Vol.1 / Issue3

Duchamp's Perspective:
The Intersection of Art and Geometry

by Craig Adcock


In addition to implying something being stopped, the word "stoppage" also suggests something being mended or repaired.  In French, "stoppage" refers to sewing or reweaving a tear in a fabric in such a way that the tear can no longer be seen.(12)  From this perspective, the individual lines in the sculpture and the network of lines in the painting can be compared with the breaks in the Large Glass.  In his early monograph, Robert Lebel pointed out that the Network of Stoppages bears a strange resemblance to the pattern of fissures in the Glass, as if the painting had somehow been a preliminary study for the subsequent breakage.(13)  When Duchamp put the Glass back together, or perhaps we could also say when he "rewove" it, he no doubt also noticed the fortuitous similarities.  The shapes of the line segments generated by the pieces of thread were random, but they seemed planned.  Likewise, the line segments caused by the Glass being smashed were determined by chance, but they also seemed necessary for its completion (or definitive incompletion).(14)  When Duchamp rebuilt the work, he was "stopping" an accidental event that had somehow made the Glass "a hundred times better."(15)  The mended cracks in the glass are not wholly invisible, but they do approach a point of disappearance--like pieces of string falling away toward some mysterious knot at infinity.  Duchamp's lines, his fractures and strands, intersect at a vanishing point in the fourth dimension, a realm that cannot be seen from our ordinary perspectives.

The Bride's "garments" and the Three Standard Stoppages can also be discussed in terms of yet another kind of "stoppage."  Glass, as a physical substance, is an insulator, and as such is often used to arrest or impede the flow of electrical current through circuits.  Duchamp may very well have been thinking of his glass plates in these kinds of terms when he was constructing the Large Glass.(16)  He also refers to the Bride's clothing as a "cooler":

(Develop the desire motor, consequence of the lubricious gearing.)  This desire motor is the last part of the bachelor machine.  Far from being in direct contact with the Bride, the desire motor is separated by an air cooler (or water).  This cooler (graphically) to express the fact that the bride, instead of being merely an asensual icicle, warmly rejects (not chastely) the bachelors' brusque offer.  This cooler will be in transparent glass.  Several plates of glass one above the other.  In spite of this cooler, there is no discontinuity between the bachelor machine and the Bride.  But the connections will be electrical and will thus express the stripping:  an alternating process.  Short circuit if necessary.(17)

In addition to the terms "vêtements de la mariée" and "refroidisseur," Duchamp uses the expression "plaques isolatrices" to describe his strips of glass.(18)  This phrase can be translated as "isolating plates" or "insulating plates."  In one of his posthumously published notes, he calls the horizontal division of the Glass a "grand isolateur," a "large insulator," and explains that it should be made using "three planes five centimeters apart in transparent material (sort of thick glass) to insulate the Hanged [Pendu] from the bachelor machine."(19)

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Figure 10
Marcel Duchamp, Draft Pistons, 1914
Figure 11
Marcel Duchamp, Travelor's Folding Item, 1916
Figure 12
Photograph of the unbroken Large Glass

Glass may play a similar exclusionary role in the workings of the Three Standard Stoppages, but in ways that are perhaps less "transparent."  While Duchamp was apparently interested in exploring a frustrated relationship between the Bride and the Bachelors, involving as it does a "short circuit," he was also trying to "delay" communication.  Whatever talking occurs, or fails to occur, between the separated Bride and Bachelors pertains to seeing or not seeing through words.  In his notes, Duchamp explains that the Bride sends her commands to the Bachelors through the "draft pistons," "triple ciphers" that use a formal alphabet constructed using the Three Standard Stoppages.  Because the chance-determined "draft pistons" (Fig. 10), which are deformed planes, are conceptually similar to the Stoppages, which are deformed lines, these interpretations again converge geometrically.  It might also be pointed out that Duchamp's readymade Traveler's Folding Item (Fig. 11) can be taken as a next logical step in this sequence:  a one-dimensional line generating a two-dimensional surface, which in its turn, generates a three-dimensional "solid"--one that can fold up.(20)  By looking somewhat further into the n-dimensional implications of these works (from the Latin implicatio, an entwining or interweaving), we may be able to ascertain how Duchamp's arrangements, his strings and fabrics, which seem to have topological insinuations, might actually operate.  Just how do the Three Standard Stoppages disappear into the Bride's clothing?

At some later point in the construction of Three Standard Stoppages, Duchamp cut the narrow strips of canvas from their stretchers, reducing them in size in the process, and then glued them down to thick pieces of plate glass.  He probably carried out this reworking when he was repairing the Large Glass at Katherine S. Dreier's home in Connecticut during the spring and summer of 1936.(21) Also at this time, he probably decided to put the various components of the Three Standard Stoppages into a specially constructed wooden case that resembles a croquet box.  Duchamp's decision to amplify the Stoppages along these lines was almost certainly connected with how he was repairing the "garments" of the Bride, which had presumably been pulverized when the Glass was accidentally broken in 1927.  From the photograph of the unbroken Large Glass taken at the Brooklyn Museum (Fig. 12), it is difficult to determine how the original "garments" were constructed, but they do not appear to have been as elaborate as the repaired strips of glass.  As pointed out earlier, Duchamp must have intended for the Stoppages and the "garments" to be related to one another because he used similarly colored strips of glass and parallel edge-on arrangements in their respective reconstructions.

Did Duchamp somehow "betray" his work by not actually dropping the pieces of string when he originally made the Three Standard Stoppages or when, over twenty years later, he further modified his original conception of the piece?  No more than he betrayed himself by learning to appreciate the breaks in the Large Glass, or by elaborating the Bride's "garments" when he repaired them.  Such operations are, I believe, commensurate with his general attitudes about such matters.(22)  Recall his statement to Katherine Kuh:  "the idea of letting a piece of thread fall on a canvas was accidental, but from this accident came a carefully planned work.  Most important was accepting and recognizing this accidental stimulation.  Many of my highly organized works were initially suggested by just such chance encounters."(23)

Dropping pieces of string was not a rule that Duchamp had to follow, but rather a point of departure in his thinking, just as the damage to the Glass wound up inspiring his admiration.(24)  His artistic approach was analogous to scientists establishing hypotheses at the beginning of a research program, but then modifying their hypotheses once work has been carried out in the laboratory.  Over the course of time, Duchamp's examples of "hasard en conserve"(25) were supplied with controls that had not been deemed necessary in the beginning.  As with the chance breakage he preserved in the Large Glass, the important thing was recognizing the accidental stimulation.  Moreover, by allowing the pieces of thread to do more than simply fall upon the canvas surfaces by actually sewing them through to the other side, Duchamp could emphasize the notion that they had intersected the canvases.  The encounter involved both chance and mathematics.

In works such as the Three Standard Stoppages, Duchamp creates physical analogues for the abstract concept of "intersection":  the one-dimensional pieces of string, the curved line segments, intersect the two-dimensional surfaces of the canvases (and they literally share points in common where they are sewn together).  The strings are thus further implicated (I am tempted to say intertwined), along geometrical lines, with the fabric of the canvas strips.  The cracks in the Glass are also a fundamental part of it.  They are "inside" the broken sheets of glass, which are, in their turn, encased inside the heavy panes of glass that Duchamp used to effect their repair.  In an analogous way, the ends of the strings in the Stoppages are sandwiched between the strips of canvas and the rectangles of glass that back them.

Duchamp's works on glass are flat, but they are nonetheless rather thick.  They are "spaces" that can be thought of, especially in this context, as rectangular solids.  Because the sheets of glass themselves have thickness, a depth that is often layered, they can be taken as three-dimensional sections out of higher-dimensional continua.  When, for example, all the configurations of the Stoppages (the strings, the templates, and the plates of glass) are considered together, their n-dimensional implications are manifest.  They are one-dimensional, two-dimensional, and three-dimensional, and they have n-dimensional possibilities.  Each configuration is related to the others through projection and intersection:  the lines can be taken as slices out of surfaces, the surfaces as slices out of solids, and the solids as slices out of hypersolids.  Esprit Pascal Jouffret, one of Duchamp's most important mathematical sources, characterized such cuts as "infinitely thin layers."(26)

Duchamp's approach--moving from lines to surfaces, and from spaces to hyperspaces--is couched in terms of perspective.  He considers how vanishing points and changing points of view would operate in 2-space, 3-space, 4-space, or any given n-space.  He suggests using "transparent glass" and "mirror" as analogues of four-dimensional perspective systems (analogues because such systems cannot actually be constructed in three-dimensional space).(27) Especially when the narrow sheets of glass are seen edge-on in the slots in their croquet box, they suggest their membership in an infinite series (reflections in mirrors can also imply infinite reiterations).  In an interview with Pierre Cabanne, Duchamp emphasized the serial characteristics of the Stoppages:  "When you've come to the word three, you have three million--it's the same thing as three.  I had decided that the things would be done three times to get what I wanted.  My Three Standard Stoppages is produced by three separate experiments, and the form of each one is slightly different.  I keep the line, and I have a deformed meter."(28)

The specifics of how Duchamp kept his line and used his deformed meter is worth exploring further.  He tells Cabanne that he had been interested in working on glass for several reasons, including the way color "is visible from the other side."  Glass was also useful in laying out its various elements:  "perspective was very important.  The Large Glass constitutes a rehabilitation of perspective, which had been completely ignored and disparaged.  For me, perspective became absolutely scientific."(29)  By using linear perspective in his design, Duchamp could arrange the Bachelors' domain in such a way that the vanishing point coincided with the horizontal division between the upper and lower panels of the Glass.

From this perspective, or from the point of view of perspective, Duchamp's saying that a "labyrinth" lies at the "central part of the stripping-bare" is significant:  the Large Glass and the Three Standard Stoppages are about occlusion.(30)  They involve unusual station points, and unusual distance points, in a perspectival system that can only be reconstructed from isolated positions outside normal space.  If Duchamp were thinking of his "strips" of glass as physical puns on the notion of "stripping" the Bride, then their structure is doubly suggestive.(31)  Because her clothing consists of transparent sections of glass that are entailed with a "point de fuite," it can be taken to include a complex set of folds, not only in the cloth of the garments, but also in the fabric of space.  Recall that Traveler's Folding Item is conceptually related to the Three Standard Stoppages.  Also, the typewriter cover has been called the "Bride's Dress."(32)  Perhaps the disappearance of the Stoppages, their dropping away toward infinity at the position of the Bride's garments, can be taken as an interdimensional folding up, a stripping bare that requires orthogonal translation into higher space.

All of the works here under discussion are related to one another through perspectivalism (and also perspectivism).  For Duchamp, the use of perspective as a system was not a matter of creating single, fixed-point ways of looking at things.  It was, on the contrary, involved in dislodging viewers from their ordinary ways of understanding.  And with this objective in mind, his choosing readymades during the same period he was working on the Stoppages can be seen as a related activity.  When Duchamp made his remark about Three Standard Stoppages being a readymade, but "not quite," he continued by saying, "it's a readymade if you wish, but a moving one."(33)  The curving pieces of string and our shifting notions of the meaning of the readymades seem to trail off from a "vanishing point" at the horizon of our own thinking.  The readymades refuse to abide by our ordinary definitions of art, and the Stoppages allude to geometries that have challenged our traditional epistemological structures.(34)  Their curvatures can be taken as references to non-Euclidean or topological geometries, complications that necessitate our reconsidering our vanishing points.  The strings, when taken as analogues for lines of sight, are transposed, or rotated, into a hidden space.

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Figure 13
Girard Desargues's discussions of perspective
Figure 14
Girard Desargues's discussions of perspective

What I have in mind here can be seen in the illustrations that accompany Girard Desargues's discussions of perspective (Figs. 13 and 14).  Desargues was the first mathematician to see connections between linear perspective and conic sections, and is generally considered to be the founder of projective geometry.(35)  He contributed to the "mathematicization" of perspective, helping to transform the practical Renaissance practice of artists into the deductive science of geometers.(36)  In the illustrations, threads from lines of sight are bunched up at the plane of the picture, as if they were lying at, or perhaps it would be better to say "in," the surface of the representation.  Rather than being part of the representations, which are behind the surface and inside the three-dimensional structure represented by the picture, they are meant to be seen as separate from it.(37)  In other words, they lie in a transparent perspectival section of our visual pyramid, the surface of the picture plane that we do not normally look at in a Renaissance picture, but through.(38)

Such lines are also connected by a technological protocol involving an "arbor."  Desargues is one of the most likely sources for Duchamp's referring to the "Bride" as an "arbor-type."(39)  The mathematician uses the term "arbre" in his discussions of perspective, as J. V. Field has explained:

"Arbre" is usually translated as "tree," but the word can equally mean "arbor" or "axle."  Like the central axle in a machine, Desargues' arbre is the member to which others are referred, that is, their relation to it is what chiefly defines their significance in the overall arrangement.  The standard metaphorical usage whereby engineers called an axle a tree might thus have suggested to Desargues an extension of the same metaphor to provide names for subsidiary elements in the geometrical scheme.(40)

In Desargues' usage, an "arbre" becomes a geometrical axis.(41)  His unusual vocabulary was probably inspired by his engineering and military experience, as Field suggests.  Desargues employs a number of other "arbor-type" terms, such as tronc (trunk), noeud (knot), rameau (branch), souche (stump), and branche (limb).  A "trunk" is a straight line that is intersected by other straight lines, "knots" are the points on the "trunk" through which the other lines pass, the other lines themselves are called "branches," a point common to a group of segments on a line is a "stump," one of these segments is a "limb," etc.(42)

Desargues' general approach of adopting an affective vocabulary for geometrical entities recalls Duchamp's practice.  For example, Desargues' term essieu (axletree) is reminiscent of Duchamp's term charnière (hinge).  "Perhaps make a hinge picture (folding yardstick, book); develop the principle of the hinge in the displacements, first in the plane, second in space.  Find an automatic description of the hinge.  Perhaps introduce it in the Pendu femelle."(43) The mechanical engineering term "axletree" refers, basically, to a fixed beam with bearings at its ends.  Because the axletree has other devices, such as wheels, branching from it, we can perhaps see why Desargues saw a comparable situation in the way geometrical projections branch off from the axes of his perspective system.  In English, the similar term "arbor" was apparently used during the seventeenth century to designate any kind of axle, but is now generally used to refer to the axles in small mechanisms such as clocks.(44)

Duchamp hints that he was familiar with these kinds of distinctions.  In one of his posthumously published notes (actually notations on a folder that originally contained several other notes), he associates the Bride, the "Pendu" (femelle), with a "standard arbor (shaft model)."(45)  In another, he connects the Bride, a "framework--standard arbor," and a "clockwork apparatus."(46)  In Desargues's way of thinking, an "arbor" or an "axletree" was analogous to an axis of rotation, a mathematical "axle," around which the elements of his transformative system revolved.  In Duchamp's descriptions of the complex workings of the Bride, "hinges" operate in comparable ways.

That Desargues was one of Duchamp's sources can be given further credence by analyzing another important iconographical element of the Bride's domain, the "nine shots," an area of the Large Glass that was also reconstructed in 1936.(47)  At a conceptual level, the "nine shots" seem to have an "Arguesian" perspectival demeanor.(48)  It has recently been noticed that a number of Duchamp's notes have been split in two.(49)  One of the most interesting instances involves the "nine shots."  A note included in his posthumously published Notes is the top part of a note published in the Green Box.  Taken together, the two parts read as follows:


Make a painting on glass so that it has neither front, nor back; neither top, nor bottom.  To use probably as a three-dimensional physical medium in a four-dimensional perspective.(50)

Shots.  From more or less far; on a target.  This target in short corresponds to the vanishing point (in perspective).  The figure thus obtained will be the projection (through skill) of the principal points of a three-dimensional body.  With maximum skill, this projection would be reduced to a point (the target).

With ordinary skill this projection will be a demultiplication of the target.  (Each of the new points [images of the target] will have a coefficient of displacement.  This coefficient is nothing but a souvenir and can be noted conventionally.  The different shots tinted from black to white according to their distance.)

In general, the figure obtained is the visible flattening (a stop on the way) of the demultiplied body.  Cannon; match with tip of fresh paint.  Repeat this operation 9 times, 3 times by 3 times from the same point:  A--3 shots; B--3 shots, C--3 shots.  A, B, and C are not in a plane and represent the schema of any object whatever of the demultiplied body.(51)

Desargues used the unusual term "ordinance" for the orthogonals in a perspective system, the sheaf of lines that recede into the distance toward a vanishing point at the horizon.  An "ordinance of lines" (ordonnance de droictes) corresponds to what we would now call a "pencil of lines" in modern geometrical parlance.(52)  Desargues, who had worked as a military engineer, may again have been prone to thinking of the trajectories of cannon shots toward a target as analogues for lines diminishing toward a vanishing point in a perspective system (or toward the vertex of a pencil of lines in a more purely geometrical representation).  His term for a vanishing point (or for the vertex in an "ordinance of lines") is "but."  He uses the expression "but d'une ordonnance," which can be translated as "butt of an ordinance," but which is probably more comprehensibly rendered as "target of an ordinance").  Duchamp's line from the note above, "This target in short corresponds to the vanishing point (in perspective)," reads in French, "Ce but est en somme une correspondance du point du fuite (en perspective)."(53)

click to enlarge
Figure 15
Marcel Duchamp, Pharmacy, 1914

Before leaving the potential influence of Desargues' vocabulary, it might be pointed out that the notion of an "arbor-type" seems to inform several of Duchamp's readymades.  Pharmacy (Fig. 15), chosen in 1914, is a tree-filled landscape with a red and green dot added by Duchamp (at vanishing points?) on the horizon line.  In addition to being a reference to the colored bottles in drugstore windows, the colors may also be a subtle reference to the techniques of anaglyphy, a practice related to stereoscopy that we know Duchamp was interested in, probably because of its n-dimensional implications.(54)  In the layout of Robert Lebel's early monograph, a design that Duchamp was largely responsible for, Pharmacy is juxtaposed to the Bottlerack (Fig. 16), also chosen in 1914.  On the facing page are the Network of Stoppages, 1914, and Cemetery of Uniforms and Liveries, No. 2 (Fig. 17), 1914, the drawing that Duchamp used to transfer the design of the "capillary tubes" and the "nine malic molds" to the Large Glass.(55)  Above Pharmacy and the Bottlerack is Cemetery of Uniforms and Liveries, No. 1 (Fig. 18), which in the more multi-layered French edition of the book, had a color image of Nine Malic Molds (Fig. 19) tipped in over it.(56) >>Next


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Figure 16
Figure 17
Marcel Duchamp, Bottle Dryer, 1914/1964
Marcel Duchamp, Cemetery of Uniforms and Liveries, No. 2, 1914
Figure 18
Figure 19
Marcel Duchamp, Cemetery of Uniforms and Liveries, No. 1, 1913
Marcel Duchamp, Nine Malic Molds, 1914-15

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12. Schwarz (cited n. 7) 444, says that Duchamp's chose his title after seeing a sign on a Parisian shop advertizing "stoppage"; see also Francis Naumann, The Mary and William Sisler Collection (New York: Museum of Modern Art, 1984) 168-71.  Jennifer Gough-Cooper and Jacques Caumont, "Ephemerides on and about Marcel Duchamp and Rrose Sélavy, 1887-1968," in Marcel Duchamp: Work and Life, ed. Pontus Hulten (Cambridge: MIT Press, 1993), in their entry for May 19, 1914, have suggested that the sign read "stoppages et talons," which would imply fixing holes in the heels (talons) of socks and stockings.

13. Robert Lebel, Marcel Duchamp, with texts by André Breton and H.-P. Roché, trans. George Heard Hamilton (New York: Grove Press, 1959) 54.

14. In an interview with James Johnson Sweeney filmed at the Philadelphia Museum of Art and broadcast as part of the "Wisdom" series on NBC television in January 1956, Duchamp himself put forward a similar argument:  "I like the cracks, the way they fall.  You remember how it happened in 1926, in Brooklyn?  They put the two panes on top of one another on a truck, flat, not knowing what they were carrying, and bounced for sixty miles into Connecticut, and that's the result!  But the more I look at it the more I like the cracks:  they are not like shattered glass.  They have a shape.  There is a symmetry in the cracking, the two crackings are symmetrically arranged and there is more, almost an intention there, an extra--a curious intention that I am not responsible for, a ready-made intention, in other words, that I respect and love."  "A Conversation with Marcel Duchamp," reprinted in Duchamp, Salt Seller (cited n. 2) 127-37, the quote is from p. 127.  The Large Glass was on view at the "International Exhibition of Modern Art" at the Brooklyn Museum between November 17, 1926, and January 9, 1927.  It thus must have been broken on its way back to Katherine S. Dreier's home in West Redding, Connecticut, in early 1927, rather than in 1926 as Duchamp says.

15. Interview with Cabanne (cited n. 11) 75:  "It's a lot better with the breaks, a hundred times better.  It's the destiny of things."  See also Mark B. Pohlad, "`Macaroni Repaired is Ready for Thursday . . .': Marcel Duchamp as Conservator," Tout-Fait: The Marcel Duchamp Studies Online Journal 1, no. 3 (December 2002) Articles <>.

16. Henderson (cited n. 6) discusses the Bride's "garments" and their relationship with the Three Standard Stoppages in terms of "telegraphy," comparing the glass plates in these works to such devices as condensers and insulators; see especially her chap. 8, "The Large Glass as a Painting of Electromagnetic Frequency."

17. Duchamp, Salt Seller (cited n. 2) 39.

18. Marcel Duchamp, Duchamp du Signe: Écrits, ed. Michel Sanouillet, with the collaboration of Elmer Peterson (Paris: Flammarion, 1975) 82-86; see also idem, Salt Seller (cited n. 2) 56-60.

19. Marcel Duchamp, Notes, ed. and trans. Paul Matisse (Paris: Centre Georges Pompidou, 1980), no. 154. 

20. For a more complete discussion of these ideas, see Craig Adcock, "Conventionalism in Henri Poincaré and Marcel Duchamp," Art Journal 44 (fall 1984): 249-58; see also idem, Marcel Duchamp's Notes (cited n. 10) 149-54.

21. Ecke Bonk, Marcel Duchamp, The Box in a Valise: de ou par Marcel Duchamp ou Rrose Sélavy, trans. David Britt (New York: Rizzoli, 1989) 216-20. See also the letters Duchamp sent to Dreier during late 1935 and early 1936 in Affectionately, Marcel: The Selected Correspondence of Marcel Duchamp, ed. Francis M. Naumann and Hector Obalk (Ghent and Amsterdam: Ludion Press, 2000) 199-207.

22. For a discussion of Duchamp's approach, along somewhat different lines, see Craig Adcock, "Duchamp's Way: Twisting Our Memory of the Past `For the Fun of It,'" in The Definitively Unfinished Marcel Duchamp, ed. Thierry de Duve (Halifax: Nova Scotia College of Art and Design; Cambridge and London: MIT Press, 1991) 311-34.

23. Interview Kuh (cited n. 4) 92.

24. Interview with Cabanne (cited 11) 75.

25. Duchamp, Duchamp du Signe (cited n. 18) 50.

26. Esprit Pascal Jouffret, Traité élémentaire de géométrie à quatre dimensions et introduction à la géométrie à n dimensions (Paris: Gauthier-Villars, 1903), xxviii.  For a more detailed discussion of Jouffret's usage and its importance for Duchamp's concept of inframince, see Adcock, Marcel Duchamp's Notes (cited n. 10) 48-55.

27. Duchamp, Salt Seller (cited n. 2), 88.  For more detailed analyses of Duchamp's use of glass and mirror as metaphors for four-dimensional perspective, see Adcock, Marcel Duchamp's Notes (cited n. 10), esp. 75-79, 146-49; also idem, "Geometrical Complication in the Art of Marcel Duchamp," Arts Magazine 58 (January 1984): 105-09

28. Interview with Cabanne (cited n. 11) 47.

29. Ibid., 38. 

30. Duchamp, Notes (cited n. 19) no. 139; see also no. 153.

31. See Henderson (cited n. 6) 63:  "The Stoppages' arrangement of one clear and two greenish glass plates parallels exactly that of the glass strips mounted on the Large Glass:  the top strip is clear and the two below are greenish in hue.  Because Duchamp located the Bride's "Clothing" at the midsection of the Glass, the gravity-drawn thread lines of the Stoppages may have become for him a metonymical sign for the fallen garment of the Bride."

32. Linde, "MARiée CELibataire" (cited n. 7) 60; Arturo Schwarz (cited n. 7, p. 463) says that Duchamp related Traveler's Folding Item to a "feminine skirt."  See also Molly Nesbit and Naomi Sawelson-Gorse, "Concept of Nothing: New Notes by Marcel Duchamp and Walter Arensberg," The Duchamp Effect: Essays, Interviews, Round Table, ed. Martha Buskirk and Mignon Nixon (Cambridge, Mass., and London: MIT Press, 1996) 131-75.  For a number of fascinating connections between Duchamp's Traveler's Folding Item and the world at large, see Rhonda Roland Shearer, "Marcel Duchamp: A Readymade Case for Collecting Objects of Our Cultural Heritage along with Works of Art," Tout-Fait: The Marcel Duchamp Studies Online Journal 1, no. 3 (December 2000) Collections <>.

33. Interview with Roberts (cited n. 1) 62.

34. Hilary Putnam, for example, has said that "the overthrow of Euclidean geometry is the most important event in the history of science for the epistemologist."  See his Mathematics, Matter and Method, 2d ed. (Cambridge: Cambridge University Press, 1979), x.

35. For one of the most complete discussions of Desargues' work and for the most reliable translations of his texts, see J. V. Field and J. J. Gray, The Geometrical Work of Girard Desargues (New York: Springer-Verlag, 1987).  Desargues' principal essay on projective geometry is Brouillon proiect d'une atteinte aux evenemens des rencontres du Cone avec un Plan (Paris, 1639); his earlier work on perspective, is entitled Exemple de l'une des manieres universelles du S.G.D.L. touchant la pratique de la perspective sans emploier aucun tiers point, de distance ny d'autre nature, qui foit hors du champ de l'ouvrage (Paris, 1636).  "S.G.D.L." is an abbreviation for "Sieur Girard Desargues Lyonnais."  This twelve page brochure included the two high-quality engraved illustrations reproduced here, which are almost certainly by Abraham Bosse (1602-1676); see J. V. Field, The Invention of Infinity: Mathematics and Art in the Renaissance (Oxford and New York: Oxford University Press, 1997) 192.  Desarques' perspective treatise was included as an appendix in Bosse's Maniere universelle de Mr. Desargues, pour pratiquer la perspective par petit-pied, comme le Geometral (Paris, 1648).

36. For a discussion of this trend, see Martin Kemp, "Geometrical Perspective from Brunelleschi to Desargues: A Pictorial Means or an Intellectual End?" Proceedings of the British Academy 70 (1984): 89-132.

37. Field (cited n. 35) 192-95.

38. Erwin Panofsky, Perspective as Symbolic Form, trans. Christopher S. Wood (New York: Zone Books, 1991); originally published as "Die Perspektive als `symbolische Form,'" in Vorträge der Bibliothek Warburg, 1924-1925 (Leipzig and Berlin, 1927) 258-330.  For a discussion of Panofsky's contributions to perspective studies, particularly strong in its analysis of sources, see Kim Veltman, "Panofsky's Perspective: A Half Century Later," in La Prospettiva rinascimentale: Codificazione e trasgressioni, vol. 1, ed. Marisa Dalai Emiliani (Florence: Centro Di, 1980) 565-84.

39. Duchamp, Salt Seller (cited n. 2) 42:  "This cinematic blossoming, which expresses the moment of the stripping, should be grafted onto an arbor-type of the bride.  This arbor-type has its roots in the desire-gears, but the cinematic effects of the electrical stripping, transmitted to the motor with quite feeble cylinders, leave (plastic necessity) the arbor-type at rest.  (Graphically, in Munich I had already made two studies of this arbor type.)  Do not touch the desire-gears, which by giving birth to the arbor-type, find within this arbor-type the transmission of the desire to the blossoming into stripping, voluntarily imagined by the bride desiring."

40. J. V. Field, "Linear Perspective and the Projective Geometry of Girard Desargues," Nuncius 2, no. 2 (1987): 3-40.

41. Henderson (cited n. 6) does not refer to Desargues in her discussion of the Bride as an "arbor-type."  She argues that because an "arbor" is an "axle," Duchamp's usage should be interpreted as a reference to such devices as the shafts in automobile transmissions or electrical generators.  I completely agree that Duchamp could have had these kinds of associations in mind along with his taking an "arbre" to refer to a geometrical axis of rotation.

42. Field and Gray (cited n. 35) 61-175.

43. Duchamp, Salt Seller (cited n. 2) 27; see also idem, Duchamp du Signe (cited n. 18) 42.

44. Field, "Linear Perspective and the Projective Geometry of Girard Desargues" (cited n. 40) 21.

45. Duchamp, Notes (cited n. 19) no. 57.

46. Ibid., no. 155.

47. There are two new sections in the upper right corner of the Large Glass with holes drilled through them to create the "nine shots."  In photographs of the Large Glass taken at the Brooklyn Museum in 1926-27, the "nine shots" are not visible.  Duchamp may have incorporated them into the Glass when he was repairing it in 1936.

48. "Arguesian" would be the adjectival counterpart of "Cartesian."  René Descartes (1596-1650) and Desargues (1593-1662) were almost exact contemporaries and communicated with one another about mathematical matters; see Field, Invention of Infinity (cited n. 35) 190-97; see also René Taton, L'Oeuvre mathématique de G. Desargues: Textes publiés et commentés avec une introduction biographique et historique, 2d rev. ed. (Lyon: Institut Interdisciplinaire d'Etudes Epistémologiques, distributed by the Librairie Philosophique J. Vrin, Paris, 1988).

49. I am indebted to Hector Obalk for drawing this connection (or reconnection) to my attention in his talk "What Is an Object? The Belated Career of the Readymade," at the interdisciplinary colloquium "Methods of Understanding in Art and Science: the Case of Duchamp and Poincaré," Harvard University, Cambridge, Massachusetts, November 7, 1999.

50. Duchamp, Notes (cited n. 19) no. 67.

51. Duchamp, Salt Seller (cited n. 2) 35.

52. Field, Invention of Infinity (cited n. 37) 197; see also Field and Gray (cited n. 35) 60-68.

53. Duchamp, Duchamp du Signe (cited n. 18) 54.

54. See Adcock, Marcel Duchamp's Notes (cited n. 10) 130-32.

55. Lebel (cited n. 13) 132-33.

56. Robert Lebel, Sur Marcel Duchamp, with texts by André Breton and H.-P. Roché (Paris and London: Éditions Trianon, 1958); a facsimile edition of this book was published by the Centre Georges Pompidou, Paris, in 1996.


Figs. 10-12, 15-19
©2003 Succession Marcel Duchamp, ARS, N.Y./ADAGP, Paris. All rights reserved.