassesMars,Jupiter,orSaturn,orispassedinitsorbitbyMercuryorVenus.

CopernicuscouldnotclaimintheCommentariolusthathisschemefittedobservationbetterthanthatofPtolemy.Forone thing, itdidn’t. Indeed, it couldn’t, since for themostpartCopernicusbasedhistheoryondatahe inferredfromPtolemy’sAlmagest, rather thanonhisownobservations.3 Insteadofappealingtonewobservations,Copernicuspointedoutanumberofhistheory’saestheticadvantages.OneadvantageisthatthemotionoftheEarthaccountedforawidevarietyofapparentmotionsofthe

Sun, stars, and the other planets. In this way, Copernicus was able to eliminate the “fine-tuning”assumedinthePtolemaictheory,thatthecenteroftheepicyclesofMercuryandVenushadtoremainalwaysonthelinebetweentheEarthandtheSun,andthatthelinesbetweenMars,Jupiter,andSaturnandthecentersoftheirrespectiveepicycleshadtoremainalwaysparalleltothelinebetweentheEarthandtheSun.InconsequencetherevolutionofthecenteroftheepicycleofeachinnerplanetaroundtheEarthandtherevolutionofeachouterplanetbyafullturnonitsepicycleallhadtobefine-tunedtotakepreciselyoneyear.CopernicussawthattheseunnaturalrequirementssimplymirroredthefactthatweviewthesolarsystemfromaplatformrevolvingabouttheSun.Another aesthetic advantage of the Copernican theory had to do with its greater definiteness

regarding the sizes of planetary orbits. Recall that the apparent motion of the planets in Ptolemaicastronomydepends,notonthesizesoftheepicyclesanddeferents,butonlyontheratiooftheradiioftheepicycleanddeferentforeachplanet.Ifoneliked,onecouldeventakethedeferentofMercurytobelargerthanthedeferentofSaturn,aslongasthesizeofMercury’sepicyclewasadjustedaccordingly.FollowingtheleadofPtolemyinPlanetaryHypotheses,astronomerscustomarilyassignedsizestotheorbits,ontheassumptionthatthemaximumdistanceofoneplanetfromtheEarthequalstheminimumdistancefromtheEarthofthenextplanetoutward.Thisfixedtherelativesizesofplanetaryorbitsforanygivenchoiceof theorderof theplanetsgoingout from theEarth,but that choicewas stillquitearbitrary.Inanycase,theassumptionsofPlanetaryHypotheseswereneitherbasedonobservationnorconfirmedbyobservation.Incontrast,fortheschemeofCopernicustoagreewithobservation,theradiusofeveryplanet’sorbit

hadtohaveadefiniteratiototheradiusoftheEarth’sorbit.*Specifically,becauseofthedifferenceinthe way that Ptolemy had introduced epicycles for the inner and outer planets (and leaving asidecomplications associatedwith the ellipticity of the orbits), the ratio of the radii of the epicycles anddeferents must equal the ratio of the distances from the Sun of the planets and Earth for the innerplanets,andequaltheinverseofthisratiofortheouterplanets.(SeeTechnicalNote13.)Copernicusdidnotpresenthisresultsthisway;hegavethemintermsofacomplicated“triangulationscheme,”whichconveyedafalseimpressionthathewasmakingnewpredictionsconfirmedbyobservation.Buthedidinfactgettherightradiiofplanetaryorbits.HefoundthatgoingoutfromtheSun,theplanetsareinorderMercury,Venus,Earth,Mars,Jupiter,andSaturn;thisispreciselythesameastheorderoftheirperiods,whichCopernicusestimatedtoberespectively3months,9months,1year,2½years,12years,and30years.Thoughtherewasasyetnotheorythatdictatedthespeedsoftheplanetsintheirorbits,itmusthaveseemedtoCopernicusevidenceofcosmicorderthatthelargertheorbitofaplanet,themoreslowlyitmovesaroundtheSun.4The theoryofCopernicusprovidesaclassicexampleofhowa theorycanbeselectedonaesthetic

criteria,withnoexperimentalevidencethatfavorsitoverother theories.Thecasefor theCopernicantheory in theCommentarioluswassimply thatagreatdealofwhatwaspeculiarabout thePtolemaictheorywasexplainedatoneblowbytherevolutionandrotationoftheEarth,andthattheCopernicantheorywasmuchmoredefinitethanthePtolemaictheoryabouttheorderoftheplanetsandthesizesoftheirorbits.CopernicusacknowledgedthattheideaofamovingEarthhadlongbeforebeenproposedby the Pythagoreans, but he also noted (correctly) that this idea had been “gratuitously asserted” bythem,withoutargumentsofthesorthehimselfwasabletoadvance.TherewassomethingelseaboutthePtolemaictheorythatCopernicusdidnotlike,besidesitsfine-

tuninganditsuncertaintyregardingthesizesandorderofplanetaryorbits.TruetoPlato’sdictumthatplanetsmoveoncirclesatconstantspeed,CopernicusrejectedPtolemy’suseofdevicesliketheequantto dealwith the actual departures from circularmotion at fixed speed.As had been done by Ibn al-Shatir,Copernicus instead introducedmore epicycles: six forMercury; three for theMoon; and foureachforVenus,Mars,Jupiter,andSaturn.HerehemadenoimprovementovertheAlmagest.This work of Copernicus illustrates another recurrent theme in the history of physical science: a

simpleandbeautiful theorythatagreesprettywellwithobservationisoftencloser to the truth thanacomplicated ugly theory that agrees better with observation. The simplest realization of the generalideasofCopernicuswouldhavebeentogiveeachplanetincludingtheEarthacircularorbitatconstantspeedwiththeSunatthecenterofallorbits,andnoepicyclesanywhere.ThiswouldhaveagreedwiththesimplestversionofPtolemaicastronomy,with justoneepicycleforeachplanet,nonefor theSunandMoon, and no eccentrics or equants. It would not have precisely agreed with all observations,becauseplanetsmovenotoncirclesbutonnearlycircularellipses; theirspeed isonlyapproximatelyconstant;andtheSunisnotatthecenterofeachellipsebutatapointalittleoff-center,knownasthefocus. (See TechnicalNote 18.) Copernicus could have done even better by following Ptolemy andintroducing an eccentric and equant for each planetary orbit, but now also including the orbit of theEarth;thediscrepancywithobservationwouldthenhavebeenalmosttoosmallforastronomersofthetimetomeasure.There is an episode in the development of quantummechanics that shows the importance of not

worrying toomuchabout small conflictswithobservation. In1925ErwinSchrödingerworkedout amethodforcalculatingtheenergiesofthestatesofthesimplestatom,thatofhydrogen.Hisresultswereingoodagreementwiththegrosspatternoftheseenergies,butthefinedetailsofhisresult,whichtookintoaccount thedeparturesof themechanicsofspecial relativity fromNewtonianmechanics,didnotagreewiththefinedetailsofthemeasuredenergies.Schrödingersatonhisresultsforawhile,untilhewiselyrealizedthatgettingthegrosspatternoftheenergylevelswasasignificantaccomplishment,wellworthpublishing,andthatthecorrecttreatmentofrelativisticeffectscouldwait.ItwasprovidedafewyearslaterbyPaulDirac.In addition to numerous epicycles, there was another complication adopted by Copernicus, one

similartotheeccentricofPtolemaicastronomy.ThecenteroftheEarth’sorbitwastakentobe,nottheSun, but a point at a relatively small distance from the Sun. These complications approximatelyaccountedforvariousphenomena,suchastheinequalityoftheseasonsdiscoveredbyEuctemon,whichare really consequences of the fact that the Sun is at the focus rather than the center of the Earth’sellipticalorbit,andtheEarth’sspeedinitsorbitisnotconstant.Another of the complications introduced by Copernicus was made necessary only by a

misunderstanding.Copernicus seems tohave thought that the revolutionof theEartharound theSunwouldgivetheaxisoftheEarth’srotationeachyeara360°turnaroundthedirectionperpendiculartothe plane of the Earth’s orbit, somewhat as a finger at the end of the outstretched arm of a dancerexecutingapirouettewouldundergoa360° turnaroundtheverticaldirectionforeachrotationof thedancer.(Hemayhavebeeninfluencedbytheoldideathattheplanetsrideonsolidtransparentspheres.)Ofcourse,thedirectionoftheEarth’saxisdoesnotinfactchangeappreciablyinthecourseofayear,soCopernicuswasforcedtogivetheEarthathirdmotion,inadditiontoitsrevolutionaroundtheSunanditsrotationarounditsaxis,whichwouldalmostcancelthisswivelingofitsaxis.Copernicusassumedthatthecancellationwouldnotbeperfect,sothattheEarth’saxiswouldswivelaroundoververymanyyears,producingtheslowprecessionof theequinoxesthathadbeendiscoveredbyHipparchus.AfterNewton’sworkitbecameclearthattherevolutionoftheEartharoundtheSuninfacthasnoinfluenceonthedirectionoftheEarth’saxis,asidefromtinyeffectsduetotheactionofthegravityoftheSunandMoonontheEarth’sequatorialbulge,andso(asKeplerargued)nocancellationofthesortarrangedby

Copernicusisactuallynecessary.Withallthesecomplications,thetheoryofCopernicuswasstillsimplerthanthatofPtolemy,butnot

dramaticallyso.ThoughCopernicuscouldnothaveknownit,histheorywouldhavebeenclosertothetruthifhehadnotbotheredwithepicycles,andhadleftthesmallinaccuraciesofthetheorytobedealtwithinthefuture.TheCommentariolusdidnotgivemuchin thewayof technicaldetails.Theseweresuppliedinhis

greatworkDeRevolutionibusOrbiumCoelestium,5commonlyknownasDeRevolutionibus,finishedin1543whenCopernicuswasonhisdeathbed.ThebookstartswithadedicationtoAlessandroFarnese,Pope Paul III. In it Copernicus raised again the old argument between the homocentric spheres ofAristotleandtheeccentricsandepicyclesofPtolemy,pointingout that theformerdonotaccountforobservations,whilethelatter“contradictthefirstprinciplesofregularityofmotion.”InsupportofhisdaringinsuggestingamovingEarth,CopernicusquotedaparagraphofPlutarch:

SomethinkthattheEarthremainsatrest.ButPhilolausthePythagoreanbelievesthat,liketheSunandMoon,itrevolvesaroundthefireinanobliquecircle.HeraclidesofPontusandEcphantusthePythagoreanmaketheEarthmove,notinaprogressivemotion,butlikeawheelinarotationfromwesttoeastaboutitsowncenter.

(In the standard edition ofDeRevolutionibus Copernicusmakes nomention of Aristarchus, but hisnamehadappearedoriginally,andhadthenbeenstruckout.)CopernicuswentontoexplainthatsinceothershadconsideredamovingEarth,hetooshouldbepermittedtotesttheidea.Hethendescribedhisconclusion:

Having thus assumed themotionswhich I ascribe to theEarth later in thevolume,by longandintensestudyIfinallyfoundthatifthemotionsoftheotherplanetsarecorrelatedwiththeorbitingof the Earth, and are computed for the revolution of each planet, not only do their phenomenafollowtherefrombutalsotheorderandsizeofalltheplanetsandspheres,andheavenitselfissolinked together that innoportionof it cananythingbe shiftedwithoutdisrupting the remainingpartsandtheuniverseasawhole.

As in theCommentariolus,Copernicuswasappealing to the fact thathis theorywasmorepredictivethanPtolemy’s;itdictatedauniqueorderofplanetsandthesizesoftheirorbitsrequiredtoaccountforobservation, while Ptolemy’s theory left these undetermined. Of course, Copernicus had no way ofconfirmingthathisorbitalradiiwerecorrectwithoutassumingthetruthofhistheory;thishadtowaitforGalileo’sobservationsofplanetaryphases.Most of De Revolutionibus is extremely technical, fleshing out the general ideas of the

Commentariolus. One point worth special mention is that in Book 1 Copernicus states an a prioricommitmenttomotioncomposedofcircles.ThusChapter1ofBookIbegins:

Firstofall,wemustnotethattheuniverseisspherical.Thereasoniseitherthat,ofallforms,thesphere is themost perfect, needing no joint and being a completewhole,which can neither beincreasednordiminished[hereCopernicussounds likePlato];or that it is themostcapaciousoffigures,bestsuitedtoencloseandretainall things[thatis, ithasthegreatestvolumeforagivensurfacearea];oreventhatalltheseparatepartsoftheuniverse,ImeantheSun,Moon,planetsandstarsareseentobeofthisshape[howcouldheknowanythingabouttheshapeofthestars?];orthatwholesstrivetobecircumscribedbythisboundary,asisapparentindropsofwaterandotherfluid bodies when they seek to be self-contained [this is an effect of surface tension, which is

irrelevant on the scale of planets]. Hence no onewill question that the attribution of this formbelongstothedivinebodies.

HethenwentontoexplaininChapter4thatinconsequencethemovementoftheheavenlybodiesis“uniform,eternal,andcircular,orcompoundedofcircularmotions.”Later inBook 1,Copernicus pointed out one of the prettiest aspects of his heliocentric system: it

explainedwhyMercuryandVenusareneverseenfarintheskyfromtheSun.Forinstance,thefactthatVenusisneverseenmorethanabout45°fromtheSunisexplainedbythefactthatitsorbitaroundtheSunisabout70percentthesizeoftheorbitoftheEarth.(SeeTechnicalNote19.)AswesawinChapter11, in Ptolemy’s theory this had required fine-tuning the motion ofMercury and Venus so that thecenters of their epicycles are always on the line between the Earth and the Sun. The system ofCopernicusalsomadeunnecessaryPtolemy’sfine-tuningofthemotionoftheouterplanets,whichkeptthelinebetweeneachplanetandthecenterofitsepicycleparalleltothelinebetweentheEarthandtheSun.TheCopernicansystemranintooppositionfromreligiousleaders,beginningevenbeforepublication

ofDeRevolutionibus.Thisconflictwasexaggeratedinafamousnineteenth-centurypolemic,AHistoryoftheWarfareofSciencewithTheologyinChristendombyCornell’sfirstpresident,AndrewDicksonWhite,6whichoffersanumberofunreliablequotationsfromLuther,Melanchthon,Calvin,andWesley.But a conflict did exist. There is a record of Martin Luther’s conversations with his disciples atWittenberg,knownasTischreden,orTableTalk.7TheentryforJune4,1539,readsinpart:

TherewasmentionofacertainnewastrologerwhowantedtoprovethattheEarthmovesandnotthe sky, theSun, and theMoon. . . . [Luther remarked,] “So it goesnow.Whoeverwants tobeclevermustagreewithnothingthatothersesteem.Hemustdosomethingofhisown.Thisiswhatthatfooldoeswhowishestoturnthewholeofastronomyupsidedown.EveninthesethingsthatarethrownintodisorderIbelieveintheHolyScriptures,forJoshuacommandedtheSuntostandstillandnottheEarth.”8

AfewyearsafterthepublicationofDeRevolutionibus,Luther’scolleaguePhilippMelanchthon(1497–1560)joinedtheattackonCopernicus,nowcitingEcclesiastes1:5—“TheSunalsorises,andtheSungoesdown,andhastenstohisplacewhereherose.”Conflictswith the literal textof theBiblewouldnaturally raiseproblems forProtestantism,which

hadreplacedtheauthorityofthepopewiththatofScripture.Beyondthis,therewasapotentialproblemforallreligions:man’shome,theEarth,hadbeendemotedtojustonemoreplanetamongtheotherfive.ProblemsaroseevenwiththeprintingofDeRevolutionibus.Copernicushadsenthismanuscripttoa

publisherinNuremberg,andthepublisherappointedaseditoraLutheranclergyman,AndreasOsiander,whosehobbywasastronomy.Probablyexpressinghisownviews,Osianderaddedapreface thatwasthoughttobebyCopernicusuntilthesubstitutionwasunmaskedinthefollowingcenturybyKepler.InthisprefaceOsianderhadCopernicusdisclaiminganyintentiontopresentthetruenatureofplanetaryorbits,asfollows:9

For it is the duty of an astronomer to compose the history of the [apparent] celestial motionsthroughcarefulandexpertstudy.Thenhemustconceiveanddevisethecausesofthesemotionsorhypothesesaboutthem.Sincehecannotinanywayattaintothetruecause,hewilladoptwhateversuppositionsenablethemotionstobecomputedcorrectlyfromtheprinciplesofgeometryforthefutureaswellasforthepast.

Osiander’sprefaceconcludes:

So far as hypotheses are concerned, let no one expect anything certain from astronomy, whichcannotfurnishit,lestheacceptasthetruthideasconceivedforanotherpurpose,anddepartfromthisstudyagreaterfoolthanwhenheenteredit.

ThiswasinlinewiththeviewsofGeminusaround70BC(quotedhereinChapter8),butitwasquitecontrarytotheevidentintentionofCopernicus,inboththeCommentariolusandDeRevolutionibus, todescribetheactualconstitutionofwhatisnowcalledthesolarsystem.Whateverindividualclergymenmayhavethoughtaboutaheliocentrictheory,therewasnogeneral

Protestant effort to suppress the works of Copernicus. Nor did Catholic opposition to Copernicusbecomeorganizeduntilthe1600s.ThefamousexecutionofGiordanoBrunobytheRomanInquisitionin1600wasnotforhisdefenseofCopernicus,butforheresy,ofwhich(bythestandardsofthetime)hewassurelyguilty.Butaswewillsee,theCatholicchurchdidintheseventeenthcenturyputinplaceaveryserioussuppressionofCopernicanideas.WhatwasreallyimportantforthefutureofsciencewasthereceptionofCopernicusamonghisfellow

astronomers. The first to be convinced by Copernicus was his sole pupil, Rheticus, who in 1540publishedanaccountoftheCopernicantheory,andwhoin1543helpedtogetDeRevolutionibus intothe hands of theNuremberg publisher. (Rheticuswas initially supposed to supply the preface toDeRevolutionibus,butwhenheleft totakeapositioninLeipzigthetaskunfortunatelyfell toOsiander.)Rheticus had earlier assisted Melanchthon in making the University of Wittenberg a center ofmathematicalandastronomicalstudies.The theory of Copernicus gained prestige from its use in 1551 by Erasmus Reinhold, under the

sponsorshipof thedukeofPrussia, to compile anewsetof astronomical tables, thePrutenicTables,which allow one to calculate the location of planets in the zodiac at any given date. These were adistinctimprovementoverthepreviouslyusedAlfonsineTables,whichhadbeenconstructedinCastilein1275atthecourtofAlfonsoX.Theimprovementwasinfactdue,nottothesuperiorityofthetheoryofCopernicus,but rather to theaccumulationofnewobservations in thecenturiesbetween1275and1551,andperhapsalsotothefactthatthegreatersimplicityofheliocentrictheoriesmakescalculationseasier.Ofcourse,adherentsofastationaryEarthcouldargue thatDeRevolutionibusprovidedonlyaconvenientschemeforcalculation,notatruepictureoftheworld.Indeed,thePrutenicTableswereusedbytheJesuitastronomerandmathematicianChristophClaviusinthe1582calendarreformunderPopeGregoryXIII that gaveusourmodernGregorian calendar, butClaviusnevergaveuphisbelief in astationaryEarth.OnemathematiciantriedtoreconcilethisbeliefwiththeCopernicantheory.In1568,Melanchthon’s

son-in-law Caspar Peucer, professor of mathematics at Wittenberg, argued in Hypotyposes orbiumcoelestium that it should be possible by a mathematical transformation to rewrite the theory ofCopernicus ina forminwhich theEarth rather than theSun isstationary.This isprecisely the resultachievedlaterbyoneofPeucer’sstudents,TychoBrahe.TychoBrahewasthemostproficientastronomicalobserverinhistorybeforetheintroductionofthe

telescope,andtheauthorofthemostplausiblealternativetothetheoryofCopernicus.Bornin1546intheprovinceofSkåne,nowinsouthernSwedenbutuntil1658partofDenmark,TychowasasonofaDanishnobleman.HewaseducatedattheUniversityofCopenhagen,wherein1560hebecameexcitedby the successful prediction of a partial solar eclipse.Hemoved on to universities inGermany andSwitzerland,atLeipzig,Wittenberg,Rostock,Basel,andAugsburg.DuringtheseyearshestudiedthePrutenic Tables and was impressed by the fact that these tables predicted the date of the 1563conjunctionofSaturnandJupiter towithina fewdays,while theolderAlfonsineTableswereoffby

severalmonths.BackinDenmark,Tychosettledforawhileinhisuncle’shouseatHerrevadinSkåne.Therein1572

heobserved in theconstellationCassiopeiawhathecalleda“newstar.” (It isnowrecognizedas thethermonuclear explosion, known as a type Ia supernova, of a preexisting star. The remnant of thisexplosionwasdiscoveredby radioastronomers in1952and found tobeatadistanceofabout9,000light-years, too far for the star to have been seen without a telescope before the explosion.) Tychoobserved thenew star formonths, using a sextant of his ownconstruction, and found that it did notexhibit any diurnal parallax, the daily shift in position among the stars of the sort that would beexpected to be caused by the rotation of the Earth (or the daily revolution around the Earth ofeverything else) if the new star were as close as theMoon, or closer. (See Technical Note 20.)Heconcluded,“Thisnewstarisnotlocatedintheupperregionsoftheairjustunderthelunarorb,norinanyplaceclosertoEarth...butfarabovethesphereoftheMoonintheveryheavens.”10Thiswasadirect contradiction of the principle ofAristotle that the heavens beyond the orbit of theMoon canundergonochange,anditmadeTychofamous.In1576theDanishkingFrederickIIgaveTychothelordshipofthesmallislandofHven,inthestrait

betweenSkåneandthelargeDanishislandofZealand,alongwithapensiontosupportthebuildingandmaintenanceofaresidenceandscientificestablishmentonHven.ThereTychobuiltUraniborg,whichincludedanobservatory,library,chemicallaboratory,andprintingpress.Itwasdecoratedwithportraitsof past astronomers—Hipparchus, Ptolemy, al-Battani, and Copernicus—and of a patron of thesciences:WilliamIV, landgraveofHesse-Cassel.OnHvenTycho trainedassistants,and immediatelybeganobservations.Alreadyin1577Tychoobservedacomet,andfoundthatithadnoobservablediurnalparallax.Not

onlydidthisshow,againcontraAristotle,thattherewaschangeintheheavensbeyondtheorbitoftheMoon.NowTychocouldalsoconclude that thepathof the cometwouldhave taken it right througheitherAristotle’ssupposedhomocentricspheresorthespheresofthePtolemaictheory.(This,ofcourse,would be a problem only if the spheres were conceived as hard solids. This was the teaching ofAristotle,whichwesawinChapter8hadbeencarriedovertothePtolemaictheorybytheHellenisticastronomersAdrastusandTheon.The ideaofhard sphereswas revived inearlymodern times,11notlongbeforeTychoruleditout.)Cometsoccurmorefrequentlythansupernovas,andTychowasabletorepeattheseobservationsonothercometsinthefollowingyears.From1583on,Tychoworkedonanewtheoryoftheplanets,basedontheideathattheEarthisat

rest, the Sun andMoon go around theEarth, and the five knownplanets go around the Sun. Itwaspublishedin1588asChapter8ofTycho’sbookonthecometof1577.Inthis theorytheEarthisnotsupposedtobemovingorrotating,soinadditiontohavingslowermotions,theSun,Moon,planets,andstarsall revolvearound theEarth fromeast towestonceaday.Someastronomersadopted insteada“semi-Tychonic” theory, in which the planets revolve around the Sun, the Sun revolves around theEarth,buttheEarthrotatesandthestarsareatrest.(Thefirstadvocateofasemi-TychonictheorywasNicolasReymersBär, although hewould not have called it a semi-Tychonic system, for he claimedTychohadstolentheoriginalTychonicsystemfromhim.)12Asmentionedseveraltimesabove,theTychonictheoryisidenticaltotheversionofPtolemy’stheory

(neverconsideredbyPtolemy)inwhichthedeferentsoftheinnerplanetsaretakentocoincidewiththeorbitof theSunaround theEarth, and theepicyclesof theouterplanetshave the same radiusas theSun’sorbitaroundtheEarth.Asfarastherelativeseparationsandvelocitiesoftheheavenlybodiesareconcerned, it is also equivalent to the theory of Copernicus, differing only in the point of view: astationarySunforCopernicus,orastationaryandnonrotatingEarthforTycho.Regardingobservations,Tycho’s theory had the advantage that it automatically predicted no annual stellar parallax,withouthaving to assume that the stars are verymuch farther fromEarth than theSunor planets (which, of

course,wenowknowtheyare).ItalsomadeunnecessaryOresme’sanswertotheclassicproblemthathadmisled Ptolemy and Buridan: that objects thrown upward would seemingly be left behind by arotatingormovingEarth.For the future of astronomy, themost important contribution ofTychowas not his theory, but the

unprecedented accuracy of his observations.When I visited Hven in the 1970s, I found no sign ofTycho’sbuildings,butthere,stillintheground,werethemassivestonefoundationstowhichTychohadanchoredhisinstruments.(Amuseumandformalgardenshavebeenputinplacesincemyvisit.)Withtheseinstruments,Tychowasabletolocateobjectsintheskywithanuncertaintyofonly1/15°.Alsoatthe site of Uraniborg stands a granite statue, carved in 1936 by Ivar Johnsson, showing Tycho in apostureappropriateforanastronomer,facingupintothesky.13Tycho’s patron Frederick II died in 1588.Hewas succeeded byChristian IV,whomDanes today

regard as one of their greatest kings, butwhounfortunately had little interest in supportingworkonastronomy.Tycho’slastobservationsfromHvenweremadein1597;hethenleftonajourneythattookhimtoHamburg,Dresden,Wittenberg,andPrague.InPrague,hebecametheimperialmathematiciantothe Holy Roman Emperor Rudolph II and started work on a new set of astronomical tables, theRudolphineTables.AfterTycho’sdeathin1601,thisworkwascontinuedbyKepler.JohannesKeplerwasthefirsttounderstandthenatureofthedeparturesfromuniformcircularmotion

thathadpuzzledastronomerssincethetimeofPlato.Asafive-year-oldhewasinspiredbythesightofthecometof1577, thefirstcomet thatTychohadstudiedfromhisnewobservatoryonHven.KeplerattendedtheUniversityofTübingen,whichundertheleadershipofMelanchthonhadbecomeeminentin theology andmathematics. At Tübingen Kepler studied both of these subjects, but becamemoreinterestedinmathematics.HelearnedaboutthetheoryofCopernicusfromtheTübingenmathematicsprofessorMichaelMästlinandbecameconvincedofitstruth.In1594KeplerwashiredtoteachmathematicsataLutheranschoolinGraz,insouthernAustria.It

was there that he published his first originalwork, theMysteriumCosmographicum (Mystery of theDescriptionof theCosmos).Aswehaveseen,oneadvantageof the theoryofCopernicuswas that itallowed astronomical observations to be used to find unique results for the order of planets outwardfrom theSun and for the sizesof their orbits.Aswas still commonat the time,Kepler in thisworkconceivedtheseorbitstobethecirclestracedoutbyplanetscarriedontransparentspheres,revolvingintheCopernican theoryaround theSun.These sphereswerenot strictly two-dimensional surfaces,butthin shellswhose inner andouter radiiwere taken tobe theminimumandmaximumdistanceof theplanet from theSun.Keplerconjectured that the radiiof these spheresareconstrainedbyanaprioricondition,thateachsphere(otherthantheoutermostsphere,ofSaturn)justfitsinsideoneofthefiveregularpolyhedrons,andeachsphere(otherthantheinnermostsphere,ofMercury)justfitsoutsideoneoftheseregularpolyhedrons.Specifically,inorderoutwardfromtheSun,Keplerplaced(1)thesphereofMercury,(2)thenanoctahedron,(3)thesphereofVenus,(4)anicosahedron,(5)thesphereofEarth,(6)adodecahedron,(7)thesphereofMars,(8)atetrahedron,(9)thesphereofJupiter,(10)acube,andfinally(11)thesphereofSaturn,allfittingtogethertightly.Thisschemedictatedtherelativesizesoftheorbitsofalltheplanets,withnofreedomtoadjustthe

results,exceptbychoosingtheorderofthefiveregularpolyhedronsthatfitintothespacesbetweentheplanets. There are 30 different ways of choosing the order of the regular polyhedrons,* so it is notsurprisingthatKeplercouldfindonewayofchoosingtheirordersothatthepredictedsizesofplanetaryorbitswouldroughlyfittheresultsofCopernicus.In fact,Kepler’soriginal schemeworkedbadly forMercury, requiringKepler todosomefudging,

andonlymoderatelywellfortheotherplanets.*But likemanyothersat thetimeof theRenaissance,KeplerwasdeeplyinfluencedbyPlatonicphilosophy,andlikePlatohewasintriguedbythetheorem

thatregularpolyhedronsexistinonlyfivepossibleshapes,leavingroomforonlysixplanets,includingtheEarth.Heproudlyproclaimed,“Nowyouhavethereasonforthenumberofplanets!”NoonetodaywouldtakeseriouslyaschemelikeKepler’s,evenifithadworkedbetter.Thisisnot

becausewe have gotten over the old Platonic fascinationwith short lists ofmathematically possibleobjects, like regularpolyhedrons.Thereareother such short lists that continue to intriguephysicists.Forinstance,itisknownthattherearejustfourkindsof“numbers”forwhichaversionofarithmeticincludingdivisionispossible:therealnumbers,complexnumbers(involvingthesquarerootof−1),andmoreexoticquantitiesknownasquaternionsandoctonions.Somephysicistshaveexpendedmuchefforttryingtoincorporatequaternionsandoctonionsaswellasrealandcomplexnumbersinthefundamentallawsofphysics.WhatmakesKepler’s schemeso foreign tous today isnothisattempt to findsomefundamental physical significance for the regular polyhedrons, but that he did this in the context ofplanetaryorbits,whicharejusthistoricalaccidents.Whateverthefundamentallawsofnaturemaybe,wecanbeprettysurenowthattheydonotrefertotheradiiofplanetaryorbits.ThiswasnotjuststupidityonKepler’spart.Inhistimenooneknew(andKeplerdidnotbelieve)that

the starswere sunswith theirown systemsofplanets, rather than just lightsona sphere somewhereoutside the sphere of Saturn. The solar system was generally thought to be pretty much the wholeuniverse,andtohavebeencreatedatthebeginningoftime.Itwasperfectlynaturalthentosupposethatthedetailedstructureofthesolarsystemisasfundamentalasanythingelseinnature.Wemaybeinasimilarpositionintoday’stheoreticalphysics.Itisgenerallysupposedthatwhatwe

calltheexpandinguniverse,theenormouscloudofgalaxiesthatweobserverushingapartuniformlyinall directions, is thewhole universe.We think that the constants of naturewemeasure, such as themasses of the various elementary particles, will eventually all be deduced from the yet unknownfundamentallawsofnature.Butitmaybethatwhatwecalltheexpandinguniverseisjustasmallpartofamuchlarger“multiverse,”containingmanyexpandingpartsliketheoneweobserve,andthattheconstantsofnaturetakedifferentvaluesindifferentpartsofthemultiverse.Inthiscase,theseconstantsareenvironmentalparametersthatwillneverbededucedfromfundamentalprinciplesanymorethanwecandeducethedistancesoftheplanetsfromtheSunfromfundamentalprinciples.Thebestwecouldhope for would be an anthropic estimate. Of the billions of planets in our own galaxy, only a tinyminorityhavetherighttemperatureandchemicalcompositiontobesuitableforlife,butit isobviousthatwhenlifedoesbeginandevolvesintoastronomers,theywillfindthemselvesonaplanetbelongingtothisminority.SoitisnotreallysurprisingthattheplanetonwhichweliveisnottwiceorhalfasfarfromtheSunastheEarthactuallyis.Inthesameway,itseemslikelythatonlyatinyminorityofthesubuniverses in themultiversewould have physical constants that allow the evolution of life, but ofcourseany scientistswill find themselves in a subuniversebelonging to thisminority.ThishadbeenofferedasanexplanationoftheorderofmagnitudeofthedarkenergymentionedinChapter8,beforedarkenergywasdiscovered.14Allthis,ofcourse,ishighlyspeculative,butitservesasawarningthatintryingtounderstandtheconstantsofnaturewemayfacethesamesortofdisappointmentKeplerfacedintryingtoexplainthedimensionsofthesolarsystem.Some distinguished physicists deplore the idea of a multiverse, because they cannot reconcile

themselvestothepossibilitythatthereareconstantsofnaturethatcanneverbecalculated.Itistruethatthemultiverse ideamaybeallwrong,andso itwouldcertainlybepremature togiveup theeffort tocalculateallthephysicalconstantsweknowabout.Butitisnoargumentagainstthemultiverseideathatitwouldmakeusunhappynottobeabletodothesecalculations.Whateverthefinallawsofnaturemaybe,thereisnoreasontosupposethattheyaredesignedtomakephysicistshappy.At Graz Kepler began a correspondence with Tycho Brahe, who had read the Mysterium

Cosmographicum.TychoinvitedKeplertovisithiminUraniborg,butKeplerthoughtitwouldbetoofartogo.TheninFebruary1600KepleracceptedTycho’sinvitationtovisithiminPrague,thecapital

since1583of theHolyRomanEmpire.ThereKepler began to studyTycho’sdata, especiallyon themotionsofMars,andfoundadiscrepancyof0.13°betweenthesedataandthetheoryofPtolemy.*KeplerandTychodidnotgetalongwell,andKeplerreturnedtoGraz.AtjustthattimeProtestants

werebeingexpelledfromGraz,andinAugust1600Keplerandhisfamilywereforcedtoleave.BackinPrague,Keplerbegana collaborationwithTycho,workingon theRudolphineTables, thenewsetofastronomicaltablesintendedtoreplaceReinhold’sPrutenicTables.AfterTychodiedin1601,Kepler’scareer problems were solved for a while by his appointment as Tycho’s successor as courtmathematiciantotheemperorRudolphII.Theemperorwasenthusiasticaboutastrology,soKepler’sdutiesascourtmathematicianincludedthe

castingofhoroscopes.Thiswasanactivity inwhichhehadbeenemployedsincehisstudentdaysatTübingen, despite his own skepticism about astrological prediction. Fortunately, he also had time topursuerealscience.In1604heobservedanewstarintheconstellationOphiuchus,thelastsupernovaseen in or near our galaxyuntil 1987. In the sameyear he publishedAstronomiaeParsOptica (TheOpticalPartofAstronomy),aworkonoptical theoryand itsapplications toastronomy, including theeffectofrefractionintheatmosphereonobservationsoftheplanets.Keplercontinuedworkonthemotionsofplanets,tryingandfailingtoreconcileTycho’sprecisedata

withCopernicantheorybyaddingeccentrics,epicycles,andequants.Keplerhadfinishedthisworkby1605, but publication was held up by a squabble with the heirs of Tycho. Finally in 1609 Keplerpublishedhis results inAstronomiaNova (NewAstronomy Founded onCauses, or Celestial PhysicsExpoundedinaCommentaryontheMovementsofMars).PartIIIofAstronomiaNovamadeamajorimprovementintheCopernicantheorybyintroducingan

equantandeccentricfortheEarth,sothatthereisapointontheothersideofthecenteroftheEarth’sorbitfromtheSunaroundwhichthelinetotheEarthrotatesataconstantrate.ThisremovedmostofthediscrepanciesthathadbedeviledplanetarytheoriessincethetimeofPtolemy,butTycho’sdataweregood enough so that Kepler could see that there were still some conflicts between theory andobservation.AtsomepointKeplerbecameconvincedthatthetaskwashopeless,andthathehadtoabandonthe

assumption,commontoPlato,Aristotle,Ptolemy,Copernicus,andTycho,thatplanetsmoveonorbitscomposedofcircles.Instead,heconcludedthatplanetaryorbitshaveanovalshape.Finally,inChapter58 (of 70 chapters) ofAstronomiaNova, Keplermade this precise. In what later became known asKepler’sfirst law,heconcludedthatplanets(includingtheEarth)moveonellipses,withtheSunatafocus, not at the center. Just as a circle is completely described (apart from its location) by a singlenumber,itsradius,anyellipsecanbecompletelydescribed(asidefromitslocationandorientation)bytwonumbers,whichcanbe takenas the lengthsof its longerandshorteraxes,orequivalentlyas thelengthof the longer axis andanumberknownas the “eccentricity,”which tellsushowdifferent themajorandminoraxesare.(SeeTechnicalNote18.)Thetwofociofanellipsearepointsonthelongeraxis,evenlyspacedaroundthecenter,withaseparationfromeachotherequaltotheeccentricitytimesthelengthofthelongeraxisoftheellipse.Forzeroeccentricity,thetwoaxesoftheellipsehaveequallength,thetwofocimergetoasinglecentralpoint,andtheellipsedegeneratesintoacircle.In fact, the orbits of all the planets known to Kepler have small eccentricities, as shown in the

followingtableofmodernvalues(projectedbacktotheyear1900):

Planet EccentricityMercury 0.205615Venus 0.006820Earth 0.016750

Mars 0.093312Jupiter 0.048332Saturn 0.055890

This iswhysimplifiedversionsof theCopernicanandPtolemaic theories(withnoepicycles in theCopernicantheoryandonlyoneepicyleforeachofthefiveplanetsinthePtolemaictheory)wouldhaveworkedprettywell.*The replacement of circles with ellipses had another far-reaching implication. Circles can be

generatedbytherotationofspheres,butthereisnosolidbodywhoserotationcanproduceanellipse.This,togetherwithTycho’sconclusionsfromthecometof1577,wentfartodiscredittheoldideathatplanets are carried on revolving spheres, an idea thatKepler himself had assumed in theMysteriumCosmographicum. Instead, Kepler and his successors now conceived of planets as traveling onfreestandingorbitsinemptyspace.ThecalculationsreportedinAstronomiaNovaalsousedwhatlaterbecameknownasKepler’ssecond

law, though this lawwasnotclearlystateduntil1621, inhisEpitomeofCopernicanAstronomy.Thesecondlawtellshowthespeedofaplanetchangesastheplanetmovesarounditsorbit.Itstatesthatasthe planetmoves, the line between theSun and the planet sweeps out equal areas in equal times.AplanethastomovefartheralongitsorbittosweepoutagivenareawhenitisneartheSunthanwhenitisfarfromtheSun,soKepler’ssecondlawhastheconsequencethateachplanetmustmovefasterthecloser it comes to theSun.Aside from tinycorrectionsproportional to the squareof theeccentricity,Kepler’ssecondlawisthesameasthestatementthatthelinetotheplanetfromtheotherfocus(theonewheretheSunisn’t)turnsataconstantrate—thatis,itturnsbythesameangleineverysecond.(SeeTechnical Note 21.) Thus to a good approximation, Kepler’s second law gives the same planetaryvelocitiesastheoldideaofanequant,apointontheoppositesideofthecenterofthecirclefromtheSun(or,forPtolemy,fromtheEarth),andatthesamedistancefromthecenter,aroundwhichthelinetotheplanetturnsataconstantrate.Theequantwasthusrevealedasnothingbuttheemptyfocusoftheellipse.OnlyTycho’ssuperbdataforMarsallowedKeplertoconcludethateccentricsandequantsarenotenough;circularorbitshadtobereplacedwithellipses.15Thesecondlawalsohadprofoundapplications,atleastforKepler.InMysteriumCosmographicum

Keplerhadconceivedof theplanetsasbeingmovedbya“motivesoul.”Butnow,with thespeedofeachplanetfoundtodecreaseasitsdistancefromtheSunincreases,KeplerinsteadconcludedthattheplanetsareimpelledintheirorbitsbysomesortofforceradiatingfromtheSun:

Ifyousubstitutetheword“force”[vis]fortheword“soul”[anima],youhavetheveryprincipleonwhichthecelestialphysicsintheCommentaryonMars[AstronomiaNova]isbased.ForIformerlybelievedcompletelythatthecausemovingtheplanetsisasoul,havingindeedbeenimbuedwiththe teaching of J.C. Scaliger* onmotive intelligences. Butwhen I recognized that thismotivecause grows weaker as the distance from the Sun increases, just as the light of the Sun isattenuated,Iconcludedthatthisforcemustbeasitwerecorporeal.16

Ofcourse,theplanetscontinueintheirmotionnotbecauseofaforceradiatingfromtheSun,butratherbecausethereisnothingtodraintheirmomentum.Buttheyareheldintheirorbitsratherthanflyingoffinto interstellar space by a force radiating from the Sun, the force of gravitation, soKeplerwas notentirelywrong.Theideaofforceatadistancewasbecomingpopularatthistime,partlyowingtotheworkonmagnetismbythepresidentoftheRoyalCollegeofSurgeonsandcourtphysiciantoElizabethI,WilliamGilbert, to whomKepler referred. If by “soul” Kepler hadmeant anything like its usualmeaning, then the transitionfroma“physics”basedonsouls toonebasedonforceswasanessential

stepinendingtheancientminglingofreligionwithnaturalscience.AstronomiaNovawasnotwrittenwiththeaimofavoidingcontroversy.Byusingtheword“physics”

in the full title, Kepler was throwing out a challenge to the old idea, popular among followers ofAristotle,thatastronomyshouldconcernitselfonlywiththemathematicaldescriptionofappearances,whilefortrueunderstandingonemustturntophysics—thatis,tothephysicsofAristotle.Keplerwasstakingoutaclaimthat it isastronomers likehimselfwhodo truephysics. Infact,muchofKepler’sthinkingwasinspiredbyamistakenphysicalidea,thattheSundrivestheplanetsaroundtheirorbits,byaforcesimilartomagnetism.Kepler also challenged all opponents of Copernicanism. The introduction to Astronomia Nova

containstheparagraph:

Advice for idiots.Butwhoever is too stupid to understand astronomical science, or tooweak tobelieve Copernicus without [it] affecting his faith, I would advise him that, having dismissedastronomicalstudies,andhavingdamnedwhateverphilosophicalstudieshepleases,hemindhisownbusinessandbetakehimselfhometoscratchinhisowndirtpatch.17

Kepler’sfirsttwolawshadnothingtosayaboutthecomparisonoftheorbitsofdifferentplanets.Thisgapwasfilledin1619inHarmonicesmundi,bywhatbecameknownasKepler’sthirdlaw:18“theratiowhichexistsbetweentheperiodictimesofanytwoplanetsispreciselytheratioofthe3/2-powerofthemeandistances.”*Thatis,thesquareofthesiderealperiodofeachplanet(thetimeittakestocompleteafullcircuitof itsorbit) isproportional to thecubeof the longeraxisof theellipse.Thus ifT is thesiderealperiodinyears,anda ishalfthelengthofthelongeraxisoftheellipseinastronomicalunits(AU),with1AUdefinedashalfthelongeraxisoftheEarth’sorbit,thenKepler’sthirdlawsaysthatT2/a3isthesameforallplanets.SincetheEarthbydefinitionhasTequalto1yearandaequalto1AU,intheseunitsithasT2/a3equalto1,soaccordingtoKepler’sthirdlaweachplanetshouldalsohaveT2/a3=1.Theaccuracywithwhichmodernvaluesfollowthisruleisshowninthefollowingtable:

(ThedeparturesfromperfectequalityofT2/a3 for thedifferentplanetsaredue to tinyeffectsof thegravitationalfieldsoftheplanetsthemselvesactingoneachother.)Never entirely emancipated fromPlatonism,Kepler tried tomake sense of the sizes of the orbits,

resurrectinghisearlieruseofregularpolyhedronsinMysteriumCosmographicum.Healsoplayedwiththe Pythagorean idea that the different planetary periods form a sort of musical scale. Like otherscientistsof the time,Keplerbelongedonly inpart to thenewworldofscience thatwas justcomingintobeing,andinpartalsotoanolderphilosophicalandpoetictradition.TheRudolphineTableswere finally completed in 1627.Based onKepler’s first and second laws,

they represented a real improvement in accuracy over the previous Prutenic Tables. The new tablespredictedthattherewouldbeatransitofMercury(thatis,thatMercurywouldbeseentopassacrosstheface of the Sun) in 1631.Kepler did not see it. Forced once again as a Protestant to leaveCatholic

Austria,Keplerdiedin1630inRegensburg.The work of Copernicus and Kepler made a case for a heliocentric solar system based on

mathematicalsimplicityandcoherence,notonitsbetteragreementwithobservation.Aswehaveseen,the simplest versions of the Copernican and Ptolemaic theories make the same predictions for theapparent motions of the Sun and planets, in pretty good agreement with observation, while theimprovements in the Copernican theory introduced by Kepler were the sort that could have beenmatchedbyPtolemyifhehadusedanequantandeccentricfortheSunaswellasfortheplanets,andifhe had added a few more epicycles. The first observational evidence that decisively favoredheliocentrismovertheoldPtolemaicsystemwasprovidedbyGalileoGalilei.WithGalileo,wecometooneof thegreatestscientistsofhistory, inaclasswithNewton,Darwin,

andEinstein.Herevolutionizedobservationalastronomywithhisintroductionanduseofthetelescope,andhisstudyofmotionprovidedaparadigmformodernexperimentalphysics.Further,toanextentthatisunique,hisscientificcareerwasattendedbyhighdrama,ofwhichwecanheregiveonlyacondensedaccount.Galileowas a patrician though not wealthy Tuscan, born in Pisa in 1564, the son of themusical

theoristVincenzoGalilei.AfterstudiesataFlorentinemonastery,heenrolledasamedicalstudentattheUniversity of Pisa in 1581. Unsurprisingly for a medical student, at this point in his life he was afollowerofAristotle.Galileo’sintereststhenshiftedfrommedicinetomathematics,andforawhilehegavemathematicslessonsinFlorence,thecapitalofTuscany.In1589GalileowascalledbacktoPisatotakethechairofmathematics.While at the University of Pisa Galileo started his study of falling bodies. Some of his work is

describedinabook,DeMotu(OnMotion),whichheneverpublished.Galileoconcluded,contrary toAristotle,thatthespeedofaheavyfallingbodydoesnotdependappreciablyonitsweight.It’sanicestory that he tested this by dropping various weights from Pisa’s Leaning Tower, but there is noevidenceforthis.WhileinPisaGalileopublishednothingabouthisworkonfallingbodies.In1591GalileomovedtoPaduatotakethechairofmathematicsatitsuniversity,whichwasthenthe

universityoftherepublicofVeniceandthemostintellectuallydistinguisheduniversityinEurope.From1597onhewasabletosupplementhisuniversitysalarywiththemanufactureandsaleofmathematicalinstruments,usedinbusinessandwar.In1597GalileoreceivedtwocopiesofKepler’sMysteriumCosmographicum.Hewrote toKepler,

acknowledgingthathe,likeKepler,wasaCopernican,thoughasyethehadnotmadehisviewspublic.KeplerrepliedthatGalileoshouldcomeoutforCopernicus,urging,“Standforth,OGalileo!”19SoonGalileocameintoconflictwiththeAristotelianswhodominatedtheteachingofphilosophyat

Padua,aselsewhereinItaly.In1604helecturedonthe“newstar”observedthatyearbyKepler.LikeTychoandKeplerhedrewtheconclusionthatchangedoesoccurintheheavens,abovetheorbitoftheMoon.Hewasattackedfor thisbyhissometimefriendCesareCremonini,professorofphilosophyatPadua.Galileo repliedwith an attack onCremonini,written in a rustic Paduan dialect as a dialoguebetween twopeasants.TheCremonini peasant argued that the ordinary rules ofmeasurement do notapplyintheheavens;andGalileo’speasantrepliedthatphilosophersknownothingaboutmeasurement:forthisonemusttrustmathematicians,whetherformeasurementoftheheavensorofpolenta.Arevolutioninastronomybeganin1609,whenGalileofirstheardofanewDutchdeviceknownasa

spyglass.Themagnifyingpropertyofglassspheresfilledwithwaterwasknowninantiquity,mentionedfor instanceby theRomanstatesmanandphilosopherSeneca.Magnificationhadbeen studiedbyal-Haitam, and in 1267 Roger Bacon had written about magnifying glasses in Opus Maius. Withimprovements in the manufacture of glass, reading glasses had become common in the fourteenthcentury.But tomagnifydistant objects, it is necessary to combine apair of lenses, one to focus theparallelraysoflightfromanypointontheobjectsothattheyconverge,andthesecondtogatherthese

light rays,eitherwithaconcave lenswhile theyarestillconvergingorwithaconvex lensafter theybegintodivergeagain,ineithercasesendingthemonparalleldirectionsintotheeye.(Whenrelaxedthelensof theeye focusesparallel raysof light toa singlepointon the retina; the locationof thatpointdependsonthedirectionoftheparallelrays.)Spyglasseswithsuchanarrangementoflenseswerebeingproduced in the Netherlands by the beginning of the 1600s, and in 1608 several Dutch makers ofspectaclesappliedforpatentsontheirspyglasses.Theirapplicationswererejected,onthegroundthatthedevicewasalreadywidelyknown.SpyglassesweresoonavailableinFranceandItaly,butcapableofmagnification by only three or four times. (That is, if the lines of sight to two distant points areseparatedbyacertainsmallangle,thenwiththesespyglassestheyseemedtobeseparatedbythreeorfourtimesthatangle.)Sometimein1609Galileoheardofthespyglass,andsoonmadeanimprovedversion,withthefirst

lensconvexonthesidefacingforwardandplanarontheback,andwithlongfocallength,*whilethesecondwasconcaveonthesidefacingthefirstlensandplanaronthebackside,andwithshorterfocallength.Withthisarrangement,tosendthelightfromapointsourceatverylargedistancesonparallelraysintotheeye,thedistancebetweenthelensesmustbetakenasthedifferenceofthefocallengths,and themagnificationachieved is the focal lengthof the first lensdividedby the focal lengthof thesecondlens.(SeeTechnicalNote23.)Galileowassoonabletoachieveamagnificationbyeightorninetimes. On August 23, 1609, he showed his spyglass to the doge and notables of Venice anddemonstratedthatwithitshipscouldbeseenatseatwohoursbeforetheybecamevisibletothenakedeye.ThevalueofsuchadevicetoamaritimepowerlikeVenicewasobvious.AfterGalileodonatedhisspyglasstotheVenetianrepublic,hisprofessorialsalarywastripled,andhistenurewasguaranteed.ByNovemberGalileohadimprovedthemagnificationofhisspyglassto20times,andhebegantouseitforastronomy.Withhisspyglass,laterknownasatelescope,Galileomadesixastronomicaldiscoveriesofhistoric

importance. The first four of these he described in Siderius Nuncius (The Starry Messenger),20publishedinVeniceinMarch1610.

1.OnNovember20,1609,GalileofirstturnedhistelescopeonthecrescentMoon.Onthebrightside,hecouldseethatitssurfaceisrough:

By oft-repeated observations of [lunar markings] we have been led to the conclusion that wecertainlyseethesurfaceoftheMoontobenotsmooth,even,andperfectlyspherical,asthegreatcrowdofphilosophershavebelievedaboutthisandotherheavenlybodies,butonthecontrary,tobeuneven, rough,andcrowdedwithdepressionsandbulges.And it is like thefaceof theEarthitself,whichismarkedhereandtherewithchainsofmountainsanddepthsofvalleys.

On thedark side, near the terminator, theboundarywith thebright side, he could see spots of light,whichheinterpretedasmountaintopsilluminatedbytheSunwhenitwasjustabouttocomeoverthelunarhorizon.Fromthedistanceofthesebrightspotsfromtheterminatorhecouldevenestimatethatsomeofthesemountainswereatleastfourmileshigh.(SeeTechnicalNote24.)Galileoalsointerpretedthe observed faint illumination of the dark side of the Moon. He rejected various suggestions ofErasmusReinholdandofTychoBrahethatthelightcomesfromtheMoonitselforfromVenusorthestars,andcorrectlyarguedthat“thismarvelousbrightness”isduetothereflectionofsunlightfromtheEarth, just as the Earth at night is faintly illuminated by sunlight reflected from the Moon. So aheavenlybodyliketheMoonwasseentobenotsoverydifferentfromtheEarth.

2.ThespyglassallowedGalileotoobserve“analmostinconceivablecrowd”ofstarsmuchdimmer

than stars of the sixthmagnitude, andhence toodim tohavebeen seenwith thenaked eye.The sixvisible starsof thePleiadeswere found tobeaccompaniedwithmore than40other stars, and in theconstellationOrionhecouldseeover500starsneverseenbefore.TurninghistelescopeontheMilkyWay,hecouldseethatitiscomposedofmanystars,ashadbeenguessedbyAlbertusMagnus.

3.Galileoreportedseeingtheplanetsthroughhistelescopeas“exactlycircularglobesthatappearaslittlemoons,”buthecouldnotdiscernanysuchimageofthestars.Instead,hefoundthat,althoughallstarsseemedmuchbrighterwhenviewedwithhistelescope,theydidnotseemappreciablylarger.Hisexplanationwasconfused.Galileodidnotknowthattheapparentsizeofstarsiscausedbythebendingof light rays in various directions by random fluctuations in the Earth’s atmosphere, rather than byanythingintrinsictotheneighborhoodofthestars.Itisthesefluctuationsthatcausestarstoappeartotwinkle.*Galileo concluded that, since itwas not possible tomake out the images of starswith histelescope,theymustbemuchfartherfromusthanaretheplanets.AsGalileonotedlater,thishelpedtoexplainwhy,iftheEarthrevolvesaroundtheSun,wedonotobserveanannualstellarparallax.

4.ThemostdramaticandimportantdiscoveryreportedinSideriusNunciuswasmadeonJanuary7,1610.Traininghis telescopeon Jupiter,Galileo saw that “three little starswerepositionednearhim,smallbutverybright.”AtfirstGalileothoughtthatthesewerejustanotherthreefixedstars,toodimtohavebeenseenbefore,thoughhewassurprisedthattheyseemedtobelinedupalongtheecliptic,twototheeastofJupiterandonetothewest.Butonthenextnightallthreeofthese“stars”weretothewestofJupiter,andonJanuary10onlytwocouldbeseen,bothtotheeast.FinallyonJanuary13,hesawthat four of these “stars” were now visible, still more or less lined up along the ecliptic. GalileoconcludedthatJupiterisaccompaniedinitsorbitwithfoursatellites,similartoEarth’sMoon,andlikeourMoonrevolvinginroughlythesameplaneasplanetaryorbits,whichareclosetotheecliptic, theplaneof theEarth’sorbitaroundtheSun.(TheyarenowknownasthefourlargestmoonsofJupiter:Ganymede,Io,Callisto,andEuropa,namedafterthegodJupiter’smaleandfemalelovers.)*ThisdiscoverygaveimportantsupporttotheCopernicantheory.Foronething,thesystemofJupiter

anditsmoonsprovidedaminiatureexampleofwhatCopernicushadconceivedtobethesystemoftheSunanditssurroundingplanets,withcelestialbodiesevidentlyinmotionaboutabodyotherthantheEarth.Also,theexampleofJupiter’smoonsputtoresttheobjectiontoCopernicusthat,iftheEarthismoving,whyis theMoonnotleftbehind?EveryoneagreedthatJupiter ismoving,andyet itsmoonswereevidentlynotbeingleftbehind.ThoughtheresultsweretoolatetobeincludedinSideriusNuncius,Galileobytheendof1611had

measuredtheperiodsofrevolutionofthefourJoviansatellitesthathehaddiscovered,andin1612hepublishedtheseresultsonthefirstpageofaworkonothermatters.21Galileo’sresultsaregivenalongwithmodernvaluesindays(d),hours(h),andminutes(m)inthetablebelow:

Joviansatellite Period(Galileo) Period(modern)Io 1d18h30m 1d18h29mEuropa 3d13h20m 3d13h18mGanymede 7d4h0m 7d4h0mCallisto 16d18h0m 16d18h5m

The accuracy of Galileo’s measurements testifies to his careful observations and precisetimekeeping.*GalileodedicatedSideriusNunciustohisformerpupilCosimoIIdiMedici,nowthegranddukeof

Tuscany, and he named the four companions of Jupiter the “Medicean stars.” Thiswas a calculatedcompliment. Galileo had a good salary at Padua, but he had been told that it would not again beincreased.Also, for this salary he had to teach, taking time away fromhis research.Hewas able tostrike an agreement with Cosimo, who named him court mathematician and philosopher, with aprofessorship atPisa that carriedno teachingduties.Galileo insistedon the title “court philosopher”becausedespitetheexcitingprogressmadeinastronomybymathematicianssuchasKepler,anddespitethe arguments of professors likeClavius,mathematicians continued to have a lower status than thatenjoyed by philosophers. Also, Galileowanted his work to be taken seriously as what philosopherscalled“physics,”anexplanationofthenatureoftheSunandMoonandplanets,notjustamathematicalaccountofappearances.In thesummerof1610Galileo leftPadua forFlorence,adecision that turnedouteventually tobe

disastrous.PaduawasintheterritoryoftherepublicofVenice,whichatthetimewaslessunderVaticaninfluencethananyotherstateinItaly,havingsuccessfullyresistedapapalinterdictafewyearsbeforeGalileo’sdeparture.MovingtoFlorencemadeGalileomuchmorevulnerabletocontrolbythechurch.Amodernuniversitydeanmight feel that thisdangerwas a justpunishment forGalileo’s evasionofteachingduties.Butforawhilethepunishmentwasdeferred.

5. In September 1610Galileomade the fifth of his great astronomical discoveries. He turned histelescope on Venus, and found that it has phases, like those of theMoon. He sent Kepler a codedmessage:“TheMotherofLoves[Venus]emulatestheshapesofCynthia[theMoon].”TheexistenceofphaseswouldbeexpectedinboththePtolemaicandtheCopernicantheories,butthephaseswouldbedifferent.InthePtolemaictheory,VenusisalwaysmoreorlessbetweentheEarthandtheSun,soitcanneverbeasmuchashalffull.IntheCopernicantheory,ontheotherhand,VenusisfullyilluminatedwhenitisontheothersideofitsorbitfromtheEarth.ThiswasthefirstdirectevidencethatthePtolemaictheoryiswrong.RecallthatthePtolemaictheory

givesthesameappearanceofsolarandplanetarymotionsseenfromtheEarthastheCopernicantheory,whateverwechooseforthesizeofeachplanet’sdeferent.ButitdoesnotgivethesameappearanceastheCopernicantheoryofsolarandplanetarymotionsasseenfromtheplanets.Ofcourse,Galileocouldnotgo toanyplanet to seehow themotionsof theSunandotherplanetsappear from there.But thephasesofVenusdid tellhimthedirectionof theSunasseenfromVenus—thebrightside is thesidefacingtheSun.OnlyonespecialcaseofPtolemy’stheorycouldgivethatcorrectly,thecaseinwhichthedeferentsofVenusandMercuryareidenticalwiththeorbitoftheSun,whichasalreadyremarkedisjustthetheoryofTycho.ThatversionhadneverbeenadoptedbyPtolemy,orbyanyofhisfollowers.

6.AtsometimeaftercomingtoFlorence,Galileofoundaningeniouswaytostudythefaceof theSun,byusingatelescopetoprojectitsimageonascreen.Withthishemadehissixthdiscovery:darkspotswereseentomoveacrosstheSun.Hisresultswerepublishedin1613inhisSunspotLetters,aboutwhichmorelater.

Therearemomentsinhistorywhenanewtechnologyopensuplargepossibilitiesforpurescience.Theimprovement of vacuum pumps in the nineteenth century made possible experiments on electricaldischarges in evacuated tubes that led to the discovery of the electron. The Ilford Corporation’sdevelopmentofphotographicemulsionsallowedthediscoveryofahostofnewelementaryparticlesinthe decade followingWorldWar II. The development of microwave radar during that war allowedmicrowaves to be used as a probe of atoms, providing a crucial test of quantum electrodynamics in1947.Andweshouldnotforgetthegnomon.ButnoneofthesenewtechnologiesledtoscientificresultsasimpressiveasthosethatflowedfromthetelescopeinthehandsofGalileo.

ThereactionstoGalileo’sdiscoveriesrangedfromcautiontoenthusiasm.Galileo’soldadversaryatPadua, Cesare Cremonini, refused to look through the telescope, as did Giulio Libri, professor ofphilosophyatPisa.Ontheotherhand,GalileowaselectedamemberoftheLinceanAcademy,foundedafewyearsearlierasEurope’sfirstscientificacademy.KeplerusedatelescopesenttohimbyGalileo,andconfirmedGalileo’sdiscoveries.(Keplerworkedoutthetheoryofthetelescopeandsooninventedhisownversion,withtwoconvexlenses.)Atfirst,Galileohadnotroublewiththechurch,perhapsbecausehissupportforCopernicuswasstill

notexplicit.CopernicusismentionedonlyonceinSideriusNuncias,near theend, inconnectionwiththequestionwhy, if theEarth ismoving, itdoesnot leave theMoonbehind.At the time, itwasnotGalileobutAristotelianslikeCremoniniwhowereintroublewiththeRomanInquisition,onmuchthesamegroundsthathadledtothe1277condemnationofvarioustenetsofAristotle.ButGalileomanagedtogetintosquabbleswithbothAristotelianphilosophersandJesuits,whichinthelongrundidhimnogood.InJuly1611,shortlyaftertakinguphisnewpositioninFlorence,Galileoenteredintoadebatewith

philosopherswho,followingwhattheysupposedtobeadoctrineofAristotle,arguedthatsolidicehadagreaterdensity(weightpervolume) than liquidwater.TheJesuitcardinalRobertoBellarmine,whohad been on the panel of theRoman Inquisition that sentencedBruno to death, tookGalileo’s side,arguing that since ice floats, itmust be less dense thanwater. In1612Galileomadehis conclusionsaboutfloatingbodiespublicinhisDiscourseonBodiesinWater.22In 1613 Galileo antagonized the Jesuits, including Christoph Scheiner, in an argument over a

peripheral astronomical issue: Are sunspots associated with the Sun itself—perhaps as cloudsimmediately above its surface, as Galileo thought, which would provide an example (like lunarmountains) of the imperfectionsof heavenlybodies?Or are they little planets orbiting theSunmorecloselythanMercury?If itcouldbeestablishedthat theyareclouds, thenthosewhoclaimedthat theSungoesaroundtheEarthcouldnotalsoclaimthattheEarth’scloudswouldbeleftbehindiftheEarthwentaroundtheSun.InhisSunspotLettersof1613,GalileoarguedthatsunspotsseemedtonarrowastheyapproachtheedgeoftheSun’sdisk,showingthatnearthedisk’sedgetheywerebeingseenataslant, and hence were being carried around with the Sun’s surface as it rotates. There was also anargumentoverwhohadfirstdiscoveredsunspots.ThiswasonlyoneepisodeinanincreasingconflictwiththeJesuits,inwhichunfairnesswasnotallononeside.23Mostimportantforthefuture,inSunspotLettersGalileoatlastcameoutexplicitlyforCopernicus.Galileo’sconflictwiththeJesuitsheatedupin1623withthepublicationofTheAssayer.Thiswasan

attackontheJesuitmathematicianOrazioGrassiforGrassi’sperfectlycorrectconclusion,inagreementwith Tycho, that the lack of diurnal parallax shows that comets are beyond the orbit of theMoon.Galileo instead offered a peculiar theory, that comets are reflections of the sun’s light from lineardisturbancesoftheatmosphere,anddonotshowdiurnalparallaxbecausethedisturbancesmovewiththeEarthasitrotates.PerhapstherealenemyforGalileowasnotOrazioGrassibutTychoBrahe,whohadpresentedageocentrictheoryoftheplanetsthatobservationcouldnotthenrefute.In these years it was still possible for the church to tolerate the Copernican system as a purely

mathematical device for calculating apparent motions of planets, though not as a theory of the realnatureoftheplanetsandtheirmotions.Forinstance,in1615BellarminewrotetotheNeapolitanmonkPaolo Antonio Foscarini with both a reassurance and a warning about Foscarini’s advocacy of theCopernicansystem:

It seems to me that Your Reverence and Signor Galileo would act prudently by contentingyourselveswithspeakinghypotheticallyandnotabsolutely,asIhavealwaysbelievedCopernicustohavespoken. [WasBellarmine taken inbyOsiander’spreface?Galileocertainlywasnot.]To

say thatbyassuming theEarth inmotionand theSun immobilesavesall theappearancesbetterthantheeccentricsandepicycleseverdidistospeakwellindeed.[Bellarmineapparentlydidnotrealize thatCopernicus likePtolemyhad employed epicycles, onlynot somany.]This holds nodangeranditsufficesforthemathematician.ButtowanttoaffirmthattheSunreallyremainsatrestattheworld’scenter,thatitturnsonlyonitselfwithoutrunningfromEasttoWest,andthattheEarthissituatedinthethirdheavenandturnsveryswiftlyaroundtheSun,thatisaverydangerousthing.Notonlymayitirritateallthephilosophersandscholastictheologians,itmayalsoinjurethefaithandrenderHolyScripturefalse.24

SensingthetroublethatwasgatheringoverCopernicanism,Galileoin1615wroteacelebratedletterabout the relationof science and religion toChristinaofLorraine, grandduchessofTuscany,whosewedding to the late grand duke Ferdinando I Galileo had attended.25 As Copernicus had in DeRevolutionibus,GalileomentionedtherejectionofthesphericalshapeoftheEarthbyLactantiusasahorribleexampleoftheuseofScripturetocontradictthediscoveriesofscience.Healsoarguedagainsta literal interpretation of the text from the Book of Joshua that Luther had earlier invoked againstCopernicustoshowthemotionoftheSun.GalileoreasonedthattheBiblewashardlyintendedasatexton astronomy, since of the five planets itmentions onlyVenus, and that just a few times.Themostfamous line in the letter to Christina reads, “I would say here something that was heard from anecclesiasticof themosteminentdegree:‘That theintentionof theHolyGhost is toteachushowonegoes to heaven, not how heaven goes.’ ” (A marginal note by Galileo indicated that the eminentecclesiasticwasthescholarCardinalCaesarBaronius,headoftheVaticanlibrary.)GalileoalsoofferedaninterpretationofthestatementinJoshuathattheSunhadstoodstill:itwastherotationoftheSun,revealed toGalileo by themotion of sunspots, that had stopped, and this in turn stopped the orbitalmotionandrotationoftheEarthandotherplanets,whichasdescribedintheBibleextendedthedayofbattle. It isnot clearwhetherGalileoactuallybelieved thisnonsenseorwasmerely seekingpoliticalcover.Against the advice of friends, Galileo in 1615went to Rome to argue against the suppression of

Copernicanism. Pope Paul V was anxious to avoid controversy and, on the advice of Bellarmine,decided to submit the Copernican theory to a panel of theologians. Their verdict was that theCopernican system is “foolish and absurd in Philosophy, and formally heretical inasmuch as itcontradictstheexpresspositionofHolyScriptureinmanyplaces.”26InFebruary1616GalileowassummonedtotheInquisitionandreceivedtwoconfidentialorders.A

signed document ordered him not to hold or defend Copernicanism. An unsigned document wentfurther, ordering him not to hold, defend, or teach Copernicanism in any way. In March 1616 theInquisition issued a public formal order, not mentioning Galileo but banning Foscarini’s book, andcalling for thewritings ofCopernicus to be expurgated.DeRevolutionibus was put on the Index ofbooksforbiddentoCatholics.InsteadofreturningtoPtolemyorAristotle,someCatholicastronomers,such as the Jesuit Giovanni Battista Riccioli in his 1651 Almagestum Novum, argued for Tycho’ssystem,whichcouldnotthenberefutedbyobservation.DeRevolutionibusremainedontheIndexuntil1835,blightingtheteachingofscienceinsomeCatholiccountries,suchasSpain.Galileo hoped for better things after 1624, when Maffeo Barberini became Pope Urban VIII.

BarberiniwasaFlorentineandanadmirerofGalileo.HewelcomedGalileotoRomeandgrantedhimhalfadozenaudiences.IntheseconversationsGalileoexplainedhistheoryofthetides,onwhichhehadbeenworkingsincebefore1616.Galileo’stheorydependedcruciallyonthemotionoftheEarth.Ineffect,theideawasthatthewaters

of the oceans slosh back and forth as the Earth rotates while it goes around the Sun, duringwhichmovementthenetspeedofaspotontheEarth’ssurfacealongthedirectionoftheEarth’smotioninits

orbit is continually increasing and decreasing. This sets up a periodic ocean wave with a one-dayperiod,andaswithanyotheroscillation,thereareovertones,withperiodsofhalfaday,athirdofaday,andsoon.Sofar,thisleavesoutanyinfluenceoftheMoon,butithadbeenknownsinceantiquitythatthehigher“spring”tidesoccuratfullandnewmoon,whilethelower“neap”tidesareatthetimesofhalf-moon.Galileo tried toexplain the influenceof theMoonby supposing that for some reason theEarth’sorbitalspeedisincreasedatnewmoon,whentheMoonisbetweentheEarthandtheSun,anddecreasedatfullmoon,whentheMoonisontheothersideoftheEarthfromtheSun.Thiswas notGalileo at his best. It’s not somuch that his theorywaswrong.Without a theoryof

gravitationtherewasnowaythatGalileocouldhavecorrectlyunderstoodthetides.ButGalileoshouldhave known that a speculative theory of tides that had no significant empirical support could not becountedasaverificationoftheEarth’smotion.The pope said that he would permit publication of this theory of tides if Galileo would treat the

motionoftheEarthasamathematicalhypothesis,notassomethinglikelytobetrue.UrbanexplainedthathedidnotapproveoftheInquisition’spublicorderof1616,buthewasnotreadytorescindit.IntheseconversationsGalileodidnotmentiontothepopetheInquisition’sprivateorderstohim.In1632Galileowasreadytopublishhistheoryofthetides,whichhadgrownintoacomprehensive

defense of Copernicanism.As yet, the church hadmade no public criticism ofGalileo, sowhen heappliedtothelocalbishopforpermissiontopublishanewbookitwasgranted.ThiswashisDialogo(DialogueConcerningtheTwoChiefSystemsoftheWorld—PtolemaicandCopernican).ThetitleofGalileo’sbookispeculiar.Therewereatthetimenottwobutfourchiefsystemsofthe

world:notjustthePtolemaicandCopernican,butalsotheAristotelian,basedonhomocentricspheresrevolvingaroundtheEarth,andtheTychonic,withtheSunandMoongoingaroundastationaryEarthbutallotherplanetsgoingaroundtheSun.WhydidGalileonotconsidertheAristotelianandTychonicsystems?About theAristotelian system, one can say that it did not agreewith observation, but it had been

knowntodisagreewithobservationfor two thousandyearswithout losingall itsadherents.Just lookbackattheargumentmadebyFracastoroatthebeginningofthesixteenthcentury,quotedinChapter10.Galileo a century later evidently thought such argumentsnotworth answering, but it is not clearhowthatcameabout.On the other hand, the Tychonic system worked too well for it to be justly dismissed. Galileo

certainlyknewaboutTycho’ssystem.Galileomayhavethoughthisowntheoryofthetidesshowedthatthe Earth does move, but this theory was not supported by any quantitative successes. Or perhapsGalileojustdidnotwanttoexposeCopernicustocompetitionwiththeformidableTycho.TheDialogotooktheformofaconversationamongthreecharacters:Salviati,astand-inforGalileo

namedforGalileo’sfriendtheFlorentinenoblemanFilippoSalviati;Simplicio,anAristotelian,perhapsnamedforSimplicius(andperhapsintendedtorepresentasimpleton);andSagredo,namedforGalileo’sVenetian friend themathematicianGiovanni Francesco Sagredo, to judgewisely between them. Thefirst threedaysof the conversation showedSalviati demolishingSimplicio,with the tidesbrought inonlyonthefourthday.ThiscertainlyviolatedtheInquisition’sunsignedordertoGalileo,andarguablythelessstringentsignedorder(nottoholdordefendCopernicanism)aswell.Tomakemattersworse,theDialogowasinItalianratherthanLatin,sothatitcouldbereadbyanyliterateItalian,notjustbyscholars.Atthispoint,PopeUrbanwasshowntheunsigned1616orderoftheInquisitiontoGalileo,perhaps

by enemies thatGalileohadmade in the earlier arguments over sunspots and comets.Urban’s angermayhavebeenamplifiedbyasuspicionthathewasthemodelforSimplicio.Itdidn’thelpthatsomeofthepope’swordswhenhewasCardinalBarberinishowedupinthemouthofSimplicio.TheInquisitionorderedsalesoftheDialogotobebanned,butitwastoolate—thebookwasalreadysoldout.

Galileo was put on trial in April 1633. The case against him hinged on his violation of theInquisition’s orders of 1616. Galileowas shown the instruments of torture and tried a plea bargain,admitting that personal vanity had led him to go too far. But he was nevertheless declared under“vehementsuspicionofheresy,”condemnedtoeternalimprisonment,andforcedtoabjurehisviewthattheEarthmovesaroundtheSun.(AnapocryphalstoryhasitthatasGalileoleftthecourt,hemutteredunderhisbreath,“Eppursimuove,”thatis,“Butitdoesmove.”)FortunatelyGalileowasnottreatedasroughlyashemighthavebeen.Hewasallowedtobeginhis

imprisonmentasaguestofthearchbishopofSiena,andthentocontinueitinhisownvillaatArcetri,near Florence, and near the convent residence of his daughters, Sister Maria Celeste and SisterArcangela.27AswewillseeinChapter12,Galileowasableduringtheseyearstoreturntohisworkontheproblemofmotion,begunahalfcenturyearlieratPisa.Galileodied in1642whilestillunderhousearrest inArcetri. Itwasnotuntil1835 thatbooks like

Galileo’sthatadvocatedtheCopernicansystemwereremovedfromtheIndexofbooksbannedbytheCatholicchurch, thoughlongbefore thatCopernicanastronomyhadbecomewidelyaccepted inmostCatholic as well as Protestant countries. Galileo was rehabilitated by the church in the twentiethcentury.28 In 1979Pope JohnPaul II referred toGalileo’sLetter toChristina as having “formulatedimportantnormsofanepistemologicalcharacter,whichare indispensable to reconcileHolyScriptureand science.”29 A commissionwas convened to look into the case of Galileo, and reported that thechurchinGalileo’stimehadbeenmistaken.Thepoperesponded,“Theerrorofthetheologiansofthetime, when they maintained the centrality of the Earth, was to think that our understanding of thephysicalworld’sstructurewas,insomeway,imposedbytheliteralsenseoftheSacredScripture.”30Myownviewisthatthisisquiteinadequate.Thechurchofcoursecannotavoidtheknowledge,now

sharedbyeveryone,thatithadbeenwrongaboutthemotionoftheEarth.Butsupposethechurchhadbeen correct and Galileo mistaken about astronomy. The church would still have been wrong tosentenceGalileo to imprisonmentand todenyhis right topublish, just as it hadbeenwrong toburnGiordano Bruno, heretic as he was.31 Fortunately, although I don’t know if this has been explicitlyacknowledgedby the church, itwouldnot todaydreamof such actions.With the exceptionof thoseIslamic countries that punish blasphemy or apostasy, theworld has generally learned the lesson thatgovernments and religious authorities have no business imposing criminal penalties on religiousopinions,whethertrueorfalse.FromthecalculationsandobservationsofCopernicus,TychoBrahe,Kepler,andGalileo therehad

emergedacorrectdescriptionof thesolarsystem,encoded inKepler’s three laws.Anexplanationofwhytheplanetsobeytheselawshadtowaitageneration,untiltheadventofNewton.

12

ExperimentsBegun

Noonecanmanipulateheavenlybodies,sothegreatachievementsinastronomydescribedinChapter11werenecessarilybasedonpassiveobservation.Fortunatelythemotionsofplanetsinthesolarsystemare simple enough so that after many centuries of observation with increasingly sophisticatedinstrumentsthesemotionscouldatlastbecorrectlydescribed.Forthesolutionofotherproblemsitwasnecessary to go beyond observation and measurement and perform experiments, in which generaltheoriesaretestedorsuggestedbytheartificialmanipulationofphysicalphenomena.Inasensepeoplehavealwaysexperimented,using trialanderror inorder todiscoverways toget

thingsdone,fromsmeltingorestobakingcakes.Inspeakinghereofthebeginningsofexperiment,Iamconcernedonlywithexperimentscarriedouttodiscoverortestgeneraltheoriesaboutnature.Itisnotpossibletobepreciseaboutthebeginningofexperimentationinthissense.1Archimedesmay

havetestedhistheoryofhydrostaticsexperimentally,buthistreatiseOnFloatingBodiesfollowedthepurelydeductivestyleofmathematics,andgavenohintoftheuseofexperiment.HeroandPtolemydidexperimentstotesttheirtheoriesofreflectionandrefraction,buttheirexamplewasnotfolloweduntilcenturieslater.Onenewthingaboutexperimentationintheseventeenthcenturywaseagernesstomakepublicuseof

its results in judging the validity of physical theories. This appears early in the century inwork onhydrostatics,asisshowninGalileo’sDiscourseonBodiesinWaterof1612.Moreimportantwasthequantitativestudyof themotionof fallingbodies,anessentialprerequisite to theworkofNewton. Itwaswork on this problem, and also on the nature of air pressure, thatmarked the real beginning ofmodernexperimentalphysics.Likemuchelse,theexperimentalstudyofmotionbeginswithGalileo.Hisconclusionsaboutmotion

appeared inDialogues Concerning Two New Sciences, finished in 1635, when he was under housearrestatArcetri.Publicationwasforbiddenbythechurch’sCongregationoftheIndex,butcopiesweresmuggledoutofItaly.In1638thebookwaspublishedintheProtestantuniversitytownofLeidenbythe firm of Louis Elzevir. The cast of Two New Sciences again consists of Salviati, Simplicio, andSagredo,playingthesamerolesasbefore.Amongmuchelse,the“FirstDay”ofTwoNewSciencescontainsanargumentthatheavyandlight

bodies fall at the same rate, contradictingAristotle’s doctrine that heavy bodies fall faster than lightones.Ofcourse,becauseofairresistance,lightbodiesdofallalittlemoreslowlythanheavyones.Indealing with this, Galileo demonstrates his understanding of the need for scientists to live withapproximations, running counter to the Greek emphasis on precise statements based on rigorousmathematics.AsSalviatiexplainstoSimplicio:2

Aristotle says, “Ahundredpound ironball falling from theheight of ahundredbracciahits thegroundbeforeoneofjustonepoundhasdescendedasinglebraccio.”Isaythattheyarriveatthesame time. You find, on making the experiment, that the larger anticipates the smaller by twoinches;thatis,whenthelargeronestrikestheground,theotheristwoinchesbehindit.Andnowyouwanttohide,behindthosetwoinches,theninety-ninebracciaofAristotle,andspeakingonlyofmytinyerror,remainsilentabouthisenormousone.

Galileoalsoshowsthatairhaspositiveweight;estimatesitsdensity;discussesmotionthroughresistingmedia;explainsmusicalharmony;andreportsonthefactthatapendulumwilltakethesametimeforeachswing,whatevertheamplitudeoftheswings.*Thisistheprinciplethatdecadeslaterwastoleadto the invention of pendulum clocks and to the accuratemeasurement of the rate of acceleration offallingbodies.The“SecondDay”ofTwoNewSciencesdealswiththestrengthsofbodiesofvariousshapes.Itison

the “Third Day” that Galileo returns to the problem of motion, and makes his most interestingcontribution.HebeginstheThirdDaybyreviewingsometrivialpropertiesofuniformmotion,andthengoesontodefineuniformaccelerationalongthesamelinesasthefourteenth-centuryMertonCollegedefinition: the speed increases by equal amounts in each equal interval of time.Galileo also gives aproofofthemeanspeedtheorem,alongthesamelinesasOresme’sproof,buthemakesnoreferencetoOresme or to the Merton dons. Unlike his medieval predecessors, Galileo goes beyond thismathematical theorem and argues that freely falling bodies undergo uniform acceleration, but hedeclinestoinvestigatethecauseofthisacceleration.AsalreadymentionedinChapter10,therewasatthetimeawidelyheldalternativetothetheorythat

bodiesfallwithuniformacceleration.Accordingtothisotherview,thespeedthatfreelyfallingbodiesacquire inany intervalof time isproportional to thedistance fallen in that interval,not to the time.*Galileogivesvariousargumentsagainstthisview,*buttheverdictregardingthesedifferenttheoriesoftheaccelerationoffallingbodieshadtocomefromexperiment.Withthedistancefallenfromrestequal(accordingtothemeanspeedtheorem)tohalf thevelocity

attained times the elapsed time, and with that velocity itself proportional to the time elapsed, thedistancetraveledinfreefallshouldbeproportionaltothesquareofthetime.(SeeTechnicalNote25.)ThisiswhatGalileosetsouttoverify.Freely falling bodiesmove too rapidly forGalileo to have been able to check this conclusion by

following how far a falling body falls in any given time, so he had the idea of slowing the fall bystudyingballsrollingdownaninclinedplane.Forthistoberelevant,hehadtoshowhowthemotionofaballrollingdownaninclinedplaneisrelatedtoabodyinfreefall.Hedidthisbynotingthatthespeedaballreachesafterrollingdownaninclinedplanedependsonlyontheverticaldistancethroughwhichtheballhasrolled,notontheanglewithwhichtheplaneistilted.*Afreelyfallingballcanberegardedasonethatrollsdownaverticalplane,andsoif thespeedofaballrollingdownaninclinedplaneisproportional to the timeelapsed, then the sameought tobe true for a freely fallingball.For aplaneinclinedatasmallanglethespeedisofcoursemuchlessthanthespeedofabodyfallingfreely(thatisthepointofusinganinclinedplane)butthetwospeedsareproportional,andsothedistancetraveledalongtheplaneisproportionaltothedistancethatafreelyfallingbodywouldhavetraveledinthesametime.InTwoNewSciencesGalileoreportsthatthedistancerolledisproportionaltothesquareofthetime.

Galileo had done these experiments at Padua in 1603 with a plane at a less than 2° angle to thehorizontal,ruledwithlinesmarkingintervalsofabout1millimeter.3Hejudgedthetimebytheequalityoftheintervalsbetweensoundsmadeastheballreachedmarksalongitspath,whosedistancesfromthestartingpointareintheratios12=1:22=4:32=9,andsoon.IntheexperimentsreportedinTwoNewSciencesheinsteadmeasuredrelativeintervalsoftimewithawaterclock.AmodernreconstructionofthisexperimentshowsthatGalileocouldverywellhaveachievedtheaccuracyheclaimed.4GalileohadalreadyconsideredtheaccelerationoffallingbodiesintheworkdiscussedinChapter11,

theDialogueConcerningtheTwoChiefWorldSystems.OntheSecondDayofthispreviousDialogue,Salviatiineffectclaimsthatthedistancefallenisproportionaltothesquareofthetime,butgivesonlyamuddledexplanation.Healsomentions that a cannonballdropped fromaheightof100braccia will

reachthegroundin5seconds.ItisprettyclearthatGalileodidnotactuallymeasurethistime,5butishere presenting only an illustrative example. If one braccio is taken as 21.5 inches, then using themodernvalueoftheaccelerationduetogravity,thetimeforaheavybodytodrop100bracciais3.3seconds, not 5 seconds. But Galileo apparently never attempted a serious measurement of theaccelerationduetogravity.The“FourthDay”ofDialoguesConcerningTwoNewSciencestakesupthetrajectoryofprojectiles.

Galileo’s ideaswere largelybasedonanexperimenthedid in16086 (discussedindetail inTechnicalNote26).Aballisallowedtorolldownaninclinedplanefromvariousinitialheights,thenrollsalongthehorizontaltabletoponwhichtheinclinedplanesits,andfinallyshootsoffintotheairfromthetableedge.Bymeasuring the distance traveledwhen the ball reaches the floor, and by observation of theball’spathintheair,Galileoconcludedthatthetrajectoryisaparabola.GalileodoesnotdescribethisexperimentinTwoNewSciences,butinsteadgivesthetheoreticalargumentforaparabola.Thecrucialpoint,whichturnedouttobeessentialinNewton’smechanics,isthateachcomponentofaprojectile’smotionisseparatelysubjecttothecorrespondingcomponentoftheforceactingontheprojectile.Onceaprojectilerollsoffatableedgeorisshotoutofacannon,thereisnothingbutairresistancetochangeitshorizontalmotion,sothehorizontaldistancetraveledisverynearlyproportionaltothetimeelapsed.On the other hand, during the same time, like any freely falling body, the projectile is accelerateddownward,sothattheverticaldistancefallenisproportionaltothesquareofthetimeelapsed.Itfollowsthat theverticaldistance fallen isproportional to thesquareof thehorizontaldistance traveled.Whatsort of curve has this property? Galileo shows that the path of the projectile is a parabola, usingApollonius’ definition of a parabola as the intersection of a conewith a plane parallel to the cone’ssurface.(SeeTechnicalNote26.)The experiments described in Two New Sciences made a historic break with the past. Instead of

limitinghimselftothestudyoffreefall,whichAristotlehadregardedasnaturalmotion,Galileoturnedtoartificialmotions,ofballsconstrainedtorolldownaninclinedplaneorprojectilesthrownforward.Inthissense,Galileo’sinclinedplaneisadistantancestoroftoday’sparticleaccelerators,withwhichweartificiallycreateparticlesfoundnowhereinnature.Galileo’sworkonmotionwascarriedforwardbyChristiaanHuygens,perhapsthemostimpressive

figureinthebrilliantgenerationbetweenGalileoandNewton.Huygenswasbornin1629intoafamilyofhighcivilservantswhohadworkedintheadministrationoftheDutchrepublicundertheHouseofOrange.From1645to1647hestudiedboth lawandmathematicsat theUniversityofLeiden,buthethen turned full-time to mathematics and eventually to natural science. Like Descartes, Pascal, andBoyle,Huygenswas a polymath,working on awide range of problems inmathematics, astronomy,statics,hydrostatics,dynamics,andoptics.Huygens’mostimportantworkinastronomywashistelescopicstudyoftheplanetSaturn.In1655he

discovereditslargestmoon,Titan,revealingtherebythatnotonlytheEarthandJupiterhavesatellites.He also explained that Saturn’s peculiar noncircular appearance, noticed by Galileo, is due to ringssurroundingtheplanet.In1656–1657Huygensinventedthependulumclock.ItwasbasedonGalileo’sobservationthatthe

timeapendulumtakesforeachswingisindependentoftheswing’samplitude.Huygensrecognizedthatthisistrueonlyinthelimitofverysmallswings,andfoundingeniouswaystopreservetheamplitude-independenceofthetimesevenforswingsofappreciableamplitudes.Whilepreviouscrudemechanicalclockswouldgainorloseabout5minutesaday,Huygens’pendulumclocksgenerallygainedorlostnomorethan10secondsaday,andoneofthemlostonlyabout½secondperday.7Fromtheperiodofapendulumclockofagivenlength,Huygensthenextyearwasabletoinferthe

value of the acceleration of freely falling bodies near the Earth’s surface. In the Horologiumoscillatorium—published later, in 1673—Huygens was able to show that “the time of one small

oscillation is related to the time of perpendicular fall from half the height of the pendulum as thecircumferenceofacircleisrelatedtoitsdiameter.”8Thatis,thetimeforapendulumtoswingthroughasmallanglefromonesidetotheotherequalsπtimesthetimeforabodytofalladistanceequaltohalfthelengthofthependulum.(NotaneasyresulttoobtainasHuygensdid,withoutcalculus.)Usingthisprinciple,andmeasuringtheperiodsofpendulumsofvariouslengths,Huygenswasabletocalculatetheaccelerationduetogravity,somethingthatGalileocouldnotmeasureaccuratelywiththemeanshehadathand.AsHuygensexpressedit,afreelyfallingbodyfalls151/12“Parisfeet”inthefirstsecond.TheratiooftheParisfoottothemodernEnglishfootisvariouslyestimatedasbetween1.06and1.08;ifwetake1Parisfoottoequal1.07Englishfeet,thenHuygens’resultwasthatafreelyfallingbodyfalls16.1feet in the first second, which implies an acceleration of 32.2 feet/second per second, in excellentagreementwiththestandardmodernvalueof32.17feet/secondpersecond.(Asagoodexperimentalist,Huygenscheckedthat theaccelerationoffallingbodiesactuallydoesagreewithinexperimentalerrorwith the acceleration he inferred from his observations of pendulums.) As we will see, thismeasurement, later repeatedbyNewton,wasessential in relating the forceofgravityonEarth to theforcethatkeepstheMooninitsorbit.TheaccelerationduetogravitycouldhavebeeninferredfromearliermeasurementsbyRiccioliofthe

timeforweightstofallvariousdistances.9Tomeasuretimeaccurately,Riccioliusedapendulumthathad been carefully calibrated by counting its strokes in a solar or sidereal day. To his surprise, hismeasurementsconfirmedGalileo’sconclusion that thedistance fallen isproportional to the squareofthetime.Fromthesemeasurements,publishedin1651,itcouldhavebeencalculated(thoughRicciolididnotdoso)that theaccelerationduetogravityis30Romanfeet/secondpersecond.It isfortunatethatRicciolirecordedtheheightoftheAsinellitowerinBologna,fromwhichmanyoftheweightsweredropped,as312Romanfeet.Thetowerstillstands,anditsheightisknowntobe323modernEnglishfeet,soRiccioli’sRomanfootmusthavebeen323/312=1.035Englishfeet,and30Romanfeet/secondper second therefore corresponds to 31 English feet/second per second, in fair agreement with themodernvalue.Indeed,ifRicciolihadknownHuygens’relationbetweentheperiodofapendulumandthetimerequiredforabodytofallhalfitslength,hecouldhaveusedhiscalibrationofpendulumstocalculatetheaccelerationduetogravity,withouthavingtodropanythingofftowersinBologna.In 1664 Huygens was elected to the newAcadémie Royale des Sciences, with an accompanying

stipend,andhemovedtoParisforthenexttwodecades.Hisgreatworkonoptics,theTreatiseonLight,waswritten in Paris in 1678 and set out the wave theory of light. It was not published until 1690,perhapsbecauseHuygenshadhopedtotranslateitfromFrenchtoLatinbuthadneverfoundthetimebeforehisdeathin1695.WewillcomebacktoHuygens’wavetheoryinChapter14.In a 1669 article in the Journal des Sçavans, Huygens gave the correct statement of the rules

governingcollisionsofhardbodies(whichDescarteshadgottenwrong):itistheconservationofwhatarenowcalledmomentumandkineticenergy.10Huygensclaimedthathehadconfirmedtheseresultsexperimentally,presumablybystudying the impactofcollidingpendulumbobs, forwhich initialandfinal velocities could be precisely calculated. And as we shall see in Chapter 14, Huygens in theHorologiumoscillatoriumcalculatedtheaccelerationassociatedwithmotiononacurvedpath,aresultofgreatimportancetoNewton’swork.TheexampleofHuygensshowshowfarsciencehadcomefromtheimitationofmathematics—from

therelianceondeductionandtheaimofcertaintycharacteristicofmathematics. In thepreface to theTreatiseonLightHuygensexplains:

Therewillbeseen[in thisbook]demonstrationsof thosekindswhichdonotproduceasgreatacertitude as those of Geometry, and which even differ much therefrom, since whereas the

Geometersprove theirPropositionsbyfixedand incontestablePrinciples,here thePrinciplesareverifiedbytheconclusionstobedrawnfromthem;thenatureofthesethingsnotallowingofthisbeingdoneotherwise.11

Itisaboutasgoodadescriptionofthemethodsofmodernphysicalscienceasonecanfind.In the work of Galileo and Huygens on motion, experiment was used to refute the physics of

Aristotle.Thesamecanbesaidof thecontemporaneousstudyofairpressure.The impossibilityofavacuumwasoneofthedoctrinesofAristotlethatcameintoquestionintheseventeenthcentury.Itwaseventuallyunderstoodthatphenomenasuchassuction,whichseemedtoarisefromnature’sabhorrenceofavacuum,actuallyrepresenteffectsofthepressureoftheair.Threefiguresplayedakeyroleinthisdiscovery,inItaly,France,andEngland.WelldiggersinFlorencehadknownforsometimethatsuctionpumpscannotliftwatertoaheight

morethanabout18braccia,or32feet.(Theactualvalueatsealeveliscloserto33.5feet.)Galileoandothers had thought that this showed a limitation on nature’s abhorrence of a vacuum. A differentinterpretationwasofferedbyEvangelistaTorricelli, aFlorentinewhoworkedongeometry,projectilemotion,fluidmechanics,optics,andanearlyversionofcalculus.Torricelliarguedthatthislimitationonsuctionpumpsarisesbecausetheweightoftheairpressingdownonthewaterinthewellcouldsupportonlyacolumnofwaternomorethan18bracciahigh.Thisweightisdiffusedthroughtheair,soanysurfacewhetherhorizontalornotissubjectedbytheairtoaforceproportionaltoitsarea;theforceperarea,orpressure,exertedbyairatrestisequaltotheweightofaverticalcolumnofair,goinguptothetop of the atmosphere, divided by the cross-sectional area of the column. This pressure acts on thesurfaceofwaterinawell,andaddstothepressureofthewater,sothatwhenairpressureatthetopofaverticalpipeimmersedinthewaterisreducedbyapump,waterrisesinthepipe,butbyonlyanamountlimitedbythefinitepressureoftheair.Inthe1640sTorricellisetoutonaseriesofexperimentstoprovethisidea.Hereasonedthatsincethe

weightofavolumeofmercury is13.6 times theweightof thesamevolumeofwater, themaximumheightofacolumnofmercuryinaverticalglasstubeclosedontopthatcanbesupportedbytheair—whetherbytheairpressingdownonthesurfaceofapoolofmercuryinwhichthetubeisstanding,orontheopenbottomofthetubewhenexposedtotheair—shouldbe18bracciadividedby13.6,orusingmoreaccuratemodernvalues,33.5feet/13.6=30inches=760millimeters.In1643heobservedthatifaverticalglasstubelongerthanthisandclosedatthetopendisfilledwithmercury,thensomemercurywillflowoutuntiltheheightofthemercuryinthetubeisabout30inches.Thisleavesemptyspaceontop, nowknown as a “Torricellian vacuum.” Such a tube can then serve as a barometer, tomeasurechangesinambientairpressure;thehighertheairpressure,thehigherthecolumnofmercurythatitcansupport.TheFrenchpolymathBlaisePascal isbestknownforhisworkofChristian theology, thePensées,

andforhisdefenseoftheJansenistsectagainsttheJesuitorder,buthealsocontributedtogeometryandto the theory of probability, and explored the pneumatic phenomena studied by Torricelli. Pascalreasonedthatifthecolumnofmercuryinaglasstubeopenatthebottomisheldupbythepressureofthe air, then the height of the column shoulddecreasewhen the tube is carried to high altitudeon amountain, where there is less air overhead and hence lower air pressure. After this prediction wasverifiedinaseriesofexpeditionsfrom1648to1651,Pascalconcluded,“Alltheeffectsascribedto[theabhorrenceofavacuum]areduetotheweightandpressureoftheair,whichistheonlyrealcause.”12PascalandTorricellihavebeenhonoredbyhavingmodernunitsofpressurenamedafterthem.One

pascalisthepressurethatproducesaforceof1newton(theforcethatgivesamassof1kilogramanaccelerationof1meterpersecondinasecond)whenexertedonanareaof1squaremeter.Onetorristhepressure thatwill support a columnof1millimeterofmercury.Standardatmosphericpressure is

760torr,whichequalsalittlemorethan100,000pascals.TheworkofTorricelliandPascalwascarriedfurtherinEnglandbyRobertBoyle.Boylewasasonof

theearlofCork, andhenceanabsenteememberof the“ascendancy,” theProtestantupper class thatdominatedIrelandinhistime.HewaseducatedatEtonCollege,tookagrandtouroftheContinent,andfoughtonthesideofParliamentinthecivilwarsthatragedinEnglandinthe1640s.Unusuallyforamember of his class, he became fascinated by science. He was introduced to the new ideasrevolutionizingastronomyin1642,whenhereadGalileo’sDialogueConcerningtheTwoChiefWorldSystems.Boyle insisted on naturalistic explanations of natural phenomena, declaring, “None ismorewilling [thanmyself] toacknowledgeandvenerateDivineOmnipotence, [but]ourcontroversy isnotaboutwhatGodcando,butaboutwhatcanbedonebynaturalagents,notelevatedabovethesphereofnature.”13But,likemanybeforeDarwinandsomeevenafter,hearguedthatthewonderfulcapabilitiesofanimalsandmenshowedthattheymusthavebeendesignedbyabenevolentcreator.Boyle’s work on air pressure was described in 1660 in New Experiments Physico-Mechanical

Touching the Spring of the Air. In his experiments, he used an improved air pump, invented by hisassistantRobertHooke, aboutwhommore inChapter14.Bypumping air out of vessels,Boylewasabletoestablishthatairisneededforthepropagationofsound,forfire,andforlife.Hefoundthatthelevelofmercuryinabarometerdropswhenair ispumpedoutof itssurroundings,addingapowerfulargument infavorofTorricelli’sconclusion thatairpressure is responsibleforphenomenapreviouslyattributedtonature’sabhorrenceofavacuum.Byusingacolumnofmercurytovaryboththepressureand thevolumeof air in a glass tube, not letting air in or out andkeeping the temperature constant,Boylewasabletostudytherelationbetweenpressureandvolume.In1662,inasecondeditionofNewExperiments,hereportedthatthepressurevarieswiththevolumeinsuchawayastokeepthepressuretimesthevolumefixed,arulenowknownasBoyle’slaw.Not evenGalileo’s experimentswith inclined planes illustrate sowell the new aggressive style of

experimentalphysicsastheseexperimentsonairpressure.Nolongerwerenaturalphilosophersrelyingon nature to reveal its principles to casual observers. InsteadMotherNaturewas being treated as adeviousadversary,whosesecretshadtobewrestedfromherbytheingeniousconstructionofartificialcircumstances.

13

MethodReconsidered

BytheendofthesixteenthcenturytheAristotelianmodelforscientificinvestigationhadbeenseverelychallenged.Itwasnaturalthentoseekanewapproachtothemethodforgatheringreliableknowledgeabout nature. The two figureswho became best known for attempts to formulate a newmethod forscience are FrancisBacon andRenéDescartes. They are, inmy opinion, the two individualswhoseimportanceinthescientificrevolutionismostoverrated.Francis Bacon was born in 1561, the son of Nicholas Bacon, Lord Keeper of the Privy Seal of

England.After an education atTrinityCollege,Cambridge, hewas called to thebar, and followedacareer in law, diplomacy, and politics. He rose to become Baron Verulam and lord chancellor ofEngland in 1618, and laterViscount St. Albans, but in 1621 hewas found guilty of corruption anddeclaredbyParliamenttobeunfitforpublicoffice.Bacon’s reputation in the history of science is largely based on his bookNovumOrganum (New

Instrument,orTrueDirectionsConcerningtheInterpretationofNature),publishedin1620.InthisbookBacon, neither a scientist nor a mathematician, expressed an extreme empiricist view of science,rejectingnotonlyAristotlebutalsoPtolemyandCopernicus.Discoveriesweretoemergedirectlyfromcareful,unprejudicedobservationofnature,notbydeductionfromfirstprinciples.Healsodisparagedany research that did not serve an immediate practical purpose. InTheNewAtlantis, he imagined acooperative research institute, “Solomon’s House,” whose members would devote themselves tocollecting useful facts about nature. In this way, manwould supposedly regain the dominance overnaturethatwaslostaftertheexpulsionfromEden.Bacondiedin1626.Thereisastorythat,truetohisempiricalprinciples,hesuccumbedtopneumoniaafteranexperimentalstudyofthefreezingofmeat.BaconandPlatostandatoppositeextremes.Ofcourse,bothextremeswerewrong.Progressdepends

onablendofobservationorexperiment,whichmaysuggestgeneralprinciples,andofdeductionsfromtheseprinciplesthatcanbetestedagainstnewobservationsorexperiments.Thesearchforknowledgeofpractical value can serve as a corrective touncontrolled speculation,but explaining theworldhasvalue in itself, whether or not it leads directly to anything useful. Scientists in the seventeenth andeighteenth centurieswould invokeBacon as a counterweight to Plato andAristotle, somewhat as anAmericanpoliticianmightinvokeJeffersonwithouteverhavingbeeninfluencedbyanythingJeffersonsaidordid. It isnotclear tome thatanyone’s scientificworkwasactuallychanged for thebetterbyBacon’swriting.GalileodidnotneedBacontotellhimtodoexperiments,andneitherIthinkdidBoyleorNewton.AcenturybeforeGalileo,anotherFlorentine,LeonardodaVinci,wasdoingexperimentsonfallingbodies,flowingliquids,andmuchelse.1Weknowaboutthisworkonlyfromapairoftreatisesonpaintingandonfluidmotionthatwerecompiledafterhisdeath,andfromnotebooksthathavebeendiscoveredfromtimetotimesincethen,butifLeonardo’sexperimentshadnoinfluenceontheadvanceofscience,atleasttheyshowthatexperimentwasintheairlongbeforeBacon.RenéDescarteswasanaltogethermorenoteworthyfigurethanBacon.Bornin1596intothejuridical

nobilityofFrance,thenoblessederobe,hewaseducatedattheJesuitcollegeofLaFlèche,studiedlawat the University of Poitiers, and served in the army of Maurice of Nassau in the Dutch war ofindependence.In1619Descartesdecidedtodevotehimselftophilosophyandmathematics,workthatbeganinearnestafter1628,whenhesettledpermanentlyinHolland.DescartesputhisviewsaboutmechanicsintoLeMonde,writtenintheearly1630sbutnotpublished

until1664, afterhisdeath. In1637hepublishedaphilosophicalwork,Discours de laméthodepourbien conduire sa raison, et chercher la véritédans les sciences (Discourseon theMethodofRightlyConductingOne’sReasonandofSeekingTruthintheSciences).Hisideaswerefurtherdevelopedinhislongestwork,thePrinciplesofPhilosophy,publishedinLatinin1644andtheninaFrenchtranslationin1647.IntheseworksDescartesexpressesskepticismaboutknowledgederivedfromauthorityorthesenses. ForDescartes the only certain fact is that he exists, deduced from the observation that he isthinkingaboutit.Hegoesontoconcludethattheworldexists,becauseheperceivesitwithoutexertinganeffortofwill.HerejectsAristotelianteleology—thingsareastheyare,notbecauseofanypurposetheymightserve.Hegivesseveralarguments(allunconvincing)fortheexistenceofGod,butrejectstheauthority of organized religion.He also rejects occult forces at a distance—things interactwith eachotherthroughdirectpullingandpushing.Descartes was a leader in bringing mathematics into physics, but like Plato he was too much

impressedbythecertaintyofmathematicalreasoning.InPartIof thePrinciplesofPhilosophy, titled“OnthePrinciplesofHumanKnowledge,”Descartesdescribedhowfundamentalscientificprinciplescould be deducedwith certainty by pure thought.We can trust in the “natural enlightenment or thefacultyofknowledgegiven tousbyGod”because“itwouldbecompletelycontradictory forHimtodeceiveus.”2ItisoddthatDescartesthoughtthataGodwhoallowedearthquakesandplagueswouldnotallowaphilosophertobedeceived.Descartesdidacceptthattheapplicationoffundamentalphysicalprinciplestospecificsystemsmight

involveuncertaintyandcallforexperimentation,ifonedidnotknowallthedetailsofwhatthesystemcontains. Inhisdiscussionof astronomy inPart IIIofPrinciplesofPhilosophy, he considersvarioushypothesesabout thenatureof theplanetarysystem,andcitesGalileo’sobservationsof thephasesofVenusasreasonforpreferringthehypothesesofCopernicusandTychotothatofPtolemy.Thisbriefsummarybarely touchesonDescartes’views.Hisphilosophywasand ismuchadmired,

especially in France and among specialists in philosophy. I find this puzzling. For someone whoclaimed to have found the truemethod for seeking reliable knowledge, it is remarkable howwrongDescarteswasaboutsomanyaspectsofnature.HewaswronginsayingthattheEarthisprolate(thatis,thatthedistancethroughtheEarthisgreaterfrompoletopolethanthroughtheequatorialplane).He,likeAristotle,waswronginsayingthatavacuumisimpossible.Hewaswronginsayingthatlightistransmitted instantaneously.*Hewaswrong in saying that space is filledwithmaterial vortices thatcarrytheplanetsaroundintheirpaths.Hewaswronginsayingthatthepinealglandistheseatofasoulresponsibleforhumanconsciousness.Hewaswrongaboutwhatquantityisconservedincollisions.Hewaswronginsayingthatthespeedofafreelyfallingbodyisproportionaltothedistancefallen.Finally,onthebasisofobservationofseverallovablepetcats,IamconvincedthatDescarteswasalsowronginsaying that animals aremachineswithout true consciousness.Voltaire had similar reservations aboutDescartes:3

Heerredonthenatureofthesoul,ontheproofsoftheexistenceofGod,onthesubjectofmatter,onthelawsofmotion,onthenatureoflight.Headmittedinnateideas,heinventednewelements,hecreatedaworld,hemademanaccordingtohisownfashion—infact,itisrightlysaidthatmanaccordingtoDescartesisDescartes’man,farremovedfrommanasheactuallyis.

Descartes’ scientificmisjudgmentswouldnotmatter in assessing theworkof someonewhowroteabout ethical or political philosophy, or even metaphysics; but because Descartes wrote about “themethodofrightlyconductingone’sreasonandofseekingtruthinthesciences,”hisrepeatedfailuretogetthingsrightmustcastashadowonhisphilosophicaljudgment.DeductionsimplycannotcarrytheweightthatDescartesplacedonit.

Eventhegreatestscientistsmakemistakes.WehaveseenhowGalileowaswrongaboutthetidesandcomets, andwewill see howNewtonwaswrong about diffraction. For all hismistakes, Descartes,unlikeBacon,didmakesignificantcontributionstoscience.ThesewerepublishedasasupplementtotheDiscourseonMethod,underthreeheadings:geometry,optics,andmeteorology.4These,ratherthanhisphilosophicalwritings,inmyviewrepresenthispositivecontributionstoscience.Descartes’ greatest contributionwas the invention of a newmathematicalmethod, now known as

analytic geometry, inwhich curves or surfaces are represented by equations that are satisfied by thecoordinatesofpointsonthecurveorsurface.“Coordinates”ingeneralcanbeanynumbersthatgivethelocationofapoint,suchaslongitude,latitude,andaltitude,buttheparticularkindknownas“Cartesiancoordinates”arethedistancesofthepointfromacenteralongasetoffixedperpendiculardirections.Forinstance,inanalyticgeometryacircleofradiusRisacurveonwhichthecoordinatesxandyaredistancesmeasuredfromthecenterofthecirclealonganytwoperpendiculardirections,andsatisfytheequationx2+y2=R2.(TechnicalNote18givesasimilardescriptionofanellipse.)Thisveryimportantuse of letters of the alphabet to represent unknown distances or other numbers originated in thesixteenth centurywith theFrenchmathematician, courtier, and cryptanalystFrançoisViète, butViètestill wrote out equations in words. Themodern formalism of algebra and its application to analyticgeometryareduetoDescartes.Usinganalyticgeometry,wecanfindthecoordinatesofthepointwheretwocurvesintersect,orthe

equation for the curvewhere two surfaces intersect, by solving the pair of equations that define thecurves or the surfaces. Most physicists today solve geometric problems in this way, using analyticgeometry,ratherthantheclassicmethodsofEuclid.In physics Descartes’ significant contributions were in the study of light. First, in his Optics,

DescartespresentedtherelationbetweentheanglesofincidenceandrefractionwhenlightpassesfrommediumAtomediumB(forexample,fromairtowater):iftheanglebetweentheincidentrayandtheperpendiculartothesurfaceseparatingthemediaisi,andtheanglebetweentherefractedrayandthisperpendicularisr,thenthesine*ofidividedbythesineofrisanangle-independentconstantn:

sineofi/sineofr=n

InthecommoncasewheremediumA is theair (or,strictlyspeaking,emptyspace),n is theconstantknownasthe“indexofrefraction”ofmediumB.Forinstance,ifA isairandB iswaterthennistheindexof refractionofwater, about 1.33. In any case like this,wheren is larger than 1, the angle ofrefractionrissmallerthantheangleofincidencei,andtherayoflightenteringthedensermediumisbenttowardthedirectionperpendiculartothesurface.Unknown to Descartes, this relation had already been obtained empirically in 1621 by the Dane

WillebrordSnellandevenearlierbytheEnglishmanThomasHarriot;andafigureinamanuscriptbythetenth-centuryArabphysicistIbnSahlsuggeststhatitwasalsoknowntohim.ButDescarteswasthefirsttopublishit.TodaytherelationisusuallyknownasSnell’slaw,exceptinFrance,whereitismorecommonlyattributedtoDescartes.Descartes’ derivation of the law of refraction is difficult to follow, in part because neither in his

accountofthederivationnorinthestatementoftheresultdidDescartesmakeuseofthetrigonometricconceptofthesineofanangle.Instead,hewroteinpurelygeometricterms,thoughaswehaveseenthesinehadbeenintroducedfromIndiaalmostsevencenturiesearlierbyal-Battani,whoseworkwaswellknowninmedievalEurope.Descartes’derivationisbasedonananalogywithwhatDescartesimaginedwouldhappenwhenatennisballishitthroughathinfabric;theballwilllosesomespeed,butthefabriccanhavenoeffectonthecomponentoftheball’svelocityalongthefabric.Thisassumptionleads(asshowninTechnicalNote27)totheresultcitedabove:theratioofthesinesoftheanglesthatthetennis

ball makes with the perpendicular to the screen before and after it hits the screen is an angle-independent constant n. Though it is hard to see this result in Descartes’ discussion, he must haveunderstood this result, because with a suitable value for n he gets more or less the right numericalanswersinhistheoryoftherainbow,discussedbelow.TherearetwothingsclearlywrongwithDescartes’derivation.Obviously, light isnota tennisball,

and thesurfaceseparatingairandwaterorglass isnota thinfabric,so this isananalogyofdubiousrelevance,especiallyforDescartes,whothoughtthatlight,unliketennisballs,alwaystravelsatinfinitespeed.5Inaddition,Descartes’analogyalsoleadstoawrongvalueforn.Fortennisballs(asshowninTechnicalNote27)hisassumptionimpliesthatnequalstheratioofthespeedoftheballvBinmediumBafteritpassesthroughthescreentoitsspeedvAinmediumAbeforeithitsthescreen.Ofcourse,theballwouldbeslowedbypassingthroughthescreen,sovBwouldbelessthanvAandtheirrationwouldbeless than1.If thisappliedtolight, itwouldmeanthat theanglebetweentherefractedrayandtheperpendicular to the surface would be greater than the angle between the incident ray and thisperpendicular.Descartes knew this, and even supplied a diagram showing the path of the tennis ballbeingbent away from theperpendicular.Descartes alsoknew that this iswrong for light, for as hadbeenobservedatleastsincethetimeofPtolemy,arayoflightenteringwaterfromtheairisbenttowardtheperpendiculartothewater’ssurface,sothatthesineofiisgreaterthanthesineofr,andhencenisgreaterthan1.InathoroughlymuddleddiscussionthatIcannotunderstand,Descartessomehowarguesthat light travelsmoreeasily inwater than inair, so that for lightn isgreater than1.ForDescartes’purposeshis failure to explain thevalueofn didn’t reallymatter, becausehe could anddid take thevalueofnfromexperiment(perhapsfromthedatainPtolemy’sOptics),whichofcoursegivesngreaterthan1.Amore convincing derivation of the law of refractionwas given by themathematician Pierre de

Fermat(1601–1665),alongthe linesof thederivationbyHeroofAlexandriaof theequal-anglesrulegoverningreflection,butnowmakingtheassumptionthatlightraystakethepathofleasttime, ratherthanof leastdistance.Thisassumption(asshowninTechnicalNote28) leads to thecorrect formula,thatnistheratioofthespeedoflightinmediumAtoitsspeedinmediumB,andisthereforegreaterthan1whenA is air andB is glassorwater.Descartes couldneverhavederived this formula forn,becauseforhimlighttraveledinstantaneously.(AswewillseeinChapter14,yetanotherderivationofthecorrectresultwasgivenbyChristiaanHuygens,aderivationbasedonHuygens’theoryoflightasatravelingdisturbance,whichdidnotrelyonFermat’saprioriassumptionthat thelightraytravelsthepathofleasttime.)Descartesmadeabrilliantapplicationofthelawofrefraction:inhisMeteorologyheusedhisrelation

betweenanglesofincidenceandrefractiontoexplaintherainbow.ThiswasDescartesathisbestasascientist.Aristotle had argued that the colors of the rainboware producedwhen light is reflectedbysmallparticlesofwatersuspendedintheair.6Also,aswehaveseeninChapters9and10,intheMiddleAgesbothal-FarisiandDietrichofFreiburghadrecognizedthatrainbowsareduetotherefractionofraysoflightwhentheyenterandleavedropsofrainsuspendedintheair.ButnoonebeforeDescarteshadpresentedadetailedquantitativedescriptionofhowthisworks.Descartesfirstperformedanexperiment,usingathin-walledsphericalglassglobefilledwithwateras

amodelofaraindrop.Heobservedthatwhenraysofsunlightwereallowedtoenter theglobealongvarious directions, the light that emerged at an angle of about 42° to the incident direction was“completely red, and incomparablymore brilliant than the rest.”He concluded that a rainbow (or atleastitsrededge)tracesthearcintheskyforwhichtheanglebetweenthelineofsighttotherainbowand thedirection from the rainbow to thesun isabout42°.Descartesassumed that the light raysarebentbyrefractionwhenenteringadrop,arereflectedfromthebacksurfaceofthedrop,andthenare

bentagainbyrefractionwhenemergingfromthedropbackintotheair.Butwhatexplainsthispropertyofraindrops,ofpreferentiallysendinglightbackatanangleofabout42°totheincidentdirection?To answer this, Descartes considered rays of light that enter a spherical drop of water along 10

different parallel lines. He labeled these rays by what is today called their impact parameter b, theclosestdistanceto thecenterof thedropthat theraywouldreachif itwentstraight throughthedropwithoutbeingrefracted.Thefirstraywaschosensothatifnotrefracteditwouldpassthecenterofthedropatadistanceequalto10percentofthedrop’sradiusR(thatis,withb=0.1R),whilethetenthraywas chosen to graze the drop’s surface (so that b =R), and the intermediate rays were taken to beequally spaced between these two. Descartes worked out the path of each ray as it was refractedenteringthedrop,reflectedbythebacksurfaceofthedrop,andthenrefractedagainasitleftthedrop,usingtheequal-angleslawofreflectionofEuclidandHero,andhisownlawofrefraction,andtakingtheindexofrefractionnofwatertobe4/3.ThefollowingtablegivesvaluesfoundbyDescartesfortheangleφ(phi)betweentheemergingrayanditsincidentdirectionforeachray,alongwiththeresultsofmyowncalculationusingthesameindexofrefraction:

b/R φ(Descartes) φ(recalculated)0.1 5°40' 5°44'0.2 11°19' 11°20'0.3 17°56' 17°6'0.4 22°30' 22°41'0.5 27°52' 28°6'0.6 32°56' 33°14'0.7 37°26' 37°49'0.8 40°44' 41°13'0.9 40°57' 41°30'1.0 13°40' 14°22'

TheinaccuracyofsomeofDescartes’resultscanbesetdowntothelimitedmathematicalaidsavailableinhistime.Idon’tknowifhehadaccesstoatableofsines,andhecertainlyhadnothinglikeamodernpocketcalculator.Still,Descarteswouldhaveshownbetterjudgmentifhehadquotedresultsonlytothenearest10minutesofarc,ratherthantothenearestminute.AsDescartesnoticed,thereisarelativelywiderangeofvaluesoftheimpactparameterbforwhich

theangleφiscloseto40°.Hethenrepeatedthecalculationfor18morecloselyspacedrayswithvaluesofbbetween80percentand100percentofthedrop’sradius,whereφisaround40°.Hefoundthattheangle φ for 14 of these 18 rays was between 40° and a maximum of 41° 30'. So these theoreticalcalculationsexplainedhisexperimentalobservationmentionedearlier,ofapreferredangleofroughly42°.Technical Note 29 gives a modern version of Descartes’ calculation. Instead of working out the

numericalvalueoftheangleφbetweentheincomingandoutgoingraysforeachrayinanensembleofrays,asDescartesdid,wederiveasimpleformulathatgivesφforanyray,withanyimpactparameterb,andforanyvalueoftherationofthespeedoflightinairtothespeedoflightinwater.Thisformulaisthen used to find the value of φ where the emerging rays are concentrated.* For n equal to 4/3 thefavoredvalueofφ,where theemerging light issomewhatconcentrated, turnsout tobe42.0°, justasfound by Descartes. Descartes even calculated the corresponding angle for the secondary rainbow,producedbylightthatisreflectedtwicewithinaraindropbeforeitemerges.

Descartessawaconnectionbetweentheseparationofcolorsthatischaracteristicoftherainbowandthecolorsexhibitedbyrefractionoflightinaprism,buthewasunabletodealwitheitherquantitatively,becausehedidnotknowthat thewhite lightof thesuniscomposedof lightofallcolors,or that theindexofrefractionoflightdependsslightlyonitscolor.Infact,whileDescarteshadtakentheindexforwatertobe4/3=1.333...,itisactuallycloserto1.330fortypicalwavelengthsofredlightandcloserto 1.343 for blue light.One finds (using the general formula derived in TechnicalNote 29) that themaximumvaluefortheangleφbetweentheincidentandemergingraysis42.8°forredlightand40.7°forblue light.This iswhyDescartessawbright red lightwhenhe lookedathisglobeofwateratanangleof42°tothedirectionoftheSun’srays.Thatvalueoftheangleφisabovethemaximumvalue40.7°oftheanglethatcanemergefromtheglobeofwaterforbluelight,sonolightfromtheblueendofthespectrumcouldreachDescartes;butitisjustbelowthemaximumvalue42.8°ofφforredlight,so(asexplainedinthepreviousfootnote)thiswouldmaketheredlightparticularlybright.TheworkofDescartesonopticswasverymuchinthemodeofmodernphysics.Descartesmadea

wildguessthatlightcrossingtheboundarybetweentwomediabehaveslikeatennisballpenetratingathinscreen,andusedittoderivearelationbetweentheanglesofincidenceandrefractionthat(withasuitable choiceof the indexof refractionn) agreedwithobservation.Next, using aglobe filledwithwaterasamodelofaraindrop,hemadeobservationsthatsuggestedapossibleoriginofrainbows,andhethenshowedmathematicallythattheseobservationsfollowedfromhistheoryofrefraction.Hedidn’tunderstandthecolorsoftherainbow,sohesidesteppedtheissue,andpublishedwhathedidunderstand.This is just aboutwhat a physicistwould do today, but aside from its application ofmathematics tophysics,whatdoesithavetodowithDescartes’DiscourseonMethod?Ican’tseeanysignthathewasfollowing his own prescriptions for “Rightly ConductingOne’s Reason and of Seeking Truth in theSciences.”I should add that in his Principles of Philosophy Descartes offered a significant qualitative

improvementtoBuridan’snotionofimpetus.7Hearguedthat“allmovementis,ofitself,alongstraightlines,” so that (contrary tobothAristotleandGalileo)a force is required tokeepplanetarybodies intheir curved orbits. ButDescartesmade no attempt at a calculation of this force.Aswewill see inChapter14,itremainedforHuygenstocalculatetheforcerequiredtokeepabodymovingatagivenspeedonacircleofgivenradius,andforNewtontoexplainthisforce,astheforceofgravitation.In 1649 Descartes traveled to Stockholm to serve as a teacher of the reigning Queen Christina.

Perhaps as a result of the cold Swedish weather, and having to get up to meet Christina at anunwontedlyearlyhour,Descartesinthenextyear,likeBacon,diedofpneumonia.FourteenyearslaterhisworksjoinedthoseofCopernicusandGalileoontheIndexofbooksforbiddentoRomanCatholics.ThewritingsofDescarteson scientificmethodhave attractedmuchattention amongphilosophers,

butIdon’tthinktheyhavehadmuchpositiveinfluenceonthepracticeofscientificresearch(oreven,asarguedabove,onDescartes’ownmostsuccessfulscientificwork).Hiswritingsdidhaveonenegativeeffect:theydelayedthereceptionofNewtonianphysicsinFrance.TheprogramsetoutintheDiscourseon Method, of deriving scientific principles by pure reason, never worked, and never could haveworked.HuygenswhenyoungconsideredhimselfafollowerofDescartes,buthecametounderstandthat scientific principles were only hypotheses, to be tested by comparing their consequences withobservation.8Ontheotherhand,Descartes’workonopticsshowsthathetoounderstoodthatthissortofscientific

hypothesisissometimesnecessary.LaurensLaudanhasfoundevidenceforthesameunderstandinginDescartes’ discussionof chemistry in thePrinciples ofPhilosophy.9This raises thequestionwhetherany scientists actually learned from Descartes the practice of making hypotheses to be testedexperimentally,asLaudan thoughtwas trueofBoyle.Myownview is that thishypotheticalpractice

waswidelyunderstoodbeforeDescartes.HowelsewouldonedescribewhatGalileodid,inusingthehypothesis that falling bodies are uniformly accelerated to derive the consequence that projectilesfollowparabolicpaths,andthentestingitexperimentally?According to the biography ofDescartes byRichardWatson,10 “Without theCartesianmethod of

analyzingmaterialthingsintotheirprimaryelements,wewouldneverhavedevelopedtheatombomb.Theseventeenth-centuryriseofModernScience,theeighteenth-centuryEnlightenment,thenineteenth-century Industrial Revolution, your twentieth-century personal computer, and the twentieth-centurydecipheringofthebrain—allCartesian.”Descartesdidmakeagreatcontributiontomathematics,butitisabsurdtosupposethatitisDescartes’writingonscientificmethodthathasbroughtaboutanyofthesehappyadvances.DescartesandBaconareonlytwoofthephilosopherswhooverthecenturieshavetriedtoprescribe

rulesforscientificresearch.Itneverworks.Welearnhowtodoscience,notbymakingrulesabouthowtodoscience,butfromtheexperienceofdoingscience,drivenbydesireforthepleasurewegetwhenourmethodssucceedinexplainingsomething.

14

TheNewtonianSynthesis

WithNewtonwecometotheclimaxofthescientificrevolution.Butwhatanoddbirdtobecastinsuchahistoricrole!NewtonnevertraveledoutsideanarrowstripofEngland,linkingLondon,Cambridge,andhisbirthplace inLincolnshire,noteventosee thesea,whosetidessomuchinterestedhim.Untilmiddleagehewasneverclosetoanywoman,noteventohismother.*Hewasdeeplyconcernedwithmatters having little to dowith science, such as the chronologyof theBookofDaniel.A catalogofNewtonmanuscriptsputonsaleatSotheby’sin1936shows650,000wordsonalchemy,and1.3millionwordson religion.With thosewhomightbecompetitorsNewtoncouldbedeviousandnasty.Yethetied up strands of physics, astronomy, andmathematics whose relations had perplexed philosopherssincePlato.WritersaboutNewtonsometimesstressthathewasnotamodernscientist.Thebest-knownstatement

alongtheselinesisthatofJohnMaynardKeynes(whohadboughtsomeoftheNewtonpapersinthe1936 auction at Sotheby’s): “Newtonwas not the first of the age of reason. Hewas the last of themagicians, the last of the Babylonians and Sumerians, the last great mindwhich looked out on thevisible and intellectual world with the same eyes as those who began to build our intellectualinheritanceratherlessthan10,000yearsago.”*ButNewtonwasnotatalentedholdoverfromamagicalpast.Neitheramagiciannoranentirelymodernscientist,hecrossed the frontierbetween thenaturalphilosophyofthepastandwhatbecamemodernscience.Newton’sachievements,ifnothisoutlookorpersonal behavior, provided the paradigm that all subsequent science has followed, as it becamemodern.Isaac Newton was born on Christmas Day 1642 at a family farm, Woolsthorpe Manor, in

Lincolnshire.Hisfather,anilliterateyeomanfarmer,haddiedshortlybeforeNewton’sbirth.Hismotherwas higher in social rank, a member of the gentry, with a brother who had graduated from theUniversityofCambridgeandbecomeaclergyman.WhenNewtonwasthreehismotherremarriedandleftWoolsthorpe,leavinghimbehindwithhisgrandmother.Whenhewas10yearsoldNewtonwenttotheone-roomKing’sSchoolatGrantham,eightmilesfromWoolsthorpe,andlivedthereinthehouseofanapothecary.AtGranthamhelearnedLatinandtheology,arithmeticandgeometry,andalittleGreekandHebrew.Attheageof17Newtonwascalledhometotakeuphisdutiesasafarmer,butforthesehewasfound

tobenotwellsuited.TwoyearslaterhewassentuptoTrinityCollege,Cambridge,asasizar,meaningthathewouldpayforhistuitionandroomandboardbywaitingonfellowsofthecollegeandonthosestudentswhohadbeenabletopaytheirfees.LikeGalileoatPisa,hebeganhiseducationwithAristotle,buthesoonturnedawaytohisownconcerns.Inhissecondyearhestartedaseriesofnotes,Questionesquandamphilosophicae,inanotebookthathadpreviouslybeenusedfornotesonAristotle,andwhichfortunatelyisstillextant.InDecember1663theUniversityofCambridgereceivedadonationfromamemberofParliament,

HenryLucas,establishingaprofessorshipinmathematics,theLucasianchair,withastipendof£100ayear.Beginningin1664thechairwasoccupiedbyIsaacBarrow,thefirstprofessorofmathematicsatCambridge,12yearsolderthanNewton.AroundthenNewtonbeganhisstudyofmathematics,partlywith Barrow and partly alone, and received his bachelor of arts degree. In 1665 the plague struckCambridge, theuniversity largelyshutdown,andNewtonwenthometoWoolsthorpe.In thoseyears,

from1664on,Newtonbeganhisscientificresearch,tobedescribedbelow.BackinCambridge,in1667NewtonwaselectedafellowofTrinityCollege;thefellowshipbrought

him£2ayearandfreeaccesstothecollegelibrary.HeworkedcloselywithBarrow,helpingtopreparewritten versions of Barrow’s lectures. Then in 1669Barrow resigned the Lucasian chair in order todevotehimselfentirelytotheology.AtBarrow’ssuggestion,thechairwenttoNewton.Withfinancialhelpfromhismother,Newtonbegantospreadhimself,buyingnewclothesandfurnishingsanddoingabitofgambling.1Alittleearlier,immediatelyaftertherestorationoftheStuartmonarchyin1660,asocietyhadbeen

formed by a few Londoners including Boyle, Hooke, and the astronomer and architect ChristopherWren,whowouldmeettodiscussnaturalphilosophyandobserveexperiments.Atthebeginningithadjustoneforeignmember,ChristiaanHuygens.Thesocietyreceivedaroyalcharterin1662astheRoyalSociety of London, and has remained Britain’s national academy of science. In 1672 Newton waselectedtomembershipintheRoyalSociety,whichhelaterservedaspresident.In1675Newtonfacedacrisis.Eightyearsafterbeginninghisfellowship,hehadreachedthepointat

whichfellowsofaCambridgecollegeweresupposedtotakeholyordersintheChurchofEngland.ThiswouldrequireswearingtobeliefinthedoctrineoftheTrinity,butthatwasimpossibleforNewton,whorejected the decision of the Council of Nicaea that the Father and the Son are of one substance.Fortunately,thedeedthathadestablishedtheLucasianchairincludedastipulationthatitsholdershouldnotbeactive in thechurch, andon thatbasisKingCharles IIwas induced to issueadecree that theholderoftheLucasianchairwouldthenceforthneverberequiredtotakeholyorders.SoNewtonwasabletocontinueatCambridge.Let’snowtakeupthegreatworkthatNewtonbeganatCambridgein1664.Thisresearchcenteredon

optics,mathematics, andwhat later came to be calleddynamics.Hiswork in anyoneof these threeareaswouldqualifyhimasoneofthegreatscientistsofhistory.Newton’s chief experimental achievementswere concernedwith optics.*His undergraduate notes,

theQuestionesquandamphilosophicae,showhimalreadyconcernedwiththenatureoflight.Newtonconcluded, contrary toDescartes, that light is not a pressure on the eyes, for if itwere then the skywould seem brighter to uswhenwe are running.AtWoolsthorpe in 1665 he developed his greatestcontribution tooptics,his theoryofcolor. Ithadbeenknownsinceantiquity thatcolorsappearwhenlightpasses throughacurvedpieceofglass,but ithadgenerallybeen thought that thesecolorsweresomehowproducedbytheglass.Newtonconjecturedinsteadthatwhitelightconsistsofallthecolors,and that the angle of refraction in glass orwater depends slightly on the color, red light being bentsomewhatlessthanbluelight,sothatthecolorsareseparatedwhenlightpassesthroughaprismoraraindrop.* This would explain what Descartes had not understood, the appearance of colors in therainbow.To test this idea,Newtoncarriedout twodecisiveexperiments.First, afterusingaprism tocreateseparateraysofblueandredlight,hedirectedtheseraysseparatelyintootherprisms,andfoundno furtherdispersion intodifferentcolors.Next,withacleverarrangementofprisms,hemanaged torecombine all the different colors produced by refraction of white light, and found that when thesecolorsarecombinedtheyproducewhitelight.Thedependenceof the angleof refractiononcolorhas theunfortunate consequence that theglass

lensesintelescopeslikethoseofGalileo,Kepler,andHuygensfocusthedifferentcolorsinwhitelightdifferently,blurring the imagesofdistantobjects.Toavoid thischromaticaberrationNewtonin1669inventeda telescopeinwhichlight is initiallyfocusedbyacurvedmirrorrather thanbyaglass lens.(Thelightraysarethendeflectedbyaplanemirroroutofthetelescopetoaglasseyepiece,sonotallchromaticaberrationwaseliminated.)Withareflectingtelescopeonlysixincheslong,hewasabletoachieve a magnification by 40 times. All major astronomical light-gathering telescopes are nowreflectingtelescopes,descendantsofNewton’sinvention.Onmyfirstvisittothepresentquartersofthe

Royal Society in Carlton House Terrace, as a treat I was taken down to the basement to look atNewton’slittletelescope,thesecondonehemade.In1671HenryOldenburg, thesecretaryandguidingspiritof theRoyalSociety, invitedNewton to

publishadescriptionofhistelescope.NewtonsubmittedaletterdescribingitandhisworkoncolortoPhilosophical Transactions of the Royal Society early in 1672. This began a controversy over theoriginality and significance of Newton’s work, especially with Hooke, who had been curator ofexperimentsat theRoyalSocietysince1662,andholderofa lectureshipendowedbySirJohnCutlersince1664.Nofeebleopponent,Hookehadmadesignificantcontributionstoastronomy,microscopy,watchmaking,mechanics,andcityplanning.Heclaimedthathehadperformedthesameexperimentson light as Newton, and that they proved nothing—colors were simply added to white light by theprism.Newtonlecturedonhistheoryoflight inLondonin1675.Heconjecturedthat light, likematter, is

composedofmanysmallparticles—contrary to theview,proposedatabout thesame timebyHookeandHuygens,thatlightisawave.ThiswasoneplacewhereNewton’sscientificjudgmentfailedhim.Therearemanyobservations,someeveninNewton’stime,thatshowthewavenatureoflight.Itistruethat in modern quantum mechanics light is described as an ensemble of massless particles, calledphotons,butinthelightencounteredinordinaryexperiencethenumberofphotonsisenormous,andinconsequencelightdoesbehaveasawave.Inhis1678TreatiseonLight,Huygensdescribed light as awaveofdisturbance in amedium, the

ether,whichconsistsofavastnumberoftinymaterialparticlesincloseproximity.Justasinanoceanwaveindeepwateritisnotthewaterthatmovesalongthesurfaceoftheoceanbutthedisturbanceofthewater,solikewiseinHuygens’theoryitisthewaveofdisturbanceintheparticlesoftheetherthatmoves ina rayof light,not theparticles themselves.Eachdisturbedparticleacts as anewsourceofdisturbance,whichcontributestothetotalamplitudeofthewave.Ofcourse,sincetheworkofJamesClerk Maxwell in the nineteenth century we have known that (even apart from quantum effects)Huygenswas only half right—light is a wave, but a wave of disturbances in electric andmagneticfields,notawaveofdisturbanceofmaterialparticles.Usingthiswavetheoryoflight,Huygenswasabletoderivetheresult that light inahomogeneous

medium(oremptyspace)behavesasifittravelsinstraightlines,asitisonlyalongtheselinesthatthewavesproducedbyall thedisturbedparticlesaddupconstructively.Hegaveanewderivationof theequal-anglesruleforreflection,andofSnell’slawforrefraction,withoutFermat’saprioriassumptionthatlightraystakethepathofleasttime.(SeeTechnicalNote30.)InHuygens’theoryofrefraction,aray of light is bent in passing at an oblique angle through the boundary between two media withdifferentlightspeedsinmuchthewaythedirectionofmarchofalineofsoldierswillchangewhentheleadingedgeofthelineentersaswampyterrain,inwhichtheirmarchingspeedisreduced.Todigressabit,itwasessentialtoHuygens’wavetheorythatlighttravelsatafinitespeed,contrary

towhat had been thought byDescartes.Huygens argued that effects of this finite speed are hard toobservesimplybecauselighttravelssofast.IfforinstanceittooklightanhourtotravelthedistanceoftheMoon from the Earth, then at the time of an eclipse of theMoon theMoonwould be seen notdirectlyoppositetheSun,butlaggingbehindbyabout33°.Fromthefactthatnolagisseen,Huygensconcludedthatthespeedoflightmustbeatleast100,000timesasfastasthespeedofsound.Thisiscorrect;theactualratioisabout1million.HuygenswentontodescriberecentobservationsofthemoonsofJupiterbytheDanishastronomer

OleRømer.Theseobservations showed that theperiodof Io’s revolutionappears shorterwhenEarthandJupiterareapproachingeachotherandlongerwhentheyaremovingapart.(AttentionfocusedonIo, because it has the shortest orbital period of any of Jupiter’s Galilean moons—only 1.77 days.)Huygensinterpretedthisaswhatlaterbecameknownasa“Dopplereffect”:whenJupiterandtheEarth

aremovingcloser togetherorfartherapart, theirseparationateachsuccessivecompletionofawholeperiod of revolution of Io is respectively decreasing or increasing, and so if light travels at a finitespeed,thetimeintervalbetweenobservationsofcompleteperiodsofIoshouldberespectivelylessorgreaterthanifJupiterandtheEarthwereatrest.Specifically,thefractionalshiftintheapparentperiodof Io shouldbe the ratioof the relative speedof Jupiter and theEarth along thedirection separatingthemtothespeedoflight,withtherelativespeedtakenaspositiveornegativeifJupiterandtheEarthare moving farther apart or closer together, respectively. (See Technical Note 31.) Measuring theapparent changes in the periodof Io andknowing the relative speedofEarth and Jupiter, one couldcalculatethespeedoflight.BecausetheEarthmovesmuchfasterthanJupiter,itischieflytheEarth’svelocitythatdominatestherelativespeed.Thescaleofthesolarsystemwasthennotwellknown,soneitherwasthenumericalvalueoftherelativespeedofseparationoftheEarthandJupiter,butusingRømer’sdataHuygenswasabletocalculatethatittakes11minutesforlighttotraveladistanceequaltotheradiusoftheEarth’sorbit,aresultthatdidnotdependonknowingthesizeoftheorbit.Toputitanotherway,sincetheastronomicalunit(AU)ofdistanceisdefinedasthemeanradiusoftheEarth’sorbit,thespeedoflightwasfoundbyHuygenstobe1AUper11minutes.Themodernvalueis1AUper8.32minutes.Therealreadywasexperimentalevidenceofthewavenatureoflightthatwouldhavebeenavailable

toNewtonandHuygens:thediscoveryofdiffractionbytheBologneseJesuitFrancescoMariaGrimaldi(a student of Riccioli), published posthumously in 1665. Grimaldi had found that the shadow of anarrowopaquerodinsunlightisnotperfectlysharp,butisborderedbyfringes.Thefringesareduetothe fact that the wavelength of light is not negligible compared with the thickness of the rod, butNewtonarguedthattheywereinsteadtheresultofsomesortofrefractionatthesurfaceoftherod.Theissueoflightascorpuscleorwavewassettledformostphysicistswhen,intheearlynineteenthcentury,ThomasYoungdiscoveredinterference,thepatternofreinforcementorcancellationoflightwavesthatarrive at given spots along different paths. As already mentioned, in the twentieth century it wasdiscovered that the twoviews are not incompatible.Einstein in 1905 realized that although light formostpurposesbehavesasawave,theenergyinlightcomesinsmallpackets,latercalledphotons,eachwithatinyenergyandmomentumproportionaltothefrequencyofthelight.Newton finally presented his work on light in his bookOpticks, written (in English) in the early

1690s.Itwaspublishedin1704,afterhehadalreadybecomefamous.Newtonwasnotonlyagreatphysicistbutalsoacreativemathematician.Hebeganin1664toread

worksonmathematics,includingEuclid’sElementsandDescartes’Geometrie.Hesoonstartedtoworkoutthesolutionstoavarietyofproblems,manyinvolvinginfinities.Forinstance,heconsideredinfiniteseries,suchasx−x2/2+x3/3−x4/4+...,andshowedthatthisaddsuptothelogarithm*of1+x.In1665Newtonbegan to thinkabout infinitesimals.He tookupaproblem: supposeweknow the

distanceD(t) traveled in any time t; how dowe find the velocity at any time?He reasoned that innonuniformmotion,thevelocityatanyinstantistheratioofthedistancetraveledtothetimeelapsedinaninfinitesimalintervaloftimeatthatinstant.Introducingthesymboloforaninfinitesimalintervaloftime,hedefinedthevelocityattimetastheratiotooofthedistancetraveledbetweentimetandtimet+o,thatis,thevelocityis[D(t+o)−D(t)]/o.Forinstance,ifD(t)=t3,thenD(t+o)=t3+3t2o+3to2+o3.Foroinfinitesimal,wecanneglectthetermsproportionaltoo2ando3,andtakeD(t+o)=t3+3t2o,sothatD(t+o)−D(t)=3t2oandthevelocityisjust3t2.Newtoncalledthisthe“fluxion”ofD(t),butitbecameknownasthe“derivative,”thefundamentaltoolofmoderndifferentialcalculus.*Newton then took up the problem of finding the areas bounded by curves. His answer was the

fundamentaltheoremofcalculus;onemustfindthequantitywhosefluxionisthefunctiondescribedbythecurve.Forinstance,aswehaveseen,3x2isthefluxionofx3,sotheareaundertheparabolay=3x2

betweenx=0andanyotherxisx3.Newtoncalledthisthe“inversemethodoffluxions,”butitbecameknownastheprocessof“integration.”Newton had invented the differential and integral calculus, but for a longwhile thiswork did not

becomewidely known.Late in 1671 he decided to publish it alongwith an account of hiswork onoptics,butapparentlynoLondonbooksellerwaswillingtoundertakethepublicationwithoutaheavysubsidy.2In 1669 Barrow gave a manuscript of Newton’sDe analysi per aequationes numero terminorum

infinitastothemathematicianJohnCollins.AcopymadebyCollinswasseenonavisittoLondonin1676by thephilosopherandmathematicianGottfriedWilhelmLeibniz,a formerstudentofHuygensandafewyearsyoungerthanNewton,whohadindependentlydiscoveredtheessentialsofthecalculusinthepreviousyear.In1676Newtonrevealedsomeofhisownresultsinlettersthatweremeanttobeseen by Leibniz. Leibniz published his work on calculus in articles in 1684 and 1685, withoutacknowledging Newton’s work. In these publications Leibniz introduced the word “calculus,” andpresenteditsmodernnotation,includingtheintegrationsign∫.Toestablishhisclaimtocalculus,Newtondescribedhisownmethodsintwopapersincludedinthe

1704editionofOpticks. InJanuary1705ananonymous reviewofOptickshinted that thesemethodsweretakenfromLeibniz.AsNewtonguessed,thisreviewhadbeenwrittenbyLeibniz.Thenin1709PhilosophicalTransactionsoftheRoyalSocietypublishedanarticlebyJohnKeilldefendingNewton’spriorityofdiscovery,andLeibnizrepliedin1711withanangrycomplainttotheRoyalSociety.In1712theRoyalSocietyconvenedananonymouscommitteetolookintothecontroversy.Twocenturieslaterthemembershipofthiscommitteewasmadepublic,anditturnedouttohaveconsistedalmostentirelyofNewton’ssupporters.In1715thecommitteereportedthatNewtondeservedcreditforthecalculus.This report had been drafted for the committee by Newton. Its conclusions were supported by ananonymousreviewofthereport,alsowrittenbyNewton.The judgment of contemporary scholars3 is that Leibniz andNewton had discovered the calculus

independently. Newton accomplished this a decade earlier than Leibniz, but Leibniz deserves greatcredit forpublishinghiswork. Incontrast, afterhisoriginaleffort in1671 to findapublisher forhistreatiseoncalculus,NewtonallowedthisworktoremainhiddenuntilhewasforcedintotheopenbythecontroversywithLeibniz.Thedecisiontogopublicisgenerallyacriticalelementintheprocessofscientificdiscovery.4Itrepresentsajudgmentbytheauthorthattheworkiscorrectandreadytobeusedbyotherscientists.Forthisreason,thecreditforascientificdiscoverytodayusuallygoestothefirsttopublish.ButthoughLeibnizwasthefirsttopublishoncalculus,asweshallseeitwasNewtonratherthanLeibnizwhoappliedcalculustoproblemsinscience.Though,likeDescartes,Leibnizwasagreatmathematician whose philosophical work is much admired, he made no important contributions tonaturalscience.ItwasNewton’stheoriesofmotionandgravitationthathadthegreatesthistoricalimpact.Itwasan

oldideathattheforceofgravitythatcausesobjectstofalltotheEarthdecreaseswithdistancefromtheEarth’s surface. Thismuchwas suggested in the ninth century by a well-traveled Irishmonk, DunsScotus(JohannesScotusErigena,orJohntheScot),butwithnosuggestionofanyconnectionof thisforcewiththemotionoftheplanets.ThesuggestionthattheforcethatholdstheplanetsintheirorbitsdecreaseswiththeinversesquareofthedistancefromtheSunmayhavebeenfirstmadein1645byaFrenchpriest,IsmaelBullialdus,whowaslaterquotedbyNewtonandelectedtotheRoyalSociety.ButitwasNewtonwhomadethisconvincing,andrelatedtheforcetogravity.Writingabout50yearslater,Newtondescribedhowhebegantostudygravitation.Eventhoughhis

statement needs a good deal of explanation, I feel I have to quote it here, because it describes inNewton’sownwordswhatseemstohavebeenaturningpointinthehistoryofcivilization.According

toNewton,itwasin1666that:

IbegantothinkofgravityextendingtotheorboftheMoon&(havingfoundouthowtoestimatetheforcewithwhich[a]globerevolvingwithinaspherepresses thesurfaceof thesphere) fromKepler’s rule of the periodical times of the Planets being in sesquialterate proportion of theirdistancesfromthecenterof theirOrbs,Ideducedthat theforceswhichkeepthePlanets in theirOrbsmust [be] reciprocally as the squares of their distances from the centers aboutwhich theyrevolve&therebycompared theMooninherOrbwith theforceofgravityat thesurfaceof theEarth & found them answer pretty nearly. All this [including his work on infinite series andcalculus]wasinthetwoplagueyearsof1665–1666.ForinthosedaysIwasintheprimeofmyageforinventionandmindedMathematicksandPhilosophymorethanatanytimesince.5

AsIsaid,thistakessomeexplaining.First, Newton’s parenthesis “having found out how to estimate the force with which [a] globe

revolvingwithinaspherepressesthesurfaceofthesphere”referstothecalculationofcentrifugalforce,acalculationthathadalreadybeendone(probablyunknowntoNewton)around1659byHuygens.ForHuygensandNewton (as forus),accelerationhadabroaderdefinition than justanumbergiving thechangeof speedper timeelapsed; it is adirectedquantity,giving thechangeper timeelapsed in thedirectionaswellasinthemagnitudeofthevelocity.Thereisanaccelerationincircularmotionevenatconstantspeed—itisthe“centripetalacceleration,”consistingofacontinualturningtowardthecenterofthecircle.HuygensandNewtonconcludedthatabodymovingataconstantspeedvaroundacircleofradiusrisacceleratingtowardthecenterofthecircle,withaccelerationv2/r,sotheforceneededtokeep itmovingon thecircle rather than flyingoff inastraight line intospace isproportional tov2/r.(SeeTechnicalNote32.)TheresistancetothiscentripetalaccelerationisexperiencedaswhatHuygenscalled centrifugal force, aswhen aweight at the end of a cord is swung around in a circle. For theweight,thecentrifugalforceisresistedbytensioninthecord.ButplanetsarenotattachedbycordstotheSun.Whatisitthatresiststhecentrifugalforceproducedbyaplanet’snearlycircularmotionaroundtheSun?Aswewillsee,theanswertothisquestionledtoNewton’sdiscoveryoftheinversesquarelawofgravitation.Next,by“Kepler’s ruleof theperiodical timesof thePlanetsbeing in sesquialterateproportionof

theirdistancesfromthecenteroftheirOrbs”NewtonmeantwhatwenowcallKepler’sthirdlaw:thatthesquareoftheperiodsoftheplanetsintheirorbitsisproportionaltothecubesofthemeanradiioftheir orbits, or in other words, the periods are proportional to the 3/2 power (the “sesquialterateproportion”)ofthemeanradii.*Theperiodofabodymovingwithspeedvaroundacircleofradiusristhecircumference2πrdividedbythespeedv,soforcircularorbitsKepler’sthirdlawtellsusthatr2/v2isproportionaltor3,andthereforetheirinversesareproportional:v2/r2isproportionalto1/r3.Itfollowsthat theforcekeepingtheplanets inorbit,which isproportional tov2/r,mustbeproportional to1/r2.Thisistheinversesquarelawofgravity.This in itselfmightberegardedas justawayof restatingKepler’s third law.Nothing inNewton’s

considerationoftheplanetsmakesanyconnectionbetweentheforceholdingtheplanetsintheirorbitsand the commonly experienced phenomena associated with gravity on the Earth’s surface. Thisconnection was provided by Newton’s consideration of the Moon. Newton’s statement that he“compared theMoon in herOrbwith the force of gravity at the surface of theEarth& found themanswer pretty nearly” indicates that he had calculated the centripetal acceleration of theMoon, andfoundthatitwaslessthantheaccelerationoffallingbodiesonthesurfaceoftheEarthbyjusttheratioonewouldexpectiftheseaccelerationswereinverselyproportionaltothesquareofthedistancefrom

thecenteroftheEarth.Tobe specific,Newton took the radius of theMoon’s orbit (well known fromobservationsof the

Moon’s diurnal parallax) to be 60Earth radii; it is actually about 60.2 Earth radii.He used a crudeestimate of the Earth’s radius,* which gave a crude value for the radius of the Moon’s orbit, andknowing that the sidereal period of theMoon’s revolution around the Earth is 27.3 days, he couldestimatetheMoon’svelocityandfromthatitscentripetalacceleration.Thisaccelerationturnedouttobe less than the acceleration of falling bodies on the surface of the Earth by a factor roughly (veryroughly)equalto1/(60)2,asexpectediftheforceholdingtheMooninitsorbitisthesamethatattractsbodieson theEarth’s surface,but reduced in accordancewith the inverse square law. (SeeTechnicalNote33.)ThisiswhatNewtonmeantwhenhesaidthathefoundthattheforces“answerprettynearly.”Thiswastheclimacticstepintheunificationofthecelestialandterrestrialinscience.Copernicushad

placedtheEarthamongtheplanets,Tychohadshownthatthereischangeintheheavens,andGalileohadseenthattheMoon’ssurfaceisrough,liketheEarth’s,butnoneofthisrelatedthemotionofplanetsto forces thatcouldbeobservedonEarth.Descarteshad tried tounderstand themotionsof thesolarsystem as the result of vortices in the ether, not unlike vortices in a pool ofwater onEarth, but histheoryhadnosuccess.NowNewtonhadshownthattheforcethatkeepstheMooninitsorbitaroundtheEarthandtheplanetsintheirorbitsaroundtheSunisthesameastheforceofgravitythatcausesanapple to fall to theground inLincolnshire,allgovernedby thesamequantitative laws.After this thedistinction between the celestial and terrestrial, which had constrained physical speculation fromAristotle on, had to be forever abandoned. But this was still far short of a principle of universalgravitation,whichwould assert that every body in the universe, not just the Earth and Sun, attractseveryotherbodywithaforcethatdecreasesastheinversesquareofthedistancebetweenthem.TherewerestillfourlargeholesinNewton’sarguments:

1.IncomparingthecentripetalaccelerationoftheMoonwiththeaccelerationoffallingbodiesonthesurfaceoftheEarth,Newtonhadassumedthattheforceproducingtheseaccelerationsdecreaseswiththe inverse square of the distance, but the distance from what? This makes little difference for themotionoftheMoon,whichissofarfromtheEarththattheEarthcanbetakenasalmostapointparticleas faras theMoon’smotion isconcerned.But foranapple falling to theground inLincolnshire, theEarthextendsfromthebottomofthetree,afewfeetaway,toapointattheantipodes,8,000milesaway.Newtonhadassumedthat thedistancerelevant to thefallofanyobjectnear theEarth’ssurface is itsdistancetothecenteroftheEarth,butthiswasnotobvious.

2.Newton’sexplanationofKepler’s third lawignoredtheobviousdifferencesbetweentheplanets.SomehowitdoesnotmatterthatJupiterismuchbiggerthanMercury;thedifferenceintheircentripetalaccelerations is just a matter of their distances from the Sun. Even more dramatically, Newton’scomparison of the centripetal acceleration of theMoon and the acceleration of falling bodies on thesurfaceof theEarth ignored theconspicuousdifferencebetween theMoonandafallingbodylikeanapple.Whydothesedifferencesnotmatter?

3.Intheworkhedatedto1665–1666,NewtoninterpretedKepler’sthirdlawasthestatementthattheproductsofthecentripetalaccelerationsofthevariousplanetswiththesquaresoftheirdistancesfromtheSunare the same for all planets.But the commonvalueof thisproduct is not at all equal to theproductofthecentripetalaccelerationoftheMoonwiththesquareofitsdistancefromtheEarth;itismuchgreater.Whataccountsforthisdifference?

4.Finally,inthisworkNewtonhadtakentheorbitsoftheplanetsaroundtheSunandoftheMoon

around the Earth to be circular at constant speed, even though as Kepler had shown they are notpreciselycircularbutinsteadelliptical,theSunandEartharenotatthecentersoftheellipses,andtheMoon’sandplanets’speedsareonlyapproximatelyconstant.

Newtonstruggledwiththeseproblemsintheyearsfollowing1666.Meanwhile,otherswerecomingto the same conclusions that Newton had already reached. In 1679 Newton’s old adversary HookepublishedhisCutlerianlectures,whichcontainedsomesuggestivethoughnonmathematicalideasaboutmotionandgravitation:

First, thatallCoelestialBodieswhatsoever,haveanattractionorgravitatingpowertowardstheirownCenters,wherebytheyattractnotonlytheirownparts,andkeepthemfromflyingfromthem,aswemayobservetheEarthtodo,butthattheydoalsoattractalltheotherCoelestialBodiesthatarewithinthesphereoftheiractivity—Thesecondsuppositionisthis,Thatallbodieswhatsoeverthatareputintoadirectandsimplemotion,willsocontinuetomoveforwardinastraightline,tillthey are by some other effectual powers deflected and bent into aMotion, describing a Circle,Ellipsis, or some other more compounded Curve Line. The third supposition is, That theseattractivepowersaresomuchthemorepowerfulinoperating,byhowmuchthenearerthebodywroughtuponistotheirownCenters.6

HookewrotetoNewtonabouthisspeculations,includingtheinversesquarelaw.Newtonbrushedhimoff, replying that he had not heard ofHooke’swork, and that the “method of indivisibles”7 (that is,calculus)wasneededtounderstandplanetarymotions.Then inAugust 1684Newton received a fateful visit inCambridge from the astronomerEdmund

Halley.LikeNewtonandHookeandalsoWren,Halleyhadseen theconnectionbetween the inversesquarelawofgravitationandKepler’sthirdlawforcircularorbits.HalleyaskedNewtonwhatwouldbetheactual shapeof theorbitofabodymovingunder the influenceofa force thatdecreaseswith theinverse squareof thedistance.Newton answered that theorbitwouldbe an ellipse, andpromised tosendaproof.LaterthatyearNewtonsubmitteda10-pagedocument,OntheMotionofBodiesinOrbit,whichshowedhowtotreatthegeneralmotionofbodiesundertheinfluenceofaforcedirectedtowardacentralbody.Three years later the Royal Society published Newton’s Philosophiae Naturalis Principia

Mathematica (Mathematical Principles of Natural Philosophy), doubtless the greatest book in thehistoryofphysicalscience.AmodernphysicistleafingthroughthePrincipiamaybesurprisedtoseehowlittleitresemblesany

of today’s treatises on physics. There are many geometrical diagrams, but few equations. It seemsalmost as if Newton had forgotten his own development of calculus. But not quite. Inmany of hisdiagrams one sees features that are supposed to become infinitesimal or infinitely numerous. Forinstance,inshowingthatKepler’sequal-arearulefollowsforanyforcedirectedtowardafixedcenter,Newton imagines that theplanet receives infinitelymany impulses toward the center, each separatedfromthenextbyan infinitesimal intervalof time.This is just thesortofcalculation that ismadenotonlyrespectablebutquickandeasybythegeneralformulasofcalculus,butnowhereinthePrincipiado thesegeneral formulasmake their appearance.Newton’smathematics in thePrincipia isnotverydifferentfromwhatArchimedeshadusedincalculatingtheareasofcircles,orwhatKeplerhadusedincalculatingthevolumesofwinecasks.ThestyleofthePrincipiaremindsthereaderofEuclid’sElements.Itbeginswithdefinitions:8

DefinitionI

Quantityofmatterisameasureofmatterthatarisesfromitsdensityandvolumejointly.

WhatappearsinEnglishtranslationas“quantityofmatter”wascalledmassainNewton’sLatin,andistodaycalled“mass.”Newtonheredefinesitastheproductofdensityandvolume.EventhoughNewtondoesnotdefinedensity,hisdefinitionofmassisstillusefulbecausehisreaderscouldtakeitforgrantedthat bodies made of the same substances, such as iron at a given temperature, will have the samedensity.AsArchimedeshadshown,measurementsofspecificgravitygivevaluesfordensityrelativetothatofwater.Newtonnotesthatwemeasurethemassofabodyfromitsweight,butdoesnotconfusemassandweight.

DefinitionII

Quantityofmotionisameasureofmotionthatarisesfromthevelocityandthequantityofmatterjointly.

WhatNewtoncalls“quantityofmotion”istodaytermed“momentum.”ItisdefinedherebyNewtonastheproductofthevelocityandthemass.

DefinitionIII

Inherentforceofmatter[visinsita]isthepowerofresistingbywhicheverybody,sofaras[it]isable,perseveresinitsstateeitherofrestingorofmovinguniformlystraightforward.

Newtongoesontoexplainthatthisforcearisesfromthebody’smass,andthatit“doesnotdifferinanywayfromtheinertiaofthemass.”Wesometimesnowdistinguishmass,initsroleasthequantitythatresistschangesinmotion,as“inertialmass.”

DefinitionIV

Impressedforceistheactionexertedonabodytochangeitsstateeitherofrestingorofuniformlymovingstraightforward.

This defines the general concept of force, but does not yet givemeaning to anynumerical valuewemightassigntoagivenforce.DefinitionsVthroughVIIIgoontodefinecentripetalaccelerationanditsproperties.Afterthedefinitionscomesascholium,orannotation,inwhichNewtondeclinestodefinespaceand

time,butdoesofferadescription:

I.Absolute,true,andmathematicaltime,inandofitself,andofitsownnature,withoutrelationtoanythingexternal,flowsuniformly....

II. Absolute space, of its own nature without relation to anything external, always remainshomogeneousandimmovable.

Both Leibniz andBishopGeorge Berkeley criticized this view of time and space, arguing that onlyrelativepositionsinspaceandtimehaveanymeaning.Newtonhadrecognizedinthisscholiumthatwenormally deal onlywith relative positions and velocities, but now he had a new handle on absolutespace: inNewton’smechanics,acceleration (unlikepositionorvelocity)hasanabsolutesignificance.Howcoulditbeotherwise?Itisamatterofcommonexperiencethataccelerationhaseffects;thereisno

needtoask,“Accelerationrelativetowhat?”Fromtheforcespressingusbackinourseats,weknowthatwearebeingacceleratedwhenweareinacarthatsuddenlyspeedsup,whetherornotwelookoutthecar’swindow.Aswewillsee,inthetwentiethcenturytheviewsofspaceandtimeofLeibnizandNewtonwerereconciledinthegeneraltheoryofrelativity.ThenatlastcomeNewton’sfamousthreelawsofmotion:

LawI

Everybodyperseveresinitsstateofbeingatrestorofmovinguniformlystraightforward,exceptinsofarasitiscompelledtochangeitsstatebyforcesimpressed.

ThiswasalreadyknowntoGassendiandHuygens.ItisnotclearwhyNewtonbotheredtoincludeitasaseparatelaw,sincethefirstlawisatrivial(thoughimportant)consequenceofthesecond.

LawII

Achangeinmotionisproportionaltothemotiveforceimpressedandtakesplacealongthestraightlineinwhichthatforceisimpressed.

By“changeofmotion”hereNewtonmeansthechangeinthemomentum,whichhecalledthe“quantityofmotion” inDefinition II. It isactually therateofchangeofmomentum that isproportional to theforce.We conventionally define the units in which force is measured so that the rate of change ofmomentumisactuallyequaltotheforce.Sincemomentumismasstimesvelocity,itsrateofchangeismasstimesacceleration.Newton’ssecondlawisthusthestatementthatmasstimesaccelerationequalstheforceproducingtheacceleration.ButthefamousequationF=madoesnotappearinthePrincipia;thesecondlawwasreexpressedinthiswaybyContinentalmathematiciansintheeighteenthcentury.

LawIII

Toanyaction there isalwaysanoppositeandequal reaction; inotherwords, theactionsof twobodiesuponeachotherarealwaysequal,andalwaysoppositeindirection.

Intruegeometricstyle,Newtonthengoesontopresentaseriesofcorollariesdeducedfromtheselaws.Notableamong themwasCorollary III,whichgives the lawof theconservationofmomentum. (SeeTechnicalNote34.)After completing his definitions, laws, and corollaries, Newton begins in Book I to deduce their

consequences.He proves that central forces (forces directed toward a single central point) and onlycentral forces give a body a motion that sweeps out equal areas in equal times; that central forcesproportional to the inverse square of the distance and only such central forces producemotion on aconicsection,thatis,acircle,anellipse,aparabola,orahyperbola;andthatformotiononanellipsesuch a force gives periods proportional to the 3/2 power of themajor axis of the ellipse (which, asmentionedinChapter11,isthedistanceoftheplanetfromtheSunaveragedoverthelengthofitspath).SoacentralforcethatgoesastheinversesquareofthedistancecanaccountforallofKepler’slaws.Newtonalsofillsinthegapinhiscomparisonoflunarcentripetalaccelerationandtheaccelerationoffallingbodies,provinginSectionXIIofBookIthatasphericalbody,composedofparticlesthateachproduceaforcethatgoesastheinversesquareofthedistancetothatparticle,producesatotalforcethatgoesastheinversesquareofthedistancetothecenterofthesphere.ThereisaremarkablescholiumattheendofSectionIofBookI,inwhichNewtonremarksthatheis

nolongerrelyingonthenotionofinfinitesimals.Heexplainsthat“fluxions”suchasvelocitiesarenottheratiosofinfinitesimals,ashehadearlierdescribedthem;instead,“Thoseultimateratioswithwhichquantitiesvanisharenotactuallyratiosofultimatequantities,butlimitswhichtheratiosofquantitiesdecreasingwithoutlimitarecontinuallyapproaching,andwhichtheycanapproachsocloselythattheirdifference is less than any given quantity.” This is essentially themodern idea of a limit, onwhichcalculusisnowbased.WhatisnotmodernaboutthePrincipiaisNewton’sideathatlimitshavetobestudiedusingthemethodsofgeometry.Book II presents a long treatmentof themotionofbodies through fluids; theprimarygoalof this

discussionwas toderive the lawsgoverning theforcesof resistanceonsuchbodies.9 In thisbookhedemolishesDescartes’ theoryofvortices.Hethengoesontocalculate thespeedofsoundwaves.HisresultinProposition49(thatthespeedisthesquarerootoftheratioofthepressureandthedensity)iscorrectonly inorderofmagnitude,becausenoonethenknewhowto takeaccountof thechanges intemperatureduringexpansionandcompression.But(togetherwithhiscalculationofthespeedofoceanwaves)thiswasanamazingachievement:thefirsttimethatanyonehadusedtheprinciplesofphysicstogiveamoreorlessrealisticcalculationofthespeedofanysortofwave.AtlastNewtoncomestotheevidencefromastronomyinBookIII,TheSystemoftheWorld.Atthe

timeofthefirsteditionofthePrincipia,therewasgeneralagreementwithwhatisnowcalledKepler’sfirst law, that the planetsmove on elliptical orbits; but there was still considerable doubt about thesecondandthirdlaws:thatthelinefromtheSuntoeachplanetsweepsoutequalareasinequaltimes,andthatthesquaresoftheperiodsofthevariousplanetarymotionsgoasthecubesofthemajoraxesoftheseorbits.NewtonseemstohavefastenedonKepler’slawsnotbecausetheywerewellestablished,butbecausetheyfittedsowellwithhistheory.InBookIIIhenotesthatJupiter’sandSaturn’smoonsobeyKepler’ssecondandthirdlaws,thattheobservedphasesofthefiveplanetsotherthanEarthshowthat they revolvearound theSun, thatall sixplanetsobeyKepler’s laws,and that theMoonsatisfiesKepler’ssecondlaw.*Hisowncarefulobservationsofthecometof1680showedthatittoomovedonaconic section: an ellipse or hyperbola, in either case very close to a parabola. Fromall this (andhisearliercomparisonofthecentripetalaccelerationoftheMoonandtheaccelerationoffallingbodiesontheEarth’ssurface),hecomestotheconclusionthatitisacentralforceobeyinganinversesquarelawbywhichthemoonsofJupiterandSaturnandtheEarthareattractedtotheirplanets,andalltheplanetsand comets are attracted to theSun.From the fact that the accelerationsproducedbygravitation areindependentofthenatureofthebodybeingaccelerated,whetheritisaplanetoramoonoranapple,dependingonlyonthenatureofthebodyproducingtheforceandthedistancebetweenthem,togetherwith the fact that theaccelerationproducedbyany force is inverselyproportional to themassof thebodyonwhichitacts,heconcludesthattheforceofgravityonanybodymustbeproportionaltothemass of that body, so that all dependence on the body’s mass cancels when we calculate theacceleration. This makes a clear distinction between gravitation and magnetism, which acts verydifferentlyonbodiesofdifferentcomposition,eveniftheyhavethesamemass.Newton then, in Proposition 7, uses his third law ofmotion to find out how the force of gravity

dependsonthenatureofthebodyproducingtheforce.Considertwobodies,1and2,withmassesm1andm2.Newtonhadshownthatthegravitationalforceexertedbybody1onbody2isproportionaltom2,andthattheforcethatbody2exertsonbody1isproportionaltom1.Butaccordingtothethirdlaw,theseforcesareequalinmagnitude,andsotheymusteachbeproportionaltobothm1andm2.Newtonwasabletoconfirmthethirdlawincollisionsbutnotingravitationalinteractions.AsGeorgeSmithhasemphasized,itwasmanyyearsbeforeitbecamepossibletoconfirmtheproportionalityofgravitationalforce to the inertial mass of the attracting as well as the attracted body. Nevertheless, Newtonconcluded, “Gravity exists in all bodies universally and is proportional to the quantity of matter in

each.”Thisiswhytheproductsofthecentripetalaccelerationsofthevariousplanetswiththesquaresoftheir distances from the Sun aremuch greater than the product of the centripetal acceleration of theMoon with the square of its distances from the Earth: it is just that the Sun, which produces thegravitationalforceontheplanets,ismuchmoremassivethantheEarth.These results of Newton are commonly summarized in a formula for the gravitational force F

betweentwobodies,ofmassesm1andm2,separatedbyadistancer:

F=G×m1×m2/r2

whereG is a universal constant, known today as Newton’s constant. Neither this formula nor theconstantGappearsinthePrincipia,andevenifNewtonhadintroducedthisconstanthecouldnothavefoundavalueforit,becausehedidnotknowthemassoftheSunortheEarth.Incalculatingthemotionof theMoonor theplanets,G appearsonlyasa factormultiplying themassof theEarthor theSun,respectively.Evenwithoutknowing thevalueofG,Newtoncouldusehis theoryofgravitation to calculate the

ratios of the masses of various bodies in the solar system. (See Technical Note 35.) For instance,knowing the ratios of the distances of Jupiter and Saturn from their moons and from the Sun, andknowingtheratiosoftheorbitalperiodsofJupiterandSaturnandtheirmoons,hecouldcalculatetheratiosof thecentripetalaccelerationsof themoonsofJupiterandSaturn toward theirplanetsand thecentripetalaccelerationoftheseplanetstowardtheSun,andfromthishecouldcalculatetheratiosofthemassesofJupiter,Saturn,andtheSun.SincetheEarthalsohasaMoon,thesametechniquecouldinprinciplebeusedtocalculatetheratioofthemassesoftheEarthandtheSun.Unfortunately,althoughthedistanceoftheMoonfromtheEarthwaswellknownfromtheMoon’sdiurnalparallax,theSun’sdiurnalparallaxwastoosmall tomeasure,andsotheratioofthedistancesfromtheEarthtotheSunand to the Moon was not known. (As we saw in Chapter 7, the data used by Aristarchus and thedistances he inferred from those datawere hopelessly inaccurate.) Newtonwent ahead anyway, andcalculatedtheratioofmasses,usingavalueforthedistanceoftheEarthfromtheSunthatwasnobetterthanalowerlimitonthisdistance,andactuallyabouthalfthetruevalue.HereareNewton’sresultsforratios of masses, given as a corollary to Theorem VIII in Book III of the Principia, together withmodernvalues:10

Ratio Newton’svalue Modernvaluem(Sun)/m(Jupiter) 1,067 1,048m(Sun)/m(Saturn) 3,021 3,497m(Sun)/m(Earth) 169,282 332,950

Ascanbeseenfromthistable,Newton’sresultswereprettygoodforJupiter,notbadforSaturn,butwayofffortheEarth,becausethedistanceoftheEarthfromtheSunwasnotknown.Newtonwasquiteawareoftheproblemsposedbyobservationaluncertainties,butlikemostscientistsuntilthetwentiethcentury,hewascavalieraboutgivingtheresultingrangeofuncertaintyinhiscalculatedresults.Also,aswe have seen with Aristarchus and al-Biruni, he quoted results of calculations to a much greaterprecisionthanwaswarrantedbytheaccuracyofthedataonwhichthecalculationswerebased.Incidentally,thefirstseriousestimateofthesizeofthesolarsystemwascarriedoutin1672byJean

Richer and Giovanni Domenico Cassini. They measured the distance to Mars by observing thedifferenceinthedirectiontoMarsasseenfromParisandCayenne;sincetheratiosofthedistancesoftheplanetsfromtheSunwerealreadyknownfromtheCopernicantheory,thisalsogavethedistanceof

the Earth from the Sun. In modern units, their result for this distance was 140 million kilometers,reasonably close to themodern value of 149.5985million kilometers for themean distance.AmoreaccuratemeasurementwasmadelaterbycomparingobservationsatdifferentlocationsonEarthofthetransitsofVenusacrossthefaceoftheSunin1761and1769,whichgaveanEarth–Sundistanceof153millionkilometers.11In1797–1798HenryCavendishwasatlastabletomeasurethegravitationalforcebetweenlaboratory

masses,fromwhichavalueofGcouldbeinferred.ButCavendishdidnotrefertohismeasurementthisway. Instead, using the well-known acceleration of 32 feet/second per second due to the Earth’sgravitational field at its surface, and the known volume of the Earth, Cavendish calculated that theaveragedensityoftheEarthwas5.48timesthedensityofwater.Thiswasinaccordwithalong-standingpracticeinphysics:toreportresultsasratiosorproportions,

rather thanasdefinitemagnitudes.For instance,aswehaveseen,Galileoshowed that thedistanceabodyfallsonthesurfaceoftheEarthisproportionaltothesquareofthetime,butheneversaidthattheconstantmultiplyingthesquareofthetimethatgivesthedistancefallenwashalfof32feet/secondpersecond.Thiswasdueat least inpart to the lackofanyuniversallyrecognizedunitof length.Galileocouldhavegiventheaccelerationduetogravityassomanybraccia/secondpersecond,butwhatwouldthismeantoEnglishmen,oreventoItaliansoutsideTuscany?Theinternationalstandardizationofunitsof length andmass12 began in 1742, when the Royal Society sent two rulersmarkedwith standardEnglishinchestotheFrenchAcadémiedesSciences;theFrenchmarkedthesewiththeirownmeasuresof length,andsentonebacktoLondon.But itwasnotuntil thegradual internationaladoptionof themetricsystem,startingin1799,thatscientistshadauniversallyunderstoodsystemofunits.TodayweciteavalueforGof66.724trillionthsofameter/second2perkilogram:thatis,asmallbodyofmass1kilogram at a distance of 1 meter produces a gravitational acceleration of 66.724 trillionths of ameter/secondpersecond.AfterlayingoutNewton’stheoriesofmotionandgravitation,thePrincipiagoesontoworkoutsome

oftheirconsequences.ThesegofarbeyondKepler’sthreelaws.Forinstance,inProposition14Newtonexplainstheprecessionofplanetaryorbitsmeasured(fortheEarth)byal-Zarqali,thoughNewtondoesnotattemptaquantitativecalculation.InProposition19Newtonnotes that theplanetsmustallbeoblate,because their rotationproduces

centrifugalforcesthatarelargestattheequatorandvanishatthepoles.Forinstance,theEarth’srotationproducesacentripetalaccelerationatitsequatorequalto0.11feet/secondpersecond,ascomparedwiththeacceleration32 feet/secondper secondof fallingbodies, so thecentrifugal forceproducedby theEarth’srotationismuchlessthanitsgravitationalattraction,butnotentirelynegligible,andtheEarthisthereforenearlyspherical,butslightlyoblate.Observationsinthe1740sfinallyshowedthat thesamependulumwillswingmoreslowlyneartheequatorthanathigherlatitudes,justaswouldbeexpectedifattheequatorthependulumisfartherfromthecenteroftheEarth,becausetheEarthisoblate.InProposition39NewtonshowsthattheeffectofgravityontheoblateEarthcausesaprecessionof

its axis of rotation, the “precession of the equinoxes” first noted by Hipparchus. (Newton had anextracurricularinterestinthisprecession;heuseditsvaluesalongwithancientobservationsofthestarsinanattempttodatesupposedhistoricalevents,suchastheexpeditionofJasonandtheArgonauts.)13InthefirsteditionofthePrincipiaNewtoncalculatesineffectthattheannualprecessionduetotheSunis6.82°(degreesofarc),andthattheeffectoftheMoonislargerbyafactor6⅓,givingatotalof50.0"(secondsofarc)peryear,inperfectagreementwiththeprecessionof50"peryearthenmeasured,andclosetothemodernvalueof50.375"peryear.Veryimpressive,butNewtonlaterrealizedthathisresultfor the precession due to the Sun and hence for the total precessionwas 1.6 times too small. In thesecondeditionhecorrectedhisresultfortheeffectoftheSun,andalsocorrectedtheratiooftheeffects

oftheMoonandSun,sothatthetotalwasagaincloseto50"peryear,stillingoodagreementwithwhatwasobserved.14Newtonhadthecorrectqualitativeexplanationoftheprecessionoftheequinoxes,andhis calculation gave the right order of magnitude for the effect, but to get an answer in preciseagreementwithobservationhehadtomakemanyartfuladjustments.ThisisjustoneexampleofNewtonfudginghiscalculationstogetanswersincloseagreementwith

observation.Alongwiththisexample,R.S.Westfall15hasgivenothers,includingNewton’scalculationof the speed of sound, and his comparison of the centripetal acceleration of the Moon with theaccelerationoffallingbodiesontheEarth’ssurfacementionedearlier.PerhapsNewtonfeltthathisrealor imagined adversaries would never be convinced by anything but nearly perfect agreement withobservation.In Proposition 24,Newton presents his theory of the tides.Gram for gram, theMoon attracts the

ocean beneath it more strongly than it attracts the solid Earth, whose center is farther away, and itattractsthesolidEarthmorestronglythanitattractstheoceanonthesideoftheEarthawayfromtheMoon.Thusthere isa tidalbulgein theoceanbothbelowtheMoon,wheretheMoon’sgravitypullswaterawayfromtheEarth,andontheoppositesideoftheEarth,wheretheMoon’sgravitypullstheEarthawayfromthewater.Thisexplainedwhyinsomelocationshightidesareseparatedbyroughly12ratherthan24hours.ButtheeffectistoocomplicatedforthistheoryoftidestohavebeenverifiedinNewton’s time.Newtonknew that theSunaswellas theMoonplaysa role in raising the tides.Thehighestandlowesttides,knownasspringtides,occurwhentheMoonisneworfull,sothattheSun,Moon, and Earth are on the same line, intensifying the effects of gravitation. But the worstcomplicationscomefromthefactthatanygravitationaleffectsontheoceansaregreatlyinfluencedbytheshapeofthecontinentsandthetopographyoftheoceanbottom,whichNewtoncouldnotpossiblytakeintoaccount.This isacommontheme in thehistoryofphysics.Newton’s theoryofgravitationmadesuccessful

predictionsforsimplephenomenalikeplanetarymotion,butitcouldnotgiveaquantitativeaccountofmorecomplicatedphenomena,likethetides.Weareinasimilarpositiontodaywithregardtothetheoryofthestrongforcesthatholdquarkstogetherinsidetheprotonsandneutronsinsidetheatomicnucleus,atheoryknownasquantumchromodynamics.Thistheoryhasbeensuccessfulinaccountingforcertainprocesses at high energy, such as the production of various strongly interacting particles in theannihilationofenergeticelectronsandtheirantiparticles,anditssuccessesconvinceusthatthetheoryiscorrect.We cannot use the theory to calculate precise values for other things thatwewould like toexplain,likethemassesoftheprotonandneutron,becausethecalculationistoocomplicated.Here,asforNewton’s theoryof the tides, theproperattitude ispatience.Physical theoriesarevalidatedwhenthey give us the ability to calculate enough things that are sufficiently simple to allow reliablecalculations,evenifwecan’tcalculateeverythingthatwemightwanttocalculate.BookIIIofPrincipiapresentscalculationsofthingsalreadymeasured,andnewpredictionsofthings

notyetmeasured,buteveninthefinalthirdeditionofPrincipiaNewtoncouldpointtonopredictionsthathadbeenverified in the40yearssince thefirstedition.Still, takenall together, theevidenceforNewton’s theories of motion and gravitation was overwhelming. Newton did not need to followAristotle and explain why gravity exists, and he did not try. In his “General Scholium” Newtonconcluded:

ThusfarIhaveexplainedthephenomenaoftheheavensandofourseabytheforceofgravity,butIhavenotyetassignedacausetogravity.Indeed,thisforcearisesfromsomecausethatpenetratesasfarasthecentersoftheSunandplanetswithoutanydiminutionofitspowertoact,andthatactsnot inproportion to thequantityof the surfacesof theparticlesonwhich it acts (asmechanicalcauses are wont to do), but in proportion to the quantity of solid matter, and whose action is

extended everywhere to immense distances, always decreasing as the inverse squares of thedistances. . . . I have not as yet been able to deduce from phenomena the reasons for thesepropertiesofgravity,andIdonot“feign”hypotheses.

Newton’sbookappearedwithanappropriateodebyHalley.Hereisitsfinalstanza:

Thenyewhonowonheavenlynectarfare,ComecelebratewithmeinsongthenameOfNewton,totheMusesdear;forheUnlockedthehiddentreasuriesofTruth:SorichlythroughhismindhadPhoebuscastTheradiusofhisowndivinity,Nearerthegodsnomortalmayapproach.

ThePrincipia established the laws of motion and the principle of universal gravitation, but thatunderstatesitsimportance.Newtonhadgiventothefutureamodelofwhataphysicaltheorycanbe:aset of simple mathematical principles that precisely govern a vast range of different phenomena.ThoughNewtonknewverywell thatgravitationwasnot theonlyphysicalforce,asfaras itwenthistheory was universal—every particle in the universe attracts every other particle with a forceproportionaltotheproductoftheirmassesandinverselyproportionaltothesquareoftheirseparation.ThePrincipianotonlydeducedKepler’srulesofplanetarymotionasanexactsolutionofasimplifiedproblem,themotionofpointmassesinresponsetothegravitationofasinglemassivesphere;itwentontoexplain(evenifonlyqualitativelyinsomecases)awidevarietyofotherphenomena:theprecessionofequinoxes,theprecessionofperihelia,thepathsofcomets,themotionsofmoons,theriseandfallofthetides,andthefallofapples.16Bycomparison,allpastsuccessesofphysicaltheorywereparochial.After the publication of the Principia in 1686–1687, Newton became famous. He was elected a

memberofparliamentfortheUniversityofCambridgein1689andagainin1701.In1694hebecamewardenof theMint,wherehepresidedovera reformof theEnglishcoinagewhile still retaininghisLucasian professorship. When Czar Peter the Great came to England in 1698, he made a point ofvisitingtheMint,andhopedtotalkwithNewton,butIcan’tfindanyaccountoftheiractuallymeeting.In 1699 Newton was appointed master of the Mint, a much better-paid position. He gave up hisprofessorship, and became rich. In 1703, after the death of his old enemy Hooke, Newton becamepresidentoftheRoyalSociety.Hewasknightedin1705.Whenin1727Newtondiedofakidneystone,hewasgivenastatefuneralinWestminsterAbbey,eventhoughhehadrefusedtotakethesacramentsoftheChurchofEngland.VoltairereportedthatNewtonwas“buriedlikeakingwhohadbenefitedhissubjects.”17Newton’s theory did not meet universal acceptance.18 Despite Newton’s own commitment to

Unitarian Christianity, some in England, like the theologian John Hutchinson and Bishop Berkeley,wereappalledbytheimpersonalnaturalismofNewton’stheory.ThiswasunfairtothedevoutNewton.Heevenargued thatonlydivine interventioncouldexplainwhy themutualgravitationalattractionoftheplanetsdoesnotdestabilizethesolarsystem,*andwhysomebodiesliketheSunandstarsshinebytheirownlight,whileothersliketheplanetsandtheirsatellitesarethemselvesdark.TodayofcourseweunderstandthelightoftheSunandstarsinanaturalisticway—theyshinebecausetheyareheatedbynuclearreactionsintheircores.Thoughunfair toNewton,HutchinsonandBerkeleywerenotentirelywrongaboutNewtonianism.

FollowingtheexampleofNewton’swork,ifnotofhispersonalopinions,bythelateeighteenthcenturyphysicalsciencehadbecomethoroughlydivorcedfromreligion.

Another obstacle to the acceptance of Newton’s work was the old false opposition betweenmathematicsandphysicsthatwehaveseeninacommentofGeminusofRhodesquotedinChapter8.NewtondidnotspeaktheAristotelianlanguageofsubstancesandqualities,andhedidnottrytoexplainthe cause of gravitation.The priestNicolas deMalebranche (1638–1715) in reviewing thePrincipiasaidthatitwastheworkofageometer,notofaphysicist.MalebrancheclearlywasthinkingofphysicsinthemodeofAristotle.WhathedidnotrealizeisthatNewton’sexamplehadrevisedthedefinitionofphysics.Themost formidablecriticismofNewton’s theoryofgravitationcamefromChristiaanHuygens.19

HegreatlyadmiredthePrincipia,anddidnotdoubtthatthemotionofplanetsisgovernedbyaforcedecreasingastheinversesquareofthedistance,butHuygenshadreservationsaboutwhetheritistruethateveryparticleofmatterattractseveryotherparticlewithsuchaforce,proportionaltotheproductoftheirmasses. In this,Huygensseemstohavebeenmisledbyinaccuratemeasurementsof theratesofpendulumsatvariouslatitudes,whichseemedtoshowthattheslowingofpendulumsneartheequatorcouldbeentirelyexplainedasaneffectofthecentrifugalforceduetotheEarth’srotation.Iftrue,thiswouldimplythattheEarthisnotoblate,asitwouldbeiftheparticlesoftheEarthattracteachotherinthewayprescribedbyNewton.StartingalreadyinNewton’slifetime,histheoryofgravitationwasopposedinFranceandGermany

by followers of Descartes and by Newton’s old adversary Leibniz. They argued that an attractionoperatingovermillionsofmilesofemptyspacewouldbeanoccultelementinnaturalphilosophy,andthey further insisted that the action of gravity should be given a rational explanation, not merelyassumed.Inthis,naturalphilosophersontheContinentwerehangingontoanoldidealforscience,goingback

to theHellenic age, that scientific theories should ultimately be founded solely on reason.We havelearnedtogivethisup.Eventhoughourverysuccessfultheoryofelectronsandlightcanbededucedfrom themodernstandardmodelofelementaryparticles,whichmay (wehope) in turneventuallybededucedfromadeeper theory,howeverfarwegowewillnevercometoafoundationbasedonpurereason.Likeme,mostphysiciststodayareresignedtothefactthatwewillalwayshavetowonderwhyourdeepesttheoriesarenotsomethingdifferent.TheoppositiontoNewtonianismfoundexpressioninafamousexchangeoflettersduring1715and

1716 between Leibniz and Newton’s disciple, the Reverend Samuel Clarke, who had translatedNewton’sOpticksintoLatin.MuchoftheirargumentfocusedonthenatureofGod:DidHeinterveneintherunningoftheworld,asNewtonthought,orhadHesetituptorunbyitselffromthebeginning?20The controversy seems to me to have been supremely futile, for even if its subject were real, it issomethingaboutwhichneitherClarkenorLeibnizcouldhavehadanyknowledgewhatever.In the end the opposition to Newton’s theories didn’t matter, for Newtonian physics went from

successtosuccess.Halleyfittedtheobservationsofthecometsobservedin1531,1607,and1682toasinglenearlyparabolicellipticalorbit, showing that thesewereall recurringappearancesof thesamecomet. UsingNewton’s theory to take into account gravitational perturbations due to themasses ofJupiterandSaturn,theFrenchmathematicianAlexis-ClaudeClairautandhiscollaboratorspredictedinNovember1758thatthiscometwouldreturntoperihelioninmid-April1759.Thecometwasobservedon Christmas Day 1758, 15 years after Halley’s death, and reached perihelion onMarch 13, 1759.Newton’stheorywaspromotedinthemid-eighteenthcenturybytheFrenchtranslationsofthePrincipiabyClairautandbyÉmilieduChâtelet,andthroughtheinfluenceofduChâtelet’sloverVoltaire.ItwasanotherFrenchman,Jeand’Alembert,whoin1749publishedthefirstcorrectandaccuratecalculationof the precession of the equinoxes, based on Newton’s ideas. Eventually Newtonianism triumphedeverywhere.ThiswasnotbecauseNewton’s theorysatisfiedapreexistingmetaphysicalcriterionforascientific

theory.Itdidn’t.ItdidnotanswerthequestionsaboutpurposethatwerecentralinAristotle’sphysics.But it provided universal principles that allowed the successful calculation of a great deal that hadpreviouslyseemedmysterious.Inthisway,itprovidedanirresistiblemodelforwhataphysicaltheoryshouldbe,andcouldbe.This is an example of a kind of Darwinian selection operating in the history of science.We get

intensepleasurewhensomethinghasbeensuccessfullyexplained,aswhenNewtonexplainedKepler’slawsof planetarymotion alongwithmuch else.The scientific theories andmethods that survive arethosethatprovidesuchpleasure,whetherornottheyfitanypreexistingmodelofhowscienceoughttobedone.TherejectionofNewton’stheoriesbythefollowersofDescartesandLeibnizsuggestsamoralforthe

practiceofscience:itisneversafesimplytorejectatheorythathasasmanyimpressivesuccessesinaccountingforobservationasNewton’shad.Successfultheoriesmayworkforreasonsnotunderstoodbytheircreators,andtheyalwaysturnouttobeapproximationstomoresuccessful theories,buttheyareneversimplymistakes.Thismoralwasnotalwaysheeded in the twentiethcentury.The1920ssawtheadventofquantum

mechanics, a radicallynewframework forphysical theory. Insteadofcalculating the trajectoriesofaplanetoraparticle,onecalculatestheevolutionofwavesofprobability,whoseintensityatanypositionand time tellsus theprobabilityof finding theplanetorparticle thenand there.Theabandonmentofdeterminism so appalled some of the founders of quantummechanics, includingMaxPlanck,ErwinSchrödinger, Louis de Broglie, and Albert Einstein, that they did no further work on quantummechanicaltheories,excepttopointouttheunacceptableconsequencesofthesetheories.SomeofthecriticismsofquantummechanicsbySchrödingerandEinsteinweretroubling,andcontinuetoworryustoday,butbytheendofthe1920squantummechanicshadalreadybeensosuccessfulinaccountingforthe properties of atoms, molecules, and photons that it had to be taken seriously. The rejection ofquantummechanicaltheoriesbythesephysicistsmeantthattheywereunabletoparticipateinthegreatprogressinthephysicsofsolids,atomicnuclei,andelementaryparticlesofthe1930sand1940s.Likequantummechanics,Newton’s theoryof thesolarsystemhadprovidedwhat latercame tobe

calledaStandardModel. I introduced this term in197121 todescribe the theoryof the structure andevolutionoftheexpandinguniverseasithaddevelopeduptothattime,explaining:

Ofcourse,thestandardmodelmaybepartlyorwhollywrong.However,itsimportanceliesnotinits certain truth, but in the commonmeetingground that it provides for an enormousvariety ofcosmologicaldata.Bydiscussingthisdatainthecontextofastandardcosmologicalmode,wecanbegintoappreciatetheircosmologicalrelevance,whatevermodelultimatelyprovescorrect.

A little later, I and other physicists started using the term Standard Model also to refer to ouremergingtheoryofelementaryparticlesandtheirvariousinteractions.Ofcourse,Newton’ssuccessorsdidnotusethistermtorefertotheNewtoniantheoryofthesolarsystem,buttheywellmighthave.TheNewtonian theory certainly provided a common meeting ground for astronomers trying to explainobservationsthatwentbeyondKepler’slaws.The methods for applying Newton’s theory to problems involving more than two bodies were

developed by many authors in the late eighteenth and early nineteenth centuries. There was oneinnovationofgreatfutureimportancethatwasexploredespeciallybyPierre-SimonLaplaceintheearlynineteenthcentury.Insteadofaddingupthegravitationalforcesexertedbyallthebodiesinanensemblelike the solar system, one calculates a “field,” a condition of space that at every point gives themagnitudeanddirectionoftheaccelerationproducedbyallthemassesintheensemble.Tocalculatethefield,onesolvescertaindifferentialequationsthatitobeys.(Theseequationssetconditionsontheway

that the field varieswhen the point at which it ismeasured ismoved in any of three perpendiculardirections.) This approach makes it nearly trivial to prove Newton’s theorem that the gravitationalforcesexertedoutsideasphericalmassgoastheinversesquareofthedistancefromthesphere’scenter.More important, as we will see in Chapter 15, the field concept was to play a crucial role in theunderstandingofelectricity,magnetism,andlight.Thesemathematicaltoolswereusedmostdramaticallyin1846topredicttheexistenceandlocation

oftheplanetNeptunefromirregularitiesintheorbitoftheplanetUranus,independentlybyJohnCouchAdamsandJean-JosephLeverrier.Neptunewasdiscoveredsoonafterward,intheexpectedplace.Small discrepancies between theory and observation remained, in themotion of theMoon and of

Halley’s and Encke’s comets, and in a precession of the perihelia of the orbit ofMercury that wasobserved tobe43" (secondsof arc) per centurygreater than couldbe accounted forbygravitationalforcesproducedby theotherplanets.Thediscrepancies in themotionof theMoonandcometswereeventually traced tonongravitational forces, but the excessprecessionofMercurywasnot explaineduntiltheadventin1915ofthegeneraltheoryofrelativityofAlbertEinstein.InNewton’stheorythegravitationalforceatagivenpointandagiventimedependsonthepositions

of allmasses at the same time, so a sudden changeof anyof thesepositions (such as a flareon thesurfaceof theSun)producesan instantaneouschange ingravitational forceseverywhere.Thiswas inconflictwith theprincipleofEinstein’s1905 special theoryof relativity, that no influence can travelfaster than light. This pointed to a clear need to seek amodified theory of gravitation. InEinstein’sgeneraltheoryasuddenchangeinthepositionofamasswillproduceachangeinthegravitationalfieldin the immediate neighborhood of the mass, which then propagates at the speed of light to greaterdistances.GeneralrelativityrejectsNewton’snotionofabsolutespaceandtime.Itsunderlyingequationsarethe

sameinallreferenceframes,whatevertheiraccelerationorrotation.Thusfar,Leibnizwouldhavebeenpleased, but in fact general relativity justifiesNewtonianmechanics. Itsmathematical formulation isbasedonaproperty that it shareswithNewton’s theory: that all bodies at agivenpointundergo thesame acceleration due to gravity.Thismeans that one can eliminate the effects of gravitation at anypoint by using a frame of reference, known as an inertial frame, that shares this acceleration. Forinstance,onedoesnot feel theeffectsof theEarth’sgravity in a freely fallingelevator. It is in theseinertialframesofreferencethatNewton’slawsapply,atleastforbodieswhosespeedsdonotapproachthatoflight.ThesuccessofNewton’streatmentofthemotionofplanetsandcometsshowsthattheinertialframes

intheneighborhoodofthesolarsystemarethoseinwhichtheSunratherthantheEarthisatrest(ormoving with constant velocity). According to general relativity, this is because that is the frame ofreferenceinwhichthematterofdistantgalaxiesisnotrevolvingaroundthesolarsystem.Inthissense,Newton’s theoryprovideda solidbasis forpreferring theCopernican theory to thatofTycho.But ingeneralrelativitywecanuseanyframeofreferencewelike,notjustinertialframes.IfweweretoadoptaframeofreferencelikeTycho’sinwhichtheEarthisatrest,thenthedistantgalaxieswouldseemtobe executing circular turns once a year, and in general relativity this enormousmotionwould createforcesakin togravitation,whichwouldacton theSunandplanetsandgive them themotionsof theTychonictheory.Newtonseemstohavehadahintofthis.Inanunpublished“Proposition43”thatdidnotmakeitintothePrincipia,NewtonacknowledgedthatTycho’stheorycouldbetrueifsomeotherforcebesidesordinarygravitationactedontheSunandplanets.22WhenEinstein’stheorywasconfirmedin1919bytheobservationofapredictedbendingofraysof

lightbythegravitationalfieldoftheSun,theTimesofLondondeclaredthatNewtonhadbeenshowntobewrong.Thiswasamistake.Newton’stheorycanberegardedasanapproximationtoEinstein’s,onethatbecomesincreasinglyvalidforobjectsmovingatvelocitiesmuchlessthanthatoflight.Notonly

doesEinstein’stheorynotdisproveNewton’s;relativityexplainswhyNewton’stheoryworks,whenitdoeswork.Generalrelativityitselfisdoubtlessanapproximationtoamoresatisfactorytheory.Ingeneralrelativityagravitationalfieldcanbefullydescribedbyspecifyingateverypointinspace

andtimetheinertialframesinwhichtheeffectsofgravitationareabsent.Thisismathematicallysimilartothefactthatwecanmakeamapofasmallregionaboutanypointonacurvedsurfaceinwhichthesurfaceappearsflat,likethemapofacityonthesurfaceoftheEarth;thecurvatureofthewholesurfacecanbedescribedbycompilinganatlasofoverlappinglocalmaps.Indeed,thismathematicalsimilarityallowsustodescribeanygravitationalfieldasacurvatureofspaceandtime.The conceptual basis of general relativity is thus different from that of Newton. The notion of

gravitationalforceislargelyreplacedingeneralrelativitywiththeconceptofcurvedspace-time.Thiswashard for somepeople to swallow. In1730AlexanderPopehadwritten amemorable epitaph forNewton:

Natureandnature’slawslayhidinnight;Godsaid,“LetNewtonbe!”Andallwaslight.

InthetwentiethcenturytheBritishsatiricalpoetJ.C.Squire23addedtwomorelines:

Itdidnotlast:theDevilhowling“Ho,LetEinsteinbe,”restoredthestatusquo.

Donot believe it. The general theory of relativity is verymuch in the style ofNewton’s theories ofmotion and gravitation: it is based on general principles that can be expressed as mathematicalequations,fromwhichconsequencescanbemathematicallydeducedforabroadrangeofphenomena,which when compared with observation allow the theory to be verified. The difference betweenEinstein’sandNewton’stheoriesisfarlessthanthedifferencebetweenNewton’stheoriesandanythingthathadgonebefore.A question remains: why did the scientific revolution of the sixteenth and seventeenth centuries

happenwhenandwhere itdid?There isno lackofpossibleexplanations.Manychangesoccurred infifteenth-century Europe that helped to lay the foundation for the scientific revolution. NationalgovernmentswereconsolidatedinFranceunderCharlesVIIandLouisXIandinEnglandunderHenryVII.ThefallofConstantinoplein1453sentGreekscholarsfleeingwestwardtoItalyandbeyond.TheRenaissanceintensifiedinterestinthenaturalworldandsethigherstandardsfortheaccuracyofancienttextsandtheirtranslation.Theinventionofprintingwithmovabletypemadescholarlycommunicationfarquickerandcheaper.ThediscoveryandexplorationofAmericareinforced the lesson that there ismuch that the ancients did not know. In addition, according to the “Merton thesis,” the ProtestantReformation of the early sixteenth century set the stage for the great scientific breakthroughs ofseventeenth-centuryEngland.ThesociologistRobertMertonsupposedthatProtestantismcreatedsocialattitudesfavorabletoscienceandpromotedacombinationofrationalismandempiricismandabeliefinan understandable order in nature—attitudes and beliefs that he found in the actual behavior ofProtestantscientists.24It is not easy to judge how important were these various external influences on the scientific

revolution.ButalthoughIcannottellwhyitwasIsaacNewtoninlate-seventeenth-centuryEnglandwhodiscoveredtheclassical lawsofmotionandgravitation,I thinkIknowwhythese lawstooktheformtheydid.Itis,verysimply,becausetoaverygoodapproximationtheworldreallydoesobeyNewton’slaws.

HavingsurveyedthehistoryofphysicalsciencefromThalestoNewton,Iwouldlikenowtooffersometentative thoughts on what drove us to the modern conception of science, represented by theachievementsofNewtonandhissuccessors.Nothinglikemodernsciencewasconceivedasagoal intheancientworldorthemedievalworld.Indeed,evenifourpredecessorscouldhaveimaginedscienceasitistoday,theymightnothavelikeditverymuch.Modernscienceisimpersonal,withoutroomforsupernatural intervention or (outside the behavioral sciences) for human values; it has no sense ofpurpose;anditoffersnohopeforcertainty.Sohowdidwegethere?Facedwith a puzzlingworld, people in every culture have sought explanations. Evenwhere they

abandonedmythology,mostattemptsatexplanationdidnotleadtoanythingsatisfying.Thalestriedtounderstand matter by guessing that it is all water, but what could he do with this idea?What newinformationdiditgivehim?NooneatMiletusoranywhereelsecouldbuildanythingonthenotionthateverythingiswater.Buteveryonceinawhilesomeonefindsawayofexplainingsomephenomenonthatfitssowelland

clarifiessomuchthatitgivesthefinderintensesatisfaction,especiallywhenthenewunderstandingisquantitative, and observation bears it out in detail. Imagine how Ptolemy must have felt when herealizedthat,byaddinganequanttotheepicyclesandeccentricsofApolloniusandHipparchus,hehadfounda theoryofplanetarymotions that allowedhim topredictwith fair accuracywhereanyplanetwouldbefoundintheskyatanytime.WecangetasenseofhisjoyfromthelinesofhisthatIquotedearlier:“WhenIsearchoutthemassedwheelingcirclesofthestars,myfeetnolongertouchtheEarth,but,sidebysidewithZeushimself,Itakemyfillofambrosia,thefoodofthegods.”Thejoywasflawed—italwaysis.Youdidn’thavetobeafollowerofAristotletoberepelledbythe

peculiarloopingmotionofplanetsmovingonepicyclesinPtolemy’stheory.Therewasalsothenastyfine-tuning:ithadtotakepreciselyoneyearforthecentersoftheepicyclesofMercuryandVenustomovearound theEarth,and forMars, Jupiter,andSaturn tomovearound theirepicycles.Foroverathousand years philosophers argued about the proper role of astronomers like Ptolemy—really tounderstandtheheavens,ormerelytofitthedata.WhatpleasureCopernicusmustthenhavefeltwhenhewasabletoexplainthatthefine-tuningand

theloopingorbitsofPtolemy’sschemearosesimplybecauseweviewthesolarsystemfromamovingEarth.Still flawed, theCopernican theorydidnotquite fit thedatawithoutuglycomplications.Howmuch then themathematically giftedKeplermust have enjoyed replacing theCopernicanmesswithmotiononellipses,obeyinghisthreelaws.So the world acts on us like a teaching machine, reinforcing our good ideas with moments of

satisfaction.Aftercenturieswelearnwhatkindsofunderstandingarepossible,andhowtofindthem.Welearnnottoworryaboutpurpose,becausesuchworriesneverleadtothesortofdelightweseek.Welearntoabandonthesearchforcertainty,becausetheexplanationsthatmakeushappyneverarecertain.We learn todoexperiments,notworryingabout the artificialityofour arrangements.Wedevelopanaestheticsensethatgivesuscluestowhattheorieswillwork,andthataddstoourpleasurewhentheydowork.Ourunderstandingsaccumulate.Itisallunplannedandunpredictable,butitleadstoreliableknowledge,andgivesusjoyalongtheway.

15

Epilogue:TheGrandReduction

Newton’s great achievement left plenty yet to be explained. The nature of matter, the properties offorces other than gravitation that act onmatter, and the remarkable capabilities of life were all stillmysterious. Enormous progresswasmade in the years afterNewton,1 far toomuch to cover in onebook,letaloneasinglechapter.Thisepilogueaimsatmakingjustonepoint,thatasscienceprogressedafterNewton a remarkable picture began to take shape: it turned out that theworld is governed bynaturallawsfarsimplerandmoreunifiedthanhadbeenimaginedinNewton’stime.Newtonhimself inBookIIIofhisOptickssketchedtheoutlineofa theoryofmatter thatwouldat

leastencompassopticsandchemistry:

Nowthesmallestparticlesofmattermaycoherebythestrongestattractions,andcomposebiggerparticles ofweaker virtue; andmany of thesemay cohere and compose bigger particleswhosevirtueisstillweaker,andsoonfordiversesuccessions,until theprogressionendsinthebiggestparticlesonwhichtheoperationsinchemistry,andthecolorsofnaturalbodiesdepend,andwhichbycoheringcomposebodiesofasensiblemagnitude.2

Healsofocusedattentionontheforcesactingontheseparticles:

Forwemustlearnfromthephenomenaofnaturewhatbodiesattractoneanother,andwhatarethelaws and properties of the attraction, before we inquire the cause by which the attraction isperform’d.Theattractionsofgravity,magnetism,andelectricity,reachtoverysensibledistances,and so have been observed by vulgar eyes, and there may be others which reach to so smalldistancesastoescapeobservation.3

Asthisshows,Newtonwaswellawarethatthereareotherforcesinnaturebesidesgravitation.Staticelectricitywasanoldstory.PlatohadmentionedintheTimaeusthatwhenapieceofamber(inGreek,electron) is rubbed it can pick up light bits ofmatter.Magnetismwas known from the properties ofnaturally magnetic lodestones, used by the Chinese for geomancy and studied in detail by QueenElizabeth’sphysician,WilliamGilbert.Newtonherealsohintsattheexistenceofforcesnotyetknownbecause of their short range, a premonition of theweak and strong nuclear forces discovered in thetwentiethcentury.In the early nineteenth century the invention of the electric battery by Alessandro Volta made it

possibletocarryoutdetailedquantitativeexperimentsinelectricityandmagnetism,anditsoonbecameknown that these are not entirely separate phenomena. First, in 1820 Hans Christian Ørsted inCopenhagen found that amagnet and awire carrying an electric current exert forces on each other.After hearing of this result, André-Marie Ampère in Paris discovered that wires carrying electriccurrents also exert forces on one another. Ampère conjectured that these various phenomena are allmuch the same: the forces exerted by and on pieces ofmagnetized iron are due to electric currentscirculatingwithintheiron.Justashappenedwithgravitation,thenotionofcurrentsandmagnetsexertingforcesoneachother

wasreplacedwiththeideaofafield, in thiscaseamagneticfield.Eachmagnetandcurrent-carrying

wirecontributestothetotalmagneticfieldatanypointinitsvicinity,andthismagneticfieldexertsaforceon anymagnetor electric current at that point.MichaelFaradayattributed themagnetic forcesproduced by an electric current to lines ofmagnetic field encircling thewire.He also described theelectric forces produced by a piece of rubbed amber as due to an electric field, pictured as linesemanating radially from the electric charges on the amber. Most important, Faraday in the 1830sshowed a connection between electric and magnetic fields: a changing magnetic field, like thatproducedby theelectriccurrent inarotatingcoilofwire,producesanelectricfield,whichcandriveelectric currents in anotherwire. It is this phenomenon that is used togenerate electricity inmodernpowerplants.Thefinalunificationofelectricityandmagnetismwasachievedafewdecadeslater,byJamesClerk

Maxwell.Maxwellthoughtofelectricandmagneticfieldsastensionsinapervasivemedium,theether,andexpressedwhatwasknownaboutelectricityandmagnetisminequationsrelatingthefieldsandtheirratesofchangetoeachother.ThenewthingaddedbyMaxwellwasthat,justasachangingmagneticfieldgeneratesanelectric field, soalsoachangingelectric fieldgeneratesamagnetic field.Asoftenhappens in physics, the conceptual basis for Maxwell’s equations in terms of an ether has beenabandoned,buttheequationssurvive,evenonT-shirtswornbyphysicsstudents.*Maxwell’stheoryhadaspectacularconsequence.Sinceoscillatingelectricfieldsproduceoscillating

magneticfields,andoscillatingmagneticfieldsproduceoscillatingelectricfields,itispossibletohaveaself-sustainingoscillationofbothelectricandmagneticfieldsintheether,oraswewouldsaytoday,inempty space.Maxwell found around 1862 that this electromagnetic oscillationwould propagate at aspeedthat,accordingtohisequations,hadjustaboutthesamenumericalvalueasthemeasuredspeedoflight. Itwasnatural forMaxwell to jump to the conclusion that light is nothingbut amutually self-sustainingoscillationofelectricandmagneticfields.Visiblelighthasafrequencyfartoohighforittobe produced by currents in ordinary electric circuits, but in the 1880s Heinrich Hertz was able togeneratewaves in accordancewithMaxwell’s equations: radiowaves that differed fromvisible lightonlyinhavingmuchlowerfrequency.Electricityandmagnetismhadthusbeenunifiednotonlywitheachother,butalsowithoptics.As with electricity and magnetism, progress in understanding the nature of matter began with

quantitative measurement, here measurement of the weights of substances participating in chemicalreactions.ThekeyfigureinthischemicalrevolutionwasawealthyFrenchman,AntoineLavoisier.Inthelateeighteenthcenturyheidentifiedhydrogenandoxygenaselementsandshowedthatwaterisacompound of hydrogen and oxygen, that air is a mixture of elements, and that fire is due to thecombinationofotherelementswithoxygen.Alsoon thebasisof suchmeasurements, itwas foundalittlelaterbyJohnDaltonthattheweightswithwhichelementscombineinchemicalreactionscanbeunderstoodonthehypothesisthatpurechemicalcompoundslikewaterorsaltconsistoflargenumbersof particles (later called molecules) that themselves consist of definite numbers of atoms of pureelements.Thewatermolecule,forinstance,consistsoftwohydrogenatomsandoneoxygenatom.Inthe following decades chemists identifiedmany elements: some familiar, like carbon, sulfur, and thecommonmetals;andothersnewlyisolated,suchaschlorine,calcium,andsodium.Earth,air,fire,andwater did not make the list. The correct chemical formulas for molecules like water and salt wereworked out, in the first half of the nineteenth century, allowing the calculation of the ratios of themasses of the atoms of the different elements from measurements of the weights of substancesparticipatinginchemicalreactions.TheatomictheoryofmatterscoredagreatsuccesswhenMaxwellandLudwigBoltzmannshowed

howheatcouldbeunderstoodasenergydistributedamongvastnumbersofatomsormolecules.Thisstep toward unification was resisted by some physicists, including Pierre Duhem, who doubted theexistenceof atomsandheld that the theoryofheat, thermodynamics,was at least as fundamental as

Newton’s mechanics andMaxwell’s electrodynamics. But soon after the beginning of the twentiethcentury several new experiments convinced almost everyone that atoms are real. One series ofexperiments,byJ.J.Thomson,RobertMillikan,andothers,showedthatelectricchargesaregainedandlost only as multiples of a fundamental charge: the charge of the electron, a particle that had beendiscoveredbyThomsonin1897.Therandom“Brownian”motionofsmallparticlesonthesurfaceofliquidswasinterpretedbyAlbertEinsteinin1905asduetocollisionsoftheseparticleswithindividualmoleculesoftheliquid,aninterpretationconfirmedbyexperimentsofJeanPerrin.Respondingtotheexperiments of Thomson and Perrin, the chemistWilhelm Ostwald, who earlier had been skepticalaboutatoms,expressedhischangeofmind in1908, inastatement that implicitly lookedall thewaybacktoDemocritusandLeucippus:“Iamnowconvincedthatwehaverecentlybecomepossessedofexperimentalevidenceofthediscreteorgrainednatureofmatter,whichtheatomichypothesissoughtinvainforhundredsandthousandsofyears.”4Butwhat are atoms?Agreat step toward the answerwas taken in 1911,when experiments in the

Manchester laboratoryofErnestRutherford showed that themassofgoldatoms is concentrated inasmallheavypositivelychargednucleus,aroundwhichrevolvelighternegativelychargedelectrons.Theelectronsareresponsibleforthephenomenaofordinarychemistry,whilechangesinthenucleusreleasethelargeenergiesencounteredinradioactivity.Thisraisedanewquestion:whatkeepstheorbitingatomicelectronsfromlosingenergythroughthe

emissionofradiation,andspiralingdownintothenucleus?Notonlywouldthisruleouttheexistenceofstableatoms; thefrequenciesof theradiationemitted in these littleatomiccatastropheswouldformacontinuum, in contradiction with the observation that atoms can emit and absorb radiation only atcertaindiscretefrequencies,seenasbrightordarklinesinthespectraofgases.Whatdeterminesthesespecialfrequencies?Theanswerswereworkedoutinthefirstthreedecadesofthetwentiethcenturywiththedevelopment

ofquantummechanics,themostradicalinnovationinphysicaltheorysincetheworkofNewton.Asitsnamesuggests,quantummechanics requiresaquantization (that is, adiscreteness)of theenergiesofvariousphysicalsystems.NielsBohrin1913proposedthatanatomcanexistonlyinstatesofcertaindefiniteenergies,andgaverulesforcalculatingtheseenergiesinthesimplestatoms.FollowingearlierworkofMaxPlanck,Einsteinhadalreadyin1905suggestedthattheenergyinlightcomesinquanta,particleslatercalledphotons,eachphotonwithanenergyproportionaltothefrequencyofthelight.AsBohrexplained,whenanatomlosesenergybyemittingasinglephoton,theenergyofthatphotonmustequal the difference in the energies of the initial and final atomic states, a requirement that fixes itsfrequency. There is always an atomic state of lowest energy, which cannot emit radiation and isthereforestable.These early steps were followed in the 1920s with the development of general rules of quantum

mechanics, rules that can be applied to any physical system.Thiswas chiefly thework ofLouis deBroglie,WernerHeisenberg,WolfgangPauli,PascualJordan,ErwinSchrödinger,PaulDirac,andMaxBorn. The energies of allowed atomic states are calculated by solving an equation, the Schrödingerequation,ofageneralmathematical type thatwasalready familiar from the studyof soundand lightwaves.Astringonamusicalinstrumentcanproducejustthosetonesforwhichawholenumberofhalfwavelengthsfitonthestring;analogously,SchrödingerfoundthattheallowedenergylevelsofanatomarethoseforwhichthewavegovernedbytheSchrödingerequationjustfitsaroundtheatomwithoutdiscontinuities. But as first recognized by Born, these waves are not waves of pressure or ofelectromagnetic fields,butwavesofprobability—aparticle ismost likely tobenearwhere thewavefunctionislargest.Quantummechanicsnotonlysolvedtheproblemofthestabilityofatomsandthenatureofspectral

lines; it also brought chemistry into the framework of physics. With the electrical forces among

electronsandatomicnucleialreadyknown,theSchrödingerequationcouldbeappliedtomoleculesaswell as to atoms, and allowed the calculation of the energies of their various states. In this way itbecamepossible in principle to decidewhichmolecules are stable andwhich chemical reactions areenergetically allowed. In 1929 Dirac announced triumphantly that “the underlying physical lawsnecessaryfor themathematical theoryofa largerpartofphysicsandthewholeofchemistryare thuscompletelyknown.”5Thisdidnotmeanthatchemistswouldhandovertheirproblemstophysicists,andretire.AsDirac

wellunderstood, forallbut thesmallestmolecules theSchrödingerequation is toocomplicated tobesolved,so thespecial toolsandinsightsofchemistryremainindispensable.Butfromthe1920son, itwouldbeunderstood thatanygeneralprincipleofchemistry, suchas the rule thatmetals formstablecompoundswith halogen elements like chlorine, is what it is because of the quantummechanics ofnucleiandelectronsactedonbyelectromagneticforces.Despite itsgreatexplanatorypower, this foundationwas itself far frombeingsatisfactorilyunified.

Therewere particles: electrons and the protons and neutrons thatmake up atomic nuclei.And therewere fields: the electromagnetic field, andwhatever then-unknown short-range fields arepresumablyresponsible for the strong forces that hold atomic nuclei together and for the weak forces that turnneutrons intoprotonsorprotons intoneutrons in radioactivity.Thisdistinctionbetweenparticles andfieldsbegantobesweptawayinthe1930s,withtheadventofquantumfieldtheory.Justasthereisanelectromagneticfield,whoseenergyandmomentumarebundledinparticlesknownasphotons,sothereisalsoanelectronfield,whoseenergyandmomentumarebundledinelectrons,andlikewiseforothertypesofelementaryparticles.Thiswasfarfromobvious.Wecandirectlyfeeltheeffectsofgravitationalandelectromagneticfields

becausethequantaofthesefieldshavezeromass,andtheyareparticlesofatype(knownasbosons)that in largenumberscanoccupy thesamestate.Thesepropertiesallow largenumbersofphotons tobuildup to formstates thatweobserveaselectricandmagnetic fields thatseemtoobey therulesofclassical (that is, non-quantum)physics.Electrons, in contrast, havemass and areparticles of a type(known as fermions) no two of which can occupy the same state, so that electron fields are neverapparentinmacroscopicobservations.In the late 1940s quantum electrodynamics, the quantum field theory of photons, electrons, and

antielectrons, scored stunning successes, with the calculation of quantities like the strength of theelectron’s magnetic field that agreed with experiment to many decimal places.* Following thisachievement, itwasnatural to try to develop a quantum field theory thatwould encompassnot onlyphotons, electrons, andantielectronsbut also theotherparticlesbeingdiscovered in cosmic rays andacceleratorsandtheweakandstrongforcesthatactonthem.Wenowhavesuchaquantumfieldtheory,knownastheStandardModel.TheStandardModelisan

expandedversionofquantumelectrodynamics.Alongwith theelectronfield there isaneutrinofield,whosequantaarefermionslikeelectronsbutwithzeroelectricchargeandnearlyzeromass.Thereisapairofquarkfields,whosequantaaretheconstituentsoftheprotonsandneutronsthatmakeupatomicnuclei.Forreasonsthatnooneunderstands,thismenuisrepeatedtwice,withmuchheavierquarksandmuchheavierelectron-likeparticlesandtheirneutrinopartners.Theelectromagneticfieldappearsinaunified “electroweak” picture along with other fields responsible for the weak nuclear interactions,whichallowprotonsandneutronstoconvertintooneanotherinradioactivedecays.Thequantaofthesefieldsareheavybosons:theelectricallychargedW+andW−,andtheelectricallyneutralZ0.Therearealsoeightmathematicallysimilar“gluon”fieldsresponsiblefor thestrongnuclear interactions,whichholdquarkstogetherinsideprotonsandneutrons.In2012thelastmissingpieceoftheStandardModelwasdiscovered:aheavyelectricallyneutralboson thathadbeenpredictedby theelectroweakpartoftheStandardModel.

TheStandardModelisnottheendofthestory.Itleavesoutgravitation;itdoesnotaccountforthe“darkmatter”thatastronomerstellusmakesupfive-sixthsofthemassoftheuniverse;anditinvolvesfartoomanyunexplainednumericalquantities,liketheratiosofthemassesofthevariousquarksandelectron-likeparticles.Butevenso,theStandardModelprovidesaremarkablyunifiedviewofalltypesofmatterandforce(exceptforgravitation)thatweencounterinourlaboratories,inasetofequationsthatcanfitonasinglesheetofpaper.WecanbecertainthattheStandardModelwillappearasatleastanapproximatefeatureofanybetterfuturetheory.TheStandardModelwouldhaveseemedunsatisfying tomanynaturalphilosophersfromThales to

Newton. It is impersonal; there is no hint in it of human concerns like love or justice.No onewhostudiestheStandardModelwillbehelpedtobeabetterperson,asPlatoexpectedwouldfollowfromthe study of astronomy. Also, contrary to what Aristotle expected of a physical theory, there is noelementofpurposeintheStandardModel.Ofcourse,weliveinauniversegovernedbytheStandardModelandcanimaginethatelectronsandthetwolightquarksarewhattheyaretomakeuspossible,butthenwhatdowemakeoftheirheaviercounterparts,whichareirrelevanttoourlives?TheStandardModelisexpressedinequationsgoverningthevariousfields,butitcannotbededuced

from mathematics alone. Nor does it follow straightforwardly from observation of nature. Indeed,quarksandgluonsareattractedtoeachotherbyforcesthatincreasewithdistance,sotheseparticlescannever be observed in isolation. Nor can the Standard Model be deduced from philosophicalpreconceptions.Rather, theStandardModel isaproductofguesswork,guidedbyaesthetic judgment,and validated by the success of many of its predictions. Though the Standard Model has manyunexplainedaspects,weexpectthatatleastsomeofthesefeatureswillbeexplainedbywhateverdeepertheorysucceedsit.The old intimacy between physics and astronomy has continued. We now understand nuclear

reactions well enough not only to calculate how the Sun and stars shine and evolve, but also tounderstandhowthelightestelementswereproducedinthefirstfewminutesofthepresentexpansionofthe universe. And as in the past, astronomy now presents physics with a formidable challenge: theexpansion of the universe is speeding up, presumably owing to dark energy that is contained not inparticlemassesandmotions,butinspaceitself.Thereisoneaspectofexperiencethatatfirstsightseemstodefyunderstandingonthebasisofany

unpurposefulphysical theory like theStandardModel.Wecannotavoid teleologyin talkingof livingthings. We describe hearts and lungs and roots and flowers in terms of the purpose they serve, atendencythatwasonlyincreasedwiththegreatexpansionafterNewtonofinformationaboutplantsandanimals due to naturalists like Carl Linnaeus and Georges Cuvier. Not only theologians but alsoscientistsincludingRobertBoyleandIsaacNewtonhaveseenthemarvelouscapabilitiesofplantsandanimalsasevidenceforabenevolentCreator.Even ifwecanavoidasupernaturalexplanationof thecapabilitiesofplantsandanimals,itlongseemedinevitablethatanunderstandingoflifewouldrestonteleologicalprinciplesverydifferentfromthoseofphysicaltheorieslikeNewton’s.Theunificationofbiologywith the restof science firstbegan tobepossible in themid-nineteenth

century,withtheindependentproposalsbyCharlesDarwinandAlfredRusselWallaceofthetheoryofevolution through natural selection. Evolution was already a familiar idea, suggested by the fossilrecord.Manyof thosewhoaccepted therealityofevolutionexplained itasaresultofa fundamentalprincipleofbiology,aninherenttendencyoflivingthingstoimprove,aprinciplethatwouldhaveruledout any unification of biology with physical science. Darwin and Wallace instead proposed thatevolutionactsthroughtheappearanceofinheritablevariations,withfavorablevariationsnomorelikelythanunfavorableones,butwith thevariations that improve the chancesof survival and reproductionbeingtheonesthatarelikelytospread.*It tooka long time fornatural selection tobeacceptedas themechanismforevolution.Noone in

Darwin’stimeknewthemechanismforinheritance,orfortheappearanceofinheritablevariations,sotherewas room for biologists to hope for amorepurposeful theory. Itwasparticularly distasteful toimaginethathumansaretheresultofmillionsofyearsofnaturalselectionactingonrandominheritablevariations.Eventuallythediscoveryoftherulesofgeneticsandoftheoccurrenceofmutationsledinthe twentiethcentury toa“neo-Darwiniansynthesis” thatput the theoryofevolution throughnaturalselection on a firmer basis. Finally this theorywas grounded on chemistry, and thereby on physics,throughtherealizationthatgeneticinformationiscarriedbythedoublehelixmoleculesofDNA.So biology joined chemistry in a unified view of nature based on physics. But it is important to

acknowledgethelimitationsofthisunification.Nooneisgoingtoreplacethelanguageandmethodsofbiology with a description of living things in terms of individual molecules, let alone quarks andelectrons.Foronething,evenmorethanthelargemoleculesoforganicchemistry,livingthingsaretoocomplicatedforsuchadescription.Moreimportant,evenifwecouldfollowthemotionofeveryatomin a plant or animal, in that immensemassof datawewould lose the things that interest us—a lionhuntingantelopeoraflowerattractingbees.Forbiology,likegeologybutunlikechemistry,thereisanotherproblem.Livingthingsarewhatthey

are not only because of the principles of physics, but also because of a vast number of historicalaccidents,includingtheaccidentthatacometormeteorhittheEarth65millionyearsagowithenoughimpact tokilloff thedinosaurs,andgoingbacktothefact that theEarthformedatacertaindistancefrom the Sun and with a certain initial chemical composition. We can understand some of theseaccidentsstatistically,butnotindividually.Keplerwaswrong;noonewilleverbeabletocalculatethedistance of the Earth from the Sun solely from the principles of physics. What we mean by theunificationof biologywith the rest of science is only that there canbeno freestandingprinciples ofbiology, any more than of geology. Any general principle of biology is what it is because of thefundamentalprinciplesofphysicstogetherwithhistoricalaccidents,whichbydefinitioncanneverbeexplained.Thepointofviewdescribedhereiscalled(oftendisapprovingly)“reductionism.”Thereisopposition

to reductionism even within physics. Physicists who study fluids or solids often cite examples of“emergence,” the appearance in the description ofmacroscopic phenomena of concepts like heat orphasetransitionthathavenocounterpartinelementaryparticlephysics,andthatdonotdependonthedetails of elementary particles. For instance, thermodynamics, the science of heat, applies in awidevarietyofsystems:notjusttothoseconsideredbyMaxwellandBoltzmann,containinglargenumbersofmolecules,butalsotothesurfacesoflargeblackholes.Butitdoesnotapplytoeverything,andwhenweaskwhetheritappliestoagivensystemandifsowhy,wemusthavereferencetodeeper,moretrulyfundamental, principles of physics. Reductionism in this sense is not a program for the reform ofscientificpractice;itisaviewofwhytheworldisthewayitis.We do not knowhow long sciencewill continue on this reductive path.Wemay come to a point

wherefurtherprogressisimpossiblewithintheresourcesofourspecies.Rightnow,itseemsthatthereisa scaleofmassaboutamillion trillion times larger than themassof thehydrogenatom,atwhichgravityandotherasyetundetectedforcesareunifiedwith theforcesof theStandardModel. (This isknownasthe“Planckmass”;itisthemassthatparticleswouldhavetopossessfortheirgravitationalattractiontobeasstrongastheelectricalrepulsionbetweentwoelectronsatthesameseparation.)Eveniftheeconomicresourcesofthehumanracewereentirelyatthedisposalofphysicists,wecannotnowconceiveofanywayofcreatingparticleswithsuchhugemassesinourlaboratories.Wemay insteadrunoutof intellectual resources—humansmaynotbesmartenough tounderstand

the really fundamental lawsofphysics.Orwemayencounterphenomena that inprinciplecannotbebrought into a unified framework for all science. For instance, although we may well come tounderstandtheprocessesinthebrainresponsibleforconsciousness,itishardtoseehowwewillever

describeconsciousfeelingsthemselvesinphysicalterms.Still,wehavecomealongwayonthispath,andarenotyetatitsend.6Thisisagrandstory—how

celestial and terrestrial physics were unified by Newton, how a unified theory of electricity andmagnetismwasdevelopedthatturnedouttoexplainlight,howthequantumtheoryofelectromagnetismwasexpandedtoincludetheweakandstrongnuclearforces,andhowchemistryandevenbiologywerebrought into a unified though incomplete view of nature based on physics. It is toward a morefundamentalphysicaltheorythatthewide-rangingscientificprincipleswediscoverhavebeen,andarebeing,reduced.

Acknowledgments

I was fortunate to have the help of several learned scholars: the classicist Jim Hankinson and thehistorians Bruce Hunt and George Smith. They read through most of the book, and I made manycorrectionsbasedontheirsuggestions.Iamdeeplygratefulforthishelp.IamindebtedalsotoLouiseWeinbergforinvaluablecriticalcomments,andforsuggestingthelinesofJohnDonnethatnowgracethisbook’sfrontmatter.Thanks,too,toPeterDear,OwenGingerich,AlbertoMartinez,SamSchweber,andPaulWoodruff for advice on specific topics. Finally, for encouragement and good advice,manythanksareduetomywiseagent,MortonJanklow;andtomyfineeditorsatHarperCollins,TimDugganandEmilyCunningham.

TechnicalNotes

The following notes describe the scientific andmathematical background formany of the historicaldevelopments discussed in this book.Readerswhohave learned some algebra andgeometry in highschool and have not entirely forgotten what they learned should have no trouble with the level ofmathematicsinthesenotes.ButIhavetriedtoorganizethisbooksothatreaderswhoarenotinterestedintechnicaldetailscanskipthesenotesandstillunderstandthemaintext.A warning: The reasoning in these notes is not necessarily identical to that followed historically.

FromThales toNewton, the style of themathematics thatwas applied to physical problemswas farmoregeometricandlessalgebraicthaniscommontoday.Toanalyzetheseproblemsinthisgeometricstylewouldbedifficult formeand tedious for the reader. In thesenotes Iwill showhow the resultsobtainedbythenaturalphilosophersof thepastdofollow(or insomecases,donotfollow)fromtheobservationsandassumptionsonwhichtheyrelied,butwithoutattemptingfaithfullytoreproducethedetailsoftheirreasoning.

Notes

1.Thales’Theorem2.PlatonicSolids3.Harmony4.ThePythagoreanTheorem5.IrrationalNumbers6.TerminalVelocity7.FallingDrops8.Reflection9.FloatingandSubmergedBodies10.AreasofCircles11.SizesandDistancesoftheSunandMoon12.TheSizeoftheEarth13.EpicyclesforInnerandOuterPlanets14.LunarParallax15.SinesandChords16.Horizons17.GeometricProofoftheMeanSpeedTheorem18.Ellipses19.ElongationsandOrbitsoftheInnerPlanets20.DiurnalParallax21.TheEqual-AreaRule,andtheEquant22.FocalLength23.Telescopes24.MountainsontheMoon25.GravitationalAcceleration26.ParabolicTrajectories27.TennisBallDerivationoftheLawofRefraction28.Least-TimeDerivationoftheLawofRefraction29.TheTheoryoftheRainbow

30.WaveTheoryDerivationoftheLawofRefraction31.MeasuringtheSpeedofLight32.CentripetalAcceleration33.ComparingtheMoonwithaFallingBody34.ConservationofMomentum35.PlanetaryMasses

1.Thales’Theorem

Thales’theoremusessimplegeometricreasoningtoderivearesultaboutcirclesandtrianglesthatisnotimmediatelyobvious.WhetherornotThaleswastheonewhoprovedthisresult,itisusefultolookatthetheoremasaninstanceofthescopeofGreekknowledgeofgeometrybeforethetimeofEuclid.Consideranycircle,andanydiameterofit.LetAandBbethepointswherethediameterintersects

the circle.Draw lines fromA andB to any other pointP on the circle. The diameter and the linesrunningfromAtoPandfromBtoPformatriangle,ABP.(Weidentifytrianglesbylistingtheirthreecornerpoints.)Thales’Theoremtellsusthatthisisarighttriangle:theangleoftriangleABPatPisarightangle,orinotherterms,90°.The trick inproving this theorem is todrawa line from thecenterCof thecircle topointP.This

divides triangleABP into two triangles,ACP andBCP. (See Figure 1.) Both of these are isoscelestriangles,thatis,triangleswithtwosidesequal.IntriangleACP,sidesCAandCParebothradiiofthecircle, which have the same length according to the definition of a circle. (We label the sides of atrianglebythecornerpointstheyconnect.)Likewise,intriangleBCP,sidesCBandCPareequal.Inanisosceles triangle the angles adjoining the two equal sides are equal, so angle α (alpha) at theintersectionofsidesAPandACisequaltotheangleattheintersectionofsidesAPandCP,whileangleβ(beta)attheintersectionofsidesBPandBCisequaltotheangleattheintersectionofsidesBPandCP.Thesumoftheanglesofanytriangleistworightangles,*orinfamiliarterms180°,soifwetakeαʹasthethirdangleoftriangleACP,theangleattheintersectionofsidesACandCP,andlikewisetakeβʹastheangleattheintersectionofsidesBCandCP,then

2α+αʹ=180°  2β+βʹ=180°

Addingthesetwoequationsandregroupingtermsgives

2(α+β)+(αʹ+βʹ)=360°

Now,αʹ+βʹistheanglebetweenACandBC,whichcometogetherinastraightline,andisthereforehalfafullturn,or180°,so

2(α+β)=360°−180°=180°

andthereforeα+β=90°.ButaglanceatFigure1showsthatα+βistheanglebetweensidesAPandBPoftriangleABP,withwhichwestarted,soweseethatthisisindeedarighttriangle,aswastobeproved.

Figure1.ProofofThales’theorem.ThetheoremstatesthatwhereverpointPislocatedonthecircle,theanglebetweenthelinesfromtheendsofthediametertoPisarightangle.

2.PlatonicSolids

InPlato’sspeculationsaboutthenatureofmatter,acentralrolewasplayedbyaclassofsolidshapesknown as regular polyhedrons, which have come also to be known as Platonic solids. The regularpolyhedrons can be regarded as three-dimensional generalizations of the regular polygons of planegeometry, and are in a sense built up from regular polygons. A regular polygon is a plane figureboundedbysomenumbernofstraightlines,allofwhichareofthesamelengthandmeetateachofthencornerswiththesameangles.Examplesaretheequilateraltriangle(atrianglewithallsidesequal)andthe square. A regular polyhedron is a solid figure bounded by regular polygons, all of which areidentical,withthesamenumberNofpolygonsmeetingwiththesameanglesateveryvertex.Themost familiar example of a regular polyhedron is the cube. A cube is bounded by six equal

squares, with three squares meeting at each of its eight vertices. There is an even simpler regularpolyhedron,thetetrahedron,atriangularpyramidboundedbyfourequalequilateraltriangles,withthreetrianglesmeetingateachofthefourvertices.(Wewillbeconcernedhereonlywithpolyhedronsthatareconvex,witheveryvertexpointedoutward,asinthecaseofthecubeandthetetrahedron.)Aswereadin theTimaeus, ithadsomehowbecameknown toPlato that these regularpolyhedronscome inonlyfivepossibleshapes,whichhetooktobetheshapesoftheatomsofwhichallmatteriscomposed.Theyare the tetrahedron,cube,octahedron,dodecahedron,and icosahedron,with4,6,8,12,and20faces,respectively.The earliest attempt to prove that there are just five regular polyhedrons that has survived from

antiquity is theclimactic lastparagraphofEuclid’sElements. InPropositions13 through17ofBookXIIIEuclidhadgivengeometricconstructionsof the tetrahedron,octahedron,cube, icosahedron,anddodecahedron.Thenhe states,* “I saynext thatnoother figure,besides the said five figures, canbeconstructedwhich is contained by equilateral and equiangular figures equal to one another.” In fact,what Euclid actually demonstrates after this statement is a weaker result, that there are only five

combinationsofthenumbernofsidesofeachpolygonalface,andthenumberNofpolygonsmeetingateachvertex,thatarepossibleforaregularpolyhedron.TheproofgivenbelowisessentiallythesameasEuclid’s,expressedinmodernterms.Thefirststepistocalculatetheinteriorangleθ(theta)ateachofthenverticesofann-sidedregular

polygon.Draw lines from thecenterof thepolygon to theverticeson theboundary.Thisdivides theinteriorofthepolygonintontriangles.Sincethesumoftheanglesofanytriangleis180°,andeachofthesetriangleshastwoverticeswithanglesθ/2,theangleofthethirdvertexofeachtriangle(theoneatthecenterofthepolygon)mustbe180°–θ.Butthesenanglesmustaddupto360°,son(180°–θ)=360°.Thesolutionis

Forinstance,foranequilateraltrianglewehaven=3,soθ=180°–120°=60°,whileforasquaren=4,soθ=180°–90°=90°.Thenextstepistoimaginecuttingalltheedgesandverticesofaregularpolyhedronexceptatone

vertex,andpushingthepolyhedrondownontoaplaneat thatvertex.TheNpolygonsmeetingat thatvertexwillthenbelyingintheplane,buttheremustbespaceleftoverortheNpolygonswouldhaveformedasingle face.SowemusthaveNθ<360°.Using theabove formula forθ anddividingbothsidesoftheinequalityby360°thengives

orequivalently(dividingbothsidesbyN),

Now,wemusthaven≥3becauseotherwisetherewouldbenoareabetweenthesidesofthepolygons,andwemusthaveN≥3becauseotherwisetherewouldbenospacebetweenthefacescomingtogetheratavertex.(Forinstance,foracuben=4becausethesidesaresquares,andN=3.)Thustheaboveinequalitydoesnotalloweither1/nor1/Ntobeassmallas1/2–1/3=1/6,andconsequentlyneithernnorNcanbeaslargeas6.Wecaneasilycheckeverypairofvaluesofwholenumbers5≥N≥3and5≥n≥3toseeiftheysatisfytheinequality,andfindthatthereareonlyfivepairsthatdo:

(a)N=3,n=3(b)N=4,n=3(c)N=5,n=3(d)N=3,n=4(e)N=3,n=5

(In thecasesn=3,n=4, andn=5 the sidesof the regularpolyhedronare respectively equilateraltriangles, squares, and regular pentagons.) These are the values of N and n that we find in thetetrahedron,octahedron,icosahedron,cube,anddodecahedron.ThismuchwasprovedbyEuclid.ButEucliddidnotprovethatthereisonlyoneregularpolyhedron

foreachpairofnandN.Inwhatfollows,wewillgobeyondEuclid,andshowthatforeachvalueofNandnwecanfinduniqueresultsfortheotherpropertiesofthepolyhedron:thenumberFoffaces,thenumberEofedges,andthenumberVofvertices.Therearethreeunknownshere,soforthispurposewe

needthreeequations.Toderivethefirst,notethatthetotalnumberofbordersofallthepolygonsonthesurfaceofthepolyhedronisnF,buteachoftheEedgesborderstwopolygons,so

2E=nF

Also,thereareNedgescomingtogetherateachoftheVvertices,andeachoftheEedgesconnectstwovertices,so

2E=NV

Finally, thereisamoresubtlerelationamongF,E,andV. Inderivingthisrelation,wemustmakeanadditionalassumption,thatthepolyhedronissimplyconnected,inthesensethatanypathbetweentwopointsonthesurfacecanbecontinuouslydeformedintoanyotherpathbetweenthesepoints.Thisisthecasefor instanceforacubeora tetrahedron,butnotforapolyhedron(regularornot)constructedbydrawingedgesandfacesonthesurfaceofadoughnut.Adeeptheoremstatesthatanysimplyconnectedpolyhedron can be constructed by adding edges, faces, and/or vertices to a tetrahedron, and then ifnecessarycontinuouslysqueezingtheresultingpolyhedronintosomedesiredshape.Usingthisfact,weshallnowshowthatanysimplyconnectedpolyhedron(regularornot)satisfiestherelation:

F−E+V=2

Itiseasytocheckthatthisissatisfiedforatetrahedron,inwhichcasewehaveF=4,E=6,andV=4,sotheleft-handsideis4–6+4=2.Now,ifweaddanedgetoanypolyhedron,runningacrossafacefromoneedgetoanother,weaddonenewfaceandtwonewvertices,soFandVincreasebyoneunitandtwounits,respectively.Butthissplitseacholdedgeattheendsofthenewedgeintotwopieces,soEincreasesby1+2=3,andthequantityF–E+Vistherebyunchanged.Likewise,ifweaddanedgethatrunsfromavertextooneoftheoldedges,thenweincreaseFandVbyoneuniteach,andEbytwounits,sothequantityF–E+Visstillunchanged.Finally,ifweaddanedgethatrunsfromonevertextoanothervertex,thenweincreasebothFandEbyoneuniteachanddonotchangeV,soagainF–E+V is unchanged. Since any simply connected polyhedron can be built up in this way, all suchpolyhedronshavethesamevalueforthisquantity,whichthereforemustbethesamevalueF–E+V=2asfora tetrahedron. (This isasimpleexampleofabranchofmathematicsknownas topology; thequantityF–E+Visknownintopologyasthe“Eulercharacteristic”ofthepolyhedron.)WecannowsolvethesethreeequationsforE,F,andV.Itissimplesttousethefirsttwoequationsto

replaceF andV in the third equation with 2E/n and 2E/N, respectively, so that the third equationbecomes2E/n–E+2E/N=2,whichgives

Thenusingtheothertwoequations,wehave

Thusforthefivecaseslistedabove,thenumbersoffaces,vertices,andedgesare:

ThesearethePlatonicsolids.

3.Harmony

ThePythagoreansdiscoveredthattwostringsofamusicalinstrument,withthesametension,thickness,andcomposition,willmakeapleasantsoundwhenpluckedatthesametime,iftheratioofthestrings’lengthsisaratioofsmallwholenumbers,suchas1/2,2/3,1/4,3/4,etc.Toseewhythisisso,wefirstneed toworkout thegeneral relationbetween the frequency,wavelength,andvelocityofanysortofwave.Anywaveischaracterizedbysomesortofoscillatingamplitude.Theamplitudeofasoundwaveis

thepressureintheaircarryingthewave;theamplitudeofanoceanwaveistheheightofthewater;theamplitudeofalightwavewithadefinitedirectionofpolarizationistheelectricfieldinthatdirection;andtheamplitudeofawavemovingalongthestringofamusicalinstrumentisthedisplacementofthestringfromitsnormalposition,inadirectionatrightanglestothestring.There isaparticularlysimplekindofwaveknownasasinewave. Ifwe takeasnapshotofasine

waveatanymoment,weseethattheamplitudevanishesatvariouspointsalongthedirectionthewaveis traveling.Concentrating foramomentonone suchpoint, ifwe look fartheralong thedirectionoftravelwewillseethattheamplituderisesandthenfallsagaintozero,thenaswelookfartheritfallstoanegativevalueandrisesagaintozero,afterwhichitrepeatsthewholecycleagainandagainaswelookstillfartheralongthewave’sdirection.Thedistancebetweenpointsatthebeginningandendofanyonecomplete cycle is a length characteristic of the wave, known as its wavelength, and conventionallydenotedbythesymbolλ(lambda).Itwillbeimportantinwhatfollowsthat,sincetheamplitudeofthewavevanishesnotonlyatthebeginningandendofacyclebutalsointhemiddle,thedistancebetweensuccessive vanishing points is half awavelength, λ/2.Any two pointswhere the amplitude vanishesthereforemustbeseparatedbysomewholenumberofhalfwavelengths.Thereisafundamentalmathematicaltheorem(notmadeexplicituntil theearlynineteenthcentury)

that virtually any disturbance (that is, any disturbance having a sufficiently smooth dependence ondistancealongthewave)canbeexpressedasasumofsinewaveswithvariouswavelengths.(Thisisknownas“Fourieranalysis.”)Eachindividualsinewaveexhibitsacharacteristicoscillationintime,aswellasindistancealongthe

wave’sdirectionofmotion.Ifthewaveistravelingwithvelocityv,thenintimetittravelsadistancevt.Thenumberofwavelengthsthatpassafixedpointintimetwillthusbevt/λ,sothenumberofcyclespersecondatagivenpointinwhichtheamplitudeandrateofchangebothkeepgoingbacktothesamevalueisv/λ.Thisisknownasthefrequency,denotedbythesymbolν(nu),soν=v/λ.Thevelocityofawaveofvibrationofastringisclosetoaconstant,dependingonthestringtensionandmass,butnearlyindependentofitswavelengthoritsamplitude,soforthesewaves(asforlight)thefrequencyissimplyinverselyproportionaltothewavelength.Nowconsiderastringofsomemusicalinstrument,withlengthL.Theamplitudeof thewavemust

vanishattheendsofthestring,wherethestringisheldfixed.Thisconditionlimitsthewavelengthsofthe individualsinewaves thatcancontribute to the totalamplitudeof thestring’svibration.Wehavenotedthatthedistancebetweenpointswheretheamplitudeofanysinewavevanishescanbeanywholenumberofhalfwavelengths.ThusthewaveonastringthatisfixedatbothendsmustcontainawholenumberNofhalfwavelengths,sothatL=Nλ/2.Thatis, theonlypossiblewavelengthsareλ=2L/N,withN=1,2,3,etc.,andsotheonlypossiblefrequenciesare*

ν=vN/2L

Thelowestfrequency,forthecaseN=1,isv/2L;allthehigherfrequencies,forN=2,N=3,etc.,areknownasthe“overtones.”Forinstance,thelowestfrequencyofthemiddleCstringofanyinstrumentis261.63cyclespersecond,butitalsovibratesat523.26cyclespersecond,784.89cyclespersecond,andsoon.Theintensitiesofthedifferentovertonesmakethedifferenceinthequalitiesofthesoundsfromdifferentmusicalinstruments.Now,supposethatvibrationsaresetupintwostringsthathavedifferentlengthsL1andL2,butare

otherwiseidentical,andinparticularhavethesamewavevelocityv.Intimetthemodesofvibrationofthe lowest frequencyof the first and secondstringswillgo throughn1=ν1t =vt/2L1 andn2 =ν2t=vt/2L2cyclesorfractionsofcycles,respectively.Theratiois

n1/n2=L2/L1

Thus,inorderforthelowestvibrationsofbothstringseachtogothroughwholenumbersofcyclesinthesametime,thequantityL2/L1mustbearatioofwholenumbers—thatis,arationalnumber.(Inthiscase,inthesametimeeachovertoneofeachstringwillalsogothroughawholenumberofcycles.)Thesoundproducedbythetwostringswillthusrepeatitself,justasifasinglestringhadbeenplucked.Thisseemstocontributetothepleasantnessofthesound.Forinstance,ifL2/L1=1/2, then thevibrationofstring2of lowest frequencywillgo through two

completecyclesforeverycompletecycleofthecorrespondingvibrationofstring1.Inthiscase,wesaythatthenotesproducedbythetwostringsareanoctaveapart.AllofthedifferentCkeysonthepianokeyboardproducefrequenciesthatareoctavesapart.IfL2/L1=2/3,thetwostringsmakeachordcalledafifth.Forexample,ifonestringproducesmiddleC,at261.63cyclespersecond,thenanotherstringthatis2/3aslongwillproducemiddleG,atafrequency3/2×261.63=392.45cyclespersecond.*IfL2/L1=3/4,thechordiscalledafourth.Theotherreasonforthepleasantnessofthesechordshastodowiththeovertones.Inorderforthe

N1thovertoneofstring1 tohave thesamefrequencyas theN2thovertoneof string2,wemusthavevN1/2L1=vN2/2L2,andso

L2/L1=N2/N1

Again,theratioofthelengthsisarationalnumber,thoughforadifferentreason.Butifthisratioisanirrationalnumber,likeπorthesquarerootof2,thentheovertonesofthetwostringscannevermatch,thoughthefrequenciesofhighovertoneswillcomearbitrarilyclose.Thisapparentlysoundsterrible.

4.ThePythagoreanTheorem

Theso-calledPythagoreantheoremisthemostfamousresultofplanegeometry.Althoughitisbelievedto be due to a member of the school of Pythagoras, possibly Archytas, the details of its origin areunknown. What follows is the simplest proof, one that makes use of the notion of proportionalitycommonlyusedinGreekmathematics.Considera trianglewithcornerpointsA,B,andP,with theangleatP a rightangle.The theorem

statesthattheareaofasquarewhosesideisAB(thehypotenuseofthetriangle)equalsthesumoftheareasof squareswhosesidesare theother twosidesof the triangle,APandBP. Inmodernalgebraicterms,wecanthinkofAB,AP,andBPasnumericalquantities,equaltothelengthsofthesesides,andstatethetheoremas

AB2=AP2+BP2

ThetrickintheproofistodrawalinefromPtothehypotenuseAB,whichintersectsthehypotenuseat a right angle, say at point C. (See Figure 2.) This divides triangle ABP into two smaller righttriangles,APCandBPC.ItiseasytoseethatbothofthesesmallertrianglesaresimilartotriangleABP—thatis,alltheircorrespondinganglesareequal.IfwetaketheanglesatthecornersAandBtobeα(alpha)andβ(beta),thentriangleABPhasanglesα,β,and90°,soα+β+90°=180°.TriangleAPChas twoanglesequal toα and90°, so tomake the sumof theangles180° its thirdanglemustbeβ.Likewise,triangleBPChastwoanglesequaltoβand90°,soitsthirdanglemustbeα.Becausethesetrianglesareallsimilar,theircorrespondingsidesareproportional.Thatis,ACmustbe

inthesameproportiontothehypotenuseAPoftriangleACPthatAPhastothehypotenuseABof theoriginaltriangleABP,andBCmustbeinthesameproportiontoBPthatBPhastoAB.WecanputthisinmoreconvenientalgebraictermsasastatementaboutratiosofthelengthsAC,AP,etc.:

ItfollowsimmediatelythatAP2=AC×AB,andBP2=BC×AB.Addingthesetwoequationsgives

AP2+BP2=(AC+BC)×AB

ButAC+BC=AB,sothisistheresultthatwastobeproved.

Figure2.Proof of thePythagorean theorem.This theorem states that the sumof the areasof twosquareswhosesidesareAPandBPequalstheareaofasquarewhosesidesarethehypotenuseAB.Toprovethetheorem,alineisdrawnfromPtoapointC,whichischosensothatthislineisperpendiculartothelinefromAtoB.

5.IrrationalNumbers

TheonlynumbersthatwerefamiliartoearlyGreekmathematicianswererational.Thesearenumbersthatareeitherwholenumbers,like1,2,3,etc.,orratiosofwholenumbers,like1/2,2/3,etc.Iftheratioof the lengths of two lines is a rational number, the lines were said to be “commensurable”—forinstance,iftheratiois3/5,thenfivetimesonelinehasthesamelengthasthreetimestheother.Itwastherefore shocking to learn that not all lineswere commensurable. In particular, in a right isoscelestriangle,thehypotenuseisincommensurablewitheitherofthetwoequalsides.Inmodernterms,sinceaccordingtothetheoremofPythagorasthesquareofthehypotenuseofsuchatriangleequalstwicethesquareofeitherofthetwoequalsides,thelengthofthehypotenuseequalsthelengthofeitheroftheothersidestimesthesquarerootof2,sothisamountstothestatementthatthesquarerootof2isnotarational number. The proof given by Euclid in Book X of the Elements consists of assuming theopposite, in modern terms that there is a rational number whose square is 2, and then deriving anabsurdity.Supposethatarationalnumberp/q(withpandqwholenumbers)hasasquareequalto2:

(p/q)2=2

Therewillthenbeaninfinityofsuchpairsofnumbers,foundbymultiplyinganygivenpandqbyanyequalwholenumbers,butletustakepandqtobethesmallestwholenumbersforwhich(p/q)2=2.Itfollowsfromthisequationthat

p2=2q2

Thisshows thatp2 isanevennumber,but theproductofany twooddnumbers isodd,sopmustbeeven.Thatis,wecanwritep=2pʹ,wherepʹisawholenumber.Butthen

q2=2pʹ2

soby thesamereasoningasbefore,q is even,andcan thereforebewrittenasq=2qʹ,whereqʹ is awholenumber.Butthenp/q=pʹ/qʹ,so

(pʹ/qʹ)2=2

withpʹandqʹwholenumbersthatarerespectivelyhalfpandq,contradictingthedefinitionofpandqasthesmallestwholenumbersforwhich(p/q)2=2.Thustheoriginalassumption,thattherearewholenumberspandqforwhich(p/q)2=2,leadstoacontradiction,andisthereforeimpossible.Thistheoremhasanobviousextension:anynumberlike3,5,6,etc.thatisnotitselfthesquareofa

wholenumbercannotbethesquareofarationalnumber.Forinstance,if3=(p/q)2,withpandq thesmallestwholenumbers forwhich thisholds, thenp2=3q2,but this is impossibleunlessp=3pʹ forsome whole number pʹ, but then q2 = 3pʹ2, so q = 3qʹ for some whole number q, so 3 = (pʹ/qʹ)2,contradictingthestatementthatpandqarethesmallestwholenumbersforwhichp2=3q2.Thus thesquarerootsof3,5,6...areallirrational.Inmodernmathematicsweaccepttheexistenceofirrationalnumbers,suchasthenumberdenoted

whose square is 2. The decimal expansion of such numbers goes on forever, without ending orrepeating;forexample, =1.414215562....Thenumbersofrationalandirrationalnumbersarebothinfinite,butinasensetherearefarmoreirrationalthanrationalnumbers,fortherationalnumberscan

belistedinaninfinitesequencethatincludesanygivenrationalnumber:

1,2,1/2,3,1/3,2/3,3/2,4,1/4,3/4,4/3,...

whilenosuchlistofallirrationalnumbersispossible.

6.TerminalVelocity

TounderstandhowobservationsoffallingbodiesmighthaveledAristotletohisideasaboutmotion,wecan make use of a physical principle unknown to Aristotle, Newton’s second law of motion. Thisprincipletellsusthattheaccelerationaofabody(therateatwhichitsspeedincreases)equalsthetotalforceFactingonthebodydividedbythebody’smassm:

a=F/m

Therearetwomainforcesthatactonabodyfallingthroughtheair.Oneistheforceofgravity,whichisproportionaltothebody’smass:

Fgrav=mg

Heregisaconstantindependentofthenatureofthefallingbody.Itequalstheaccelerationofafallingbodythatissubjectonlytogravity,andhasthevalue32feet/secondpersecondonandneartheEarth’ssurface.Theotherforceistheresistanceoftheair.Thisisaquantityf(v)proportionaltothedensityofthe air, which increases with velocity and also depends on the body’s shape and size, but does notdependonitsmass:

Fair=−f(v)

Aminussignisputinthisformulafortheforceofairresistancebecausewearethinkingofaccelerationin a downward direction, and for a falling body the force of air resistance acts upward, sowith thisminussignintheformula,f(v)ispositive.Forinstance,forabodyfallingthroughasufficientlyviscousfluid,theairresistanceisproportionaltovelocity

f(v)=kv

withkapositiveconstantthatdependsonthebody’ssizeandshape.Forameteororamissileenteringthethinairoftheupperatmosphere,wehaveinstead

f(v)=Kv2

withKanotherpositiveconstant.UsingtheformulasfortheseforcesinthetotalforceF=Fgrav+FairandusingtheresultinNewton’s

law,wehave

a=g−f(v)/m

Whenabody is first released, its velocityvanishes, so there is no air resistance, and its accelerationdownward is just g. As time passes its velocity increases, and air resistance begins to reduce itsacceleration.Eventuallythevelocityapproachesavaluewheretheterm–f(v)/mjustcancelsthetermgintheformulaforacceleration,andtheaccelerationbecomesnegligible.This is theterminalvelocity,definedasthesolutionoftheequation:

f(vterminal)=gm

Aristotleneverspokeofterminalvelocity,butthevelocitygivenbythisformulahassomeofthesamepropertiesthatheattributedtothevelocityoffallingbodies.Sincef(v)isanincreasingfunctionofv,theterminalvelocityincreaseswiththemassm.Inthespecialcasewheref(v)=kv,theterminalvelocityissimplyproportionaltothemassandinverselyproportionaltotheairresistance:

vterminal=gm/k

Butthesearenotgeneralpropertiesofthevelocityoffallingbodies;heavybodiesdonotreachterminalvelocityuntiltheyhavefallenforalongtime.

7.FallingDrops

Stratoobservedthatfallingdropsgetfartherandfartherapartastheyfall,andconcludedfromthisthatthesedropsacceleratedownward. Ifonedrophas fallen farther thananother, then ithasbeen fallinglonger, and if thedrops are separating, then theone that is falling longermust alsobe falling faster,showingthatitsfallisaccelerating.ThoughStratodidnotknowit,theaccelerationisconstant,andasweshallsee,thisresultsinaseparationbetweendropsthatisproportionaltothetimeelapsed.AsmentionedinTechnicalNote6,ifairresistanceisneglected,thentheaccelerationdownwardof

any falling body is a constantg,which in the neighborhood of theEarth’s surface has the value 32feet/secondpersecond.Ifabodyfallsfromrest,thenafteratimeintervalτ(tau)itsvelocitydownwardwillbegτ.Henceifdrops1and2fallfromrestfromthesamedownspoutattimest1andt2, thenatalatertimetthespeeddownwardofthesedropswithbev1=g(t–t1)andv2=g(t–t2),respectively.Thedifferenceintheirspeedswillthereforebe

v1−v2=g(t−t1)−g(t−t2)=g(t2−t1)

Althoughbothv1andv2are increasingwith time, theirdifference is independentof the time t, so theseparationsbetweenthedropssimplyincreasesinproportiontothetime:

s=(v1−v2)t=gt(t1−t2)

Forinstance,iftheseconddropleavesthedownspoutatenthofasecondafterthefirstdrop,thenafterhalfasecondthedropswillbe32×1/2×1/10=1.6feetapart.

8.Reflection

ThederivationofthelawofreflectionbyHeroofAlexandriawasoneoftheearliestexamplesofthemathematical deduction of a physical principle from a deeper, more general, principle. Suppose anobserveratpointAseesthereflectioninamirrorofanobjectatpointB.IftheobserverseestheimageoftheobjectatapointPonthemirror,thelightraymusthavetraveledfromBtoPandthentoA.(Heroprobablywouldhavesaid that the light traveled from theobserveratA to themirrorand then to theobjectatB,asiftheeyereachedouttotouchtheobject,butthismakesnodifferenceintheargumentbelow.)Theproblemofreflectionis:whereonthemirrorisP?Toanswerthisquestion,Heroassumedthatlightalwaystakestheshortestpossiblepath.Inthecase

ofreflection,thisimpliesthatPshouldbelocatedsothatthetotallengthofthepathfromBtoPandthentoAistheshortestpaththatgoesfromBtoanywhereonthemirrorandthentoA.Fromthis,heconcludedthatangleθi(thetai)betweenthemirrorandtheincidentray(thelinefromB tothemirror)equalsangleθrbetweenthemirrorandthereflectedray(thelinefromthemirrortoA).Hereistheproofoftheequal-anglesrule.DrawalineperpendiculartothemirrorfromBtoapointBʹ

that is as far behind themirror asB is in front of themirror. (See Figure 3.) Suppose that this lineintersectsthemirroratpointC.SidesBʹCandCPofrighttriangleBʹCPhavethesamelengthsassidesBCandCPofrighttriangleBCP,sothehypotenusesBʹPandBPofthesetwotrianglesmustalsohavethesamelength.ThetotaldistancetraveledbythelightrayfromBtoPandthentoAisthereforethesameasthedistancethatwouldbetraveledbythelightrayifitwentfromBʹtoPandthentoA.TheshortestdistancebetweenpointsBʹandAisastraightline,sothepaththatminimizesthetotaldistancebetween theobject and theobserver is theone forwhichP ison the straight linebetweenBʹ andA.When two straight lines intersect, the anglesonopposite sidesof the intersectionpoint are equal, soangleθbetweenlineBʹPandthemirrorequalsangleθrbetweenthereflectedrayandthemirror.Butbecause the two right trianglesBʹCPandBCPhave thesamesides,angleθmust also equal angleθibetweentheincidentrayBPandthemirror.So,sincebothθiandθrareequal toθ, theyareequal toeachother.Thisisthefundamentalequal-anglesrulethatdeterminesthelocationPonthemirroroftheimageoftheobject.

Figure3.ProofofHero’stheorem.ThistheoremstatesthattheshortestpathfromanobjectatBtothemirrorandthentoaneyeatAisoneforwhichanglesθiandθrareequal.Thesolidlinesmarkedwith

arrowsrepresentthepathofalightray; thehorizontal lineis themirror;andthedashedlineisalineperpendicular to themirror that runs fromB toapointB' on theother sideof themirror at anequaldistancefromit.

9.FloatingandSubmergedBodies

InhisgreatworkOnFloatingBodies,Archimedesassumedthatifbodiesarefloatingorsuspendedinwaterinsuchawaythatequalareasatequaldepthsinthewaterarepresseddownbydifferentweights,thenthewaterandthebodieswillmoveuntilallequalareasatanygivendeptharepresseddownbythesame weight. From this assumption, he derived general consequences about both floating andsubmergedbodies,someofwhichwereevenofpracticalimportance.First,considerabodylikeashipwhoseweightislessthantheweightofanequalvolumeofwater.

Thebodywillfloatonthesurfaceofthewater,anddisplacesomequantityofwater.Ifwemarkoutahorizontalpatchinthewateratsomedepthdirectlybelowthefloatingbody,withanareaequaltotheareaofthebodyatitswaterline,thentheweightpressingdownonthissurfacewillbetheweightofthefloatingbodyplustheweightofthewaterabovethatpatch,butnotincludingthewaterdisplacedbythebody,asthiswaterisnolongerabovethepatch.Wecancomparethiswiththeweightpressingdownonanequalareaatanequaldepth,awayfromthe locationof thefloatingbody.Thisofcoursedoesnotincludetheweightofthefloatingbody,butitdoesincludeallthewaterfromthispatchtothesurface,withnowaterdisplaced.Inorderforbothpatchestobepresseddownbythesameweight,theweightofthewaterdisplacedby the floatingbodymustequal theweightof the floatingbody.This iswhy theweightofashipisreferredtoasits“displacement.”Nextconsiderabodywhoseweightisgreaterthantheweightofanequalvolumeofwater.Sucha

bodywillnotfloat,butitcanbesuspendedinthewaterfromacable.Ifthecableisattachedtoonearmof a balance, then in this way we can measure the apparent weight Wapparent of the body whensubmergedinwater.Theweightpressingdownonahorizontalpatchinthewateratsomedepthdirectlybelow the suspended bodywill equal the trueweightWtrue of the suspended body, less the apparentweightWapparent,whichiscanceledbythetensioninthecable,plustheweightofthewaterabovethepatch,whichofcoursedoesnotincludethewaterdisplacedbythebody.Wecancomparethiswiththeweight pressing down on an equal area at an equal depth, a weight that does not includeWtrue or–Wapparent, butdoes include theweightof all thewater from thispatch to the surface,withnowaterdisplaced.Inorderforbothpatchestobepresseddownbythesameweight,wemusthave

Wtrue−Wapparent=Wdisplaced

whereWdisplacedistheweightofthewaterdisplacedbythesuspendedbody.Sobyweighingthebodywhen suspended in thewater andweighing itwhenoutof thewater,wecan findbothWapparent andWtrue,andinthiswayfindWdisplaced.IfthebodyhasvolumeV,then

Wdisplaced=ρwaterV

whereρwater(rhowater)isthedensity(weightpervolume)ofwater,closeto1grampercubiccentimeter.(Ofcourse,forabodywithasimpleshapelikeacubewecouldinsteadfindVbyjustmeasuringthedimensionsofthebody,butthisisdifficultforanirregularlyshapedbodylikeacrown.)Also,thetrue

weightofthebodyis

Wtrue=ρbodyV

whereρbodyisthedensityofthebody.ThevolumecancelsintheratioofWtrueandWdisplaced,sofromthemeasurementsofbothWapparentandWtruewecanfindtheratioofthedensitiesofthebodyandofwater:

Thisratioiscalledthe“specificgravity”ofthematerialofwhichthebodyiscomposed.Forinstance,ifthebodyweighs20percentlessinwaterthaninair,thenWtrue–Wapparent=0.20×Wtrue,soitsdensitymustbe1/0.2=5timesthedensityofwater.Thatis,itsspecificgravityis5.Thereisnothingspecialaboutwaterinthisanalysis;ifthesamemeasurementsweremadeforabody

suspendedinsomeotherliquid,thentheratioofthetrueweightofthebodytothedecreaseinitsweightwhensuspendedintheliquidwouldgivetheratioofthedensityofthebodytothedensityofthatliquid.Thisrelationissometimesusedwithabodyofknownweightandvolumetomeasurethedensitiesofvariousliquidsinwhichthebodymaybesuspended.

10.AreasofCircles

Tocalculatetheareaofacircle,Archimedesimaginedthatapolygonwithalargenumberofsideswascircumscribedoutsidethecircle.Forsimplicity,let’sconsideraregularpolygon,allofwhosesidesandanglesareequal.Theareaof thepolygon is thesumof theareasofall the right triangles formedbydrawinglinesfromthecentertothecornersofthepolygon,andlinesfromthecentertothemidpointsofthesidesofthepolygon.(SeeFigure4,inwhichthepolygonistakentobearegularoctagon.)Thearea of a right triangle is half the product of its two sides around the right angle, because two suchtriangles can be stacked on their hypotenuses tomake a rectangle,whose area is the product of thesides.Inourcase, thismeansthat theareaofeachtriangleishalf theproductofthedistancer to themidpointoftheside(whichisjusttheradiusofthecircle)andthedistancesfromthemidpointofthesidetothenearestcornerofthepolygon,whichofcourseishalfthelengthofthatsideofthepolygon.Whenweaddupalltheseareas,wefindthattheareaofthewholepolygonequalshalfofrtimesthetotalcircumferenceofthepolygon.Ifweletthenumberofsidesofthepolygonbecomeinfinite,itsareaapproachestheareaofthecircle,anditscircumferenceapproachesthecircumferenceofthecircle.Sotheareaofthecircleishalfitscircumferencetimesitsradius.Inmodern terms,wedefine a numberπ = 3.14159 . . . such that the circumference of a circle of

radiusris2πr.Theareaofthecircleisthus

1/2×r×2πr=πr2Thesameargumentworksifweinscribepolygonswithinthecircle,ratherthancircumscribingthem

outside the circle as inFigure 4. Since the circle is always between an outer polygon circumscribedarounditandaninnerpolygoninscribedwithinit,usingpolygonsofbothsortsallowedArchimedestogiveupperandlowerlimitsfortheratioofthecircumferenceofacircletoitsradius—inotherwords,for2π.

Figure 4. Calculation of the area of a circle. In this calculation a polygon with many sides iscircumscribedaboutacircle.Here,thepolygonhaseightsides,anditsareaisalreadyclosetotheareaofthecircle.Asmoresidesareaddedtothepolygon,itsareabecomescloserandclosertotheareaofthecircle.

11.SizesandDistancesoftheSunandMoon

AristarchususedfourobservationstodeterminethedistancesfromtheEarthtotheSunandMoonandthe diameters of the Sun and Moon, all in terms of the diameter of the Earth. Let’s look at eachobservationinturn,andseewhatcanbelearnedfromit.Below,dsanddmare thedistances fromtheEarthtotheSunandMoon,respectively;andDs,Dm,andDedenotethediametersoftheSun,Moon,andEarth.Wewillassumethatthediametersarenegligiblecomparedwiththedistances,sointalkingof the distance from theEarth to theMoonor Sun, it is unnecessary to specify points on theEarth,Moon,orSunfromwhichthedistancesaremeasured.

Observation1

WhentheMoonishalffull,theanglebetweenthelinesofsightfromtheEarthtotheMoonandtotheSunis87°.WhentheMoonishalffull,theanglebetweenthelinesofsightfromtheMoontotheEarthandfrom

theMoontotheSunmustbejust90°(seeFigure5a),sothetriangleformedbythelinesMoon–Sun,Moon–Earth, andEarth–Sun is a right triangle,with theEarth–Sun line as thehypotenuse.The ratiobetweenthesideadjacenttoanangleθ(theta)ofarighttriangleandthehypotenuseisatrigonometricquantityknownas thecosineofθ,abbreviatedcosθ,whichwecan lookup in tablesor findonanyscientificcalculator.Sowehave

dm/ds=cos87°=0.05234=1/19.11

andthisobservationindicatesthattheSunis19.11timesfartherfromtheEarththanistheMoon.Notknowing trigonometry,Aristarchuscouldonlyconclude that thisnumber isbetween19and20. (Theangleactuallyisnot87°,but89.853°,andtheSunisreally389.77timesfartherfromtheEarththanistheMoon.)

Figure5.ThefourobservationsusedbyAristarchustocalculatethesizesanddistancesoftheSunandMoon.(a)ThetriangleformedbytheEarth,Sun,andMoonwhentheMoonishalffull.(b)TheMoonjustblottingoutthediskoftheSunduringatotaleclipseoftheSun.(c)TheMoonpassingintotheshadowoftheEarthduringaneclipseoftheMoon.ThespherethatjustfitsintothisshadowhasadiametertwicethatoftheMoon,andPistheterminalpointoftheshadowcastbytheEarth.(d)LinesofsighttotheMoonspanninganangleof2°;theactualangleiscloseto0.5°.

Observation2

TheMoonjustcoversthevisiblediskoftheSunduringasolareclipse.ThisshowsthattheSunandMoonhaveessentiallythesameapparentsize,inthesensethattheangle

betweenthelinesofsightfromtheEarthtooppositesidesofthediskoftheSunisthesameasfortheMoon. (See Figure 5b.) This means that the triangles formed by these two lines of sight and the

diametersof theSunandMoonare “similar”; that is, theyhave the same shape.Hence the ratiosofcorrespondingsidesofthesetwotrianglesarethesame,so

Ds/Dm=ds/dm

Usingtheresultofobservation1thengivesDs/Dm=19.11,whiletheactualratioofdiametersisreallycloseto390.

Observation3

TheshadowoftheEarthatthepositionoftheMoonduringalunareclipseisjustwideenoughtofitaspherewithtwicethediameteroftheMoon.LetP be the pointwhere the cone of shadow cast by theEarth ends.Thenwe have three similar

triangles:thetriangleformedbythediameteroftheSunandthelinesfromtheedgesoftheSun’sdisktoP;thetriangleformedbythediameteroftheEarthandthelinesfromtheedgesoftheEarth’sdisktoP; and the triangle formedby twice the diameter of theMoon and the lines froma spherewith thatdiameter at thepositionof theMoonduring a lunar eclipse toP. (SeeFigure5c.) It follows that theratiosofcorrespondingsidesofthesetrianglesareallequal.SupposethatpointPisatdistanced0fromtheMoon.ThentheSunisatdistanceds+dm+d0fromPandtheEarthisatdistancedm+d0fromP,so

Therestisalgebra.Wecansolvethesecondequationford0,andfind:

InsertingthisresultinthefirstequationandmultiplyingwithDeDs(De–2Dm)gives

ThetermsdmDs×(–2Dm)and2DmdmDsontheright-handsidecancel.Theremainderontheright-handsidehasafactorDe,whichcancelsthefactorDeontheleft-handside,leavinguswithaformulaforDe:

Ifwenowusetheresultofobservation2,thatds/dm=Ds/Dm, thiscanbewrittenentirely in termsofdiameters:

IfweusethepriorresultDs/Dm=19.1,thisgivesDe/Dm=2.85.Aristarchusgavearangefrom108/43=2.51and60/19=3.16,whichnicelycontainsthevalue2.85.Theactualvalueis3.67.ThereasonthatthisresultofAristarchuswasprettyclosetotheactualvaluedespitehisverybadvalueforDs/DmisthattheresultisveryinsensitivetotheprecisevalueofDsifDs>>Dm.Indeed,ifweneglectthetermDminthedenominatoraltogethercomparedwithDs, thenalldependenceofDscancels,andwehavesimplyDe=3Dm,whichisalsonotsofarfromthetruth.Ofmuchgreaterhistorical importance is the fact that ifwecombine the resultsDs/Dm=19.1and

De/Dm=2.85,wefindDs/De=19.1/2.85=6.70.TheactualvalueisDs/De=109.1,buttheimportantthing is that the Sun is considerably bigger than the Earth. Aristarchus emphasized the point bycomparing the volumes rather than the diameters; if the ratio of diameters is 6.7, then the ratio ofvolumesis6.73=301.Itisthiscomparisonthat,ifwebelieveArchimedes,ledAristarchustoconcludethattheEarthgoesaroundtheSun,nottheSunaroundtheEarth.TheresultsofAristarchusdescribedsofaryieldvaluesforallratiosofdiametersoftheSun,Moon,

andEarth,andtheratioofthedistancestotheSunandMoon.Butnothingsofargivesustheratioofanydistancetoanydiameter.Thiswasprovidedbythefourthobservation:

Observation4

TheMoonsubtendsanangleof2°.(See Figure 5d.) Since there are 360° in a full circle, and a circle whose radius is dm has a

circumference2πdm,thediameteroftheMoonis

AristarchuscalculatedthatthevalueofDm/dmisbetween2/45=0.044and1/30=0.033.Forunknownreasons Aristarchus in his surviving writings had grossly overestimated the true angular size of theMoon; it actually subtends an angle of 0.519°, givingDm/dm = 0.0090. As we noted in Chapter 8,ArchimedesinTheSandReckonergaveavalueof0.5°fortheanglesubtendedbytheMoon,whichisquiteclosetothetruevalueandwouldhavegivenanaccurateestimateoftheratioofthediameteranddistanceoftheMoon.With his results from observations 2 and 3 for the ratioDe/Dm of the diameters of the Earth and

Moon,andnowwithhisresultfromobservation4fortheratioDm/dmofthediameteranddistanceoftheMoon,AristarchuscouldfindtheratioofthedistanceoftheMoontothediameteroftheEarth.Forinstance,takingDe/Dm=2.85andDm/dm=0.035wouldgive

(Theactualvalueisabout30.)Thiscouldthenbecombinedwiththeresultofobservation1fortheratiods/dm=19.1ofthedistancestotheSunandMoon,givingavalueofds/De=19.1×10.0=191fortheratio of the distance to the Sun and the diameter of the Earth. (The actual value is about 11,600.)MeasuringthediameteroftheEarthwasthenexttask.

12.TheSizeoftheEarth

Eratosthenesusedtheobservationthatatnoononthesummersolstice,theSunatAlexandriais1/50ofafullcircle(thatis,360°/50=7.2°)awayfromthevertical,whileatSyene,acitysupposedlyduesouthofAlexandria, theSunatnoononthesummersolsticewasreportedtobedirectlyoverhead.BecausetheSunissofaraway,thelightraysstrikingtheEarthatAlexandriaandSyeneareessentiallyparallel.Theverticaldirectionatanycityis just thecontinuationofa linefromthecenterof theEarthtothatcity, so theanglebetween the lines from theEarth’s center toSyeneand toAlexandriamust alsobe7.2°,or1/50ofafullcircle.(SeeFigure6.)HenceonthebasisoftheassumptionsofEratosthenes,theEarth’scircumferencemustbe50timesthedistancefromAlexandriatoSyene.

Figure6.TheobservationusedbyEratosthenestocalculatethesizeoftheEarth.Thehorizontallinesmarkedwitharrowsindicateraysofsunlightatthesummersolstice.ThedottedlinesrunfromtheEarth’scentertoAlexandriaandSyene,andmarktheverticaldirectionateachplace.

SyeneisnotontheEarth’sequator,asmightbesuggestedbythewaythefigureisdrawn,butratherclose to the Tropic ofCancer, the line at latitude 231⁄2°. (That is, the angle between lines from theEarth’scentertoanypointontheTropicofCancerandtoapointduesouthontheequatoris231⁄2°.)Atthe summer solstice theSun isdirectlyoverheadatnoonon theTropicofCancer rather thanon theequatorbecausetheEarth’saxisofrotationisnotperpendiculartotheplaneofitsorbit,buttiltedfromtheperpendicularbyanangleof231⁄2°.

13.EpicyclesforInnerandOuterPlanets

PtolemyintheAlmagestpresentedatheoryoftheplanetsaccordingtowhich,initssimplestversion,eachplanetgoesonacirclecalledanepicyclearoundapointinspacethatitselfgoesaroundtheEarthonacircleknownastheplanet’sdeferent.ThequestionbeforeusiswhythistheoryworkedsowellinaccountingfortheapparentmotionsoftheplanetsasseenfromEarth.Theanswerfortheinnerplanets,MercuryandVenus,isdifferentfromthatfortheouterplanets,Mars,Jupiter,andSaturn.First, consider the inner planets,Mercury andVenus.According toourmodernunderstanding, the

Earth and each planet go around the Sun at approximately constant distances from the Sun and atapproximatelyconstantspeeds.Ifwedonotconcernourselveswiththelawsofphysics,wecanjustaswell change our point of view to one centered on the Earth. From this point of view, the Sun goesaroundtheEarth,andeachplanetgoesaroundtheSun,allatconstantspeedsanddistances.ThisisasimpleversionofthetheoryduetoTychoBrahe,whichmayalsohavebeenproposedbyHeraclides.Itgives the correct apparentmotions of the planets, apart from small corrections due to the facts thatplanets actuallymove on nearly circular elliptical orbits rather than on circles, the Sun is not at thecentersoftheseellipsesbutatrelativelysmalldistancesfromthecenters,andthespeedofeachplanetvariessomewhatastheplanetgoesarounditsorbit.It isalsoaspecialcaseofthetheoryofPtolemy,though one never considered by Ptolemy, in which the deferent is nothing but the orbit of the SunaroundtheEarth,andtheepicycleistheorbitofMercuryorVenusaroundtheSun.Now,asfarastheapparentpositionintheskyoftheSunandplanetsisconcerned,wecanmultiply

thechangingdistanceofanyplanetfromtheEarthbyaconstant,withoutchangingappearances.Thiscanbedone, for instance,bymultiplying the radiiofboth theepicycleand thedeferentby thesameconstantfactor,chosenindependentlyforMercuryandVenus.Forinstance,wecouldtaketheradiusofthedeferentofVenustobehalfthedistanceoftheSunfromtheEarth,andtheradiusofitsepicycletobehalftheradiusoftheorbitofVenusaroundtheSun.Thiswillnotchangethefactthatthecentersofthe planets’ epicycles always stay on the line between theEarth and theSun. (SeeFigure 7a,whichshowstheepicycleanddeferentforoneoftheinnerplanets,notdrawntoscale.)TheapparentmotionofVenusandMercuryintheskywillbeunchangedbythistransformation,aslongaswedon’tchangethe ratio of the radii of each planet’s deferent and epicycle. This is a simple version of the theoryproposedbyPtolemyfortheinnerplanets.Accordingtothistheory,theplanetgoesarounditsepicyclein the same time that it actually takes to go around the Sun, 88 days forMercury and 225 days forVenus,whilethecenteroftheepicyclefollowstheSunaroundtheEarth,takingoneyearforacompletecircuitofthedeferent.Specifically,sincewedonotchangetheratiooftheradiiofthedeferentandepicycle,wemusthave

rEPI/rDEF=rP/rE

whererEPIandrDEFaretheradiioftheepicycleanddeferentinPtolemy’sscheme,andrPandrEaretheradiioftheorbitsoftheplanetandtheEarthinthetheoryofCopernicus(orequivalently,theradiioftheorbitsoftheplanetaroundtheSunandtheSunaroundtheEarthinthetheoryofTycho).Ofcourse,PtolemyknewnothingofthetheoriesofTychoorCopernicus,andhedidnotobtainhistheoryinthisway. The discussion above serves to show only why Ptolemy’s theory worked so well, not how hederivedit.Now,letusconsidertheouterplanets,Mars,Jupiter,andSaturn.Inthesimplestversionofthetheory

ofCopernicus (orTycho) eachplanet keeps a fixeddistance not only from theSun, but also fromamoving pointCʹ in space, which keeps a fixed distance from the Earth. To find this point, draw aparallelogram(Figure7b),whosefirstthreeverticesinorderarounditarethepositionSoftheSun,thepositionE of theEarth, and thepositionPʹ of oneof theplanets.ThemovingpointCʹ is the emptyfourth corner of the parallelogram. Since the line betweenE andS has a fixed length, and the linebetweenPʹandCʹistheoppositesideoftheparallelogram,ithasanequalfixedlength,sotheplanetstaysatafixeddistancefromCʹ,equal tothedistanceoftheEarthfromtheSun.Likewise,sincethelinebetweenS andpʹ has a fixed length, and the line betweenE andCʹ is the opposite side of theparallelogram,ithasanequalfixedlength,sopointCʹstaysatafixeddistancefromtheEarth,equaltothedistanceoftheplanetfromtheSun.ThisisaspecialcaseofthetheoryofPtolemy,thoughacaseneverconsideredbyhim,inwhichthedeferentisnothingbuttheorbitofpointCʹaroundtheEarth,and

theepicycleistheorbitofMars,Jupiter,orSaturnaroundCʹ.

Figure7.AsimpleversionoftheepicycletheorydescribedbyPtolemy.(a)Thesupposedmotionofoneoftheinnerplanets,MercuryorVenus.(b)ThesupposedmotionofoneoftheouterplanetsMars,Jupiter,orSaturn.PlanetPgoesonanepicyclearoundpointC inoneyear,withthelinefromCtoPalways parallel to the line from the Earth to the Sun, while pointC goes around the Earth on thedeferentinalongertime.(ThedashedlinesindicateaspecialcaseofthePtolemaictheory,forwhichitisequivalenttothatofCopernicus.)

Onceagain,as faras theapparentposition in theskyof theSunandplanets isconcerned,wecanmultiply the changing distance of any planet from the Earth by a constant without changingappearances,bymultiplyingtheradiiofboththeepicycleandthedeferentbyaconstantfactor,chosenindependentlyforeachouterplanet.Althoughwenolongerhaveaparallelogram,thelinebetweentheplanetandCremainsparalleltothelinebetweentheEarthandSun.Theapparentmotionofeachouterplanetintheskywillbeunchangedbythistransformation,aslongaswedon’tchangetheratiooftheradiiofeachplanet’sdeferentandepicycle.ThisisasimpleversionofthetheoryproposedbyPtolemyfortheouterplanets.Accordingtothistheory,theplanetgoesaroundConitsepicyclein1year,whileCgoesaroundthedeferentinthetimethatitactuallytakestheplanettogoaroundtheSun:1.9yearsforMars,12yearsforJupiter,and29yearsforSaturn.Specifically,sincewedonotchangetheratiooftheradiiofthedeferentandepicycle,wemustnow

have

rEPI/rDEF=rE/rP

whererEPIandrDEFareagaintheradiioftheepicycleanddeferentinPtolemy’sscheme,andrPandrEaretheradiioftheorbitsoftheplanetandtheEarthinthetheoryofCopernicus(orequivalently, theradiioftheorbitsoftheplanetaroundtheSunandtheSunaroundtheEarthinthetheoryofTycho).Once again, the abovediscussiondescribes, not howPtolemyobtainedhis theory, but onlywhy thistheoryworkedsowell.

14.LunarParallax

Suppose that thedirection to theMoonisobservedfrompointOon thesurfaceof theEarth tobeatangleζʹ(zetaprime)tothezenithatO.TheMoonmovesinasmoothandregularwayaroundthecenteroftheEarth,sobyusingtheresultsofrepeatedobservationsoftheMoonitispossibletocalculatethedirection from the centerC of the Earth to theMoonM at the same moment, and in particular tocalculateangleζbetweenthedirectionfromCtotheMoonandthedirectionofthezenithatO,whichisthesameas thedirectionof the linefromthecenterof theEarth toO.AnglesζandζʹdifferslightlybecausetheradiusreoftheearthisnotentirelynegligiblecomparedwiththedistancedoftheMoonfromthecenteroftheEarth,andfromthisdifferencePtolemycouldcalculatetheratiod/re.PointsC,O,andMformatriangle,inwhichtheangleatCisζ,theangleatOis180°–ζʹ,and(since

thesumoftheanglesofanytriangleis180°)theangleatMis180°–ζ–(180°–ζʹ)=ζʹ–ζ.(SeeFigure8.)Wecancalculate the ratiod/re from theseanglesmuchmoreeasily thanPtolemydid,byusingatheoremofmoderntrigonometry:thatinanytrianglethelengthsofsidesareproportionaltothesinesoftheoppositeangles.(SinesarediscussedinTechnicalNote15.)TheangleoppositethelineoflengthrefromCtoOisζʹ–ζ,andtheangleoppositethelineoflengthdfromCtoMis180°–ζʹ,so

Figure8.UseofparallaxtomeasurethedistancetotheMoon.Hereζ’istheobservedanglebetweenthelineofsighttotheMoonandtheverticaldirection,andζisthevaluethisanglewouldhaveiftheMoonwereobservedfromthecenteroftheEarth.

OnOctober1,AD135,PtolemyobservedthatthezenithangleoftheMoonasseenfromAlexandriawasζʹ = 50°55’, and his calculations showed that at the samemoment the corresponding angle thatwouldbeobservedfromthecenteroftheEarthwasζ=49°48’.Therelevantsinesare

sinζʹ=0.776   sin(ζʹ−ζ)=0.0195

FromthisPtolemycouldconcludethatthedistancefromthecenteroftheEarthtotheMooninunitsoftheradiusoftheEarthis

Thiswasconsiderablylessthantheactualratio,whichonaverageisabout60.Thetroublewasthatthedifferenceζʹ–ζwasnotactuallyknownaccuratelybyPtolemy,butatleastitgaveagoodideaoftheorderofmagnitudeofthedistancetotheMoon.Anyway,PtolemydidbetterthanAristarchus,fromwhosevaluesfortheratioofthediametersofthe

Earth andMoon and of the distance and diameter of theMoon he could have inferred that d/re isbetween215/9=23.9and57/4=14.3.ButifAristarchushadusedacorrectvalueofabout1/2°fortheangulardiameteroftheMoon’sdiskinsteadofhisvalueof2°hewouldhavefoundd/retobefourtimesgreater,between57.2and95.6.Thatrangeincludesthetruevalue.

15.SinesandChords

Themathematiciansandastronomersofantiquitycouldhavemadegreatuseofabranchofmathematicsknownastrigonometry,whichistaughttodayinhighschools.Givenanyangleofarighttriangle(otherthantherightangleitself)trigonometrytellsushowtocalculatetheratiosofthelengthsofallthesides.Inparticular,thesideoppositetheangledividedbythehypotenuseisaquantityknownasthe“sine”ofthatangle,whichcanbe foundby looking itup inmathematical tablesor typing theangle inahandcalculatorandpressing“sin.”(Thesideofthetriangleadjacenttoanangledividedbythehypotenuseisthe “cosine” of the angle, and the side opposite divided by the side adjacent is the “tangent” of theangle,butitwillbeenoughforustodealherewithsines.)ThoughnonotionofasineappearsanywhereinHellenisticmathematics,Ptolemy’sAlmagestmakesuseofarelatedquantity,knownasthe“chord”ofanangle.Todefinethechordofanangleθ(theta),drawacircleofradius1(inwhateverunitsoflengthyou

find convenient), and draw two radial lines from the center to the circumference, separated by thatangle.Thechordoftheangleisthelengthofthestraightline,orchord,thatconnectsthepointswherethetworadiallinesintersectthecircumference.(SeeFigure9.)TheAlmagestgivesatableofchords*inaBabyloniansexigesimalnotation,withanglesexpressedindegreesofarc,runningfrom1/2°to180°.Forinstance,thechordof45°isgivenas451519,orinmodernnotation

whilethetruevalueis0.7653669....The chord has a natural application to astronomy. Ifwe imagine the stars as lying on a sphere of

radiusequalto1,centeredonthecenteroftheEarth,thenifthelinesofsighttotwostarsareseparatedbyangleθ,theapparentstraight-linedistancebetweenthestarswillbethechordofθ.Toseewhatthesechordshavetodowithtrigonometry,returntothefigureusedtodefinethechordof

angleθ,anddrawaline(thedashedlineinFigure9)fromthecenterofthecirclethatjustbisectsthechord.Wethenhavetworighttriangles,eachwithanangleatthecenterofthecircleequaltoθ/2,andasideoppositethisanglewhoselengthishalfthechord.Thehypotenuseofeachofthesetrianglesistheradiusofthecircle,whichwearetakingtobe1,sothesineofθ/2,inmathematicalnotationsin(θ/2),ishalfthechordofθ,or:

chordofθ=2sin(θ/2)

Henceanycalculationthatcanbedonewithsinescanalsobedonewithchords,thoughinmostcaseslessconveniently.

Figure9.Thechordofanangleθ.Thecircleherehasaradiusequalto1.Thesolidradiallinesmakeanangleθat thecenterof thecircle; thehorizontal line runsbetween the intersectionsof these lineswiththecircle;anditslengthisthechordofthisangle.

16.Horizons

Normallyourvisionoutdoorsisobstructedbynearbytreesorhousesorotherobstacles.Fromthetopofahill,onaclearday,wecanseemuchfarther,buttherangeofourvisionisstillrestrictedbyahorizon,beyondwhichlinesofsightareblockedbytheEarthitself.TheArabastronomeral-BirunidescribedaclevermethodforusingthisfamiliarphenomenontomeasuretheradiusoftheEarth,withoutneedingtoknowanydistancesotherthantheheightofthemountain.

Figure10.Al-Biruni’suseofhorizonstomeasurethesizeoftheEarth.Oisanobserveronahillofheighth;Histhehorizonasseenbythisobserver;thelinefromHtoOistangenttotheEarth’ssurfaceatH,andthereforemakesarightanglewiththelinefromthecenterCoftheEarthtoH.

AnobserverOontopofahillcanseeouttoapointHontheEarth’ssurface,wherethelineofsightistangenttothesurface.(SeeFigure10.)ThislineofsightisatrightanglestothelinejoiningHtotheEarth’scenterC,sotriangleOCHisarighttriangle.Thelineofsightisnotinthehorizontaldirection,butbelowthehorizontaldirectionbysomeangleθ,whichissmallbecausetheEarthis largeandthehorizonfaraway.Theanglebetweenthelineofsightandtheverticaldirectionatthehillisthen90°–θ,sosincethesumoftheanglesofanytrianglemustbe180°,theacuteangleofthetriangleatthecenteroftheEarthis180°–90°–(90°–θ)=θ.ThesideofthetriangleadjacenttothisangleisthelinefromCtoH,whoselengthistheEarth’sradiusr,whilethehypotenuseofthetriangleisthedistancefromCtoO,whichisr+h,wherehistheheightofthemountain.Accordingtothegeneraldefinitionofthecosine,thecosineofanyangleistheratiooftheadjacentsidetothehypotenuse,whichheregives

Tosolvethisequationforr,notethatitsreciprocalgives1+h/r=1/cosθ,sobysubtracting1fromthisequationandthentakingthereciprocalagainwehave

Forinstance,onamountaininIndiaal-Birunifoundθ=34’,forwhichcosθ=0.999951092and1/cosθ–1=0.0000489.Hence

Al-Birunireportedthattheheightofthismountainis652.055cubits(aprecisionmuchgreaterthanhecould possibly have achieved),which then actually gives r = 13.3million cubits,while his reportedresultwas12.8millioncubits.Idon’tknowthesourceofal-Biruni’serror.

17.GeometricProofoftheMeanSpeedTheorem

Supposewemakeagraphofspeedversustimeduringuniformacceleration,withspeedontheverticalaxisand timeon thehorizontalaxis.Thegraphwillbeastraight line, risingfromzerospeedatzerotimetothefinalspeedatthefinaltime.Ineachtinyintervaloftime,thedistancetraveledistheproductofthespeedatthattime(thisspeedchangesbyanegligibleamountduringthatintervaliftheintervalisshort enough) times the time interval. That is, the distance traveled is equal to the area of a thinrectangle,whoseheightistheheightofthegraphatthattimeandwhosewidthisthetinytimeinterval.(SeeFigure11a.)Wecanfilluptheareaunderthegraph,fromtheinitialtothefinaltime,bysuchthinrectangles,andthetotaldistancetraveledwillthenbethetotalareaofalltheserectangles—thatis,theareaunderthegraph.(SeeFigure11b.)

Figure11.Geometricproofofthemeanspeedtheorem.Theslantedlineisthegraphofspeedversustime for abodyuniformlyaccelerated from rest. (a)Thewidthof the small rectangle is a short timeinterval; itsarea iscloseto thedistancetraveledin that interval. (b)Timeduringaperiodofuniformacceleration,brokenintoshortintervals;asthenumberofrectanglesisincreasedthesumoftheareasoftherectanglesbecomesarbitrarilyclosetotheareaundertheslantedline.(c)Theareaundertheslantedlineishalftheproductoftheelapsedtimeandthefinalspeed.

Of course, however thinwemake the rectangles, it is only an approximation to say that the areaunder thegraphequals the totalareaof therectangles.Butwecanmake therectanglesas thinaswelike,andinthiswaymaketheapproximationasgoodaswelike.Byimaginingthelimitofaninfinitenumberofinfinitelythinrectangles,wecanconcludethatthedistancetraveledequalstheareaunderthegraphofspeedversustime.So far, this argumentwould be unchanged if the accelerationwas not uniform, inwhich case the

graphwouldnotbe a straight line. In fact,wehave just deduceda fundamental principleof integralcalculus:thatifwemakeagraphoftherateofchangeofanyquantityversustime,thenthechangeinthis quantity in any time interval is the area under the curve.But for a uniformly increasing rate ofchange,asinuniformacceleration,thisareaisgivenbyasimplegeometrictheorem.Thetheoremsaysthattheareaofarighttriangleishalftheproductofthetwosidesadjacenttothe

rightangle—that is, the twosidesother than thehypotenuse.This follows immediately from the factthatwecanputtwoofthesetrianglestogethertoformarectangle,whoseareaistheproductofitstwosides.(SeeFigure11c.)Inourcase,thetwosidesadjacenttotherightanglearethefinalspeedandthetotaltimeelapsed.Thedistancetraveledistheareaofarighttrianglewiththosedimensions,orhalftheproductof thefinalspeedandthetotal timeelapsed.Butsincethespeedis increasingfromzeroataconstantrate,itsmeanvalueishalfitsfinalvalue,sothedistancetraveledisthemeanspeedmultipliedbythetimeelapsed.Thisisthemeanspeedtheorem.

18.Ellipses

Anellipseisacertainkindofclosedcurveonaflatsurface.Thereareatleastthreedifferentwaysofgivingaprecisedescriptionofthiscurve.

Figure12.Theelementsofanellipse.Themarkedpointswithintheellipseareitstwofoci;aandbarehalf the longer and shorter axes of the ellipse; and the distance fromeach focus to the center of theellipse isea.The sumof the lengths r+andr– of the two lines from the foci to a pointP equals 2awhereverPisontheellipse.Theellipseshownherehasellipticitye 0.8.

FirstDefinition

Anellipseisthesetofpointsinaplanesatisfyingtheequation

wherexisthedistancefromthecenteroftheellipseofanypointontheellipsealongoneaxis,yisthedistanceofthesamepointfromthecenteralongaperpendicularaxis,andaandbarepositivenumbersthatcharacterize thesizeandshapeof theellipse,conventionallydefinedsothata≥b.Forclarityofdescription it is convenient to think of the x-axis as horizontal and the y-axis as vertical, though ofcourse theycan lie alongany twoperpendiculardirections. It follows fromEq. (1) that thedistance

ofanypointontheellipsefromthecenteratx=0,y=0satisfies

soeverywhereontheellipse

Notethatwheretheellipseintersectsthehorizontalaxiswehavey=0,sox2=a2,andthereforex=±a;thusEq.(1)describesanellipsewhoselongdiameterrunsfrom–ato+aalongthehorizontaldirection.Also,wheretheellipseintersectstheverticalaxiswehavex=0,soy2=b2,andthereforey=±b,andEq.(1)thereforedescribesanellipsewhoseshortdiameterrunsalongtheverticaldirection,from–bto+b.(SeeFigure12.)Theparameteraiscalledthe“semimajoraxis”oftheellipse.Itisconventionaltodefinetheeccentricityofanellipseas

Theeccentricityisingeneralbetween0and1.Anellipsewithe=0isacircle,withradiusa=b.Anellipsewithe=1issoflattenedthatitjustconsistsofasegmentofthehorizontalaxis,withy=0.

SecondDefinition

Anotherclassicdefinitionofanellipseisthatitisthesetofpointsinaplaneforwhichthesumofthedistances to twofixedpoints(thefociof theellipse) isaconstant.For theellipsedefinedbyEq. (1),thesetwopointsareatx=±ea,y=0,wheree is theeccentricityasdefinedinEq.(3).Thedistancesfromthesetwopointstoapointontheellipse,withxandysatisfyingEq.(1),are

sotheirsumisindeedconstant:

Thiscanberegardedasageneralizationoftheclassicdefinitionofacircle,asthesetofpointsthatareallthesamedistancefromasinglepoint.Sincethereiscompletesymmetrybetweenthetwofocioftheellipse,theaveragedistances and

ofpointsontheellipse(witheverylinesegmentofagivenlengthontheellipsegivenequalweightintheaverage)fromthetwofocimustbeequal: = ,andthereforeEq.(5)gives

Thisisalsotheaverageofthegreatestandleastdistancesofpointsontheellipsefromeitherfocus:

ThirdDefinition

TheoriginaldefinitionofanellipsebyApolloniusofPergaisthatitisaconicsection,theintersectionof a conewith a plane at a tilt to the axis of the cone. Inmodern terms, a conewith its axis in theverticaldirection is the setofpoints in threedimensions satisfying thecondition that the radiiof thecircularcrosssectionsoftheconeareproportionaltodistanceintheverticaldirection:

whereuandymeasuredistancealongthetwoperpendicularhorizontaldirections,zmeasuresdistanceintheverticaldirection,andα(alpha)isapositivenumberthatdeterminestheshapeofthecone.(Ourreasonforusinguinsteadofxforoneofthehorizontalcoordinateswillbecomeclearsoon.)Theapexofthiscone,whereu=y=0,isatz=0.Aplanethatcutstheconeatanobliqueanglecanbedefinedasthesetofpointssatisfyingtheconditionthat

whereβ(beta)andγ(gamma)aretwomorenumbers,whichrespectivelyspecifythetiltandheightoftheplane.(Wearedefiningthecoordinatessothattheplaneisparalleltothey-axis.)CombiningEq.(9)withthesquareofEq.(8)gives

u2+y2=α2(βu+γ)2

orequivalently

ThisisthesameasthedefiningEq.(1)ifweidentify

Notethatthisgivese=αβ,sotheeccentricitydependsontheshapeoftheconeandonthetiltoftheplanecuttingthecone,butnotontheplane’sheight.

19.ElongationsandOrbitsoftheInnerPlanets

Oneof thegreatachievementsofCopernicuswastoworkoutdefinitevaluesfor therelativesizesofplanetaryorbits.AparticularlysimpleexampleisthecalculationoftheradiioftheorbitsoftheinnerplanetsfromthemaximumapparentdistanceoftheseplanetsfromtheSun.Considertheorbitofoneoftheinnerplanets,MercuryorVenus,intheapproximationthatitandthe

Earth’sorbitarebothcircleswiththeSunatthecenter.Atwhatiscalled“maximumelongation,” theplanetisseenatthegreatestangulardistanceθmax(thetamax)fromtheSun.Atthistime,thelinefromtheEarthtotheplanetistangenttotheplanet’sorbit,sotheanglebetweenthislineandthelinebetweentheSunandtheplanetisarightangle.ThesetwolinesandthelinefromtheSuntotheEarththusformarighttriangle.(SeeFigure13.)ThehypotenuseofthistriangleisthelinebetweentheEarthandtheSun,sotheratioofthedistancerPbetweentheplanetandtheSuntothedistancerEoftheEarthfromtheSun is thesineofθmax.Here isa tableof theanglesofmaximumelongation, theirsines,and theactualorbitalradiirPofMercuryandVenus,inunitsoftheradiusrEoftheEarth’sorbit:

Figure13.ThepositionsoftheEarthandaninnerplanet(MercuryorVenus)whentheplanetis

atitsmaximumapparentdistancefromtheSun.ThecirclesaretheorbitsoftheEarthandplanet.

ThesmalldiscrepanciesbetweenthesineofθmaxandtheobservedratiosrP/rEoftheorbitalradiioftheinnerplanetsandtheEarthareduetothedepartureoftheseorbitsfromperfectcircleswiththeSunatthecenter,andtothefactthattheorbitsarenotinpreciselythesameplane.

20.DiurnalParallax

Considera“newstar”oranotherobjectthateitherisatrestwithrespecttothefixedstarsormovesverylittlerelativetothestarsinthecourseofaday.SupposethatitismuchclosertoEarththanthestars.OnecanassumethattheEarthrotatesonceadayonitsaxisfromeasttowest,orthatthisobjectandthestarsrevolvearoundtheEarthonceadayfromwesttoeast;ineithercase,becauseweseetheobjectindifferent directions at different times of night, its position will seem to shift during every eveningrelative to thestars.This iscalled the“diurnalparallax”of theobject.Ameasurementof thediurnalparallaxallowsthedeterminationofthedistanceoftheobject,orifitisfoundthatthediurnalparallaxistoosmalltobemeasured,thisgivesalowerlimittothedistance.Tocalculate theamountof thisangularshift,consider theobject’sapparentposition relative to the

starsseenfromafixedobservatoryonEarth,whentheobjectjustrisesabovethehorizon,andwhenitishighestinthesky.Tofacilitatethecalculation,wewillconsiderthecasethatissimplestgeometrically:theobservatoryisontheequator,andtheobjectisinthesameplaneastheequator.Ofcourse,thisdoesnotaccuratelygivethediurnalparallaxofthenewstarobservedbyTycho,butitwillindicatetheorderofmagnitudeofthatparallax.Thelinetotheobjectfromthisobservatorywhentheobjectjustrisesabovethehorizonistangentto

theEarth’ssurface,sotheanglebetweenthislineandthelinefromtheobservatorytothecenteroftheEarthisarightangle.Thesetwolines,togetherwiththelinefromtheobjecttothecenteroftheEarth,thusformarighttriangle.(SeeFigure14.)Angleθ(theta)ofthistriangleattheobjecthasasineequaltotheratiooftheoppositeside,theradiusrEoftheEarth,tothehypotenuse,thedistancedoftheobjectfromthecenteroftheEarth.Asshowninthefigure,thisangleisalsotheapparentshiftofthepositionoftheobjectrelativetothestarsduringthetimebetweenwhenitrisesabovethehorizonandwhenitishighest in thesky.The total shift in itsposition fromwhen it risesabove thehorizon towhen it setsbelowthehorizonis2θ.Forinstance,ifwetaketheobjecttobeatthedistanceoftheMoon,thend 250,000miles,whilerE4,000miles,sosinθ 4/250,andthereforeθ 0.9°,andthediurnalparallaxis1.8°.FromatypicalspotonEarth,suchasHven,toanobjectatatypicallocationintheskylikethenewstarof1572,thediurnalparallaxissmaller,butstillofthesameorderofmagnitude,intheneighborhoodof1°.Thisismore than large enough for it to have been detected by a naked-eye astronomer as expert as TychoBrahe,butTychocouldnotdetectanydiurnalparallax,sohewasabletoconcludethatthenewstarof1572 is farther than theMoon. On the other hand, there was no difficulty inmeasuring the diurnalparallaxoftheMoonitself,andinthiswayfindingthedistanceoftheMoonfromtheEarth.

Figure14.UseofdiurnalparallaxtomeasurethedistancedfromtheEarthtosomeobject.HeretheviewisfromapointabovetheEarth’snorthpole.Forsimplicity,theobserverissupposedtobeontheequator,andtheobjectisinthesameplaneastheequator.Thetwolinesseparatedbyanangleθarethe lines of sight to the object when it just arises above the horizon and six hours later, when it isdirectlyabovetheobserver.

21.TheEqual-AreaRule,andtheEquant

AccordingtoKepler’sfirstlaw,theplanets,includingtheEarth,eachgoaroundtheSunonanellipticalorbit,buttheSunisnotatthecenteroftheellipse;itisatanoff-centerpointonthemajoraxis,oneofthetwofocioftheellipse.(SeeTechnicalNote18.)Theeccentricityeoftheellipseisdefinedsothatthedistanceofeachfocusfromthecenteroftheellipseisea,wherea ishalf the lengthof themajoraxisoftheellipse.Also,accordingtoKepler’ssecondlaw,thespeedofeachplanetinitsorbit isnotconstant,butvaries in suchaway that the line from theSun to theplanet sweepsoutequalareas inequaltimes.Thereisadifferentapproximatewayofstatingthesecondlaw,closelyrelatedtotheoldideaofthe

equantusedinPtolemaicastronomy.InsteadofconsideringthelinefromtheSuntotheplanet,considertheline to theplanetfromtheother focusof theellipse, theemptyfocus.Theeccentricityeofsomeplanetaryorbitsisnotnegligible,bute2isverysmallforallplanets.(ThemosteccentricorbitisthatofMercury,forwhiche=0.206ande2=0.042;fortheEarth,e2=0.00028.)Soitisagoodapproximationincalculatingthemotionsoftheplanetstokeeponlytermsthatareindependentoftheeccentricityeorproportionaltoe,neglectingalltermsproportionaltoe2orhigherpowersofe. In thisapproximation,Kepler’s second law is equivalent to the statement that the line from the empty focus to the planetsweepsoutequalanglesinequaltimes.Thatis,thelinebetweentheemptyfocusoftheellipseandtheplanetrotatesaroundthatfocusataconstantrate.Specifically,weshowbelowthatif istherateatwhichareaissweptoutbythelinefromtheSunto

theplanet,and (dottedphi)istherateofchangeoftheangleϕbetweenthemajoraxisoftheellipseandthelinefromtheemptyfocustotheplanet,then

whereO(e2)denotes termsproportional toe2 orhigherpowersofe,andR is a numberwhose valuedependsontheunitsweusetomeasureangles.Ifwemeasureanglesindegrees,thenR=360°/2π=57.293...°,anangleknownasa“radian.”Orwecanmeasureanglesinradians,inwhichcasewetakeR=1.Kepler’ssecondlawtellsusthatinagiventimeintervaltheareasweptoutbythelinefromtheSun to the planet is always the same; this means that is constant, so ϕ is constant, up to terms

proportional toe2.So toagoodapproximation inagiven time interval theangle sweptout from theemptyfocusoftheellipsetotheplanetisalsoalwaysthesame.Now,inthetheorydescribedbyPtolemy,thecenterofeachplanet’sepicyclegoesaroundtheEarth

onacircularorbit, thedeferent,but theEarth isnotat thecenterof thedeferent. Instead, theorbit iseccentric—theEarthisatapointasmalldistancefromthecenter.Furthermore,thevelocityatwhichthecenteroftheepicyclegoesaroundtheEarthisnotconstant,andtherateatwhichthelinefromtheEarthtothiscenterswivelsaroundisnotconstant.Inordertoaccountcorrectlyfortheapparentmotionoftheplanets,thedeviceoftheequantwasintroduced.ThisisapointontheothersideofthecenterofthedeferentfromtheEarth,andatanequaldistancefromthecenter.Thelinefromtheequant(ratherthan from the Earth) to the center of the epicyclewas supposed to sweep out equal angles in equaltimes.Itwillnotescapethereader’snoticethatthisisverysimilartowhathappensaccordingtoKepler’s

laws.Ofcourse,therolesoftheSunandEartharereversedinPtolemaicandCopernicanastronomy,buttheemptyfocusof theellipse in the theoryofKeplerplays thesameroleas theequant inPtolemaicastronomy, and Kepler’s second law explains why the introduction of the equant worked well inexplainingtheapparentmotionoftheplanets.Forsomereason,althoughPtolemyintroducedaneccentrictodescribethemotionoftheSunaround

the Earth, he did not use an equant in this case. With this final equant included (and with someadditionalepicyclesintroducedtoaccountforthelargedepartureofMercury’sorbitfromacircle),thePtolemaictheorycouldaccountverywellfortheapparentmotionsoftheplanets.

HereistheproofofEq.(1).DefineθastheanglebetweenthemajoraxisoftheellipseandthelinefromtheSuntotheplanet,andrecallthatϕisdefinedastheanglebetweenthemajoraxisandthelinefromtheemptyfocustotheplanet.AsinTechnicalNote18,definer+andr–asthelengthsoftheselines—that is, thedistances fromtheSun to theplanetandfromtheemptyfocus to theplanet, respectively,given(accordingtothatnote)by

wherexisthehorizontalcoordinateofthepointontheellipse—thatis,itisthedistancefromthispointto a line cutting through theellipsealong itsminor axis.Thecosineof anangle (symbolizedcos) isdefinedintrigonometrybyconsideringarighttrianglewiththatangleasoneofthevertices;thecosineof the angle is the ratio of the side adjacent to that angle to the hypotenuse of the triangle. Hence,referringtoFigure15,

Figure15.Ellipticalmotionofplanets.Theorbit’sshapehereisanellipse,which(as inFigure12)hasellipticity0.8,muchlargerthantheellipticityofanyplanetaryorbitinthesolarsystem.Thelinesmarkedr+andr–gorespectivelytotheplanetfromtheSunandfromtheemptyfocusoftheellipse.

Wecansolvetheequationattheleftforx:

Wetheninsertthisresultintheformulaforcosϕ,obtaininginthiswayarelationbetweenθandϕ:

Sincebothsidesofthisequationareequalwhateverthevalueofθ,thechangeintheleft-handsidemustequal the change in the right-hand side when we make any change in θ. Suppose we make aninfinitesimalchangeδθ (deltatheta)inθ.Tocalculatethechangeinϕ,wemakeuseofaprinciple incalculus,thatwhenanyangleα(suchasθorϕ)changesbyanamountδα(deltaalpha),thechangeincosα is–(δα/R)sinα.Also,whenanyquantity f, suchas thedenominator inEq. (5),changesbyaninfinitesimalamountδf,thechangein1/fis–δf/f2.EquatingthechangesinthetwosidesofEq.(5)thusgives

Nowweneedaformulafortheratioofsinϕandsinθ.Forthispurpose,notefromFigure15thattheverticalcoordinateyofapointon theellipse isgivenbyy=r+ sinθandalsobyy=r- sinϕ, sobyeliminatingy,

UsingthisinEq.(6),wehave

Now,whatistheareasweptoutbythelinefromtheSuntotheplanetwhentheangleθischangedbyδθ?Ifwemeasureanglesindegrees,thenitistheareaofanisoscelestrianglewithtwosidesequaltor+,andthethirdsideequaltothefraction2πr+×δθ/360°ofthecircumference2πr+ofacircleofradiusr+.Thisareais

(AminussignhasbeeninsertedherebecausewewantdA tobepositivewhenϕ increases;butaswehavedefinedthem,ϕincreaseswhenθdecreases,soδϕispositivewhenδθ isnegative.)ThusEq.(8)maybewritten

TakingδAandδϕ tobetheareaandanglesweptoutinaninfinitesimaltimeintervalδt,anddividingEq.(10)byδt,wefindacorrespondingrelationbetweentheratesofsweepingoutareasandangles:

Sofar,thisisallexact.Nowlet’sconsiderhowthislookswheneisverysmall.ThenumeratorofthesecondfractioninEq.(11)is(1–ecosθ)2=1–2ecosθ+e2cos2θ, so the termsofzerothandfirstorderinthenumeratoranddenominatorofthisfractionarethesame,thedifferencebetweennumeratoranddenominatorappearingonlyintermsproportionaltoe2.Equation(11)thusimmediatelyyieldsthedesiredresult,Eq.(1).Tobealittlemoredefinite,wecankeepthetermsinEq.(11)ofordere2:

whereO(e3)denotestermsproportionaltoe3orhigherpowersofe.

22.FocalLength

Consideraverticalglasslens,withaconvexcurvedsurfaceinfrontandaplanesurfaceinback,likethe

lens that Galileo andKepler used as the front end of their telescopes. The curved surfaces that areeasiest to grind are segments of spheres, andwewill assume that the convex front of the lens is asegmentofasphereofradiusr.Wewillalsoassumethroughoutthatthelensisthin,withamaximumthicknessmuchlessthanr.Supposethatarayoflighttravelinginthehorizontaldirection,paralleltotheaxisofthelens,strikes

thelensatpointP,andthatthelinefromthecenterCofcurvature(behindthelens)toPmakesanangleθ(theta)withthecenterlineofthelens.Thelenswillbendtherayoflightsothatwhenitemergesfromthebackofthelensitmakesadifferentangleϕwiththecenterlineofthelens.TheraywillthenstrikethecenterlineofthelensatsomepointF.(SeeFigure16a.)Wearegoingtocalculatethedistancefofthispointfromthelens,andshowthatitisindependentofθ,sothatallhorizontalraysoflightstrikingthe lens reach the centerline at the same pointF. Thuswe can say that the light striking the lens isfocusedatpointF;thedistancefofthispointfromthelensisknownasthe“focallength”ofthelens.First, note that the arcon the frontof the lens from the centerline toP is a fractionθ/360°of the

wholecircumference2πrofacircleofradiusr.Ontheotherhand,thesamearcisafractionϕ/360°ofthewholecircumference2πfofacircleofradiusf.Sincethesearcsarethesame,wehave

andtherefore,cancelingfactorsof360°and2π,

Sotocalculatethefocallength,weneedtocalculatetheratioofϕtoθ.Forthispurpose,weneedtolookmorecloselyatwhathappenstothelightrayinsidethelens.(See

Figure16b.)ThelinefromthecenterofcurvatureCtothepointPwhereahorizontallightraystrikesthe lens is perpendicular to the convex spherical surface of the lens atP, so the angle between thisperpendicular and the light ray (that is, the angle of incidence) is justθ.Aswas known toClaudiusPtolemy, ifθ is small (as itwillbe fora thin lens), then theangleα (alpha)between the rayof lightinsidetheglassandtheperpendicular(thatis,theangleofrefraction)willbeproportionaltotheangleofincidence,sothat

α=θ/n

Figure16.Focallength.(a)Definitionoffocallength.Thehorizontaldashedline is theaxisof thelens.Horizontallinesmarkedwitharrowsindicateraysoflightthatenterthelensparalleltothisaxis.OnerayisshownenteringthelensatpointP,wheretheraymakesasmallangleθ toalinefromthecenterofcurvatureCthatisperpendiculartotheconvexsphericalsurfaceatP;thisrayisbentbythelenstomakeanangleϕ tothelensaxisandstrikesthisaxisatfocalpointF,atadistance f fromthelens.Thisisthefocallength.Withϕproportionaltoθ,allhorizontalraysarefocusedtothispoint.(b)Calculationoffocallength.Shownhereisasmallpartofthelens,withtheslantedhatchedsolidlineonthe left indicating a short segment of the convex surface of the lens. The solid linemarkedwith anarrowshowsthepathofarayoflightthatentersthelensatP,whereitmakesasmallangleθtothenormaltotheconvexsurface.Thisnormalisshownasaslanteddottedline,asegmentofthelinefromPtothecenterofcurvatureofthelens,whichisbeyondthebordersofthisfigure.Insidethelensthisrayisrefractedsothatitmakesanangleαwiththisnormal,andthenisrefractedagainwhenitleavesthelenssothatitmakesanangleϕwiththenormaltotheplanarbacksurfaceofthelens.Thisnormalisshownasadottedlineparalleltotheaxisofthelens.

wheren>1isaconstant,knownasthe“indexofrefraction,”thatdependsonthepropertiesofglassandthesurroundingmedium,typicallyair.(ItwasshownbyFermatthatn is thespeedoflightinairdivided by the speed of light in glass, but this information is not needed here.) The angle β (beta)betweenthelightrayinsidetheglassandthecenterlineofthelensisthen

β=θ−a=(1−1/n)θ

Thisistheanglebetweenthelightrayandthenormaltotheflatbacksurfaceofthelenswhenthelightray reaches this surface.On the other hand,when the light ray emerges from the backof the lens itmakesadifferentangleϕ(phi)tothenormaltothesurface.Therelationbetweenϕandβisthesameasifthelightweregoingintheoppositedirection,inwhichcaseϕwouldbetheangleofincidenceandβtheangleofrefraction,sothatβ=ϕ/n,andtherefore

ϕ=nβ=(n−1)θ

Soweseethatϕissimplyproportionaltoθ,andtherefore,usingourpreviousformulaforf/r,wehave

This is independentofθ, soaspromisedallhorizontal light raysentering the lensare focused to thesamepointonthecenterlineofthelens.If the radiusofcurvaturer isvery large, then thecurvatureof the frontsurfaceof the lens isvery

small,sothatthelensisnearlythesameasaflatplateofglass,withthebendingoflightonenteringthelensbeingnearlycanceledbyitsbendingonleavingthelens.Similarly,whatevertheshapeofthelens,iftheindexofrefractionniscloseto1,thenthelensbendsthelightrayverylittle.Ineithercasethefocal length isvery large,andwesay that the lens isweak.Astrong lens isone thathasamoderateradiusofcurvatureandanindexofrefractionappreciablydifferentfrom1,asforinstancealensmadeofglass,forwhichn 1.5.Asimilarresultholdsifthebacksurfaceofthelensisnotplane,butasegmentofasphereofradius

r’.Inthiscasethefocallengthis

Thisgivesthesameresultasbeforeifr’ismuchlargerthanr,inwhichcasethebacksurfaceisnearlyflat.Theconceptoffocallengthcanalsobeextendedtoconcavelenses,likethelensthatGalileousedas

theeyepieceofhistelescope.Aconcavelenscantakeraysoflightthatareconvergingandspreadthemout so that they are parallel, or even diverge. We can define the focal length of such a lens byconsideringconvergingraysoflightthataremadeparallelbythelens;thefocallengthisthedistancebehindthelensofthepointtowhichsuchrayswouldconvergeifnotmadeparallelbythelens.Thoughitsmeaningisdifferent,thefocallengthofaconcavelensisgivenbyaformulaliketheonewehavederivedforaconvexlens.

23.Telescopes

AswesawinTechnicalNote22,athinconvexlenswillfocusraysoflightthatstrikeitparalleltoitscentralaxistoapointFonthisaxis,atadistancebehindthelensknownasthefocallengthfofthelens.Parallel rays of light that strike the lens at a small angle γ (gamma) to the central axiswill also befocusedbythelens,buttoapointthatisalittleoffthecentralaxis.Toseehowfaroff,wecanimaginerotatingthedrawingoftheraypathinFigure16aaroundthelensbyangleγ.Thedistancedofthefocalpointfromthecentralaxisofthelenswillthenbethesamefractionofthecircumferenceofacircleofradiusfthatγisof360°:

andtherefore

(Thisworksonlyforthinlenses;otherwisedalsodependsontheangleθintroducedinTechnicalNote22.)IftheraysoflightfromsomedistantobjectstrikethelensinarangeΔγ(deltagamma)ofangles,theywillbefocusedtoastripofheightΔd,givenby

(Asusual,thisformulaissimplerifΔγismeasuredinradians,equalto360°/2π,ratherthanindegrees;inthiscaseitreadssimplyΔd=fΔγ.)Thisstripoffocusedlightisknownasa“virtual image.”(SeeFigure17a).Wecannotsee thevirtual image justbypeeringat it,becauseafter reaching this image theraysof

lightdivergeagain.Tobefocusedtoapointontheretinaofarelaxedhumaneye,raysof lightmustenterthelensoftheeyeinmoreorlessparalleldirections.Kepler’stelescopeincludedasecondconvexlens,knownastheeyepiece,tofocusthedivergingraysoflightfromthevirtualimagesothattheyleftthe telescope alongparallel directions.By repeating the above analysis, butwith thedirectionof thelight rays reversed, we see that for the rays of light from a point on the light source to leave thetelescope on parallel directions, the eyepiecemust be placed at a distance f’ from thevirtual image,wheref’isthefocallengthoftheeyepiece.(SeeFigure17b.)Thatis,thelengthLofthetelescopemustbethesumofthefocallengths

L=f+fʹ

The rangeΔγ' of directions of the light rays from different points on the source entering the eye isrelatedtothesizeofthevirtualimageby

Figure17.Telescopes.(a)Formationofavirtual image.The twosolid linesmarkedwitharrowsareraysoflightthatenterthelensonlinesseparatedbyasmallangleΔγ.Theselines(andothersparalleltothem)arefocusedtopointsatadistanceffromthelens,withaverticalseparationΔdproportionaltoΔγ.(b)ThelensesinKepler’stelescope.Thelinesmarkedwitharrowsindicatethepathsoflightraysthatenteraweakconvexlensfromadistantobject,onessentiallyparalleldirections;arefocusedbythelenstoapointatadistanceffromthelens;divergefromthispoint;andarethenbentbyastrongconvexlenssothattheyentertheeyeonparalleldirections.

Theapparentsizeofanyobjectisproportionaltotheanglesubtendedbyraysoflightfromtheobject,sothemagnificationproducedbythetelescopeistheratioofthisanglewhentheraysentertheeyetotheangletheywouldhavespannediftherewerenotelescope:

BytakingtheratioofthetwoformulaswehavederivedforsizeΔdofthevirtualimage,weseethatthemagnificationis

Togetasignificantdegreeofmagnification,weneedthelensatthefrontofthetelescopetobemuchweakerthantheeyepiece,withf>>f ’.Thisisnotsoeasy.AccordingtotheformulaforthefocallengthgiveninTechnicalNote22,tohave

a strong glass eyepiece with short focal length f', it is necessary for it to have a small radius ofcurvature,whichmeans either that itmust be very small, or that itmust not be thin (that is,with athicknessmuchlessthantheradiusofcurvature),inwhichcaseitdoesnotfocusthelightwell.Wecaninsteadmakethelensatthefrontweak,withlargefocallengthf,butinthiscasethelengthL=f+f’ f

ofthetelescopemustbelarge,whichisawkward.IttooksometimeforGalileotorefinehistelescopetogiveitamagnificationsufficientforastronomicalpurposes.Galileousedasomewhatdifferentdesigninhistelescope,withaconcaveeyepiece.Asmentionedin

TechnicalNote22,ifaconcavelensisproperlyplaced,convergingraysoflightthatenteritwillleaveonparalleldirections;thefocallengthisthedistancebehindthelensatwhichtherayswouldconvergeifnotforthelens.InGalileo’stelescopetherewasaweakconvexlensinfrontwithfocallengthf,withastrongconcavelensoffocallengthf’behindit,atadistancef’infrontoftheplacewheretherewouldbeavirtualimageifnotfortheconcavelens.Themagnificationofsuchatelescopeisagaintheratiof/f ’,butitslengthisonlyf–f’insteadoff+f ’.

24.MountainsontheMoon

ThebrightanddarksidesoftheMoonaredividedbyalineknownasthe“terminator,”wheretheSun’sraysarejusttangenttotheMoon’ssurface.WhenGalileoturnedhistelescopeontheMoonhenoticedbrightspotsonthedarksideoftheMoonneartheterminator,andinterpretedthemasreflectionsfrommountainshighenoughtocatchtheSun’srayscomingfromtheothersideoftheterminator.Hecouldinfer theheightof thesemountainsbyageometricalconstructionsimilar to thatusedbyal-Biruni tomeasure the size of the Earth. Draw a triangle whose vertices are the center C of the Moon, amountaintopMonthedarksideoftheMoonthatjustcatchesarayofsunlight,andthespotTontheterminatorwherethisraygrazesthesurfaceoftheMoon.(SeeFigure18.)Thisisarighttriangle;lineTMistangenttotheMoon’ssurfaceatT,sothislinemustbeperpendiculartolineCT.ThelengthofCTis just the radius r of themoon,while the length ofTM is the distanced of themountain from theterminator.Ifthemountainhasheighth,thenthelengthofCM(thehypotenuseofthetriangle)isr+h.AccordingtothePythagoreantheorem,wethenhave

(r+h)2=r2+d2

andtherefore

d2=(r+h)2−r2=2rh+h2

SincetheheightofanymountainontheMoonismuchlessthanthesizeoftheMoon,wecanneglecth2comparedwith2rh.Dividingbothsidesoftheequationby2r2thengives

SobymeasuringtheratiooftheapparentdistanceofamountaintopfromtheterminatortotheapparentradiusoftheMoon,Galileocouldfindtheratioofthemountain’sheighttotheMoon’sradius.

Figure 18. Galileo’smeasurement of the height ofmountains on theMoon. The horizontal linemarkedwith an arrow indicates a ray of light that grazes theMoon at the terminatorTmarking theboundarybetweentheMoon’sbrightanddarksides,andthenstrikesthetopMofamountainofheighthatadistancedfromtheterminator.GalileoinSideriusNunciusreportedthathesometimessawbrightspotsonthedarksideoftheMoon

atanapparentdistancefromtheterminatorgreaterthan1/20theapparentdiameteroftheMoon,soforthesemountains d/r > 1/10, and therefore according to the above formula h/r > (1/10)2 /2 = 1/200.Galileoestimated the radiusof theMoon tobe1,000miles,*so thesemountainswouldbeat least5mileshigh.(Forreasonsthatarenotclear,Galileogaveafigureof4miles,butsincehewastryingonlytosetalowerboundonthemountainheight,perhapshewasjustbeingconservative.)Galileothoughtthat thiswas higher than anymountain on theEarth, but nowwe know that there aremountains onEarththatarealmost6mileshigh,soGalileo’sobservationsindicatedthattheheightsofmountainsontheMoonarenotverydifferentfromtheheightsofterrestrialmountains.

25.GravitationalAcceleration

Galileoshowedthatafallingbodyundergoesuniformacceleration—thatis,itsspeedincreasesbythesameamountineachequalintervaloftime.Inmodernterms,abodythatfallsfromrestwillaftertimethaveavelocityvgivenbyaquantityproportionaltot:

v=gt

wheregisaconstantcharacterizingthegravitationalfieldatthesurfaceoftheEarth.AlthoughgvariessomewhatfromplacetoplaceontheEarth’ssurface,itisneververydifferentfrom32feet/secondpersecond,or9.8meters/secondpersecond.Accordingto themeanspeedtheorem, thedistance thatsuchabodywill fall fromrest in time t is

vmeant,wherevmeanistheaverageofgtandzero;inotherwords,vmean=gt/2.Hencethedistancefallenis

Inparticular,inthefirstsecondthebodyfallsadistanceg(1second)2/2=16feet.Thetimerequiredto

falldistancedisingeneral

Thereisanother,moremodernwayoflookingatthisresult.Thefallingbodyhasanenergyequaltothesumofakineticenergytermandapotentialenergyterm.Thekineticenergyis

wheremisthebody’smass.Thepotentialenergyismgtimestheheight(measuredfromanyarbitraryaltitude),soifthebodyisdroppedfromrestataninitialheighth0andfallsdistanced,then

Epotential=mgh=mg(h0−d)

Hencewithd=gt2/2,thetotalenergyisaconstant:

E=Ekinetic+Epotential=mgh0

Wecanturnthisaround,andderivetherelationbetweenvelocityanddistancefallenbyassuming theconservationofenergy.IfwesetEequaltothevaluemgh0thatithasatt=0,whenv=0andh=h0,thentheconservationofenergygivesatalltimes

fromwhichitfollowsthatv2/2=gd.Sincevistherateofincreaseofd,thisisadifferentialequationthatdeterminestherelationbetweendandt.Ofcourse,weknowthesolutionofthisequation:itisd=gt2/2,forwhichv=gt.Sobyusingtheconservationofenergywecangettheseresultswithoutknowinginadvancethattheaccelerationisuniform.This is an elementary example of the conservation of energy,whichmakes the concept of energy

useful in awidevarietyof contexts. Inparticular, theconservationof energy shows the relevanceofGalileo’sexperimentswithballsrollingdowninclinedplanestotheproblemoffreefall,thoughthisisnotanargumentusedbyGalileo.Foraballofmassmrollingdownaplane,thekineticenergyismv2/2,where v is now the velocity along the plane, and the potential energy ismgh, whereh is again theheight.Inadditionthereisanenergyofrotationoftheball,whichtakestheform

wherer is theradiusof theball,ν (nu) is thenumberofcomplete turnsof theballpersecond,andζ(zeta) is a number that depends on the shape and mass-distribution of the ball. In the case that isprobablyrelevanttoGalileo’sexperiments,thatofasoliduniformball,ζhasthevalueζ=2/5.(Iftheballwerehollow,wewouldhaveζ =2/3.)Now,when theballmakesone complete turn it travels adistanceequaltoitscircumference2πr,sointimetwhenitmakesνtturnsittravelsdistanced=2πrνt,andthereforeitsvelocityisd/t=2πνr.Usingthisintheformulafortheenergyofrotation,weseethat

Dividingbymand1+ζ,theconservationofenergythereforerequiresthat

This is the same relation between speed and distance fallend =h0 –h that holds for a body fallingfreely,except thatghasbeenreplacedwithg/(1+ζ).Asidefromthischange, thedependenceof thevelocityoftheballrollingdowntheinclinedplaneontheverticaldistancetraveledisthesameasthatforabodyinfreefall.Hencethestudyofballsrollingdowninclinedplanescouldbeusedtoverifythatfreely falling bodies experience uniform acceleration; but unless the factor 1/(1 + ζ) is taken intoaccount,itcouldnotbeusedtomeasuretheacceleration.Byacomplicatedargument,HuygenswasabletoshowthatthetimeittakesapendulumoflengthL

toswingthroughasmallanglefromonesidetotheotheris

Thisequalsπtimesthetimerequiredforabodytofalldistanced=L/2,theresultstatedbyHuygens.

26.ParabolicTrajectories

Supposeaprojectile is shothorizontallywithspeedv.Neglectingair resistance, itwill continuewiththishorizontalcomponentofvelocity,butitwillacceleratedownward.Henceaftertimet itwillhavemovedahorizontaldistancex=vtandadownwarddistancezproportional to thesquareof the time,conventionally written as z = gt2/2, with g = 32 feet/second per second, a constant measured afterGalileo’sdeathbyHuygens.Witht=x/v,itfollowsthat

z=gx2/2v2

Thisequation,givingonecoordinateasproportionaltothesquareoftheother,definesaparabola.Note that if the projectile is fired from a gun at height h above the ground, then the horizontal

distancextraveledwhentheprojectilehasfallendistancez=handreachedthegroundis .Evenwithout knowingv org,Galileo could have verified that the path of the projectile is a parabola, bymeasuringthedistancetraveleddforvariousheightsfallenh,andcheckingthatdisproportionaltothesquarerootofh.ItisnotclearwhetherGalileoeverdidthis,butthereisevidencethatin1608hedidacloselyrelatedexperiment,mentionedbrieflyinChapter12.AballisallowedtorolldownaninclinedplanefromvariousinitialheightsH,thenrollsalongthehorizontaltabletoponwhichtheinclinedplanesits,andfinallyshootsoffintotheairfromthetableedge.AsshowninTechnicalNote25,thevelocityoftheballatthebottomoftheinclinedplaneis

whereasusualg=32feet/secondpersecond,andζ(zeta)istheratioofrotationaltokineticenergyofthe ball, a number depending on the distribution ofmasswithin the rolling ball. For a solid ball ofuniformdensity,ζ=2/5.Thisisalsothevelocityoftheballwhenitshootsoffhorizontallyintospacefromtheedgeofthetabletop,sothehorizontaldistancethattheballtravelsbythetimeitfallsheighthwillbe

Galileo did not mention the correction for rotational motion represented by ζ, but he may havesuspected that somesuchcorrectioncould reduce thehorizontaldistance traveled,because insteadofcomparingthisdistancewiththevalued= expectedintheabsenceofζ,heonlycheckedthatforfixedtableheighth,distancedwasindeedproportionalto ,toanaccuracyofafewpercent.Foronereasonoranother,Galileoneverpublishedtheresultofthisexperiment.

Formanypurposesofastronomyandmathematics,itisconvenienttodefineaparabolaasthelimitingcaseofanellipse,whenonefocusmovesveryfarfromtheother.Theequationforanellipsewithmajoraxis2aandminoraxis2bisgiveninTechnicalNote18as:

inwhichforconveniencelateronwehavereplacedthecoordinatesxandyusedinNote18withz–z0andx,withz0aconstantthatcanbechosenaswelike.Thecenterofthisellipseisatz=z0andx=0.AswesawinNote18,thereisafocusatz–z0=–ae,x=0,whereeistheeccentricity,withe2≡1–b2/a2,andthepointofclosestapproachofthecurvetothisfocusisatz–z0=–aandx=0.Itwillbeconvenienttogivethispointofclosestapproachthecoordinatesz=0andx=0bychoosingz0=aandinwhichcasethenearbyfocusisatz=z0–ea=(1–e)a.Wewant to letaandbbecomeinfinitelylarge,sothattheotherfocusgoestoinfinityandthecurvehasnomaximumxcoordinate,butwewanttokeepthedistance(1–e)aofclosestapproachtothenearerfocusfinite,soweset

1−e=ℓ/a

withℓheldfixedasagoes to infinity.Sinceeapproachesunity in this limit, thesemiminoraxisb isgivenby

b2=a2(1−e2)=a2(1−e)(1+e)→2a2(1−e)=2ℓa

Usingz0=aandthisformulaforb2,theequationfortheellipsebecomes

The terma2/a2on the leftcancels the1on theright.Multiplying theremainingequationwitha thengives

Foramuchlargerthanx,y,orℓ,thefirsttermmaybedropped,sothisequationbecomes

Thisisthesameastheequationwederivedforthemotionofaprojectilefiredhorizontally,providedwetake

sothefocusFoftheparabolaisatdistanceℓ=v2/2gbelowtheinitialpositionoftheprojectile.(SeeFigure19.)Parabolascan,likeellipses,beregardedasconicsections,butforparabolastheplaneintersectingthe

coneisparalleltothecone’ssurface.Takingtheequationofaconecenteredonthe =α(z+z0),andtheequationofaplaneparalleltotheconeassimplyy=α(z–z0),withz0arbitrary,theintersectionoftheconeandtheplanesatisfies

Figure 19.Theparabolic path of aprojectile that is fired fromahill in ahorizontal direction.PointFisthefocusofthisparabola.

Aftercancelingthetermsα2z2andα2z20,thisreads

whichis thesameasourpreviousresult,providedwetakez0=ℓ/α2.Note thataparabolaofagivenshapecanbeobtainedfromanycone,withanyvalueof theangularparameterα (alpha),because theshapeofanyparabola(asopposedtoitslocationandorientation)isentirelydeterminedbyaparameterℓwith the units of length;we do not need to know separately any unit-free parameter likeα or theeccentricityofanellipse.

27.TennisBallDerivationoftheLawofRefraction

Descartesattemptedaderivationofthelawofrefraction,basedontheassumptionthatarayoflightisbentinpassingfromonemediumtoanotherinthesamewaythatthetrajectoryofatennisballisbentinpenetratinga thinfabric.Supposea tennisballwithspeedvA strikesa thinfabricscreenobliquely. Itwill losesomespeed,so thatafterpenetrating thescreen itsspeedwillbevB<vA,butwewouldnotexpecttheball’spassagethroughthescreentomakeanychangeinthecomponentoftheball’svelocityalong the screen.We can draw a right triangle whose sides are the components of the ball’s initialvelocityperpendiculartothescreenandparalleltothescreen,andwhosehypotenuseisvA.Iftheball’soriginal trajectory makes an angle i to the perpendicular to the screen, then the component of itsvelocity along the direction parallel to the screen is vA sin i. (See Figure 20.) Likewise, if afterpenetratingthescreentheball’strajectorymakesananglerwiththeperpendiculartothescreen,thenthe component of its velocity along the direction parallel to the screen is vB sin r. UsingDescartes’assumptionthatthepassageoftheballthroughthescreencouldchangeonlythecomponentofvelocityperpendiculartotheinterface,nottheparallelcomponent,wehave

vAsini=vBsinr

andtherefore

wherenisthequantity:

Figure20.Tennisballvelocities.Thehorizontal linemarksascreenpenetratedbyatennisballwithinitial speed vA and final speed vB.The solid linesmarkedwith arrows indicate themagnitude anddirectionofthevelocitiesoftheballbeforeandafteritpenetratesthescreen.Thisfigureisdrawnwiththeball’spathbenttowardtheperpendiculartothescreen,asisthecaseforlightraysenteringadensermedium.Itshowsthatinthiscasepassageoftheballthroughthescreengreatlyreducesthecomponentofitsvelocityalongthescreen,contrarytotheassumptionofDescartes.

Equation(1)isknownasSnell’slaw,anditisthecorrectlawofrefractionforlight.Unfortunately,theanalogybetweenlightandtennisballsbreaksdownwhenwecometoEq.(2)forn.Sincefortennisballs,vBislessthanvA,Eq.(2)givesn<1,whilewhenlightpassesfromairtoglassorwater,wehaven>1.Notonlythat;thereisnoreasontosupposethatfortennisballsvB/vAisactuallyindependentofanglesiandr,sothatEq.(1)isnotusefulasitstands.AsshownbyFermat,whenlightpassesfromamediumwhereitsspeedisvAintoonewhereitsspeed

isvB,theindexofrefractionnactuallyequalsvA/vB,notvB/vA.Descartesdidnotknowthatlighttravelsatafinitespeed,andofferedahand-wavingargumenttoexplainwhynisgreaterthanunitywhenAisairandBiswater.Forseventeenth-centuryapplications,likeDescartes’theoryoftherainbow,itdidn’tmatter,becausenwasassumedtobeangle-independent,whichiscorrectforlightifnotfortennisballs,anditsvaluewastakenfromobservationsofrefraction,notfrommeasurementsofthespeedoflightinvariousmedia.

28.Least-TimeDerivationoftheLawofRefraction

HeroofAlexandriapresentedaderivationofthelawofreflection,thattheangleofreflectionequalsthe

angleofincidence,fromtheassumptionthatthepathofthelightrayfromanobjecttothemirrorandthentotheeyeisasshortaspossible.Hemightjustaswellhaveassumedthatthetimeisasshortaspossible, since the time taken for light to travel anydistance is thedistancedividedby the speedoflight,and in reflection thespeedof lightdoesnotchange.On theotherhand in refractiona light raypasses through the boundary between media (such as air and glass) in which the speed of light isdifferent,andwehave todistinguishbetweenaprincipleof leastdistanceandoneof least time. Justfromthefactthatarayoflightisbentwhenpassingfromonemediumtoanother,weknowthatlightbeingrefracteddoesnottakethepathofleastdistance,whichwouldbeastraightline.Rather,asshownbyFermat, thecorrect lawof refractioncanbederivedbyassuming that light takes thepathof leasttime.Tocarryoutthisderivation,supposethatarayoflighttravelsfrompointPAinmediumAinwhich

thespeedoflightisvAtopointPBinmediumBinwhichthespeedoflightisvB.Tomakethiseasytodescribe,supposethatthesurfaceseparatingthetwomediaishorizontal.Lettheanglebetweenthelightrays inmediaA andB and the vertical direction be i and r, respectively. If pointsPA andPB are atverticaldistancesdAanddBfromtheboundarysurface,thenthehorizontaldistanceofthesepointsfromthepointwheretheraysintersectthissurfacearedAtanianddBtanr, respectively,wheretandenotesthetangentofanangle,theratiooftheoppositetotheadjacentsideinarighttriangle.(SeeFigure21.)Althoughthesedistancesarenotfixedinadvance,theirsumisthefixedhorizontaldistanceLbetweenpointsPAandPB:

L=dAtani+dBtanr

TocalculatethetimetelapsedwhenthelightgoesfromPAtoPB,wenotethatthedistancestraveledinmediaAandBaredA/cosianddB/cosr,respectively,wherecosdenotesthecosineofanangle,theratiooftheadjacentsidetothehypotenuseinarighttriangle.Timeelapsedisdistancedividedbyspeed,sothetotaltimeelapsedhereis

We need to find a general relation between angles i and r (independent ofL,dA, and dB) that issatisfiedbythatangleiwhichmakestimetaminimum,whenrdependsoniinsuchawayastokeepLfixed.Forthispurpose,considerδi,aninfinitesimalvariationδ(delta)oftheangleofincidencei.ThehorizontaldistancebetweenPAandPBisfixed,sowhenichangesbyamountδi,theangleofrefractionrmustalsochange,saybyamountδr,givenbytheconditionthatLisunchanged.Also,attheminimumoftthegraphoftversusimustbeflat,foriftisincreasingordecreasingatsomeitheminimummustbeatsomeothervalueofiwheretissmaller.Thismeansthatthechangeintcausedbyatinychangeδimustvanish,atleasttofirstorderinδi.SotofindthepathofleasttimewecanimposetheconditionthatwhenwevarybothiandrthechangesδLandδtmustbothvanishatleasttofirstorderinδiandδr.

Figure21.Pathofalightrayduringrefraction.Thehorizontallinemarkstheinterfacebetweentwotransparent media A and B, in which light has different speeds vA and vB, and angles i and r aremeasuredbetweenthelightrayandthedashedverticallineperpendiculartotheinterface.ThesolidlinemarkedwitharrowsrepresentsthepathofalightraythattravelsfrompointPAinmediumAtopointPontheinterfacebetweenthemediaandthentopointPBinmediumB.

Toimplementthiscondition,weneedstandardformulasfromdifferentialcalculusforthechangesδtanθ(theta)andδ(1/cosθ)whenwemakeaninfinitesimalchangeδθinangleθ:

whereR=360°/2π=57.293 . . . ° ifθ ismeasured indegrees. (Thisangle iscalleda radian. Ifθ ismeasuredinradiansthenR=1.)Usingtheseformulas,wefindthechangesinLandtwhenwemakeinfinitesimalchangesδiandδrinanglesiandr:

TheconditionthatδL=0tellsusthat

sothat

Forthistovanish,wemusthave

orinotherwords

withtheindexofrefractionngivenbytheangle-independentratioofvelocities:

n=vA/vB

Thisisthecorrectlawofrefraction,withthecorrectformulaforn.

29.TheTheoryoftheRainbow

SupposethatarayoflightarrivesatasphericalraindropatapointP,whereitmakesanangleitothenormaltothedrop’ssurface.Iftherewerenorefraction,thelightraywouldcontinuestraightthroughthedrop.Inthiscase,thelinefromthecenterCofthedroptothepointQofclosestapproachoftheraytothecenterwouldmakearightanglewiththelightray,sotrianglePCQwouldbearighttrianglewithhypotenuse equal to the radiusR of the circle, and the angle atP equal to i. (See Figure 22a.) Theimpactparameterbisdefinedasthedistanceofclosestapproachoftheunrefractedraytothecenter,soitisthelengthofsideCQofthetriangle,givenbyelementarytrigonometryas

b=Rsini

Wecanequallywellcharacterizeindividuallightraysbytheirvalueofb/R,aswasdonebyDescartes,orbythevalueoftheangleofincidencei.Becauseofrefraction,theraywillactuallyenterthedropatanglertothenormal,givenbythelawof

refraction:

wheren 4/3istheratioofthespeedoflightinairtoitsspeedinwater.Theraywillcrossthedrop,andstrikeitsbacksideatpointp'.SincethedistancesfromthecenterCofthedroptoPandtoP'arebothequaltotheradiusRofthedrop,thetrianglewithverticesC,P,andP'isisosceles,sotheanglesbetweenthelightrayandthenormalstothesurfaceatPandatP'mustbeequal,andhencebothequaltor.Someof the lightwillbe reflectedfromthebacksurface,andby the lawof reflection theanglebetweenthereflectedrayandthenormaltothesurfaceatP'willagainber.ThereflectedraywillcrossthedropandstrikeitsfrontsurfaceatapointP",againmakingananglerwiththenormaltothesurfaceatP".Someofthelightwillthenemergefromthedrop,andbythelawofrefractiontheanglebetweentheemergentrayandthenormaltothesurfaceatP"willequaltheoriginalincidentanglei.(SeeFigure22b.Thisfigureshowsthepathofthelightraythroughaplaneparalleltotheray’soriginaldirectionthatcontainsthecenteroftheraindropandtheobserver.Onlyraysthatimpingeonthedrop’ssurfacewhereitintersectsthisplanehaveachanceofreachingtheobserver.)

Figure22.Thepathofarayofsunlightinasphericaldropofwater.Therayisindicatedbysolidlinesmarkedwitharrows,andentersthedropatpointP,whereitmakesanangleiwiththenormaltothesurface.(a)Pathoftherayiftherewerenorefraction,withQthepointoftheray’sclosestapproachtothecenterCofthedropinthiscase.(b)TherayrefractedonenteringthedropatP,reflectedfromthebacksurfaceofthedropatP",andthenrefractedagainonleavingthedropatP".DashedlinesrunfromthecenterCofthedroptopointswheretheraymeetsthesurfaceofthedrop.

Inthecourseofallthisbouncingaround,thelightraywillhavebeenbenttowardthecenterofthedropbyananglei–rtwiceonenteringandleavingthedrop,andbyanangle180°–2rwhenreflectedbythebacksurfaceofthedrop,andhencebyatotalangle

2(i−r)+180°−2r=180°−4r+2i

If the lightraybouncedstraightbackfromthedrop(as is thecasefor i=r=0) thisanglewouldbe180°, and the initial and final light rays would be along the same line, so the actual angle φ (phi)betweentheinitialandfinallightraysis

φ=4r−2i

Wecanexpressrintermsofi,as

where for any quantity x, the quantity arcsin x is the angle (usually taken between –90° and +90°)whosesineisx.Thenumericalcalculationforn=4/3reportedinChapter13showsthatφrisesfromzeroati=0toamaximumvalueofabout42°andthendropstoabout14°ati=90°.Thegraphofjversusiisflatatitsmaximum,solighttendstoemergefromthedropatadeflectionangleφcloseto42°.If we look up at a misty sky with the sun behind us, we see light reflected back primarily from

directions in the skywhere theanglebetweenour lineof sightand the sun’s rays isnear42°.Thesedirectionsformabow,usuallyrunningfromtheEarth’ssurfaceupintotheskyandthendownagaintothe surface. Because n depends slightly on the color of light, so does the maximum value of thedeflectionangleφ,sothisbowisspreadoutintodifferentcolors.Thisistherainbow.

Itisnotdifficulttoderiveananalyticformulathatgivesthemaximumvalueofφforanyvalueoftheindexofrefractionn.Tofindthemaximumofφ,weusethefactthatthemaximumoccursatanincidentangleiwherethegraphofφversusiisflat,sothatthevariationδφ(deltaphi)inφproducedbyasmallvariationδiinivanishestofirstorderinδi.Tousethiscondition,weuseastandardformulaofcalculus,whichtellsusthatwhenwemakeachangeδxinx,thechangeinarcsinxis

whereifarcsinxismeasuredindegrees,thenR=360°/2π.Thuswhentheangleofincidencevariesbyanamountδi,theangleofdeflectionchangesby

or,sinceδsini=cosiδi/R,

Hencetheconditionforamaximumofφisthat

Squaring both sides of the equation and using cos2i = 1 – sin2i (which follows from the theoremofPythagoras),wecanthensolveforsini,andfind

Atthisangle,φtakesitsmaximumvalue:

Forn=4/3,themaximumvalueofφisreachedforb/R=sini=0.86,forwhichi=59.4°,wherer=40.2°,andφmax=42.0°.

30.WaveTheoryDerivationoftheLawofRefraction

Thelawofrefraction,whichasdescribedinTechnicalNote28canbederivedfromanassumptionthatrefractedlightraystakethepathofleasttime,canalsobederivedonthebasisofthewavetheoryoflight. According to Huygens, light is a disturbance in a medium, which may be some transparentmaterialorspacethatisapparentlyempty.Thefrontofthedisturbanceisaline,whichmovesforwardinadirectionatrightanglestothefront,ataspeedcharacteristicofthemedium.

Figure 23. Refraction of a light wave. The horizontal line againmarks the interface between twotransparentmedia,inwhichlighthasdifferentspeeds.Thecrosshatchedlinesshowasegmentofawavefrontattwodifferenttimes—whentheleadingedgeandwhenthetrailingedgeofthewavefrontjusttouchtheinterface.Thesolidlinesmarkedwitharrowsshowthepathstakenbytheleadingandtrailingedgesofthewavefront.

Considerasegmentofthefrontofsuchadisturbance,whichisoflengthL inmedium1, travelingtoward an interfacewithmedium 2. Let us suppose that the direction ofmotion of the disturbance,whichisatrightanglestothisfront,makesangle iwiththeperpendiculartothisinterface.WhentheleadingedgeofthedisturbancestrikestheinterfaceatpointA,thetrailingedgeBisstillatadistance(along the direction the disturbance is traveling) equal to L tan i. (See Figure 23.) Hence the timerequiredforthetrailingedgetoreachtheinterfaceatpointDisLtani/v1,wherev1isthevelocityofthedisturbanceinmedium1.Duringthistimetheleadingedgeofthefrontwillhavetraveledinmedium2atananglertotheperpendicular,reachingapointCatadistancev2Ltani/v1fromA,wherev2isthevelocityinmedium2.Atthistimethewavefront,whichisatrightanglestothedirectionofmotioninmedium2,extendsfromCtoD,sothatthetrianglewithverticesA,C,andDisarighttriangle,witha90°angleatC.Thedistancev2Ltani/v1fromAtoCisthesideoppositeanglerinthisrighttriangle,whilethehypotenuseisthelinefromAtoD,whichhaslengthL/cosi.(Again,seeFigure23.)Hence

Recallingthattani=sini/cosi,weseethatthefactorsofcosiandLcancel,sothat

sinr=v2sini/v1

orinotherwords

whichisthecorrectlawofrefraction.It is not an accident that thewave theory, asworked out byHuygens, gives the same results for

refractionastheleast-timeprincipleofFermat.Itcanbeshownthat,evenforwavespassingthroughaheterogeneousmedium inwhich the speed of light changes gradually in various directions, not justsuddenlyataplaneinterface,thewavetheoryofHuygenswillalwaysgivealightpaththattakestheshortesttimetotravelbetweenanytwopoints.

31.MeasuringtheSpeedofLight

Supposeweobservesomeperiodicprocessoccurringatsomedistancefromus.Fordefinitenesswewillconsideramoongoingaroundadistantplanet,buttheanalysisbelowwouldapplytoanyprocessthatrepeatsperiodically.Supposethatthemoonreachesthesamestageinitsorbitattwoconsecutivetimest1andt2;forinstancethesemightbetimesthatthemoonconsecutivelyemergesfrombehindtheplanet.IftheintrinsicorbitalperiodofthemoonisT,thent2–t1=T.Thisistheperiodweobserve,providedthe distance between us and the planet is fixed.But if this distance is changing, then the periodweobservewillbeshiftedfromT,byanamountthatdependsonthespeedoflight.Supposethatthedistancesbetweenusandtheplanetattwosuccessivetimeswhenthemoonisatthe

samestageinitsorbitared1andd2.Wethenobservethesestagesintheorbitattimes

tʹ1=t1+d1/ctʹ2=t2+d2/c

wherecisthespeedoflight.(Weareassumingherethatthedistancebetweentheplanetanditsmoonmaybeneglected.)Ifthedistancebetweenusandthisplanetischangingatratev,eitherbecauseitismovingorbecauseweareorboth,thend2–d1=vT,andsotheobservedperiodis

(This derivation depends on the assumption that v should change very little in a time T, which istypicallytrueinthesolarsystem,butvmaybechangingappreciablyoverlongertimescales.)Whenthedistantplanetismovingtowardorawayfromus,inwhichcasevisrespectivelynegativeorpositive,itsmoon’sapparentperiodwillbedecreasedorincreased,respectively.WecanmeasureTbyobservingtheplanetatatimewhenv=0,andthenmeasurethespeedoflightbyobservingtheperiodagainatatimewhenvhassomeknownnonzerovalue.

ThisisthebasisofthedeterminationofthespeedoflightbyHuygens,basedonRømer’sobservationof the changingapparentorbitalperiodof Jupiter’smoon Io.Butwith the speedof lightknown, thesamecalculationcantellustherelativevelocityvofadistantobject.Inparticular,thelightwavesfromaspecificlineinthespectrumofadistantgalaxywilloscillatewithsomecharacteristicperiodT,relatedtoitsfrequencyν(nu)andwavelengthλ(lambda)byT=1/ν=λ/c.Thisintrinsicperiodisknownfromobservations of spectra in laboratories on Earth. Since the early twentieth century the spectral linesobserved in very distant galaxies have been found to have longer wavelengths, and hence longerperiods,fromwhichwecaninferthatthesegalaxiesaremovingawayfromus.

32.CentripetalAcceleration

Accelerationistherateofchangeofvelocity,butthevelocityofanybodyhasbothamagnitude,knownas the speed, and a direction.The velocity of a bodymoving on a circle is continually changing itsdirection, turning toward the circle’s center, so even at constant speed it undergoes a continualaccelerationtowardthecenter,knownasitscentripetalacceleration.Letuscalculatethecentripetalaccelerationofabodytravelingonacircleofradiusr,withconstant

speed v. During a short time interval from t1 to t2, the body will move along the circle by a smalldistancevΔt,whereΔt(deltat)=t2–t1,andtheradialvector(thearrowfromthecenterofthecircletothe body) will swivel around by a small angleΔθ (delta theta). The velocity vector (an arrowwithmagnitudev in thedirectionof thebody’smotion) isalways tangent to thecircle,andhenceat rightanglestotheradialvector,sowhile theradialvector’sdirectionchangesbyanangleΔθ, thevelocityvector’sdirectionwillchangebythesamesmallangle.Sowehavetwotriangles:onewhosesidesarethe radial vectors at times t1 and t2 and the chord connecting the positions of the body at these twotimes;and theotherwhosesidesare thevelocityvectorsat times t1and t2, and thechangeΔv in thevelocitybetweenthesetwotimes.(SeeFigure24.)ForsmallanglesΔθ,thedifferenceinlengthbetweenthechordandthearcconnectingthepositionsofthebodiesattimest1andt2,isnegligible,sowecantakethelengthofthechordasvΔt.Now, these trianglesaresimilar (that is, theydiffer insizebutnot inshape)because theyareboth

isoscelestriangles(eachhastwoequalsides)withthesamesmallangleΔθbetweenthetwoequalsides.Sotheratiosoftheshortandlongsidesofeachtriangleshouldbethesame.Thatis

andtherefore

ThisisHuygens’formulaforthecentripetalacceleration.

Figure24.Calculationofcentripetalacceleration.Top:Velocitiesofaparticlemovingonacircleattwotimes,separatedbyashorttimeintervalΔt.Bottom:Thesetwovelocities,broughttogetherintoatrianglewhoseshortsideisthechangeofvelocityinthistimeinterval.

33.ComparingtheMoonwithaFallingBody

The ancient supposed distinction between phenomena in the heavens and on Earth was decisivelychallenged byNewton’s comparison of the centripetal acceleration of theMoon in its orbitwith thedownwardaccelerationofafallingbodynearthesurfaceoftheEarth.From measurements of the Moon’s diurnal parallax, its average distance from the Earth was

accuratelyknowninNewton’stimetobe60timestheradiusoftheEarth.(Theactualratiois60.27.)TocalculatetheEarth’sradiusNewtontook1'(minuteofarc)attheequatortobeamileof5,000feet,sowith360°foracircleand60’to1°,theearth’sradiuswas

(The mean radius is actually 20,926,300 feet. This was the greatest source of error in Newton’scalculation.)TheorbitalperiodoftheMoon(thesiderealmonth)wasaccuratelyknowntobe27.3days,or2,360,000seconds.ThevelocityoftheMooninitsorbitwasthen

Thisgivesacentripetalaccelerationof

Accordingtotheinversesquarelaw,thisshouldhaveequaledtheaccelerationoffallingbodiesonthesurfaceoftheEarth,32feet/secondpersecond,dividedbythesquareoftheratiooftheradiusoftheMoon’sorbittotheradiusoftheearth:

Itisthiscomparison,ofan“observed”lunarcentripetalaccelerationof0.0073foot/secondpersecondwiththevalueexpectedfromtheinversesquarelaw,0.0089foot/secondpersecond,towhichNewtonwasreferringwhenhesaidthatthey“answerprettynearly.”Hedidbetterlater.

34.ConservationofMomentum

Supposetwomovingobjectswithmassesm1andm2collidehead-on.Ifinashorttimeintervalδt(deltat)object1exertsforceFonobject2,theninthistimeintervalobject2willexperienceanaccelerationa2thataccordingtoNewton’ssecondlawobeystherelationm2a2=F.Itsvelocityv2willthenchangebyanamount

δv2=a2δt=Fδt/m2

AccordingtoNewton’sthirdlaw,particle2willexertonparticle1aforce–Fthatisequalinmagnitudebut(asindicatedbytheminussign)oppositeindirection,sointhesametimeintervalthevelocityv1ofobject1willundergoachangeintheoppositedirectiontoδv2,givenby

δv1=a1δt=−Fδt/m1

Thenetchangeinthetotalmomentumm1v1+m2v2isthen

m1δv1+m2δv2=0

Ofcourse,thetwoobjectsmaybeincontactforanextendedperiod,duringwhichtheforcemaynotbeconstant,butsincethemomentumisconservedineveryshortintervaloftime,itisconservedduringthewholeperiod.

35.PlanetaryMasses

InNewton’stimefourbodiesinthesolarsystemwereknowntohavesatellites:JupiterandSaturnaswellastheEarthwereknowntohavemoons,andalltheplanetsaresatellitesoftheSun.AccordingtoNewton’slawofgravitation,abodyofmassMexertsaforceF=GMm/r2onasatelliteofmassmatdistance r (whereG is a constant of nature), so according to Newton’s second law of motion thecentripetalaccelerationof thesatellitewillbea=F/m=GM/r2.Thevalueof theconstantGand theoverallscaleofthesolarsystemwerenotknowninNewton’stime,buttheseunknownquantitiesdonotappearintheratiosofmassescalculatedfromratiosofdistancesandratiosofcentripetalaccelerations.

IftwosatellitesofbodieswithmassesM1andM2arefoundtobeatdistancesfromthesebodieswithaknownratior1/r2and tohavecentripetalaccelerationswithaknownratioa1/a2, then the ratioof themassescanbefoundfromtheformula

Inparticular,forasatellitemovingatconstantspeedvinacircularorbitofradiusrtheorbitalperiodisT = 2πr/v, so the centripetal acceleration v2/r is a = 4π2r/T2, the ratio of accelerations is a1/a2 =(r1/r2)/(T2/T1)2,andtheratioofmassesinferredfromorbitalperiodsandratiosofdistancesis

By1687all theratiosof thedistancesof theplanetsfromtheSunwerewellknown,andfromtheobservationoftheangularseparationofJupiterandSaturnfromtheirmoonsCallistoandTitan(whichNewtoncalled the“Huygeniansatellite”) itwasalsopossible toworkout theratioof thedistanceofCallistofromJupitertothedistanceofJupiterfromtheSun,andtheratioofthedistanceofTitanfromSaturntothedistanceofSaturnfromtheSun.ThedistanceoftheMoonfromtheEarthwasquitewellknownasamultipleofthesizeoftheEarth,butnotasafractionofthedistanceoftheEarthfromtheSun,whichwasthennotknown.NewtonusedacrudeestimateoftheratioofthedistanceoftheMoonfromtheEarthandthedistanceoftheEarthfromtheSun,whichturnedouttobebadlyinerror.Asidefrom this problem, the ratios of velocities and centripetal accelerations could be calculated from theknownorbitalperiodsofplanetsandmoons.(NewtonactuallyusedtheperiodofVenusratherthanofJupiterorSaturn,but thiswasjustasusefulbecause theratiosof thedistancesofVenus,Jupiter,andSaturnfromtheSunwereallwellknown.)AsreportedinChapter14,Newton’sresultsfortheratiosofthemassesofJupiterandSaturntothemassoftheSunwerereasonablyaccurate,whilehisresultfortheratioofthemassoftheEarthtothemassoftheSunwasbadlyinerror.

Endnotes

PARTI:GREEKPHYSICS1.MatterandPoetry

1.Aristotle,Metaphysics,BookI,Chapter3,983b6,20(Oxfordtrans.).HereandbelowIfollowthestandardpracticeofcitingpassagesfromAristotlebyreferringtotheirlocationinI.Bekker’s1831Greekedition.By“Oxford trans.,” Imean that theEnglish languageversion is taken fromThe Complete Works of Aristotle—The Revised Oxford Translation, ed. J. Barnes (PrincetonUniversity Press, Princeton, N.J., 1984), which uses this convention in citing passages fromAristotle.2. Diogenes Laertius, Lives of the Eminent Philosophers, Book I, trans. R. D. Hicks (LoebClassicalLibrary,HarvardUniversityPress,Cambridge,Mass.,1972),p.27.3.From J.Barnes,ThePresocraticPhilosophers, rev. ed. (Routledge andKegan Paul, London,1982), p. 29. The quotations in this work, hereafter cited as Presocratic Philosophers, aretranslations into English of the fragmentary quotations in the standard sourcebook byHermannDielsandWalterKranz,DieFragmentederVorsokratiker(10thed.,Berlin,1952).4.PresocraticPhilosophers,p.53.5. From J. Barnes,EarlyGreek Philosophy (Penguin, London, 1987), p. 97. Hereafter cited asEarlyGreekPhilosophy.AsinPresocraticPhilosophers,thesequotationsaretakenfromDielsandKranz,10thed.6. From K. Freeman, The Ancilla to the Pre-Socratic Philosophers (Harvard University Press,Cambridge,Mass.,1966),p.26.HereaftercitedasAncilla.ThisisatranslationintoEnglishofthequotationsinDiels,FragmentederVorsokratiker,5thed.7.Ancilla,p.59.8.EarlyGreekPhilosophy,p.166.9.Ibid.,p.243.

10.Ancilla,p.93.11.Aristotle,Physics,BookVI,Chapter9,239b5(Oxfordtrans.).12.Plato,Phaedo, 97C–98C.Here and below I follow the standard practice of citing passages from

Plato’sworksbygivingpagenumbersinthe1578StephanosGreekedition.13.Plato,Timaeus,54A–B,fromDesmondLee,trans.,TimaeusandCritias(PenguinBooks,London,

1965).14.Forinstance,intheOxfordtranslationofAristotle’sPhysics,BookIV,Chapter6,213b1–2.15.Ancilla,p.24.16.EarlyGreekPhilosophy,p.253.17.Ihavewrittenaboutthispointatgreaterlengthinthechapter“BeautifulTheories”inDreamsofaFinal Theory (Pantheon, New York, 1992; reprinted with a new afterword, Vintage, New York,1994).

2.MusicandMathematics

1.Fortheprovenanceofthesestories,seeAlbertoA.Martínez,TheCultofPythagoras—ManandMyth(UniversityofPittsburghPress,Pittsburgh,Pa.,2012).2.Aristotle,Metaphysics,BookI,Chapter5,985b23–26(Oxfordtrans.).

3.Ibid.,986a2(Oxfordtrans.).4.Aristotle,PriorAnalytics,BookI,Chapter23,41a23–30.5.Plato,Theaetetus,147D–E(Oxfordtrans.).6.Aristotle,Physics,215p1–5(Oxfordtrans.).7.Plato,TheRepublic,529E,trans.RobinWakefield(OxfordUniversityPress,Oxford,1993),p.261.8.E.P.Wigner,“TheUnreasonableEffectivenessofMathematics,”CommunicationsinPureandAppliedMathematics13(1960):1–14.

3.MotionandPhilosophy

1. J. Barnes, in The Complete Works of Aristotle—The Revised Oxford Translation (PrincetonUniversityPress,Princeton,N.J.,1984).2. R. J. Hankinson, in The Cambridge Companion to Aristotle, ed. J. Barnes (CambridgeUniversityPress,Cambridge,1995),p.165.3.Aristotle,Physics,BookII,Chapter2,194a29–31(Oxfordtrans.,p.331).4.Ibid.,Chapter1,192a9(Oxfordtrans.,p.329).5.Aristotle,Meteorology,BookII,Chapter9,396b7–11(Oxfordtrans.,p.596).6.Aristotle,OntheHeavens,BookI,Chapter6,273b30–31,274a,1(Oxfordtrans.,p.455).7.Aristotle,Physics,BookIV,Chapter8,214b12–13(Oxfordtrans.,p.365).8.Ibid.,214b32–34(Oxfordtrans.,p.365).9.Ibid.,BookVII,Chapter1,242a50–54(Oxfordtrans.,p.408).

10.Aristotle,OntheHeavens,BookIII,Chapter3,301b25–26(Oxfordtrans.,p.494).11.ThomasKuhn,“RemarksonReceivingtheLaurea,”inL’AnnoGalileiano(EdizioniLINT,Trieste,

1995).12.DavidC.Lindberg, inTheBeginningsofWesternScience (UniversityofChicagoPress,Chicago,

Ill.,1992),pp.53–54.13.DavidC.Lindberg, inTheBeginningsofWesternScience,2nded. (UniversityofChicagoPress,

Chicago,Ill.,2007),p.65.14.MichaelR.Matthews,inIntroductiontoTheScientificBackgroundtoModernPhilosophy(Hackett,

Indianapolis,Ind.,1989).

4.HellenisticPhysicsandTechnology

1.Here I borrow the title of the leadingmodern treatise on this age:PeterGreen,Alexander toActium(UniversityofCaliforniaPress,Berkeley,1990).2.IbelievethatthisremarkisoriginallyduetoGeorgeSarton.3.ThedescriptionofStrato’sworkbySimpliciusispresentedinanEnglishtranslationbyM.R.CohenandI.E.Drabkin,ASourceBookinGreekScience(HarvardUniversityPress,Cambridge,Mass.,1948),pp.211–12.4. H. Floris Cohen,HowModern Science Came into the World (Amsterdam University Press,Amsterdam,2010),p.17.5. For the interaction of technologywith physics research inmodern times, seeBruce J.Hunt,PursuingPower and Light: Technology andPhysics from JamesWatt to Albert Einstein (JohnsHopkinsUniversityPress,Baltimore,Md.,2010).6.Philo’sexperimentsaredescribedinaletterquotedbyG.I.Ibry-MassieandP.T.Keyser,Greek

ScienceoftheHellenisticEra(Routledge,London,2002),pp.216–19.7.The standard translation intoEnglish isEuclid,TheThirteenBooksof theElements, 2nd ed.,trans.ThomasL.Heath(CambridgeUniversityPress,Cambridge,1925).8. This is quoted in a Greek manuscript of the sixth century AD, and given in an EnglishtranslationinIbry-MassieandKeyser,GreekScienceoftheHellenisticEra.9.SeeTableV.1,p.233,of thetranslationofPtolemy’sOpticsbyA.MarkSmith in“Ptolemy’sTheory of Visual Perception,” Transactions of the American Philosophical Society 86, Part 2(1996).

10.Quoteshereare fromT.L.Heath, trans.,TheWorksofArchimedes (CambridgeUniversityPress,Cambridge,1897).

5.AncientScienceandReligion

1.Plato,Timaeus, 30A, trans.R.G.Bury, inPlato,Volume9 (LoebClassicalLibrary,HarvardUniversityPress,Cambridge,Mass.,1929),p.55.2.ErwinSchrödinger,ShearmanLecturesatUniversityCollegeLondon,May1948,publishedasNatureandtheGreeks(CambridgeUniversityPress,Cambridge,1954).3.AlexandreKoyré,From theClosedWorld to the InfiniteUniverse (JohnsHopkinsUniversityPress,Baltimore,Md.,1957),p.159.4.Ancilla,p.22.5.Thucydides,Historyof thePeloponnesianWar, trans.RexWarner(Penguin,NewYork,1954,1972),p.511.6. S. Greenblatt, “The Answer Man: An Ancient Poem Was Rediscovered and the WorldSwerved,”NewYorker,August8,2011,pp.28–33.7.EdwardGibbon,TheDeclineandFalloftheRomanEmpire,Chapter23(Everyman’sLibrary,NewYork,1991),p.412.HereaftercitedasGibbon,DeclineandFall.8.Ibid.,Chapter2,p.34.9. Nicolaus Copernicus,On the Revolutions of Heavenly Spheres, trans. Charles Glenn Wallis(Prometheus,Amherst,N.Y.,1995),p.7.

10. Lactantius, Divine Institutes, Book 3, Section 24, trans. A. Bowen and P. Garnsey (LiverpoolUniversityPress,Liverpool,2003).

11.Paul,EpistletotheColossians2:8(KingJamestranslation).12.Augustine,Confessions,BookIV,trans.A.C.Outler(Dover,NewYork,2002),p.63.13.Augustine,Retractions, Book I, Chapter 1, trans.M. I. Bogan (Catholic University of America

Press,Washington,D.C.,1968),p.10.14.Gibbon,DeclineandFall,ChapterXL,p.231.

PARTII:GREEKASTRONOMY6.TheUsesofAstronomy

1.Thischapterisbasedinpartonmyarticle“TheMissionsofAstronomy,”NewYorkReviewofBooks56, 16 (October 22, 2009): 19–22; reprinted in The Best American Science and NatureWriting,ed.FreemanDyson(HoughtonMifflinHarcourt,Boston,Mass.,2010),pp.23–31,andinTheBestAmericanScienceWriting,ed.JeromeGroopman(HarperCollins,NewYork,2010),pp.272–81.2.Homer,Iliad,Book22,26–29.QuotationfromRichmondLattimore,trans.,TheIliadofHomer

(UniversityofChicagoPress,Chicago,Ill.,1951),p.458.3. Homer,Odyssey, Book V, 280–87. Quotations from Robert Fitzgerald, trans., The Odyssey(Farrar,StrausandGiroux,NewYork,1961),p.89.4.DiogenesLaertius,LivesoftheEminentPhilosophers,BookI,23.5. This is the interpretation of some lines of Heraclitus argued by D. R. Dicks, Early GreekAstronomytoAristotle(CornellUniversityPress,Ithaca,N.Y.,1970).6.Plato’sRepublic,527D–E,trans.RobinWakefield(OxfordUniversityPress,Oxford,1993).7.Philo,On theEternityof theWorld, I (1).Quotation fromC.D.Yonge, trans.,TheWorksofPhilo(HendricksonPeabody,Mass.,1993),707.

7.MeasuringtheSun,Moon,andEarth

1. The importance of Parmenides andAnaxagoras as founders ofGreek scientific astronomy isemphasized byDanielW.Graham,Science Before Socrates—Parmenides, Anaxagoras, and theNewAstronomy(OxfordUniversityPress,Oxford,2013).2.Ancilla,p.18.3.Aristotle,OntheHeavens,BookII,Chapter14,297b26–298a5(Oxfordtrans.,pp.488–89).4.Ancilla,p.23.5.Aristotle,OntheHeavens,BookII,Chapter11.6.Archimedes,OnFloatingBodies, inT.L.Heath,trans.,TheWorksofArchimedes(CambridgeUniversityPress,Cambridge,1897),p.254.HereaftercitedasArchimedes,Heathtrans.7.AtranslationisgivenbyThomasHeathinAristarchusofSamos(Clarendon,Oxford,1923).8.Archimedes,TheSandReckoner,Heathtrans.,p.222.9.Aristotle,OntheHeavens,BookII,14,296b4–6(Oxfordtrans.).

10.Aristotle,OntheHeavens,BookII,14,296b23–24(Oxfordtrans.).11.Cicero,DeRePublica,1.xiv§21–22,inCicero,OntheRepublicandOntheLaws,trans.ClintonW.

Keys(LoebClassicalLibrary,HarvardUniversityPress,Cambridge,Mass.,1928),pp.41,43.12. This work has been reconstructed by modern scholars; see Albert van Helden,Measuring theUniverse—CosmicDimensions fromAristarchus toHalley (University ofChicago Press,Chicago,Ill.,1983),pp.10–13.

13.Ptolemy’sAlmagest, trans.andannotatedG.J.Toomer(Duckworth,London,1984).ThePtolemystarcatalogisonpages341–99.

14.Foracontraryview,seeO.Neugebauer,AHistoryofAncientMathematicalAstronomy (Springer-Verlag,NewYork,1975),pp.288,577.

15.Ptolemy,Almagest,BookVII,Chapter2.16. Cleomedes,Lectures on Astronomy, ed. and trans. A. C. Bowen and R. B. Todd (University of

CaliforniaPress,BerkeleyandLosAngeles,2004).

8.TheProblemofthePlanets

1.G.W.Burch,“TheCounter-Earth,”Osiris11,267(1954).2. Aristotle,Metaphysics, Book I, Part 5, 986a 1 (Oxford trans.). But in Book II of On theHeavens, 293b 23–25,Aristotle says that the counter-Earthwas supposed to explainwhy lunareclipsesaremorecommonthansolareclipses.3.TheparagraphquotedhereisasgivenbyPierreDuheminToSavethePhenomena—AnEssayontheIdeaofPhysicalTheoryfromPlatotoGalileo,trans.E.DolanandC.Machler(University

ofChicagoPress,Chicago,Ill.,1969),p.5,hereaftercitedasDuhem,ToSavethePhenomena.AmorerecenttranslationofthispassagefromSimpliciusisgivenbyI.Mueller:seeSimplicius,OnAristotle’s “On the Heavens 2.10–14” (Cornell University Press, Ithaca, N.Y., 2005), 492.31–493.4,p.33.Wedon’tknowifPlatoeveractuallyproposedthisproblem.SimpliciuswasquotingSosigenesthePeripatetic,aphilosopherofthesecondcenturyAD.4.ForveryclearillustrationsshowingthemodelofEudoxus,seeJamesEvans,TheHistoryandPracticeofAncientAstronomy(OxfordUniversityPress,Oxford,1998),pp.307–9.5.Aristotle,Metaphysics,BookXII,Chapter8,1073b1–1074a1.6.ForatranslationbyI.Mueller,seeSimplicius,OnAristotle“OntheHeavens3.1–7” (CornellUniversityPress,Ithaca,N.Y.,2005),493.1–497.8,pp.33–36.7.Thiswasthework,in1956,ofthephysicistsTsung-DaoLeeandChen-NingYang.8.Aristotle,Metaphysics,BookXII,Section8,1073b18–1074a14(Oxfordtrans.).9. These references are given by D. R. Dicks, Early Greek Astronomy to Aristotle (CornellUniversityPress, Ithaca,N.Y.,1970),p.202.Dicks takesadifferentviewofwhatAristotlewastryingtoaccomplish.

10.Mueller,Simplicius,OnAristotle’s“OntheHeavens2.10–14,”519.9–11,p.59.11.Ibid.,504.19–30,p.43.12.SeeBook IofOttoNeugebauer,AHistoryofAncientMathematicalAstronomy (Springer-Verlag,

NewYork,1975).13.G.Smith,privatecommunication.14.Ptolemy,Almagest,trans.G.J.Toomer(Duckworth,London,1984),BookV,Chapter13,pp.247–

51.Also seeO.Neugebauer,AHistory of AncientMathematical Astronomy, PartOne (Springer-Verlag,Berlin,1975),pp.100–3.

15. Barrie Fleet, trans., Simplicius on Aristotle “Physics 2” (Duckworth, London, 1997), 291.23–292.29,pp.47–48.

16.QuotedbyDuhem,ToSavethePhenomena,pp.20–21.17.Ibid.18. For comments on themeaning of explanation in science, and references to other articles on this

subject, see S.Weinberg, “Can Science Explain Everything? Anything?” inNew York Review ofBooks48,9(May31,2001):47–50.Reprints:AustralianReview(2001);inPortuguese,FolhadaS.Paolo (2001); inFrench,LaRecherche (2001);TheBestAmericanScienceWriting,ed.M.RidleyandA.Lightman(HarperCollins,NewYork,2002);TheNortonReader(W.W.Norton,NewYork,December 2003); Explanations—Styles of Explanation in Science, ed. John Cornwell (OxfordUniversityPress,London,2004),23–38; inHungarian,Akadeemia176,No.8:1734–49(2005);S.Weinberg,LakeViews—ThisWorldandtheUniverse(HarvardUniversityPress,Cambridge,Mass.,2009).

19. This is not from theAlmagest but from theGreek Anthology, verses compiled in the ByzantineEmpire around AD 900. This translation is from Thomas L. Heath, Greek Astronomy (Dover,Mineola,N.Y.,1991),p.lvii.

PARTIII:THEMIDDLEAGES9.TheArabs

1.ThisletterisquotedbyEutychius,thenpatriarchofAlexandria.ThetranslationhereisfromE.M.Forster,PharosandPharillon(Knopf,NewYork,1962),pp.21–22.AlesspithytranslationisgivenbyGibbon,DeclineandFall,Chapter51.

2.P.K.Hitti,HistoryoftheArabs(Macmillan,London,1937),p.315.3. D. Gutas, Greek Thought, Arabic Culture—The Graeco-Arabic Translation Movement inBaghdadandEarly‘AbbāsidSociety(Routledge,London,1998),pp.53–60.4.Al-Biruni,Book of theDetermination atCoordinates of Localities, Chapter 5, excerpted andtrans.J.LennartBerggren,inTheMathematicsofEgypt,Mesopotamia,China,India,andIslam,ed.VictorKatz(PrincetonUniversityPress,Princeton,N.J.,2007).5.QuotedinP.Duhem,ToSavethePhenomena,p.29.6.QuotedbyR.ArnaldezandA.Z.IskandarinTheDictionaryofScientificBiography (Scribner,NewYork,1975),Volume12,p.3.7.G.J.Toomer,Centaurus14,306(1969).8.MosesbenMaimon,GuidetothePerplexed,Part2,Chapter24,trans.M.Friedländer,2nded.(Routledge,London,1919),pp.196,198.9.BenMaimonisherequotingPsalms115:16.

10.SeeE.Masood,ScienceandIslam(Icon,London,2009).11.N.M.Swerdlow,ProceedingsoftheAmericanPhilosophicalSociety117,423(1973).12.ThecasethatCopernicuslearnedofthisdevicefromArabsourcesismadebyF.J.Ragep,HistoryofScience14,65(2007).

13. This is documented by Toby E. Huff, Intellectual Curiosity and the Scientific Revolution(CambridgeUniversityPress,Cambridge,2011),Chapter5.

14.Theseareverses13,29,and30ofthesecondversionofFitzgerald’stranslation.15.QuotedinJimal-Khalili,TheHouseofWisdom(Penguin,NewYork,2011),p.188.16.Al-Ghazali’sTahafut al-Falasifah, trans. SabihAhmadKamali (Pakistan PhilosophicalCongress,

Lahore,1958).17. Al-Ghazali, Fatihat al-‘Ulum, trans. I. Goldheizer, in Studies on Islam, ed. Merlin L. Swartz

(OxfordUniversityPress,1981),quotation,p.195.

10.MedievalEurope

1.See,e.g.,LynnWhiteJr.,MedievalTechnologyandSocialChange (OxfordUniversityPress,Oxford,1962),Chapter2.2.PeterDear,RevolutionizingtheSciences—EuropeanKnowledgeandItsAmbitions,1500–1700,2nded.(PrincetonUniversityPress,Princeton,N.J.,andOxford,2009),p.15.3.ThearticlesofthecondemnationaregiveninatranslationbyEdwardGrantinASourceBookinMedievalScience,ed.E.Grant(HarvardUniversityPress,Cambridge,Mass.,1974),pp.48–50.4.Ibid.,p.47.5.QuotedinDavidC.Lindberg,TheBeginningsofWesternScience(UniversityofChicagoPress,Chicago,Ill.,1992),p.241.6.Ibid.7. Nicole Oresme, Le livre du ciel et du monde, in French and trans. A. D. Menut and A. J.Denomy(UniversityofWisconsinPress,Madison,1968),p.369.8. Quoted in “Buridan,” inDictionary of Scientific Biography, ed. Charles Coulston Gillespie(Scribner,NewYork,1973),Volume2,pp.604–5.9.SeethearticlebyPiagetinTheVoicesofTime,ed.J.T.Fraser(Braziller,NewYork,1966).

10.Oresme,Lelivre.11.Ibid.,pp.537–39.12. A. C. Crombie, Robert Grosseteste and the Origins of Experimental Science—1100–1700

(Clarendon,Oxford,1953).13.Forinstance,seeT.C.R.McLeish,Nature507,161–63(March13,2014).14.QuotedinA.C.Crombie,MedievalandEarlyModernScience (DoubledayAnchor,GardenCity,

N.Y.,1959),Volume1,p.53.15.TranslationbyErnestA.Moody,inASourceBookinMedievalScience,ed.E.Grant,p.239.Ihave

takenthelibertyofchangingtheword“latitude”inMoody’stranslationto“incrementofvelocity,”whichIthinkmoreaccuratelyindicatesHeytesbury’smeaning.

16.DeSotoisquotedinanEnglishtranslationbyW.A.Wallace,Isis59,384(1968).17.QuotedinDuhem,ToSavethePhenomena,pp.49–50.

PARTIV:THESCIENTIFICREVOLUTION

1.HerbertButterfield,TheOriginsofModernScience,rev.ed.(FreePress,NewYork,1957),p.7.2.Forcollectionsofessaysonthistheme,seeReappraisalsoftheScientificRevolution,ed.D.C.LindbergandR.S.Westfall(CambridgeUniversityPress,Cambridge,1990),andRethinking theScientificRevolution,ed.M.J.Osler(CambridgeUniversityPress,Cambridge,2000).3.StevenShapin,TheScientificRevolution(UniversityofChicagoPress,Chicago,Ill.,1996),p.1.4. PierreDuhem,The System of theWorld:AHistory ofCosmologicalDoctrines fromPlato toCopernicus(Hermann,Paris,1913).

11.TheSolarSystemSolved

1.ForanEnglish translation,seeEdwardRosen,ThreeCopernicanTreatises (Farrar,StrausandGiroux,NewYork,1939),orNoelM.Swerdlow,“TheDerivationandFirstDraftofCopernicus’sPlanetaryTheory:ATranslation of theCommentariolus with Commentary,”Proceedings of theAmericanPhilosophicalSociety117,423(1973).2.Forareview,seeN.Jardine,JournaloftheHistoryofAstronomy13,168(1982).3.O.Neugebauer,AstronomyandHistory—SelectedEssays (Springer-Verlag,NewYork,1983),essay40.4.TheimportanceofthiscorrelationforCopernicusisstressedbyBernardR.Goldstein,JournaloftheHistoryofAstronomy33,219(2002).5. For anEnglish translation, seeNicolasCopernicusOn theRevolutions, trans.EdwardRosen(PolishScientificPublishers,Warsaw,1978; reprint, JohnsHopkinsUniversityPress,Baltimore,Md., 1978); orCopernicus—On the Revolutions of theHeavenly Spheres, trans.A.M.Duncan(BarnesandNoble,NewYork,1976).QuotationsherearefromRosen.6.A.D.White,AHistoryoftheWarfareofSciencewithTheologyinChristendom(Appleton,NewYork, 1895), Volume 1, pp. 126–28. For a deflation of White, see D. C. Lindberg and R. L.Numbers, “Beyond War and Peace: A Reappraisal of the Encounter Between Christianity andScience,”ChurchHistory58,3(September1986):338.7.ThisparagraphhasbeenquotedbyLindbergandNumbers,“BeyondWarandPeace,”andbyT.Kuhn,TheCopernicanRevolution (HarvardUniversity Press,Cambridge,Mass., 1957), p. 191.Kuhn’ssourceisWhite,AHistoryoftheWarfareofSciencewithTheology.TheGermanoriginalisSämtlicheSchriften,ed.J.G.Walch(J.J.Gebauer,Halle,1743),Volume22,p.2260.8.Joshua10:12.9.ThisEnglishtranslationofOsiander’sprefaceistakenfromRosen,trans.,NicolasCopernicusOntheRevolutions.

10.QuotedinR.Christianson,Tycho’sIsland(CambridgeUniversityPress,Cambridge,2000),p.17.11.Onthehistoryoftheideaofhardcelestialspheres,seeEdwardRosen,“TheDissolutionoftheSolid

Celestial Spheres,” Journal of the History of Ideas 46, 13 (1985). Rosen argues that Tychoexaggeratedtheextenttowhichthisideahadbeenacceptedbeforehistime.

12. For claims toTycho’s system and for its variations, seeC. Schofield, “TheTychonic and Semi-TychonicWorldSystems,”inPlanetaryAstronomyfromtheRenaissancetotheRiseofAstrophysics—Part A: Tycho Brahe to Newton, ed. R. Taton and C. Wilson (Cambridge University Press,Cambridge,1989).

13. For a photograph of this statue, taken by Owen Gingerich, see the frontispiece of my essaycollectionFacingUp—ScienceandItsCulturalAdversaries(HarvardUniversityPress,Cambridge,Mass.,2001).

14.S.Weinberg,“AnthropicBoundontheCosmologicalConstant,”PhysicalReviewLetters59,2607(1987); H. Martel, P. Shapiro, and S. Weinberg, “Likely Values of the Cosmological Constant,”AstrophysicalJournal492,29(1998).

15.J.R.VoelkelandO.Gingerich,“GiovanniAntonioMagini’s‘Keplerian’Tablesof1614andTheirImplicationsfortheReceptionofKeplerianAstronomyintheSeventeenthCentury,”JournalfortheHistoryofAstronomy32,237(2001).

16.Quoted inRobert S.Westfall,TheConstruction ofModern Science—Mechanism andMechanics(CambridgeUniversityPress,Cambridge,1977),p.10.

17.This is the translationofWilliamH.Donahue, inJohannesKepler—NewAstronomy (CambridgeUniversityPress,Cambridge,1992),p.65.

18. JohannesKepler,EpitomeofCopernicanAstronomyandHarmonies of theWorld, trans.CharlesGlennWallis(Prometheus,Amherst,N.Y.,1995),p.180.

19. Quoted by Owen Gingerich in Tribute to Galileo in Padua, International Symposium a curadell’UniversitadiPadova,2–6dicembre1992,Volume4(EdizioniLINT,Trieste,1995).

20.Quotations fromGalileoGalilei,SideriusNuncius, or The SiderealMessenger, trans.Albert vanHelden(UniversityofChicagoPress,Chicago,Ill.,1989).

21.GalileoGalilei,DiscorseeDimostrazioneMatematiche.Forafacsimileofthe1663translationbyThomasSalusbury,seeGalileoGalilei,DiscourseonBodiesinWater,withintroductionandnotesbyStillmanDrake(UniversityofIllinoisPress,Urbana,1960).

22. For a modern edition of a seventeenth-century translation, see Galileo,Discourse on Bodies inWater,trans.ThomasSalusbury,intro.andnotesbyStillmanDrake.

23.Fordetailsofthisconflict,seeJ.L.Heilbron,Galileo(OxfordUniversityPress,Oxford,2010).24.Thisletteriswidelycited.ThetranslationquotedhereisfromDuhem,ToSavethePhenomena,p.

107.Afuller translation isgiven inStillmanDrake,DiscoveriesandOpinionsofGalileo (Anchor,NewYork,1957),pp.162–64.

25.AtranslationoftheentireletterisgiveninDrake,DiscoveriesandOpinionsofGalileo,pp.175–216.

26.QuotedinStillmanDrake,Galileo(OxfordUniversityPress,Oxford,1980),p.64.27. The letters ofMaria Celeste to her father fortunately survive.Many are quoted in Dava Sobel,Galileo’sDaughter(Walker,NewYork,1999).Alas,Galileo’sletterstohisdaughtersarelost.

28.SeeAnnibaleFantoli,Galileo—ForCopernicanismandfortheChurch,2nded.,trans.G.V.Coyne(University of Notre Dame Press, South Bend, Ind., 1996); Maurice A. Finocchiaro, RetryingGalileo,1633–1992(UniversityofCaliforniaPress,BerkeleyandLosAngeles,2005).

29.QuotedinDrake,Galileo,p.90.30.QuotedbyGingerich,TributetoGalileo,p.343.31.ImadeastatementtothiseffectatthesamemeetinginPaduawhereKuhnmadetheremarksabout

Aristotle cited in Chapter 4 andwhere Gingerich gave the talk about Galileo fromwhich I havequotedhere.SeeS.Weinberg,inL’AnnoGalileiano(EdizioniLINT,Trieste,1995),p.129.

12.ExperimentsBegun

1.SeeG.E.R.Lloyd,Proceedingsof theCambridgePhilosophicalSociety,N.S.10,50(1972),reprinted inMethods and Problems inGreek Science (CambridgeUniversity Press, Cambridge,1991).2. Galileo Galilei, Two New Sciences, trans. Stillman Drake (University of Wisconsin Press,Madison,1974),p.68.3.StillmanDrake,Galileo(OxfordUniversityPress,Oxford,1980),p.33.4.T.B.Settle,“AnExperimentintheHistoryofScience,”Science133,19(1961).5.ThisisDrake’sconclusionintheendnotetop.259ofGalileoGalilei,DialogueConcerningtheTwo Chief World Systems: Ptolemaic and Copernican, trans. Stillman Drake (Modern Library,NewYork,2001).6. Our knowledge of this experiment is based on an unpublished document, folio 116v, inBiblioteca Nazionale Centrale, Florence. See Stillman Drake, Galileo at Work—His ScientificBiography (University of Chicago Press, Chicago, Ill., 1978), pp. 128–32; A. J. Hahn, “ThePendulumSwingsAgain:AMathematicalReassessment ofGalileo’sExperimentswith InclinedPlanes,”ArchivefortheHistoryoftheExactSciences56,339(2002),withareproductionofthefolioonp.344.7.CarloM.Cipolla,ClocksandCulture1300–1700(W.W.Norton,NewYork,1978),pp.59,138.8. Christiaan Huygens, The Pendulum Clock or Geometrical Demonstrations Concerning theMotionofPendulaasAppliedtoClocks,trans.RichardJ.Blackwell(IowaStateUniversityPress,Ames,1986),p.171.9.ThismeasurementwasdescribedindetailbyAlexandreKoyréinProceedingsoftheAmericanPhilosophical Society 97, 222 (1953) and 45, 329 (1955). Also see Christopher M. Graney,“Anatomy of a Fall: Giovanni Battista Riccioli and the Story of g,”Physics Today, September2012,pp.36–40.

10. On the controversy over these conservation laws, see G. E. Smith, “The Vis-Viva Dispute: AControversyattheDawnofMathematics,”PhysicsToday,October2006,p.31.

11.ChristiaanHuygens,TreatiseonLight, trans.SilvanusP.Thompson(UniversityofChicagoPress,Chicago,Ill.,1945),p.vi.

12.QuotedbyStevenShapininTheScientificRevolution (UniversityofChicagoPress,Chicago,Ill.,1996),p.105.

13.Ibid.,p.185.

13.MethodReconsidered

1.SeearticlesonLeonardoinDictionaryofScientificBiography,ed.CharlesCoulstonGillespie(Scribner,NewYork,1970),Volume8,pp.192–245.2. Quotations are fromRenéDescartes,Principles of Philosophy, trans. V. R.Miller and R. P.Miller(D.Reidel,Dordrecht,1983),p.15.3. Voltaire, Philosophical Letters, trans. E. Dilworth (Bobbs-Merrill Educational Publishing,Indianapolis,Ind.,1961),p.64.4.ItisoddthatmanymodernEnglishlanguageeditionsofDiscourseonMethod leaveoutthese

supplements,asif theywouldnotbeofinteresttophilosophers.Foraneditionthatdoesincludethem,seeRenéDescartes,DiscourseonMethod,Optics,Geometry,andMeteorology,trans.PaulJ.Olscamp(Bobbs-Merrill,Indianapolis,Ind.,1965).TheDescartesquoteandthenumericalresultsbelowarefromthisedition.5.ItisarguedthatthetennisballanalogyfitswellwithDescartes’theoryoflightasarisingfromthedynamicsofthetinycorpusclesthatfillspace;seeJohnA.Schuster,“DescartesOpticien—TheConstructionoftheLawofRefractionandtheManufactureofItsPhysicalRationales,1618–29,”inDescartes’NaturalPhilosophy,ed.S.Graukroger,J.Schuster,andJ.Sutton(Routledge,LondonandNewYork,2000),pp.258–312.6.Aristotle,Meteorology,BookIII,Chapter4,374a,30–31(Oxfordtrans.,p.603).7.Descartes,PrinciplesofPhilosophy,trans.V.R.MillerandR.P.Miller,pp.60,114.8. On this point, see Peter Dear, Revolutionizing the Sciences—European Knowledge and ItsAmbitions,1500–1700, 2nd ed. (PrincetonUniversity Press, Princeton,N.J., andOxford, 2009),Chapter8.9. L. Laudan, “The Clock Metaphor and Probabilism: The Impact of Descartes on EnglishMethodologicalThought,”AnnalsofScience22,73(1966).ContraryconclusionswerereachedinG.A.J.Rogers,“DescartesandtheMethodofEnglishScience,”AnnalsofScience29,237(1972).

10.RichardWatson,CogitoErgoSum—TheLifeofRenéDescartes(DavidR.Godine,Boston,Mass.,2002).

14.TheNewtonianSynthesis

1.ThisisdescribedinD.T.Whiteside,ed.,GeneralIntroductiontoVolume20,TheMathematicalPapersofIsaacNewton(CambridgeUniversityPress,Cambridge,1968),pp.xi–xii.2.Ibid.,Volume2,footnote,pp.206–7;andVolume3,pp.6–7.3.See,forexample,RichardS.Westfall,NeveratRest—ABiographyofIsaacNewton(CambridgeUniversityPress,Cambridge,1980),Chapter14.4.PeterGalison,HowExperimentsEnd(UniversityofChicagoPress,Chicago,Ill.,1987).5.QuotedinWestfall,NeveratRest,p.143.6.Quoted inDictionary of Scientific Biography, ed. Charles CoulstonGillespie (Scribner,NewYork,1970),Volume6,p.485.7.QuotedinJamesGleick,IsaacNewton(Pantheon,NewYork,2003),p.120.8.QuotationsfromI.BernardCohenandAnneWhitman,trans.,IsaacNewton—ThePrincipia,3rded. (University of California Press, Berkeley and Los Angeles, 1999). Before this version, thestandard translation was The Principia—Mathematical Principles of Natural Philosophy(UniversityofCaliforniaPress,BerkeleyandLosAngeles,1962),trans.FlorianCajori(1792),rev.trans.AndrewMotte.9. G. E. Smith, “Newton’s Study of Fluid Mechanics,” International Journal of EngineeringScience36,1377(1998).

10.Modernastronomicaldata in thischapterarefromC.W.Allen,AstrophysicalQuantities,2nded.(Athlone,London,1963).

11.ThestandardworkonthehistoryofthemeasurementofthesizeofthesolarsystemisAlbertvanHelden,Measuring the Universe—Cosmic Dimensions from Aristarchus to Halley (University ofChicagoPress,Chicago,Ill.,1985).

12. See Robert P. Crease, World in the Balance—The Historic Quest for an Absolute System ofMeasurement(W.W.Norton,NewYork,2011).

13.SeeJ.Z.BuchwaldandM.Feingold,NewtonandtheOriginofCivilization (PrincetonUniversityPress,Princeton,N.J.,2014).

14.SeeS.Chandrasekhar,Newton’sPrincipiafortheCommonReader(Clarendon,Oxford,1995),pp.472–76;Westfall,NeveratRest,pp.736–39.

15.R.S.Westfall,“NewtonandtheFudgeFactor,”Science179,751(1973).16. SeeG. E. Smith, “HowNewton’sPrincipia Changed Physics,” in InterpretingNewton: CriticalEssays,ed.A.JaniakandE.Schliesser(CambridgeUniversityPress,Cambridge,2012),pp.360–95.

17. Voltaire, Philosophical Letters, trans. E. Dilworth (Bobbs-Merrill Educational Publishing,Indianapolis,Ind.,1961),p.61.

18.TheoppositiontoNewtonianismisdescribedinarticlesbyA.B.Hall,E.A.Fellmann,andP.Casiniin“Newton’sPrincipia:ADiscussionOrganized andEditedbyD.G.King-Hele andA.R.Hall,”MonthlyNoticesoftheRoyalAstronomicalSociety42,1(1988).

19. Christiaan Huygens, Discours de la Cause de la Pesanteur (1690), trans. Karen Bailey, withannotationsbyKarenBaileyandG.E.Smith,availablefromSmithatTuftsUniversity(1997).

20.Shapinhasarguedthat thisconflictevenhadpolitical implications:StevenShapin,“OfGodsandKings:NaturalPhilosophyandPoliticsintheLeibniz-ClarkeDisputes,”Isis72,187(1981).

21.S.Weinberg,GravitationandCosmology(Wiley,NewYork,1972),Chapter15.22.G.E.Smith,tobepublished.23. Quoted in A Random Walk in Science, ed. R. L. Weber and E. Mendoza (Taylor and Francis,

London,2000).24. Robert K. Merton, “Motive Forces of the New Science,”Osiris 4, Part 2 (1938); reprinted inScience,Technology,andSocietyinSeventeenth-CenturyEngland(HowardFertig,NewYork,1970),andinOnSocialStructureandScience,ed.PiotrySztompka(UniversityofChicagoPress,Chicago,Ill.,1996),pp.223–40.

15.Epilogue:TheGrandReduction

1.Ihavegivenamoredetailedaccountofsomeof thisprogress inTheDiscoveryofSubatomicParticles,rev.ed.(CambridgeUniversityPress,Cambridge,2003).2.IsaacNewton,Opticks,orATreatiseoftheReflections,Refractions,Inflections,andColoursofLight(Dover,NewYork,1952,basedon4thed.,London,1730),p.394.3.Ibid.,p.376.4.ThisisfromOstwald’sOutlinesofGeneralChemistry,andisquotedbyG.Holton,inHistoricalStudiesinthePhysicalSciences9,161(1979),andI.B.Cohen,inCriticalProblemsintheHistoryofScience,ed.M.Clagett(UniversityofWisconsinPress,Madison,1959).5. P.A.M.Dirac, “QuantumMechanics ofMany-Electron Systems,”Proceedings of theRoyalSocietyA123,713(1929).6.Toforestallaccusationsofplagiarism,IwillacknowledgeherethatthislastparagraphisariffonthelastparagraphofDarwin’sOntheOriginofSpecies.

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Index

Thepaginationofthiselectroniceditiondoesnotmatchtheeditionfromwhichitwascreated.Tolocateaspecificentry,pleaseuseyoure-bookreader’ssearchtools.

Abbasidcaliphate,104–8,116,118–19Abraham,Max,34AbuBakr,caliph,103AcadémieRoyaledesSciences,196,240Academy(Athens)

Neoplatonic,51,104Plato’s,18–19,22,50–51,85

acceleration,139,191,193,195–97,226–27,234–35,237,251,286–89,339–42.SeealsocentripetalaccelerationAchillini,Alessandro,141Actium,battleof(31BC),31Adams,JohnCouch,250AdelardofBath,126AdrastusofAphrodisias,97,99,110,160aestheticcriteria(beauty),150–52,255,265Aëtius,63,78,85air

densityandweightof,Galileo,191aselement,Greeks,5–6,10,12,64,259fallingbodiesand,190,287–88fallingdropsand,289motionofprojectiles,Aristotle,25–27,133Philo’sexperimentson,35projectilesand,Galileo,194shapeofatoms,Plato,10,12

airpressure,134,190,197–200airpump,180,200al-Ashari,121al-Battani(Albatenius),107,114,116–17,159,207AlbertofSaxony,135AlbertusMagnus,127,128,176al-Biruni,108–9,116,119–21,137,239,311–13al-Bitruji,112,117alchemy,110–11,115,215,218nAlexanderofAphrodisias,96AlexandertheGreat,22,30–32,56Alexandria,32–41,48,50,62,66,72–75,76n,104,301–2,308

MuseumandLibary,32–33,35,50,75,88,105al-Farghani(Alfraganus),107,126Al-Farisi,117,128,209AlfonsineTables,158–59AlfonsoX,kingofCastile,158algebra,15,40–41,106,109,115,139,206al-Ghazali(Algazel),121–22,127,130–31algorithm,107,115al-Haitam(Alhazen),110,116,137,174Ali,fourthcaliph,103–4al-Khwarizmi,106–7,114,126al-Kindi(Alkindus),111Almagest(Ptolemy),51,74n,88,91n,94,107,114,126,135,149,151,303,309–10AlmagestumNovum(Riccioli),184al-Mamun,Abbasidcaliph,104,105,120al-Mansur,Abbasidscaliph,104Almohadcaliphate,112,114,116,123Almoraviddynasty,114,116AlQanum(IbnSina),112al-Rashid,Harun,Abbasidcaliph104–5

al-Razi(Rhazes),111,119–20,126al-Shirazi,117al-Sufi(Azophi),108,120Al-Tusi,116–17,120Al-Zahrawi(Abulcasis),112Al-Zarqali(Arzachel),113–14,241amber,electricityand,257–58AmbroseofMilan,50Ampère,André-Marie,257Amrou,Arabgeneral,104Anaxagoras,10,47,63,65Anaximander,4–5,7,13,45,57–58,65,111Anaximenes,5,35,65Andromedagalaxy,108AntikytheraMechanism,71nApollonius,xv,21,39–40,51,87,90,91n,97,194,254,318–19apriorireasoning,9,64,94,154–54Aquinas,Thomas,27,127–30,136Arabicnumbers,107,126Arabscience,xiv,103–27,141

Aristotleand,27,141astronomer/mathematiciansvs.philosopher/physicians,106–11,126chemists,110–11declineinscience,116,118goldenageof,104–6humorismand,42medievalEuropeand,115,124,126–27religionand,118–23Ptolemyand,88,141

Archimedes,19,21,37–39,41,51,66,68–72,126,129,189,232,291–95,300ArchytasofTarentum,17,18,283Aristarchus,51,66–70,72,75,85–86,94,109,143,154,239,295–301,309Aristophanes,10Aristotle,xiv,4,8,12,22–30,141

airand,25,27,35Arabsand,27,105–6,111–16,121,141banned,inmedievalEurope,129–32,181challengedbyendof16thcentury,201Copernicusand,148,153Descartesand,203–4,209,212earlyChristiansand,49–51Earth’ssphericalshapeand,63–66elementsand,10fallingbodiesand,25–29,49,51,64–66,71,129,133,173,190,194,286–88Galileoand,172–73,181,185–86,190,194,197gravitationand,66Hellenisticperiodand,33–36influenceof,27–28judging,bymodernstandards,28–30Keplerand,167,169n,170mathematicsand,19medievalEuropeand,27–28,124,129–35,137–38,141–43,181naturalvs.artificialand,24–25Newtonand,216–17,243,246,248planetaryorbitsandhomocentrismof,10,70–71,78–80,83–87,94–97,131,142–43,148,153,159–60,167,185–86,212,255Ptolemaicmodelsvs.,95–99,106,112–14,128–35,137–38,141–43Pythagoreansand,16–17,78–79rainbowand,209scientificrevolutionand,201–4teleologyand,23–24,26,36,203,264unchangeableheavensand,159–60,173vacuumand,129–31,134,197–98,204

arithmetic,xiv,15,18,125,163Arrian,56

artificial,vs.natural.,24–25Assayer,The(Galileo),40,182astrology,42–43,99–100,106,108,110,116,135,146,166AstronomaiaeParsOptica(Kepler),166AstronomiaNova(Kepler),166–70astronomicaltables,106,109,114,116,158,161,172astronomicalunit(AU),171,222Athens,7–10,22,32–33,46–47,50–52,62,63,75atmosphere,79,166,176atmosphericpressure,199atoms

Greekson,7,10–12,46,275nuclei,243,249,260,262–63quantummechanicsand,180,249structureandenergystatesof,259–63

Attacks(ZenoofElea),8AugustineofHippo,50,127Averroes.SeeIbnRushdAvicenna.SeeIbnSinaAyurveda,42

Babylonians,xiv,1,4,7,15,57,58,60,77,99,107,310Bacon,Francis,201–2,212,214Bacon,Nicholas,201Bacon,Roger,137–38,174Baghdad,104–7,111,116Bär,NicholasReymers,160Barnes,J.,367,369barometer,199–200Baronius,CardinalCaesar,183Barrow,Isaac,217,224Barton,Catherine,215nBaytal-Hikmah(HouseofWisdom),105–6,120BeginningofSciences,The(al-Ghazali),122–23Bellarmine,Roberto,181,182,184Berkeley,George,233,245–46Bible,135–36,146,156,183,187

Daniel,215Ecclesiastes,156Genesis,125,136Joshua,135,183

bigbang,131biology,xiii,9,115,265–68bisectedeccentric,92n.Seealsoequantblackholes,267blood,circulationof,118BoethiusofDacia,124–25,128Bohr,Niels,261Bokhara,sultanof,111Bologna,Universityof,127,147Boltzmann,Ludwig,259–60,267Bonaventure,Saint,129BookoftheFixedStars(al-Sufi),108Born,Max,261–62bosons,263,264Boyle,Robert,194,199–200,202,213,217,265Boyle’slaw,200Bradwardine,Thomas,138Brahe,Tycho.SeeTychoBraheBroglie,Louisde,248,261Brownianmotion,260Bruno,Giordano,157,181,188Bullialdus,Ismael,226

Buridan,Jean,71,132–35,137,161,212BurningSphere,The(al-Haitam),110Butterfield,Herbert,145ByzantineEmpire,103–4,116

Caesar,Julius,31,50,60calculus,15,195,223–26,231–32,236,315,327

differential,223–25integral,39,223–25limitsin,236

calendarsAntikytheraMechanismfor,71nArabsand,109,116,118Greeksand,56,58–61Gregorian,61,158Julian,61Khayyamand,109Moonvs.Sunasbasisof,59–61

Callimachus,57Callippus,81–87,95,97,142Callisto(moonofJupiter),177–78,364

periodof,178Calvin,John,155Cambridge,Universityof,217–18,245CanterburyTales(Chaucer),27–28Cartesiancoordinates,205–6Cassini,GiovanniDomenico,239–40Cassiopeia,supernovain,159catapults,35–36,41cathedralschools,125–27Catholics.SeeChristianity;RomanCatholicchurchCatoptrics(Hero),36Catoptrics(Pseudo-Euclidian),35–36Cavendish,Henry,240celestialequator,57–58celestiallatitudeandlongitude,73–74centrifugalforce,226–27centripetalacceleration,227–30,233,235,237–39,242,359–62,364Cesarini,Virginio,40nChaeronea,battleof,22Chalcidius(Calcidius),86changelessness,9,23,56Charlemagne,105,125CharlesII,kingofEngland,218CharlesVII,kingofFrance,253Châtelet,Émiliedu,248Chaucer,Geoffrey,27chemistry,xiii–xiv,11,110–11,115,213,218n,256–57,259

alchemyvs.,110–11biologyand,266–68quantummechanicsand,262

China,xiv,1,257chords,sinesand,107,309–11Christianity,26–27,116,118–19.SeealsoRomanCatholicchurch;Protestantism

Aristotlebannedby,127–32Copernicusand,156–57early,andimpactonscience,48–52Galileoand,183–88

ChristianIV,kingofDenmark,161Christina,queenofSweden,212ChristinaofLorraine,183,187Cicero,13,17,39,71circle

areaof,39,294–95definitionof,167,318

civilengineering,41Clairaut,Alexis-Claude,237n,247–48Clarke,Samuel,247classification,byAristotle,24,26Clavius,Christoph,158,179Cleomedes,75–76Cleopatra,31clocks,58

pendulum,191,195sundial,58water,35,193

Clouds,The(Aristophanes),10Cohen,Floris,33,369Cohen,I.B.,383Collins,John,224color,theoryof,218–19Columbus,65n,107comets

distancefromEarth,40nEncke’s,250Galileoand,182,186,205Halley’s,247,250Keplerand,161Newtonand,237,244,247Tychoand,159–60,168

commensurablelines,282–85Commentariolus(Copernicus),117,148–51,153–54,157complexnumbers,163cone,volumeof,19conicsection,40,194,235,237

ellipseand,318–19parabolasand,344–45

conservationofenergy,340–41conservationofmomentum,197,235,362–63ConstantineI,emperorofRome,48,49Constantinople,104,253constellations,56–57Copernicus,Nicolaus,72,134,141,146,307

Arabsand,107,117astronomers’receptionof,157–58Descartesand,204FrancisBaconand,201Galileoand,173,177–79,181–88Keplerand,162–63,166–73,255Newtonand,237n,251planetarymotionand,48–49,85–86,90–91,95,117–18,124,141n,148–63,172,228,240Ptolemaictheoryand,304–7,325relativesizesofplanetaryorbitsand,320–21religiousoppositionto,155–57,181,184–88,213Tycho’salternativeto,158–61

Córdoba,112,114,123CosimoIIdiMedici,178cosine,296,309,313cosmicrays,263counter-Earth,78Crease,R.P.,381Cremonini,Cesare,173,180Crombie,A.C.,137,375Ctesibius,35,41cube,10,12,17,162,163n,275,278–79cubicequations,109Cutler,SirJohn,220Cuvier,Georges,265

CyrilofAlexandria,50–51

d’Alembert,Jean,248Dalton,John,11,259Damascus,104,117,118darkenergy,83,165,265darkmatter,9,264Darwin,Charles,24,172,200,248,265–66,383daysoftheweek,77nDeanalysiperaequationesnumberterminoruminfinitas(Newton),224Dear,Peter,125,269,373,380deduction,xv,19–21,132,164,189,197,201–3,205,247,264–65,289deferents,88–92,93,97,110,149,150,160,180,303–6,324–24DemetriusofPhaleron,32Democritus,7,11–14,44,46–47,65,110–11,260DeMotu(OnMotion)(Galileo),173density

ofEarthvs.water240Newtonand,232

DeRevolutionibus(Copernicus),153–58,183–84derivative,223Descartes,René,37,40,141,194,201,202–14,218,223,229,236,246,248,342,346–48Descartes’law,37,207DialogueConcerningtheTwoChiefSystemsoftheWorld(Galileo),185–87,193,199DialogueConcerningTwoNewSciences(Galileo),190,193–94Dicks,D.R.,373DietrichofFreiburg,128,209diffraction,205,222DiogenesLaertius,4,44,57,63DionysiusII,ofSyracuse,10,18Diophantus,40,107Dioscorides,105Dirac,Paul,152,261–62,383DiscourseonBodiesinWater(Galileo),181,190DiscourseonLight(al-Haitam),110DiscourseonMethod(Descartes),203,205,212–13displacement,38–39,292–93DNA,266dodecahedron,10,12,162,163n,275,279Dominicans,127–30,140Donne,John,vii,42Dopplereffect,221DoubtsconcerningGalen(al-Razi),111Drake,S.,378,379Dreyer,J.L.E.,84Droysen,JohannGustav,31Duhem,Pierre,99,146,260,378DunsScotus(JohannesScotusErigena,JohntheScot),225

EarthAristotleon,24–25,49,64–66,70–71,132,143axisofrotation,58,148,152–53,302axiswobbleof,74–75Copernicuson,86,95,148–56densityof,240distancetoMoon,63,66–68,70,72–73,75,83,295–301distancetoplanets,149–50distancetostars,70distancetoSun,63,66–68,70,75,83,90,164,295–301epicyclesofplanetsand,303–7Eratostheneson,75–76,301–2Galileoon,184–88Greeksonmotionof,10,70–72,79–86,89,153–54

Heraclideson,89medievalEuropeandmotionofplanetsaround,124,132,136,143moonassatelliteof,178,181,363(seealsoMoon)motionaroundSun,debated,89,95,143NewtonandratioofmassesofSunand,239orbitof,circularvs.elliptical,53,59,91,152orbitof,eccentricity,167,324orbitof,Kepleron,162,166–67orbitof,speed,59,152precessionoforbit,al-Zarqalimeasures,113–14Ptolemyonmotionofplanetsaround,88–92,94–95,143,324–25Pythagoreansonmotionof,72,78,151,153–54relativespeedofJupiterand,221–22rotationof,70–71,85–86,108,134–36,148,151,153–54,241,246shapeof,Descartesand,204shapeof,equatorialbulge,153shapeof,flat,65,78shapeof,oblate,241,246shapeof,spherical,49,63–66siderealperiodand,171sizeof,53,63,65n,67–68,70,75–76,107–9,228n,301–2,311–13,362tidesand,184–85

earth,aselement,6,10,12,64–66,259Easter,dateof,60–61eccentricity,167–69,317–19,324eccentrics,87–88,91–95,98,112,142,151–53,166,169,254eclipses

lunar,59,63–64,94,298planets,94solar,4,63,66–67,72–73,83,94,298

ecliptic,57,73–74,177Ecphantus,153Egypt,xiv,1,4,7,31–33,46,55,104–5,116.SeealsoAlexandriaEinstein,Albert,34,172,204n,222–23,248–53,260–61electromagnetism,250,257–64electrons,9,34,180,247,257,260–64electroweaktheory,263–64elementaryparticles,9,11,14,21,180,247,249,262–63,267elements

alchemistson,11Aristotleon,10,64–66chemical,identified,11,259Greeksonfour,6,10,12medievalEuropeand,125Platoon,10,18,45

Elements(Euclid),15,17–19,35,47,51,69,126,223,232,275,285ElizabethI,queenofEngland,170,257ellipses,40,167,235,255,316–19

fociof,167,316,318parabolaand,343

ellipticalorbitsofplanets,117–18,167,231,324–28Empedocles,6–7,10,12,45,111empiricism,132–33,201–2,253Encke’scomet,250EpicurusofSamos,22,46,48epicycles,xv,87,91n,98,254

Arabsand,110,112,117Copernicusand,151–53,182equal-anglesandequantsand,324–25innerandouterplanetsand,303–7Keplerand,166medievalEuropeand,141–42Ptolemaicmodeland,88–95,97–98,149–53,155,168,254–55Tychoand,160

EpitomeofCopernicanAstronomy(Kepler),168EpitomeoftheAlmagest(Regiomontanus),141equal-anglesrule,36–37,208–10,221,290equal-arearule,168–69,231,323–28equant,87,92–95,117,151–52,166,169,254,323–28equinoxes,58,60–61

precessionof,74–75,107,118,153,241–42,244,248Eratosthenes,51,75–76,107,301–2ether,10,258Euclid,15,17–19,35,37,47,51,69,105,119,126,206,210,223,232,272,275–77,285Euctemon,59–60,81,152Eudoxus,18–19,51,80–87,95,97,142Eulercharacteristic,278Europa,177–78Europe,medieval,1,105,124–43

scientificrevolutionand,253Eusebius,bishopofCaesarea,58evolution,24,164–65,265–66experiment,xv,255

Aristotleand,29Descartesand,204,208–9,213developmentof,189–202Galileoand,192–94,200,213Greeksand,24,29,35,69Huygensand,195–97hypothesisand,213medicineand,41–43medievalEuropeand,133,137Newtonand,218–19predictionand,146

explanation,descriptionvs.,xiii,99

fallingbodiesAristotleand,25–29,51,64–66,71,129,133,173,190daVinciand,202Descartesand,204Earth’srotationand,135experimentswithinclinedplanes,192–94,200,339–43fallingdrops,33,288–89Galileoand,173,190–94,200,213,240,339–46Greeksand,25–27,33Huygensand,195–96,342meanspeedtheoremand,140medievalEuropeand,135–37,140Moon’sorbitand,228–30,235,242,361–62Newtonand,190,225–26,228–30,235,242,245Oresmeand,136–37terminalvelocityand,25–26,286–88thrownobjects(projectile),27,51,71,133–35,156,161,170,194,342–46Tychoand,161

Fantoli,A.,378Faraday,Michael,258Fermat,Pierrede,37,208,221,331,348,358fermions,263fieldconcept,21,250,262fine-tuning,82–83,85–86,89–90,149–51,155,255Finocchiaro,M.A.,378fire

Boyleand,200aselement,6,10,12,64,259Lavoisierand,259

firstmover,doctrineof,26firstprinciples,27,97

Fitzgerald,Edward,109floatingandsubmergedbodies,38–39,181,291–94fluxion.Seederivativefocallength,174–75,329–33,336focus.Seealsoellipse,fociiof

planet’sorbitaround,59,151–52,167–68forces,Newtonandconceptof,233,257.SeealsospecificforcesFormationofShadows,The(al-Haitam),110Foscarini,PaoloAntonio,182–84Fourieranalysis,280Fracastoro,Girolomo,141–42,186Franciscans,127,129,138FrederickII,kingofDenmark,159,161Freeman,K.,367FerdinandoI,granddukeofTuscany,183frequency,279–82FulbertofChartres,126fundamentalprinciples,99,163–65,203–4

galaxiesfirstobservationof,108relativevelocityand,359

Galen,42,105,111,126GalileoGalilei,106,134,202,216

astrologyand,146Boyleand,199cometsand,40n,182,205conflictwithchurchandtrialof,181–88,190,213church’srehabilitationof,187Descartesand,204fallingbodiesand,173,190–97,200,240,339–42floatingbodiesand,181,190geometryand,40–41Huygensand,194–95Jupiter’smoonsand,177–78Keplerand,173,179–81Moons’surfaceand,175–76,228,337–39pendulumand,191,195phasesofVenusand,87,143,154,179–80,204planetarysystemandorbitofEartharoundSun,72,172–88,199,204,212projectilesand,193–95,213,342–46Saturnand,194starsvs.planetsand,176–77sunspotsand,180–83telescopeand,79,174–80,219,329–33,336tidesand,184–85,205Tychoand,186vacuumand,198

Galison,Peter,204n,381Ganymede(moonofJupiter),177–78GardenofEpicurus,22,46–47Gassendi,Pierre,46,234Geminus,95–97,157,246generaltheoryofrelativity,14,234,250–53genetics,266GeographicMemoirs(Eratosthenes),75Geometrie(Descartes),223geometry,xiv,125,139,197,199

algebravs.,40–41analytic,40,205–7Galileoand,40–41Greeksand,4,15–20,35,39–40,67meanspeedtheoremand,139,313–15

Newtonand,236,246Platonicsolidsand,274–79Pythagoreantheorem,283–84Thalesand,4,272–74

GerardofCremona,126–27Gibbon,Edward,47,48,51–52Gilbert,William,170,257Gingerich,O.,269,377,378,379gluons,264–65gnomon,58,75,79,81,180God

Descartesand,203firstmoverargumentfor,26freedomof,andscience,130–31Newtonand,146,247voidand,26

gold,8,39,41,110Goldstein,B.R.,376“GoodMorrow,The”(Donne),42Graham,David,371Grassi,Orazio,182gravitation,264.Seealsofallingbodies

Aristotleand,66,286curvedspace-timeand,252Earth’sbulgeand,153Galileoandaccelerationdueto,193,196,240,339–42generalrelativityand,251–53Hookeand,230–31Huygensandaccelerationdueto,195–96,212Huygens’criticismof,246inversesquarelawof,227,230,237Keplerand,169,171laboratorymassesand,240Moon’smotionand,93,196,228Newtonand,99,196,212,225–31,236–47,250–54,257,286–87,363–65Oresmeand,136–37planetaryorbitsand,53,93,134,148,169,171,212,225–28,363–65planetaryorbitsandfallingbodies,228–31,236–37precessionofEarth’sorbitand,113–14precessionoftheequinoxesand,74,241–42siderealperiodandKepler’sthirdlaw,171StandardModelforcesand,264,268Sun’sandplanetaryorbits,Keplerand,169terminalvelocityand,286–87tidesand,185,242–43wobbleofEarth’saxisand,74–75

gravitationalfield,250–52,263Greeks.SeealsoAristotle;Plato;andotherspecificindividualsandtheories

Arabsand,101,104–5,108,111,121,126–27Classicalvs.Hellenisticagevs.,31–34Dorian,6,12Ionian,3–10,63mathematicsand,15–21,40matterand,xiv,3–14measurementofEarth,Sun,andMoon,63–77medievalEuropeand,101,124,126–27,129,137motionand,22–30planetsand,77–101poetryand,12–13religionand,44–47sphericalshapeofEarthand,63–66

Green,Peter,369Greenblatt,Stephen,46,370GregoryIX,Pope,127

GregoryXIII,Pope,61,158Grimaldi,FrancescoMaria,222Grosseteste,Robert,137–38GuidetothePerplexed(Maimonides),114Gutas,Dimitri,105,374

Halley,Edmund,231,244,247,250HaloandtheRainbow,The(al-Haitam),110HandyTables(Ptolemy),114Hankinson,R.J.,23,269,369Harmonicesmundi(Kepler),170–71harmony,279–82Harriot,Thomas,207Hartley,L.P.,xiiHartmann,Georg,141nheat,259–60,267Heath,Thomas,84,370,372,374heavens,changeabilityof,159–60,173,228Heaviside,Oliver,258nHeidelberg,Universityof,140Heilbron,J.L.,378Heisenberg,Werner,261heliocentricism,70–72,118,155–58,172.SeealsoCopernicus;SunHellenisticperiod,xv,1,31–43,46–48,53,87,99,160.SeealsoAlexandria;andspecificindividuals

fluidstaticsand,38–39lightand,35–36technologyand,34–41

HenryofHesse,142HenryVII,kingofEngland,253Heraclides,85–86,89,124,134,153,303Heraclitus,6,8,13,57Heraclius,Byzantineemperor,103Hermes(Eratosthenes),75Herodotus,45–46,58HeroofAlexandria,36–37,41,51,137,189,208,210,289–91,348Hertz,Heinrich,259Hesiod,13,47,55–56Heytesbury,Williamof,138–39Hipparchus,xv,48,51,72–75,87–90,93–94,97,107,137,142,153,159,254Hippasus,17Hippocrates,42,105,111,119,121Hippolytus,63hippopede,81Hisahal-Jabrw-al-Muqabalah(al-Khwarizmi),106–7HistoryofAncientMathematicalAstronomy,A(Neugebauer),64nHistoryoftheWarfareofSciencewithTheologyinChristendom,A(White),155Hitti,Philip,105,374Holton,G.,383Homer,6n,13,55–56homocentricmodels,80–87,95,97,98,112–13,128,142,153,160,185–86Hooke,Robert,200,217,220,230–31,245Horace,99nhorizon

curveofEarthand,65measuringEarthwith,108–9,311–13

Horologiumoscillatorium(Huygens),195,197Huff,Toby,374HughCapet,kingofFrance,126HuleguKhan,116Hunt,Bruce,269,370humors,four,42–43,111HunaynibnIshq,104–5Hussein,grandsonofMuhammad,104

Hutchinson,John,245,246Huygens,Christiaan,36,194–97,208,212–13,217,220,226–27,234,341–42,356,358–60

accelerationduetogravityandfallingbodies,195–96centrifugalforceand,226,227centripetalaccelerationand,212,360constantgandacceleration,342pendulumclockand,195Saturn’smoonTitanand,195speedoflightand,359wavetheoryoflightand,36,196–97,208,356,358

hydrogen,11,152,259hydrostatics,33,189,190Hypatia,51Hypotyposesorbiumcoelestium(Peucer),158

Ibnal-Nafis,118Ibnal-Shatir,117,151,237nIbnBajjah(Avempace),112,115IbnRushd(Averroes),27,112–13,115,117,119,121,128,130IbnSahl,110,207IbnSina(Avicenna),111–12,116,120,126IbnTufayl(Abubacer),112,115icosahedron,10,12,162,163n,275,279Iliad(Homer),47impactparameterb,209–11impetus,51,133–34,212.SeealsomomentumIncoherenceofIncoherence(IbnRushd),121IncoherenceofthePhilosophers,The(al-Ghazali),121incommensurablelines,12,285indexofrefraction(n),206–8,210–12,331–32,348,351,355India,xiv,1,57,104,107–8,116,207infiniteseries,8,223,226infinitesimals,223,224n,231–32,236InnocentIV,Pope,129intelligentdesign,44InventionsofthePhilosophers(al-Ghazali),121inversesquarelawofgravitation,227–31,235,237Io(moonofJupiter),177–78,221–22,359irrationalnumbers,17–18,284–86Isfahan,108–10Islam,xiv,26,101–23

sciencevs.religionand,118–23,131,188Sunnivs.Shiite,104

JabiribnHayyan,110–11,218nJardine,N.,376Jefferson,Thomas,46–47Jesuits,40n,158,181–82Jews,61,105,114,126JohnofDumbleton,138–39JohnofPhiloponus,51,133JohnPaulII,Pope,187Johnsson,Ivar,161JohnXXI,Pope,129JohnXXII,Pope,130Jordan,Pascual,261JournaldesSçavans,197Julian,emperorofRome,48Jupiter,77,167

Aristotleand,84–85conjunctionofSaturnand,159Copernicusand,148–51distancefromSun,163n

epicyclesand,303–6Halley’scometand,247Keplerand,162,163n,171moonsof,177–78,221–22,236–37,359,363–64Newtonand,236–39Ptolemyand,89,94,149,255

Justinian,emperorofRome,51,104

Keill,John,225Kepler,Johannes,40,79,141,146,153,267

Copernicusand,156,170,172,255ellipticalorbitsand,59,91–92,95,161–73,325equal-arearuleand,231–32Galileoand,173,179–81Newtonand,99,226–32,235–37,241,248,249supernovaand,166telescope,180–81,219,329,334–35Tychoand,161,165–69

Kepler’slawsofplanetarymotion,99,188,227n,235–37first,167,172,188,227n,236,323second,168–70,172,188,227n,236–37,323–25third,170–72,188,227–31,236

Keynes,JohnMaynard,216Khayyam,Omar,109–10,119Kilwardy,Robert,129kineticenergy,197,339–40kosmos,6n,10,12,64–66Koyré,Alexandre,45,370Kuhn,Thomas,28–29,369,377,378

Lactantius,49,66,183Laplace,Pierre-Simon,250Laskar,Jacques,245nLaudan,Laurens,213,380Lavoisier,Antoine,11,259Laws(Plato),47leapyear,60Leibniz,GottfriedWilhelm,224–25,233,246–48,251lenses,79,174–75,329–33,336LeonardodaVinci,202LettertoChristina(Galileo),183,187Leucippus,7,44,260Leverrier,Jean-Joseph,250Libri,Giulio,180LightoftheMoon,The(al-Haitam),110LightoftheStars,The(al-Haitam),110light.Seealsooptics;reflection;refraction

Descartesand,206–8,348Einsteinand,252electromagnetismand,268energyand,261fieldconceptand,250Greeksand,35–37Grossetesteand,137Huygensand,196–97,208,220–23Newtonand,218,220–23speedof,37,204,207–8,221–22,259,348,358–59wavetheoryof,36,196–97,208,220–23,221–22,279,281,356–58

limits,224n,236,315Lindberg,David,29–30,132,369,375,377Linnaeus,Carl,265Livy,39Lloyd,G.E.R.,379

logarithm,223nLorentz,Hendrik,34LouisXI,kingofFrance,253Lucas,Henry,217Lucretius,46luminosity,87

magnitudeand,88nLuther,Martin,155–56,183Lyceum,22,32–33,66,75

M31(galaxy),108Machiavelli,Niccolò,46magneticfield,109,220,250,257–58,263magnetism,xiv,170,237,257–59,268.Seealsoelectromagnetismmagnification,174–75,219,334–36magnitudeofstars,88nMaimon,Mosesben(Maimonides),98,114–15Malebranche,Nicolasde,122,246Marcellus,ClaudiusMarcus,39,71MariaCeleste,Sister,187mariners,65,175MarriageofMercuryandPhilology,The(MartianusCapella),124Mars,77,245n

apparentretrogrademotionof,90,148brightnessof,87Copernicusand,148–51distanceto,239–40eccentricityoforbit,167epicyclesand,303–6Greeksand,81–82,84,87–90,94asidealtestcase,165nKeplerand,162,165,169Ptolemyand,89,90,94,255siderealperiodof,171

MartianusCapella,124–25Martinez,A.M.,269,368mass,232–33,237–38mathematics,1.Seealsoalgebra;calculus;geometry;andspecificindividualsandtheories

Arabsand,105–7,111,117,123Babylonian,15Copernicusand,158–59Descartesand,203,213–14Einsteinand,253fieldapproachand,250Galileoand,172,179Glaber,Raoul(Radulfus),125Greeksand,15–21,35,39–40,47,63,65–70,79,105Keplerand,161–62,255medievalEuropeand,126,137–40Neoplatonistsand,47Newtonand,218,223–25,246,253Ptolemaicmodelsand,79–80,88–99Pythagoreansand,16–18roleof,inscience,xv,19–21,79,101,140,146,197

matteralchemistsand,11Aristotleand,64–65atomictheoryof,259–60darkmatterand,9earlyGreeksand,4–14,44–45Newtonand,256–57Platoand,10,13

Matthews,Michael,30,369

Maxwell,JamesClerk,220,258–60,267Maxwell’sequations,258nMayr,Simon,177nmeanspeedtheorem,138–41,191–92,313–15Mechanicesyntaxism(Philo),35medicine,41–43,106,111–12,114–16,118,141medievalEurope,xiv,26–28,101,124–43Melanchthon,Philipp,155,157,158,161mercury,11,198–200

Mercury,77,165n,245n,250apparentretrogrademotionof,148Aristotleand,84–85Copernicusand,86,148–51,155eccentricityoforbit,167,324elongationsandorbitof,320–21epicyclesand,303–5Greekmodelsand,81–82,84–86,88–91,94,124Keplerand,162–63,171–72Ptolemyand,88–91,94,149,155,255

Mersenne,Marin,16Merton,Robert,253,382Merton,Walterde,138MertonCollege,Oxford,138–41,191Mertonthesis,253Mesopotamia,104,107,110Metaphysics(Aristotle),4,16,83–84Meteorology(Aristotle),127Meteorology(Descartes),208Metoniccycle,60–61MetonofAthens,60metricsystem,240–41microwaveradar,180MidsummerNight’sDream,A(Shakespeare),34Miletus,3–4,7–8,11,33,254MilkyWay,128,176Millikan,Robert,260mirrors,35–38,289–91,348

curved,37–38,79telescopeand,79,219

modernscience,xiii–xiv,254–55beginningof,in17thcentury,189–200Descartesand,212earlyGreeksand,11–12Galileoand,172,190Huygensand,197impersonalnatureof,254Newtonand,216

molecules,249,259,262,266momentum,133–34,232,234–35.Seealsoimpetus

conservationof,362–63Monde,Le(Descartes),203Montaigne,Michelde,46Moon

Arabsand,114,117Aristarchusonsizeanddistanceof,295–301Aristotleon,10,84,10,159brightsideof,63calendarand,59–60Copernicusand,151distancefromEarth,53,63,66–68,72–73,83,94,239,364Earth’sequatorialbulgeand,153eclipsesof,59n,63–64Galileoand,175–76,337–39Greeksand,10,53,53,57,59,77,79–82,84

Keplerand,237Newtonand,196,228–30,235,237,242–44,250,361–62parallaxand,239,323,307–9phasesof,59,66,179–80Ptolemyand,88,91,93–94,117Pythagoreansand,78sizeof,63,68–69,75,83,295–301solareclipseand,298sphericalshapeof,66surfaceof,128,175–76,337–39terminatorand,175–76,337–38tidesand,184–85,242–43Tychoand,160

More,Thomas,46Morison,SamuelEliot,65nMorocco,104–5,116motion.Seealsofallingbodies;momentum;planetarymotion;andspecifictypes

Aristotleon,19,25–29,51,129,133Galileoandexpermentalstudyof,172,187,190–95Greekson,8,19,25–29,51,129,133Huygensand,194–97medievalEuropeand,134–35Newtonandlawsof,225–26,234–36,243–44,254ZenoofEleaon,8

Muhammad,prophet,103,104Müller,Johann.SeeRegiomontanusmultiverse,164–65music,125,171,191

Pythagoreansand,15–17technicalnoteonharmony,279–82

MysteriumCosmographicum(Kepler),162,165,169,171,173

natural,vs.artificial,24–25NaturalQuestions(Adelard),126naturalselection,24,248,265–66navigation,56,75neo-Darwiniansynthesis,266Neoplatonists,5,47,51,80,97–98,127Neptune,250Neugebauer,O.,64,372,373,376neutrinos,9,263neutrons,243,262–64NewAtlantis,The(FrancisBacon),202NewExperimentsPhysico-MechanicalTouchingtheSpringoftheAir(Boyle),200Newton,Isaac,29–30,40,46,69,91,106,146,172,188,199,202,205,212–13,215–55,265,268

backgroundof,215–18calculusand,223–25causeofgravityand,243–44celestialandterrestrialphysicsunifiedby,228,249,260,268centripetalaccelerationand,196,226–30,361–62Descartesand,213diffractionand,205Earth’saxis,74,153Earth’sradiusand,361–62Galileoand,194generalrelativityand,250–53gravitationalconstant(G),238,240–41,287,289,363Huygensand,196–97importanceof,244–45,247–49mechanicsof,152,260momentumand,133Moon’smotionand,93,196,226–30,361–62motionandgravitationand,99,133,136,190,225–45,254,363

oppositiontotheoriesof,245–48opticsand,218–20,222–23planetaryorbitsand,212,225–31,236–41,244–45ratioofmassesofplanetsandSun,238–39,364–65religionand,245–46rotationofEarthandplanetsand,241–42telescopeand,79,219theoryofmatterof,256–57tidesand,242–43Tychoand,251–52

Newton’slawsofmotionfirst,234second,234–35,286–88,363third,234–35,237–38,363

newton(unitofforce),199Nicaea,Councilof,60,218NicholasofCusa,140–41Nicias,46Nicomachus,23Nineveh,battleof,103–4nominalists,132northcelestialpole,56–57,74–75Novara,DomenicoMaria,147NovumOrganum(Bacon),201–2Numbers,R.L.,377

observationAristotleand,24–25,27,64,113,115,185–86benMaimonon,115Copernicusand,91,149–55,158,162,172errorsin,andAristarchus,69–70experimentvs.,189FrancisBaconand,201Galileoandheliocentrism,172–73generalprinciplesanddeductionsblendedwith,202Greektheoriesofmotionsofplanetsand,90,91Grossetesteand,137homocentricmodelsand,86–87Keplerand,166–67,172mathematicsand,20,99medicinevs.physicsand,115–16medievalEuropeand,vs.deductivenaturalscience,132–34moderntheoreticalphysicists,97Newtonand,242,248,250–51planetarymotionand,189Platoand,61–62predictionand,146Ptolemyand,88,90–93,95,115smallconflictswith,151–53,190successofexplanationand,248,254Tychoandaccuracyof,160–61

observatories,118,142occasionalism,121–22,131octahedron,10,12,162,163n,275,279octonions,163Odyssey(Homer),47,56Oldenburg,Henry,219Omar,caliph,103OnArchitecture(Vitruvius),35OnFloatingBodies(Archimedes),19,38–39,66,189,291–94OnNature(Empedocles),6OnParaboloidalBurningMirrors(al-Haitam),110OnSpeeds(Eudoxus),80

OntheEquilibriumofBodies(Archimedes),38OntheForms(Democritus),14OntheHeavensandtheEarth(Oresme),135OntheHeavens(Aristotle),25,27,64,80,127OntheHeavens(Cleomedes),75OntheMeasurementoftheEarth(Eratosthenes),75OntheMotionofBodiesinOrbit(Newton),231OntheNatureofThings(Lucretius),46OntheRepublic(Cicero),17,71OntheRevolutionsoftheHeavenlyBodies(Copernicus),48OntheSizesandDistancesoftheSunandtheMoon(AristarchusofSamos),66–67Ophiuchus,supernovain,166ophthalmology,118Opticks(Newton),223–24,245n,247,256–57Optics(al-Haitam),110Optics(Descartes),206–7Optics(Euclid),35Optics(Ptolemy),37,208optics.Seealsolight;reflection;refraction

Arabsand,110–11,117Descartesand,206–12electricityandmagnetismand,259FrancisBaconand,138Greeksand,33,35–37Grossetesteand,137Huygensand,196–97Keplerand,166medievalEuropeand,137–38Newtonand,218,256–57

OpusMaius(Bacon),174orbitalperiods,364orbits,53,79.Seealsoplanetarymotions;andspecificmoonsandplanets

circularvs.elliptical,8,95,154–55,165nsizesof,149–50,154

Oresme,Nicole,71,132,135–37,139–40,161,191Orion,176orrery,71–72Ørsted,HansChristian,257Orthodoxchurch,61Osiander,Andreas,156–57,182Ostwald,Wilhelm,260Othman,caliph,103OttoIII,emperorofGermany,126OttomanTurks,116overtones,16,281–82OxfordUniversity,131,137–41oxygen,259

Padua,Universityof,134,140–41,147,173,178–80,193Pakistan,104,123Palestine,116parabola,40,194

orbitofcometsand,247trajectoryofprojectilesand,194,342–46

parallax,94,148annual,70,148,160–61,177diurnal,159–60,182,321–23,361lunar,307–9

Paramegmata(calendarsofstars),56Paris,Universityof,125,127–35,138Parmenides,7–9,12,23,63–64PartsofAnimals(Aristotle),24Pascal,Blaise,194,199

pascal(unit),199Paul,Saint,49–50Pauli,Wolfgang,261PaulIII,Pope,48,153PaulV,Pope,183–84Pecham,John,129pendulum,191,195–96,241,246,341–42Pensées(Pascal),199periods

ofplanets,150,227ofJovianmoons,178

Perrin,Jean,260Persia,5,7,103–6,108–11,116PetertheGreat,czarofRussia,245Peucer,Caspar,158Peurbach,Georgvon,141Phaedo(Plato),10phasetransition,267PhilipII,kingofMacedon,22Philolaus,78,153PhiloofAlexandria,62PhiloofByzantium,35PhilosophiaeNaturalisPrincipiaMathematica(Newton),227n,231–47PhilosophicalTransactionsoftheRoyalSociety,219–20,224–25philosophy,3,27–28,101,106,112,141,146,179Philus,LuciusFurius,71Phoenicians,13,56–57photographicemulsions,180photons,220,223,249,261–62Physics(Aristotle),19,23–26,96,127physics.Seealsospecificconcepts

Arabsand,110Aristotleand,23–29,115–16,128,131biologyand,266Galileoand,179Greeksand,19Keplerand,170mathematicsand,20–21,246medievalEuropeand,128,131,137modernscienceand,xiii–xiv,146,170Newtonanddefinitionof,246

physiologi,12pi,39,294–95Piaget,Jean,132Picard,Jean-Félix,228npions,82Pisa,Universityof,134,172,173,178,180Planck,Max,248,261Planckmass,268PlanetaryHypotheses(Ptolemy),94,107,149–50planetsandplanetarysystem.SeealsoEarth;Sun;andspecificconcepts;individualscientists;andplanets

ancientmechanicalmodelof,71–72apparentbrightnessand,86–87,90,142–43,176n–77napparentretrogrademotionand,90,148Arabsand,107,112–14,117–18Aristotleand,78–80,83–87,94–98,131,142–43circularorbitsand,91–93,95,151Copernicusand,90–91,94–95,117–18,142–43,148–58,162,168,172,179–80,182–83,188,255Descartesand,204,212difficultyofunderstanding,98distancesof,89,94,142–43,171n,240,364,365eccentricityoforbits,167–68,168nellipticalorbitsand,151,161–69,172,255,323–28elongationsandorbitsofinner,320–21

emergenceofcorrectdescriptionof,188epicyclictheoryof,xv,303–7equal-arealawandequantsand,323–28Eudoxusand,80–84Galileoand,72,172–88,199,204,212generalrelativityand,251–52gravitationand,134,363–64Greeksand,45,48,53,77–101,151,153–54homocentricmodel,86–87impetusand,133–34instabilityof,245nKeplerand,161–73,188,236–37,248,255Marsastestfortheoriesof,165nmassesof,technicalnoteon,363–64medievalEuropeand,131,133–34,137,142–43Newtonand,225–31,236–45,248–49,251–52oblateshapeof,241orderof,150,154,162paththroughzodiac,57phasesof,142,154,237Ptolemyand,88–98,94,107,118,142–43,149–55,168,172,254–55Pythagoreansand,78–79siderealperiodsand,170–71sizeof,162,176n–77n,239–40sizesoforbitsand,171–72speedoforbitsand,77–78,91,95,150–51,168–69,230,323StandardModeland,249Tycho,160,182,188

Plato,8–13,34,85,138,146Arabsand,105,111,121Aristotlevs.,22–23astronomyand,19–20,51,61,91,117,151,155,264deductionvs.observationand,132FrancisBaconand,202homeworkproblem,79–80Keplerand,163,167,171magnetismand,257mathematicsand,17–20,203matterand,10–13,111,274–79medievalEuropeand,124,127,132religionand,45,47,50

Plato’sUniverse(Vlastos),6nPleiades,55,176Plotinus,47Plutarch,70,153–54Pneumatics(Philo),35poetry,1,12–14Pogson,Norman,88nPolaris(NorthStar),75polygon,275–76,294–95polyhedrons,regular,10–12,15,18,162–64,171,274–79Pope,Alexander,252–53precession

ofapparentorbitofSunaroundEarth,113–14ofequinoxes,74–75,107,118,153,241–42,244,248ofperihelia,241,244,250

prediction,154,242–43,265Priestley,Joseph,11PrinciplesofPhilosophy(Descartes),203–4,212–13PriorAnalytics(Aristotle),17prism,211,218–20probability,theoryof,199Proclus,51,97–98projectiles,thrownobjectsand,27,51,71,133,135,161,193–94,213,342–46

proportions,theoryof,17Protestantism,156–57,166,253protons,243,262–64PrutenicTables,158,166,172Ptolemy,Claudius,48,51,79,330

Arabsand,105–7,110,112–14,117–18,141,160Aristotelianmodelsvs.,95–99,106,112–14,141–43,160chordsvs.sinesand,309–11Copernicusvs.,149–55,255Descartesand,204Earth’srotationand,135epicyclesand,87,255,303–7equal-anglesandequants,37,87,254–55,324–25experimentand,189FrancisBaconand,201Galileoand,172–73,179–80,185Kepler,165,167–68lunarparallaxand,237n,307–9medievalEuropeand,126,128–29,141planetarymotionand,51,71–74,87–96,100,137,254–55refractionand,37,79,137,330Tychoand,159–61,165

PtolemyI,31–32,35PtolemyII,32PtolemyIII,40,75PtolemyIV,40,75PtolemyXV,31pulmonarycirculation,118PunicWar,Second,39Pythagoras,15,47,72Pythagoreans,15–20,72,78–79,111,141n,151,153–54,279–82Pythagoreantheorem,17,283–84

quadraticequations,15quantumchromodynamics,243quantumelectrodynamics,180,263,268quantumfieldtheory,262,263–64quantummechanics,21,34,152,220,248–49,261–65,268quarks,243,263–65quaternions,163Questionesquandamphilosophicae(Newton),217,218Qutb,Sayyid,123

radiation,261radioactivity,260–62,264radioastronomers,159radiowaves,259Ragep,F.J.,374rainbows,25,117,128,207–12,219,342–56rationalnumbers,284realnumbers,163reductionism,267–68reflection,35–37,79,189

Hero’slawof,37,208,210,289–91,348refraction,37,79,189

Arabsand,110,117Descartesand,208–12Fermatand,208Huygens’and,221Keplerand,166lawof,37,206–10,221lawof,least-timederivationof,348–51,358lawof,tennisballderivation,346–48

lawof,wavetheoryderivationof,356–58Newtonand,218–19rainbowand,342–56

Regiomontanus(JohannMüller),141,147Regulaesolvendisophismata(Heytesbury),138Regulus,74nReinhold,Erasmus,158,166,176religion,sciencevs.,1,101

biblicalliteralismand,133–36Copernicansystemand,155–57,170,182–83Descartesand,203earlyChristianityand,48–52Galileoand,182–88Greeksand,44–48Islamicworldand,118–23Keplerand,170medievalcondemnationofAristotleand,127–32Newtonand,215,218,245–46scientificrevolutionand,146

Republic(Plato),13,19–20,61retinaoftheeye,112,174retrogrademotionofplanets,148Rheticus,GeorgJoachim,157Riccioli,GiovanniBattista,184,196,222Richer,Jean,239–40RobertGrossetesteandtheOriginsofExperimentalScience(Crombie),137RomanCatholicchurch,157,179,181

dateforEaster,61Indexofbooks,184,187–88,190,213

RomanInquisition,157,181,181,184–88Romannumerals,107Rome,ancient,1,22,31–32,39,42,47,87,88,125

declineofscienceatendof,48,50–52,118Rømer,Ole,221–22,359Rosen,E.,376,377Ross,W.D.,84RoyalSocietyofLondon,217,219–20,225–26,231,240,245Rubaiyat,The(Khayyam),110,119RudolphII,HolyRomanemperor,161,166RudolphineTables,161,166,172Rushdie,Salman,113Rutherford,Ernest,260

Sagredo,GiovanniFrancesco,186,190Salam,Abdus,123Salviati,Filippo,186,190SandReckoner,The(Archimedes),68–70,300–301Sarton,G.,369Saturn,77–78,83,85,89,94

apparentretrogrademotionof,148Aristotleand,83,85conjunctionofJupiterand,159Copernicusand,148–49,151distancefromSun,163neccentricityoforbit,167epicyclesand,303–6Halley’scometand,247Keplerand,162,163n,171,236–37moonof,195,363,364Newtonand,236–39orbitlength,78Ptolemyand,89,94,149,255ringsof,195

siderealperiodof,171Scaliger,JuliusCaesar,169Scheiner,Christoph,181Schrödinger,Erwin,45,152,248–49,261Schrödingerequation,261Schuster,J.A.,380Schweber,S.,269scientificmethod,xiv,213–14scientificrevolution,xiv,xvi

aftermathof,256–68Copernicusand,145–58criticismsofnotionof,145–46Descartesand,201,203–14experimentationand,189–20014thcenturyandrevoltvs.Aristotelianism,28,132FrancisBaconand,200–202,214Galileoand,172–88,190–94Huygensand,194–97Keplerand,161–72locationandtimingof,118,253–55Newtonasclimaxof,215–5516thto17thcentury,40,145–46Tychoand,158–61,165–66

ScipioAfricanus,17screws,37,38seasons,58–60,81,107,152Seleucus,72Seneca,174Settle,T.B.,379sextant,159sexualselection,266nShakespeare,William,46ShapeoftheEclipse,The(al-Haitam),110Shapin,Steven,145,376,379,380,382ShiiteMuslims,104,120siderealperiod,171

day,85nlunarmonth,59n,228year,118

SideriusNuncius(Galileo),175–78,181,338–39SigerofBrabent,128simplicity,150n,151–53,155,172Simplicius,5,51,80,85–87,90,96,104,186,190sine,107,206–7,210,307

chordsand,309–11wave,280–81

Sirius,55,88Smith,George,93,238,269,373,379,381,382Snell,Willebrord,207Snell’slaw,37,207,221,347–48Socrates,9–10,19–20,61–62,121Solon,4solstices,58,60,75,76nSosigenes,86–87,142Soto,Domingode,140sound,16,44,200

harmonyandwavesof,279–82speedof,24,236,242

spaceandtime,111,233–34,251–52Spain,Islamic,104–5,112–14,116,123,126specialtheoryofrelativity,34,152,204n,251specificgravity,39,41,108,232,293spectrallines,262,359Spengler,Oswald,xiv

sphere,volumeof,39sphericalgeometry,116Spica,74squareroots,17–18Squire,J.C.,252–53StandardModel,21,249,263–65starcatalogs,73–74,88stars,10,55–58,64,80,82–84,107,115,160

annualparallax,70brightnessof,55,79,88,176–77,245–46,265Copernicusand,148–49distanceof,148,161,177Galileoand,176–78Keplerand,164magnitudeof,88n

staticelectricity,257steamengine,Hero’s,36,41Stoics,22,75Strato,32–33,35,64,66,70,140,288stringtheory,14,131strongforces,257,262suction,35,197–98sulfur,11,259Sulla,LuciusCornelius,22SummaTheologica(Aquinas),128Sun

Arabsand,107,114,117–18Aristarchusondistanceof,66–70,75,239,295–301Aristarchusonplanetaryorbitsaround,70,72,85–86,143Aristotleand,84–85Copernicusand,59,86,90–92,95,124,143,148–49,152,172diameterof,83distancefromEarth,53,63,66–68,83,90,94,148,239,240,267,295–301,364diurnalparallax,239Earth’sequatorialbulgeand,153eclipsesof,63eclipticof,throughzodiac,57epicyclesand,90,303–7equinoxesand,74Galileoand,179–83gravitationalfieldof,250–52Greeksandmotionof,10,47,53,55,57–59,77,79–82,84–86,88–95,124,143Heraclidesonplanetaryorbitsaround,85–86,89,124,134,153Keplerandellipticalorbitsaround,167–70,172Newtonandratioofmassesplanetstomassof,238–39Ptolemyand,88–94,172Pythagoreansand,78rotationof,183seasonsand,81shiningof,245–46,265sizeof,53,63,67–70,75,295–301stationary,churchand,vs.Copernicus,183timeofdayand,58Tychoand,160

SunspotLetters(Galileo),180–82sunspots,181–83,186supernova,159,166,173surfacetension,66Swerdlow,N.M.,374,376Swineshead,Richard,138Syene,75–76.301–2SylvesterII,Pope,126symmetry,9,82synodiclunarmonth,59n

Syracuse,37–39,46SystemoftheWorld,The(Newton),236

TablesofToledo(al-Zarqali),114tangent,108,309tauandtheta,decaymodes,82–83taxonomy,24technology,1,34–41,58–59,71n,138,180teleology,23–24,26,36,97,203,265telescope

Arabsand,118focallengthof,329–33Galileoand,172,174–80,336Keplerand,180–81,334magnificationand,333–36Newtonand,79,219

temperingofmusicalscales,282nTempier,Étienne,129terminalvelocity,25–26,286–88terminator,175–76,337Tertullian,50Tetrabiblos(Ptolemy),100tetrahedron,10,12,162,275,278–79Thales,3–5,8,11,15,44,57,254Thales’theorem,272–74Theaetetus,18theodolite,36Theodorus,18TheodosiusI,emperorofRome,48,60–61Theogony(Hesiod),47TheonofAlexandria,51,110TheonofSmyrna,97,99thermodynamics,260,267ThierryofChartres,125–26Thomas,Dylan,13Thomson,J.J.,260ThousandandOneNights,A(al-Rashid),104thrownobjects,27,51,71,133,135,161,193–94,213,342–46Thucydides,46tides,184–86,205,215,242–44Timaeus(Plato),10–12,45,86,97–98,124–25,257,275TimesofLondon,252Timocharis,74Timuriddynasty,118TimurLenk(Tamburlain),118Tischreden(Luther),156Titan(moonofSaturn),195,364Toomer,G.J.,373,374Torricelli,Evangelista,198–200torr(unitofairpressure),199Toynbee,Arnold,xivtranslations,medievalEuropeand,125–27,132TreatiseonLight(Huygens),196–97,220TreatiseontheSmallPoxandMeasles,A(Al-Razi),111triangles,11–13,15,283–84

equilateral,275–77isoscelesright,12,285

triangulation,150TricksoftheProphets(al-Razi),119trigonometry,15,67,68n–69n,107–8,207

distancesofSunandMoonand,296–301lunarparallaxand,307sinesandchordsand,309–11

TropicofCancer,76n,302Tusicouple,117TychoBrahe,40n,107,158–60,165–67,169,176,180,182,184–86,188,204,251–52,303–4,307,322–23Tychonicsystem,185–86,251–52

Ummayadcaliphate,114,116uncertainty,69,197,203–4,239,255unificationofforces,34,228–29,260,266–68unitsofmeasure,240–41universe

expanding,69,83,164,249,265solarsystemseenas,164

universities,127,131Uraniborgobservatory,159–61uranium,11Uranus,250UrbanVII,Pope,184–86UrsaMajor,56–57UrsaMinor,57

vacuum(void),25–26,131,134,134,180,197–200,204vanHelden,A.,372,378,381Vatican,128,179,183velocity

accelerationand,138–39distanceand,223,224nmomentumand,232

Venus,77,245napparentbrightness,87,142–43apparentretrogrademotionof,148Aristotleon,84–85Copernicusand,86,148–51,155distancefromEarth,142–43eccentricityoforbit,167elongationsandorbitof,320–21epicyclesanddeferentsand,303–5Galileoand,179–80,204Greeksand,81–82,84–90,92,94,124Keplerand,162,171phasesof,142–43,179–80,204Ptolemyand,88–90,92,94,155,255siderealperiod,171transitacrossSun,240

Viète,François,206Vitruvius,35Vlastos,Gregory,6nVolta,Alessandro,257Voltaire,204,215n,245,248

Wallace,AlfredRussel,24,265–66water

accelerationoffalling,33bodiessubmergedin,35,38,291–94chemicalcompositionof,259aselement,4,6,10,12,64–66,259falling,33,288–89Platoon,10,12

Watson,Richard,213,381wave,amplitudeandvelocityof,279–81weakforces,257,262Weinberg,L.,269Weinberg,S.,368,371,372,373,377,379,382Westfall,R.S.,242,376,377,381,382

White,AndrewDickson,155,376,377Whiteside,D.T.,381Wigner,Eugene,20,368WilliamIV,landgraveofHesse-Cassel,159WilliamofMoerbeke,129Witten,Edward,20Woodruff,P.,269WorksandDays(Hesiod),55Wren,Christopher,217,231

Xenophanes,6,12,14,46,65

year,lengthof,59–60,107Young,Thomas,222

Zeno,8,11,23zero,126zodiac,57–58,77,81,114

AbouttheAuthor

PhotobyMattValentine

STEVENWEINBERGisatheoreticalphysicistandwinneroftheNobelPrizeinPhysics,theNationalMedalofScience, theLewisThomasPrize for theScientistasPoet,andnumeroushonorarydegreesandotherawards.HeisamemberoftheNationalAcademyofScience,theRoyalSocietyofLondon,the American Philosophical Society, and other academies. A longtime contributor to The New YorkReviewofBooks,he is theauthorofTheFirstThreeMinutes,DreamsofaFinalTheory,FacingUp,andLakeViews,aswellasleadingtreatisesintheoreticalphysics.HeholdstheJoseyRegentalChairinScienceattheUniversityofTexasatAustin.

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LecturesonQuantumMechanicsLakeViewsCosmology

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*AspointedoutbyGregoryVlastos,inPlato’sUniverse(UniversityofWashingtonPress,Seattle,1975),anadverbialformofthewordkosmoswasusedbyHomertomean“sociallydecent”and“morallyproper.”ThisusesurvivesinEnglishintheword“cosmetic.”ItsusebyHeraclitusreflectstheHellenicviewthattheworldisprettymuchwhatitshouldbe.ThewordappearsinEnglishalsointhecognates“cosmos”and“cosmology.”

*Infact(asdiscussedinTechnicalNote2),whatevermayhavebeenprovedbyTheaetetus,Elementsdoesnotprovewhatitclaimstoprove, that there are only five possible convex regular solids. Elements does prove that for regular polyhedrons, there are just fivecombinationsofthenumberofsidesofeachfaceofapolyhedronandofthenumberoffacesthatmeetateachvertex,butitdoesnotprovethatforeachcombinationofthesenumbersthereisjustonepossibleconvexregularpolyhedron.

*TheGreekword kineson,which is usually translated as “motion,” actually has amore general significance, referring to any sort ofchange.ThusAristotle’sclassificationoftypesofcauseappliednotonlytochangeofposition,buttoanychange.TheGreekwordforarefersspecificallytochangeoflocation,andisusuallytranslatedas“locomotion.”

*Itwasgenerallysupposedintheancientworldthatwhenweseesomethingthelighttravelsfromtheeyetotheobject,asifvisionwereasortoftouchingthatrequiresustoreachouttowhatisseen.InthefollowingdiscussionIwilltakeforgrantedthemodernunderstanding,thatinvisionlighttravelsfromtheobjecttotheeye.Fortunately,inanalyzingreflectionandrefraction,itmakesnodifferencewhichwaythelightisgoing.

*TheAssayerisapolemicagainstGalileo’sJesuitadversaries,takingtheformofalettertothepapalchamberlainVirginioCesarini.AswewillseeinChapter11,GalileoinTheAssayerwasattackingthecorrectviewofTychoBraheandtheJesuitsthatcometsarefartherfromEarth than theMoon is. (The quotation here is taken from the translation byMauriceA. Finocchiaro, inTheEssentialGalileo,Hackett,Indianapolis,Ind.,2008,p.183.)

*PierreGassendiwasaFrenchpriestandphilosopherwhotriedtoreconciletheatomismofEpicurusandLucretiuswithChristianity.

*Tobemoreprecise,thisisknownasthe“synodic”lunarmonth.The27-dayperiodfortheMoontoreturntothesamepositionrelativetothefixedstarsisknownasthe“sidereal”lunarmonth.

*Thisdoesnothappeneverymonth,becausetheplaneoftheorbitoftheMoonaroundtheEarthisslightlytiltedwithrespecttotheplaneoftheorbitoftheEartharoundtheSun.TheMooncrossestheplaneoftheEarth’sorbittwiceeverysiderealmonth,butthishappensatfullmoon,whentheEarthisbetweentheSunandtheMoon,onlyaboutonceevery18years.

*TheequinoxisthemomentwhentheSuninitsmotionagainstthebackgroundofstarscrossesthecelestialequator.(Inmodernterms,itis themomentwhen the line between the Earth and the Sun becomes perpendicular to the Earth’s axis.)At points on the Earthwithdifferentlongitude,thismomentoccursatdifferenttimesofday,sotheremaybeaone-daydifferenceinthedatethatdifferentobserversreporttheequinox.SimilarremarksapplytothephasesoftheMoon.

*Ithasbeenargued(inO.Neugebauer,AHistoryofAncientMathematicalAstronomy,Springer-Verlag,NewYork,1975,pp.1093–94),thatAristotle’sreasoningabout theshapeof theEarth’sshadowontheMoonis inconclusive,sinceaninfinitevarietyof terrestrialandlunarshapeswouldgivethesamecurvedshadow.

*SamuelEliotMorisoncitedthisargumentinhisbiographyofColumbus(AdmiraloftheOceanSea,LittleBrown,Boston,Mass.,1942)toshow,contrarytoawidespreadsupposition,thatitwaswellunderstoodbeforeColumbussetsailthattheEarthisasphere.Thedebatein thecourtofCastileoverwhether tosupport theproposedexpeditionofColumbusconcernednot theshapeof theEarth,but itssize.ColumbusthoughttheEarthwassmallenoughsothathecouldsailfromSpaintotheeastcoastofAsiawithoutrunningoutoffoodandwater.HewaswrongaboutthesizeoftheEarth,butofcoursewassavedbytheunexpectedappearanceofAmericabetweenEuropeandAsia.

*ThereisafascinatingremarkbyArchimedesinTheSandReckoner,thatAristarchushadfoundthatthe“Sunappearedtobeabout1/720partofthezodiac”(TheWorksofArchimedes,trans.T.L.Heath,CambridgeUniversityPress,Cambridge,1897,p.223).Thatis,theanglesubtendedonEarthbythediskoftheSunis1/720times360°,or0.5°,notfarfromthecorrectvalue0.519°.Archimedesevenclaimedthathehadverifiedthisbyhisownobservations.Butaswehaveseen,inhissurvivingworkAristarchushadgiventheanglesubtendedbythediskoftheMoonthevalue2°,andhehadnotedthatthedisksoftheSunandMoonhavethesameapparentsize.WasArchimedesquotinga later measurement by Aristarchus, of which no report has survived? Was he quoting his own measurement, and attributing it toAristarchus?Ihaveheardscholarssuggestthatthesourceofthediscrepancyisacopyingerrororamisinterpretationofthetext,butthisseemsveryunlikely.Asalreadynoted,AristarchushadconcludedfromhismeasurementoftheangularsizeoftheMoonthatitsdistancefromtheEarthmustbebetween30and45/2timesgreaterthantheMoon’sdiameter,aresultquiteincompatiblewithanapparentsizeofaround0.5°.Modern trigonometry tellsuson theotherhand that if theMoon’sapparentsizewere2°, then itsdistance fromtheEarthwouldbe28.6timesitsdiameter,anumberthatisindeedbetween30and45/2.(TheSandReckonerisnotaseriousworkofastronomy,butademonstrationbyArchimedesthathecouldcalculateverylargenumbers,suchasthenumberofgrainsofsandneededtofillthesphereofthefixedstars.)

* There is a famous ancient device known as the Antikythera Mechanism, discovered in 1901 by sponge divers off the island ofAntikythera,intheMediterraneanbetweenCreteandmainlandGreece.Itisbelievedtohavebeenlostinashipwrecksometimearound150to100BC.ThoughtheAntikytheraMechanismisnowacorrodedmassofbronze,scholarshavebeenabletodeduceitsworkingsbyX-raystudiesofitsinterior.Apparentlyitisnotanorrerybutacalendricaldevice,whichtellstheapparentpositionoftheSunandplanetsin thezodiaconanydate.Themost important thingabout it is that its intricategearworkprovidesevidenceof thehighcompetenceofHellenistictechnology.

* The celestial latitude is the angular separation between the star and the ecliptic. While on Earth we measure longitude from theGreenwichmeridian, the celestial longitude is the angular separation, on a circle of fixed celestial latitude, between the star and thecelestialmeridianonwhichliesthepositionoftheSunatthevernalequinox.

*OnthebasisofhisownobservationsofthestarRegulus,PtolemyinAlmagestgaveafigureof1°inapproximately100years.

*Eratostheneswaslucky.SyeneisnotpreciselyduesouthofAlexandria(itslongitudeis32.9°E,whilethatofAlexandriais29.9°E)andthenoonSunat thesummersolstice isnotpreciselyoverheadatSyene,butabout0.4° fromthevertical.The twoerrorspartlycancel.WhatEratostheneshadreallymeasuredwastheratioofthecircumferenceoftheEarthtothedistancefromAlexandriatotheTropicofCancer (called the summer tropical circlebyCleomedes), thecircleon theEarth’s surfacewhere thenoonSunat the summer solsticereallyisdirectlyoverhead.Alexandriaisatalatitudeof31.2°,whilethelatitudeoftheTropicofCanceris23.5°,whichislessthanthelatitude ofAlexandria by 7.7°, so the circumference of theEarth is in fact 360°/7.7°= 46.75 times greater than the distance betweenAlexandriaandtheTropicofCancer,justalittlelessthantheratio50givenbyEratosthenes.

*Forthesakeofclarity,whenIrefertoplanetsinthischapter,Iwillmeanjustthefive:Mercury,Venus,Mars,Jupiter,andSaturn.

*WecanseethecorrespondenceofdaysoftheweekwithplanetsandtheassociatedgodsinthenamesofthedaysoftheweekinEnglish.Saturday,Sunday,andMondayareobviouslyassociatedwithSaturn,theSun,andtheMoon;Tuesday,Wednesday,Thursday,andFridayarebasedonanassociationofGermanicgodswithsupposedLatinequivalents:TyrwithMars,WotanwithMercury,ThorwithJupiter,andFriggawithVenus.

*Inayearof365¼days,theEarthactuallyrotatesonitsaxis366¼times.TheSunseemstogoaroundtheEarthonly365¼timesinthisperiod,becauseatthesametimethattheEarthisrotating366¼timesonitsaxis,itisgoingaroundtheSunonceinthesamedirection,giving365¼apparentrevolutionsoftheSunaroundtheEarth.Sinceittakes365.25daysof24hoursfortheEarthtospin366.25timesrelativetothestars,thetimeittakestheEarthtospinonceis(365.25×24hours)/366.25,or23hours,56minutes,and4seconds.Thisisknownasthesiderealday.

*TheapparentluminosityofstarsincatalogsfromPtolemy’stimetothepresentisdescribedintermsoftheir“magnitude.”Magnitudeincreaseswithdecreasingluminosity.Thebrighteststar,Sirius,hasmagnitude-1.4,thebrightstarVegahasmagnitudezero,andstarsthatarejustbarelyvisibletothenakedeyeareofsixthmagnitude.In1856theastronomerNormanPogsoncomparedthemeasuredapparentluminosityofanumberofstarswiththemagnitudesthathadhistoricallybeenattributedtothem,andonthatbasisdecreedthatifonestarhasamagnitudegreaterthananotherby5units,itis100timesdimmer.

*Inoneofthefewhintstotheoriginoftheuseofepicycles,PtolemyatthebeginningofBookXIIoftheAlmagestcreditsApolloniusofPergawithprovingatheoremrelatingtheuseofepicyclesandeccentricsinaccountingfortheapparentmotionoftheSun.

*InthetheoryoftheSun’smotionaneccentriccanberegardedasasortofepicycle,onwhichthelinefromthecenteroftheepicycletotheSunisalwaysparalleltothelinebetweentheEarthandthecenteroftheSun’sdeferent,thusshiftingthecenteroftheSun’sorbitawayfromtheEarth.SimilarremarksapplytotheMoonandplanets.

*Theterm“equant”wasnotusedbyPtolemy.Hereferredinsteadtoa“bisectedeccentric,”indicatingthatthecenterofthedeferentistakentobeinthemiddleofthelineconnectingtheequantandtheEarth.

*Thesameistruewheneccentricsandequantsareadded;observationcouldhavefixedonlytheratiosofthedistancesoftheEarthandtheequantfromthecenterofthedeferentandtheradiiofthedeferentandepicycle,separatelyforeachplanet.

*TheassociationofastrologywiththeBabyloniansisillustratedinOdeXIofBook1ofHorace:“Donotinquire(wearenotallowedtoknow)whatendsthegodshaveassignedtoyouandme,Leoconoe,anddonotmeddlewithBabylonianhoroscopes.Howmuchbettertoendurewhateveritprovestobe.”(Horace,OdesandEpodes,ed.andtrans.NiallRudd,LoebClassicalLibrary,HarvardUniversityPress,Cambridge,Mass.,2004,pp.44–45).ItsoundsbetterinLatin:“Tunequaesieris—scirenefas—quemmihi,quemtibi,finemdidederint,Leuconoë,necBabyloniostemptarisnumerous,utmelius,quidquiderit,pati.”

*His fullname isAbūAbdallāhMuhammadibnMūsāal-Khwārizmī.FullArabnames tend tobe long,soIwillusually justgive theabbreviatednamebywhichthesepersonsaregenerallyknown.Iwillalsodispensewithdiacriticalmarkssuchasbarsovervowels,asinā,whichhavenosignificanceforreaders(likemyself)ignorantofArabic.

*Alfraganusisthelatinizednamebywhichal-FarghanibecameknowninmedievalEurope.Inwhatfollows,thelatinizednamesofotherArabswillbegiven,ashere,inparentheses.

*Al-Biruniactuallyusedamixeddecimalandsexigesimalsystemofnumbers.Hegavetheheightofthemountainincubitsas652;3;18,thatis,652plus3/60plus18/3,600,whichequals652.055inmoderndecimalnotation.

*ButseethefootnoteinChapter10.

*Alaterwriter,GeorgHartmann(1489–1564),claimedthathehadseenaletterbyRegiomontanuscontainingthesentence“Themotionof thestarsmustvarya tinybitonaccountof themotionof theEarth”(DictonaryofScientificBiography,Scribner,NewYork,1975,VolumeII,p.351).Ifthisistrue,thenRegiomontanusmayhaveanticipatedCopernicus,thoughthesentenceisalsoconsistentwiththePythagoreandoctrinethattheEarthandSunbothrevolvearoundthecenteroftheworld.

*Butterfieldcoinedthephrase“theWhiginterpretationofhistory,”whichheusedtocriticizehistorianswhojudgethepastaccordingtoitscontributiontoourpresentenlightenedpractices.Butwhenitcametothescientificrevolution,ButterfieldwasthoroughlyWhiggish,asamI.

*AsmentionedinChapter8,thereisonlyonespecialcaseofthesimplestversionofPtolemy’stheory(withoneepicycleforeachplanet,andnonefortheSun)thatisequivalenttothesimplestversionoftheCopernicantheory,differingonlyinpointofview:itisthespecialcaseinwhichthedeferentsoftheinnerplanetsareeachtakentocoincidewiththeorbitoftheSunaroundtheEarth,whiletheradiioftheepicyclesoftheouterplanetsallequalthedistanceoftheSunfromtheEarth.TheradiioftheepicyclesoftheinnerplanetsandtheradiiofthedeferentsoftheouterplanetsinthisspecialcaseofthePtolemaictheorycoincidewiththeradiiofplanetaryorbitsintheCopernicantheory.

*Thereare120waysofchoosing theorderofany fivedifferent things;anyof the fivecanbe first,anyof the remainingfourcanbesecond,anyoftheremainingthreecanbethird,andanyoftheremainingtwocanbefourth,leavingonlyonepossibilityforthefifth,sothenumberofwaysofarrangingfivethingsinorderis5×4×3×2×1=120.Butasfarastheratioofcircumscribedandinscribedspheresisconcerned,thefiveregularpolyhedronsarenotalldifferent;thisratioisthesameforthecubeandtheoctahedron,andfortheicosahedronandthedodecahedron.Hencetwoarrangementsofthefiveregularpolyhedronsthatdifferonlybytheinterchangeofacubeandanoctahedron,orofanicosahedronandadodecahedron,givethesamemodelofthesolarsystem.Thenumberofdifferentmodelsistherefore120/(2×2)=30.

*Forinstance,ifacubeisinscribedwithintheinnerradiusofthesphereofSaturn,andcircumscribedabouttheouterradiusofthesphereof Jupiter, then the ratio of theminimumdistance of Saturn from theSun and themaximumdistance of Jupiter from theSun,whichaccordingtoCopernicuswas1.586,shouldequalthedistancefromthecenterofacubetoanyofitsverticesdividedbythedistancefromthecenterofthesamecubetothecenterofanyofitsfaces,or√3=1.732,whichis9percenttoolarge.

*ThemotionofMarsistheidealtestcaseforplanetarytheories.UnlikeMercuryorVenus,Marscanbeseenhighinthenightsky,whereobservations are easiest. In any given span of years, itmakesmanymore revolutions in its orbit than Jupiter or Saturn.And its orbitdepartsfromacirclemorethanthatofanyothermajorplanetexceptMercury(whichisneverseenfarfromtheSunandhenceisdifficulttoobserve),sodeparturesfromcircularmotionatconstantspeedaremuchmoreconspicuousforMarsthanforotherplanets.

*ThemaineffectoftheellipticityofplanetaryorbitsisnotsomuchtheellipticityitselfasthefactthattheSunisatafocusratherthanthecenteroftheellipse.Tobeprecise,thedistancebetweeneitherfocusandthecenterofanellipseisproportionaltotheeccentricity,whilethevariationinthedistanceofpointsontheellipsefromeitherfocusisproportionaltothesquareoftheeccentricity,whichforasmalleccentricitymakesitmuchsmaller.Forinstance,foraneccentricityof0.1(similartothatoftheorbitofMars)thesmallestdistanceoftheplanetfromtheSunisonly½percentsmallerthanthelargestdistance.Ontheotherhand,thedistanceoftheSunfromthecenterofthisorbitis10percentoftheaverageradiusoftheorbit.

*ThisisJuliusCaesarScaliger,apassionatedefenderofAristotleandopponentofCopernicus.

*AsubsequentdiscussionshowsthatbythemeandistanceKeplermeant,notthedistanceaveragedovertime,butrathertheaverageoftheminimumandmaximumdistancesoftheplanetfromtheSun.AsshowninTechnicalNote18,theminimumandmaximumdistancesof a planet from theSun are (1 -e)a and (1+e)a,wheree is the eccentricity anda is half the longer axis of the ellipse (that is, thesemimajoraxis),sothemeandistanceisjusta.ItisfurthershowninTechnicalNote18thatthisisalsothedistanceoftheplanetfromtheSun,averagedoverthedistancetraveledbytheplanetinitsorbit.

*Focallengthisalengththatcharacterizestheopticalpropertiesofalens.Foraconvexlens,itisthedistancebehindthelensatwhichrays thatenter the lens inparalleldirectionsconverge.Foraconcave lens thatbendsconverging rays intoparalleldirections, the focallengthisthedistancebehindthelensatwhichtherayswouldhaveconvergedifnotforthelens.Thefocallengthdependsontheradiusofcurvatureofthelensandontheratioofthespeedsoflightinairandglass.(SeeTechnicalNote22.)

*TheangularsizeofplanetsislargeenoughsothatthelinesofsightfromdifferentpointsonaplanetarydiskarefartherapartastheypassthroughtheEarth’satmospherethanthesizeoftypicalatmosphericfluctuations;asaresult,theeffectsofthefluctuationsonthelightfromdifferentlinesofsightareuncorrelated,andthereforetendtocancelratherthanaddcoherently.Thisiswhywedonotseeplanetstwinkle.

*ItwouldhavepainedGalileotoknowthatthesearethenamesthathavesurvivedtothepresent.TheyweregiventotheJoviansatellitesin1614bySimonMayr,aGermanastronomerwhoarguedwithGalileooverwhohaddiscoveredthesatellitesfirst.

*PresumablyGalileowasnotusingaclock,butratherobservingtheapparentmotionsofstars.Sincethestarsseemtogo360°aroundtheEarthinasiderealdayofnearly24hours,a1°changeinthepositionofastarindicatesapassageoftimeequalto1/360times24hours,or4minutes.

*This isactually trueonly for swingsof thependulum throughsmallangles, thoughGalileodidnotnote thisqualification. Indeedhespeaksofswingsof50°or60°(degreesofarc)takingthesametimeasmuchsmallerswings,andthissuggeststhathedidnotactuallydoalltheexperimentsonthependulumthathereported.

*Takenliterally,thiswouldmeanthatabodydroppedfromrestwouldneverfall,sincewithzeroinitialvelocityattheendofthefirstinfinitesimalinstantitwouldnothavemoved,andhencewithspeedproportionaltodistancewouldstillhavezerovelocity.Perhapsthedoctrinethatthespeedisproportionaltothedistancefallenwasintendedtoapplyonlyafterabriefinitialperiodofacceleration.

*OneofGalileo’sargumentsisfallacious,becauseitappliestotheaveragespeedduringanintervaloftime,nottothespeedacquiredbytheendofthatinterval.

*ThisisshowninTechnicalNote25.Asexplainedthere,thoughGalileodidnotknowit,thespeedoftheballrollingdowntheplaneisnotequal tothespeedofabodythatwouldhavefallenfreelythesameverticaldifference,becausesomeoftheenergyreleasedbytheverticaldescentgoesintotherotationoftheball.Butthespeedsareproportional,soGalileo’squalitativeconclusionthatthespeedofafallingbodyisproportionaltothetimeelapsedisnotchangedwhenwetakeintoaccounttheball’srotation.

*Descartescomparedlighttoarigidstick,whichwhenpushedatoneendinstantaneouslymovesattheotherend.Hewaswrongaboutstickstoo,thoughforreasonshecouldnotthenhaveknown.Whenastickispushedatoneend,nothinghappensattheotherenduntilawaveofcompression(essentiallyasoundwave)hastraveledfromoneendofthesticktotheother.Thespeedofthiswaveincreaseswiththerigidityofthestick,butEinstein’sspecialtheoryofrelativitydoesnotallowanythingtobeperfectlyrigid;nowavecanhaveaspeedexceeding that of light. Descartes’ use of this sort of comparison is discussed by Peter Galison, “Descartes Comparisons: From theInvisibletotheVisible,”Isis75,311(1984).

*Recallthatthesineofanangleisthesideoppositethatangleinarighttriangle,dividedbythehypotenuseofthetriangle.Itincreasesastheangleincreasesfromzeroto90°,inproportiontotheangleforsmallangles,andthenmoreslowly.

*Thisisdonebyfindingthevalueofb/Rwhereaninfinitesimalchangeinbproducesnochangeinφ,sothatatthatvalueofφthegraphofφversusb/Risflat.Thisisthevalueofb/Rwhereφreachesitsmaximumvalue.(Anysmoothcurvelikethegraphofφagainstb/Rthatrisestoamaximumandthenfallsagainmustbeflatatthemaximum.Apointwherethecurveisnotflatcannotbethemaximum,sinceifthecurveatsomepointrisestotherightorlefttherewillbepointstotherightorleftwherethecurveishigher.)Valuesofφintherangewherethecurveofφversusb/Risnearlyflatvaryonlyslowlyaswevaryb/R,sotherearerelativelymanyrayswithvaluesofφinthisrange.

*Inhisfifties,Newtonhiredhishalfsister’sbeautifuldaughter,CatherineBarton,ashishousekeeper,butthoughtheywereclosefriendstheydonotseemtohavebeenromanticallyattached.Voltaire,whowasinEnglandatthetimeofNewton’sdeath,reportedthatNewton’sdoctor and “the surgeon inwhose armshe died” confirmed toVoltaire thatNewtonnever had intimacieswith awoman (seeVoltaire,Philosophical Letters,Bobbs-Merrill Educational Publishing, Indianapolis, Ind., 1961, p. 63).Voltaire did not say how the doctor andsurgeoncouldhaveknownthis.

*Thisisfromaspeech,“Newton,theMan,”thatKeyneswastogiveatameetingattheRoyalSocietyin1946.Keynesdiedthreemonthsbeforethemeeting,andthespeechwasgivenbyhisbrother.

*Newtondevotedacomparableefforttoexperimentsinalchemy.Thiscouldjustaswellbecalledchemistry,asbetweenthetwotherewasthennomeaningfuldistinction.AsremarkedinconnectionwithJabiribnHayyaninChapter9,untilthelateeighteenthcenturytherewasnoestablishedchemical theory thatwould ruleout theaimsofalchemy, like the transmutationofbasemetals intogold.AlthoughNewton’sworkonalchemythusdidnotrepresentanabandonmentofscience,itledtonothingimportant.

*Aflatpieceofglassdoesnotseparatethecolors,becausealthougheachcolorisbentbyaslightlydifferentangleonenteringtheglass,theyareallbentbacktotheiroriginaldirectiononleavingit.Becausethesidesofaprismarenotparallel,lightraysofdifferentcolorthatarerefracteddifferentlyonenteringtheglassreachtheprism’ssurfaceonleavingtheprismatanglesthatarenotequaltotheanglesofrefractiononentering,sowhentheseraysarebentbackonleavingtheprismthedifferentcolorsarestillseparatedbysmallangles.

*Thisisthe“naturallogarithm”of1+x,thepowertowhichtheconstante=2.71828. . .mustberaisedtogivetheresult1+x.Thereason for this peculiar definition is that the natural logarithmhas some properties that aremuch simpler than those of the “commonlogarithm,”inwhich10takestheplaceofe.Forinstance,Newton’sformulashowsthatthenaturallogarithmof2isgivenbytheseries1-½+⅓-¼+...,whiletheformulaforthecommonlogarithmof2ismorecomplicated.

*Theneglectoftheterms3to2ando3inthiscalculationmaymakeitseemthatthecalculationisonlyapproximate,butthatismisleading.In the nineteenth centurymathematicians learned to dispensewith the rather vague idea of an infinitesimalo, and to speak insteadofpreciselydefinedlimits:thevelocityisthenumbertowhich[D(t+o)-D(t)]/ocanbemadeascloseaswelikebytakingosufficientlysmall.Aswewillsee,Newtonlatermovedawayfrominfinitesimalsandtowardthemodernideaoflimits.

*Kepler’sthreelawsofplanetarymotionwerenotwellestablishedbeforeNewton,thoughthefirstlaw—thateachplanetaryorbitisanellipsewiththeSunatonefocus—waswidelyaccepted.ItwasNewton’sderivationoftheselawsinthePrincipiathatledtothegeneralacceptanceofallthreelaws.

*ThefirstreasonablyprecisemeasurementofthecircumferenceoftheEarthwasmadearound1669byJean-FélixPicard(1620–1682),andwasusedbyNewtonin1684toimprovethiscalculation.

*Newtonwasunabletosolvethethree-bodyproblemoftheEarth,Sun,andMoonwithenoughaccuracytocalculatethepeculiaritiesinthemotionoftheMoonthathadworriedPtolemy,Ibnal-Shatir,andCopernicus.Thiswasfinallyaccomplishedin1752byAlexis-ClaudeClairaut,whousedNewton’stheoriesofmotionandgravitation.

* InBook III of theOpticks,Newton expressed the view that the solar system is unstable, and requires occasional readjustment.Thequestionofthestabilityofthesolarsystemremainedcontroversialforcenturies.Inthelate1980sJacquesLaskarshowedthatthesolarsystemischaotic;itisimpossibletopredictthemotionsofMercury,Venus,Earth,andMarsformorethanabout5millionyearsintothefuture.Someinitialconditionsleadtoplanetscollidingorbeingejectedfromthesolarsystemafterafewbillionyears,whileothersthatarenearlyindistinguishabledonot.Forareview,seeJ.Laskar,“IstheSolarSystemStable?,”www.arxiv.org/1209.5996(2012).

*Maxwellhimselfdidnotwriteequationsgoverningelectricandmagneticfieldsintheformknowntodayas“Maxwell’sequations.”Hisequationsinsteadinvolvedotherfieldsknownaspotentials,whoseratesofchangewithtimeandpositionaretheelectricandmagneticfields.ThemorefamiliarmodernformofMaxwell’sequationswasgivenaround1881byOliverHeaviside.

*HereandinwhatfollowsIwillnotciteindividualphysicists.Somanyareinvolvedthatitwouldtaketoomuchspace,andmanyarestillalive,sothatIwouldriskgivingoffensebycitingsomephysicistsandnotothers.

*Iamherelumpingsexualselectiontogetherwithnaturalselection,andpunctuatedequilibriumalongwithsteadyevolution;andIamnotdistinguishingbetweenmutationsandgeneticdriftasasourceofinheritablevariations.Thesedistinctionsareveryimportanttobiologists,buttheydonotaffectthepointthatconcernsmehere:thereisnoindependentlawofbiologythatmakesinheritablevariationsmorelikelytobeimprovements.

*ThismaynothavebeenknowninthetimeofThales,inwhichcasetheproofmustbeofalaterdate.

*ThisisfromthestandardtranslationbyT.L.Heath(Euclid’sElements,GreenLionPress,SantaFe,N.M.,2002,p.480).

*Forapianostringtherearesmallcorrectionsduetothestiffnessofthestring;thesecorrectionsproducetermsinvproportionalto1/L3.Iwillignorethemhere.

*InsomemusicalscalesmiddleGisgivenaslightlydifferentfrequencyinordertomakepossibleotherpleasantchordsinvolvingmiddleG.Theadjustmentoffrequenciestomakeasmanychordsaspossiblepleasantiscalled“tempering”thescale.

*ThistableappearsinthetranslationoftheAlmagestbyG.J.Toomer(Ptolemy’sAlmagest,Duckworth,London,1984,pp.57–60).

*Galileousedadefinitionof“mile”thatisnotverydifferentfromthemodernEnglishmile.Inmodernunits,theradiusoftheMoonisactually1,080miles.