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

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Springer Science & Business Media, 23 de nov. 2008 - 814 pàgines
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Key Issues ver since the late 1970s when Pia Holdt, a student of mine at the time, and Jed Buchwald, a colleague normally working in another field, made E me aware of how fascinating the history of perspective constructions is, I have wanted to know more. My studies have resulted in the present book, in which I am mainly concerned with describing how the understanding of the geometry behind perspective developed and how, and to what extent, new insights within the mathematical theoryof perspective influenced the way the discipline was presented in textbooks. In order to throw light on these aspects of the history of perspective, I have chosen to focus upon a number of key questions that I have divided into two groups. Questions Concerning the History of Geometrical Perspective • How did geometrical constructions of perspective images emerge? • How were they understood mathematically? • How did the geometrical constructions give rise to a mathematical theory of perspective? • How did this theory evolve? Inconnectionwith the last question it is natural to takeup the following themes.
 

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Continguts

Chapter I The Birth of Perspective
1
I2 The Origin of Perspective
2
I3 Four Stimuli
3
Representation of Special Lines
4
A Search for Mathematical Rules
10
I4 Brunelleschi
11
Brunelleschis Conception of Perspective
13
Brunelleschis Success
14
Amato
383
VIII5 Pozzos Influential Textbook
386
Pozzos Virtual Dome
388
A Vault As Picture Plane
389
VIII6 A Special Approach to Perspective Costa
394
VIII7 Mathematical Approaches to Perspective
397
Stellini
398
Torelli
399

I5 Perspective Before the Renaissance?
15
Chapter II Alberti and Piero della Francesca
17
Albertis Views on the Art of Painting
18
II3 Albertis Model
19
Albertis Two Methods of Producing a Perspective Image
21
II4 Albertis Construction
22
An Open Window
23
A Scaled Unit
24
Placement of the Centric Point
25
Choice of Parameters
28
II5 Albertis Theoretical Reflections and His Diagonal Rule
29
II6 The Third Dimension in Albertis Construction
33
II7 Albertis Construction in History
34
De Prospectiva Pingendi
36
II9 The Theoretical Foundation of De Prospectiva
37
Foreshortening of Orthogonals and Line Segments Parallel to 𝜋
38
Piero on Visual Distortion
40
Piero on the Correctness of the Construction
42
Filarete and Francesco di Giorgio
43
II11 Pieros Diagonal Construction
44
II12 Pieros Distance Point Construction
46
Piero on the Correctness of His Distance Point Construction
48
II13 The Division Theorem
50
II15 The Column Problem
51
Equidistant Line Segments
53
Columns on Square Bases
54
Cylindrical Columns
56
II16 Pieros Plan and Elevation Construction
59
Pieros Construction
60
II17 Pieros Cube
64
Pieros Idea
66
Pieros Heads
71
II19 Pieros Use of Perspective
75
II20 Pieros Influence
79
Chapter III Leonardo da Vinci
81
Leonardos Trattato
82
Leonardos Approach to Perspective
83
Outline of This Chapter
84
Linear Perspective Versus Other Concepts of Perspective
85
Natural Versus Accidental Perspective
86
Composite and Simple Perspective
87
III3 Visual Appearances and Perspective Representations
88
III4 Leonardo on Visual Appearances of Lengths
89
The Law of Inverse Proportionality
90
Pacioli and the Law of Inverse Proportionality
94
The Law of Inverse Proportionality and Euclids Theory
95
Leonardo on the Appearance of a Rectangle
96
The Appearance of the Vertical Boundaries
97
The Appearance of Collinear Line Segments
98
III5 Leonardo on Perspective Representations
100
The Perspective Images of Particular Line Segments
101
The Perspective Images of Collinear Line Segments
102
Leonardo and the Column Problem
105
Leonardos Appeal for a Large Viewing Distance
107
III7 Leonardos Doubts and Their Consequences
111
Leonardos Use of Perspective
112
Chapter IV Italy in the Cinquecento
115
IV2 The Architectural Painting and Sculpting Traditions
116
Sirigatti Cataneo and Peruzzi
122
Lomazzo
124
IV3 A Mathematical Approach to Perspective The Contributions by Vignola and Danti
125
Vignolas Plan and Elevation Construction
126
Vignolas Distance Point Construction
128
Vignolas Comparison of His Two Methods
130
Danti on Convergence Points
136
IV4 Connection Between Perspective and Another Central Projection Commandinos Contributions
138
Commandinos Constructions
141
Commandinos Influence
145
IV5 Another Mathematical Approach Benedettis Contributions
146
Benedetti on Pointwise Constructions
147
Benedetti and Convergence Points
149
Benedettis Influence
152
Barbaro on the Regular Polyhedra
155
IV7 The Italian Pre1600 Contributions to Perspective
158
Chapter V North of the Alps Before 1600
161
Viator
162
Ringelberg
166
Cerceau
169
V3 Cousin
172
Cousins Introduction of a Distance Point Construction
175
Cousins Use of Points of Convergence
178
Cousin on the Column Problem
182
V4 Dürer
183
Dürers Books
188
Dürers Plan and Elevation Construction
194
Dürers Enigmatic Method
197
The Second Method as Described
199
The Second Method as Illustrated
200
The Second Method and Alberti Constructions
201
The Second Method and a Distance Point Construction
202
The Diagrams Illustrating the Second Method
204
Construction of the Side fg
205
Dürers Programme
206
The Lesson of Dürers Mistakes
207
Dürers Diagonal Method
210
Dürers Influence on the Development of Perspective
212
Perspective Touched Upon by a Painter
213
Hirschvogel and Lautensack
217
Ryff Taking Up the Italian Tradition
222
Jamnitzer Lencker Stör and Hass
224
Pfinzing
230
V7 The SixteenthCentury NonItalian Tableau
236
Chapter VI The Birth of the Mathematical Theory of Perspective Guidobaldo and Stevin
237
Guidobaldos Struggle with Perspective
238
The Contents of Perspectivae Libri Sex
240
VI2 Guidobaldos Theory of Perspective
241
Line Segments Parallel to the Picture Plane
242
The Main Theorem of Perspective
244
Guidobaldos Proofs of the Main Theorem
246
Vanishing Lines
249
VI3 Guidobaldos TwentyThree Methods
250
Guidobaldos Rabatment
251
The Tenth Method
255
The TwentyFirst Method
256
Untraditional Picture Planes
257
Inverse Problems of Perspective
259
Direct Constructions
261
VI5 Guidobaldos Role in the History of Perspective
262
VI6 Stevin and His Work on Perspective
265
Stevins Path to Perspective
268
The Contents of Van de Verschaeuwing
269
VI7 The Foundation of Stevins Theory
270
The Invariance Theorem
271
VI8 Stevins Practice of Perspective
273
Stevins Rabatment
276
Stevins Examples
277
VI9 Stevin and Inverse Problems of Perspective
279
VI10 Further Issues in Stevins Work
282
The Column Problem
284
A Perspective Instrument
285
VI11 Stevins Influence
287
The Knowledge of Stevins Work Abroad
288
The Knowledge of Stevins Work at Home
289
Chapter VII The Dutch Development after Stevin
291
VII2 The Theory and Practice of Perspective
296
VII3 The Work by Marolois
297
Maroloiss Theory and Practice of Perspective
298
Maroloiss Method of Construction
301
Maroloiss Instrument
302
Shadows and Inverse Problems of Perspective
304
The Column Problem
308
Arithmetical Calculations
309
The LeftHand and RightHand Side Panels
313
The Bottom Panel
314
The Top Panel
316
The Back Panel
317
Van Schootens Intention and Inspiration
319
Georg Mohr
323
Abraham de Graaf
324
Hendrik van Houten
327
VII6 The Problems of Reversing and Scaling
328
The Problem of Reversing
330
The Problem of Scaling
334
Reduced Distance
336
VII7 sGravesandes Essay on Perspective
338
The Contents of sGravesandes Work
339
Camerae Obscurae
340
The Basic Theory
342
The TurnedIn Eye Point
343
A Particular Line
345
sGravesandes Basic Constructions
348
Oblique Picture Planes
351
sGravesandes Examples
354
Shadows
357
Response to sGravesandes Work
359
VII8 Traces of Desarguess Method in Dutch Perspective
360
VII9 Jelgerhuis and the Choice of Parameters
363
The Parameters of a Picture
364
Jelgerhuiss Choice
367
Chapter VIII Italy after Guidobaldo
369
VIII2 Perspective in Textbooks on Architecture
370
The GalliBibienas and Piranesi
371
VIII3 Perspective in Other Textbooks
372
A Textbook on the Theory of Vision Diano
374
VIII4 The Prospettiva Pratica Tradition
375
Contino
377
Torricelli
379
Troili
381
Chapter IX France and the Southern Netherlands after 1600
401
Perspective and Projective Geometry
402
IX2 The Theory of Perspective Taught
403
Hérigone
404
Bourdin Dechales and Tacquet
406
Rohault and Ozanam
407
The Encyclopedias
409
IX3 The Works of de Caus and Vaulezard
410
Vaulezard on Cylindrical Mirror Anamorphoses
413
Vaulezard on Perspective
415
IX4 The Work of Aleaume and Migon
418
Introduction of a Perspective Grid
420
Introduction of an Angle Scale
422
Methods Independent of Vanishing Points
424
Further Issues Treated by Aleaume and Migon
426
IX5 Desarguess Perspective Method
427
Desarguess Avoidance of Vanishing Points
433
Theoretical Reflections in La perspective
436
Theoretical Reflections in Aux théoriciens
437
Conclusion on Desargues and Vanishing Points
441
Points at Infinity in Desarguess Work on Perspective
442
IX6 Brouillon project and Perspective
445
Cross Ratios
446
Two Traditions
447
IX7 Perspectivists at War and the Work of Dubreuil
448
Desargues and Dubreuil
449
Dubreuils ComradesinArms
451
IX8 The Work of Niceron
452
Nicerons Construction of an Anamorphic Grid
454
IX9 Second Act of the Desargues Drama
457
Desarguess Supporter Bosse
460
IX10 The 1660s and 1670s
465
Le Clerc
466
Bourgoing
467
IX11 Perspective and the Educated Mathematician
470
IX12 French EighteenthCentury Literature on Perspective
471
Bretez Courtonne Deidier and Roy
474
Petitot and Curel
477
Lacaille
479
Jeaurat
482
Michel
484
Valenciennes
485
Chapter X Britain
489
Wren Moxon and Salmon
490
Ditton
492
X3 Taylor and His Work on Perspective
494
Taylors Inspiration
496
Taylors Two Books on Perspective
498
X4 Taylors Fundamental Concepts and Results
502
Vanishing Points and Lines
503
The Directing Plane
506
X5 Taylors Basic Constructions
508
Taylors Inspiration from sGravesande
510
X6 Taylors Contributions to Plane Perspective Geometry
511
Taylors Solution to Problem 1
512
X7 Taylors Contributions to Solid Perspective Geometry
515
X8 Taylors Examples of Drawing Figures in Perspective
519
Constructions as an Intellectual Experiment
524
X10 Taylor on Reflections
529
X11 Taylor on Inverse Problems of Perspective
534
Problems Concerning the Shape of an Original Figure
536
Determining the Eye Point as Well as the Shape
537
X12 The Immediate Response to Taylors Work
538
X13 Taylors Work in History
540
X14 Hamiltons Comprehensive Work on Perspective
541
Perspective and Conic Sections
542
Hamiltons Influence
546
X15 Kirby and Highmore
547
Kirbys Publications on Perspective
548
Kirbys Main Work on Perspective
552
Kirbys Inspiration
554
Kirby on the Practice of Perspective
555
Kirby and the Column Problem
557
Kirbys Service to Taylor
561
Highmore
562
X16 The Taylor Tradition Continued
568
Fournier and Cowley Addressing Students at Military Academies
570
Emerson the Textbook Writer
571
The Scientist Priestley Entering the Field
573
Noble Attempting to Bridge the Gap Between Theory and Practice
577
Malton and Son
579
Clarke Presenting Perspective for Young Gentlemen
584
Wood Writing for Painters
585
Taylors Influence on the Drawing of Chairs
587
X17 Perspective in Textbooks on Mathematics
588
Martin
589
Muller
591
X18 British Individualists
592
Hooper
594
Ferguson
595
Adams
596
X19 British Mathematicians and Perspective
597
X20 The British Chapter
598
Chapter XI The GermanSpeaking Areas after 1600
599
Faulhaber and Bramer
600
Brunn and Scheiner
602
Halt
603
Hartnack
604
Bischoff and Bürja
605
Kircher and Schott
609
Leupold
611
Mathematischer Lust und Nutzgarten
612
XI4 Perspective Presented for Practitioners
614
The Architect and Drawer Schübler
615
The Engraver Werner
617
Gericke and Weidemann Professors at the Academy of Art
618
The Theologian Horstig
619
The Wolffian Tradition
620
Weidler
623
Hennert and Lorenz
624
Segner and Bürja
625
Kästners Analytical Approach
626
Kästners General Theory
628
Kärstens Mastodon
629
XI6 Traces of Lambert
631
XI7 Perspective in the German Countries
633
Chapter XII Lambert
635
XII3 Early Approach to Perspective
642
XII4 The Contents of Freye Perspektive
647
Lamberts Possible Sources
648
XII5 Constructing Polygons in the Picture Plane
650
XII6 Oblique Figures
655
Comparing Some of Taylors and Lamberts Ideas
658
The Applicability of the Theory of Oblique Planes
660
XII7 Shadows
661
XII8 Reflections
664
Lamberts Room
665
Determination of Areas that Can Be Seen Reflected
669
Reflection in Curved Surfaces
671
XII9 Parallel Projections
674
A Precursor of Pohlkes Theorem
678
XII10 Inverse Problems of Perspective
679
XII11 Lamberts Practice of Perspective
682
Trompe LŒils
685
Rainbows Fountains a Starry Sky and Perspective Pictures
686
XII12 Ruler Geometry
689
The Steiner Circle
691
Lamberts Examples
692
Points on a Conic
694
A Specific Application of the Perspective Freedom
695
A Line Through an Inaccessible Point
698
Perspective Ruler Geometry and Projective Geometry
702
XII13 Lamberts Impact
703
Chapter XIII Monge Closing a Circle
707
Monges Descriptive Geometry
708
XIII2 Monge and Linear Perspective
709
Monges Influence on Teaching Perspective
711
Chapter XIV Summing Up
713
XIV2 Local Approaches to Perspective
714
The French and Belgian Development
715
The Dutch Development
716
Interplay Between Perspective and Other Geometrical Disciplines
717
The Status of the Theory of Perspective
718
XIV4 The Theory and Practice of Perspective
719
The Usefulness of the Theory of Perspective
720
Appendix One On Ancient Roots of Perspective
723
The Remoteness Theorem
724
The Convergence Theorem
725
Optics and Perspective in Harmony
727
Ptolemys Planisphaerium
728
Conclusion
730
Appendix Two The Appearance of a Rectangle à la Leonardo da Vinci
731
The Curves for Three Different Distances
732
The Angle Between the Line Segments
734
Appendix Three sGravesande Taking Recourse to the Infinitesimal Calculus to Draw a Column Base in Perspective
735
The First Step
736
The Infinitesimal and Limit Situation
737
The Perspective Image of the Visible Part of the Column Base
738
Appendix Four The Perspective Sources Listed Countrywise in Chronological Order
739
France and the Southern Netherlands
741
Germany Austria and Switzerland
742
The Northern Netherlands
744
Britain
745
First Bibliography PreNineteenth Century1 Publications on Perspective
747
Second Bibliography Supplementary Literature
771
Index
795
Illustration Credits
811
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