Did Jan Van Eyck Have a Perspectival System

On the Arnolfini Portrait and the Lucca Madonna: Did Jan van Eyck Have a PerspectivalSystem?Author(s): James ElkinsSource: The Art Bulletin, Vol. 73, No. 1 (Mar., 1991), pp. 53-62Published by: College Art AssociationStable URL: http://www.jstor.org/stable/3045778Accessed: 07/10/2008 13:52

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On the Arnolfini Portrait and the Lucca Madonna: Did Jan van Eyck Have a Perspectival System?

James Elkins

Exchanges on the topic of Jan van Eyck's pictorial constructions have been going on intermittently since the turn of the century. The most recent hypothesis, put forward in 1982-83, has if anything clouded the issue further by proposing an entirely new "elliptical perspective." The ensuing debate, which appeared in The Art Bulletin, raises fruitful questions for further research: the problem of knowing how accurate a reconstruction needs to be, and of how reconstructed lines should be interpreted. The present essay has two purposes. It attempts to settle the question of Jan van Eyck's perspective, at least in the case of the Arnolfini Portrait and the Lucca Madonna, and to introduce a new, higher level of accuracy and reproducibility for perspectival reconstructions in general.'

It has long been known that the inception of perspective in the Low Countries can be credited to Petrus Christus,2 but the role played by Jan van Eyck continues to be disputed. A central case, the Arnolfini Portrait, has at- tracted no less than six different hypotheses.3 It may be useful to review the literature on the Arnolfini Portrait, and then, after sorting out the various accounts, to turn to the Lucca Madonna to see if those findings apply to other works.

Karl Doehlemann, in an article published in a mathe-

This study is dedicated to E. E. Rosenthal, who taught me the value of careful analysis per fas et nefas. I would like to thank him and an

anonymous reader for comments on previous versions of this paper. 2 See K. Doehlemann, "Die Perspektive der Briider van Eyck," Zeitschrift fir Mathematik und Physik, LII, 1905, 419-425; "Die Entwicklung der

Perspektive in der altniederlandischen Kunst," Repertoriumfiur Kunstwis- senschaft, xxxiv, 1911, 500-535; and his "Nochmals die Perspektive bei den Briidern Van Eyck," Zeitschrift fir Mathematik und Physik, xxxv, 1912, 262-267; J.G. Kern, "Die Kritik der perspektivischen Zeichnung und ihre Bedeutung ftir die Kunstgeschichte," Kunstgeschichtliche Gesell- schaft, Berlin, Sitzungsberichte, vi, 1905,37-46; idem, "Eine perspektivische Kreiskonstruktion bei Sandro Botticelli," Jahrbuch des preussischen Kunst-

sammlungen, 1905, 137ff; and J.M. Collier, "Perspective in the Arnolfini Portrait," Art Bulletin, XLV, 1983, 691.

It might be reiterated in this context that Petrus Christus's "mastery" of perspective extended only to the knowledge that orthogonals should be drawn to a single point and to the guide provided by a diagonal drawn through foreshortened squares. 3 I am not considering informal observations about Jan's perspective, for example, that it is "not mathematically accurate," that it is "influenced by Italian theory," or that it employs a small "vanishing area." It is not that such observations should be excluded a priori in favor of more exact remarks, it is that an attentive observer can discern something more than a single "vanishing area" even without drawing lines on reproductions. That quality of attentive looking, it seems to me, has analytic precedence over more cursory responses.

matics journal, doubted that the orthogonals in the Arnolfini Portrait converge to vanishing points at all and concluded that Jan understood only vanishing areas. To him, Jan was an experimenter whose "errors" led from medieval parallel perspective to a kind of empirical vanishing-area perspective, which was, nevertheless, decisively different from the mathematically correct solu- tion of Petrus Christus. In Doehlemann's account, Jan looked at surfaces like walls or ceilings and painted "bands" of vanishing points: one for the ceiling, another below it for the floor, and so forth.4

Doehlemann's analysis was challenged by Joseph Kern, who saw the Arnolfini Portrait differently (Fig. 1). In his opinion, one vanishing point belongs to each surface, and Doehlemann's "bands" of vanishing points are really different horizon lines. Kern was a fastidi- ous worker, and noted Jan's omission of a single line

4 Doehlemann, "Die Perspektive," 423 and 425:

Die Tiefenlinien der Decke [of the Arnolfini Portrait] zeigen nach der Kernschen Tafel keinen gemeinsamen Fluchtpunkt (die Decke ist so dunkel, dap ich auf der Photographie diese Linien nicht mehr sehe).... Dann bleibt mir aber kaum etwas anderes ubrig, als anzunehmen, dap Jan das Gesetz vom Fluchtpunkt der Tiefenlinien einer Ebene, das hei3t dessen mathematisch prazisen Ausdruck, iiberhaupt nicht gekannt hat.... Wenn dann Jan an Stelle eines Augpunktes ein ganzes Gebiet annahm, so trat naturgemai an Stelle eines Horizontes ein ganzer Streifen und so erklare ich die verschiedenen Horizonte, die Kern konstatiert.... [Doehlemann concludes:] Man wird Jan van Eyck in bezug auf die Perspektive als einen Praktiker bezeichnen miissen, der mit ungewohnlich scharfer Beobachtungsgabe ausgestattet, aber doch nur auf empirischer Grundlage, das parallelperspektivische System seiner Vorganger verliep und die perspektivische Zeichnung insofer verbesserte, als er die Bilder paralleler Geraden um einen Fluchtpunktbezirk sich drehen liep.

54 THE ART BULLETIN MARCH 1991 VOLUME LXXIII NUMBER 1

h

i

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1 Jan van Eyck, Arnolfini Portrait, after the reconstruction by J. Kern

(an orthogonal from a), but found the concurrence as

perfect as possible in a painting.5 Panofsky, in the course of a more general comparison

of Northern and Italian rendering of light, found the Arnolfini Portrait had four "central" vanishing points, and

thought of it not simply as a slightly "incorrect" construc- tion, but more profoundly as the result of what, in the

5J.G. Kern, "Perspektive und Bildarchitektur bei Jan van Eyck," Repertorium fir Kunstwissenschaft, xxxv, 1912, 29:

In Wirklichkeit ist fur ein Bild die Ubereinstimmung vollkommen. Doehlemann wirde wohl zu demselben Schlusse gelangt sein, wenn er neben meiner Zeichnung das Originalbild statt der Photo-

graphie das Bild eine erhebliche Grope aufweist.... Meine Tafel [pl. 1, top left] ergibt in Ubereinstimmung mit dem Original, nach der sie angefertigt ist, das typische Bild einer perspektivischen Konstruk- tion einzelner Ebenen nach gesonderten Fluchtpunkten.... Vermut- lich stiitzt sich Doehlemann bei der Ablehnung meiner Hypothese

Italian view, would have to be called a wrongheaded approach to perspective in general: that is, it reflected the idea that individual objects and regions need not be subservient to a sense of whole space.6 Panofsky proba-

auf die Zeichnung des dritten Deckenbalkens von rechts, der sich in der Tat in das System der ubrigen Balken nicht einfugt.

Kern opens with a five-point conclusion, of which the first item is: "1. Jan van Eyck konstruiert einzelne Begrenzungsebenen des Raumes unter Anwendung des Fluchtpunktes fur diese einzelnen Ebenen" (ibid., 27). G. ten Doesschate, Perspective: Fundamentals, Controversials, History, Nieuwkoop, 1964,140-41, agrees with Kern's reconstruction of three vanishing points, but does not follow Kern's explanation of

multiple horizons.

6E. Panofsky, Early Netherlandish Painting, Its Origins and Character, Cambridge, 1966, 3 and 7: "It matters little ... that ... the Arnolfini

portrait is not fairly ['correctly'] constructed and has four central

vanishing points instead of one."

DID JAN VAN EYCK HAVE A PERSPECTIVAL SYSTEM? 55

2 Arnolfini Portrait, after the reconstruction by D. Carleton 3 Arnolfini Portrait, after the reconstruction by J. Collier

bly got his four vanishing points from Kern's plate by adding lines from the canopy (pl. 1, top left: b, c and d).7 His term "vanishing points," given the informal nature of his passage, can be safely read as "vanishing areas." In an earlier article, also influenced by Kern's analysis, Panofsky had filled in the area between his four "points" to make a lozenge-shaped vanishing area, or Fluchtregion, a kind of generalization of the four points.8

More recently, David Carleton, a mathematician, has proposed that there are two vanishing areas, Kern's F' and F" (Fig. 2), and that they were generated in "elliptical perspective" by the use of a convex mirror.9

To John Ward, Carleton's reconstructions were too

7 The suggestion, made by D. Carleton, "A Mathematical Analysis of the Perspective of the Arnolfini Portrait and Other Similar Interior Scenes by Jan van Eyck," Art Bulletin, XLIV, 1982, 119, is reasonable in light of the more detailed reconstructions. 8 E. Panofsky, "Once More the Friedsam Annunciation and the Problem of the Ghent Altarpiece," Art Bulletin, xx, 1938, 419. See also L. Brion-Guerry, Jean Pelerin Viator ..., Paris, 1962, 94-95, where Panof- sky's opinion is accepted. 9Carleton (as in n. 7), 119. The questions raised by hypotheses like Carleton's, which purport to find anachronistit (or, in this case, entirely new) perspective methods in paintings, cannot be addressed here. Accounts like his are not uncommon in the literature, and involve a combination of terminological, conceptual, and historical difficulties. In this case, there are (a) terms new to perspective, such as "elliptical perspective," and "elliptical cut-section," and they need to be related to

simple, and he found more vanishing areas. As he pointed out, the way Carleton drew the orthogonals from h, i, and j (Fig. 2) makes it ambiguous whether they are meant to be parts of the upper or lower vanishing areas, since he drew them as interrupted lines after they cross the two major vanishing areas.l0 Ward drew several conclusions, among them that the Arnolfini Portrait was not done with the aid of an eccentric perspective theory (such as that posited by Carleton), but rather with a general awareness of, and partial insouciance about, the canonical theory requiring one vanishing point. In Ward's view, Jan's lines have variable accuracy because he was

existing terms. How is "elliptical perspective" different from inaccurate linear perspective with two principal vanishing points? There are also (b) problems of application: it does not accord with traditional perspective, for example, that a "cut section" needs to include all the objects visible in the representation, since objects can also be in front of the plane of projection. If Carleton would claim the opposite, then his new denotation would require mathematical justification. Historically, (c) Carleton's technical dialogue derives not from the originating texts of perspective, but from his own modern mathematical training, and when he invents and reinvents terms and methods, his discourse requires a translation into normative perspectival terms. I consider the exchange in more detail in a work in progress on art-historical writing. 0 J. Ward, "On the Mathematics of the Perspective of the Arnolfini

Portrait and Similar Works of Jan van Eyck," Art Bulletin, XLV, 1983, 680-686, esp. 681.


concerned chiefly to estimate how much accuracy would be needed for a line to appear accurate."

At present, the last word in the debate is James Collier's; he found "five, six, seven, or more vanishing points" (Fig. 3), and concluded that Jan's method was

purely "empirical," since he did not draw orthogonals to single vanishing points even within single surfaces, such as the floor and ceiling.12

On the Detail of Reconstructions The reconstruction of perspectival configurations in works of art has suffered a decline since its beginnings just before the turn of the century. Some of the most accurate reconstructions were made-of other artists' works-by Doehlemann, who remained sensitive to the way that

orthogonals and diagonals tend not to recede to perfectly geometric vanishing points. Since then, although the term "vanishing area" is in wide use, perspective reconstructions have become less empirical and more idealized. Art historians tend to use small reproductions, and it has become conventional to assume a single vanishing point and draw toward it for clarity's sake. An illogical result of this practice is that art historians continue to

disagree on the perspective methods of a number of

major paintings. They disagree not only on the kind of

perspective that should legitimately be found in a painting, but also on the directions of the lines that are involved: a curious occurrence, since the paintings are static objects, and only one direction of a line is possible.13 (Note, for example, the varying directions given to the

ceiling beams in Figs. 1-3.) It is, however, possible to achieve a greater consensus

by attempting a greater degree of accuracy. There are

n Ibid., 686:

In summary, all available evidence points to the conclusion that Van

Eyck did not paint the Arnolfini Portrait with the aid of a convex

mirror, ... did not have a perspective theory, and did not consis-

tently converge his orthogonals in two vanishing areas positioned at the foci of an ellipse. A careful analysis of the perspective in all of his pictures indicates, first, that he recognized early on that orthogonals in a single plane converge to a point; second, that he only bothered to construct this convergence carefully when its accuracy was visually significant, and, third, that this approach to perspective varied with the expressive requirements of a given picture and of a

given area of a picture. See also ibid., 681 and 683, concerning Ward's claim of a "very close station point." (This last claim cannot be mathematically adjudi- cated-no geometrical evidence exists for it in the painting-and so

belongs to the realm of comparative stylistics. On those criteria, the

painting does seem to be done from a close distance, but not abnor-

mally so for the period.) 12 Collier (as in n. 2), 691.

13 To be precise, it is necessary to distinguish four kinds of similarity or difference between two alternate reconstructions: (a) the existence of the lines (which ones the author chose to draw); (b) the positions of the lines

(measured according to their point of intersection or the distance between them if they are parallel); (c) the directions of the lines (their angles, measured relative to a horizontal line); and (d) the extent of the lines (since they are conventionally pictured as segments, with begin- ning and ending, rather than rays, which would be mathematically if not historically more appropriate).

many rewards for such an apparently inappropriate exactitude. If reconstructions were routinely done with the accuracy I attempt here, then the analyses could agree on where the lines are and concentrate on more difficult methodological issues. Elsewhere, I have proposed a reproducible procedure for placing and number- ing the lines.14 When lines are not perfect, as painted lines cannot be, then it is possible to use line- and curve-fitting techniques (such as the least-squares method), and to record the margins of error so that future historians may compare their hypotheses to see if they can be accommo- dated to what the painting allows. In these ways, future researchers need not start from scratch, producing con-

flicting or incommensurate analyses; instead they can choose to agree or disagree with the levels of accuracy and completeness of previous analyses. It is not possible to go into those methods here; instead I reproduce two

examples (Figs. 4-5).15

Analyzing the Arnolfini Portrait A closer inspection of the Arnolfini Portrait reveals many more lines than previous analyses have shown (Fig. 4).16 Even Collier's reconstruction is considerably idealized,

14 This is discussed in chapter 3 of my dissertation, "Linear Perspective in Renaissance Painting and in Modern Scholarship," University of

Chicago, 1989. See further n. 19. 15 In pls. 4 and 5, small direction arrows (>) indicate a segment that is bowed out or hinged away from a neighboring segment; double arrows

(>>) indicate strong bending; and circles with dots signify points used to define the lines. Such points are usually (1) the end points of the

painted segments and (2) the places where the segment diverges most from a line drawn between its endpoints. They are indicated on the

plates in order that a rival explanation might say why it diverges from the lines I have drawn.

Vanishing lines are numbered sequentially along each side of the

painting, with very short lines numbered separately. This is done for the same reason: numbered lines are easier to compare, especially in cases like the ceiling in the Arnolfini Portrait, where judicious choice of lines might support a range of hypotheses.

Interrupted lines signify not uncertainty in the line itself but

uncertainty generated by other factors, for example, the interrupted vanishing orthogonal from the floor labeled 5.

16 In such reproductions, I have found it useful to number the lines, so

they may be more easily compared with other theories. The ceiling has eleven beams, and since each is seen from one side

except the central one, there are theoretically thirty-two lines, not eleven as drawn by Kern or seven as drawn by Collier. Of the

thirty-two, I could only see twenty-eight. The left wall and window, including its saddle bars, yields nine major

orthogonals (drawn in the plate, in which I have given numbers 3, 4, 5, 6, 7, 12, and 15 for reference), as opposed to the two usually drawn, not

including six lines implied between the rows of bottle glass, one

(numbered 4) in the grating, and twenty-three more in the very short

segments of stained glass along the far margin of the window-frame- for a total of thirty-nine orthogonals.

Nine clearly visible lines recede from the near shutters to a lateral

vanishing point outside the picture (numbers 1-9 along the left

margin); the farther shutter yields three more lines receding to a second lateral vanishing point (shown, unnumbered, above the brim of Arnolfini's hat); the rug contains five orthogonals (three of which are

drawn); and the canopy of the bed has four orthogonals (all of which are drawn and numbered 1-4; the tassels are drawn very neatly in three almost parallel lines).


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for example, in the first six lines between the floorboards, which he draws receding to a single point. Some lines in the original are quite accurate,17 and others are off by wide margins. The fifth floorboard line, for instance, can be drawn in two quite different directions, depending on whether the segment below or above the terrier is taken as standard.18 Its two possible directions X and Y differ by approximately three degrees, providing a margin of error that may be used to judge other elements of the picture.19 The tangle of lines can be pared away by discounting shorter segments, those with the greatest margin of error,20 and those in darkness.21 The most reliable orthogonals by these criteria intersect in four small vanishing areas, in accord with Panofsky's observation. Two of the

17 The third floorboard line is both long and almost perfectly straight. 18 The same is true of lines 6 and 7, and it is interesting that line 5 does not diverge more strongly than the four lines of the border of the rug, which are meant not to look straight. 19

"Margins of error" needs to be given several senses, depending on what is being measured. In the references in this article, I use it to denote the angle between two alternate reconstructed lines, each of which is meant to trace a single line on the original. In this sense, the term corresponds to the relative direction of the lines (see n. 13, heading [c]).

In a full account, which I have not found necessary here (but which might be called upon for future debate), other error terms are required- other ways of measuring deviations from ideal geometry. For example, some reconstructed lines may be eliminated from further consideration using several (named) criteria such as the ratio:

length of the line segment distance of its endpoint from the vanishing area

The smaller this fraction, the less the confidence we might have in the reliability of the segment; and numerical values can be collected. Values > 1 standard deviation from the norm can be discarded, as I do here. Whatever the exact method, it is essential that the criteria be given in print. This is discussed in Elkins (as in n. 14), and see further n. 31 below. 20 "Shorter segments" can be given as precise a value as desired, or the deleted segments can be listed for comparison with other accounts. Here the ten orthogonals on the floor have relative lengths as follows:

1 1 2 2.5 3 4.5 4 2+1 5 1.25 + 1 6 0.25 + 3.5 7 1+.3 8 0.6 9 0.5

10 0.9

where lengths such as 0.25 + 3.5 denote lines interrupted by objects such as the terrier. Here I have taken the mean length, 1.45, and omitted any line with a total length less than 2/3 of the mean. Hence lines 1, 7, 8, 9, and 10 are omitted because they are too short to be reliably measured. (This is taking a slight liberty from the standard deviation of 1.2 to allow for longer lines that are interrupted. Cutting off figures beyond the standard deviation would also omit lines 4 and 5.)

Next, lines with large margins of error are omitted. Most of the lines have errors of < 1?. Lines 6 and 7 have errors of approximately 1?; line 5 has an error of approximately 3?; and line 1, if its short left-hand segment is taken into account, has an error in excess of 5?. This eliminates line 5, and yields a list of four lines, numbers 2, 3, 4, and 6, which intersect in six points, all within the small area marked by the ellipse at H,.

vanishing areas are near the picture's vertical axis (H1 for the floorboards and H2 for the rafters) and two are disposed at either side (H3 for the principal lines from the window and H4, which continues beyond the border of the picture, for the canopy).2

The analysis can help distinguish between the theories we have considered. Since orthogonals like 5, 6, and 7 on the floor have large margins of error, Kern's thesis that Jan drew to vanishing points instead of to vanishing areas is unsupportable: we simply cannot be sure that lines converge with that accuracy. The same margin of error makes it impossible to say if Jan's painting contains "five, six, seven, or more vanishing points," as Collier found, because the points are not separate from one another by a greater distance than the margin of error of the individual lines.23 Doehlemann, Ward, and Carleton, who claimed multiple vanishing areas (and Panofsky, who claimed four vanishing "points"), are more in accord with the evidence since the analysis reveals four significantly different vanishing areas.2

The Lucca Madonna Following his theory of "elliptical perspective" Carleton also found two principal vanishing points in the Lucca Madonna (Fig. 6, H1 and H2). He hypothesized that Jan van Eyck projected symmetrical pairs of orthogonals, so

21 In the roof, darkness swallows many of the cross-beams and it is hard to tell their shadowed sides from the shadows they cast on the ceiling. Those that are visible have much higher margins of error than the floorboards, and a thorough error analysis (which I have not carried out here) would need to set a higher limit on the acceptable error than for the floorboards. Beams 17 and 18, for example (only no. 18 is drawn here), differ by 10?, but many others differ by 4? or 5?. The mean deviation is approximately 3?, with a higher standard deviation than in the floorboards.

Of the fifteen statistically most reliable lines, two-thirds fall within the ellipse H2 (lines 18, 20, 22, 24, and 25 miss the ellipse).

22By inspection of the four ellipses surrounding H1-H4, we can conclude that the H1 and H, are arranged on a vertical axis, and that it may correspond to the central axis of the picture: the ellipses H1 and H2 are above one another, and the picture's central axis runs just inside each one, near their right sides. The lateral vanishing areas H3 and H, are far less accurate, but it is possible to conclude they are lower than the midpoint of a line between H, and H2.

The vanishing area H4 continues beyond the border since line 2 (from the bed canopy) does not intersect lines 3 and 4.

2 Collier's mention of "five, six, seven, or more vanishing points" is not meant to imply that Jan had some system or rule that required drawing to so many points; he suggests that "While Jan appears to have had some notion that straight lines in any planar surface appear to converge, his works reveal that even internal lines within the confines of a single surface do not always align to the same vanishing point. This is not a systematic perspective 'system,' but, as Panofsky pointed out, one based upon empirical observation." Collier (as in n. 2), 691. But it is important to note that, regardless of whether or not Jan intended multiple vanishing points, the inaccuracy inherent in the picture does not permit the conclusion that there exist "five, six, seven, or more vanishing points."

4"Significantly different" since they are separated by more than the accumulated errors of the lines.


6 Lucca Madonna, reconstruction by D. Carleton

that the upper vanishing area H2 records the meeting places of three pairs of symmetrically disposed lines.25

Some lines do not fit well in this schema, including the shelf on the right-hand window and the receding edges of the base of the Virgin's throne (Fig. 6, a and b).26 Ward, in reply to Carleton's general claim that Jan used two vanishing points, found little coherent perspective (Fig. 7).27 Since Carleton, in his reply, accepted the lines drawn by Ward,28 their analyses differ only in which intersec-

25 Two lines from the windowsill and niche, and two pairs from the

canopy of the cloth of honor. The lower vanishing area H, is formed by five pairs of lines from the rug (whose borders are f and g) and one pair from the edges of the floor (d and e). 26 Carleton made the plausible suggestion that Jan wished to avoid the visual pun that would have resulted if he had drawn the bases of the

Virgin's throne as continuations of the edges of the floor. See D. Carleton, "Reply [to J. Ward]," Art Bulletin, XLV, 1983,687. 27 It was pointed out by a reader of this essay that Ward's conclusions, even though he finds the perspective "quite inconsistent" and disputes Carleton's claim of a "consistent application of a mathematical theory of perspective," nevertheless are consistent with, and may be read as

implying, a result closely analogous to Panofsky's hypothesis of four

vanishing areas. (Ward [as in n. 10], 680.) Thus Ward acknowledges that there is "a tendency... for the orthogonals [in the Arnolfini Portrait and other paintings Carleton analyzes] that belong to a side wall or vertical

receding plane to cross the vertical axis passing through the vanishing area of the floor or ceiling orthogonals before they meet," which would

produce a version of Panofsky's hypothesis. Ward later refers to the

perspective "systems" of the floor and cloth of honor of the Lucca Madonna (ibid., 681).

7 Lucca Madonna, reconstruction by J. Ward

tions they considered significant. Ward, for example, paired the shelf with the sill of the niche and got a vanishing point well to the left of center (Fig. 7, c). By pairing the two windowsills, Carleton obtained a vanishing point close to the midline of the picture (Fig. 6, H2). Carleton defended his method by appealing to a classifi- cation of important and unimportant orthogonals:

In order to analyze the perspective of Jan's interior scenes meticulously, agreement must be reached as to which orthogonals are of primary value and which are of secondary value. Ward agrees with this premise, saying that certain orthogonals are: ". .. shorter, fewer, more scattered, or otherwise less in need of treatment of a single system." I classify Jan's secondary orthogonals as those which were probably not used in the actual structuring of the rooms.... It then follows that a primary orthogonal is one that plays a dominant role in determining the structure of the parallelepiped that defines the space of the room. Examples are corner edges, floor tiles, roof lines, and certain window edges. Jan generally uses these orthogonals in symmetrically matching pairs.29

28 See ibid., 686, where he says he used a digitized version of Ward's own diagram to create his version. 29 Ibid.


Accordingly Carleton drew the matching orthogonals d and e separately from f and g (Fig. 6), and Ward drew d with f and e with g (Fig. 7). Carleton accused Ward of circular reasoning, saying Ward found a random pattern of orthogonals because "a random selection of pairs of

orthogonals will produce a vanishing area only if a single vanishing point was used."

Ward was-according to Carleton-expecting to find either a single vanishing point or none at all, and so he

paired adjacent orthogonals. His method could have been sufficient to locate a single vanishing point, but since there is only a general vanishing area, it produces incoherent results. Carleton defended his own method

against circularity by noting that Jan disposed the orthogonals in symmetrical pairs (an observation that applies particularly well to this painting and not at all to others, including the Arnolfini Portrait), and that the natural way to create a perspective picture is to work from the

perspective box "inwards, pairing symmetrically placed orthogonals on the left and right sequentially." It might be objected that while an artist will often begin with the

perspective box, it is not usual to work from the sides inward: there may be many reasons not to do so, and an interest in some portion of the picture may override a desire to maintain symmetry. But beyond this, both Carleton's and Ward's methods, despite the former's

objection, are circular in the sense that they analyze in such a way as to disclose a particular pattern (or lack of it). The only really defensible analysis when the perspective is unknown or uncertain is a reconstruction of the kind I have outlined, which includes an error analysis and does not-in the initial drawing-choose which lines to include, but includes all of them.

Such an analysis of the Lucca Madonna, using forty-six of the painting's ninety-eight orthogonals,30 reveals four

significantly different vanishing areas, as in the Arnolfini Portrait (Fig. 5). The vanishing areas from the side walls, H, and H4 are precise,31 and the other two much less so, but the presence of four areas is consonant with Panof- sky's generalized claim that Jan used four vanishing points in the Arnolfini Portrait. The finding weakens

30 As in the Arnolfini Portrait, there are six major orthogonals in the left wall, and a number of others produced by the bottle glass (numbers 3-6, 8, 9, 11, 12, 14, 15, 17, and 18) and the small segments along the far windowframe (numbers 19-39, not illustrated).

The floor has a pattern that includes a number of orthogonals deemed too faint to include here (numbers 2-6, 10, 26-30, and 32-36). (Orthogonals 2 and 3 are strongly out of line with 4 and 5, more so than

corresponding orthogonals in the rug.) The other lines not included in the analysis are the eight orthogonals from the capitals atop the arms of the throne (interrupted lines). 31 The ellipse drawn around H3 could be drawn larger if line 7 and the

remaining shorter lines are taken into account (i.e., lines 3-6, 8, 9, 11, 12, 14, 15, 17, 18). Such a revaluation of the analysis of pl. 5 would have to introduce a new criterion for the inclusion of lines whose painted length (that is, in the window, between rows of bottle glass) is small

compared to their projected length (the distance they have to be drawn to intersect). Such a criterion should be made in quantitative fashion ("I accept for analysis all lines where such ratio is > 1/5") and applied throughout the painting. See n. 19.

Ward's hypothesis that Jan's paintings do not have any consistent system of vanishing areas and also Carleton's claim that Jan used a perspective system based on two

vanishing areas. (The vanishing areas from either side

actually contain more orthogonals than those from the floor and ceiling, but since the walls do not have rafters and floorboards and since only a sliver of the window is visible, they are less prominent.)

At the same time, some details of the analysis substan- tiate parts of Ward's thesis. In particular, there is an

intriguing concurrence of lines from small portions of the

picture, as against Carleton's assertion: lines from the floor at the lower left converge to a different place than lines from the adjacent section of rug: the former con-

verge off to the right, the latter closer to the midline of the painting. Together they define the vanishing area HL. In addition, lines from the right side of the rug and the adjacent floor converge to still another area (labeled HIR). There is still a discrete lower vanishing area for all

orthogonals from the floor, but it is quite large and diffuse (it would encompass HIL and HIR). Similarly, lines from the left canopy converge differently from lines from the right canopy (H2L and HR).32 There is immediate visual truth in this, since adjacent orthogonals are apt to have been painted at the same time, and they are also taken in with single glances, so our eyes are sensitive to the coherence of their lines.

Conclusions The statistical analysis weeds out the remaining theories: it shows that Carleton's hypothesis calling for two

vanishing areas is in need of greater support, since both

paintings have four points of equal weight; and it

partially supports Ward's method of drawing lines from local areas of the painting rather than from symmetrically disposed orthogonals. Panofsky's informal claim of four vanishing "points" again fares best.

Each surface, whether wall, floor, or ceiling, was seen

separately, and the vanishing areas of ceiling and floors do diverge, as Carleton noted; but there is also interest in small sections within surfaces, such as the bare floor and

adjacent rug border in the Lucca Madonna.33 Seeing is indeed episodic; small portions of the painting, such as the adjacent orthogonals in the lower corners, were seen together, and so were larger chunks like the rug. Instead

32 Ward's diagram (Fig. 7) does not make this latter claim, and his lines are often oriented differently than those I have measured (compare, for

example, line e in Fig. 7 with line 37 in pl. 5 and line g with line 31): nevertheless his general method of drawing lines is supported by closer

analysis. 33 A reader points out that the bed in the Arnolfini Portrait appears, as

closely as may be judged from its ungeometric contour, to be pointing toward H2 and not to H4 (Fig. 4). It is therefore a separate unit, and evidence that Jan did not always think in terms of surfaces, but also in terms of objects and parts of objects. Stress on the concept of a "surface"- that is, a wall, floor, or ceiling-is part and parcel of interest in, and often assumption of, the primacy of the perspective box, with its five

intersecting planes. Without that notion, it is easier to conceive different units of attention-for example, the right-hand border of a rug, the edge of a bed, a canopy.


of "episodic seeing," this might be called "Chinese-box seeing," with increasingly larger parts perceived as wholes. The smallest units at the lower left of the Lucca Madonna are the two sides of the rug's dark border, which are seen, and were possibly painted, in two groups (Fig. 5, 11-12 and 13-14). Their four lines form a unit, and that unit is in turn related to lines from the tile floor to its left, which in turn consists of two groups, the lines around 7-9 and those around 1-3.34 All lines in the corner can also be seen as a whole, distinct from the more vertical lines of the central swatch of the rug. The largest "Chinese box" is the perspective box of the room itself.

The separation of the four vanishing areas H1-H4 can best be accounted for by a desire to minimize the distractions of disharmonious diminutions: with the vanishing area of the ceiling moved up, and that of the floor down, and those of each wall extended "too far" toward the opposite side, there is a maximum uniformity in the deceleration of the various rates of foreshortening. The only other way to attain such a mean would be to draw in medieval parallel projection. If Jan had drawn his composition in correct linear perspective, some of his gentler recessions would have been lost: relative to the adjacent rear wall and throne, the window would be

squeezed; relative to the basin below, the shelf would tilt

uncomfortably; and in relation to the canopy, the floor would heave up steeply (Fig. 8).35

Jan van Eyck presumably had no interest or awareness of these analytic finesses, and we need not assume he

34 The two sets 11-12 and 13-14 form a unit since they converge at the

upper portion of H1, and the tile lines 1-10 form a separate and

adjoining unit since they converge a little higher (to the right of the label H2,). No chronology is possible in these cases; one might say instead that lines 1-6 and 7-9 are a unit, which is conjoined to the unit

composed of lines 11-14.

35 The effects shown in Fig. 8 depend on the placement of the principal point (open circle) and on the relative sizes of the Madonna and Child, throne, and back wall. Here I have retained all such proportions and varied only the objects between them and the picture border-the window, niche, canopy, and floor. A full comparison of this type would be best done on computer, with a high-resolution scanned image of the

painting in order to keep a semblance of the original textures and

lighting. This increased uniformity of foreshortening does not necessarily

imply a greater relaxation or calmness in the scene as a whole. It should be kept in mind that apparent steepness (or "velocity" or "acceleration") of converging objects is a two-edged sword: when a tabletop recedes

rapidly, so that it is seen nearly edge-on, it may appear quite static; on the other hand, a "softened" recession, in which a great deal of the

tabletop is shown and it seems to tilt, may appear imbalanced and

8 Lucca Madonna, corrected perspective by the author

had any "system" in mind-the differences between the two paintings are great enough to argue against that. Yet he accomplished by eye, and with consistency between

paintings, a compromise between medieval and Renais- sance sensitivities.

James Elkins received his Ph.D. degree from the University of Chicago. He is the author of numerous studies on aspects of art and science, including an earlier Art Bulletin article (LXIX, 1987), "Piero della Francesca and the Renaissance Proof of Linear Perspective" [The School of the Art Institute of Chicago, Chicago, Ill. 60603].

vertiginous. What I claim in this instance is that Jan's arrangement achieves more uniform deceleration of the walls and objects, which would be inconsistently foreshortened in proper linear perspective since some are nearer, and some farther, from the principal point.

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Did Jan Van Eyck Have a Perspectival System