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A History of Mathematics This page intentionally left blank A History of Mathematics From Mesopotamia to Modernity Luke Hodgkin 1 3 GreatClarendonStreet,OxfordOX26DP OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland CapeTown DaresSalaam HongKong Karachi KualaLumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto Withofficesin Argentina Austria Brazil Chile CzechRepublic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore SouthKorea Switzerland Thailand Turkey Ukraine Vietnam OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork ©OxfordUniversityPress,2005 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2005 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India PrintedinGreatBritain onacid-freepaperby AntonyRoweLtd.,Chippenham,Wiltshire ISBN0–19–852937–6(Hbk) 978–0–19–852937–8 1 3 5 7 9 10 8 6 4 2 Preface This book has its origin in notes which I compiled for a course on the history of mathematics at King’s College London, taught for many years before we parted company. My major change in outlook (which is responsible for its form) dates back to a day ten years ago at the University of Warwick, whenIwascomparingnotesonteachingwiththelateDavidFowler. Heexplainedhis own history of mathematics course to me; as one might expect, it was detailed, scholarly, and encouragedstudentstodoresearchof theirown,particularlyontheGreeks.ItoldhimthatIgave what I hoped was a critical account of the whole history of mathematics in a series of lectures, trying to go beyond what they would find in a textbook. David was scornful. ‘What’, he said, ‘doyoumeanthatyoustandupinfrontof thosestudentsandtellstories?’ Ihadtoacknowledge thatIdid. David’sapproachmeantthatstudentsshouldbetaughtfromthestartnottoacceptanystoryat facevalue,andtobeinterestedinquestionsratherthannarrative.It’scertainlydesirableasregards theGreeks,andit’sagoodapproachingeneral,evenif itmaysometimesseemtoodifficultandtoo purist.Ihopehewouldnotbetoohardonmyattemptsatacompromise.Theaimsof thebookin this, itsultimateform, aresetoutintheintroduction; briefly, Ihopetointroducestudentstothe history,orhistoriesof mathematicsasconstructionswhichwemaketoexplainthetextswhichwe have,andtorelatethemtoourownideas.Suchconstructionsareoftencontroversial,andalways provisional;butthatisthenatureof history. TheoriginalimpulsetowritecamefromDavidRobinson,mycollaboratoronthecourseatKing’s, whosuggested(unsuccessfully)thatIshouldturnmycoursenotesintoabook;andprovidentially fromAlisonJonesof theOxfordUniversityPress, whoturnedupatKing’swhenIwasataloose endandaskedif Ihadabooktopublish.Iproducedaproposal;shepersuadedthepresstoaccept itandkeptmewriting.Withoutherconstantfeedbackandinvolvementitwouldneverhavebeen completed. Iamgratefultoanumberof friendsforadviceandencouragement. JeremyGrayreadanearly draft and promoted the project as a referee; the reader is indebted to him for the presence of exercises.GeoffreyLloydgaveexpertadviceontheGreeks;Iamgratefulforallof it,evenif Ionly paidattentiontosome.JohnCairns,FelixPiraniandGervaseFletcherreadpartsof themanuscript and made helpful comments; various friends and relations, most particularly Jack Goody, John Hope,JessicaHinesandSamandJoeGoldHodgkinexpressedawishtoseethefinishedproduct. Finally, I’m deeply grateful to my wife Jean who has supported the project patiently through writingandrevision.Toher,andtomyfatherThomaswhoIhopewouldhaveapproved,thisbook isdedicated. This page intentionally left blank Contents Listof figures xi PictureCredits xiv Introduction 1 Whythisbook? 1 Ontexts,andonhistory 2 Examples 5 Historicismand‘presentism’ 6 Revolutions,paradigms,andallthat 8 Externalversusinternal 10 Eurocentrism 12 1. Babylonianmathematics 14 1. Onbeginnings 14 2. Sourcesandselections 17 3. Discussionof theexample 20 4. Theimportanceof number-writing 21 5. Abstractionanduselessness 24 6. Whatwentbefore 27 7. Someconclusions 30 AppendixA.Solutionof thequadraticproblem 30 Solutionstoexercises 31 2. Greeksand‘origins’ 33 1. PlatoandtheMeno 33 2. Literature 35 3. Anexample 36 4. Theproblemof material 39 5. TheGreekmiracle 42 6. Tworevolutions? 44 7. Drowningintheseaof Non-identity 45 8. Onmodernizationandreconstruction 47 9. Onratios 49 AppendixA.FromtheMeno 51 AppendixB.Onpentagons,goldensections,andirrationals 52 Solutionstoexercises 54 viii Contents 3. Greeks,practicalandtheoretical 57 1. Introduction,andanexample 57 2. Archimedes 60 3. HeronorHero 63 4. Astronomy,andPtolemyinparticular 66 5. OntheunculturedRomans 69 6. Hypatia 71 AppendixA.FromHeron’sMetrics 73 AppendixB.FromPtolemy’sAlmagest 75 Solutionstoexercises 76 4. Chinesemathematics 78 1. Introduction 78 2. Sources 80 3. Aninstanthistoryof earlyChina 80 4. TheNineChapters 82 5. Countingrods—whoneedsthem? 85 6. Matrices 88 7. TheSongdynastyandQinJiushao 90 8. On‘transfers’—when,andhow? 95 9. Thelaterperiod 98 Solutionstoexercises 99 5. Islam,neglectanddiscovery 101 1. Introduction 101 2. Onaccesstotheliterature 103 3. Twotexts 106 4. Thegoldenage 108 5. Algebra—theorigins 110 6. Algebra—thenextsteps 115 7. Al-Samaw’alandal-Ka¯sh¯i 117 8. Theusesof religion 123 AppendixA.Fromal-Khwa¯rizm¯i’salgebra 125 AppendixB.Tha¯bitibnQurra 127 AppendixC.Fromal-Ka¯sh¯i,TheCalculator’sKey,book4,chapter7 128 Solutionstoexercises 130 6. Understandingthe‘scientificrevolution’ 133 1. Introduction 133 2. Literature 134 3. Scholasticsandscholasticism 135 4. Oresmeandseries 138 5. Thecalculatingtradition 140 6. Tartagliaandhisfriends 143 7. Onauthority 146 Contents ix 8. Descartes 149 9. Infinities 151 10. Galileo 153 AppendixA 155 AppendixB 156 AppendixC 157 AppendixD 158 Solutionstoexercises 159 7. Thecalculus 161 1. Introduction 161 2. Literature 163 3. Theprioritydispute 165 4. TheKeralaconnection 167 5. Newton,anunknownwork 169 6. Leibniz,aconfusingpublication 172 7. ThePrincipiaanditsproblems 176 8. Thearrivalof thecalculus 178 9. Thecalculusinpractice 180 10. Afterword 182 AppendixA.Newton 183 AppendixB.Leibniz 185 AppendixC.FromthePrincipia 186 Solutionstoexercises 187 8. Geometriesandspace 189 1. Introduction 189 2. Firstproblem:thepostulate 194 3. Spaceandinfinity 197 4. Sphericalgeometry 199 5. Thenewgeometries 201 6. The‘time-lag’question 203 7. Whatrevolution? 205 AppendixA.Euclid’spropositionI.16 207 AppendixB.Theformulaeof sphericalandhyperbolictrigonometry 209 AppendixC.FromHelmholtz’s1876paper 210 Solutionstoexercises 210 9. Modernityanditsanxieties 213 1. Introduction 213 2. Literature 214 3. Newobjectsinmathematics 214 4. Crisis—whatcrisis? 217 5. Hilbert 221 6. Topology 223