The Psychology of Perspective and Renaissance Art

Michael Kubovy

ii

Edition 1.1, October 6, 2003c©Michael Kubovy

Contents

1 The Arrow in the Eye 1

2 The elements of perspective 17

3 Brunelleschi invents perspective 27

4 Brunelleschi’s peepshow 31

5 The robustness of perspective 41

6 Illusion, delusion, collusion, & paradox 49

7 Perceive the window to see the world 61

8 Marginal distortions 73

9 The Brunelleschi window abandoned 87

10 The psychology of egocenters 101

11 Perspective & the evolution of art 107

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iv CONTENTS

List of Figures

1.1 Mantegna, Archers Shooting at Saint Christopher . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Mantegna, Archers Shooting at Saint Christopher, detail . . . . . . . . . . . . . . . . . . . . . 31.3 Taddeo Gaddi, The Presentation of the Virgin . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Piero della Francesca, Flagellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Masaccio, Tribute Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Piero della Francesca, Brera altar-piece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Domenico Veneziano, Martyrdom of Saint Lucy . . . . . . . . . . . . . . . . . . . . . . . . . . 71.8 Raphael, Dispute Concerning the Blessed Sacrament . . . . . . . . . . . . . . . . . . . . . . . 81.9 Domenico Veneziano, La Sacra Conversazione . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.10 Pietro Perugino, Virgin Appearing to Saint Bernard . . . . . . . . . . . . . . . . . . . . . . . 101.11 Copy after Mantegna, Archers Shooting at Saint Christopher . . . . . . . . . . . . . . . . . . 111.12 Mantegna, Saint Christopher’s Body Being Dragged Away after His Beheading . . . . . . . . 121.13 Alberti, Tempio Malatestiano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.14 Alberti, Tempio Malatestiano, niche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.15 Mantegna, detail of Figure 1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.16 Alberti, Self-portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Masaccio, Trinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Alberti’s window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Camera obscura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Geometry of the camera obscura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Main features of central projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Jan van Eyck, Annunciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8 Mantegna, Martyrdom of Saint James . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 The Flying Fish of Tyre (ca. 1170) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.9 Vanishing points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.10 Definition of the horizon line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.11 Plan and elevation of Masaccio’s Trinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.12 Perspective representation of a pavement consisting of square tiles . . . . . . . . . . . . . . . 252.13 Leonardo da Vinci, Alberti’s construzione legittima . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Depiction of Brunelleschi’s first experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 Wheatstone’s stereoscopic drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

v

vi LIST OF FIGURES

4.2 Fra Andrea Pozzo, St. Ignatius Being Received into Heaven . . . . . . . . . . . . . . . . . . . 334.3 Mantegna, ceiling fresco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Peruzzi’s Salla delle Prospettive seen from center of room . . . . . . . . . . . . . . . . . . . . 354.5 Peruzzi’s Salla delle Prospettive seen from center of projection . . . . . . . . . . . . . . . . . 364.6 Focus and depth of field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Experimental apparatus for Smith and Smith’s experiment. . . . . . . . . . . . . . . . . . . . 39

5.1 La Gournerie’s inverse projection problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Jan Vredeman de Vries, architectural perspective . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Stimuli in the Rosinski et al. (1980) experiments . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Displays in the Rosinski et al. (1980) experiments . . . . . . . . . . . . . . . . . . . . . . . . . 455.5 Data of Experiment 1 of Rosinski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.6 Modified data of Experiment 1 of Rosinski et al. . . . . . . . . . . . . . . . . . . . . . . . . . 475.7 Data of Experiment 2 of Rosinski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.8 Stimulus for Goldstein’s (1979) experiment: Rousseau, The Village of Becquigny (1857) . . . 485.9 Data from Goldstein’s (1979) experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.1 Stimulus for observing Emmert’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.2 A classification of trompe l’œil pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.3 Carlo Crivelli (attrib.), Two saints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4 Antonello da Messina, Salvatore Mundi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.5 Jan van Eyck, Portrait of a Young Man . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.6 Francisco de Zurbaran, Saint Francis in Meditation . . . . . . . . . . . . . . . . . . . . . . . . 546.7 Laurent Dabos, Peace Treaty between France and Spain . . . . . . . . . . . . . . . . . . . . . 546.8 Jacob de Wit, Food and Clothing of Orphans . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.9 Cornelis Gijsbrechts, Easel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.10 Jean-Baptiste Chardin, The White Tablecloth . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.11 J. van der Vaart (attrib.), Painted Violin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.12 Jacopo de’Barbari, Dead Partridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.13 Edward Collier, Quod Libet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.14 Samuel van Hoogstraten, Still Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.15 Trompe l’œil (early nineteenth century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.16 Drawing used by Kennedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.17 The vase-face reversible figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.18 A Necker cube formed by phenomenal contours . . . . . . . . . . . . . . . . . . . . . . . . . . 586.19 The vertical-horizontal illusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.20 The double dilemma of picture perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.1 Donatello The Feast of Herod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.2 Perspective drawing of a figure and determination of center of projection . . . . . . . . . . . . 637.3 How to project a transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.4 Photograph of a photograph (Time, March 29, 1968) . . . . . . . . . . . . . . . . . . . . . . . 657.5 We can only compensate for one surface at a time: stimulus . . . . . . . . . . . . . . . . . . . 667.6 We can only compensate for one surface at a time: what you see . . . . . . . . . . . . . . . . 667.7 Plan of Ames distorted room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.8 Distorted room as seen by subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

LIST OF FIGURES vii

7.9 Views of John Hancock Tower, Boston. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.10 Drawing of unfamiliar object that we perceive to have right angles . . . . . . . . . . . . . . . 687.11 Drawing of impossible object that we perceive to have right angles . . . . . . . . . . . . . . . 687.12 Drawing of cube indicating angles comprising fork juncture and arrow juncture . . . . . . . . 697.13 Drawing that does not look rectangular and does not obey Perkins’s laws . . . . . . . . . . . 697.14 Irregular shape seen as a mirror-symmetric — it obeys an extension of Perkins’s laws . . . . . 697.15 Figure that looks irregular because it does not obey extension of Perkins’s laws . . . . . . . . 697.17 Shepard and Smith stimulus specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.16 Objects used in the Shepard and Smith experiment . . . . . . . . . . . . . . . . . . . . . . . . 717.18 Results of the Shepard and Smith experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.1 Two central projections of a church & cloister . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.2 Oblique cubes under normal perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.3 Oblique cubes under exaggerated perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.4 Marginal distortions of cubes seen from above . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.5 Four displays and response keys used by Sanders (1963) . . . . . . . . . . . . . . . . . . . . . 758.6 Median reaction time for Sanders (1963) experiment . . . . . . . . . . . . . . . . . . . . . . . 758.7 How Finke and Kurtzman (1981) measured the extent of the visual field . . . . . . . . . . . . 768.9 Raphael, The School of Athens (1510–1) Fresco. Stanza della Segnatura, Vatican, Rome. . . . 778.8 Marginal distortion in spheres and human bodies . . . . . . . . . . . . . . . . . . . . . . . . . 788.10 Detail of Figure 8.9 showing Ptolemy, Euclid, and others. . . . . . . . . . . . . . . . . . . . . 788.11 Marginal distortions in columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.12 Paolo Uccello, Sir John Hawkwood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.13 Diagram illustrating argument about perspective made by Goodman . . . . . . . . . . . . . . 84

9.1 Edgerton’s depiction of Brunelleschi’s second experiment . . . . . . . . . . . . . . . . . . . . . 879.2 Droodle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889.3 Kenneth Martin, Chance and Order Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . 899.5 Marcel Duchamp, Bottlerack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909.4 Jean Tinguely, Homage to New York (remnant) . . . . . . . . . . . . . . . . . . . . . . . . . . 919.6 Advertisement for a 3-D (stereoscopic) film . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.7 Andrea Mantegna, Saint James Led to Execution . . . . . . . . . . . . . . . . . . . . . . . . . 949.8 Central projection in Mantegna’s Saint James Led to Execution . . . . . . . . . . . . . . . . . 949.9 Leonardo da Vinci, The Last Supper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.10 Perspective construction of Leonardo’s The Last Supper . . . . . . . . . . . . . . . . . . . . . 979.11 Plan and elevation of room represented in Leonardo’s The Last Supper . . . . . . . . . . . . . 989.12 Leonardo’s Last Supper seen from eye level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.13 How the architecture of the refectory relates to Leonardo’s Last Supper . . . . . . . . . . . . 999.14 Leonardo’s Last Supper, cropped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009.15 Leonardo’s Last Supper, cropped, top only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

10.1 Definitions of two elementary camera movements: pan and tilt . . . . . . . . . . . . . . . . . 10210.2 The moving room of Lee and Aronson (1974) . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.3 Predictions for speed of “reading” letters traced on the head . . . . . . . . . . . . . . . . . . . 10510.4 The Parthenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.5 Horizontal curvature of Parthenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

viii LIST OF FIGURES

11.1 Paolo Uccello, Perspective Study of a Chalice . . . . . . . . . . . . . . . . . . . . . . . . . . . 11011.4 Kasimir Malevich, two Suprematist drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.5 Piero della Francesca (?), Perspective of an Ideal City . . . . . . . . . . . . . . . . . . . . . . 11311.6 Gentile Bellini, Procession of the Relic of the True Cross . . . . . . . . . . . . . . . . . . . . . 11311.2 Sol LeWitt, untitled sculpture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11411.3 Leonardo da Vinci, A War Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Tables

11.1 Gablik: cognitive development & megaperiods of art history . . . . . . . . . . . . . . . . . . . 111

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x LIST OF TABLES

List of Boxes

2.1 Drawback of the pinhole camera . . . . . . 192.2 The distance between the vanishing point

and a distance point equals the distance

between the center of projection and the

picture plane . . . . . . . . . . . . . . . 264.1 Photographing illusionistic walls . . . . . . 344.2 Viewing from the center of projection vs.

the removal of flatness information . . . . 377.1 How the visual system might infer the cen-

ter of projection . . . . . . . . . . . . . . 639.1 The aleatory process that generated Figure

9.3 . . . . . . . . . . . . . . . . . . . . 89

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xii LIST OF TABLES

Chapter 11

The invention of perspective and theevolution of art

. . . the jury wrote down all three dates ontheir slates and then added them up, andreduced the answer to shillings and pence.

Lewis Carroll, from “Alice’s Adventures inWonderland,” 1865 (Carroll, 1976, p. 117)

In this last chapter, I will discuss three views ofthe place of perspective in the history of art: thoseof Panofsky, Goodman, and Gablik. The first twoare relativists and claim that perspective is a con-vention of representation adopted during the Renais-sance. Gablik has proposed an interesting parallelbetween the development of cognitive abilities in chil-dren and the evolution of art.

In his book on The Renaissance Rediscovery ofLinear Perspective, Samuel Edgerton wrote a mas-terly exposition of Panofsky’s seminal article “DiePerspektive als ‘symbolische Form’ “ (Panofsky,1924/25) and of its reception among scholars inter-ested in perspective. I will quote extensively from hisdiscussion because it serves so well to introduce thepoints I wish to make in conclusion.

This article created extraordinary interestin subsequent decades [after its publicationin 1927] because the author argued that lin-ear perspective by no means conclusively de-fined visual reality, rather that it was only aparticular constructional approach for rep-resenting pictorial space, one which hap-pened to be peculiar to the culture of the

Italian Renaissance.Art historians, trying at that time to jus-

tify the rise and spread of modern abstractart, were pleased because Panofsky seemedto be saying that linear perspective was notthe last word in pictorial truth, that it, too,could pass away as had all earlier artisticconventions. . . Such a notion has since beenexpressly defended by various writers on artand psychology, among them Rudolph Arn-heim [1974], Gyorgy Kepes [1944], and Nel-son Goodman [1976 as well as Francastel,1951, and Suzi Gablik, 1976].

However, Panofsky’s essay did containone egregious error. With ingenious rea-soning, the author tried to show that theancient Greeks and Romans — Euclid andVitruvius in particular — conceived of thevisual world as curved, and that since theretina is in fact a concave surface, we do in-deed tend to see straight lines as curved. . . .

Panofsky’s essay, particularly in recentyears, has come under criticism from scien-tists, as well as from E. H. Gombrich [1969,1976, 1980] and other scientific-minded arthistorians. Writers on optics and percep-tual psychology such as James J. Gibson[1971], G. ten Doesschate [1964], and M. H.Pirenne [1952–3] have challenged Panofskyfor his subjective curvature hypothesis and

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denial that linear perspective has a catholicor “ultimate” veracity. They are especiallyput off by Panofsky’s reference to perspec-tive as a “symbolic form,” which is to say,a mere convention. . . Unfortunately, Panof-sky never explained definitively just whathe meant by the phrase “symbolic form.”However, he certainly has in mind a moresubtle meaning than a “system of conven-tions [like]1 versification in poetry.” [Thisis how Pirenne summarized Panofsky’s the-ory.] Indeed, Professor Pirenne and otherscientist critics misunderstand the ingenu-ity of Panofsky’s approach as much as theyfind Panofsky himself misunderstood classi-cal optics and modern perceptual psychol-ogy. (1975, pp. 153–5)

Edgerton proceeds to show how Panofsky’s notionof symbolic form is inspired by Ernst Cassirer’s Kan-tian philosophy, which he capsulates as follows:

The symbols man uses to communicateideas about the objective world have an au-tonomy all their own. Indeed, the humanmind systematizes these symbols into struc-tures that develop quite independently ofwhatever order might exist in the naturalworld to begin with. . .

The real thrust of [Panofsky’s] essay wasnot to prove that the ancients believed thevisual world was curved or that Renaissanceperspective was a mere artistic convention,but that each historical period in West-ern civilization had its own special “perspec-tive,” a particular symbolic form reflectinga particular Weltanschauung. Thus linearperspective was the peculiar answer of theRenaissance period to the problem of repre-senting space. . .

In the 15th century, there emergedmathematically ordered “systematic space,”infinite, homogeneous, and isotropic, mak-ing possible the advent of linear perspec-tive. . . Linear perspective, whether “truth”

1Edgerton’s interpolation.

or not, thus became the symbolic form ofthe Italian Renaissance because it reflectedthe general world view of the Italian peopleat this particular moment in history. (1975,pp. 156, 157–8)

As Edgerton so well explains, Panofsky’s positionwas not blithely relativistic: It is more important tounderstand why the artists of the Renaissance wereinterested in perspective than to determine whetherit is the “correct” method of representation. In thisbook, I have attempted to convey the variety as wellas subtlety of the reasons why Renaissance artistswere interested in perspective. I hope I have per-suaded the reader that “truth” was not at stakehere. To be sure, perspective was a system thatenabled artists to represent space according to ge-ometric rules. Mainly, however, it was a frameworkwithin which originality without arbitrariness2 couldbe achieved.

Nelson Goodman took the issue a step further bymarshaling all his philosophical arguments in supportof the relativistic conception of perspective. Good-man’s sustained analysis of the notions of represen-tation, realism, and resemblance is also an impas-sioned defense of the argument that perspective isnot an absolute standard of fidelity, that it is butone of many methods of representation. According toGoodman, depictions are analogous to descriptions,and descriptions need not resemble the things theydescribe. Indeed, sometimes they cannot resemblethe thing they are describing because that thing sim-ply doesn’t exist (e.g., a unicorn). Why then do wethink that a picture should resemble the thing it rep-resents? Goodman answers that conventions of rep-resentation are responsible for this misapprehension.From the correct observation that a picture usuallyresembles other pictures of the same kind of thing,we tend to infer that a picture resembles the kind ofthing it represents. The key argument is here: Good-man asks himself whether

the most realistic picture is the one that pro-vides the greatest amount of pertinent infor-mation. But this hypothesis can be quickly

2The term is Wimsatt’s (1968 p. 80).

109

and completely refuted. Consider a realis-tic picture, painted in ordinary perspectiveand normal color, and a second picture justlike the first except that the perspective isreversed and each color is replaced by itscomplementary. The second picture, appro-priately interpreted, yields exactly the sameinformation as the first. . . . The alert abso-lutist will argue that for the second picturebut not the first we need a key. Rather, thedifference is that for the first the key is al-ready at hand. For proper reading of thesecond picture, we have to discover rules ofinterpretation and apply them deliberately.Reading of the first is by virtually automatichabit; practice has rendered the symbols sotransparent that we arc not aware of any ef-fort, of any alternatives, or of making anyinterpretation or all. (1976, pp. 35–6)

I believe that I have provided us with the tools torefute Goodman’s radical relativism.3 I have shownthat perspective is not a thoroughgoing, arbitrary ap-plication of the geometric system of central projec-tion. Rather, it is a geometric system tempered bywhat perception can or cannot do. It has evolvedinto a system adapted to the capabilities of our per-ceptual system. To respond, Goodman would have toclaim that what perception can do depends on whatit learned to do, and that there is no limit to whatperception can learn. But that argument is false.There are clear limits to the extent of perceptual rear-rangement (induced by wearing prisms, mirrors, andother devices that modify the form of the optical in-formation reaching our eyes) to which human beingscan adapt. We cannot arbitrarily change the way weperceive optical information, nor can we arbitrarilychange our motor responses to it, regardless of theamount of time or effort we might invest in doing so(Welch, 1978, pp. 277–9).

We have seen that Panofsky’s view on the conven-tionality of perspective may not have been as extremeas some have interpreted it to be because it does notexaggerate the importance of the role played by the

3See also Gombrich’s (1982) broader attack on Goodman’sconventionalistic position.

“correct” representation of space in Renaissance art.We ham also seen that Goodman’s view, on the otherhand, is the most radical position on this matter thatone can take precisely because it makes the “correct-ness” of perspective into a central issue, thereby im-poverishing our understanding of perspective in Re-naissance art rather than enriching it. We turn nowto a third view, which shares some of the featuresof Goodman’s approach. Suzi Gablik, in her bookProgress in Art, has presented a cultural analog ofthe classical embryological law, “ontogeny recapitu-lates phylogeny,” according to which an embryo, inthe course of its maturation, goes through stages dur-ing which it takes on the appearances of its evolu-tionary ancestors. Gablik has proposed a similar lawfor the evolution of art, which I call “sophogeny re-capitulates ontogeny,” namely, that the evolution ofcultural wisdom parallels the development of the in-dividual. I will argue that Gablik, to make her point,emphasizes only one of the goals of Renaissance per-spective — the representation of objects in space —and that she implies that art cannot achieve this goalwithout being rigid and inflexible, rule-bound andlacking in true conceptual autonomy.

Gablik’s point of departure is the theory of cogni-tive development of Jean Piaget, the celebrated Swisspsychologist. Piaget proposed that it is possible todiscover milestones in the development of thinking,perception, problem solving, and all the other cog-nitive abilities. He distinguished three major stagesin cognitive development. In the preoperational stage(which ends at about 5 years of age), children have avery poor grasp of causality and reversibility. Forinstance, if you pour a liquid from a tall, narrowglass to fill a squat, short one of equal capacity, re-fill the tall glass with liquid, and then ask a preop-erational child which glass contains more liquid, thechild will say that the taller glass contains more. Thechild does not understand the concepts of conserva-tion (the amount of fluid) and of compensation (thetrade-off of height for area of the cross section), whichare physical expressions of the formal concept of re-versibility. In the concrete-operational stage (whichruns to about the age of 10), children understandthe reversibility underlying certain physical opera-tions but are unable to deal with the logical con-

110 CHAPTER 11. PERSPECTIVE & THE EVOLUTION OF ART

cepts that are their abstract representation. Finally,in the formal-operational stage, children can under-stand abstract logical and mathematical structuresthat underly reality.

At this point, we should let Gablik to speak forherself:

According to our own cognitive map [Ta-ble 11.1]. . . it would seem that a fully devel-oped formal-operational stage has not ap-peared in the art of any culture except thatof post-Renaissance Western art. . . . Now ifdefining the history of art in terms of cog-nitive stages is of any value, it is to the ex-tent that it may contribute to explaining theimportance of this development specifically,of an increase in the autonomy of forms tothe point where even abstract forms devoidof content can be constructed and manipu-lated. (Compare, in this regard, Uccello’sdrawing of a chalice [Figure 11.1] with SolLeWitt’s open modular cubes [Figure 11.2],or Leonardo’s War Machine [Figure 11.3]with Malevich’s Suprematist Elements [Fig-ure 11.4]4. . . In making the sseemingly para-doxical assertion that these contemporaryworks, which when viewed on their own ap-pear to be visually much simpler than a Re-naissance painting, are in reality more com-plex, I refer to the complexity which is occa-sioned by the Modern paradigm viewed as awhole, and to the infinite number of systemswhich it is able to generate. The Renais-sance paradigm derives from a single, closedlogical system — perspective — which is re-peated over and over again in every picturein much the same way, so that every pic-ture is srigidly bound and dictated by therules of the system. The Modern paradigmis characterized by its openness and by the

4I suppose that Gablik wants us to compare the two squaresand the circle in the Malevich to the divided box and the wheelin the Leonardo. There is something odd in this comparison:We arc being asked to compare two juxtaposed paintings byMalevich to one drawing by Leonardo. I fail to see how sucha comparison can possibly be meaningful.

infinite number of possibilities and positionswhich can be taken. (1976, pp. 44–5)

Figure 11.1: Paolo Uccello, Perspective Study of aChalice (1430-40). Pen and ink. Gabinetto dei Dis-egni a Stampe, Florence.

Gablik can make her case only if she can demon-strate that Renaissance artists used perspectiverigidly and concretely:

The belief that the universe is ordered andrationally explicable in terms of geometrywas part of a deterministic world-picturewhich viewed nature as stable and unchang-ing, and considered that mastery of it couldbe achieved by universal mathematical prin-ciples. The spatial illusionism of one-pointperspective reflected a world which was per-manent and fixed in its ways, modeled onan absolute space and time unrelated to anyoutward circumstance. One has only to lookat [paintings by] Piero della Francesca (seeFigure 11.5] or. . . Bellini (see Figure 11.6] to

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Table 11.1: Stages of cognitive development and megaperiods of art history. (Source: Gablik, 1976, p. 43.)

Stages of cognitive development Spatial characteristics Megaperiods of art historyenactive mode

Preoperational stage: Topological relations: Ancient & MedievalThe stage at which representationsare characterized by static imageryand space is subjectively organized.Psychical and physical ideas are notyet dissociated.

Distance between objects is basedon their proximity to one anotheron a two-dimensional plane whichonly takes height and breadth intoaccount. Absence of depth, no uni-fied global space which conservessize and distance.

(including Græco-Byzantine, ancient Ori-ental, Egyptian, archaicGreek, and early medieval)

iconic modeConcrete-operational stage: Projective & Euclidean relations: The RenaissanceThe stage at which representationcan arrange all spatial figures in co-ordinate systems. Representation isstill attached to its perceptual con-tent, however. The emergence ofperspective as a formal logic, ap-plicable to any content whatsoever,but still confined to empirical real-ity and to the concrete features ofthe perceptual world.

Based on the static viewpoint of asingle observer. Separation of ob-server and world.

symbolic modeFormal-operational stage: Indeterminate atmospheric space The Modern periodThe stage at which hypothetical de-ductive, logico-mathematical, andpropositional systems emerge, con-structed and manipulated as inde-pendent relational entities withoutreference to empirical reality.

(late Monet, Cubism, Rothko):Space as an all-over extension inwhich all points are of equal sta-tus and are relative to each other.No dominance of volume over void.(Pollock)

(including late Impression-ism, Cubism, Formalism,Serial art, art governedby logical systems and bypropositional thinking)

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Figure 11.4: Kasimir Malevich: Left: Suprematist Elements: Squares (1915). Right: Suprematist Element:Circle (1913). Pencil. Sheet: 18-1/2 × 14-3/8”. Composition: 11-1/2 × 11-1/8”. Collection, The Museumof Modern Art, New York.

sense this immutability of things: a worldis portrayed in which chance and indeter-minacy play no part. From this vantagepoint, we can sec how a totally mathema-tized philosophy of nature was the dominantinfluence on the course of Western painting,and how these processes of mathematics of-fer themselves as a bridge from one stage inthe development of art to the next.

In the Renaissance, geometry was truthand all nature was a vast geometrical sys-tem. (The book of nature, Galileo wrote,is written in geometrical characters.) Per-spective images were based on observation,but they were rationalized and structuredby mathematics. For Alberti in 1435, thefirst requirement of a painter was to knowgeometry; and Piero, in De Prospettiva Pin-gendi, virtually identified painting with per-spective, writing three treatises to show how

the visible world could be reduced to math-ematical order by the principles of perspec-tive and solid geometry. (1976, p. 70)

These views stress the rigidity, the rationality, andthe immutability of the laws of perspective. Un-doubtedly, there is some truth in Gablik’s portraitof an era fascinated by geometry. But fascinationis not fetishism. During the Renaissance, geometrywas always subordinate to perception: I have shownhow the geometry of central projection was routinelyviolated to counteract its perceptually unacceptableeffects. We have seen that perspective was far frombeing a single, closed, logical system that was re-peated over and over. Gablik has produced a cari-cature of Renaissance art, which even with regard toits use of perspective was far from being rigid anduncompromising. To be sure, perspective was usedfor a representational purpose, and in that respectit remained tied to the concrete objects it served torepresent. But it also served to explore other aspects

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Figure 11.5: Piero della Francesca (attrib. doubtful), Perspective of an Ideal City (ca. 1470). Panel. GalleriaNazionale delle Marche, Palazzo Ducale, Urbino.

Figure 11.6: Gentile Bellini, Procession of the Relic of the True Cross (1496). Canvas. Accademia, Venice.

114 CHAPTER 11. PERSPECTIVE & THE EVOLUTION OF ART

Figure 11.2: Sol LeWitt, untitled (1969). Bakedenamel and aluminum. John Weber Gallery, NewYork.

Figure 11.3: Leonardo da Vinci, A War Machine(Codex Atlanticus, Folio 387r). Drawing.

of experience. Indeed, it is possible to make a caseagainst Gablik’s position by applying a slightly dif-ferent set of Piagetian concepts. Taking my analysisof the effects of perspective as a point of departure,one might argue that the Renaissance artists were ex-ploring the nature of egocentrism and ways of usingperspective to free oneself from one’s special vantagepoint. To do so is a sign of one’s ability to transcendegocentrism. One might argue that the Cubists wereengaged in a similar exploration, but can one say thatthey were, in this respect, more advanced than werethe Renaissance artists? And certainly one would notclaim that Sol LeWitt’s sculpture is part of such aninvestigation. I am convinced that by carefully select-ing the dimensions along which comparisons betweendifferent periods of art were made, one could developan argument that any period in art is more advancedthan all the others.5

We have disagreed with Goodman; perspective isnot mere convention. We have disagreed with Gab-lik; sophogeny does not recapitulate ontogeny. AndPanofsky was mistaken on some matters. But Panof-sky had an extremely useful formulation of the im-portance of perspective: It served as symbolic form.Even though perspective has a very sturdy geometricand perceptual foundation, which makes it, in somesense, the best method to represent space on a flatsurface, the question of whether perspective is “true”is far less important than the inquiry about how per-spective was put to use by Renaissance artists in anartistic context. I have tried to answer this ques-tion and to show that these uses were far removedfrom the oversimplified view of perspective as a pro-crustean system in the service of crass illusionism.Perspective often enabled the Renaissance artist tocast the deeply religious contents of his art in a formthat could produce in the viewer spiritual effects thatcould not have been achieved by any other formalmeans. In that sense, perspective should be viewed

5A similar thesis was presented by Gowans (1979), appar-ently formulated without knowledge of Gablik’s book. As onewho disagrees with this theory, I find some satisfaction in not-ing a 700-year discrepancy between their chronologies. Accord-ing to Gowans, the Piagetian stage of formal operations wasattained by the Romanesque period (twelfth century), whereasaccording to Gablik it wasn’t attained until late Impressionism(late nineteenth century).

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as “symbolic form.”