Ibn al-Haytham’s theory of conics, geometrical constructions and practical geometry. A history of Arabic sciences and mathematics 3

760 Pages·2013·3.09 MB·English

by  Roshdi Rashed

IBN AL-HAYTHAM’S THEORY
OF CONICS,
GEOMETRICAL CONSTRUCTIONS
AND PRACTICAL GEOMETRY
This book provides a unique primary source on the history and philosophy
of mathematics and science from the medieval Arab world. The present text
is complemented by two preceding volumes of A History of Arabic Sciences
and Mathematics, which focused on founding figures and commentators in the
ninthandtenthcenturies,andthehistoricalandepistemologicaldevelopmentof
‘infinitesimal mathematics’ as it became clearly articulated in the oeuvre of Ibn
al-Haytham.
Thisvolumeexaminestheincreasingtendency,aftertheninthcentury,toexplain
mathematical problems inherited from Greek times using the theory of conics.
RoshdiRashedarguesthatIbnal-Haythamcompletesthetransformationofthis
‘areaofactivity’intoapartofgeometryconcernedwithgeometricalconstructions,
dealingnotonlywiththemetricalpropertiesofconicsectionsbutalsowithways
ofdrawingthemandpropertiesoftheirpositionandshape.
Includingextensivecommentaryfromoneoftheworld’sforemostauthorities
onthesubject,thisbookcontributesamoreinformedandbalancedunderstanding
of the internal currents of the history of mathematics and the exact sciences in
Islam,andoftheiradaptiveinterpretationandassimilationintheEuropeancon-
text.Thisfundamentaltextwillappealtohistoriansofideas,epistemologistsand
mathematiciansatthemostadvancedlevelsofresearch.
Roshdi Rashed is one of the most eminent authorities on Arabic mathematics
and the exact sciences. A historian and philosopher of mathematics and science
andahighlycelebratedepistemologist,heiscurrentlyEmeritusResearchDirector
(distinguishedclass)attheCentreNationaldelaRechercheScientifique(CNRS)
in Paris, and is the Director of the Centre for History of Medieval Science and
PhilosophyattheUniversityofParis(DenisDiderot,ParisVII).Healsoholdsan
HonoraryProfessorshipattheUniversityofTokyoandanEmeritusProfessorship
attheUniversityofMansourahinEgypt.
J. V. Field is a historian of science and is a Visiting Research Fellow in
the Department of History of Art and Screen Media, Birkbeck, University of
London,UK.
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
andcivilisations.Comprisingoriginalmonographsaswellasscholarlysurveys,it
coverstopicsinthefieldsofMiddleEasternliterature,archaeology,law,history,
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ralityofviews,theseriespresentsseriousscholarshipinalucidandstimulating
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PREVIOUSLYPUBLISHEDBYCURZON
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PERSIANLYRICPOETRY
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2. THEHANBALISCHOOLOFLAWANDIBNTAYMIYYAH
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Annotatedtranslationofal-T.a¯hiral-H.adda¯d’sImra’tuna¯ fi‘l-shar¯ı awa
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’l-mujtama,withanintroduction
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SelectedEasternwritings
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15. EARLYISLAMICSPAIN
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16. GERMANORIENTALISM
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19. AL-GHAZALI,AVERROESANDTHEINTERPRETATION
OFTHEQUR’AN
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20. EASTERNCHRISTIANITYINTHEMODERNMIDDLEEAST
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25. THEUAEANDFOREIGNPOLICY
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Barany,underthesupervisionofNaderEl-Bizri)
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32. THEOLOGICALAPPROACHESTOQUR’ANICEXEGESIS
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33. IBNAL-HAYTHAMANDANALYTICALMATHEMATICS
AhistoryofArabicsciencesandmathematics,Volume2
RoshdiRashed(translatedbySusanGlynnandRogerWareham)
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CONSTRUCTIONSANDPRACTICALGEOMETRY
AhistoryofArabicsciencesandmathematics,Volume3
RoshdiRashed,translatedbyJ.V.Field
IBN AL-HAYTHAM’S THEORY
OF CONICS,
GEOMETRICAL CONSTRUCTIONS
AND PRACTICAL GEOMETRY
A history of Arabic sciences and mathematics
Volume 3
Roshdi Rashed
Translated by J. V. Field
Firstpublished2013
byRoutledge
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CONTENTS
Foreword .................................................................................. xiii
Preface ..................................................................................... xv
INTRODUCTION: CONIC SECTIONS AND GEOMETRICAL
CONSTRUCTIONS .......................................................................... 1
CHAPTER I: THEORY OF CONICS AND GEOMETRICAL CONSTRUCTIONS:
‘COMPLETION OF THE CONICS’
1.1. INTRODUCTION ....................................................................... 9
1.1.1. Ibn al-Haytham and Apollonius’ Conics ...................................... 9
1.1.2. The eighth book of the Conics .................................................. 10
1.1.3. The Completion of the Conics: the purpose of the enterprise .............. 27
1.1.4. History of the text ................................................................ 32
1.2. MATHEMATICAL COMMENTARY ................................................ 38
1.3. TRANSLATED TEXT: On the Completion of the Conics ........................ 171
CHAPTER II: CORRECTING THE BANª MªSÆ’S LEMMA FOR
APOLLONIUS’ CONICS
2.1. INTRODUCTION ....................................................................... 247
2.2. MATHEMATICAL COMMENTARY ................................................ 248
2.3. HISTORY OF THE TEXT ............................................................. 269
2.4. TRANSLATED TEXT: On a Proposition of the Banº Mºsæ ..................... 273
CHAPTER III: PROBLEMS OF GEOMETRICAL CONSTRUCTION ............... 289
3.1. THE REGULAR HEPTAGON ........................................................ 289
3.1.1. Introduction ....................................................................... 289
3.1.2. The traces of a work by Archimedes on the regular heptagon .............. 292
3.1.3. A priority dispute: al-Sijzî against Abº al-Jºd ............................... 300
3.1.4. The lemmas for the construction of the heptagon: the division
of a segment ....................................................................... 314
3.1.4.1. Archimedes’ division (D ) .............................................. 315
1
3.1.4.1.1. First stage: the division in the text attributed to Archimedes.. 316
3.1.4.1.2. Second stage: Ibn Sahl ............................................ 318
3.1.4.1.3. Third stage: al-Qºhî and al-∑æghænî ............................. 321
3.1.4.1.3.1. Al-Qºhî: the first treatise .................................... 321
3.1.4.1.3.2. Al-∑æghænî(cid:3)..................................................... 326
3.1.4.1.3.3. Al-Qºhî: the second treatise .............................. 333
3.1.4.2. The range studied by Abº al-Jºd and al-Sijzî (D ) ................... 341
2
3.1.4.3. Abº al-Jºd’s range (D ) ................................................. 351
3
3.1.4.4. Comparing the ranges: Abº al-Jºd, al-Shannî,
Kamæl al-Dîn ibn Yºnus ........................................................... 353
3.1.4.5. Ibn al-Haytham’s ranges (D and D ) ................................. 360
4 5