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Three Standard Stoppages
Linda Dalrymple Henderson
Marcel Duchamp and the New Geometries (Part 2)
In a 1964 lecture Duchamp described the painting as follows: "Using again the technique of demultiplication in my interpretation of Cubist theory, I painted the heads of my two brothers playing chess, not in a garden this time, but in indefinite space." Indeed, the "indefinite space" of Portrait of Chess Players may well represent Duchamp's major essay in the Cubist approach 'to the fourth dimension. A similar ambiguity produced by Cubist faceting had elicited Princet's initial suggestion of the similarity of Cubist painting to n-dimensional geometry. Artists like Metzinger had then worked specifically at breaking objects into facets in order to destroy any clear sense of three-dimensionality and to "insert" an extra dimension. It was, in fact, at the time Duchamp was working on the Portrait of Chess Players that Princet is supposed to have introduced him to Picasso and Braque, the original sources for the Cubist mode reinterpreted in terms of the fourth dimension by Princet and Metzinger.
If Duchamp's Portrait of Chess Players was somehow connected with Jouffret's suggestion that the operation of the chess player's mind is similar to the visualization of the fourth dimension, this painting was already more sophisticated than the majority of Cubist paintings seeking to portray the fourth dimension. Whatever the case, Duchamp was not satisfied for long with the Cubist technique for evoking the fourth dimension. During the fall of 1912 he was to embark on a serious study of the question of dimensions and the related problem of perspective.
Before that step and Duchamp's simultaneous rejection of conventional oil painting, however, he may have tried to incorporate the fourth dimension in painting by means of motion studies similar to those of Kupka. As discussed in Chapter 2, Kupka seems to have associated his own paintings of movement, based on the chronophotography of Marey, with the "four-dimensional displacement" described by Pawlowski in his 1912 "Voyage au pays de la quatrieme dimension." If Kupka had discovered the hyperspace philosophy of Hinton through Theosophical publications, Duchamp's reading of Jouffret, who quoted Hinton at the beginning of his blindfold chess discussion, would have encouraged him to look further at Hinton's approach.
Duchamp's studies of the "static representation of movement" began in December 1911 with the schematic action of the Coffee Mill (Collection Mrs. Robin Jones, Rio de Janeiro), painted for Duchamp-Vilion's kitchen. That same month he painted Sad Young Man on a Train (Peggy Guggenheim Foundation, Venice) and the first of the two versions of Nude Descending a Staircase (Philadelphia Museum of Art, The Louise and Walter Arensberg Collection). Nude Descending a Staircase, No. 2 followed in January 1912, with The King and Queen Surrounded by Swift Nudes (Philadelphia Museum of Art, The Louise and Walter Arensberg Collection) completed in May 1912. Speaking to Cabanne of Sad Young Man on a Train, Duchamp described the procedure used in these works: "Then, there is the distortion of the young man - I had called this elementary parallelism. It was a formal decomposition; that is, linear elements following each other like parallels and distorting the object. The object is completely stretched out, as if elastic. The lines follow each other in parallels, while changing subtly to form the movement, or the form of the young man in question. I also used this procedure in the Nude Descending a Staircase."
Duchamp at times used the term demultiplication to describe this process as well, as in his discussion of the Portrait of Chess Players quoted above. However, in the end "elementary" or "elemental parallelism" was to be a more useful approach. In its purest geometrical sense elemental parallelism would figure in some of the more complex theorizing in the notes for the Large Glass. One such note provides a clearer statement of Duchamp's interpretation of the term: "Elemental parallelism: repetition of a line equivalent to an elemental line (in the sense of similar at any point) in order to generate the surface. Same parallelism when passing from plane to volume: Sort of parallel multiplication of the n-dim'l continuum to form the n + 1 dim'l continuum."
With an "element" understood in mathematics as, for example, every position of a straight line as it moves to generate a surface, Duchamp's talk of "elemental parallelism" comes close to Hinton's method of generating a hypercube by moving a cube through space in a new fourth direction. That Duchamp in a conversation with Cabanne also connected elemental parallelism with Marey confirms that he and Kupka were thinking of their recording of successive phases of movement as a means to create higher dimensional "virtual volumes." As Duchamp had explained earlier in the same conversation, "The movement of form in time inevitably ushered us into geometry and mathematics."
Both Kupka and Duchamp, however, became dissatisfied with this approach and turned to depictions of complex space in order to evoke the fourth dimension. During July and August 1912 Duchamp traveled to Munich, where he made his first studies for The Bride Stripped Bare by Her Bachelors, Even and painted The Passage from Virgin to Bride (The Museum of Modem Art, New York). In Munich he also made the first of the large group of notes for the Large Glass, which he would later publish in the collections, the Box of 1914, the Green Box (1934)' and A l'infinitif (1966). When Duchamp returned from Munich, he painted The Bride in August 1912, where he later claimed he "first glimpsed the fourth dimension in his work." This is a surprising statement, considering the theoretical connections with the fourth dimension that lay behind the Portrait of Chess Players and the 1911-1912 motion studies. Although there is a dark and elusive depth within The Bride, it seems more probable that its four-dimensional association for Duchamp in later years was simply an extension of his subsequent theorizing on the four-dimensional Bride of the Large Glass.
Marcel Duchamp, The Bride, 1912.
Before his departure for Munich, Duchamp had been a regular participant in the discussions of the Puteaux artists and had formed his friendship with Princet. However, Duchamp's relationship with Gleizes and the other more orthodox Cubists cooled, at least temporarily, after Gleizes and others requested that Duchamp remove his Nude Descending a Staircase, No.2 from the 1912 Salon des Independants. Even if it attempted to evoke the fourth dimension, Duchamp's painting of motion must have made the elder Cubists worry that their style would be confused with that of the rival Italian Futurists. After Duchamp's return from Munich, he spent more time with Picabia and Apollinaire than with his companions of the earlier part of the year, except, perhaps, for Princet. Nevertheless, he did participate in the crowning achievement of the Puteaux group, the 'Salon de La Section d'Or' of 10-20 October 1912, and on this occasion his fellow artists were willing to include the Nude Descending a Staircase No.2.
Duchamp's post-Munich rethinking of dimensions and perspective was accompanied by a reevaluation of the content and technique of painting, which led him to give up conventional oil painting altogether toward the end of 1912. Reacting against an art that he believed had become purely "retinal," Duchamp wished to put painting once again "at the service of the mind." By November 1912 he had taken a job as a librarian at the Bibliotheque Sainte-Genevieve in order to support himself. But the atmosphere of a library and the ready availability of such a vast amount of information cannot have been unwelcome for one who was determined to "intellectualize" painting.
It is difficult to estimate the rate of Duchamp's progress in the new geometries during the course of 1912, until the notes for the Large Glass begin. Even then, dating remains a major problem. Undoubtedly, throughout 1912 Princet continued to coach Duchamp and to suggest sources for his study. From the A l'infinitif notes, it is also clear that Duchamp did take advantage of the resources at Sainte-Genevieve. His enthusiasm for his new project is reflected in a 1913 letter from Gertrude Stein to Mabel Dodge, in which Stein described meeting the young Duchamp, who talked "very urgently about the fourth dimension."
The Large Glass
The Bride Stripped Bare by Her Bachelors, Even (The Large Glass) is customarily dated 1915,1923, for although Duchamp prepared a large number of notes and studies for it before 1915, actual work on the glass panels did not begin until after his move to New York in 1915. The Large Glass is a masterful statement of the conceptual approach to art that Duchamp had adopted at the end of 1912: in it both technique and content have been revolutionized. The sense of the artist's hand has been replaced, as Duchamp wished, by the look of machine fabrication in a variety of materials on glass. Not only is the style of the Large Glass machine-oriented; the subject matter of this work is set forth in creaturely machine images.
Marcel Duhamp; The Bride Stripped Bare by Her Bachelors, Even (The Large Glass), 1915-23, Oil, varnish, lead foil, lead wire, and dust on two glass panels, 277.5×175.9 cm.
The notes that were published as the Green Box in 1934 have been essential for art historians in deciphering the iconography of The Bride Stripped Bare by Her Bachelors, Even. With the help of these notes and Duchamp's preparatory studies, the title is now understood to describe a lovemaking machine made up of a "Bride" above and a complex "Bachelor Apparatus" below. There is a clear separation between upper and lower sections, however, and Duchamp's statement is ultimately a pessimistic one about the futility of all attempts at meaningful human contact. Sexual fulfillment is replaced by "Slow Life-Vicious Circle-Onanism," in what is for Duchamp the archetypal love relationship.
Unlike the notes of the Green Box, which relate primarily to the iconography of the Large Glass, A l'infinitif records the evolution of the theory behind the forms of the Large Glass. There are a number of extraneous notes in the initial sections, as well as a section on "Color," but the notes collected under "Appearance and Apparition," "Perspective," and "The Continuum" are devoted to the questions of perspective and the representation of higher dimensions. With few exceptions the geometry of the A l'infinitif notes is n-dimensional geometry. Although Poincare, the major proponent of non-Euclidean geometry in France during the early years of the twentieth century, was a major source for Duchamp, it is Poincare's work on higher dimensions that dominates the A l'infinitif notes.
Nevertheless, in connection with his preparation for the Large Glass Duchamp did create what is surely the purest expression of non-Euclidean geometry in early twentieth-century art: the Three Standard Stoppages of 1913-1914. Notes referring to the Stoppages appear in the Box of 1914 and the Green Box, but Duchamp nowhere mentions non-Euclidean geometry, which was the source of his idea for the Stoppages. The four notes read as follows:
3 Standard Stops =
The Idea of the Fabrication
-If a straight horizontal thread one meter long falls from a height of one meter straight on to a horizontal plane twisting as it pleases and creates a new image of the measure of length.-
-3 patterns obtained in more or less similar conditions: considered in their relation to one another they are an approximate reconstitution of the measure of length.
The 3 standard stoppages are the meter diminished.
To create the Three Standard Stoppages Duchamp performed the action described in the "Idea of the Fabrication" note above three pieces of painted canvas. The fallen threads were fastened to the canvases with varnish, and the canvases were finally glued onto strips of plate glass. Wooden templates were also made as "meter sticks" of the new measure, and Duchamp then used each template three times to determine the network of tubes connecting the "Nine Malic Molds" of the Large Glass.
Marcel Duhamp; Three Standard Stoppages, 1913-14.
Beneath the playfulness of Duchamp's new physics is a demonstration of the basic principle of a non-Euclidean geometry that rejects Euclid's assumption of the inde-formability of figures in movement. Unlike the non-Euclidean geometries associated with the names of Lobachevsky and Riemann, which discarded Euclid's parallel postulate, this alternative to Euclidean geometry involves figures that readily change their shape when they are moved about. This type of non-Euclidean geometry had grown out of nineteenth-century studies of congruence by mathematicians such as Riemann and Helmholtz and is most easily visualized on surfaces of irregular curvature. In Duchamp's Stoppages, it is simply the movement of a line (the thread) from one area of space to another, which illustrates that geometrical figures do not necessarily retain their shape when moved about, as Euclid and geometers for two thousand years after him had assumed they would.
It was this same association of non-Euclidean geometry with deformability that had made this type of geometry so attractive to Metzinger and Gleizes in Du Cubisme, and Duchamp undoubtedly received his first introduction to these ideas at the Puteaux gatherings. Despite all their talk in Du Cubisme, however, neither Metzinger nor Gleizes ever produced a work as overtly non-Euclidean as the Three Standard Stoppages. Although Metzinger may have consciously incorporated curvature into certain of his paintings, he and Gleizes looked to non-Euclidean geometry primarily as a theoretical justification for Cubist distortion and their conventionalist view of art history.
The non-Euclidean Three Standard Stoppages established the forms of the tubes connecting the Nine Malic Molds (the Bachelors of the Large Glass), but Duchamp's overriding theoretical concern in the Large Glass was the question of dimensions, the interrelation of the second, third, and fourth dimensions. Early in his initial planning for the Large Glass, Duchamp had determined that there should be a fundamental difference between the upper and lower sections: the Bride would represent a four, dimensional Bride, while the three-dimensionality of the Bachelors would be emphasized by a strict application of the canons of Renaissance perspective.
Although his continuing interest in the fourth dimension was a tribute to the influence of his Cubist friends in the Puteaux group, the idea of reintroducing perspective went directly against a basic tenet of Cubism. From its beginnings in the art of Picasso and Braque, Cubism had sought deliberately to avoid all traces of traditional perspective, "that miserable tricky perspective," "that fourth dimension in reverse," "that infallible device for making all things shrink," as Apollinaire had described it in 1913.
Duchamp's reevaluation of painting in terms of the relationship of dimensions was much more fundamental than Cubism's recent rejection of one-point perspective in favor of a new kind of pictorial space. His concern was "scientific" and, if the passage from two to three dimensions was to be studied, a scientific method for depicting three dimensions would best serve his purposes. As he later recalled the situation in his interviews with Cabanne,
Duchamp: The "Large Glass" constitutes a rehabilitation of perspective, which had then been completely ignored and disparaged. For me perspective became absolutely scientific.
Cabanne: It was no longer realistic perspective.
Duchamp: No. It's a mathematical, scientific perspective.
Cabanne: Was it based on calculations?
Duchamp: Yes, and on dimensions. These were the important elements.
Duchamp also explained to Cabanne his final solution for incorporating higher dimensions in the Large Glass:
Since I found that one could make a cast shadow from a three-dimensional thing, any object whatsoever-just as the projecting of the sun on the earth makes two dimensions-I thought that, by simple intellectual analogy, the fourth dimension could project an object of three dimensions, or, to put it another way, any three-dimensional object, which we see dispassionately, is a projection of something four-dimensional, something we're not familiar with.
It was a bit of sophism, but still it was possible. "The Bride" in the "Large Glass" was based on this, as if it were the projection of a four-dimensional object.
Even if Duchamp did "glimpse" the fourth dimension in his August 1912 painting The Bride, as he later remarked, the lengthy process chronicled in the A l'infinitif notes would be required in order to provide a theoretical basis for the Bride's four-dimensionality.
From the beginning, however, Duchamp may have associated the term epanouissement, which he used for the "blossoming" of the Bride, with the fourth dimension. Later in life his advocacy of eroticism as a way beyond the "isms" of art often led Duchamp to incorporate sex in his explanations of the fourth dimension, which by that time were less strictly geometrical. Nevertheless, in the early years of the century geometry and sex had been blended in the word epanouissement, which was frequently used in discussions of higher dimensions. In addition to Jouffret's use of the term in an explanation of higher dimensional geometrical figures, Pierre Valin had employed it in the concluding installment of his 1909 La Phalange article, "L'Evolution de la philosophie du 19e au 20e siecle." There Valin writes, "It is necessary to recognize nevertheless that there are absolute mathematical reasons for demonstrating that the syntheses created by the fourth dimension are realized by expansion [epanouissement]; that, in the higher dimensions, beings and influences will be of a much more subtle fluidity, and surroundings, of a capacity more and more radiant."
In the Large Glass Duchamp emphasized just such a distinction between the "fluidity" of the Bride's Domain and the strictly measured three-dimensional perspective of the Bachelor Apparatus. The visual representation of the "blossoming" is a hazy, cloudlike "milky way." This quality of the upper portion of the glass is further described and contrasted to the Bachelor Apparatus in two nearly identical notes in the Green Box and A l'infinitif. Unlike the "mensurated" forms of the Bachelor Apparatus ("rectangle, circle, square"), those of the Bride are ideal and thus beyond specific measurements. Of the parabolas and hyperbolas in the Bride, Duchamp writes, "In the Bride-the principal forms will be more or less large or small, no longer have mensurability in relation to their destination: a sphere in the Bride will have any radius (the radius given to represent it is 'fictitious and dotted').
Once the relevant notes of A l'infinitif have been explored, the essential difference between the Bride and the Bachelor Apparatus will become even clearer. If the Bride's epanouissement was somehow connected with the fourth dimension in Duchamp's mind, that was not enough to make her four-dimensional. Her final form was to be the result of a long process of geometrical investigation, which can be retraced, at least in part, in the A l'infinitif notes for the Large Glass.
(Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Princeton: Princeton University Press, 1983), Chapter 3, "Marcel Duchamp and the New Geometries," pp. 126-135.)
Marcel Duchamp and the New Geometries (Part 3)
Marcel Duchamp and the New Geometries (Part 4)
Marcel Duchamp and the New Geometries (Part 5)
 Duchamp, "A Propos of Myself," unpublished lecture delivered at the City Art Museum of St. Louis, Missouri, 24 Nov. 1964. For this quotation, see d'Harnoncourt and McShine, eds., Marcel Duchamp, p. 254.
 Golding, Cubism, p. 164.
 Duchamp, as quoted in Sweeney, "Eleven Europeans in America," p. 20.
 One earlier work of 1911, Portrait (Dulcinea) (Philadelphia Museum of Art, The Louise and Walter Arensberg Collection), depicts five discrete positions of a walking woman (in progressive stages of undress). Portrait was completed in time for the Salon d'Automne of 1911 and may well represent Duchamp's initial response to the Cubist work of salle 41 in the spring of 1911, or even to Kupka's motion studies. Dee has suggested that another work of 1911, Yvonne and Magdeleine Tom in Tatters (Philadelphia Museum of Art, The Louise and Walter Arensberg Collection), may have been influenced by Hinton and hyperspace philosophy's suggestion that the passage of time is simply our lower dimensional reading of four-dimensional reality ("Ce Façonnement symetrique," p. 364). If these works were indeed motivated by interest in the fourth dimension, it is most likely that La Fresnaye or another artist brought word of the Mercereau circle Cubists' explorations to Puteaux before the 1911 Salon d'Automne. See again Chapter 2, n. 180, and the related text for the problems with Dee's earlier date for Kupka's artistic interest in the fourth dimension.
 Duchamp, as quoted in Cabanne, Dialogues, p. 29. Duchamp discusses elementary parallelism in connection with Marey in ibid., p. 34.
 Duchamp, A l'infinitif, in Salt Seller, p. 92.
 Dee has noted Jouffret's use of the term "parallelism" in discussing this generational process ("Ce Façonnement symetrique," p. 7).
 Duchamp, in Cabanne, Dialogues, pp. 34, 31. Ulf Linde, in his essay "La Roue de bicyclette," in L'Oeuvre de Marcel Duchamp (III, 36·37), provides a good discussion of "demultiplication" as a means to indicate an n + 1 dimensional unity, beyond n-dimensional multiplicity. Linde also quotes from a chapter of Pawlowski's Voyage au pays de la quatrieme dimension, "La Diligence innombrable," which Clair had cited in Marcel Duchamp ou le grand fictif (pp. 103-5). As an illustration of the magic of the fourth dimension, Pawlowski had given the example of a stagecoach route on which the coach was replaced by a modem automobile. If the speed of the automobile were increased to infinity, Pawlowski reasoned, the same auto might then be at all points of the route at the same time. Impossible in the world of three dimensions, this "simultaneity" could easily be realized in the fourth dimension.
Although Linde's use of the passage in comparison with Duchamp's 1913 Bicycle Wheel is acceptable, problems of chronology make it inapplicable for Duchamp's studies of motion during 1911-1912. "La Diligence innombrable" was not among the chapters of Voyage au pays de la quatrieme dimension serialized in Comoedia between February and December 1912. Thus, Duchamp would not have seen Pawlowski's text until it was published in book form in December of that year, long after he had completed the series The King and Queen Surrounded by Swift Nudes (Mar.-May 1912) to which Clair compares the passage.
 Duchamp, 1956 interview, as paraphrased in Lawrence D. Steefel, "The Position of La Mariee mise d nu par ses celibataires, meme (1915-1923) in the Stylistic and Iconographic Development of the Art of Marcel Duchamp" (Ph.D. dissertation, Princeton University, 1960), p. 110. See also Steefel's article, "The Art of Marcel Duchamp: Dimension and Development in Le Passage de la Vierge a la Mariee," Art Journal, XXII (Winter 1962-63), 78.
 In Marcel Duchamp ou Ie grand fictif (pp. 113-14) Clair compares the Bride to a passage from Pawlowski's Voyage au pays de la quatrieme dimension about the "grave disorders" that came about in the first stages of the introduction of the fourth dimension: "Without apparent wound or visible opening, certain organs found themselves transported outside of the body and, under the natural pressure of the muscles, grouped themselves in an indescribable mass, avoiding every known rule and defying established anatomy." However, this chapter of Pawlowski's novel, "Au-dela des forces nature lies," did not appear in Comoedia until 11 Nov. 1912, three months after Duchamp painted the first version of the Bride in August. Clair's comparison is cited numerous times in essays in L'Oeuvre de Marcel Duchamp, including Clair's own otherwise excellent essay, "Marcel Duchamp et la tradition des perspecteurs" (III, 144). There, he also speaks of a four-dimensional being's ability to see inside a three-dimensional body. In contrast to the notion of a body turning inside out, the idea of seeing through an object has more of a historical basis in the popular tradition of "the fourth dimension," which often compared four-dimensional vision to a three-dimensional being's ability to see the interior of a square in Flatland.
 In October 1912 Duchamp traveled by car with Apollinaire, Picabia, and Gabrielle Buffet to the Jura mountains; see Camfield, Picabia: His Art, p. 34. According to Gil Bias, 2 Sept. 1912, p. 4, Duchamp was planning to submit a work entitled Section d'Or to the Salon d'Automne of 1912. For Duchamp's ideas on the Golden Section, see Marcel Duchamp: Notes, particularly the notes numbered 70, 84, and 183.
 Duchamp, as quoted in Sweeney, "Eleven Europeans in America," p. 20. On this decision, see Lebel, Marcel Duchamp, p. 27, and Schwarz, Complete Works, pp. 19-21.
 Jennifer Gough-Cooper and Jacques Caumont, "Plan pour ecrire une vie de Marcel Duchamp," in L'Oeuvre de Marcel Duchamp, I, 15.
 Undated Stein letter to Dodge, Mabel Dodge Luhan Archive, Yale University. See Chapter 4, n. 141, for the dating of this letter.
 The Large Glass was abandoned in 1923 in as "finished" a state as Duchamp wished. The cracks in the glass date from 1927, when it was shattered in transit from an exhibition at the Brooklyn Museum. For a description of the damage to the glass and Duchamp's repair of it in 1936, see Schwarz, Complete Works, pp. 148-49. The two metal bars which now divide the upper and lower panels were added at the time of its repair. Originally, the horizon line was augmented by three "narrow plates of mirrored glass." See Jean Suquet, Le Gueridon et la virgule (Paris: Christian Bourgois, 1976), p. 58; also, pp. 9-82 on the "Handler of Gravity," who was to have bridged the two areas of the glass.
 Duchamp, Green Box, in Salt Seller, p. 56.
 The most obvious exception to this statement is a drawing labeled "Pseudo sphere (Projections from the center)" included in the "Perspective" section of A l'infinitif. The drawing and its mistaken label are discussed below. Adcock argues that other of the notes in A l'infinitif may be related to non-Euclidean geometry ("Marcel Duchamp's Notes," ch. 4).
 Duchamp, Green Box, in Salt Seiler, p. 33. Duchamp's description of the Three Standard Stoppages as "canned chance" and his general fascination with the idea of chance suggests another reason Poincare would have appealed to the artist. Poincare lectured on probability theory at the Universite de Paris and published his Calcul des probabilites in 1896, with a second edition appearing in 1912. Jouffret had also published a book on this subject, Sur la probabilite du tir des bouches a feu et la methode des moindre carres of 1875.
 Duchamp, Box of 1914, in Salt Seller, p. 22.
 Duchamp was also to employ the Three Standard Stoppages in Tu m’ of 1918, his last oil painting on canvas.
 Linde has suggested that the three "Draft Pistons" within the "Top Inscription" or "Milky Way" representing the Bride's "cinematic blossoming" may allude to non-Euclidean geometry. Duchamp determined their forms by photographing small squares of cloth as they were "deformed" by a breeze. See Linde, "Piston de courant d'air," in L'Oeuvre de Marcel Duchamp, II, 80.
 Apollinaire, Les Peintres Cubistes, p. 68; The Cubist Painters, p. 45.
 Cabanne, Dialogues, p. 38.
 Duchamp, as quoted in Cabanne, Dialogues, p. 40. Duchamp had explained the Bride in this way as early as 1954 in a letter to Andre Breton (4 Oct. 1954), published in Medium, no. 4 (Jan. 1955), p. 33.
 See Green Box, in Salt Seller, pp. 42-44.
 On Duchamp and eroticism, see Cabanne, Dialogues, p. 88. Lebel in Marcel Duchamp writes that Duchamp "considered the sexual act the pre-eminent fourth-dimensional situation" (p. 28). This erotic connection has been analyzed by Steefel in "The Position of La Mariee mise Ii nu par ses celibataires, meme," pp. 311-12, and in "The Art of Marcel Duchamp: Dimension and Development in Le Passage de la Vierge Ii la Mariee," pp. 72-79. See also Suquet, Le Gueridon et la virgule, pp. 63-64. Suquet quotes Pawlowski on the subject of love and the fourth dimension, but it must be emphasized that Pawlowski's idealist vision of human love as the key to a future four-dimensional era is far different from Duchamp's cynical caricaturing of sex. See Pawlowski, Voyage au pays de la quatrieme dimension (1912), ch. 45.
Countering Lebel, Schwarz argues against the association of Duchamp's fourth dimension with sex as too -"metaphorical." However, the Duchamp statement he cites is hardly clearcut: "I would not say that sex is the fourth dimension; far from it, I would never say that. Sex is three-dimensional as well as four-dimensional. There is, however, an expression beyond sex which can be transferred into a fourth dimension. But the fourth dimension is not sex as such. Sex is only an attribute, which can be transferred into a fourth dimension, but it is not the definition or the status of the fourth dimension. Sex is sex." See Schwarz, Complete Works, p. 36, n. 4.
 On Jouffret's use of epanouissement (Traite elementaire, pp. 62-63), see Adcock, "Marcel Duchamp's Notes," pp. 295-96.
 See again Valin, "L'Evolution de la philosophie du 1ge au 20e siecle," p. 401.
 See Green Box, in Salt Seller, p. 36. The first chapter of Book 4 of Poincare's Science et Methode of 1908 is entitled "The Milky Way and the Theory of Gases." In it Poincare discusses the greater likeness of the Milky Way to Crookes's "fourth state of matter," radiant matter, than to gaseous matter. Because Crookes's "fourth state of matter" was at times connected with the fourth dimension by its popularizers, the Milky Way may have possessed an additional four-dimensional quality for Duchamp.
 Duchamp, A l'infinitif, in Salt Seller, p. 83. See also Green Box, in Salt Seller, pp. 44-45.