Symbolic meaning in Leonardo's and Raphael's painted architecture

WELL \ AEA SYMBOLIC MEANING IN LEONARDO'S AND RAPHAEL’S PAINTED ARCHITECTURE By WARMAN WELLIVER Perspective is the curb and guide of painting. Leonardo, Treatise on Painting T# study of Leonardo’s Cenacolo and Raphael’s Disputa and School of Athens, together with one on Titian’s Pesaro Altarpiece to be published elsewhere, form a sequel to my earlier work on the architecture in some paintings by Domenico Veneziano and Piero della Francesca: I wanted to sec how the greatest High Renaissance painters dealt with the problems and opportunities of painted architecture, and particularly whether their work continued what I had called a tradition of symbolic architecture. One especially interesting aspect of these later paintings, in contrast to the earlier ones, was that, being still in their original positions, they would make it possible to reconstruct more fully the intentions of the artist and the experience of the viewer. Since this is not a very well known area of inguiry, I will begin with a few general observations. First, this kind of study, as I conceive it, rests on the following chain of reasoning. There is good cause to believe that many postBrunelleschan painters of architecture first designed a floor plan and clevation and then, choosing an appropriate viewing distance and eye level, translated these to the flat surface of their paintings.? These plans and clevations, if we could see them, would be interesting and perhaps valuable evidence of the artist’s thought. Since the historical enterprise consists largely in rediscovering and reenacting the thought of historical figures, the historian concerned with this kind of painting ought to retrieve these plans and clevations if he can. In the light of this reasoning, I would like to make two comments on present scholarly practice. The first is that most of what has been written about the use of space in fifteenth-century Italian painting takes little or no account of the precise shape and size of that space—in John White's Birth and Rebirth of Pictorial Space, for instance, onc looks in vain for even a single floor plan of painted

architecture? If the principles stated above are correct, then this reluctance to come to grips with the concrete problems of painted architectural space is a plex, and by fitting into a consistent and likely account of the painter's thought, then the chances are good that it is correct. mistake. My second comment pertains to the reconstructions of painted architecture LEONARDO'S AND HIS PREDECESSORS ARCHITECTURAL SETTINGS that have been made by a few other scholars. Almost all these reconstructions OF THE CENACOLO are partly inconsistent with the premises just mentioned because, although they do retrieve a floor plan of the painting, they cannot succeed in retrieving the painter’s plan since they do not correctly define it—or perhaps it would be Now this insistence on seeking the painter's unit of measure, and therefore the numbers on his floor plan, is not, I hope, mere pedantry. In searching for a Much has been made of the difficulty of deciphering the perspective scheme in Leonardo's Cenacolo (Fig. 1), and one scholar has even gone so far as to maintain that we should not try: it would be “neither pious nor scientific.” But the same scholar called the painting “che most completely excogitated picture in the history of art”; and surely such completeness included the rather elementary excogitation of choosing a depth, width and height for the room and a viewing distance (which of course had to be decided on before the room could be translated to the painting). If this is so, then it is both the duty and opportunity of the historian to try to discover them instead of succumbing to inhibitions which would surely not have hindered an artist so mathematically and geometrically inclined as Leonardo.® Thanks to the coffered ceiling, the task is not really so very difficult. Let us continue the cciling to its intersection with the picture plane; assume that che coffers are square (seen foreshortened there is no way of knowing), and that painting’s meaning and purpose, for the thought of the painter and perhaps of Leonardo used the same system of measurement as was used in the cities where fairer to say that they do not retrieve it since they do not set out to find it. We must remember that a floor plan consists of lines and numbers, a number representing the length of a line in a given unit of measure; the painter’s floor plan must have shown lengths in the unit of measure that he chose to use. But most modern reconstructions determine lengths in metric or Anglo-American units of measure, which obviously cannot have been the unit used by a fifteenthcentury Italian artist.* Therefore the modern reconstructions, although they may show us the correct shape of the painter’s floor plan, cannot tell us the numbers that he used to indicate its dimensions. his patron or an adviser, we make inferences from every available datum— he was born and worked, that is, a unit about one-third the height of a man, from the style and composition, from the persons, actions, objects and places painted, from other works of the painter and of other artists, from literary and called a braccio; compare the coffer squares with the figures below; and estimate the size of a square as 2 by 2 braccia in the scale of the painting (Fig. 2).” Doing this, we find the hall to be 28 braccia deep by 12 wide by 9 high, the “correct” viewer to be standing 12 braccia from the painting, and the sum of painted and real space (converting the metric dimensions of the refectory to braccia of the painting) to be 72 by 11, and then 12, braccia (Fig. 3). Assuming that we now know the floor plan and viewing distance which Leonardo chose for the scene, two subjects of speculation present themselves: what can we learn about the painting by comparing Leonardo's solution of the other written sources, from the biographies of the painter and his patron, and from the historical situation at the time. We never know beforehand which data will prove the more enlightening and which the less. Obviously, numbers, too, can support inference. Modern perspective studies, in failing to scarch for the painter's unit of measure and his dimensions, may therefore be missing a crucial clue to the understanding of the work and its maker. The reader should remember that the following argument, like all exegetical thought, is hypothetical. It starts from two assumptions, the shape of a floor or ceiling quadrilateral and the painter’s unit of measure; works out the resulting floor plan; tests the plan, and thereby the assumptions on which ie is based, by tests intrinsic and extrinsic (I have limited probative discussion to the notes); architectural problem with the solutions chosen by his predecessors? and what, beyond this general relation of the painting to the traditional treatment ofthe scene, were the particular merits of Leonardo’s solution for this subject and site? I will creat these two questions in that order. and concludes by considering clements of the painter’s thought suggested by Let us first survey the problems that a painter faced in painting a Cenacolo his design of the architecture.5 If this interpretation wins adherents and withstands attack by explaining aspects of the painting chat would otherwise peron the end wall of a refectory and then examine the solutions devised by Leonardo’s immediate predecessors. Scenes of cating were naturally a favorite

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Fig. 5. Refectory, S. Maria delle Grazie, Milan Fig. 6. ANDREA DA CASTAGNO, Cenacolo. S. Apollonia, Florence Fig. 8. PERUGINO, Cenacolo. S. Onofrio, Florence

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Fig. 16. RAPHAEL, Disputa. Vatican City PYTHACORAS'S TETRACTYS DIS PUTA T ' , L NG bows I / CHRIST CD PROPHETS *% SAINTS. 33 L I La 22 #4 “ H >“ Stres al "Io Gop pui Er rs > L I TI I_ Host_To VIEWER | x 19 EPA A MEET FLooR v 6) v | Fig. 17. Pythagoras’s tetractys and the dimensions of Fig. 16 1 A X &) v (d) L'ALTEZTA Wa Fig. 18. (a) Disposition of figures in Fig. 12 (b) Major groups of figures in (a) (c) Lincar abstraction of (b) VT \ N (d) Euclid’s (?) design (c) Euclidean design of figures in Fig. 16 \ EEN \ Vv {2} pu, Fig. 19. RAPHAEL, scale drawing of section through front ov of Fig. 12. Biblioteca Ambrosiana, Milan —

subject for this position: of the five scenes bordering the central crucifixion which Taddeo Gaddi painted about 1335 on the west wall of the refectory of S. Croce, no less than four show diners at table, the Cenacolo occupyin eh whole of the lowest register. At least six versions of the Cenacolo Rte before Leonardo's survive on the end walls of Florentine refectories > But if for the prior the subject was ideal, for the painter following Brunelleschi the site was far from being so. The viewer would naturally think of the painted scen as an extension of the real scene of monks assembled for a meal: both ainted and real figures eat and drink, use similar, if not identical, table settings sit at tables similar in appearance and perhaps symmetrical in position. It would be a natural expectation, then, that the space depicted in the painting should prolong the real space of the refectory.!° The oblong shape of a refecto how. ever, makes a convincing pictorial extension of its space almost impossible for three reasons, which correspond to the viewer’s location in three dimensions Perhaps worst of all, a refectory greatly aggravates the contradiction between the single, fixed viewing station required by Brunclleschan perspective and the reality of viewers who are free to move in depth. A viewer natural stands at least a width away from a painting in order to take in its full width comfortably and without rotating his head; and chapels, being usually not far from square, prevent or discourage viewing from a much greater distance But a refectory! It is generally between three and four times as long as it is wide, and the viewer can vary his station over a range two or three mes the width of the painting." If this is an interior scene, like the Cenacolo, and the viewer moves very far forward or backward from the “correct” station the relation of foreground and background objects, which is reasonable for that station, may become incongruous: particularly, the angle between the fi shortened floor, side walls or ceiling, and the end wall in the painti may sensibly diverge from the normal ninety degrees which a viewer infers when standing at the “correct” station. A good example of this effect is the aspect of the ceiling in that milestone of perspective painting, Masaccio’s Trinity, . from a distance of about a small refectory’s length, or some three times the prescribed viewing distance (Fig. 4): the ceiling seems to slope d à stecp angle from the horizontal. PE COR A The second problem posed by a refectory arises from the viewer’s frecdo to move laterally. One might think that at least the narrowness of the ro: m would give the painter an advantage, since it prevents the viewer from stra ing far from the axis through the center of the painting, somewhere slong 50 which his ideal station is customarily placcd; and this would indeed be true, if walls of a refectory he could be restrained by invisible walls. But the long side in the painting: as depth into ng runni have a most unfortunate effect on walls apparent width the and ory refect the the viewer moves toward a side wall of side of the same the on wall r simila a of that wall consequently diminishes, es in a behav wall, real the ding exten painting, perhaps even a wall ostensibly (I swell to seems It ing. shrink of d distressingly opposite way, swelling instea its from r farthe and it to r neare ly suppose) because the viewer moves slight he sees it full-face but counterpart on the opposite side of the painting, because with the se of its contrast the opposite wall slightly foreshortened, and becau much of the width of a shrinking real wall. Where a painted side wall takes up be fatal to any illucan effect s painting or joins the real wall, this incongruou the center axis, that on not and ng sion of a viewer very distant from the painti the painted space continues the real space. location, the one Lastly, a problem arises on account of the viewer’s vertical rdo's time the Leona By move. to dimension in which he is ordinarily not free height of a the to ng painti a in gencral problem of the relation of the horizon experience ce: practi in least at ed, viewer’s cye above the floor had been resolv “correct” a eye 's viewer the had shown that for narrative paintings above painting, the in floor the below y horizon at his eye level, and therefore usuall tomed to suspending their was not satisfactory, and viewers were well accus aded that they were sense of horizontality cnough to allow their being persu For a viewer at the up.!? ng looki were looking horizontally when in fact they since the angle by easier, even is far end of a long refectory this suspension ingly smaller. But spond corre is ntal which his line of sight rises above horizo ce from the paintdistan r greate this of at the same time, and precisely because the painting reof on horiz the and ing, the difference between his eye level takes in the side now view of field his on: sults in a serious threat to his illusi walls of the building and their orthogonals, which aim down at his personal the juncture of the vanishing point, and he sees them harshly interrupted at the latter suddenly bend side wall and the painting, where the orthogonals of 5). up to aim at its higher vanishing point (Fig. 98), his predecessors In the half-century before Leonardo’s painting (1495- on of continuity as had evolved a strategy for protecting the viewer's illusi shape of a refectory. Anmuch as possible against these threats posed by the santi, 1480) and Peru(Ognis o andai Ghirl drea da Castagno (1447), Domenico gruity between near gino (1490) all simply forestalled the possibility of incon

and more distant objects by screening out the space beyond the diners— Andrea all of it, Domenico and Perugino enough to prevent the juxtaposition of the figures with anything more distant (Figs. 6, 7 and 8). All three limited their side walls to a small proportion of the width of the painting, and Perugino even went so far as to avoid any appearance of foreshortening at the top by locating the horizon virtually at that level.!3 True, he did venture some timid architecture in the background, but the very fact that it consists only of columns is one morc testimony to his and his predecessors’ horror of foreshortened walls, Both he and Andrea took the additional precaution of keeping the side walls in their paintings well away from those of the refectory, thus making the walls’ opposite behavior for the lateral viewer less conspicuous and avoiding any obvious break between the orthogonals of building and painting."4 In the matter of the viewing distance cach painter chose a different solution, hair as roughly one-third the width of the door, in our scale some 2 feet, and the length of Thomas’s head as about 24,.'® Instead of mitigating the inevitable distortion of the side walls for che distant viewer by restricting their width and separating them from the walls of the refectory, Leonardo stretched both of them across one-third of the painting and carried them virtually to a junction with the real walls. Consequently, the distant viewer gets the strong impression that they meet the rear wall at an angle of one hundred twenty degrees or more and that, seen from a lateral position, they contradict the appearance of the real walls (Fig. 5).2° To the still greater disturbance of the distant viewer, Leonardo carried the conspicuous lateral orthogonals defined by the tops of the tapestrics almost to a junction with the refectory walls at a point where their inconsistency with the orthogonals formed by the frieze decorations of the refectory would be most apparent (Fig. 5).2! And in adand cach solution had some merit as a defense of illusion. Andrea chose a disdition to all this he violated an elementary principle of illusion, which his not appreciably exceed it.!5 Ghirlandaio, instead, chose two-fifths of the refectory’s length, thus reducing the average deviation nearly to a minimum." Perugino went to the opposite extreme from Andrea and put his “correct” viewer so close to the painting—one width away—that he would no longer sce the foreshortencd side walls of the refectory at all.!7 For this viewer there life size: if Christ were to rise and advance to the picture plane, He would stand some 3.6 meters, or 12 feet, tall.22 tance so great—four-fifths the length of the refectory—that a viewer could would be little if any incongruity between real and painted space, since the few meters of real space left could easily seem but the extension of the painted space. In the case of the more distant viewer, for whom the refectory’s side walls would increasingly define the real space, Perugino’s concealment and avoidance of foreshortening in the painting would mitigate any contradiction between its space and that of the refectory. The components of this defensive strategy developed by Leonardo's immepredecessors had been carcful to observe, by making his figures about twice Why so uncompromising a rejection of the defensive strategy formulated by his predecessors, we ask —of the safeguards which they had developed against the threats to illusion posed by the extreme mobility of viewers in a refectory? The answer, I think, is that their defensive way required above all things shallow architectural space, whereas one has only to look at Leonardo’s earlier paintings, the Annunciation and Adoration, at his writings, and at the works of Domenico Veneziano and Picro della Francesca—his true predecessors in the use of architectural space—to see that what interests him is deep space.” Domenico and Piero, however, did not have to contend with refectories, as far as we know; how were the special demands of these to be reconciled with diate predecessors constitute, almost without exception, a list of the things which Leonardo did not do.'® Choosing, as had Perugino, nearly the shortest possible viewing distance, 12/11 of the painting’s width, he showed no such the deep architectural space which Leonardo preferred? The answer was simple and bold: he would combine deep space with a short viewing distance and thus exploit all the visual discomfort of the distant viewer, which his predecesconcern as Perugino’s for the visual comfort of the distant viewer. Far from screening out distant objects, he posed his figures at the near end of a long hall and opened a door and windows in the far wall. Instead of avoiding incongruity between the figures and these distant objects, he positively invited it by framing Christ’s and Thomas’s heads in the door and a window: the viewer at sors had worked so hard to alleviate, as a magnet to pull him closer to the painting. What the distant viewer secs is troubling, is out ofjoint; then let him the far end of the refectory who senses how much farther he is from Christ follow the straight and narrow axis of the painting to the proper viewing distance, to the point where there can no longer be any disappointed expectation that the space of Christ should continue the space of the refectory, because there is hardly any refectory left to see. Let him look up at Christ and the than Christ is from the wall beyond, will read the width of Christ’s head and disciples in all their double-life-sized majesty —and at nothing else.

One wonders also whether there may not have been in this exhortation to the distant viewer an admonition ad hominem. Goethe tells us (and we could reach the same conclusion from archaeological evidence as well) that the prior's table was at the far end of the refectory from the painting —the prior had one of the worst seats in the house.?4 Was Leonardo's message preeminently for him and perhaps for the illustrious guests he must have entertained from time to time? Were they to take their meals where their eyes could find no peace and relieve their discomfort only by coming among the lowliest ofthe monks?25 By some such reasoning Leonardo was emboldened to design a hall for the scene even wider than the refectory instead of narrower, and over twice as deep as it was wide, a hall which came close to adding a second refectory to the first (Fig. 3). Instead of struggling against the long architectural view and trying to wall it in as closely as possible beyond the picture plane, as his predecessors had done, he sympathized with it and generously extended the architecture from a length four times its width to six. The painting was a revolutionary break with the long tradition of shallow, closed space on the end wall of a refectory. Now let us look for the reasons which may have recommended to Leonardo the particular dimensions that he chose for his deep space. The basic dimensions which confronted him as he attacked the problem were the width and length of the refectory: it is fifteen Milanese braccia wide by sixty long. As it happens, fifteen is a number that seems quite irrelevant both to Christianity and to the Last Supper; one is hardly surprised that, working in a strong tradition of numerical symbolism, Leonardo decided to impose different dimensions on his space by devising a different scale. The scale which he chose made his painted hall twelve braccia wide, and this is, of course, a most appropriate number for the feast of the twelve disciples. On this scale the length of the refectory becomes 44 braccia; a great virtue of 28 braccia as the length of the painted hall thus appears to be that it results in a total length of real and painted space which is a multiple of twelve. If we consider along with these dimensions, that the viewing distance is twelve braccia, we can suspect that Leonardo designed his floor plan and viewer's station with an eye to their permeation by the number of Christ's disciples.?? dow. The painter has to design his architecture not only for itself but also for the two-dimensional appearance it will have when the images of objects at different distances are all intercepted by this one fixed plane. Incorporating deeper and shallower forms in one surface design, the painter unifies his work, attains the compositional interest that is the particular domain of painting, and maintains the authority of the surface over the recession of decp space — In this blending of the flat and the deep Leonardo especially exercised his geometric and symbolic talents. The whole scene he restrained in a twodimensional composition of rectangles with proportions of 4:3, 3:2, 2:1, 1:1, and 1:4 (12 by 9, 9 by 6, 6 by 3 and 12 by 6, 6 by 6, and 3 by 12 braccia; Fig. 9).# As if balancing the two sides of the composition on a Christian-Pythagorean fulcrum, he defined with Christ's arms a wedge consisting of two right triangles with sides in the ratio of 3:4:5 (Fig. 9). The deepest element in the painting (except for the landscape), the far wall, he conjoined to the surface in various ways. For one thing, when Christ’s feet were still visible against a center floor stripe, the intimation of a monumental twelfth- or thirteenth-century crucifix awaiting Him must have been strong indeed (Fig. 10).% Another link between the far wall and the foreground is that the upper edge of the table virtually coincides with the (invisible) bottom of the wall. Finally, the design of the wall itself is an intricate concatenation of dimensions in braccia of the picture plane (B,, Fig. 11) and braccia at the depth of the wall (B,). One detail of this design particularly shows the stamp of Leonardo s thought: the top of the pediment which finishes the frame around Christ’s upper body and is the only curved architectural line in the painting. Leonardo placed the intersection of this arc with the center axis at one picture-plane braccio above the vanishing point, but drew the arc with a radius equal to three braccia at the depth of the far wall. As a result, the pediment not only epitomizes chat mysterious union of three dimensions with two which, for Leonardo, was painting, but symbolizes the far more mysterious union of three and one which is Christianity.?! THE PYTHAGOREAN AND EUCLIDEAN STRUCTURE OF RAPHAEL'S “DISPUTA” AND “SCHOOL” though related, matters: the painting is, after all, a flat, vertical surface, and The School ofAthens (Fig. 12) is an apotheosis of chought: in groups around the two masters of ancient thought (Plato, on the left, and Aristotle) the figures on it the elements of the architecture will be seen not as in a floor plan and elevation but as they would form a two-dimensional design traced on a winwrite, read, expound, calculate, or follow intently the thinking of the seers. On either side in the foreground the inquiry is conspicuously mathematical But the design of architecture and the design of a painting are two different,

and geometrical, Pythagoras, his table of numbers and musical ratios held before him for ready reference, writes in a space ruled off from the text for arithmetical or geometrical illustration (Fig. 13). At the right Euclid (or perhaps Archimedes) measures with a divider on a six-pointed design (Fig. 14). It is a foreground to set a viewer measuring and calculating, too; what are the shape and proportions and dimensions of this noble space and what, to follow the example of Pythagoras, their meaning? The painting being in Rome, the most likely unit of measure is the standard the third module on the plan, as a painter-architect would surely do cither on the drawing board or in his mind, we can recognize a correspondence of the ratios 1:2(:3):4 and 1:10 with Pythagoras’s tetractys (1+2+3-+4=10; Fig. 13)25 On the wall opposite the School Raphael painted the so-called Disputa (Fig. 16).3 It, too, is an architectural painting at the bottom —the floor, steps and platform provide a stage for the figures similar to chat in the School —and what we might call the dimensions of the upper part are very much in tune with Roman unit (in the Cinquecento) of the palmo, a choice perhaps further recomthose of the School: the viewer stands 44 palmi from the host on the altar; the and by the location of the vanishing point close to Plato's left hand. Since the and therefore (taking 22/7 as the value of =) 33 palmi in circumference; Christ mended by the prominence of that part of the body in the two central figures standing figures in the foreground are considerably taller than the width of one foor rectangle (defined by imaginary lines bisecting the continuous white stripes), it seems reasonable to assume for these rectangles a width of threcfourths the height of a standard man, or six palmi.2? On the supposition that the depth of a floor rectangle is 44, palmi, the viewing distance proves to be 25 palmi (in the scale of the painting), or 6.75 meters, about what one would expect where the maximum viewing distance was probably no more than 7.65 meters (the room is 8.40 meters wide and originally seems to have had bookcases along the east and west walls).>> The viewing distance conjectured in this way, it is easy to follow the floor plan back (Fig. 15): from the viewer to the bottom of the steps is 34 palmi (25 plus 9), to the top of the steps (assuming treads 2 palmi deep) 40, and to the front of the pylons 54% (calculated from the convergence of orthogonals on the upper floor); and from there to the far end of the farther barrel vault all distances can be calculated from the diminishing apparent width of the interval between the pylons and then of the vaults. The painting provides no data for finding the distance to the triumphal arch, but by the time we have extended the plan to the far end of the vaulted aisle, the modular scheme has become sufficiently clear to allow extrapolation of a likely distance of 222 palmi to the near, and 250 to the far, side of the arch.34 The modular scheme is elegant mathematics: the module of 624, palmi is the sum of two key dimensions, the length of a vault and the width of the crossing prophets and saints sit in a celestial semicircle about 21 palmí in inner diameter occupies a disc 22 palmi in circumference; the structure culminates in the one God.37 Once again the dimensions seem a variation on Pythagoras’s tetractys (Figs. 13 and 17; the ten numbered rectangles are those of which no part is occupied or overlapped). As if to confirm this relation, Moses sits opposite Pythagoras and holds tablets that are the same shape as the Pythagorean diagram of the musical ratios and that contain the Ten Commandments.® The underlying structure of the architecture in both paintings, then, scems to be essentially Pythagorean, and Pythagoras clearly deserves his place as the foremost sage in the School.? Instructed by his significance, we might expect a spccial importance to attach also to Euclid or Archimedes, his almost equally prominent counterpart on the right, and to the six-pointed shape which is the counterpart of his numbers and ratios (Fig. 14).4 If on the floor plan of the School we plot the positions of the figures (Fig. 18a), they can be seen to divide into the two ranks beside Plato and Aristotle, the three figures on the steps, and five other groups (Fig. 18b; I have omitted four minor pairs of figures and the seated Michelangelo, who was not in the original composition).* When the five other groups and the intersection of the viewer’s axis with the picture plane are regarded as the apices of triangles, the resulting figure is not unlike the Euclidean design (Figs. 18c and d). Just as the plan of the architecture is Pythagorean, so is the plan of the figures Euclidean. him to the far side of the arch ten times the viewing distance. If we mark off Having found both this Pythagorean-Euclidean counterpoint in the architecture and figures of the School and a Pythagorean theme in the architecture of the Disputa, we would do well to look also for a Euclidean theme in the figures of the Disputa. And since there the Pythagorean element is essentially vertical, in contrast to its horizontal manifestation in the School, we might expect any Euclidean arrangement of the figures to be vertical also. Including again 56 57 (or the length and width of a vault); it places the center of the crossing at the triply unitary distance of 111 palmi from the viewer; it results in the proportion 1:2:4 for the distances from him to the near side of the near vault, the near side of the farther vault, and the far side of the arch; and it makes the distance from

the intersection of the center vertical axis wich the bottom edge of the painting (this point in an elevation is identical with che intersection of the viewer's axis with the picture plane on a plan), we again find six peripheral points which considered as the apices of triangles, lead to a design like Euclid’s (Fig. 18e). And just as Moses with his Ten Commandments sits opposite Pythagoras’s numbers, so do we now find David sitting opposite the classical geomcter. looking at him as he surveys his two triangles, and realizing, of course that they form a Star of David.4? | | Pythagoras and Euclid, then, provide us with a fourfold key to the construction of the paintings—Pythagoras to the architecture, horizontal in the School largely vertical in the Disputa; Euclid to the figures, horizontal in the School, vertical in the Disputa, The Judaeo-Christian scene, primarily vertical, stands in harmonious and amicable dialogue with the classical scene, primarily horizontal, just as Plato and Aristotle conduct their friendly dialogue, the one pointing up, the other ahead.” In the end the School proves to be not only an apotheosis of thought but part of a larger apotheosis of agreement and harmony: in this little room the viewer finds himself equally incorporated in both the heights of Judaco-Christian revelation and the breadth and depth of the searching mind of antiquity.** Thus did Raphael, fresh from four years of apprenticeship to the Florentines, achieve in Rome the monumental expression mony of all traditions. of the heritage of Pico and Ficino, of the Florentine faith in the essential har- BLOOMINGTON, INDIANA October, 1976. 1 am ve in the Pata Pesan 5 ! Leonardo, Treatise on Painting, ed. A. McMahon, Princeton, 1956, Vol. I ; Welliver, “The Symbolic Architecture of Domenico Veneziano ond Piero della Francs “ 2 ‘on » Le to be published in the acts of the Convegno “ Tiziano e Venezia,” held in Venice in XXXVI, 1973, PP. 1-30; “The Buried Treasure of Titian’s Perspective: The Architecture H much indebted to Philipp Fehl, whose suggestion that I study the perspective scheme of the Pala DP vane {which it would probably never have occurred to me to do) led me to begin also the work on the oth zint ings that I had contemplated for some time, to Howard Saalman for his valuable cc ument, and to Jerrold Lancs for many editorial suggestions, omy 2 See note 35 below and Fig. 19. | know of no earlier scale drawings to be painted, This is not the kind of drawing that would be likely dp rat sels 1964, p. 246. ing relation of size and position between nearer and more distant elements; and second, when, having reconstructed a floor plan from some elements in a painting, we use it to predict where other clements should appear on the picture plane, we find that the artist did in face put them there (Welliver, “Symbolic Architecture,” p. 28, note 27). 3 Second edition, New York, 1972. See note 6 below and A. Parronchi, Studi su la dolce prospettiva, Milan, 4 For previous reconstructions see Welliver, “Symbolic Architecture,” p. 25, note 3, and p. 30, note 56. Krautheimer, Janson and Jones have used units that might have been used in the fifteenth century; ibid., p. 25, note 4; p. 26, notes 8 and 13; and p. 30, note $7. But only Janson seems to be aware that he might be discovering the artist’s dimensions. 5 Probably the most important intrinsic tests of the likelihood chat a unit of measure has been correctly identinumbers and only the simplest fractions; and, second, of whole fied are, first, the prevalence in the dimensions to the subject. The teader can judge for himself whether ions ” of the di the “symbolic appropri these tests are met by the units of measure postulated here. The most important extrinsic test is the identificaanother painting; that the palmo gives good results in both the in tion of the same or a similar unit of School and Disputa increases the likelihood that it is correct. For further probative evidence sce notes 25 (the perspective construction, The danger in not finding a floor plan and viewing distance at once is that the interscale of Andrea del Sarto's Cenacolo) and 35 (Pythagorcan numerology; the architectural sketch on Raphael's cartoon; diameters of vaults and of circle at the crossing). The fact that in all four paintings treated by Welliver, "Symbolic Architecture,” there appears a foot parallel to the picture plane which serves as a satisfactory unit of measure, means that any three of the paintings are extrinsic tests of the unit used in the fourth. Titian's Pesero Altarpiece is a striking example of the value of an extrinsic test. The unit of measure appears to be the Roman piede, but this is a priori most unlikely in a Venctian altarpiece. However, in a smaller altarpiece that Titian painted some years carlier for Jacopo Pesaro, the most likely unit of measure proves to be the Roman palmo. Hence we can be more confident that the identification of the Roman unit in the later painting is correct. 6 L. Steinberg, “Leonardo's Last Supper," Art Quarterly, XXXVI, 1973, pp. 346 (“Leonardo's perspective, .. „in Pedretti’s words, ‘has puzzled generations of scholars’ "), 372 and 369. This reputation for difficulty seems to me to be undeserved and to tell more about the scholars than the perspective—I have found Raphacl's and Titian’s paintings far more difficult to decipher. The inhibition regarding measurement and analysis is thoroughly uaaesthetic pref modern gh as a bl which restrains Steinberg, though Florentine. Ie was a stroke of luck that Steinberg's essay appeared shortly before I began work on the painting: it has a great deal to say about the architectural setting and the site that is new and thoughtful, and lowe much to a dialogue with it. But it is not, think, 2 satisfactory interpretation of the painted space for the reason adduced at the beginning of this study as the defect of most such discussions: it docs not proceed from an analysis of the preter will prematurely soar into subjective and mystical exegesis—to paraphrase Leonardo, perspective is a curb for the interpreter as well as for the painter. It is also an invaluable guide and helper; time and again Steinberg’s argument cries out for the solid ground of a floor plan and viewing distance (pp. 350, 357, 360. 375 and 376). The whole discussion of the contradictions and ambiguity of Leonardo’s space (pp. 359€ and 376-78) is based on photographs taken, and on a hypothetical viewer looking, from a point at least three times as far from the work as the “correct” viewing station. This is not to say, of course, that the too-distant view should be ignored, but only that its implications cannot be understood without our knowing where the artist intended that the “correct” viewer should stand. (Steinberg does show a plan in his fig. 39, but it serves only to illustrate the position of the piers at the frant corners ofthe hall, is not accurate even for the refectory, the dimensions of which we know, and is accompanied by no explanation of how the depth of the hall was determined. See note 19 below.) ining references to Steinberg’s essay in order that the reader may under} prepend this critique to the stand the point of view, including my feelings of gratitude, from which they are written and because an essay of such importance, and to part of which this study constitutes a demur, deserves a general as well as specifie comments. right and left. We can estimate the width of the hall (equal to 6 coffers) very approximately from the height of the left- É peerd must have been lost: we have those mentioned in note 35 only because they survived piggy-back on th ar toon for the School. OFDomenico Veneziano and Picro della Francesca we have hardly any drawings of any kind—a dozen or so attributed to Domenico and none of Piero’s except the illustrations in his De os diva pingendi. The arguments for the use of architectural plans are, first, that after artists lcamed how à tran lat them to a painting in the early fifteenth century, this was the obvious way to obatin an accutate and convine. 7 Lam indebted to Steinberg, pp. 342 and 351, and fig. 33, for che reconstruction of the simulated pilasters at

most apostle, St. Bartholomew, On colorplate s$ of F. Hartt, History of Italian Renaissance Art, New York, 1969, his half-height is about 45 nim. By the customary rule that a man is 3 braccia tall (sce note 32 below), a braccio at this depth in the painting is 30% mm. long. The far side of the table, at approximately the same depth, is 306-2 mm. long, af 10+ braccia, Allowing a little less than a braccio for the space between table and wall (sce note 8 below), we get a width of 12 braccia for the hall and of 2 brarria for a coffer. The doubts that would ordinarily arise concerning so approximate a calculation are in this case forestalled by the extreme likelihood that the length of one side of a coffer would be either a whole number of a mixed number with Va as the fraction. The widths of the hall, resulting from the latter possibility, of 9 (1%, braccia coffers) or 15 braccia (2'/, braccia coffers}, seem excluded by the calculation based on St. Bartholomew, even taking into account its large margin of error. The width of the wall is about 8.90 m., which equals 15 Milanese braccia of 0.595 m. (8.92 m.). One braccio of the painting at che picture plane (8.902 11) is 0.81 m. * There is à moderate margin of error in the determination of the depth of the hall: the most accurate way to state it, assuming a viewing distance of 12 braccia, would probably be 28.3 braccia ste .7. I choose 28 because it avoids fractional coffers and because 29, having no factors, is nut reduceable to simple proportions. Thedimensions of the refectory (8.87 X 35.5 m.) are given by L. Heydenreich, Leonardo: The Last Supper, Londan, 1974, p. 16. The size and location of the table have always posed a problem, many scholars— most recently and particularly Steinberg (p. 303. but see p. 351} maintaining chat the space between it and the walls is too small to accommodate the figures which scem to occupy it. Steinberg is mistaken in thinking that Môller's diagram (Steinberg’s fig. 5) demonstrates that “the extremities of the board run so close to the side walls that they leave insufficient floor space for St. Simon's seat at far right . . .""; this conclusion overlooks the possibility that the board extends nearer to the picture plane than the trestles. As Leonardo writes, “L'ochio non può giudicare dove la cosa alta debe discendere” (Jean Paul Richter, The Literary Works of Leonardo da Vinci, and cd., London, 1939, no. 105). To calculate the position of the board we can begin with the legs of the trestles: if their original position has survived the mutilation of the painting, then they are some 11/, breccia beyond the picture plane. One of the most authoritative copies shows the hanging tablecloth casting a shadow on the trestles almost halfway from the cloth to the floor (Steinberg, p. 345. fig. 27; cf. fig. 32). Assuming a source from which light would fall on the table at an angle of about forty-five degrecs from the vertical (as light coming from the nearest refectory window would do), the shadow indicates that the table projects about 14 braccio forward of the legs of the trestle (see Fig. 9). C quently, on the that the near edge of the table is one braccio beyond the picture plane, its length proves to be approximately 10% braccia, and each of the two figures at its ends has available some % braccio of space—about 19 inches or 48 cm. in our scales, This is not generous, to be sure, but it is enough to make the relation of table, disciple and wall a rational one and so to avoid the impasse which has troubled previous scholars. From the length of the table, its width can be calculated as about 27; braccia; thus its proportions are 4:4, the same as those of the refectory. ? 5. Spirito (Orcagna?), S. Croce, S. Apollonia (Andrea da Castagno), Ognissanti (Ghirlandaio), S. Marca (Ghirlandaio), $. Onoftio (called di Foligno; Perugino); sec L. Vertova, I cenacali fiorentini, Turin, 1965. 19 Since writing this study I have seen C. Gilbert's article, “Last Suppers and Their Refectories,” in The Pursuit of Holiness in Late Medieval and Renaissance Religion, ed. C. Trinkaus, Leiden, 1974, pp. 371, Gilbert also remarks the frequency of paintings of the Last Supper in Florentine refectories, reasons that they were intended to prolong realistically the space of the refectory (p- 372), and notes the large number of meals depicted an the end wall of the S, Croce tefectory (p. 376). 11 Approximate ratios of length to width are: S. Apollonia 3, Ognissanti 4, 5. Onoftio 4.3, S. Maria delle Grazie 4. 12 Two objections to a vanishing point below the floor of a painting are, that the deeper the figures stand within the painted space, the higher are their bodies cut aff by the front cdge of the floor, and that the viewer's inability to sec the whole scene dampens his sense of presence at, or participation in, it. The conventional solution was to draw the scene as if the viewer's eye were some distance above the floor of the painting (at the point to which cameras are elevated for modern photographs), even though this might be far above the point 15 The width of Andrea's painting is 7.94 m. (the interior of the painted room), the viewing disance about a efen‘ di rani a viewing 2 22,3 m., the length of the refectory 28.1 m. . ( (for my method of calculating rhe possildepth of the i 1 asserts that f., followed by Hartt, pp. 222f,er i pp. 198f., i paragraph, below). White, third of the number Ne indicationdeep: ee Iis no convincing the cause there because pai fs painting for Andrea's i iewi distance late a viewing ; d i w as twice is it that indications three are there But width, its to ion i loops in the fers decoration (noted by White), the number of spine curves a he goe seup behind c L i could be accommodated on the st i and the number of figures which diners, approximatesKlijn nee) Hart, by noted recede, they as depth apparent in diminish to rectangles img ) pp. Gil ade by equal divisions of a shallow foreshortened surface seen from a great distance. i to mine, except as regar dineusses Andreas perspective arrangement and makes some observations similar calı instead of f calculating. i i he appears to estimate which di distance, i ing 13 m. and the length of the 16 The wich of Ghitlandaio's painting is 8.12 m., the viewing distance about ak of Perugino’s painting (measured from one refectory wall he to the other) is 8.10 m., the viewing, Co m. and the length of the refectory 34.5 m. .§ i recognize its implications because gent s«60, sees this early but does not, I believe, correctly | . is failure to distinguish between the “correct” and the too-distant viewer. the 2 That a normal head with long hair should obscure half the width ofa doorway suchas “arequiresan Ye than one-third as far from the head as the head is from the doorway. i side wallsgive hm a IA Viewer or els judge the latter distance, but the tapestries on the wi Tour for a space betheir a nen high, from this dimension ia high, 3 braccia i are a little less than 8 Milanese idea: they . hi mption that they are oblong and hung vertically), an 5 ‘ ; conjectured as between 27%: and ae aca thse ofhalf theie width, the depth of the hall could be of 33 braccia and amg eg Mika (Steinberg, fig. 39, chooses a depth of 31 braccia). Taking the mean ciaEven fore lane braccia beyond the picture plane, we get a viewing distance of de Milanese painting over twice as far istance Bae shoulaon iewi ist’ proves to be), the viewing ich is about what Christ's size (which | panel e h t from braccia 60 some is viewer the refectory the of end far he ia. life size. Ifiach reasoning fron Chet a8 Christ is from the doorway—and the head will seem three times in, i must bsr ep seems more conscious and complex than that which viewers are likely to indulge only occasionally, that Leonardo painted primarily not for viewers me ourselves, wee see the work . . — uld see it two or thece times a day for years on end. wide . he geleen, his wel Fe 1 When it was clearer than it is now that the painted anhi : lle iot i of real and painted side walls may have ben slightly less ar beren 352f. and fig, 38. In Leonardo’s time the interruption of the wall orthogonalsas even 2 Das Abendm harsher, if the floor level was then one m. lower than at present; E. Möller, + 1952, p. 50, fig. 49- > n'aPeak (The Murat Painters of Tuscany from res to Andres de Sart, London, 1900,Li 2) cst 6 Oe 5a a Whe. OF cose, i CenacoloJ to be one and ; one-ha i io’ Ognissanti i mates the figures in Ghirlandaio’s fe plane this ¢ on the picture Fee and can ulating the height of a standing figure on the method of calculating depends i estimate ı lane heig das's height (standing) to be 1.83 m.; converted to i ee series toe times the 3 braccia (69 in.) which was a rule of thumb for a man’s height . ni U . bel space in Andrea's, Ghirlandaio’s, Perugino’s and Leonardo s Cenacoli, alin breccia of ofthe rmemen by 12; Leona(9dez rn 12 Perugino, 15; by 9 Ghitlandaio, 14; by 7 Andrea, first): (depth are the painting, eed Picro del M ran The proportions of the space in the four paintings of Domenico Veneziano and of Flagellation), an by Weller, “Symbolic Architecture.” arc 7 by 4, 2 by 1. 3 by 1 (left side only 5 > U "3 Perugino further disguised the comers by keeping the change in value and color to a minimum. "4 In Andrea's painting the space of almost one m, between the side walls of the painting and of the refectory is taken up by the face of a pilaster and by a simulated brick wall parallel to the picture plane. . 7 b continuous. hare he = er Lande "Annunciation (1475-78) we make the lines of the floor pattern | assume drangles to be squares, take the side ofa square as the uniti of measure, and ass i 3 braccia; viewer to neat side nn we brain the following distances; viewer to picture plane, woul ? a line ’ 8 wall, 9; viewer to far side of wall, 13.5 (it comes as a surprise to find that the wall theIch aofbe in oe feet "thick in life scale). The lateral distance from the building a the righ to 2 © pi throug! f Gabriel's hand) is 6 braccia. Prolonging the left end of the wall i braccia; picture plane to wall ned mn three u es: viewer to picture plane, 3 by 6=18 square 60 from which he would actually look.

6 by 636; top of wall, 4% by 6227. The painting is 36 sides of a square, or 6 braccia, wide: its real width is recorded as 2.17 m., which equals 3.72 Florentine braccia. 34 Johann von Goethe, Werke, Munich, 1973, Val. XII, p. 165. Möller, pp. 49ff., gives a plan and discussion of the kitchen and doors of the refectory. 35 A possible hint of Leonardo's feelings about the prior is provided by the anecdote in Giorgio Vasari, Le vite dei pits ecceltenti pittori, scultori e architetti, Milan, 1945, Vol. I, pp. asf. lt is of some interest that the scale of the architecture in Andrea del Sarto's Cenarolo (S. Salvi, Florence) is the same as in Leonardo's, the width of the fresco being 15 Florentine braccia and 11 braccia of the painting (using the edge of the floor as the baseline and 11/, braccia as the width of a floor square). The viewing distance is unusually short—8 braccia of the painting, or only #1 of its width. The refectory is also unusually short, 23.8 by 9.3 m., and the viewing distance (6.4 in.) is little more than one-fourth the len gth (the prior presumably sat at the same end as the fresco, the main entrance and kitchen being at the opposite end). In screening off the space beyond the diners, Andrea reverted to the defensive tactics of Leonardo's predecessors. 26 See notes 7 and 8 above. 27 Another scene of Christ and the apostles in an architectural setting based on a dimension of 12 braccia is mentioned by Welliver, “Symbolic Architecture,” p. 26, note 6. At 24 braccia from the far wall, Christ marks one-third the length of the total space. The proportions of the total space, 6:1, are the same as those of a transverse row of coffers. If, before the mutilation of the lower part of the painting, the design of the floor matched that of the ceiling, as seems likely, this identity would then have been emphasized by one conspicuous trans verse row of six floor squares; Steinberg, pp. 312f. and 345f. 28 Sce Welliver, “Symbolic Architecture,” Figs. 3 and 6. 29 The bottom edge of the present painted surface is very uneven. I have used a baseline defined by the lowest point on this surface (bencath the right far corner of the hall). Any baseline higher than this results in a ceiling correspondingly less than 9 bracaia high. Other studies of the proportions of the painting have been made by T. Brachert, “A Musical Canon of Proportion in L. da Vinci's Last Supper,” Art Bulletin, 53, 1971, pp. 461-67, and the authors listed in his note $, p. 464. It is next to impossible for me to follow Brachert's argument because of both the complexity of his diag German. and the obscurity of his text, which is a translation from the 20 Steinberg, pp. 312, 329 and 345f. 31 The values I have used for By and By on Hartt’s colorplate $$ are, respectively, 37 and 11,1 mm. 32 The Roman palmo was .222 m., the less used piede .296 m. (1% palmo), the Florentine braccio „584 m. (about 2 piedi). Thus, Alberti’s standard man 3 breccia tall was also 6 piedi, or 8 palmi, tall (about 69 in. or 175 cm.). For 6 piedi as a usual height see Viteuvius, HI, 1: and Alberti, Kleinere kunsthistorische Schriften, ed. H. Janitschek, Vienna, 1877, pp. 107 and 181. 33 Measurements of the room and all the wall paintings are given by D. Redig de Campos, Raffaello nelle Stanze, Milan, n.d. (first reprinting), pp. 83-8. The width of the Scuola is 8.14 m.; taking five full rectangles, equal to 30 palmi, for the width, onc palmo of the painting equals 27 cm. We have na way of knowing the depth of a rectangle, but it is probably safe to assume that it is in a simple proportion to the width, either one to onc ar three to four or two to three or one to two (that is, 6, 4%, 4, or 3 palmi deep). The viewing distances corresponding to these four possibilities are, respectively, 33.2, 24.9, 22.1, and 16.6 palmi. The width of the room from east to west (the School is on the east wall) is about 3t palmi in the scale of the painting, and bookcases along the west wall probably limited the maximum viewing distance to about 28 palmi; hence the viewing distance of 33.2 palmi, corresponding to a rectangle 6 palmi deep. is too great and can be climinated. Since the School is 30 palmi wide and a painting cannot be comfortably viewed as a whole from a distance that is much less than its width, we can also climinate the shortest viewing distance (16.6) and prefer the greater of the remaining two (24.9), which was surely 25 and corresponds to rectangles 4'/, palmi deep. (Since the viewing distance would repeatedly be used to calculate the apparent size of objects at varying distances, it would be much more convenient to use a whole than a mixed number, The difference between 24.9 and 25.0 palıni is 0.495 and is surely well within the margin of error compounded of my measurements and the probable imprecision of the artist—he was, after all, making a painting and not an architectural drawing.) I calculate the viewing distance (vp) by solving the equation, vn:vo plus depth of two rectangles :: apparent rectangle width at bottom step (measured on a photograph) : apparent rectangle width at bottom of painting. This is simply an application of the rule that the apparent size of an object varies in inverse proportion to its distance from the viewer. The viewing distance can also be approximated graphically by extending to the 62 , presumably, by the four comers at the horizon a diagonal through the square formed crossing. Using this ( to the efe in pictur seatedrlyfigure perpendicula ee oe less weISre Fre nose af figure sFfoot if: aue ii o ibility in this painting is the left, shod foot mi foot scems wo be somewhat longer than re This . angelo Michel ly probab i who plans De oad. pa foot some 2 piedi (1. potes) deep er Michel| angelo's rt po palmt theis 2.4 rectangle Pen pie he small (depth)to i mende u dep Now due a of ratio A . stature al phoric his of for even great tt 100, seems to el which siae one ja: ubrary. An and his height to no less than 8, ah ri reduces the foot to 14, piedi, . library is Be by enn ir men that the room was a wie ane oot he reoblem and N |Academy) à par ie Britis the of ings Proceed tion,” and Decora Shearman “The Vatican Stanze: Functions bes tS sie have ruct const heavy of and omate 79-83 Assuming that the bookcases were ee hae aan = Fin inl em. 22 by 33 or palmo, 1 by s the painting are about 1%2 5 a (ie bookin ind die note 5 à on ng. Painti the of scale the in palmi 4 vam viewing distance of 7.65 m., or 28% for the Dispu ent to 6.75 im. for the Sctool and 6.97 m. (equival i Raphael chose che arch equals the width of walls. both along ure furnit deep ) below) arc Cent with there being > aasnewrangles, from the assumption that the width o : e | x Approximately the same distances result width of the foregi the as , same the is l interva c iglypti | . At . this distance the intertr tl he Verde . 9 18 er round foreg the in ding steps symmetrical with those trad the gallery there were descen ly at che third modular virtual . be | would riser t farthes the ng), painti xign of Tock steps in the 1 . palm module aa). g. though perhaps only coincidental, relation between the ung" onthe eind a Mari te depth h estima en , we can pee) course Of pn i the first step to the far side oft id2 oe ance ology: from ane ja dt the superior cubic number (V as en je , cs. number (6°) held by the Pythagorcans to be which numberPoss si Duran Palace pati, 250 is arch the of side far the to from the viewer ge , 1486; see ©. Hersey, and another facet of Pythagorean numer script of Vitruvius and was printed in the fiest edition (Rome the upper foor berendan ng adds point to the Socrates group, on de Ne dorenn element in che painti m ps es. (ant speech and scene ng openi the think, moeren »ythagoras group. They are enacting, | . From left to right the figures are imacus s the book which Plato carries de : pe Dof he Timaen ing on his fingers, Socrates has reached four, as much as to say, a c on, Count \ ) Soer speech Critias, One, two hree . . but, my dear Timaeus, where is the fousth of my genof yenerday se mus bei D Be ehi zende pen answers that the fourth would not willingly have been absent and on ly K ey ariye ce ae roll: the call to s finger his on count to had t m Raphael thought Socrates og do be: e an illustration of the tetrattys which is alm: th fi A ures. At the same time they provid directly abave Pythage saat Bi athe almost are They digi. ten the of four first enun eration of the the cuous standing figures on the lower floor, carry thro wanding figures between, the only conspi i ns augur wel for the anf i eratio veen of the modular scheme, two consid à per de Ee (now in Mi jan) are few ore scene the for Raphael's cartoon hin he . sttuction On the lower part of and Lamberto Vitali, Raffor ot 1 onPe uber Oberh d Konra es; sketch l ectura A \ sible archit P aon i or calle XXI1. (iam indebted to Alessandro XXX~X ations illustr 1972, Milan, marge iere ecn the post-Brunelleschan tra ina Les for ting interes ely extrem are They ion enten attent ing scale or quasi-scale draw ing archi ecture Bh ont hey are ‘as far as 1 know, the earliest surviv ed ed guess chu x O which ion elevat an is 19) (my Fig. of these ) One above. h the foreground floor, steps, and upperfloor as Aaash cent m a sectio n throug note pai ) to see show at 2the botto : ‘ted: er innn ill, xxxut, except that! ave a same is theting ed onFig. lons. (My oft the19protec glassOberhu some ber's 10 cm, from the papet and therefore approxive RA we Cth re measur ànn) The 3 an the same as one palme in the painting (note de ht and largest dimension, 27 cm. is of the pa mo of epainting ) me hs?} (ewelft ns fractio in d divide stick ae go stat Ray tael used a measuring ln as tne Entes inte Pand in the pn ons (27 and 18 cm.) areshould cher n Owe divisidepth be only 7'/, palmt acco floor ofthe up er he 2 lmi floor). the larger pylon which would leave 3 7'/,-pa os2 ifthe planLee Fig. 1 jrhed line, for

Parity might well cast doubt on the accuracy of the reconstructed plan, werc it not chat an anomal in it betrays a difficulty chat must have arisen as a result of enlarging the upper floor. The space between pilasters o! the foreshortened walls of the pylons should be as wide as that on the front walls, since both contain niches and statues of the sane height and presumably the same width. On the front walls this space is 7/, palmi wide, but on the side walls only 4/4 (a wall of 8 pafmi less pilasters of 21, and 1%, palmt). Obviously a niche and statue designed for a 7'/,-pabui space cannot fit into 41/, palmi. It seems a reasonable inference that between drawin and painting Raphael devided to deepen the floor by 7 palmi and was able to gain part of this additional de; < by reducing the size of the pylons, but was unwilling to gain the rest and provide the proper inter den space by moving the front of the gallery 3%, palmi back, because this would have disrupted his dur scheiic. Why deepen the floor? Perhaps to give the figures more space and, by moving the pylon back and reducin its width, to gain room beneath the frame for che statues of Apollo and Athena (hence the importance of “altezza) and reduce the mass and prominence of pylons extending all the way to the side frames. (For other changes from the sketches or cartoon and other anomalies in the architecture, see the last three ragraphs of Paragraphs o this note.) The second consideration that supports the accuracy of the present reconstruction is the fact that the barrel vaults and the circle at the crossing turn out to have diameters of 28 pabni, From at least the time of Beunelleschi and Masaccio, it sents to have been thought desirable that the diameters of circles and semicircles should be integral multiples of 7. The reason for this is obvious: before the advent of decimals the value used for x was 22/7; hence, if the diameter of an arch or circle was divisible by 7, its circumference proved to be a multiple of st. In particular, if che diameter of an arch, say, was 21, its circumference was 33, 2 most appropriate dimension for a Chriuian arch. (Sec Welliver, “Symbolic Architecture,” p. 18 and note 5 she diameters of the circles and arches at the crossing in Brunelleschi’s S. Lorenzo and S. Spirito arc about a braccia, The diameter of the cupola in the Pazzi Chapel is about 17%, braccia, which results in a circumference of 55 braccia for the baw of the cupola and 27, braccia for the arches, doubtless thought of as 27, or 3°) In the present case the cirvumference of the arch is 44 pahni, or (the palo equalling ¥/, piede) 33 ‘edi 3 4 the circumference of the car be at the crossing is 88 pahni or 66 piedi or, to use the Florentine unit mere 33 braccia. The center af thas circle is 104 pal from the viewer, and Plato's one straight index finger is ointe ing (two-dimensionally) at (he corresponding center of the dome. In reconstructing the architecture we have gone from the 4?/, pahni by 4%. piedi Boor rectangles at the bottom ofthe painting, at which Aristotle's (ina sense) 42% fingers are pomting, to the unity at the center of the mystic 33-hraceia circle, to which Plato is urging us at the top. (The right hands of Plato, whose head is thought to be a portrait of Leonardo, and of Ane totle may be quotations of the right hands of Thomas and Christ in the Cenacele.) Other changes besides that of the pylons were made between sketches and j painting: the risers of the steps were increased from Y to sure pale, aud oblique pilastroni at the crossing, to be seen in one of the sketches are not found in the painting (pce Oberhuber, p. 36). One change occurred even after the cartoon: en the right third of it a Moor pattern of small squares is drawn, similar to the Hoor in the adjoining scene, The Laws, on the south wall of the room, The interval between orthogonals in this pattem at the depth of the rightmost foot in the wene, which is at the bottom of the cartoon, is about 24.5 cm. and world be about 27 can, or one palme, at dhe baseline. Since the floor pattern, like the rest of the cartoon, is pricked for transfer to the wall by spolvers, we must assume that it was painted and later erased. A likely explanation for this change is that Raphael had virtually decided on a rectangular pattern similar ro that in the alread completed Disputa, but wished to we a sample of this variant before the final decision, The variant not tie. fying, he erased it and painted the present rectangles, but then found chem and the vacant center foreground so repetitive of the Dispute that he inserted the huge figure of Michelangelo, which is not in the cartoon The crossing, like the pybeum, on anomalous in the extreme. The section of foreshortened circle immediatel above the vault and medallions implies a dome with a base wider than the width of the vault. Barring the unlikely case of an elliptical donne, this would mean that the distance between the two vaults shown in the painting should be greater by the same amount than the width of a vault (28 paiui); but the floor plan shows it to be equal to the width of a vault. Furthermore, the medallions in the spandrels (as well as the wider dome) imply the presence of he oblique pilastroné which were eliminated between sketches and painting. As things are now, the bases of the spandrels--the only supports of the dome—would themselves have no support a short distance to the right or lett of the arch. The anomatous pylons and crowing suggest that three trains of thought combined and were reconciled 64 to reach the final solution: (1) the free and preliminary sketches of the desired architectural forms (Oberhuber. illustrations XXX-Xxxn); (2) the requir of the two-di ional CORP ition; and (3) the modular floor plan which, once determined, remained intact. The pylons were treated as described above, Fig. 19 being part of the reconciliation of (5) with (2). At the crossing, the 28-palmi square of (3) would require that the lateral vaults spring from the same points as the longitudinal vaults, but this would spoil the light and spacious ness desired at the crossing and so foreshorten the medallions as to render them almost useless for decoration. Therefore (as with the pylons) Raphael left the essential floor plan unchanged but painted spandrels of a dome wider than 28 palmi and, contradicting these, lower walls perpendicular to the gallery which would both catch full light (in other words, spandrels and a dome which the floor plan did not permit and lower walls which would not support them). 36 The title is derived from Vasari's mistaken description of the subject as a dispute (“tutti i savi del mondo che dispurano . . ."}; Vol. u, pp. 133f. 37 L assume the floor rectangles to be 3 palmi deep by 4 wide (the same proportion as in the School) and obtain a viewing distance of 27 palmi. This equals 6.97 m. (one palmo equals .258 m.) and is consistent with the view= ing distance for the Schaal of 6.75 m. and with the distance from the painting to the putative bookcase along the opposite wall of about 7.65 m. Distances from the viewer are: to the bottom step, 33 palmi; to the upper floor, 37; to the altar, 43. The altar is 4 palmi high by 6 wide; 2 palmi would be a reasonable depth (proportions of 3:2:3); hence the host placed at the center of the table would be 44 palmi from the viewer. With a little imagination and a few sketches it is easy to sec how the half-cllipse of the saints and prophets. the half-circle around Christ, and the half-figure of God the Father could have evolved from the drum, the dome and the lantern of a cupola. Thus the forms of the upper part may resemble the central clement in the architecture of the School, just as the lower part resembles its foreground. The conical shape suggested by the rays of the sun reenforces the roundness of the celestial architecture. The rays converge at a point abour 28 palmi above the floor. 38 Deut. 4:13. Henry Fischel has kindly examined the painted text and says that, as is usually tue of Hebrew inscriptions in paintings of this epoch, it makes no sense. He also tells me that it is usual in synagogucs—presumably this was ruc also in Raphael’s time—to display two such tablets, cach with a word or two on five lines ding for the C d On each of Raphael’s tablets there are five full lines of characters (counting a line beneath Mases’s left hand), although the tops of a few characters in sixth lines can be seen above his robe. 39 All the ratios in Pythagoras's table of the musical tones can be formed from the floor rectangles of both paintings as follows: 6:8, di four rectangles, Disputa 6X8 palmi 6:9, dispente two rectangles, School 6x9 6:12, diapason six rectangles, Disputa 6X12 8:9, epagdoön six rectangles, Disputa 8x9 8:12, diapente eight rectangles, Disputa 8x12 9:12, di on four gles, School 9xXı2 The rectangles themselves, 3 by 4 palmi in the Disputa and 4%, by 6 in the School, ate reductions of the two ratios of the diatessaron. 40 5, Valtieri interprets this passage as Bramante’s illustration of a new method for drawing architecture in perspective; “La Scuola d'Atene: ‘Bramante’ suggerisce un nuovo metodo per costruire in prospettiva un’architettura armoniosa,” Mitteilung des Kunsthistorischen Instituts Florenz, XVI, 1973, pp. 63-72. Vasari says that the figure may be a portrait of Bramante; Vol. U, p. 134. #1 Two of the omitted pairs of figures have the obvious function of defining the comers of the pylons. Of the other two, one is in the right distance, facing somewhat away fram the viewer, and the other is just to the right of, and beyond, Socrates. 42 Both Moses and David are in che fourth place from the end of the semicircle of figures. The six-pointed star was not an exclusively Jewish symbol in antiquity: the earliest denomination of the device as the Shield of David is to be found in the tradition of Jewish Cabala and magic in the fourteenth century; G. Scholem, “The Curious History of the Six-Pointed Star,” Commentary, 8, 1949, pp. 243-51. As a symbol of Jewish magic and mysticism, it is well paired with the numbers and musical ratios of Pythagoras. © It is often said that Plato and Aristotle are walking together. In fact only Plato is walking; Aristotle, unlike his disciples, is clearly not peripatetic. Perhaps the contrast in their movement, no less than that between their

vertical and horizontal gestures, was meant as 2 symbol of the difference between the twa philosophers. 4% What a perfect foil for the spirit af amity and optimisin pervading the works is the brooding giant, Michelangelo, the solitary pessimist at work {in life) on the profoundly pessimistic Sistine Ceiling! QUARTERLY Photo credits: Figs, 2, $, 12, 13, 14 and 16, Alinari; Figs. 4. 6,7 and 8, Gabinetto Fotografica, Soprintendenza alle Gallerie, Florence. in. ART