PRINT March 1978

Duchamp and the Classical Perspectivists

TIME AND AGAIN, DUCHAMP INSISTED that the Large Glass (fig. 1) was also (perhaps even in the first place) a consideration on perspective. When Pierre Cabanne asked him how he had arrived at the idea, he replied, “Perspective was very important. The Large Glass is actually a rehabilitation of perspective, which had been completely neglected and decried. With me, perspective became absolutely scientific . . . It was scientific mathematical perspective . . . based on calculations and measurements.”1 To Richard Hamilton he likewise admitted: “The projection [of each part of the Glass] in perspective [on the Glass] is a perfect example of classical perspective, I mean that I imagined the various elements of the bachelor machine first of all as arranged behind the Glass, on the ground, rather than as distributed over a surface in two dimensions.”2 We know that Duchamp drew up several perspective diagrams in this way, to situate the various pieces of his Bachelor Apparatus—now on a reduced scale, now life-size—before they were outlined on the surface of the Glass.

Finally an observation in the notes to A l’Infinitif makes it fairly plain which way his mind was leading: “See catalogue of the Bibliothèque Sainte-Geneviève under ‘Perspective: Nicéron (le père J.Fr.) Thaumaturgus Opticus.’”3

This remark probably dates back to the time when Duchamp was employed at the Bibliothèque Sainte-Geneviève, from the end of 1913 through the whole of 1914. As luck would have it, the possessions of this library, which came from the former convent of the Order of St. Geneviève, were (and still are) particularly rich in works on perspective. As well as Nicéron, these include the major treatises of Abraham Bosse, Emmanuel Maignan, the Père du Breuil, Athanasius Kircher, and also an extremely rare edition of the first treatise on perspective printed in Europe, the De Artificiali Perspective (Toul, 1509) of Jean Pélerin Viator. As a librarian at Sainte-Geneviève, Duchamp thus had access to the most complete body of works he could hope to find on the very problem which preoccupied him at the time.

An obvious fact which needs to be stressed is that by substituting a plate of glass for an opaque canvas spread on a stretcher as a support, Duchamp was doing no more than applying the analyses of the classical perspectivists to the letter in making a real pariete di vetro (wall of glass). Their procedure was based on the definition of the picture plane as a plane intersecting the pyramid formed by the visual rays, whose apex is the eye of the observer, and whose base is the object to be represented (fig. 2). As Panofsky explained:

We shall speak of a treatment of space as “perspective” in the full sense when, and only when, there is not merely a foreshortening of single objects such as houses or pieces of furniture, but the entire picture . . . is transformed into a window as it were, through which we look into the space beyond, when therefore the material surface of the painting is negated as such and becomes simply a “picture plane” on which is projected the whole of that space seen beyond it and containing within itself all separate objects.4

More specifically still, Abraham Bosse reports in the Manière universelle de M. Desargues pour pratiquer la perspective:

The flat surface that we understand to cross the radiation of sight is called by some transparency, by others glass, or section, and by others by another name. But Desargues calls it picture [tableau] . . . So, when it is a matter of considering the nature of the portrait of a subject, one can conceive of a kind of Table of glass, thin, flat, even and transparent, on the front of which one imagines that the eye sees the subject. . . Now this Table and the Picture [tableau] in which the portrait of a subject is rendered are evidently one and the same thing.5

In saying this, Bosse was simply taking up the comparison established by Alberti in 1436 between a picture and an “open window through which I look at what will be depicted there” (una fenestra aperta per donde io miri quello que quivi sarà dipinto) or better still, Leonardo’s likening of the picture to a “pariete di vetro.”

We must not forget that one of the points of departure for the Large Glass was Duchamp’s fascination with the spectacle of a grinder seen through the window of a chocolate-maker in a street in Rouen. The machine Duchamp saw was moving below street-level, at an angle Nicéron would no doubt have called “catoptric ”(cf. fig. 3).

So glass, window, and frontage were his constant obsessions. Under the generic heading of “speculations,” in fact, he developed considerations on the specular—a “speculum” of which that of Etant donnés, directed toward a woman’s genitals, was to be the last phase. He wrote, still toward 1913: “The question of the frontages. Be cross-examined by frontages. The demands of the frontage . . . The sorrow is to cut the glass.”6 Duchamp gave a few examples of this “sorrow,” breaking the glass in amorous play (envisaging a spectacle of desire) before the “permanent incompletion” of the Large Glass—all so many preliminary comments on the Bride.

Fresh Widow, in 1920 (fig. 4), was presented to the eye as a pariete di vetro. But the glass is rendered opaque by the black leather panels veiling the panes; the erotic exasperation of the frontage is wearing mourning, so to speak. Conversely, in the following year, the French window of the Bagarre d’Austerlitz was spruced up with whiting (Blanc d’Espagne), a prelude to a more vivid brilliance of its transparency.

These windows are all ironical plays on the definitions of the picture proposed by Alberti, Leonardo or Desargues, but Duchamp regarded them as independent works of art: “My windows,” he confided to Sidney Janis, “in the same way as one might say ‘my etchings..’ ”7 He even offered a final version, later than the Glass—the Door for Gradiva, of 1937 (fig. 5). There the projection of a three-dimensional body, perpendicular to the picture plane, literally cuts into the partition of glass. The object is silhouetted in the glass thickness at the points of passage of the rays of the visual pyramid. Also, perhaps, here the partition no longer separates a Bride and Bachelors, but merges the two lovers in the instantaneous projected cutout, united at last.

If the Large Glass, by definition, takes the laws of perspective literally, it also seems to illustrate them. One cannot fail to be struck by the fact that the very peculiar architecture of the Glass—that is, two glass plates of unequal dimensions (the same width but different heights), one above the other yet separated by a boundary marked in three horizontal lines—that this architecture exactly repeats the articulation of the plates illustrating the treatises on perspective.

In Abraham Bosse’s Manière universelle (fig. 6), for example, the first thing one notices is that the proportions of Bosse’s plates are practically identical with those of the Large Glass; the average ratio of height to width is 1:1.60 as in the Glass (1:1.567). This figure is significant because it approximates the proportions of the Golden Section. Thus, Large Glass is proportioned according to the Golden Section. Better still, it shows one “picture” above another, the respective proportions of which—unequal above and below—are well known to the classical painters. Nor is this arrangement confined to the Large Glass. It recurs at least twice, in two other major works by Duchamp: in the proportions of The Bride, a picture painted in 1912 whose general design was to be repeated in the Glass, and more significantly still, in the preparatory study for the large final environment on which Duchamp worked in secret for 20 years, from 1946 to 1966, the Etant donnés 1° la Chute d’eau 2° le gaz d’éclairage. Again in that study the ratio of length to width is that of the Golden Section. More pertinent still, however, is the fact that the female sexual organs portrayed in both works—the “sex cylinder” in the Glass and the woman’s genitals in Etant donnés—are placed on a horizontal line at the same level as the horizontal line indicating the robes of the Bride in the Large Glass: very precisely at the vanishing point of the horizon line, exactly as the half-open genitals of the Philadelphia environment are placed at the vanishing point of the visual pyramid, the apex being the stereoscopic peephole(s) in the door.

But this is going too fast. To return to Abraham Bosse and his Manière universelle de pratiquer la perspective, each plate illustrating the treatise is likewise divided into two zones of unequal length superimposed and referring, respectively, to what the perspectivists called “the perspective” and “the geometral” of a single figure: “The representation of a body by three figures, full, profile and elevation, is called Geometral or small Geometral Foot. And the representation of the same body by a portion of its surface, seen at a single glance, is called Perspective or else the portrait of that body. . . .”8

Let us take plate 26 of the Manière universelle as an example: “The two practices,” writes Bosse, “of the small geometral foot and the perspective . . . are carried out by using a grid, of different form and size above and below.”9 In connection with plate 29 (fig. 7), Bosse again observes: “This plate is separated as if into two.” But here the division of the glass screen into two, referring to what Duchamp regarded as the Bride’s domain, or the domain above, and the domain of the Bachelors below, is marked by three horizontal lines, identical with those denoting the Bride’s robes in the Glass: Bosse’s ZCX indicates the horizon line just as, in Duchamp, the lower horizontal line indicates the horizon of the bachelor domain; and EV indicates the boundary line of the upper domain. Between these two straight lines, a third line distinguishes the two incompatible spaces of the perspective and the “geometral.”

Now, when we take a look at another treatise of classical perspective, La Perspective pratique nécessaire à tous les peintres . . . (Paris, 1663) by the Père Du Breuil, we find similar associations in the plates illustrating it. In Treatise III, the plate depicting Practice LXXVII (77) (fig. 8), “To elevate furniture placed at random,” shows two heterogeneous spaces, one above the other: in the upper space, the volumes are rendered in perspective and in the lower space the same volumes are brought down to a double geometrical projection, on the one hand onto a plane and on the other onto a perspective plane, “or the same iconographical plane placed in perspective.” As in the Large Glass, the two spaces are separated by three horizontal lines—a straight dotted line, GK, marking the horizon of the lower plane. Again, Du Breuil uses the term “scenography” elsewhere to denote the inclusion of a volume according to an illusionist rendering in the upper part—which he compares with the projections of that same body in the lower part, according to the iconographical plane and the orthographical plane (i.e. its elevation).

It is easy to draw a parallel with the device imagined by Duchamp, his division of the Glass into a domain “above” given over to illusionist rendering insofar as the true form of the Bride, which is four-dimensional, can ever be known, and a domain “below” treated as a mathematical diagram. But Duchamp, as a man of the early 20th century who was attracted by all the esthetic—and also, above all, scientific—novelties of the day—as much an engineer of vision as an artist, as interested in the scientific phenomena of optics as in purely retinal sensations and, in that, showing a sensibility very close to Leonardo da Vinci’s—this man singularly complicated the issues raised by classical perspective.

The classical perspectivists were aware of the problem of finding the semblance of objects when they were recorded on a transparent medium at the points where the visual rays that emanate from those objects intersected with the medium. In other words, they had sought methods that would allow them to pass correctly from the two-dimensional projection of a body (the “geometral”) to its three-dimensional representation (the “perspective”). In other words, again, the plates of the treatises of perspective—on either side of a line acting as the horizon line of a lower space in one direction and as the boundary line of an upper space in the other—articulate two distinct areas, the one two-dimensional and the other three-dimensional, the laws of classical Euclidean geometry allowing exact congruency to be conceived in between the division of the two. The problem Duchamp tackled was how to pass correctly, by means of the same device but according this time to the laws of modern non-Euclidean geometry (that is, speculating on worlds with more than three dimensions), from the three-dimensional representation of a body to the apprehension of a “four-dimensional” being. What the Large Glass articulates, then, at the juncture of its own horizon, is the portrayal of a three-dimensional area (the Bachelor Machine) and a four-dimensional area (the domain of the Bride).

Let us turn back briefly to the theories of n-dimensional geometry such as Duchamp could have read in Henri Poincaré and Elie Jouffret. If a point cuts a straight line, if a straight line cuts a plane and if a plane cuts a space, the space in turn, by analogy, cuts an expanse (“étendue”). In the most general way, an n-dimensional universe is cut by an (n - 1)-dimensional universe. Or again, as Duchamp said, the semblance of an n-dimensional object is its appearance in an (n - 1)-dimensional universe. To translate: in the perspective treatises the semblance of a solid body in three dimensions is its appearance on the two-dimensional plane of the pariete di vetro, at the points where the plane glass cuts the three-dimensional visual pyramid. By analogy: if a glass cuts a space and, according to the art of perspective, offers the semblance of a volumetric object on a plane area, surely there might be a transparent medium capable of “cutting” a four-dimensional expanse?

But in the meantime let us look at simpler things and try to see to what extent Duchamp was actually inspired by the figures he found in the treatises on perspective and their arrangements for the iconography of the Glass. I shall take the lower part first and then the upper part.

The nine small standing Malic Molds (fig. 9), apparently huddling one against the other on the surface of the Glass, in reality enter into a large parallelepiped measuring about 70.8 cm. in width and 110.2 cm. in depth. A common horizontal plane cuts across them at the level of the genitals, the “sex polygon” or plane. This arrangement can be associated with the peculiar outline of the figures drawn by Jean Pélerin Viator in his De Artificiali Perspective (fig. 10). The uniforms of this ecclesiastical confraternity, the face and the mass of the hair, the cap and ostrich feather, are all brought down to a basic envelope which turns the characters into a bunch of naive stylized figurines, near relations of those forming the “Cemetery of Uniforms and Liveries.” We must also notice the very special viewpoint from which the molds are perceived, conforming in every detail with the description Viator gives of the viewpoint “of the elevated chair.”

The Glider (fig. 11), with its regular volume, a parallelepiped, is drawn according to a perspective which “forces the design.”10 In fact, the volume is anamorphosed in the same way as a chair (fig. 12) in Jean-François Nicéron’s Thaumaturgus Opticus, seu perspectivae curiosae intelligentiam et praxis necessarium (Paris, 1646). The Glider moves forward to splinter the surface of the Glass, whereas at the back, if its edges are extended, the lines meet at the vanishing point, just as in Nicéron’s plate the fleeing lines of the chair meet at point Q on the horizon line QR.

Cones, regular bodies again, are favorite figures in treatises on perspective, and the ways of representing them are legion (fig. 13). Moreover, in the anamorphosis cabinets they were frequently used to straighten the images; in the Bachelor Machines, we know that as the Sieves (fig. 14) they serve to “straighten” the gas flowing through them.

Jurgis Baltrusaitis has given an illuminating analysis of the close relations that existed in Paris, in the intellectual circles of the Convent of the Minims, between Mersenne, Nicéron, Maignan and Salomon de Caus and Descartes.11 In particular, he shows to what extent the perspectivists were influenced by the theory of automata in Descartes. In the Discours de la Méthode the automaton and its clockwork movement explains the bodies of organic beings, “which will in no way seem strange to those who, knowing how many and various are the automata or moving machines the industry of man can make, while employing very few pieces in comparison with the great multitude of bones, muscles, nerves, arteries, veins and all other parts in the body of each animal, consider this body as a machine.” More precisely still, man is soon compared to a hydraulic mechanism: “The nerves of the machine I am describing to you [man] can be compared very well with the pipes of the machines of these fountains; his muscles and his tendons with the various devices and springs designed to work them; the animal spirits with the water that stirs them. . . . What is more, respiration and other such acts . . . are like the movements of a clock or a mill which the ordinary course of the water can render continuous.” Hidden levers, lastly, animated the entire organic apparatus, analogous, Descartes said, to those animating the automata that turned and danced in gardens and grottoes.

“In taking inspiration from these moving machines in his meditations on the structure and function of living organisms, Descartes was not only a logician,” writes Baltrusaitis. “He was also an imaginative man, who considered the world as a theatre where the secrets of nature were revealed by the toys constructed by man.”12 Speculation of this kind had been anticipated by Salomon de Caus who, in his Traité de l'homme of 1615 portrayed hidden hydraulic mechanisms, operating in the lower section of the plates, which controlled automata representing human beings or, more frequently, mythological divinities, in the upper section. Later, the fascination of the calculated mechanism dominated Nicéron’s perspective games. The specular machinery was set in motion and by turns revealed the visible and the invisible, the concealed and the apparent to the wondering eye. After him, and after Maignan, Athanasius Kircher took up the speculations on the Magiae naturae et artis. In his Musurgia universalis, published in Rome in 1650, he taught the art of constructing musical automata: a mill wheel, turned by a fall of water, operated a drum which, by means of valves and levers, worked small figurines and organ music (figs. 15, 16, 17).

One wonders whether Duchamp might not have had the opportunity at the Bibliothèque Sainte-Geneviève to glance through this superb folio volume and other works13 and to find in Athanasius Kircher a person whose curiosity was akin to his own. Note Kircher’s interest in optics (which led him to invent the magic lantern), his interest in acoustics and mechanical music and lastly his passion for the physical and mathematical games that made his Ars magna sciendi a kind of lettered ancestor of what at the turn of the last century became La Science amusante de Tom Tit (a popular French children’s book of “playful physics”).

After all, what is a Bachelor Machine, if not, as in Descartes, the assimilation of an organic body and an automaton through which gas flows, the comparison between man and machine having been taken so far at this point that it becomes a matter of indifference whether the bachelors have been mechanized or the machines eroticized.

We know that the Oculist Witnesses take up some of the elements of the more complex pattern of the small glass entitled A regarder d’un oeil, de près, pendant presque une heure. (To be Looked at, with One Eye, Close to, for Almost an Hour, fig. 18). We examine the constituent parts of that small glass, with its transparency, measurements and functional intent—always remembering the instruction of the title to apply one eye closely to it (which is not unreminiscent of the perspective instrument called a portillon, a stylus placed upright in the center of a graduated dial, with a lens at its tip). Since the parts thus described are related to the perfectly serious accompanying instructions which, in a way, explain the use of this “scopic” machinery, it is hard not to recognize . . . a gnomon.

In his notes Duchamp often came back to the problems of his obsession with clocks. He illustrated one of these, “the clock in profile and the space inspector,” in a completed work in 1964, which he entitled Pendule de profil (Clock in profile). “When a clock is seen from the side [in profile],” Duchamp stated in 1958, “it no longer tells the time.” This clock in profile, which depended on the inclination of its face for its utility, could just as easily be regarded as the sundial of the Oculist Witnesses (fig. 19).

Now a famous gnomonic work, the Perspectiva horaria sive de horographia gnomonica tum theoretica (fig. 20) by Emmanuel Maignan, figures under the heading of “perspective” in the Bibliothèque Sainte-Geneviève. Like Nicéron, Maignan was a priest of the Minim Order and, like him too, a specialist in odd perspectives; he published this treatise in 1648, two years after the Thaumaturgus opticus quoted by Duchamp. Considerations on gnomonics often accompanied the treatises of perspective. As a matter of fact, the triangulation exercises necessary to establish the perspective diagrams of bodies in space seem little different from those necessary to set sidereal clocks.

Looking at the various plates of the Perspectiva horaria, we shall thus not be too surprised to find more elements of the Oculist Witnesses. The plate corresponding to the instruction “Gnomonem ad angulos rectos, in dato plano erigere . . .” can thus be related to the pattern of A regarder d’un oeil de près: a stylus is erected vertically in the center of a circle, KO and KA being the projections of shadows cast by the sun at two points in its course, their respective lengths being equal and symmetrical in relation to each other: the bisecting line of the angle QKO, formed by the corresponding directions of the shadow of the gnomon, marks the meridian or north-south line. Similarly, in Duchamp’s Glass the tip of the stylus standing upright in the center of the dial cuts the bisecting line of an angle formed by the two arms of the Scissors. In the plate illustrating the projection of a gnomonic optical sphere onto a plane (fig. 21), the eye of the observer is conventionally shown as opening at the tip of the stylus and in the meridian plane NOFG, just as Duchamp prescribes looking with one eye at a time, from the tip of the obelisk and “close to.” (In fact, just as the lens focuses the solar rays and projects them, so the eye, substituting for it exactly, projects the visual rays. This lens, planned to be stuck to the Large Glass, was left out in the end.)

It is certainly far harder to associate the portrayal of the Bride with the treatises on perspective since it escapes the laws of classical perspective by definition, as the three-dimensional projection of a four-dimensional being. I have explained elsewhere why the Bride was regarded as a kind of flayed figure.14 Most of those who have written about the fourth dimension—Pawlowski, and also Jouffret, Howard Hinton, Ouspensky—have always imagined that, in the eyes of a four-dimensional being, our body would have neither outside nor inside, and would consequently be presented, as it were, in transparency, as if it were flayed. From there, the inference is that, while the regulated bodies of the Bachelor Machine have their “envelope,” the Bride has no envelope, in the geometrical and biological meaning of the word (figs. 22, 23).

What is more, “Le Pendu femelle in ordinary perspective assumes the form of a hanged woman, whose true form one could perhaps try to find.”15 The three-dimensional appearance of the Bride, whose semblance is four-dimensional, can in fact be achieved by the conventional rendering of shadow and light, just as the two-dimensional appearance of three-dimensional semblances is achieved with flat tints in plane geometry.

One aspect of the Bride, however, is very striking for our study. The jagged capricious contour circumscribing the Milky Way could scarcely be more unusual: “Louis XV Clouds,” Duchamp wrote.16 The delineation suggests the form that might be taken by a very fine piece of thread lying loose, and, consequently, conjures up the images of the Standard Stoppages, which we know were outlined by a thread falling at random and giving a “new portrayal of the unit of length.”17 Now, if the Milky Way was the Bride’s “cinematic fulfillment”—her “halo,” her culmination in the realm of the fourth dimension—then her appearance must borrow, in a way, from a “distended physics,” with the risky contours of the Stoppages as the standard rule.

Consider the conventional way of portraying volumes in the treatises of perspective: sheaves materializing the visual rays wrap the form of the objects like nets, then come to meet at one point, the eye of the beholder (fig. 24). (To accentuate the parallel with a net, the observer himself grasps the bundle of threads and raises it to his eye.) Ordinary perspective commands where these threads are stretched: the viewpoint is “forced,” as Duchamp said. But beyond the clenched fist holding the net, beyond the eye enveloping the object “according to a certain viewpoint and a certain distance,” the threads are represented as floating freely in space, describing a meandering pattern oddly like that of the Milky Way. They portray a universe beyond the perspectivist’s world; they envelop a body invisible to our eye, obeying the unknown laws of a distended physics.

Only four-dimensional sight could encompass the Bride in the plurality of her forms. But just as, by analogy, from the passage from the “geometral” to the perspective one can infer the existence of a fourth dimension, so, from the passage from monocular vision (or “flat” vision, in two dimensions) to binocular vision (in relief, voluminous), one can infer the existence of what Duchamp called “a four-dimensional eye.” This led him, it appears, to compose a whole hierarchy of ways of looking, overlapping or duplicating the hierarchy of the possible worlds of n-dimensions.

The point of departure was the “things to be looked at with one eye. With the left eye, and with the right eye. . . . One could set down a series of things to be seen with one eye.”18 The note dates from 1913, when Duchamp was working at the Bibliothèque Sainte-Geneviève. Now this library still possesses a highly unusual opuscule on a subject too near to Duchamp’s interests for him not at least to have glanced through it: the Discours by Sébastien Le Clerc, published in Paris in 1679, “touching on the viewpoint where it is proved that the things one sees distinctly are seen only with one eye” (figs. 25, 26). Le Clerc, as a fervent Cartesian, upheld the thesis that clear and distinct vision of an object could only be monocular. In the case of an object C considered by two eyes, A and B, through a glass plate OLMN, the animal spirits, crowding, according to Le Clerc, along the two optical nerves, “transmit a second image” of C to the brain, so that “one eye must be closed so as not to see double.” In fact, closing eye A, one sees C in E and closing eye B, one sees it in D.19 Another demonstration is shown in plate 37: “In the visual cone of the right eye A, is the arrow CD, which cannot be seen with that eye, but only with the left eye B. I say that the arrow CD, although seen with the left eye alone, cannot be seen clearly.” And a similar demonstration is repeated for plate 49 (fig. 27): “. . . eye B sees the lines CD, EF, one beside the other, and eye A sees them one above the other, so that the image of the right eye is a regular trapezoid and the image of the left eye an irregular trapezoid.”20 These various demonstrations allow Le Clerc to conclude: “The rules of perspective are truly founded on a single viewpoint.” Also, “The images of the objects come to the eye in dwindling size”21 (fig. 28).

One work by Duchamp seems entirely devoted to the monocular and binocular optical tricks (fig. 29) performed by Le Clerc, the Tu m’, Duchamp’s last oil (fig. 30), which can, in many respects, be regarded as a recapitulation of all the perspectivist “deceits” allowed by an opaque support. The strange diagram filling the right third of the picture is not the least of its mysteries. A rectangle is seen in perspective; double lines, meandering just like those of the Standard Stoppages, are drawn horizontally from its four corners. A kind of range of color-samples and circles is threaded along the straight lines. The outline of the whole becomes a parallelepiped volume, with its longest, floating, edges apparently obeying the new aleatory distended physics. The circles succeed one another, overlapping, in like fashion to those which illustrate the gradual dwindling of the image of an object as recorded in Le Clerc’s plates.

Duchamp’s entire production could thus be classed according to the various viewpoints governing each of his works:

Monocular vision: To be Looked at with One Eye, Close to, for Almost an Hour.
Monocular vision of right eye: parallelepiped anamorphosed on the right side of Tu m’.
Binocular vision of left eye: shadows cast by the ready-mades on the left side of Tu m’.
Monocular vision left and right alternately: Wilson-Lincoln effect prism (not executed).
Binocular vision:
Handmade Stereopticon Slides, Pharmacy, Precision Optics, anaglyphic film, “Anaglyphic Chimney,” Etant Donnés. etc.

Monocular vision sees a flat universe with no relief, a universe of two-dimensional beings. A sphere crossing the plane where such beings move would never be grasped in its reality as a three-dimensional body, but would be seen instead as a point gradually widening to become a circle of the same diameter as the sphere, then dwindling anew until it disappeared. Or again, a square revolving on its axis would be seen as a succession of diamonds, flattening more and more until they became no more than a line. And so on. The phenomenon would be experienced more as a chronology than a morphology, the dimension of time being precisely the “third dimension” of the phenomenon envisaged.

Now, from the “geometral” to the perspective, from monocular to binocular vision, there is no break in continuity: from a certain angle, the straight line is reduced to a dot, the square reduced to a line, the cube reduced to a square. By analogy, the hyper-cube, seen from a certain angle, is reduced to a cube. In other words, a four-dimensional body can pass through a single dot (the “aim” in Duchamp’s vocabulary): there is an angle at which the object and its projection, the being and its image, the object and, its shadow, merge, whatever the number of dimensions envisaged.

The “optical wonders” achieved by Nicéron and Maignan offered the same singularities in their effects. The astonishing St. François de Paul of Santa Trinità dei Monti, which when seen from a certain angle presents a landscape of sea and mountains strewn with tiny figurines, becomes transformed into the majestic figure of St. Francis kneeling at prayer when one stands at the end of the gallery where the work is placed and sees it from the side (figs. 31, 32).

Anamorphotic mechanisms of Nicéron and Maignan thus demonstrate what Duchamp said in the White Box: “All form is the projection of another form according to a certain vanishing point and a certain distance.”22 One can say, by analogy, that all solid bodies here below—trees, houses, human beings—constitute the possible projection of an infinity of four-dimensional entities. If the fresco of St. John the Evangelist at Patmos executed by Nicéron represents, for monocular vision playing on the two/three-dimensional articulation, and depending on the movement of the eye of the observer (since here it is literally the beholder who makes the picture), now a Saint in ecstasy, now a landscape—then for binocular vision playing on the three/four-dimensional articulation, and again depending on the point of view adopted, all solid bodies will be seen now as an “inkstand,” now as a “house,” now as a “captive balloon,” etc.23 So much so that the observer almost “loses the possibility of recognizing two similar things.”24 The “Rule of Coincidence” is none other than the rule of a “scopic” machine making everything the virtual projection of everything else, “the continuous in n dimensions being essentially the mirror of the continuous in three dimensions.”

In Duchamp’s eyes, an image virtually refers to a large number of singular objects, the entire visible domain being simply an incessant flow of anamorphoses enmeshed one with another, to infinity.

Let us look at Tu m’ again: when we face the picture squarely, the shadows cast by the “readymades” stretch out on the left and the “floating” edges of the parallelepiped on the right; the brush in the center, as a real object, is reduced to a dot because it pierces the surface of the canvas perpendicular to its plane (fig. 30). But if we move to the side (fig. 33), the shadows of the readymades and the design of the parallelepiped straighten up; at the same time, the brush tends to release itself from the canvas. Finally, if we stand along the same plane as the canvas, we find that it has been reduced to no more than a line. Conversely, the brush springs out with all the incongruence of an object piercing the picture. It serves the same function as the famous skull in Holbein’s Ambassadors, of course: to expose the vanity of the painting. But this time of all paintings. There is no humanist moral, just reasoning pushed to its limit. The real is shown where the canvas fades out, where the pictorial surface is effaced, because the picture has been made to revolve around an axis until it becomes reduced to a line (fig. 35; cf fig. 34 ), not, as in the Large Glass, because a transparent support has been substituted for an opaque one. Where painting vanishes, the readymade makes its appearance. Painting (the art of painting) is a matter of viewpoint: “It is the beholder who makes the picture.” Once anamorphosed, it leaves room for the manufactured object (antipainting, antiart, and so on).

So now let us go further. If all form is the projection of another form, if all appearance here below represents or repeats a semblance which in its turn is but the appearance of another form according to the circular concatenation of anamorphoses, without our ever being able to say which form is primary and which derived, which principal and which secondary, then the entire visual domain is identical in its working with what Wittgenstein said of language: an infinite tautology.

It was not by accident that Duchamp mentioned to his friends on several occasions that he was interested in the Viennese philosopher and in logical positivism25 and, more generally, in all theories of language sharing in the tradition of Anglo-Saxon nominalism.26 Especially since Duchamp’s conception of the sign was totally empirical, admitting that a word may be capable of separating from the visible figure with which it is linked by its “meaning” to fit onto another.

Hence his fascination with puns, spoonerisms, metagrams, the entire rotation of words in the volume of language, turning about themselves and tracing the full circle of possibles around a fixed point—identical with the rotation of figures in visual space and the appearance of their various possibilities on the glass plane. The rotation was so identical for Duchamp that in his film Anemic Cinema he made the representation of discs bearing puns alternate exactly with discs bearing drawings, both texts and figures biting their own tails, revolving endlessly, not only owing to the real movement impressed on the disc, but also to the inward circular movement impressed on the sentences and the drawings. The one type reads indifferently, either way, the other following on with no visible beginning or end in a reduplication (for a moment allowing just the shimmer of an unthinkable bottomless pit, which is never more than a linguistic effect here and an optical effect there). Hence the fact that in Duchamp the anagram is to speech what the anamorphosis is to figure: the duplication of language from a single element and by the unremitting game of combinations, a self-alienation of language ceaselessly giving rise to other figures.

It seems strange, one must admit, if we take care to step back a little at the end of this analysis, to observe that Duchamp’s work, which is so eagerly and over-hastily seen as the harbinger of contemporary iconoclasm, in actual fact pursues a project of restoration. And in this respect it is closer than has been observed to the work of Jacques Villon and the preoccupations of the Puteaux group. It is a project to give back to perspective some of the importance which the history of modern painting as a whole, from Manet and Cézanne onward, has been bent on denying; to make the pariete di vetro (partition of glass) and its handling once more the conceptual tool par excellence of the art of painting, alone capable of reviving the intelligence which that art had lost (painters had become “stupid” ever since they abandoned the perspectivist habit, which was a metaphysical habit, and indulged only in the delights of a retinal surface and in physical delights)—so that it would automatically become, as Leonardo had wanted, purely cosa mentale.

It is also strange to see that in the very same year of 1913, when Duchamp returned to the teaching of the perspectivists, the second great artist of the century, Henri Matisse, painted his monochrome Studios, where the illusionist depth of the image tended to merge with the material plane of the picture, thus closing Alberti’s window and achieving the equation of the picture as an opaque flat support with the subject it represented. It is paradoxical to observe that while an entire major trend of present art up to Pollock, Newman and their successors has persisted in making the picture an a-focal flat area—the notion of the picture as a material support—Marcel Duchamp thought only of going against the grain of all this “modernity,” asserting the notion of the picture as a transparent plane. Whether or not this meant a reversion to the past, or a need to react, would need very careful consideration, coming from an artist whom one is accustomed to presenting as the epitome of all avant-gardes and the herald of all breaks with tradition.

Etant Donnés . . . , his last piece, on which he worked for 20 years and in the deepest secrecy, is still too close in time for us to put forward more than hypotheses about its significance. In this cubic space pierced with two peepholes, one is certainly tempted to recognize the optical cabinets manufactured by Samuel Van Hoogstraeten and the 17th-century Dutch perspectors, and the transposition to binocular vision of what the Wonderlijke Perspectijfkas (perspective wonder-box) was to monocular vision. We are even more likely to remember that the keyhole for watching scenes of an erotic character was in frequent use among the anamorphosis-makers (Erhard Schön, for example). More unusual, perhaps, is the fact that the sight offered here should have its various elements carefully arranged on black-and-white squares of linoleum, like figures on a perspective chessboard. It is quite legitimate, as has been suggested, to see these elements as so many realistic paraphrases of what, in the Large Glass, remained an abstract statement. The Bride is really there, stripped bare, reclining, her left hand raising an Auer gas burner, like the torches the small characters in the engravings of Abraham Bosse carry in their fists to denote the shadows. At the back of a landscape, a waterfall flows tirelessly with the same scintillating repeated movement; somewhere it animates an invisible mill-wheel, working the automata of a sempiternal Theatrum Mundi.

The eye does not plunge directly into this landscape. In addition to the door another barrier separates it from this strange puppet-show, a brick wall pierced with a hole. From this wall with its two peepholes a space stretches out, distinct from the space where the scene itself takes place, a somber area which Duchamp compared to a kind of “dark room.” The linking of two spaces in this way is not without resemblance to another piece of optic apparatus, also life-size: the large camera obscura built in Amsterdam by Kircher in 1671 (fig. 36). In this last construction, as in Plato’s cave, the images projected by a bright world of trees, houses, etc., through a monocular peephole opened in the partition of the outer box, are depicted before the eyes of a beholder standing still in the center of the two boxes. Now in Etant Donnés everything takes place as if the device were working in reverse: the beholder is thrust back to the other side of the two circumscribed spaces, outside the brick wall and outside the door, as if into the darkness of an outer world. Conversely, the real, bright, visible world, with the trees, the water and the woman, is entirely contained inside the box. While the classical camera obscura was an instrument which articulated the three-dimensional and the two-dimensional, by the intermediary of a monocular opening, it appears—this is a pure hypothesis—as if Etant donnés might be an instrument capable of projecting a three-dimensional image, a perfect simulacrum of our world, through a dark room and a twin peephole, outside into a mysterious expanse.

It is my turn now to close the circle on itself. Consider the engraving by Dürer of a nude woman reclining against a peaceful country scene, with a lake and trees (fig. 37). The woman’s thighs are opened in the direction of the person looking at her. Between the eye of this witness and his viewpoint, which happens to be her sex, stands a perspective device, which we now recognize unmistakably: it consists of a stylus, to the tip of which the eye must be applied, “close to,” and, at a distance, an open grille whose coordinates locate the projecting visual rays of the reclining body. At right angles to this grille, related to it as the (male) bottom of the Large Glass is related to the (female) top, is a sheet of paper of the same size as the grille, and likewise gridded. On the plane where the sheet of paper rests, also serving as its horizon line, lies the garment that the woman has cast off. On the paper itself a hand holding a pencil traces—very much in “bachelor” fashion, as befits the hand of an artist—the possible projection of this body spread-eagled around its vanishing point.

Legend has it that Roudaki, who died in 950 in Samarkand, dedicated one of his poems to a woman so beautiful that her image passed through a pinhole in a curtain and was depicted on the opposite wall. The image was inverted so that Allah could contemplate it properly from the skies.



This is a revised version of an article that original appeared as “Duchamp et la tradition des perspecteurs,” Abécédaire, Catalogue Marcel Duchamp, Editions du Musée National d’Art Moderne, Centre Georges Pompidou, Paris, 1977, Vol.

1. Pierre Cabanne, Entretiens avec Marcel Duchamp, Paris, 1967, p. 65.

2. Unpublished interview, 1959.

3. Marcel Duchamp, Duchamp du Signe, Ecrits de Marcel Duchamp, ed. M Santouillet, Paris. 1976; hereafter referred to as DDS.

4. Erwin Panofsky, “Die Perspektive als Symbolische Form,” Vorträge der Bibliothek Warburg, 1924–25, pp. 258ff.

5. Abraham Bosse, Manière universelle de M. Desargues, pour pratiquer la perspective par petit-pied comme le géométral, Paris, 1647, pp. 45–46.

6. DDS, p 105.

7. Unpublished conversation.

8. Bosse, p. 25

9. Bosse, p. 84.

10. DDS, p 69 (“en force le dessin”).

11. J. Baltrusaitis, Anamorphoses; ou magie artificielle des effets merveilleux, Paris, 1969, ch. iv, pp. 61–70.

12. Baltrusaitis, p. 66.

13. His Ars magna lucis et umbrae in decem libros digesta (Rome, 1646), for instance.

14. J. Clair, Marcel Duchamp ou le grand fictif, Paris, 1975.

15. DDS, p. 69.

16. Not “Louis XV images,” as mistakenly transcribed by M. Sanouillet; DDS, p. 108.

17. DDS, p. 50.

18. DDS, pp. 107–08.

19. S. Le Clerc, Discours touchant le point de veuë dans lequel il est prové que les chosen qu’on voit distinctement, ne sont veuës que d’un oeil, Paris, 1679, pp. 14–16.

20. Le Clerc, p. 48.

21. Le Clerc, p. 66.

22. DDS, p. 69.

23. DDS, p. 135.

24. DDS, p. 47.

25. For example: “I do not believe in language which, instead of explaining unconscious thoughts, in reality creates the thought by and after the word. (I declare myself an eager ‘nominalist,’ at least in this simplified form.) . . . As a good nominalist, I suggest the word, patatautology, which, after frequent repetition, will create the concept of what I am trying to explain in this letter by these execrable means: subject, verb, object, etc.” (Unpublished letter to M. Mayoux, 8 March 1956). “The Vienna logisticians have elaborated a system whereby, as far as I have understood, everything is tautology, that is to say, repetition of premises. . . All is tautology.” (Entretiens avec Pierre Cabanne, p. 204). “Once, I took an interest in this group of English philosophers, those who claim that all language tends to become tautological and consequently meaningless. I even tried to read the book they wrote, The Meaning of Meaning . . . . I agree with their ideas . . .” (Calvin Tomkins, The Bride and the Bachelors, New York, 1965, pp. 31–32); etc.

26. It is legitimate to wonder whether the immense success of Duchamp’s work in Anglo-Saxon countries might not have been due precisely to the philosophical presuppositions, from Condillac to Bertrand Russell, on which it is based.

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