FOUNDING FIGURES AND COMMENTATORS IN ARABIC MATHEMATICS Inthisuniqueinsightintothehistoryandphilosophyofmathematicsandscienceinclassical Islamiccivilisation,theeminentscholarRoshdiRashedilluminatesthevarioushistorical, textual and epistemic threads that underpinned the history of Arabic mathematical and scientific knowledge up to the seventeenth century. The first of five wide-ranging and comprehensivevolumes,thisbookprovidesadetailedexplorationofArabicmathematics andsciencesintheninthandtenthcenturies. ExtensiveanddetailedanalysesandannotationssupportanumberofkeyArabictexts, which are translated here into English for the first time. In this volume Rashed focuses on the traditions of celebrated polymaths from the ninth- and tenth century ‘School of Baghdad’ – such as the Banu¯ Mu¯sa¯, Tha¯bit ibn Qurra, Ibra¯h¯ım ibn Sina¯n, Abu¯ Ja‘far al-Kha¯zin, Abu¯ Sahl Wayjan ibn Rusta¯m al-Qu¯h¯ı – and eleventh-century Andalusian mathematicians such as Abu¯ al-Qa¯sim ibn al-Samh. and al-Mu’taman ibn Hu¯d. The Archimedean–Apolloniantraditionsofthesepolymathsarethematicallyexploredtoillus- trate the historical and epistemological development of ‘infinitesimal mathematics’ as it becamemoreclearlyarticulatedintheeleventh-centuryinfluentiallegacyofal-Hasanibn al-Haytham(‘Alhazen’). Contributingtoamoreinformedandbalancedunderstandingoftheinternalcurrentsof thehistoryofmathematicsandtheexactsciencesinIslam,andofitsadaptiveinterpretation andassimilationintheEuropeancontext,thisfundamentaltextwillappealtohistoriansof ideas,epistemologistsandmathematiciansatthemostadvancedlevelsofresearch. Roshdi Rashed is one of the most eminent authorities on Arabic mathematics and the exact sciences. A historian and philosopher of mathematics and science and a highly celebratedepistemologist,heiscurrentlyEmeritusResearchDirector(distinguishedclass) attheCentreNationaldelaRechercheScientifique(CNRS)inParis,andistheDirector oftheCentreforHistoryofMedievalScienceandPhilosophyattheUniversityofParis (DenisDiderot,ParisVII). NaderEl-BizriisaReaderattheUniversityofLincoln,andaChercheurAssociéatthe CentreNationaldelaRechercheScientifiqueinParis(CNRS,UMR7219).Hehaslectured on‘ArabicSciencesandPhilosophy’attheUniversityofCambridgesince1999.Heheld aVisitingProfessorshipattheUniversityofLincoln(2007–2010),andsince2002hehas beenaseniorResearchAssociateaffiliatedwithTheInstituteofIsmailiStudies,London. [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:i i–viii CULTURE AND CIVILIZATION IN THE MIDDLE EAST GeneralEditor:IanRichardNetton ProfessorofIslamicStudies,UniversityofExeter This series studies the Middle East through the twin foci of its diverse cultures andcivilizations.Comprisingoriginalmonographsaswellasscholarlysurveys,it coverstopicsinthefieldsofMiddleEasternliterature,archaeology,law,history, philosophy,science,folklore,art,architectureandlanguage.Whilethereisaplu- ralityofviews,theseriespresentsseriousscholarshipinalucidandstimulating fashion. PREVIOUSLYPUBLISHEDBYCURZON THEORIGINSOFISLAMICLAW TheQur’an,theMuwatta’andMadinanAmal YasinDutton AJEWISHARCHIVEFROMOLDCAIRO ThehistoryofCambridgeUniversity’sGenizahcollection StefanReif THEFORMATIVEPERIODOFTWELVERSHI’ISM HadithasdiscoursebetweenQumandBaghdad AndrewJ.Newman QUR’ANTRANSLATION Discourse,textureandexegesis HusseinAbdul-Raof CHRISTIANSINAL-ANDALUS711–1000 AnnRosemaryChristys FOLKLOREANDFOLKLIFEINTHEUNITEDARABEMIRATES SayyidHamidHurriez [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:ii i–viii THEFORMATIONOFHANBALISM Pietyintopower NimrodHurvitz ARABICLITERATURE Anoverview PierreCachia STRUCTUREANDMEANINGINMEDIEVALARABICAND PERSIANLYRICPOETRY Orientpearls JulieScottMeisami MUSLIMSANDCHRISTIANSINNORMANSICILY Arabic-speakersandtheendofIslam AlexanderMetcalfe MODERNARABHISTORIOGRAPHY Historicaldiscourseandthenation-state YoussefChoueiri THEPHILOSOPHICALPOETICSOFALFARABI,AVICENNA ANDAVERROES TheAristotelianreception SalimKemal PUBLISHEDBYROUTLEDGE 1. THEEPISTEMOLOGYOFIBNKHALDUN ZaidAhmad 2. THEHANBALISCHOOLOFLAWANDIBNTAYMIYYAH Conflictorconcilation AbdulHakimIAl-Matroudi 3. ARABICRHETORIC Apragmaticanalysis HusseinAbdul-Raof 4. ARABREPRESENTATIONSOFTHEOCCIDENT East–WestencountersinArabicfiction RasheedEl-Enany [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:iii i–viii 5. GODANDHUMANSINISLAMICTHOUGHT Abdal-Jabba¯r,IbnS¯ına¯ andal-Ghaza¯l¯ı MahaElkaisy-Friemuth 6. ORIGINALISLAM MalikandtheMadhhabofMadina YasinDutton 7. AL-GHAZALIANDTHEQUR’AN Onebook,manymeanings MartinWhittingham 8. BIRTHOFTHEPROPHETMUHAMMAD DevotionalpietyinSunniIslam MarionHolmesKatz 9. SPACEANDMUSLIMURBANLIFE AtthelimitsofthelabyrinthofFez SimonO’Meara 10. ISLAMANDSCIENCE TheintellectualcareerofNizamal-Dinal-Nizaburi RobertG.Morrison 11. IBN‘ARABΖTIMEANDCOSMOLOGY MohamedHajYousef 12. THESTATUSOFWOMENINISLAMICLAWANDSOCIETY Annotatedtranslationofal-T.a¯hiral-H.adda¯d’sImra’tuna¯ fi (cid:2) (cid:2) ‘l-shar¯ı awa‘l-mujtama,withanintroduction RonakHusniandDanielL.Newman 13. ISLAMANDTHEBAHA’IFAITH AcomparativestudyofMuhammad‘Abduhand‘Abdul-Baha‘Abbas OliverScharbrodt 14. COMTEDEGOBINEAUANDORIENTALISM SelectedEasternwritings TranslatedbyDanielO’DonoghueEditedbyGeoffreyNash 15. EARLYISLAMICSPAIN ThehistoryofIbnal-Qu¯t.¯ıya DavidJames [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:iv i–viii 16. GERMANORIENTALISM ThestudyoftheMiddleEastandIslamfrom1800to1945 UrsulaWokoeck 17. MULLA¯ S.ADRA¯ ANDMETAPHYSICS Modulationofbeing SajjadH.Rizvi 18. SCHOOLSOFQUR’ANICEXEGESIS Genesisanddevelopment HusseinAbdul-Raof 19. AL-GHAZALI,AVERROESANDTHEINTERPRETATION OFTHEQUR’AN CommonsenseandphilosophyinIslam AvitalWohlman,translatedbyDavidBurrell 20. EASTERNCHRISTIANITYINTHEMODERNMIDDLEEAST EditedbyAnthonyO’MahonyandEmmaLoosley 21. ISLAMICREFORMANDARABNATIONALISM ExpandingthecrescentfromtheMediterraneantothe IndianOcean(1880s–1930s) AmalN.Ghazal 22. ISLAMICETHICS DivinecommandtheoryinArabo-Islamicthought Mariamal-Attar 23. MUSLIMFORTRESSESINTHELEVANT BetweenCrusadersandMongols KateRaphael 24. BEINGHUMANINISLAM Theimpactoftheevolutionaryworldview DamianHoward 25. THEUAEANDFOREIGNPOLICY Foreignaid,identitiesandinterests KhalidS.Almezaini 26. AHISTORYOFEARLYAL-ANDALUS TheAkhba¯rmajmu¯’a DavidJames [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:v i–viii 27. INSPIREDKNOWLEDGEINISLAMICTHOUGHT Al-Ghazali’stheoryofmysticalcognitionanditsAvicennianfoundation AlexanderTreiger 28. SHI’ITHEOLOGYINIRAN Thechallengeofreligiousexperience OriGoldberg 29. FOUNDINGFIGURESANDCOMMENTATORSINARABIC MATHEMATICS AhistoryofArabicsciencesandmathematics,Volume1 RoshdiRashed(translationseditedbyNaderEl-Bizri) 30. THEMUSLIMCONQUESTOFIBERIA MedievalArabicnarratives NicolaClarke 31. ANGELSINISLAM Jalalal-Dinal-Suyuti’sal-Haba’ikfiakhbaral-mala’ik StephenBurge [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:vi i–viii FOUNDING FIGURES AND COMMENTATORS IN ARABIC MATHEMATICS A history of Arabic sciences and mathematics Volume 1 Roshdi Rashed Edited by Nader El-Bizri Translated by Roger Wareham, with Chris Allen and Michael Barany [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:vii i–viii Firstpublished2012 byRoutledge 2ParkSquare,MiltonPark,Abingdon,OX144RN SimultaneouslypublishedintheUSAandCanada byRoutledge 711ThirdAvenue,NewYork,NY10017(8thFloor) RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness ©2012RoutledgeandCentreforArabUnityStudies TherightofRoshdiRashedtobeidentifiedastheauthorhasbeenasserted inaccordancewithsections77and78oftheCopyright,Designsand PatentsAct1988. Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedor utilisedinanyformorbyanyelectronic,mechanical,orothermeans,now knownorhereafterinvented,includingphotocopyingandrecording,orin anyinformationstorageorretrievalsystem,withoutpermissioninwriting fromthepublishers. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationData Acataloguerecordforthisbookhasbeenrequested ISBN978-0-415-58217-9(hbk) ISBN978-0-203-63659-6(ebk) TypesetinTimesNewRoman byCenveoPublisherServices Thisbookwaspreparedfrompress-readyfilessuppliedbytheeditor. [13:4031/8/20115689-Rashed-prelims.tex] JobNo:5517 Rashed:SecondLanguageAcquisition Page:viii i–viii CONTENTS Editor’s Foreword ........................................................................ xiii Preface...................................................................................... xix Note ........................................................................................xxiv CHAPTER I : BANª MªSÆ AND THE CALCULATION OF THE VOLUME OF THE SPHERE AND THE CYLINDER 1.1. INTRODUCTION ....................................................................... 1 1.1.1. The Banº Mºsæ: dignitaries and learned ...................................... 1 1.1.2. The mathematical works of the Banº Mºsæ ................................... 7 1.1.3. Treatise on the measurement of plane and spherical figures: a Latin translation and a rewritten version by al-™ºsî ........................ 10 1.1.4. Title and date of the Banº Mºsæ treatise ....................................... 34 1.2. MATHEMATICAL COMMENTARY ................................................ 38 1.2.1. Organization and structure of the Banº Mºsæ book .......................... 38 1.2.2. The area of the circle ............................................................. 40 1.2.3. The area of the triangle and Hero’s formula .................................. 46 1.2.4. The surface area of a sphere and its volume .................................. 47 1.2.5. The two-means problem and its mechanical construction ................... 60 1.2.6a. The trisection of angles and Pascal’s Limaçon .............................. 66 1.2.6b. Approximating cubic roots .................................................... 69 1.3. TRANSLATED TEXT: On the Knowledge of the Measurement of Plane and Spherical Figures........................................................................ 73 CHAPTER II: THÆBIT IBN QURRA AND HIS WORKS IN INFINITESIMAL MATHEMATICS 2.1. INTRODUCTION ....................................................................... 113 2.1.1. Thæbit ibn Qurra: from Îarræn to Baghdad ................................... 113 2.1.2. The works of Thæbit ibn Qurra in infinitesimal mathematics ............... 122 2.1.3. History of the texts and their translations ..................................... 124 2.2. MEASURING THE PARABOLA ................................................... 130 2.2.1. Organization and structure of Ibn Qurra’s treatise ........................... 130 2.2.2. Mathematical commentary ....................................................... 133 2.2.2.1. Arithmetical propositions .............................................. 133 2.2.2.2. Sequence of segments and bounding ................................ 142 2.2.2.3. Calculation of the area of a portion of a parabola ................... 154 2.2.3. Translated text: On the Measurement of the Conic Section Called Parabola ......................................................................... 169 2.3. MEASURING THE PARABOLOID .................................................. 209 2.3.1. Organization and structure of Ibn Qurra’s treatise ............................ 209 2.3.2. Mathematical commentary ....................................................... 214 2.3.2.1. Arithmetical propositions .............................................. 214 2.3.2.2. Extension to sequences of segments ................................. 218 2.3.2.3. Volumes of cones, rhombuses and other solids .................... 223 2.3.2.4. Property of four segments ............................................ 230 2.3.2.5. Arithmetical propositions .............................................. 231