Sourcebook in the Mathematics of Medieval Europe and North Africa

592 Pages·2016·6.81 MB·English

by  Victor J. Katz et al. (eds.)

Sourcebook in the
Mathematics of Medieval
Europe and North Africa

Sourcebook in the
Mathematics of Medieval
Europe and North Africa
Edited by
VICTOR J. KATZ
MENSO FOLKERTS
BARNABAS HUGHES
ROI WAGNER
J. LENNART BERGGREN
PRINCETON UNIVERSITY PRESS • PRINCETON AND OXFORD
Copyright(cid:2)c 2016byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,
Woodstock,OxfordshireOX201TR
press.princeton.edu
AllRightsReserved
ISBN978-0-691-15685-9
LibraryofCongressControlNumber:2016935714
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinTimesRomanandArial
Printedonacid-freepaper.∞
PrintedintheUnitedStatesofAmerica
TypesetbyNovaTechsetPvtLtd,Bangalore,India
1 3 5 7 9 10 8 6 4 2
Contents
Preface xi
Permissions xiii
GeneralIntroduction 1
Chapter 1. The Latin Mathematics of Medieval Europe
4
MensoFolkertsandBarnabasHughes
Introduction 4
ILatinSchools,800–1140 6
I-1.BriefSelections 6
1.TheQuadriviumofMartianusCapella 6
2.TheQuadriviumofCassiodorus 9
3.TheQuadriviumofIsidoreofSeville 10
4.ANewCreation:RulesforAdditionofSignedNumbers 11
I-2.Numbering 12
1.Romannumerals 12
2.Fingerreckoning 13
3.IsidoreofSeville,Libernumerorum(BookofNumbers) 18
4.Hindu-Arabicnumerals 20
I-3.Arithmetic 22
1.Boethius,Deinstitutionearithmetica(IntroductiontoArithmetic) 22
2.Computus 29
3.“Gerbert’sjump” 36
4.PandulfofCapua,Liberdecalculatione 39
I-4.Geometry 44
1.Gerbert,Geometry 44
2.Gerbert:Areaofanequilateraltriangle 48
3.Unitsofmeasurement 49
vi Contents
4.FrancoofLiège,Dequadraturacirculi 50
5.HughofSt.Victor,Practicageometriae 53
I-5.RecreationalMathematics 55
1.Numberpuzzles 55
2.Thoughtproblems 57
3.TheJosephusProblem 58
4.Themostimportantmedievalnumbergame:theRithmimachia 59
II.ASchoolBecomesaUniversity:1140–1480 64
II-1.Translations 66
1.Translatorsandtranslations 66
2.Translations:Apracticalillustration 70
II-2.Arithmetic 75
1.Al-Khwa¯rizm¯ı,Arithmetic 75
2.LeonardoofPisa(Fibonacci),Liberabbaci(BookonCalculation) 79
3.JohnofSacrobosco,Algorismusvulgaris 85
4.JohannesdeLineriis,Algorismusdeminuciis 93
5.JordanusdeNemore(Nemorarius),Deelementisarithmeticeartis 96
6.Combinatoricsandprobability,DeVetula 100
II-3.Algebra 103
1.Al-Khwa¯rizm¯ı,Algebra 103
2.LeonardoofPisa,Liberabbaci(BookonCalculation) 106
3.LeonardoofPisa,BookofSquares 112
4.JordanusdeNemore,Denumerisdatis(OnGivenNumbers) 116
5.NicoleOresme,Algorismusproportionum(AlgorithmofRatios) 119
6.NicoleOresme,Deproportionibusproportionum(OntheRatioof
Ratios) 121
II-4.Geometry 123
1.Banu¯Mu¯sa¯ibnSha¯kir,TheBookoftheMeasurementofPlaneand
SphericalFigures 123
2.Abu¯Bakr,Libermensurationum(OnMeasurement) 126
3.LeonardoofPisa,Depracticageometrie(PracticalGeometry) 130
4.JohnofMurs,Deartemensurandi 140
5.JordanusdeNemore,Liberphilotegni 143
6.DominicusdeClavasio,Practicageometriae 146
II-5.Trigonometry 148
1.Ptolemy,OntheSizeofChordsinaCircle 149
2.LeonardoofPisa,Depracticageometrie(PracticalGeometry) 153
3.JohannesdeLineriis,Canones 155
4.RichardofWallingford,Quadripartitum 157
5.GeoffreyChaucer,ATreatiseontheAstrolabe 160
6.Regiomontanus,OnTriangles 162
Contents vii
II-6.Mathematicsoftheinfinite 173
1.Angleofcontingence 174
2.ThomasBradwardine,Tractatusdecontinuo(OntheContinuum) 178
3.JohnDunsScotus,IndivisiblesandTheology 180
4.Doeslighttravelinstantaneouslyorovertime? 182
5.NicoleOresme,QuestionessupergeometriamEuclidis(Questionsonthe
GeometryofEuclid) 184
II-7.Statics,Dynamics,andKinematics 185
1.RobertGrosseteste,Delineis,angulisetfiguris(Onlines,anglesand
figures) 185
2.JordanusdeNemore,Derationeponderis(OntheTheoryofWeights) 186
3.ThomasBradwardine,Tractatusdeproportionibus 189
4.WilliamHeytesbury,Regulesolvendisophismata(RulesforSolving
Sophisms) 191
5.GiovannidiCasali,Develocitatemotusalterationis(OntheVelocityof
MotionofAlteration) 194
6.NicoleOresme,Deconfigurationibusqualitatumetmotuum(Onthe
ConfigurationsofQualitiesandMotions) 197
III.AbbacistSchools:1300–1480 207
III-1.ForeignExchange 208
III-2.Geometry 209
III-3.Algebra 210
1.GiliodaSiena,ALectureinIntroductoryAlgebra 210
2.PaoloGirardi,LibrodiRagioni 211
3.JacobodaFirenze,Tractatusalgorismi 212
4.MasterDardi,Newequationssolved 213
Sources 216
References 221
Chapter 2. Mathematics in Hebrew in Medieval Europe
224
RoiWagner
Introduction 224
I.PracticalandScholarlyArithmetic 227
1.AbrahamibnEzra,SeferHamispar(TheBookofNumber) 227
2.AaronbenIsaac,Arithmetic 235
3.ImmanuelbenJacobBonfils,Ondecimalnumbersandfractions 237
(cid:2)
4.JacobCanpant.on,BarNotenT.aam 239
5.ElijahMizrah.i,SeferHamispar(TheBookofNumber) 244
(cid:2)
6.LevibenGershon,MaaseH.oshev(TheArtoftheCalculator) 253
viii Contents
II.Numerology,Combinatorics,andNumberTheory 268
(cid:3)
1.AbrahamibnEzra,SeferHaeh.ad(TheBookofOne) 269
(cid:2)
2.AbrahamibnEzra,SeferHaolam(BookoftheWorld) 271
(cid:2)
3.LevibenGershon,MaaseH.oshev(TheArtoftheCalculator) 273
4.LevibenGershon,OnHarmonicNumbers 277
5.QalonymosbenQalonymos,SeferMelakhim(BookofKings) 283
6.DonBenvenistebenLavi,Encyclopedia 284
7.AaronbenIsaac,Arithmetic 285
III.MeasurementandPracticalGeometry 286
1.AbrahamibnEzra(?),SeferHamidot(TheBookofMeasure) 287
2.AbrahambarH.iyya,H.iburHameshih.aVehatishboret(TheTreatiseon
MeasuringAreasandVolumes) 296
3.RabbiShlomoIs.h.aqi(Rashi),OntheMeasurementsofthe
TabernacleCourt 313
4.SimonbenS.emah.,Responsa165concerningSolomon’sSea 315
5.LevibenGershon,Astronomy 320
IV.ScholarlyGeometry 326
1.LevibenGershon,CommentaryonEuclid’sElements 326
2.LevibenGershon,TreatiseonGeometry 335
3.QalonymosbenQalonymos,OnPolyhedra 337
4.ImmanuelbenJacobBonfils,MeasurementoftheCircle 339
5.SolomonbenIsaac,OntheHyperbolaandItsAsymptote 340
(cid:2)
6.AbnerofBurgos(AlfonsodiValladolid),SeferMeyasher Aqov
(BookoftheRectifyingoftheCurved) 345
V.Algebra 354
1.Quadraticwordproblems 354
2.SimonMot.ot.,Algebra 358
3.Ibnal-Ah.dab,IgeretHamispar(TheEpistleoftheNumber) 362
Sources 374
References 376
Chapter 3. Mathematics in the Islamic World in
Medieval Spain and North Africa
381
J.LennartBerggren
Introduction 381
I.Arithmetic 385
(cid:3)
1.Ibnal-Banna¯ ,Arithmetic 385
(cid:2)
2. Al¯ıb.Muh.ammadal-Qalas.a¯d¯ı,RemovingtheVeilfromtheScienceof
Calculation 398
3.Muh.ammadibnMuh.ammadal-Fulla¯n¯ıal-Kishna¯w¯ı,Onmagicsquares 407
Contents ix
II.Algebra 410
(cid:3)
1.Ah.madibnal-Banna¯ ,Algebra 410
2.Muh.ammadibnBadr,AnAbridgementofAlgebra 422
III.Combinatorics 427
(cid:2)
1.Ah.madibnMunim,Fiqhal-h.isa¯b(OntheScienceofCalculation) 427
(cid:3)
2.Ibnal-Banna¯ onCombinatorics,RaisingtheVeil 446
3.Shiha¯bal-D¯ınibnal-Majd¯ı,Onenumeratingpolynomialequations 449
IV.Geometry 452
(cid:2) (cid:2)
1.Abu¯ AbdAllahMuh.ammadibn Abdu¯n,OnMeasurement 452
2.Abu¯ al-Qa¯simibnal-Samh.,ThePlaneSectionsofaCylinderandthe
DeterminationofTheirAreas 456
(cid:2) (cid:2)
3.Abu¯ AbdAllahMuh.ammadibnMua¯dhal-Ja¯yya¯n¯ı,Onratios 468
(cid:3)
4.Al-MutamanibnHu¯d,Kita¯bal-Istikma¯l(BookofPerfection) 478
5.Muh.y¯ıal-D¯ınibnAb¯ıal-Shukral-Maghrib¯ı,RecensionofEuclid’s
Elements 494
V.Trigonometry 502
(cid:2) (cid:2)
1.Abu¯ AbdAllahMuh.ammadibnMua¯dhal-Jayya¯n¯ı,BookofUnknowns
ofArcsoftheSphere 502
(cid:2) (cid:2)
2.Abu¯ AbdAllahMuh.ammadibnMua¯dhal-Jayya¯n¯ı,OnTwilightandthe
RisingofClouds 520
(cid:2) (cid:2)
3.Abu¯ AbdAllahMuh.ammadibnMua¯dhal-Jayya¯n¯ı,Ontheqibla 530
4.Ibra¯h¯ımibnal-Zarqa¯lluh,Onauniversalastrolabe 533
5.Abu¯ Muh.ammadJa¯biribnAflah.,CorrectionoftheAlmagest 539
Sources 544
References 546
Appendices
Appendix1.ByzantineMathematics 549
1.MaximusPlanudes,TheGreatCalculationAccordingtotheIndians 551
2.ManuelMoschopoulos,OnMagicSquares 554
3.IsaacArgyros,OnSquareRoots 559
4.Anonymousfifteenth-centurymanuscriptonarithmetic 561
Sources 562
Appendix2.DiophantusArithmetica,BookI,#24 563
Appendix3.FromtheGanitasa¯rasangrahaofMahavira 563
Appendix4.TimeLine 564
EditorsandContributors 567
Index 571