The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge

836 Pages·2007·20.45 MB·English

by  Kirsti Andersen (auth.)

Sources and Studies
in the History of Mathematics and
Physical Sciences
EditorialBoard
IZ. Buchwald I Liitzen I Hogendijk
Advisory Board
P.I Davis T. Hawkins
A.E. Shapiro D. Whiteside
SourcesandStudiesinthe
HistoryofMathematicsandPhysicalSciences
K.Andersen
BrookTaylor'sWorkonLinearPerspective
H.IM.Bos
Redefming GeometricalExactness: Descartes'Transformationofthe Early ModernConceptof
Construction
I Cannon/SoDostrowsky
TheEvolutionofDynamics:VibrationTheoryFrom1687to1742
B.ChandlerlW.Magnus
TheHistoryofCombinatorialGroupTheory
AI.Dale
HistoryofInverseProbability:FromThomasBayestoKarlPearson,SecondEdition
AI.Dale
Pierre-SimondeLaplace,PhilosophicalEssayonProbabilities,Translatedfrom thefifth French
editionof1825,withNotesbytheTranslator
A Dale
MostHonourableRemembrance:TheLifeandWorkofThomasBayes
P.IFederico
DescartesOnPolyhedra:AStudyoftheDeSolidorumElementa
B.R.Goldstein
TheAstronomyofLeviBenGerson(1288-1344)
H.H.Goldstine
AHistoryofNumericalAnalysisfromthe16thThroughthe19thCentury
H.H.Goldstine
AHistoryoftheCalculusofVariationsFromthe17thThroughthe19thCentury
G.GraBhoff
TheHistoryofPtolemy'sStarCatalogue
A HermannlK.vonMeyennfv.F.Weisskopf(Eds.)
WolfgangPauli:ScientificCorrespondenceI:1919-1929
A HermannlK.vonMeyennfv.F.Weisskopf(Eds.)
WolfgangPauli:ScientificCorrespondenceIT:1930-1939
C.C.Heyde/E.Seneta,I.J.
Bienayme:StatisticalTheoryAnticipated
IP.Hogendijk
IbnA1-Haytham'sCompletionoftheConics
I H0yrup
Length,Widths,Surfaces:APortraitofOldBabylonianAlgebraandItsKin
A.Jones
PappusofAlexandria,Book7oftheCollection
ContinuedafterthelIIustrationCredits
Kirsti Andersen
The Geometry of an Art
The History of the Mathematical
Theory of Perspective from Alberti
to Monge
~
Springer
KirstiAndersen
DepartmentofHistoryofScience
TheStenoInstitute
UniversityofAarhus
Denmark
SourcesandSeriesEditor:
JesperLiitzen
InstituteforMathematicalSciences
UniversityofCopenhagen
DK-2100Copenhagen
Denmark
LibraryofCongressControlNnmber:2005927076
ISBN10:0-387-25961-9
ISBN13:978-0387-25961-1
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Contents
Introduction xix
KeyIssues xix
QuestionsConcerningtheHistoryofGeometricalPerspective xix
QuestionsConcerningTextbooksonPerspective xix
TheWord'Perspective' xx
OtherPublications xxi
ThePeriodandRegionsExamined XXI
TheSourcesandHowTheyAreUsed xxi
ContextsandRestrictions xxii
Conclusions xxiii
Acknowledgements xxv
Colleagues,Students,andFriends xxv
Institutions xxvi
SourcesofFunding xxvi
Libraries xxvii
Notestothe Reader xxix
DrawingsandNotation xxix
ConceptsRelatedtotheEyePointandthePicturePlane xxix
ConceptsRelatedtoImagesofPoints,Lines,andPlanes xxx
Orthogonals,Transversals,andVerticals XXXi
Rabatment xxxii
MathematicalTerminology, Results, andTechniques xxxii
LinesandLineSegments xxxii
Resultsfrom theTheoryofProportion xxxiii
MathematicalTechniques xxxiii
ThePlacementoftheMathematicalExplanations xxxiv
Bibliographies xxxiv
TwoBibliographies xxxiv
References,Orthography,andOrderingofLetters xxxiv
Biographies xxxv
DatesfortheProtagonists xxxvi
MyText xxxvi
QuotationsandParaphrases xxxvi
UseofMyEarlierPublications xxxvii
vii
viii Contents
ChapterI. TheBirthofPerspective 1
11. TheFirstWrittenAccountofGeometricalPerspective 1
12. TheOriginofPerspective 2
1.3. FourStimuli 3
PaintingaView 3
RepresentationofSpecialLines 4
A SearchforMathematicalRules 10
InspirationfromOptics 10
1.4. Brunelleschi 11
FourPossibleTechniques 11
Brunelleschi'sConceptionofPerspective 13
NoConclusion 13
Brunelleschi'sSuccess 14
1.5. PerspectiveBeforetheRenaissance? 15
ChapterII. AlbertiandPierodellaFrancesca 17
Ill. TheTwoEarliestAuthors 17
112. AlbertiandHisWork 17
Alberti'sViewsontheArtofPainting 18
11.3. Alberti'sModel 19
Alberti'sTwoMethodsofProducingaPerspectiveImage 21
114. Alberti'sConstruction 22
TheRepresentationofOrthogonals 22
AnOpenWindow 23
A ScaledUnit 24
PlacementoftheCentricPoint 25
TheImagesoftheTransversals 25
ChoiceofParameters 28
Alberti'sUseofa PerspectiveGrid 28
11.5. Alberti'sTheoreticalReflectionsandHisDiagonalRule 29
11.6. TheThirdDimensioninAlberti'sConstruction 33
11.7. Alberti'sConstructioninHistory 34
11.8. PierodellaFrancescaandHisWork 34
DeProspectivaPingendi 36
11.9. TheTheoreticalFoundationofDeProspectiva 37
TheAngleAxiom 37
ForeshorteningofOrthogonalsandLineSegmentsParallelto n 38
PieroonVisualDistortion 40
11.10. PieroandAlberti'sConstruction 40
Piero'sRabatment 40
PieroontheCorrectnessoftheConstruction 42
FilareteandFrancescodiGiorgio 43
11.11. Piero'sDiagonalConstruction 44
11.12. Piero'sDistancePointConstruction 46
TheOriginofDistancePointConstructions 46
PieroontheCorrectnessofHisDistancePointConstruction 48
1113. TheDivisionTheorem 50
11.14. Piero'sTreatmentoftheThirdDimension 50
Contents ix
11.15. TheColumnProblem 51
EquidistantLineSegments 53
ColumnsonSquareBases 54
CylindricalColumns 56
Piero'sConsiderations 56
11.16. Piero'sPlanandElevationConstruction 59
TheOriginofthePlanandElevationTechnique 59
Piero'sConstruction 60
11.17. Piero'sCube 64
Piero'sIdea 66
Piero'sIllustrations 66
Piero'sHeads 71
11.18. Piero'sAnamorphoses 71
11.19. Piero'sUseofPerspective 75
11.20. Piero'sInfluence 79
ChapterIII. Leonardoda Vinci 81
111.1. LeonardoandtheHistoryofPerspective 81
Leonardo's Trattato 82
Leonardo'sApproachtoPerspective 83
OutlineofThisChapter 84
111.2. Leonardo'sVariousConceptsofPerspective 84
LinearPerspectiveVersusOtherConceptsofPerspective 85
NaturalVersusAccidentalPerspective 86
CompositeandSimplePerspective 87
111.3. VisualAppearancesandPerspectiveRepresentations 88
IlIA. LeonardoonVisualAppearancesofLengths 89
Leonardo'sAxiomandtheAngleAxiom 89
TheLawofInverseProportionality 90
PacioliandtheLawofInverseProportionality 94
TheLawofInverseProportionalityandEuclid'sTheory 95
LeonardoontheAppearanceofa Rectangle 96
TheAppearanceoftheVerticalBoundaries 97
TheAppearanceofCollinearLineSegments 98
111.5. LeonardoonPerspectiveRepresentations 100
ThePerspectiveImagesofParticularLineSegments 101
ThePerspectiveImagesofCollinearLineSegments 102
LeonardoandtheColumnProblem 105
Leonardo'sAppealforaLargeViewingDistance 107
111.6. LeonardoandCurvilinearPerspective 107
111.7. Leonardo'sDoubtsandTheirConsequences 111
PerspectiveandVisualImpressions 111
FixedEyePoint 111
Leonardo'sUseofPerspective 112
ChapterIV. ItalyintheCinquecento 115
IV.I. TheItalianSixteenth-CenturyPerspectivists 115
x Contents
IY.2. TheArchitectural,Painting,andSculptingTraditions 116
Gaurico 116
Serlio 116
Sirigatti,Cataneo,andPeruzzi 122
Lomazzo 124
IY.3. A MathematicalApproachtoPerspective
TheContributionsbyVignolaandDanti 125
TheFirstEditionofVignola'sWorkonPerspective 125
Vignola'sPlanandElevationConstruction 126
Vignola'sDistancePointConstruction 128
Vignola'sComparisonofHisTwoMethods 130
DantionConvergencePoints 136
IY.4. ConnectionBetweenPerspectiveandAnotherCentral
Projection- Commandino'sContributions 138
TheContextofCommandino'sWork 138
Commandino'sConstructions 141
Commandino'sInfluence 145
IY.5. AnotherMathematicalApproach- Benedetti'sContributions 146
Benedetti'sAlbertiConstruction 146
BenedettionPointwiseConstructions 147
BenedettiandConvergencePoints 149
Benedetti'sInfluence 152
IY.6. AnEncyclopediaonPerspective- Barbaro'sBook 152
Barbaro'sSources 152
BarbaroontheRegularPolyhedra 155
IY.7. TheItalianPre-1600ContributionstoPerspective 158
ChapterV; NorthoftheAlpsBefore1600 161
Y.I. TheIntroductionofPerspectiveNorthoftheAlps 161
Y.2. ViatorandHisFollowers 161
Viator 162
Ringelberg 166
Cerceau 169
Y.3. Cousin 172
Cousin'sIntroductionofa DistancePointConstruction 175
Cousin'sUseofPointsofConvergence 178
CousinontheColumnProblem 182
Y.4. Durer 183
Durer'sIntroductionto Perspective 183
Durer'sBooks 188
Diirer'sPlanandElevationConstruction 194
Diirer'sEnigmaticMethod 197
TheSecondMethodasDescribed 199
TheSecondMethodasIllustrated 200
TheSecondMethodandAlbertiConstructions 201
TheSecondMethodanda DistancePointConstruction 202
TheDiagramsIllustratingtheSecondMethod 204
FinishingtheImagebytheSecondMethod 204
ConstructionoftheSidefg 205
Contents xi
Diirer'sProgramme 206
TheLessonofDiirer'sMistakes 207
RestrictionsInducedbyPerspective 207
Diirer'sPracticalMethods 207
Diirer'sDiagonalMethod 210
Diirer'sInfluenceontheDevelopmentofPerspective 212
V5. Diirer'sGermanSuccessors 212
PerspectiveTouchedUponbyaPainter 213
PerspectivePresentedbyaCount 213
HirschvogelandLautensack 217
RyffTakingUptheItalianTradition 222
Jamnitzer,Lencker, Stor,andHass 224
Pfinzing 230
V6. VredemandeVries 230
V7. TheSixteenth-CenturyNon-ItalianTableau 236
ChapterVI. TheBirthoftheMathematicalTheory
ofPerspectiveGuidobaldoandStevin 237
VI.1. GuidobaldoandHisWorkonPerspective 237
Guidobaldo'sStrugglewithPerspective 238
TheContentsofPerspectivaeLibriSex 240
VI.2. Guidobaldo'sTheoryofPerspective 241
LineSegmentsParalleltothePicturePlane 242
TheMainTheoremofPerspective 244
Guidobaldo'sProofsoftheMainTheorem 246
VanishingLines 249
VI.3. Guidobaldo'sTwenty-ThreeMethods 250
Guidobaldo'sRabatment 251
TheSixthMethod 251
TheTenthMethod 255
TheTwenty-FirstMethod 256
VIA. NewThemesinGuidobaldo'sWork 256
UntraditionalPicturePlanes 257
InverseProblemsofPerspective 259
DirectConstructions 261
VI.5. Guidobaldo'sRoleintheHistoryofPerspective 262
VI.6. StevinandHisWorkonPerspective 265
Stevin'sPathtoPerspective 268
TheContentsof Van de Verschaeuwing 269
VI.7. TheFoundationofStevin'sTheory 270
TheInvarianceTheorem 271
VI.8. Stevin'sPracticeofPerspective 273
Stevin'sBasicConstructions 273
Stevin'sRabatment 276
Stevin'sExamples 277
VI.9. StevinandInverseProblemsofPerspective 279
VLlO. FurtherIssuesinStevin'sWork 282
TheContentsofStevin'sAppendix 282
TheColumnProblem 284