Duchamprsquos Perspective TheIntersection of Art andGeometry

click to enlarge

Figure 1Marcel Duchamp Three Standard Stoppages 1913

Marcel Duchamprsquos readymade but ldquonot quiterdquo as he called theThree Standard Stoppages(Fig 1) is a highly ramified work of

art(1)The pieces of string used in its construction are relatedto sight lines and to vanishing points In addition to theirostensive references to perspective and projective geometrythe Stoppages allude to happenstance They are perhaps theartistrsquos best known work that incorporates uncertain outcomesinto its operation (In one of his Green Box notes Duchamp

says that the Stoppages are ldquocanned chancerdquo)(2) To make thework he glued three pieces of string to three narrow canvasespainted solid Prussian blue (Each string had a differentrandomly generated curvature) He then cut three wooden

templates to match the shapes of these ldquodiminished metersrdquo(3)

click still images to enlarge

Figure 2Marcel Duchamp Network of Stoppages 1914

Figure 3Marcel Duchamp Bride Stripped Bareby Her Bachelors Even 1915-23

As this description indicates the piece was quite unusualphysically and it was conceptually unprecedented In terms ofhis personal development Duchamp said the work had beencrucial ldquohellip it opened the wayndashthe way to escape from thosetraditional methods of expression long associated with arthellipFor me the Three Standard Stoppages was a first gesture

liberating me from the pastrdquo(4)

Duchamp used the Stoppages to design the pattern of lines inhis painting Network of Stoppages (Fig 2) and then afterrendering this plan view in perspective transferred it to TheBride Stripped Bare by Her Bachelors Even (Fig 3) In theLarge Glass as the Bride Stripped Bare is also knownthe ldquonetworkrdquo comprises the ldquocapillary tubesrdquo iconographical

elements that connect the ldquonine malic moldsrdquo(5) The ThreeStandard Stoppages the Network of Stoppages and the LargeGlass are associated with one another through geometrical

projection and section Duchamprsquos approach with respect toestablishing their mutual relationships is complex He notonly redrew the Networkof Stoppages in perspective so that he could incorporate thescheme into the imagery of the Glass he also recast physicalcounterparts of the Stoppages into the actual structure of theGlass thethree plates used in the Three Standard Stoppages areconceptually related to the three narrow sections of glassused to construct the ldquogarmentsrdquo of the Bride (Fig 4) Ineach work two plates

are in green glass and one is in white glass(6) The strips ofglass at the horizon line of the Large Glass are seen edge-onan arrangement comparable to looking down into the box of theThreeStandard Stoppages with the sheets of glass inserted intotheir slots To my knowledge this relationship was firstpointed out by Ulf Linde

The Bridersquos Clothes are to be found on the horizonndashthe linethat governs the Bachelor Apparatusrsquo perspective and which isin the far distance Thus the Clothes seem to be the sourceof the waterfall Moreover the Clothes are undoubtedly thehiding-placeof the Standard Stoppages as well For this part as it isexecuted on the Glass looks exactly like the glass plates asthey appear set in the croquet casendashas if the Clothes simplyrepeated the three glass plates in profile One might say that

it is the three threads that set the Chariot in motion(7)

click still images to enlarge

Figure 4Marcel Duchamp Bride StrippedBare by Her Bachelors Even (DetailThe ldquogarmentsrdquoof the Bride) 1915-23

Figure 5Marcel DuchampChocolateGrinder No 2 1914

Although some of what Linde says here is unclear at least tome it is nonetheless suggestive especially his propositionthat the Stoppages are hidden in the Bridersquos clothingDuchamprsquos use of different colored glass in just the same wayin both applications (and the colors are more apparent whenthe glass plates are seen edge-on) indicates that he somehowmeant for the Stoppages and the Bridersquos ldquogarmentsrdquo to belinked together I believe that their most importantaffiliation is perspectival the vanishing point at thehorizon line of the Glass is tied to the ldquogarmentsrdquo throughgeometry

In a note from the Box of 1914 that was subsequentlyrepublished in the Green Box Duchamp explains that pieces ofstring one meter long were to be dropped from a height of one

Figure 4Marcel Duchamp Bride StrippedBare by Her Bachelors Even (DetailThe ldquogarmentsrdquoof the Bride) 1915-23

Figure 5Marcel DuchampChocolateGrinder No 2 1914

Although some of what Linde says here is unclear at least tome it is nonetheless suggestive especially his propositionthat the Stoppages are hidden in the Bridersquos clothingDuchamprsquos use of different colored glass in just the same wayin both applications (and the colors are more apparent whenthe glass plates are seen edge-on) indicates that he somehowmeant for the Stoppages and the Bridersquos ldquogarmentsrdquo to belinked together I believe that their most importantaffiliation is perspectival the vanishing point at thehorizon line of the Glass is tied to the ldquogarmentsrdquo throughgeometry

In a note from the Box of 1914 that was subsequentlyrepublished in the Green Box Duchamp explains that pieces ofstring one meter long were to be dropped from a height of one

meter twisting ldquoas they pleasedrdquo during their fall Thechance-generated curvatures would create ldquonew

configurations of the unit of lengthrdquo(8) Although we do notknow exactly how he constructed the work we do know that healmost certainly did not use this method The ends of thepieces of string in the Stoppages are sewn through thesurfaces of the canvases and are attached to them from

behind(9) Presumably Duchamp sewed down the strings leavingthem somewhat loose jiggled and jostled them back and forthuntil he obtained three interesting curves and then glued thesegments to the canvases using varnish Sewing would not havebeen out of keeping with his general working methodsespecially since he was also at this time (1914) sewing threadto his painting Chocolate Grinder No 2 (Fig 5)

Duchamp wanted to relate his various works to each other Themoving segments of thread in the Three Standard Stoppages areconceptually similar to the moving lines and shapes in hiscubo-futurist paintings They are also conceptually similar tothe parallel lines on the drums of the ldquochocolate grinderrdquowhich can in their turn also be related to thechronophotographic sources of the earlier paintingsChronophotography was among Duchamprsquos primary interests during

this period(10) What I have in mind here can be seen bycomparing Duchamprsquos works with Eacutetienne-Jules Mareyrsquos images ofmoving lines Figs 6 and 7) These kinds of time-exposurephotographs not only recall such paintings as Sad Young Man ona Train (Fig 8) and Nude Descending a Staircase No 2 (Fig9) but also the Three Standard Stoppages and Chocolate

Grinder No 2(11)

click images to enlarge

Figure 6Eacutetienne-JulesMarey Image of moving linesFigure 7Eacutetienne-JulesMarey Image of moving linesFigure 8Marcel DuchampSad Young Manon a Train 1911

Figure 6Eacutetienne-JulesMarey Image of moving linesFigure 7Eacutetienne-JulesMarey Image of moving linesFigure 8Marcel DuchampSad Young Manon a Train 1911

Figure 9Marcel DuchampNude Descendinga Staircase No 2 1912PAGES

In addition to implying something being stopped the wordldquostoppagerdquo also suggests something being mended or repairedIn French ldquostoppagerdquo 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 sculptureand the network of lines in the painting can be compared withthe breaks in the Large Glass In his early monograph RobertLebel pointed out that the Network of Stoppages bears astrange resemblance to the pattern of fissures in the Glassas if the painting had somehow been a preliminary study for

the subsequent breakage(13) When Duchamp put the Glass backtogether or perhaps we could also say when he ldquorewoverdquo it heno doubt also noticed the fortuitous similarities The shapesof the line segments generated by the pieces of thread wererandom but they seemed planned Likewise the line segmentscaused by the Glass being smashed were determined by chancebut they also seemed necessary for its completion (or

definitive incompletion)(14)

When Duchamp rebuilt the work he was ldquostoppingrdquo an accidentalevent that had somehow made the Glass ldquoa hundred times

betterrdquo(15) The mended cracks in the glass are not whollyinvisible but they do approach a point of disappearancendashlikepieces of string falling away toward some mysterious knot at

infinity Duchamprsquos lines his fractures and strandsintersect at a vanishing point in the fourth dimension arealm that cannot be seen from our ordinary perspectives

The Bridersquos ldquogarmentsrdquo and the Three Standard Stoppages canalso be discussed in terms of yet another kind of ldquostoppagerdquoGlass as a physical substance is an insulator and as suchis oftenused to arrest or impede the flow of electrical currentthrough circuits Duchamp may very well have been thinking ofhis glass plates in these kinds of terms when he was

constructing the Large Glass (16) He also refers to the Bridersquosclothing as a ldquocoolerrdquo

(Develop the desire motor consequence of the lubriciousgearing) This desire motor is the last part of the bachelormachine Far from being in direct contact with the Bride thedesire motor is separated by an air cooler (or water) Thiscooler (graphically) to express the fact that the brideinstead of being merely an asensual icicle warmly rejects(not chastely) the bachelorsrsquo brusque offer This cooler willbe in transparent glass Several plates of glass one above theother In spite of this cooler there is no discontinuitybetween the bachelor machine and the Bride But theconnections will be electrical and will thus express thestripping an alternating process Short

circuit if necessary(17)

In addition to the terms ldquovecirctements de la marieacuteerdquo andldquorefroidisseurrdquo Duchamp uses the expression ldquoplaquesisolatricesrdquo to describe his strips of glass (18)

This phrase can be translated as ldquoisolating platesrdquo orldquoinsulating platesrdquo In one of his posthumously publishednotes he calls the horizontal division of the Glass a ldquograndisolateurrdquoa ldquolarge insulatorrdquo and explains that it should be made using

ldquothree planes five centimeters apart in transparent material(sort of thick glass) to insulate the Hanged [Pendu] from the

bachelor machinerdquo(19)

click to enlarge

Figure 10Marcel DuchampDraft Pistons 1914

Figure 11Marcel DuchampTravelorrsquos Folding Item 1916

Figure 12Photograph ofthe unbroken Large Glass

ldquothree planes five centimeters apart in transparent material(sort of thick glass) to insulate the Hanged [Pendu] from the

bachelor machinerdquo(19)

click to enlarge

Figure 10Marcel DuchampDraft Pistons 1914

Figure 11Marcel DuchampTravelorrsquos Folding Item 1916

Figure 12Photograph ofthe unbroken Large Glass

Glass may play a similar exclusionary role in the workings ofthe Three Standard Stoppages but in ways that are perhapsless ldquotransparentrdquo While Duchamp was apparently interested inexploring a frustrated relationship between the Bride and theBachelors involving as it does a ldquoshort circuitrdquo he was alsotrying to ldquodelayrdquo communication Whatever talking occurs orfails to occur betweenthe separated Bride and Bachelors pertains to seeing or notseeing through words In his notes Duchamp explains that theBride sends her commands to the Bachelors through the ldquodraftpistonsrdquoldquotriple ciphersrdquo that use a formal alphabet constructed usingthe Three Standard Stoppages Because the chance-determinedldquodraft pistonsrdquo (Fig 10) which are deformed planes areconceptually similar to the Stoppages which are deformedlines these interpretations again converge geometrically Itmight also be pointed out that Duchamprsquos readymade TravelerrsquosFolding Item (Fig 11) can be taken as a next logical step inthis sequence a one-dimensionalline generating a two-dimensional surface which in its turn

generates a three-dimensional ldquosolidrdquondashone that can fold up(20)

By looking somewhat further into the n-dimensionalimplicationsof these works (from the Latin implicatio an entwining orinterweaving) we may be able to ascertain how Duchamprsquosarrangements his strings and fabrics which seem to havetopological insinuations might actually operate Just how dothe Three Standard Stoppages disappear into the Bridersquosclothing

At some later point in the construction of Three StandardStoppages Duchamp cut the narrow strips of canvas from theirstretchers reducing them in size in the process and thenglued them down to thick pieces of plate glass He probablycarried out this reworking when he was repairingthe Large Glass at Katherine S Dreierrsquos home in Connecticut

during the spring and summer of 1936(21) Also at this time heprobably decided to put the various components of the ThreeStandard Stoppages into a specially constructed wooden casethat resembles a croquet box Duchamprsquos decision to amplifythe Stoppages along these lines was almost certainly connectedwith how he was repairing the ldquogarmentsrdquo of the Bride whichhad presumably been pulverized when the Glass was accidentallybroken in 1927 From the photograph of the unbroken LargeGlass taken at the Brooklyn Museum

(Fig 12)

it is difficult to determine how the original ldquogarmentsrdquo wereconstructed but they do not appear to have been as elaborateas the repaired strips of glass As pointed out earlierDuchamp must have intended for the Stoppages and theldquogarmentsrdquo to be related to one another because he usedsimilarly colored strips of glass and parallel edge-onarrangements in their respective reconstructions

Did Duchamp somehow ldquobetrayrdquo his work by not actually droppingthe pieces of string when he originally made the ThreeStandard Stoppages or when over twenty years later hefurther modified his original conception of the piece No morethan he betrayed himself by learning to appreciate the breaksin the Large Glass or by elaborating the Bridersquos ldquogarmentsrdquowhen he repaired them Such operations are I believe

commensurate with his general attitudes about such matters(22)

Recall his statement to Katherine Kuh ldquothe idea of letting apiece of thread fall on a canvas was accidental but from thisaccident came a carefully planned work Most important wasaccepting and recognizing this accidental stimulation Many ofmy highly organized works were initially suggested by just

such chance encountersrdquo(23)

Dropping pieces of string was not a rule that Duchamp had tofollow 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 establishinghypotheses at the beginning of a research program but thenmodifying their hypotheses once work has been carried out inthe laboratory Over the course of time Duchamprsquos examples of

ldquohasard en conserverdquo (25)were supplied with controls that hadnot been deemed necessary in the beginning As with the chancebreakage he preserved in the Large Glass the important thingwas recognizing the accidental stimulation Moreover byallowing the pieces of thread to do more than simply fall uponthe canvas surfaces by actually sewing them through to theother side Duchamp could emphasize the notion that they hadintersected the canvases The encounter involved both chanceand mathematics

In works such as the Three Standard Stoppages Duchamp createsphysical analogues for the abstract concept of ldquointersectionrdquothe one-dimensional pieces of string the curved linesegments intersect the two-dimensional surfaces of thecanvases (and they literally share points in common where theyare sewn together) The strings are thus further implicated (Iam tempted to say intertwined) along geometrical lines withthe fabric of the canvas strips The cracks in the Glass arealso a fundamental part of it They are ldquoinsiderdquo the brokensheets of glass which are in their turn encased inside theheavy panes of glass that Duchamp used to effect their repairIn an analogous way the ends of the strings in the Stoppagesare sandwiched between the strips of canvas and the rectanglesof glass that back them

Duchamprsquos works on glass are flat but they are nonethelessrather thick They are ldquospacesrdquo that can be thought ofespecially in this context as rectangular solids Because thesheets of glass themselves have thickness a depth that is

often layered they can be taken as three-dimensional sectionsout of higher-dimensional continua When for example all theconfigurations of the Stoppages (the strings the templatesand the plates of glass) are considered together their n-dimensional implications are manifest They are one-dimensional two-dimensional and three-dimensional and theyhave n-dimensional possibilities Each configuration isrelated to the others through projection and intersection thelines can be taken as slices out of surfaces the surfaces asslices out of solids and the solids as slices out ofhypersolids Esprit Pascal Jouffret one of Duchamprsquos mostimportant mathematical sources characterized such cuts as

ldquoinfinitely thin layersrdquo (26)

Duchamprsquos approachndashmoving from lines to surfaces and fromspaces to hyperspacesndashis couched in terms of perspective Heconsiders how vanishing points and changing points of viewwould operate in 2-space 3-space 4-space or any given n-space He suggests using ldquotransparent glassrdquo and ldquomirrorrdquo asanalogues of four-dimensional perspective systems (analoguesbecause such systems cannot actually be constructed in three-dimensional space)(27)

Especially when the narrow sheets of glass are seen edge-on inthe slots in their croquet box they suggest their membershipin an infinite series (reflections in mirrors can also implyinfinite reiterations) In an interview with Pierre CabanneDuchamp emphasized the serial characteristics of theStoppages ldquoWhen yoursquove come to the word three you have threemillionndashitrsquos the same thing as three I had decided that thethings would be done three times to get what I wanted MyThree Standard Stoppages is produced by three separateexperiments and the form of each one is slightly different Ikeep the line and I have a deformed meterrdquo(28)

he specifics of how Duchamp kept his line and used his

deformed meter is worth exploring further He tells Cabannethat he had been interested in working on glass for severalreasons including the way color ldquois visible from the othersiderdquo Glass was also useful in laying out its variouselements ldquoperspective was very important The Large Glassconstitutes a rehabilitation of perspective which had beencompletely ignored and disparaged For me perspective became

absolutely scientificrdquo(29)

y using linear perspective in his design Duchamp couldarrange the Bachelorsrsquo domain in such a way that the vanishingpoint coincided with the horizontal division between the upperand lower panels of the Glass

From this perspective or from the point of view ofperspective Duchamprsquos saying that a ldquolabyrinthrdquo lies at theldquocentral part of the stripping-barerdquo is significant the Large

Glass and the Three Standard Stoppages are about occlusion(30)

They involve unusual station points and unusual distancepoints in a perspectival system that can only bereconstructed from isolated positions outside normal space IfDuchamp were thinking of his ldquostripsrdquo of glass as physicalpuns on the notion of ldquostrippingrdquo the Bride then their

structure is doubly suggestive(31) Because her clothing consistsof transparent sections of glass thatare entailed with a ldquopoint de fuiterdquo it can be takento include a complex set of folds not only in the cloth ofthe garments but also in the fabric of space Recall thatTravelerrsquos Folding Item is conceptually related to the ThreeStandard StoppagesAlso the typewriter cover has been called

the ldquoBridersquos Dressrdquo (32)Perhaps the disappearance of theStoppages their dropping away toward infinity at the positionof the Bridersquos garments can be taken as an interdimensionalfolding up a stripping bare thatrequires orthogonaltranslation into higher space

Perhaps the disappearance of the Stoppages their droppingaway toward infinity at the position of the Bridersquos garmentscan be taken as an interdimensional folding up a strippingbare that requires orthogonal translation into higher space

All of the works here under discussion are related to oneanother through perspectivalism (and also perspectivism) ForDuchamp the use of perspective as a system was not a matterof creating single fixed-point ways of looking at things Itwas on the contrary involved in dislodging viewers fromtheir ordinary ways of understanding And with this objectivein mind his choosing readymades during the same period he wasworking on the Stoppagescan be seen as a related activityWhen Duchamp made his remark about Three Standard Stoppagesbeing a readymade but ldquonot quiterdquo he continued by saying

ldquoitrsquos a readymade if you wish but a moving onerdquo(33)

The curving pieces of string and our shifting notions of themeaning of the readymades seem to trail off from a ldquovanishingpointrdquoat the horizon of our own thinking The readymadesrefuse to abideby our ordinary definitions of art and the Stoppagesallude to geometries that have challenged our traditionalepistemological structures(34)

Their curvatures can be taken as references to non-Euclideanor topological geometries complications that necessitate ourreconsidering our vanishing points The strings when taken asanalogues for lines of sight are transposed or rotated intoa hidden space

click to enlarge

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich are behind the surface and inside the three-dimensional

during the spring and summer of 1936(21) Also at this time heprobably decided to put the various components of the ThreeStandard Stoppages into a specially constructed wooden casethat resembles a croquet box Duchamprsquos decision to amplifythe Stoppages along these lines was almost certainly connectedwith how he was repairing the ldquogarmentsrdquo of the Bride whichhad presumably been pulverized when the Glass was accidentallybroken in 1927 From the photograph of the unbroken LargeGlass taken at the Brooklyn Museum

(Fig 12)

it is difficult to determine how the original ldquogarmentsrdquo wereconstructed but they do not appear to have been as elaborateas the repaired strips of glass As pointed out earlierDuchamp must have intended for the Stoppages and theldquogarmentsrdquo to be related to one another because he usedsimilarly colored strips of glass and parallel edge-onarrangements in their respective reconstructions

Did Duchamp somehow ldquobetrayrdquo his work by not actually droppingthe pieces of string when he originally made the ThreeStandard Stoppages or when over twenty years later hefurther modified his original conception of the piece No morethan he betrayed himself by learning to appreciate the breaksin the Large Glass or by elaborating the Bridersquos ldquogarmentsrdquowhen he repaired them Such operations are I believe

commensurate with his general attitudes about such matters(22)

Recall his statement to Katherine Kuh ldquothe idea of letting apiece of thread fall on a canvas was accidental but from thisaccident came a carefully planned work Most important wasaccepting and recognizing this accidental stimulation Many ofmy highly organized works were initially suggested by just

such chance encountersrdquo(23)

Dropping pieces of string was not a rule that Duchamp had tofollow 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 establishinghypotheses at the beginning of a research program but thenmodifying their hypotheses once work has been carried out inthe laboratory Over the course of time Duchamprsquos examples of

ldquohasard en conserverdquo (25)were supplied with controls that hadnot been deemed necessary in the beginning As with the chancebreakage he preserved in the Large Glass the important thingwas recognizing the accidental stimulation Moreover byallowing the pieces of thread to do more than simply fall uponthe canvas surfaces by actually sewing them through to theother side Duchamp could emphasize the notion that they hadintersected the canvases The encounter involved both chanceand mathematics

In works such as the Three Standard Stoppages Duchamp createsphysical analogues for the abstract concept of ldquointersectionrdquothe one-dimensional pieces of string the curved linesegments intersect the two-dimensional surfaces of thecanvases (and they literally share points in common where theyare sewn together) The strings are thus further implicated (Iam tempted to say intertwined) along geometrical lines withthe fabric of the canvas strips The cracks in the Glass arealso a fundamental part of it They are ldquoinsiderdquo the brokensheets of glass which are in their turn encased inside theheavy panes of glass that Duchamp used to effect their repairIn an analogous way the ends of the strings in the Stoppagesare sandwiched between the strips of canvas and the rectanglesof glass that back them

Duchamprsquos works on glass are flat but they are nonethelessrather thick They are ldquospacesrdquo that can be thought ofespecially in this context as rectangular solids Because thesheets of glass themselves have thickness a depth that is

often layered they can be taken as three-dimensional sectionsout of higher-dimensional continua When for example all theconfigurations of the Stoppages (the strings the templatesand the plates of glass) are considered together their n-dimensional implications are manifest They are one-dimensional two-dimensional and three-dimensional and theyhave n-dimensional possibilities Each configuration isrelated to the others through projection and intersection thelines can be taken as slices out of surfaces the surfaces asslices out of solids and the solids as slices out ofhypersolids Esprit Pascal Jouffret one of Duchamprsquos mostimportant mathematical sources characterized such cuts as

ldquoinfinitely thin layersrdquo (26)

Duchamprsquos approachndashmoving from lines to surfaces and fromspaces to hyperspacesndashis couched in terms of perspective Heconsiders how vanishing points and changing points of viewwould operate in 2-space 3-space 4-space or any given n-space He suggests using ldquotransparent glassrdquo and ldquomirrorrdquo asanalogues of four-dimensional perspective systems (analoguesbecause such systems cannot actually be constructed in three-dimensional space)(27)

Especially when the narrow sheets of glass are seen edge-on inthe slots in their croquet box they suggest their membershipin an infinite series (reflections in mirrors can also implyinfinite reiterations) In an interview with Pierre CabanneDuchamp emphasized the serial characteristics of theStoppages ldquoWhen yoursquove come to the word three you have threemillionndashitrsquos the same thing as three I had decided that thethings would be done three times to get what I wanted MyThree Standard Stoppages is produced by three separateexperiments and the form of each one is slightly different Ikeep the line and I have a deformed meterrdquo(28)

he specifics of how Duchamp kept his line and used his

deformed meter is worth exploring further He tells Cabannethat he had been interested in working on glass for severalreasons including the way color ldquois visible from the othersiderdquo Glass was also useful in laying out its variouselements ldquoperspective was very important The Large Glassconstitutes a rehabilitation of perspective which had beencompletely ignored and disparaged For me perspective became

absolutely scientificrdquo(29)

y using linear perspective in his design Duchamp couldarrange the Bachelorsrsquo domain in such a way that the vanishingpoint coincided with the horizontal division between the upperand lower panels of the Glass

From this perspective or from the point of view ofperspective Duchamprsquos saying that a ldquolabyrinthrdquo lies at theldquocentral part of the stripping-barerdquo is significant the Large

Glass and the Three Standard Stoppages are about occlusion(30)

They involve unusual station points and unusual distancepoints in a perspectival system that can only bereconstructed from isolated positions outside normal space IfDuchamp were thinking of his ldquostripsrdquo of glass as physicalpuns on the notion of ldquostrippingrdquo the Bride then their

structure is doubly suggestive(31) Because her clothing consistsof transparent sections of glass thatare entailed with a ldquopoint de fuiterdquo it can be takento include a complex set of folds not only in the cloth ofthe garments but also in the fabric of space Recall thatTravelerrsquos Folding Item is conceptually related to the ThreeStandard StoppagesAlso the typewriter cover has been called

the ldquoBridersquos Dressrdquo (32)Perhaps the disappearance of theStoppages their dropping away toward infinity at the positionof the Bridersquos garments can be taken as an interdimensionalfolding up a stripping bare thatrequires orthogonaltranslation into higher space

Perhaps the disappearance of the Stoppages their droppingaway toward infinity at the position of the Bridersquos garmentscan be taken as an interdimensional folding up a strippingbare that requires orthogonal translation into higher space

All of the works here under discussion are related to oneanother through perspectivalism (and also perspectivism) ForDuchamp the use of perspective as a system was not a matterof creating single fixed-point ways of looking at things Itwas on the contrary involved in dislodging viewers fromtheir ordinary ways of understanding And with this objectivein mind his choosing readymades during the same period he wasworking on the Stoppagescan be seen as a related activityWhen Duchamp made his remark about Three Standard Stoppagesbeing a readymade but ldquonot quiterdquo he continued by saying

ldquoitrsquos a readymade if you wish but a moving onerdquo(33)

The curving pieces of string and our shifting notions of themeaning of the readymades seem to trail off from a ldquovanishingpointrdquoat the horizon of our own thinking The readymadesrefuse to abideby our ordinary definitions of art and the Stoppagesallude to geometries that have challenged our traditionalepistemological structures(34)

Their curvatures can be taken as references to non-Euclideanor topological geometries complications that necessitate ourreconsidering our vanishing points The strings when taken asanalogues for lines of sight are transposed or rotated intoa hidden space

click to enlarge

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich are behind the surface and inside the three-dimensional

as the damage to the Glass wound up inspiring his

admiration(24)

His artistic approach was analogous to scientists establishinghypotheses at the beginning of a research program but thenmodifying their hypotheses once work has been carried out inthe laboratory Over the course of time Duchamprsquos examples of

ldquohasard en conserverdquo (25)were supplied with controls that hadnot been deemed necessary in the beginning As with the chancebreakage he preserved in the Large Glass the important thingwas recognizing the accidental stimulation Moreover byallowing the pieces of thread to do more than simply fall uponthe canvas surfaces by actually sewing them through to theother side Duchamp could emphasize the notion that they hadintersected the canvases The encounter involved both chanceand mathematics

In works such as the Three Standard Stoppages Duchamp createsphysical analogues for the abstract concept of ldquointersectionrdquothe one-dimensional pieces of string the curved linesegments intersect the two-dimensional surfaces of thecanvases (and they literally share points in common where theyare sewn together) The strings are thus further implicated (Iam tempted to say intertwined) along geometrical lines withthe fabric of the canvas strips The cracks in the Glass arealso a fundamental part of it They are ldquoinsiderdquo the brokensheets of glass which are in their turn encased inside theheavy panes of glass that Duchamp used to effect their repairIn an analogous way the ends of the strings in the Stoppagesare sandwiched between the strips of canvas and the rectanglesof glass that back them

Duchamprsquos works on glass are flat but they are nonethelessrather thick They are ldquospacesrdquo that can be thought ofespecially in this context as rectangular solids Because thesheets of glass themselves have thickness a depth that is

often layered they can be taken as three-dimensional sectionsout of higher-dimensional continua When for example all theconfigurations of the Stoppages (the strings the templatesand the plates of glass) are considered together their n-dimensional implications are manifest They are one-dimensional two-dimensional and three-dimensional and theyhave n-dimensional possibilities Each configuration isrelated to the others through projection and intersection thelines can be taken as slices out of surfaces the surfaces asslices out of solids and the solids as slices out ofhypersolids Esprit Pascal Jouffret one of Duchamprsquos mostimportant mathematical sources characterized such cuts as

ldquoinfinitely thin layersrdquo (26)

Duchamprsquos approachndashmoving from lines to surfaces and fromspaces to hyperspacesndashis couched in terms of perspective Heconsiders how vanishing points and changing points of viewwould operate in 2-space 3-space 4-space or any given n-space He suggests using ldquotransparent glassrdquo and ldquomirrorrdquo asanalogues of four-dimensional perspective systems (analoguesbecause such systems cannot actually be constructed in three-dimensional space)(27)

Especially when the narrow sheets of glass are seen edge-on inthe slots in their croquet box they suggest their membershipin an infinite series (reflections in mirrors can also implyinfinite reiterations) In an interview with Pierre CabanneDuchamp emphasized the serial characteristics of theStoppages ldquoWhen yoursquove come to the word three you have threemillionndashitrsquos the same thing as three I had decided that thethings would be done three times to get what I wanted MyThree Standard Stoppages is produced by three separateexperiments and the form of each one is slightly different Ikeep the line and I have a deformed meterrdquo(28)

he specifics of how Duchamp kept his line and used his

deformed meter is worth exploring further He tells Cabannethat he had been interested in working on glass for severalreasons including the way color ldquois visible from the othersiderdquo Glass was also useful in laying out its variouselements ldquoperspective was very important The Large Glassconstitutes a rehabilitation of perspective which had beencompletely ignored and disparaged For me perspective became

absolutely scientificrdquo(29)

y using linear perspective in his design Duchamp couldarrange the Bachelorsrsquo domain in such a way that the vanishingpoint coincided with the horizontal division between the upperand lower panels of the Glass

From this perspective or from the point of view ofperspective Duchamprsquos saying that a ldquolabyrinthrdquo lies at theldquocentral part of the stripping-barerdquo is significant the Large

Glass and the Three Standard Stoppages are about occlusion(30)

They involve unusual station points and unusual distancepoints in a perspectival system that can only bereconstructed from isolated positions outside normal space IfDuchamp were thinking of his ldquostripsrdquo of glass as physicalpuns on the notion of ldquostrippingrdquo the Bride then their

structure is doubly suggestive(31) Because her clothing consistsof transparent sections of glass thatare entailed with a ldquopoint de fuiterdquo it can be takento include a complex set of folds not only in the cloth ofthe garments but also in the fabric of space Recall thatTravelerrsquos Folding Item is conceptually related to the ThreeStandard StoppagesAlso the typewriter cover has been called

the ldquoBridersquos Dressrdquo (32)Perhaps the disappearance of theStoppages their dropping away toward infinity at the positionof the Bridersquos garments can be taken as an interdimensionalfolding up a stripping bare thatrequires orthogonaltranslation into higher space

Perhaps the disappearance of the Stoppages their droppingaway toward infinity at the position of the Bridersquos garmentscan be taken as an interdimensional folding up a strippingbare that requires orthogonal translation into higher space

All of the works here under discussion are related to oneanother through perspectivalism (and also perspectivism) ForDuchamp the use of perspective as a system was not a matterof creating single fixed-point ways of looking at things Itwas on the contrary involved in dislodging viewers fromtheir ordinary ways of understanding And with this objectivein mind his choosing readymades during the same period he wasworking on the Stoppagescan be seen as a related activityWhen Duchamp made his remark about Three Standard Stoppagesbeing a readymade but ldquonot quiterdquo he continued by saying

ldquoitrsquos a readymade if you wish but a moving onerdquo(33)

The curving pieces of string and our shifting notions of themeaning of the readymades seem to trail off from a ldquovanishingpointrdquoat the horizon of our own thinking The readymadesrefuse to abideby our ordinary definitions of art and the Stoppagesallude to geometries that have challenged our traditionalepistemological structures(34)

Their curvatures can be taken as references to non-Euclideanor topological geometries complications that necessitate ourreconsidering our vanishing points The strings when taken asanalogues for lines of sight are transposed or rotated intoa hidden space

click to enlarge

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich are behind the surface and inside the three-dimensional

often layered they can be taken as three-dimensional sectionsout of higher-dimensional continua When for example all theconfigurations of the Stoppages (the strings the templatesand the plates of glass) are considered together their n-dimensional implications are manifest They are one-dimensional two-dimensional and three-dimensional and theyhave n-dimensional possibilities Each configuration isrelated to the others through projection and intersection thelines can be taken as slices out of surfaces the surfaces asslices out of solids and the solids as slices out ofhypersolids Esprit Pascal Jouffret one of Duchamprsquos mostimportant mathematical sources characterized such cuts as

ldquoinfinitely thin layersrdquo (26)

Duchamprsquos approachndashmoving from lines to surfaces and fromspaces to hyperspacesndashis couched in terms of perspective Heconsiders how vanishing points and changing points of viewwould operate in 2-space 3-space 4-space or any given n-space He suggests using ldquotransparent glassrdquo and ldquomirrorrdquo asanalogues of four-dimensional perspective systems (analoguesbecause such systems cannot actually be constructed in three-dimensional space)(27)

Especially when the narrow sheets of glass are seen edge-on inthe slots in their croquet box they suggest their membershipin an infinite series (reflections in mirrors can also implyinfinite reiterations) In an interview with Pierre CabanneDuchamp emphasized the serial characteristics of theStoppages ldquoWhen yoursquove come to the word three you have threemillionndashitrsquos the same thing as three I had decided that thethings would be done three times to get what I wanted MyThree Standard Stoppages is produced by three separateexperiments and the form of each one is slightly different Ikeep the line and I have a deformed meterrdquo(28)

he specifics of how Duchamp kept his line and used his

deformed meter is worth exploring further He tells Cabannethat he had been interested in working on glass for severalreasons including the way color ldquois visible from the othersiderdquo Glass was also useful in laying out its variouselements ldquoperspective was very important The Large Glassconstitutes a rehabilitation of perspective which had beencompletely ignored and disparaged For me perspective became

absolutely scientificrdquo(29)

y using linear perspective in his design Duchamp couldarrange the Bachelorsrsquo domain in such a way that the vanishingpoint coincided with the horizontal division between the upperand lower panels of the Glass

From this perspective or from the point of view ofperspective Duchamprsquos saying that a ldquolabyrinthrdquo lies at theldquocentral part of the stripping-barerdquo is significant the Large

Glass and the Three Standard Stoppages are about occlusion(30)

They involve unusual station points and unusual distancepoints in a perspectival system that can only bereconstructed from isolated positions outside normal space IfDuchamp were thinking of his ldquostripsrdquo of glass as physicalpuns on the notion of ldquostrippingrdquo the Bride then their

structure is doubly suggestive(31) Because her clothing consistsof transparent sections of glass thatare entailed with a ldquopoint de fuiterdquo it can be takento include a complex set of folds not only in the cloth ofthe garments but also in the fabric of space Recall thatTravelerrsquos Folding Item is conceptually related to the ThreeStandard StoppagesAlso the typewriter cover has been called

the ldquoBridersquos Dressrdquo (32)Perhaps the disappearance of theStoppages their dropping away toward infinity at the positionof the Bridersquos garments can be taken as an interdimensionalfolding up a stripping bare thatrequires orthogonaltranslation into higher space

Perhaps the disappearance of the Stoppages their droppingaway toward infinity at the position of the Bridersquos garmentscan be taken as an interdimensional folding up a strippingbare that requires orthogonal translation into higher space

All of the works here under discussion are related to oneanother through perspectivalism (and also perspectivism) ForDuchamp the use of perspective as a system was not a matterof creating single fixed-point ways of looking at things Itwas on the contrary involved in dislodging viewers fromtheir ordinary ways of understanding And with this objectivein mind his choosing readymades during the same period he wasworking on the Stoppagescan be seen as a related activityWhen Duchamp made his remark about Three Standard Stoppagesbeing a readymade but ldquonot quiterdquo he continued by saying

ldquoitrsquos a readymade if you wish but a moving onerdquo(33)

The curving pieces of string and our shifting notions of themeaning of the readymades seem to trail off from a ldquovanishingpointrdquoat the horizon of our own thinking The readymadesrefuse to abideby our ordinary definitions of art and the Stoppagesallude to geometries that have challenged our traditionalepistemological structures(34)

Their curvatures can be taken as references to non-Euclideanor topological geometries complications that necessitate ourreconsidering our vanishing points The strings when taken asanalogues for lines of sight are transposed or rotated intoa hidden space

click to enlarge

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich are behind the surface and inside the three-dimensional

deformed meter is worth exploring further He tells Cabannethat he had been interested in working on glass for severalreasons including the way color ldquois visible from the othersiderdquo Glass was also useful in laying out its variouselements ldquoperspective was very important The Large Glassconstitutes a rehabilitation of perspective which had beencompletely ignored and disparaged For me perspective became

absolutely scientificrdquo(29)

y using linear perspective in his design Duchamp couldarrange the Bachelorsrsquo domain in such a way that the vanishingpoint coincided with the horizontal division between the upperand lower panels of the Glass

From this perspective or from the point of view ofperspective Duchamprsquos saying that a ldquolabyrinthrdquo lies at theldquocentral part of the stripping-barerdquo is significant the Large

Glass and the Three Standard Stoppages are about occlusion(30)

They involve unusual station points and unusual distancepoints in a perspectival system that can only bereconstructed from isolated positions outside normal space IfDuchamp were thinking of his ldquostripsrdquo of glass as physicalpuns on the notion of ldquostrippingrdquo the Bride then their

structure is doubly suggestive(31) Because her clothing consistsof transparent sections of glass thatare entailed with a ldquopoint de fuiterdquo it can be takento include a complex set of folds not only in the cloth ofthe garments but also in the fabric of space Recall thatTravelerrsquos Folding Item is conceptually related to the ThreeStandard StoppagesAlso the typewriter cover has been called

the ldquoBridersquos Dressrdquo (32)Perhaps the disappearance of theStoppages their dropping away toward infinity at the positionof the Bridersquos garments can be taken as an interdimensionalfolding up a stripping bare thatrequires orthogonaltranslation into higher space

Perhaps the disappearance of the Stoppages their droppingaway toward infinity at the position of the Bridersquos garmentscan be taken as an interdimensional folding up a strippingbare that requires orthogonal translation into higher space

All of the works here under discussion are related to oneanother through perspectivalism (and also perspectivism) ForDuchamp the use of perspective as a system was not a matterof creating single fixed-point ways of looking at things Itwas on the contrary involved in dislodging viewers fromtheir ordinary ways of understanding And with this objectivein mind his choosing readymades during the same period he wasworking on the Stoppagescan be seen as a related activityWhen Duchamp made his remark about Three Standard Stoppagesbeing a readymade but ldquonot quiterdquo he continued by saying

ldquoitrsquos a readymade if you wish but a moving onerdquo(33)

The curving pieces of string and our shifting notions of themeaning of the readymades seem to trail off from a ldquovanishingpointrdquoat the horizon of our own thinking The readymadesrefuse to abideby our ordinary definitions of art and the Stoppagesallude to geometries that have challenged our traditionalepistemological structures(34)

Their curvatures can be taken as references to non-Euclideanor topological geometries complications that necessitate ourreconsidering our vanishing points The strings when taken asanalogues for lines of sight are transposed or rotated intoa hidden space

click to enlarge

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich are behind the surface and inside the three-dimensional

Perhaps the disappearance of the Stoppages their droppingaway toward infinity at the position of the Bridersquos garmentscan be taken as an interdimensional folding up a strippingbare that requires orthogonal translation into higher space

All of the works here under discussion are related to oneanother through perspectivalism (and also perspectivism) ForDuchamp the use of perspective as a system was not a matterof creating single fixed-point ways of looking at things Itwas on the contrary involved in dislodging viewers fromtheir ordinary ways of understanding And with this objectivein mind his choosing readymades during the same period he wasworking on the Stoppagescan be seen as a related activityWhen Duchamp made his remark about Three Standard Stoppagesbeing a readymade but ldquonot quiterdquo he continued by saying

ldquoitrsquos a readymade if you wish but a moving onerdquo(33)

The curving pieces of string and our shifting notions of themeaning of the readymades seem to trail off from a ldquovanishingpointrdquoat the horizon of our own thinking The readymadesrefuse to abideby our ordinary definitions of art and the Stoppagesallude to geometries that have challenged our traditionalepistemological structures(34)

Their curvatures can be taken as references to non-Euclideanor topological geometries complications that necessitate ourreconsidering our vanishing points The strings when taken asanalogues for lines of sight are transposed or rotated intoa hidden space

click to enlarge

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich are behind the surface and inside the three-dimensional

Figure 13Girard Desarguesrsquos discussionsof perspective

Figure 14Girard Desarguesrsquos discussionsof perspective

What I have in mind here can be seen in the illustrations thataccompany Girard Desarguesrsquos discussions of perspective (Figs13 and 14) Desargues was the first mathematician to seeconnections between linear perspective and conic sections andis generally considered to be the founder of projective

geometry(35) He contributed to the ldquomathematicizationrdquo ofperspectivehelping to transform the practical Renaissance practice of

artists into the deductive science of geometers(36)

In the illustrations threads from lines of sight are bunchedup at the plane of the picture as if they were lying at orperhaps it would be better to say ldquoinrdquo the surface of therepresentation Rather than being part of the representationswhich 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 sectionof our visual pyramid the surface of the picture plane thatwe do not normally look at in a Renaissance picture but

through(38)

Such lines are also connected by a technological protocolinvolving an ldquoarborrdquo Desargues is one of the most likelysources for Duchamprsquos referring to the ldquoBriderdquo as an ldquoarbor-

typerdquo(39) The mathematician uses the term ldquoarbrerdquo in hisdiscussions of perspective as J V Field has explained

ldquoArbrerdquo is usually translated as ldquotreerdquo but the word canequally mean ldquoarborrdquo or ldquoaxlerdquo Like the central axle in amachine Desarguesrsquo arbre is the member to which others arereferred that is their relation to it is what chieflydefines their significance in the overall arrangement Thestandard metaphorical usage whereby engineers called an axle atree might thus have suggested to Desargues an extension ofthe same metaphor to provide names for subsidiary elements inthe geometrical scheme(40)

In Desarguesrsquo usage an ldquoarbrerdquo becomes a geometrical axis(41)

His unusual vocabulary was probably inspired by hisengineering and military experience as Field suggestsDesargues employs a number of other ldquoarbor-typerdquo terms suchas tronc (trunk) noeud (knot) rameau (branch) souche(stump) and branche (limb) A ldquotrunkrdquo is a straight line thatis intersected by other straight lines ldquoknotsrdquo are the pointson the ldquotrunkrdquo through which the other lines pass the otherlines themselves are called ldquobranchesrdquo a point common to agroup of segments on a line is a ldquostumprdquo one of these

segments is a ldquolimbrdquo etc(42)

Desarguesrsquo general approach of adopting an affectivevocabulary for geometrical entities recalls Duchamprsquospractice For example Desarguesrsquo term essieu (axletree) isreminiscent of Duchamprsquos term charniegravere (hinge) ldquoPerhaps makea hinge picture (folding yardstick book) develop theprinciple of the hinge in the displacements first in theplane second in space Find an automatic description of the

hinge Perhaps introduce it in the Pendu femellerdquo(43) Themechanical engineering term ldquoaxletreerdquo refers basically to afixed beam with bearings at its ends Because the axletree hasother devices such as wheels branching from it we canperhaps see why Desargues saw a comparable situation in theway geometrical projections branch off from the axes of hisperspective system In English the similar term ldquoarborrdquo wasapparently used during the seventeenthcentury to designate any kind of axle but is now generallyused to refer to the axles in small mechanisms such as

clocks(44)

Duchamp hints that he was familiar with these kinds ofdistinctions In one of his posthumously published notes(actually notations on a folder that originally containedseveral other notes) he associates the Bride the ldquoPendurdquo(femelle) with a ldquostandard arbor (shaft model)rdquo(45)

In another he connects the Bride a ldquoframeworkndashstandardarborrdquo and a ldquoclockwork apparatusrdquo(46)

In Desarguesrsquos way of thinking an ldquoarborrdquo or an ldquoaxletreerdquowas analogous to an axis of rotation a mathematical ldquoaxlerdquoaround which the elements of his transformative systemrevolved InDuchamprsquos descriptions of the complex workings of the Brideldquohingesrdquo operate in comparable ways

That Desargues was one of Duchamprsquos sources can be given

further credence by analyzing another important iconographicalelement of the Bridersquos domain the ldquonine shotsrdquo an area of

the Large Glass that was also reconstructed in 1936(47) At aconceptual level the ldquonine shotsrdquo seem to have an ldquoArguesianrdquo

perspectival demeanor(48) It has recently been noticed that a

number of Duchamprsquos notes have been split in two(49) One of themost interesting instances involves the ldquonine shotsrdquoA note included in his posthumously published Notes is the toppart of a note published in the Green Box Taken together thetwo parts read as follows

Make a painting on glass so that it has neither front norback 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 inshort corresponds to the vanishing point (in perspective) Thefigure thus obtained will be the projection (through skill) ofthe principal points of a three-dimensional body With maximumskill this projection would be reduced to a point (thetarget)With ordinary skill this projection will be a demultiplicationof the target (Each of the new points [images of the target]will have a coefficient of displacement This coefficient isnothing but a souvenir and can be noted conventionally Thedifferent shots tinted from black to white according to theirdistance)In general the figure obtained is the visible flattening (astop on the way) of the demultiplied body Cannon match withtip of fresh paint Repeat this operation 9 times 3 times by3 times from the same point Andash3 shots Bndash3 shots Cndash3 shotsA B and C are not in a plane and represent the schema of anyobject whatever of the demultiplied body

(51)

Desargues used the unusual term ldquoordinancerdquo for theorthogonals in a perspective system the sheaf of lines thatrecede into the distance toward a vanishing point at thehorizon An ldquoordinance of linesrdquo (ordonnance de droictes)corresponds to what we would now call a ldquopencil of linesrdquo in

modern geometrical parlance(52)

Desargues who had worked as a military engineer may againhave been prone to thinking of the trajectories of cannonshots toward a target as analogues for lines diminishingtoward a vanishing point in a perspective system (or towardthe vertex of a pencil of lines in a more purely geometricalrepresentation) His term for a vanishing point (or for thevertex in an ldquoordinance of linesrdquo) is ldquobutrdquo He uses theexpression ldquobut drsquoune ordonnancerdquo which can be translated asldquobutt of an ordinancerdquo but which is probably morecomprehensibly rendered as ldquotarget of an ordinancerdquo)Duchamprsquos line from the note above ldquoThis target in shortcorresponds to the vanishing point (in perspective)rdquo reads inFrench ldquoCe but est en somme une correspondance du point dufuite (en perspective)rdquo

(53)

click to enlarge

Figure 15Marcel Duchamp Pharmacy 1914

Before leaving the potential influence of Desarguesrsquo

Desarguesrsquo general approach of adopting an affectivevocabulary for geometrical entities recalls Duchamprsquospractice For example Desarguesrsquo term essieu (axletree) isreminiscent of Duchamprsquos term charniegravere (hinge) ldquoPerhaps makea hinge picture (folding yardstick book) develop theprinciple of the hinge in the displacements first in theplane second in space Find an automatic description of the

hinge Perhaps introduce it in the Pendu femellerdquo(43) Themechanical engineering term ldquoaxletreerdquo refers basically to afixed beam with bearings at its ends Because the axletree hasother devices such as wheels branching from it we canperhaps see why Desargues saw a comparable situation in theway geometrical projections branch off from the axes of hisperspective system In English the similar term ldquoarborrdquo wasapparently used during the seventeenthcentury to designate any kind of axle but is now generallyused to refer to the axles in small mechanisms such as

clocks(44)

Duchamp hints that he was familiar with these kinds ofdistinctions In one of his posthumously published notes(actually notations on a folder that originally containedseveral other notes) he associates the Bride the ldquoPendurdquo(femelle) with a ldquostandard arbor (shaft model)rdquo(45)

In another he connects the Bride a ldquoframeworkndashstandardarborrdquo and a ldquoclockwork apparatusrdquo(46)

In Desarguesrsquos way of thinking an ldquoarborrdquo or an ldquoaxletreerdquowas analogous to an axis of rotation a mathematical ldquoaxlerdquoaround which the elements of his transformative systemrevolved InDuchamprsquos descriptions of the complex workings of the Brideldquohingesrdquo operate in comparable ways

That Desargues was one of Duchamprsquos sources can be given

further credence by analyzing another important iconographicalelement of the Bridersquos domain the ldquonine shotsrdquo an area of

the Large Glass that was also reconstructed in 1936(47) At aconceptual level the ldquonine shotsrdquo seem to have an ldquoArguesianrdquo

perspectival demeanor(48) It has recently been noticed that a

number of Duchamprsquos notes have been split in two(49) One of themost interesting instances involves the ldquonine shotsrdquoA note included in his posthumously published Notes is the toppart of a note published in the Green Box Taken together thetwo parts read as follows

Make a painting on glass so that it has neither front norback 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 inshort corresponds to the vanishing point (in perspective) Thefigure thus obtained will be the projection (through skill) ofthe principal points of a three-dimensional body With maximumskill this projection would be reduced to a point (thetarget)With ordinary skill this projection will be a demultiplicationof the target (Each of the new points [images of the target]will have a coefficient of displacement This coefficient isnothing but a souvenir and can be noted conventionally Thedifferent shots tinted from black to white according to theirdistance)In general the figure obtained is the visible flattening (astop on the way) of the demultiplied body Cannon match withtip of fresh paint Repeat this operation 9 times 3 times by3 times from the same point Andash3 shots Bndash3 shots Cndash3 shotsA B and C are not in a plane and represent the schema of anyobject whatever of the demultiplied body

(51)

Desargues used the unusual term ldquoordinancerdquo for theorthogonals in a perspective system the sheaf of lines thatrecede into the distance toward a vanishing point at thehorizon An ldquoordinance of linesrdquo (ordonnance de droictes)corresponds to what we would now call a ldquopencil of linesrdquo in

modern geometrical parlance(52)

Desargues who had worked as a military engineer may againhave been prone to thinking of the trajectories of cannonshots toward a target as analogues for lines diminishingtoward a vanishing point in a perspective system (or towardthe vertex of a pencil of lines in a more purely geometricalrepresentation) His term for a vanishing point (or for thevertex in an ldquoordinance of linesrdquo) is ldquobutrdquo He uses theexpression ldquobut drsquoune ordonnancerdquo which can be translated asldquobutt of an ordinancerdquo but which is probably morecomprehensibly rendered as ldquotarget of an ordinancerdquo)Duchamprsquos line from the note above ldquoThis target in shortcorresponds to the vanishing point (in perspective)rdquo reads inFrench ldquoCe but est en somme une correspondance du point dufuite (en perspective)rdquo

(53)

click to enlarge

Figure 15Marcel Duchamp Pharmacy 1914

Before leaving the potential influence of Desarguesrsquo

further credence by analyzing another important iconographicalelement of the Bridersquos domain the ldquonine shotsrdquo an area of

the Large Glass that was also reconstructed in 1936(47) At aconceptual level the ldquonine shotsrdquo seem to have an ldquoArguesianrdquo

perspectival demeanor(48) It has recently been noticed that a

number of Duchamprsquos notes have been split in two(49) One of themost interesting instances involves the ldquonine shotsrdquoA note included in his posthumously published Notes is the toppart of a note published in the Green Box Taken together thetwo parts read as follows

Make a painting on glass so that it has neither front norback 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 inshort corresponds to the vanishing point (in perspective) Thefigure thus obtained will be the projection (through skill) ofthe principal points of a three-dimensional body With maximumskill this projection would be reduced to a point (thetarget)With ordinary skill this projection will be a demultiplicationof the target (Each of the new points [images of the target]will have a coefficient of displacement This coefficient isnothing but a souvenir and can be noted conventionally Thedifferent shots tinted from black to white according to theirdistance)In general the figure obtained is the visible flattening (astop on the way) of the demultiplied body Cannon match withtip of fresh paint Repeat this operation 9 times 3 times by3 times from the same point Andash3 shots Bndash3 shots Cndash3 shotsA B and C are not in a plane and represent the schema of anyobject whatever of the demultiplied body

(51)

Desargues used the unusual term ldquoordinancerdquo for theorthogonals in a perspective system the sheaf of lines thatrecede into the distance toward a vanishing point at thehorizon An ldquoordinance of linesrdquo (ordonnance de droictes)corresponds to what we would now call a ldquopencil of linesrdquo in

modern geometrical parlance(52)

Desargues who had worked as a military engineer may againhave been prone to thinking of the trajectories of cannonshots toward a target as analogues for lines diminishingtoward a vanishing point in a perspective system (or towardthe vertex of a pencil of lines in a more purely geometricalrepresentation) His term for a vanishing point (or for thevertex in an ldquoordinance of linesrdquo) is ldquobutrdquo He uses theexpression ldquobut drsquoune ordonnancerdquo which can be translated asldquobutt of an ordinancerdquo but which is probably morecomprehensibly rendered as ldquotarget of an ordinancerdquo)Duchamprsquos line from the note above ldquoThis target in shortcorresponds to the vanishing point (in perspective)rdquo reads inFrench ldquoCe but est en somme une correspondance du point dufuite (en perspective)rdquo

(53)

click to enlarge

Figure 15Marcel Duchamp Pharmacy 1914

Before leaving the potential influence of Desarguesrsquo

Desargues used the unusual term ldquoordinancerdquo for theorthogonals in a perspective system the sheaf of lines thatrecede into the distance toward a vanishing point at thehorizon An ldquoordinance of linesrdquo (ordonnance de droictes)corresponds to what we would now call a ldquopencil of linesrdquo in

modern geometrical parlance(52)

Desargues who had worked as a military engineer may againhave been prone to thinking of the trajectories of cannonshots toward a target as analogues for lines diminishingtoward a vanishing point in a perspective system (or towardthe vertex of a pencil of lines in a more purely geometricalrepresentation) His term for a vanishing point (or for thevertex in an ldquoordinance of linesrdquo) is ldquobutrdquo He uses theexpression ldquobut drsquoune ordonnancerdquo which can be translated asldquobutt of an ordinancerdquo but which is probably morecomprehensibly rendered as ldquotarget of an ordinancerdquo)Duchamprsquos line from the note above ldquoThis target in shortcorresponds to the vanishing point (in perspective)rdquo reads inFrench ldquoCe but est en somme une correspondance du point dufuite (en perspective)rdquo

(53)

click to enlarge

Figure 15Marcel Duchamp Pharmacy 1914

Before leaving the potential influence of Desarguesrsquo

vocabulary it might be pointed out that the notion of anldquoarbor-typerdquo seems to inform several of Duchamprsquos readymadesPharmacy (Fig 15) chosen in 1914 is a tree-filled landscapewith a red and green dot added by Duchamp (at vanishingpoints) on the horizon line In addition to being a referenceto the colored bottles in drugstore windows the colors mayalso be a subtle reference to the techniques of anaglyphy apractice related to stereoscopy that we know Duchamp wasinterested in probably because of its n-dimensional

implications(54) In the layout of Robert Lebelrsquos earlymonograph a design that Duchamp was largely responsible forPharmacy is juxtaposed to the Bottlerack (Fig 16)also chosen in 1914 On the facing page are the Network ofStoppages 1914 and Cemetery of Uniforms and Liveries No2(Fig 17) 1914 the drawing that Duchamp used to transferthe design of the ldquocapillary tubesrdquo and the ldquonine malic moldsrdquo

to the Large Glass(55) Above Pharmacy and the Bottlerack isCemetery of Uniforms and Liveries No 1 (Fig 18) which inthe more multi-layered French edition of the book had a color

image of Nine Malic Molds (Fig 19) tipped in over it(56)

click images to enlarge

Figure 16Figure 17

Marcel DuchampBottle Dryer 19141964Marcel DuchampCemetery of Uniformsand Liveries No 2 1914

click images to enlarge

Figure 18Figure 19

Marcel DuchampCemetery of Uniformsand Liveries No 1 1913Marcel DuchampNineMalic Molds 1914-15

PAGES

click to enlarge

Figure 20Photograph of Duchamp 1942

With Desarguesrsquo terminology such as ldquotreerdquo ldquotrunkrdquo ldquobranchrdquoand ldquolimbrdquo in mind these works look positively geometricalInNetwork of Stoppages for example the pattern of linesresemble branches especially if the painting is rotatedninety degrees clockwise In the background the nude woman inldquoYoung Man and Girl in Springrdquo the first layer of Network ofStoppages is then centered in the boughs of the tree Fromthis perspective she becomes a precursor for the Bride as anldquoarbor-typerdquo In theBottlerack the prongs appear to berotated around a central axis (anarbre) and suggest reiteratedline segments (rameaux or branches) That theseinterpretations can be taken seriously is reinforced by aninteresting photograph of Duchamp taken in 1942 showing himstanding in front of a tree that has been provided with prongsso that it can act as a bottle dryer (Fig 20) A number ofbottles which have been hung upon this ldquoarbre-seacutechoirrdquo canbe seen behind Duchamp and he has a network of linearshadows which have been cast from the branches of the tree

falling across his face(57)

The various connections here under discussion can perhaps bemade more evident in the sense of our being able to ldquoseerdquointo Duchamprsquos n-dimensional realm by bringing his importantpainting Tu mrsquo (Fig 21) into the discussion

click to enlarge

Figure 21Marcel Duchamp Tu mrsquo1918

This work has ldquoanamorphicrdquo aspects and is closely related tothe Three Standard Stoppages which were used to draw a number

of its curving shapes(58) The shadows of readymadesndashthe BicycleWheel the Corkscrew and the Hat Rackndashstretch out across thesurface of the picture plane suggesting an anamorphictransformation At one level of course Tu mrsquo is about the

ldquoshadowyrdquo existence of art objects(59) The Corkscrew in factexists only as a shadow on this painting Buton more important levels the work is about geometryndashbothEuclidean and non-Euclidean geometry In addition to thesegeometries of constant curvature Duchamp may also have beenthinking about topology some elements in the painting seem tobe stretched and pulled as if they

were elastic(60)

The shadows of the readymades are themselves distortedtransformations and they are cast onto a surface that seemsto be warped and curved and the space behind the surface isfilled with strangely bent geometrical objects

On the right-hand side of the canvas there is an irregularopen-sided rectangular ldquosolidrdquo The left side of this solid isa white surface that recedes into the space of the canvasaccording to one-point perspective From each corner of thewhite surface two lines drawn with the templates of theThree Standard Stoppages extend at more or less right anglestoward the right One of each of these is black and the otherred The black lines at all four edges are drawn with the sametemplate Each set of lines at the upper boundary of the solid

cross one another at two points and each set are drawn in thesame way The two lines at the lower edges of the solid do notcross one another and they are rotated and inverted withrespect to one another

There are also a series of color bands (twenty-four in all)extending orthogonally back into the space of the ldquosolidrdquo orinto its virtual shape They seem to continue on behind itThese bands are connected to the curved line segments thatcomprise the ambiguous edges of the transparent solid avolume we could think of as a 3-space with fluctuanttransparent faces Each of the color bands is surrounded by anumber of concentric circles that also recede back into thepaintingrsquos virtual space according to one-point perspectiveThe vanishing point coincides with the bottom edge of thecanvas just to the right of center below the indexical handwhich incidentally is a hand-painted readymade elementexecuted by a certain A Klang a sign painter Duchamp hiredto carry out this task Klangrsquos minuscule signature is visiblenear the sleeve

Duchamprsquos complex geometrical arrangement is made even morecomplex by the shadow of the Hat Rack which occupies the sameregion of the canvas as the ldquosolidrdquo On one level the HatRack resembles a tree and the shadows cast from its multiplebranches suggest yet another ldquoarbor-typerdquo We know that theBride is based in part on the idea of the cast shadow ldquoas

if it were the projection of a four-dimensional objectrdquo(61)

The way the Hat Rack interacts with the ldquosolidrdquo is indicativeof the complexities that would be involved in such spaces Thelines and color bands seem to overlay the shadow but theshadow seems to overlay the white rectangle at the left sideof the ldquosolidrdquo The shadow can thus be read as both in frontof and behind the chunk of space outlined and bounded by theelements of Duchamprsquos design

The spatial complexities of Tu mrsquo can also be seen in the

recession of its orthogonals They plunge backward in a waythat is comparable to the convergence of orthogonals in theLarge Glass In the former the lines come together just atthe lower edge of the painting in the latter just at theupper boundary of the Bachelorsrsquo domain In Tu mrsquo thevanishing point is where the ldquosolidrdquo (and also its edges drawnwith the Three Standard Stoppages) would disappear In theLarge Glass the point is at the center of the three plates ofglass running across the Bridersquos horizon It is where theseldquolinesrdquo would disappear if rotated ninety degrees TheBridersquos garments when thus folded up can be taken asorthogonals to a point of intersectionndashthe intersection ofparallel lines at infinity

In Euclidean geometry parallel lines do not intersect Themathematical convention that they do intersect at infinity wasone of Desarguesrsquo important contributions (Parallel lines doseem to intersect at the vanishing point of a perspectivesystem which may have given Desargues his idea) Thinking ofparallel lines as meeting at infinity eventually contributedto the development of non-Euclidean geometries in the

nineteenth century(62)

The conceptual point where parallel lines meet cannot be seenany more than the curvature of space can be perceiveddirectly If the curved lines in theThree StandardStoppagesare taken as references to non-Euclidean lines ofsight then they are fundamentally hidden in ldquogarmentsrdquo of theBride just as the vanishing point in Tu mrsquoseems to disappearoff the edge of its hyperspatial expanse

The left side of Tu mrsquo is also complicated In addition to theshadows of the Bicycle Wheel and the Corkscrew lines drawnwith the templates of the Three Standard Stoppages are placedat the lower left-hand side of the canvas Each of these linesegments is at the edge of three curved surfaces that seem tofall back into the space of the canvas If these irregular

planes are thought of as a ldquopencil of surfacesrdquo (Desarguesuses the term ldquoordonnance de plansldquo) they would withdrawdownward at more or less right angles to the space of thecanvas toward a line of intersection located at an infinitedistance (Desargues says that a sheaf of parallel planes canbe imagined converging at an ldquoessieurdquo an ldquoaxlerdquo just as anldquoordinance of linesrdquo can be imagined intersecting at a ldquopointagrave une distance infinierdquo)

(63)

The edge of the upper member of this pencil of planes isblack and it is drawn with the same ldquostoppagerdquo that was usedat each edge of the rectangular ldquosolidrdquo on the right side ofthe canvas The edge of the line segment in the middleregister was used as the other line at the edges of the upperboundary and the edge of the line segment in the lowerregister was used as the other line at the edges of the lowerboundary of the ldquosolidrdquo The shadow of the Bicycle Wheel seemsto overlay this arrangement of superposed curved surfacesThere is also a sequence of flat color squares recedingaccording to a plunging perspective back from the center ofthe canvas into an infinite space at the upper left corner ofthe canvas This arrangement of color squares seems to overlaythe shadow of the Bicycle Wheel In contrast the shadow ofthe Corkscrew which seems to spiral out from the axle of thewheel overlays the color squares Reading the shadows asriding on the surface of the actual canvas is thus complicatedby their relationships with objects occupying the virtualspace depicted ldquoinsiderdquo the canvas Duchamp further emphasizesthe spatial oddities of his picture by using various forms ofldquointersectionrdquo The corkscrew intersects the canvas by seemingto spiral into it the safety pins pierce the surface of thecanvas and the bottle brush and the bolt go through the frontside of the picture and are fastened to it from behind

click to enlarge

Figure 22Marcel Duchamp Tu mrsquo 1918(side view)

Duchamp is obviously playing with real and represented objectsand with real and represented space in Tu mrsquo To furthercomplicate the issues he paints a trompe lrsquooeiltear in thesurface of the canvas which is held together by the realsafety pins In addition to these ready-made elements thebottle brush juts out from the tear at right angles to thecanvas As an actual object a readymade the bottle brushcasts actual shadows that can be contrasted with the virtualshadows of the Bicycle Wheel the Corkscrew and the Hat Rackwhich Duchamp traced onto the surface with pencil In terms ofits geometry the bottle brush is really only visible when welook at Tu mrsquo from the side at an oblique angle (Fig 22)When we view the canvas straight on all we see is the end ofthe brush Looking at the canvas from the side also allows usto see the other elements of the painting and they seem lessstretched out less constrained by the plunging perspectiveThe shift is particularly apparent in the sequence of colorsquares at the upper left side of the canvas In fact we nownotice that these shapes are not really squares butparallelograms that look more ldquonaturalrdquo from the side thanfrom the front

click to enlarge

recession of its orthogonals They plunge backward in a waythat is comparable to the convergence of orthogonals in theLarge Glass In the former the lines come together just atthe lower edge of the painting in the latter just at theupper boundary of the Bachelorsrsquo domain In Tu mrsquo thevanishing point is where the ldquosolidrdquo (and also its edges drawnwith the Three Standard Stoppages) would disappear In theLarge Glass the point is at the center of the three plates ofglass running across the Bridersquos horizon It is where theseldquolinesrdquo would disappear if rotated ninety degrees TheBridersquos garments when thus folded up can be taken asorthogonals to a point of intersectionndashthe intersection ofparallel lines at infinity

In Euclidean geometry parallel lines do not intersect Themathematical convention that they do intersect at infinity wasone of Desarguesrsquo important contributions (Parallel lines doseem to intersect at the vanishing point of a perspectivesystem which may have given Desargues his idea) Thinking ofparallel lines as meeting at infinity eventually contributedto the development of non-Euclidean geometries in the

nineteenth century(62)

The conceptual point where parallel lines meet cannot be seenany more than the curvature of space can be perceiveddirectly If the curved lines in theThree StandardStoppagesare taken as references to non-Euclidean lines ofsight then they are fundamentally hidden in ldquogarmentsrdquo of theBride just as the vanishing point in Tu mrsquoseems to disappearoff the edge of its hyperspatial expanse

The left side of Tu mrsquo is also complicated In addition to theshadows of the Bicycle Wheel and the Corkscrew lines drawnwith the templates of the Three Standard Stoppages are placedat the lower left-hand side of the canvas Each of these linesegments is at the edge of three curved surfaces that seem tofall back into the space of the canvas If these irregular

planes are thought of as a ldquopencil of surfacesrdquo (Desarguesuses the term ldquoordonnance de plansldquo) they would withdrawdownward at more or less right angles to the space of thecanvas toward a line of intersection located at an infinitedistance (Desargues says that a sheaf of parallel planes canbe imagined converging at an ldquoessieurdquo an ldquoaxlerdquo just as anldquoordinance of linesrdquo can be imagined intersecting at a ldquopointagrave une distance infinierdquo)

(63)

The edge of the upper member of this pencil of planes isblack and it is drawn with the same ldquostoppagerdquo that was usedat each edge of the rectangular ldquosolidrdquo on the right side ofthe canvas The edge of the line segment in the middleregister was used as the other line at the edges of the upperboundary and the edge of the line segment in the lowerregister was used as the other line at the edges of the lowerboundary of the ldquosolidrdquo The shadow of the Bicycle Wheel seemsto overlay this arrangement of superposed curved surfacesThere is also a sequence of flat color squares recedingaccording to a plunging perspective back from the center ofthe canvas into an infinite space at the upper left corner ofthe canvas This arrangement of color squares seems to overlaythe shadow of the Bicycle Wheel In contrast the shadow ofthe Corkscrew which seems to spiral out from the axle of thewheel overlays the color squares Reading the shadows asriding on the surface of the actual canvas is thus complicatedby their relationships with objects occupying the virtualspace depicted ldquoinsiderdquo the canvas Duchamp further emphasizesthe spatial oddities of his picture by using various forms ofldquointersectionrdquo The corkscrew intersects the canvas by seemingto spiral into it the safety pins pierce the surface of thecanvas and the bottle brush and the bolt go through the frontside of the picture and are fastened to it from behind

click to enlarge

Figure 22Marcel Duchamp Tu mrsquo 1918(side view)

Duchamp is obviously playing with real and represented objectsand with real and represented space in Tu mrsquo To furthercomplicate the issues he paints a trompe lrsquooeiltear in thesurface of the canvas which is held together by the realsafety pins In addition to these ready-made elements thebottle brush juts out from the tear at right angles to thecanvas As an actual object a readymade the bottle brushcasts actual shadows that can be contrasted with the virtualshadows of the Bicycle Wheel the Corkscrew and the Hat Rackwhich Duchamp traced onto the surface with pencil In terms ofits geometry the bottle brush is really only visible when welook at Tu mrsquo from the side at an oblique angle (Fig 22)When we view the canvas straight on all we see is the end ofthe brush Looking at the canvas from the side also allows usto see the other elements of the painting and they seem lessstretched out less constrained by the plunging perspectiveThe shift is particularly apparent in the sequence of colorsquares at the upper left side of the canvas In fact we nownotice that these shapes are not really squares butparallelograms that look more ldquonaturalrdquo from the side thanfrom the front

click to enlarge

planes are thought of as a ldquopencil of surfacesrdquo (Desarguesuses the term ldquoordonnance de plansldquo) they would withdrawdownward at more or less right angles to the space of thecanvas toward a line of intersection located at an infinitedistance (Desargues says that a sheaf of parallel planes canbe imagined converging at an ldquoessieurdquo an ldquoaxlerdquo just as anldquoordinance of linesrdquo can be imagined intersecting at a ldquopointagrave une distance infinierdquo)

(63)

The edge of the upper member of this pencil of planes isblack and it is drawn with the same ldquostoppagerdquo that was usedat each edge of the rectangular ldquosolidrdquo on the right side ofthe canvas The edge of the line segment in the middleregister was used as the other line at the edges of the upperboundary and the edge of the line segment in the lowerregister was used as the other line at the edges of the lowerboundary of the ldquosolidrdquo The shadow of the Bicycle Wheel seemsto overlay this arrangement of superposed curved surfacesThere is also a sequence of flat color squares recedingaccording to a plunging perspective back from the center ofthe canvas into an infinite space at the upper left corner ofthe canvas This arrangement of color squares seems to overlaythe shadow of the Bicycle Wheel In contrast the shadow ofthe Corkscrew which seems to spiral out from the axle of thewheel overlays the color squares Reading the shadows asriding on the surface of the actual canvas is thus complicatedby their relationships with objects occupying the virtualspace depicted ldquoinsiderdquo the canvas Duchamp further emphasizesthe spatial oddities of his picture by using various forms ofldquointersectionrdquo The corkscrew intersects the canvas by seemingto spiral into it the safety pins pierce the surface of thecanvas and the bottle brush and the bolt go through the frontside of the picture and are fastened to it from behind

click to enlarge

Figure 22Marcel Duchamp Tu mrsquo 1918(side view)

Duchamp is obviously playing with real and represented objectsand with real and represented space in Tu mrsquo To furthercomplicate the issues he paints a trompe lrsquooeiltear in thesurface of the canvas which is held together by the realsafety pins In addition to these ready-made elements thebottle brush juts out from the tear at right angles to thecanvas As an actual object a readymade the bottle brushcasts actual shadows that can be contrasted with the virtualshadows of the Bicycle Wheel the Corkscrew and the Hat Rackwhich Duchamp traced onto the surface with pencil In terms ofits geometry the bottle brush is really only visible when welook at Tu mrsquo from the side at an oblique angle (Fig 22)When we view the canvas straight on all we see is the end ofthe brush Looking at the canvas from the side also allows usto see the other elements of the painting and they seem lessstretched out less constrained by the plunging perspectiveThe shift is particularly apparent in the sequence of colorsquares at the upper left side of the canvas In fact we nownotice that these shapes are not really squares butparallelograms that look more ldquonaturalrdquo from the side thanfrom the front

click to enlarge

Figure 22Marcel Duchamp Tu mrsquo 1918(side view)

Duchamp is obviously playing with real and represented objectsand with real and represented space in Tu mrsquo To furthercomplicate the issues he paints a trompe lrsquooeiltear in thesurface of the canvas which is held together by the realsafety pins In addition to these ready-made elements thebottle brush juts out from the tear at right angles to thecanvas As an actual object a readymade the bottle brushcasts actual shadows that can be contrasted with the virtualshadows of the Bicycle Wheel the Corkscrew and the Hat Rackwhich Duchamp traced onto the surface with pencil In terms ofits geometry the bottle brush is really only visible when welook at Tu mrsquo from the side at an oblique angle (Fig 22)When we view the canvas straight on all we see is the end ofthe brush Looking at the canvas from the side also allows usto see the other elements of the painting and they seem lessstretched out less constrained by the plunging perspectiveThe shift is particularly apparent in the sequence of colorsquares at the upper left side of the canvas In fact we nownotice that these shapes are not really squares butparallelograms that look more ldquonaturalrdquo from the side thanfrom the front

click to enlarge

Figure 23Jean-Franccedilois NiceacuteronThaumaturgus opticus1646

Duchamp probably learned something about these kinds ofanamorphic effects during the period he was working at theBibliothegraveque Sainte-Geneviegraveve in Paris One of his notes forthe Large Glass which he wrote at this time suggestsconsulting the libraryrsquos collection ldquoPerspective See thecatalogue of the Bibliothegraveque Sainte-Geneviegraveve The wholesection on perspective Niceacuteron (Father J-F) Thaumaturgus

opticusrdquo(64) Many of the books on perspective available toDuchamp at the library deal with the unusual or ldquoaberrantrdquosystems used in anamorphosis These include works by FatherJean-Franccedilois Niceacuteron whom Duchamp mentions by name in his

note(65)

One of Niceacuteronrsquos images from Thaumaturgus opticus (Fig 23) isevocative of Tu mrsquo especially if thesketch is fully extended (the left-hand side of the upper partcontinues at the right-hand side of the lower part)(66)

Thus reconnected the long narrow dimensions of the imageapproximate those of Tu mrsquo Duchamp may also have seen asimilarity here between the string held by the assistant inthe left-hand part of the drawing and the segments of stringin Three Standard Stoppages In Niceacuteronrsquos illustration as inperspective drawings generally the curling end of the line ismeant to indicate that it is a thread used in the constructionof the image rather than being an integral element of theimagery

click to enlarge

Figure 24Hans Holbein the YoungerThe French Ambassadors of KingHenri II at the court of theEnglish King Henry VIII 1533

Duchamprsquos thread is more complex The strings in theThreeStandard Stoppagesare themselves spaces one-dimensionalspaces and they are intended to indicate a more difficultgeometry than the one Niceacuteron had in mind But Duchamprsquosmanner of taking an oblique view and his interest in observinga scene through a visual system rotated away from normalspace is very similar to the way Niceacuteron turns hisoutstretched images onto the wall Duchamprsquos (and Niceacuteronrsquos)procedure is also reminiscent of Hans Holbeinrsquos famousportrait The French Ambassadors (Fig 24) in which adistended skull crosses the picture plane at more or lessright-angles to the orthogonals of the perspective system used

to construct the painting(67)The French Ambassadorsis afavoriteimage among postmodernists primarily because it bringstogether two different ways of looking at objects in one

picture(68)The primary visual order the three-dimensionalspace of the scientific perspective is undermined by theanomalous skull falling across it The abnormal space of thedeathrsquos head interpenetrates the normal space where theambassadors live casting a shadow across their existence Italso displaces the dominant viewing subject from a position in

front of the painting to one at the sidendashto a position that is

essentially outside the picturersquos frame of reference(69)

As the skull comes into adjustment the painting becomesdistorted and vice versa Jean Clair has discussed Tu mrsquo interms comparable to those just used to describe Holbeinrsquospainting He points out that when looked at obliquely ldquotheshadows of the readymades and the design of the parallelepiped

straighten uprdquo(70) He also notices the way in which the bottlebrush seems to rotate out from the surface of the canvaschanging from a ldquodotrdquo or point into ldquono more than a linerdquoAccording to Clair the function of the bottle brush issimilar to that of the skull in Holbeinrsquos picture namely ldquotoexpose the vanity of the paintingBut this time of all

paintingsrdquo(71)

We can amplify Clairrsquos remarks by pointing out that as wemove to the side of Tu mrsquo the surface of the picture isvisually rotated If we were able to continue on around thepicture in order to look at it edge on the surface would bereduced to a line segment from which the ldquoline segmentrdquo ofthe bottle brush would extend at a right angle The bottlebrush is a readymade a counterpart of an orthogonal one thatcomes out into our space rather than receding into the spaceof the painting The sequence of color squares apparentlyattached to the surface of the canvas with the bolt wouldpresumably be receding in the opposite direction along theaxis of the shaft (the axle) of the bolt back into the spaceof the canvas which as we move to the side is not onlyflattened into a two-dimensional surface but further reducedto a one-dimensional line segment Clairrsquos statement that asthe ldquopainting vanishes the readymade makes its appearancerdquois quite true We could also say that the actual readymade(the bottle brush) makes its appearance as the virtualreadymades and their shadows disappear And vice versa as thereal elements of the work vanish the virtual elementsreappear

A similar language could be used to describe the intersectionof the strings with the glass plates of the Three StandardStoppages They trail off at right-angles as it were alonglines that are orthogonal to the canvas strips as if they hadbeen rotated out of the virtual space of the ldquoPrussian bluerdquointo the actual space of the canvases If the strings areanalogous to ldquolines of sightrdquo they are like threads lyingldquoinrdquo the surface of the perspectival plane as we have seen inDesarguesrsquo perspective renderings (Figs 13 and 14) or inNiceacuteronrsquos illustration (Fig 23) In this sense the stringscan be taken as anamorphic lines crossing the representationalspace of the sheets of glass Recall what Duchamprsquos space wasintended to show his glass has ldquoneither front nor backneither top nor bottomrdquo and it can be used as a ldquothree-dimensional physical mediumrdquo in the construction of a ldquofour-dimensional perspectiverdquo In the Large Glass and the ThreeStandard Stoppages Duchamp was both literally andfiguratively boxing and encasing the geometrical elements ofhis iconographyndashinside glass and inside an n-dimensionalprojective system With Tu mrsquo he was also enclosing the basicelements of his own working method and indeed the basicelements of painting as a general practice inside a complexpictorial space one with unusual curvatures

Duchamprsquos works such as the ones I have discussed in thispaper with their various projections and intersections eachin their turn folding up into the next suggest that he wasthinking about different kinds of geometries Henri Poincareacuteamong the artistrsquos most likely mathematical sources often

discusses the interrelationships of geometries(72)

Projective geometry which was prefigured in Renaissanceperspective and initially elaborated in the work of suchseventeenth-century mathematicians as Desargues and Blaise

Pascal(73)

was later during the nineteenth century recognized as beingcentral to mathematics in general By the end of the century

both Euclidean and non-Euclidean geometry had been subsumedunder the principles of projective geometry(74)

Projective geometry deals with properties of geometricalfigures that remain invariant under transformation It studiesmappings of one figure onto another brought about byprojection and section and it tries to find qualities thatremain fixed during these procedures (Desarguesrsquo Theorem andPascalrsquos Theorem describe famous examples) Twentieth-centurymathematicians have invented methods of transformation thatare even more general than projection and section One of themost important of these approaches topology considersgeometrical properties of figures that are unchanged whilethese figures undergo deformations such as stretching andbending Especially in the context of the present discussionPoincareacute can be thought of as the ldquofather

of modern topologyrdquo (75) a subject that he referred to asanalysis situs (Latin for ldquoanalysis of the siterdquo ldquotopologyrdquocoming from the Greek equivalent for ldquostudy of the placerdquo) Hepoints out that this geometry ldquogives rise to a series oftheorems just as closely interconnected as those of Euclidrdquo(76)

Duchamprsquos Tu mrsquo can very nearly serve as an illustration forPoincareacutersquos arguments As pointed out earlier the elongatedshadows can be taken as anamorphic deformations and thus asreferences to topological transformations with four-dimensional or more generally n-dimensional ramifications(branchings) particularly insofar as anamorphic projectionsseem to intersect normal space at oblique angles In ways thatare like Holbeinrsquos famous skull the cast shadows in Tu mrsquoseem to traverse the space of the picture and in this sensethey are orthogonal to it (shadows are literally orthogonal tothe surfaces on which they are cast) From the perspective ofthe fourth dimension the strings in Three Standard Stoppagescan also be interpreted as falling away from normal space

along perpendicular lines at least insofar as they plummettoward the horizon of the Bride Duchamprsquos cast shadows andperhaps his cast segments of strings are projective analogiesfor higher-dimensional spaces His general approach can beseen in the following note

For an ordinary eye a point in a three-dimensional spacehides conceals the fourth direction of the continuumndashwhich isto say that this eye can try to perceive physically thisfourth direction by going around the said point From whateverangle it looks at the point this point will always be theborder line of the fourth directionndashjust as an ordinary eyegoing around a mirror will never be able to perceive anythingbut the reflected three-dimensional image and nothing from

behind(77)

Looked at ldquoedge-onrdquo in the sense of being seen undergoing ann-dimensional rotation the individual ldquostoppagesrdquo can betaken as trailing off into the fourth direction of whatDuchamp

calls the ldquoeacutetenduerdquo(78)From such a perspective they would beperceived as points The viewer equipped with a four-dimensional visual system to use Duchamprsquos words would beable to ascertain that a ldquopointrdquo is always a ldquoborder linerdquo ofthis ldquofourth directionrdquo At the center of the Bridersquosgarments the Stoppages recede anamorphically into thelabyrinth of the fourth dimension a space that isorthogonal to normal space Duchamp was probably aware that indescriptions of n-dimensional geometry when n is greater than3 the convention is to say that planes intersect at pointsunlike what happens in three-dimensional space where of

course they intersect along lines(79) The curvature of thestring does not really affect this n-dimensional argumentsince curvature depends upon whether or not the space

is Euclidean non-Euclidean or whatever(80) We can in a sense

choose the space to have any curvature we want(81)

In Tu mrsquo readymades cast shadows onto the surface of thepainting but these shadows do more than ride on the surfaceAs we have seen they are interlocked in curious ways with theentities depicted in the space of the picture convolutionsthat indicate Duchamp was interested in the readymades andtheir shadows as geometrical objects The shadows themselveshave perspectival implications and topological associationsand they are obviously seen differently under changing anglesof view As we walk ldquoaroundrdquo the picture it presents shiftingaspects In Tu mrsquo and indeed in most of his works Duchampwas interested in exploring both actual viewpoint andphilosophical point of view as well as the effects of the twoacting together

Such consequences were apparently on Duchamprsquos mind when hechose readymades bicycle wheels corkscrews and hat rackswere works of art depending upon how they were perceived Hewas involved with a discourse of surface (and reflectivesurface) in many of his works (often using glass and mirror intheir construction) Because projective analogies such asshadows and falling pieces of string can be related to severaldifferent geometries not just to n-dimensional Euclidean orfor that matter n-dimensional non-Euclidean geometry Duchampcan entail other regimes of meaning into his system Withinany given framework one which might say be used tointerpret theThree Standard Stoppages Network of StoppagesTu mrsquo the Large Glass Nine Malic Molds or the readymadesDuchamp understood that the implications of choosing onestandpoint over another were manifold (and the etymological

associations of this last term are germane here)(82)

Duchamp believed that just as how we use a particulargeometry to interpret the shape of the world is largely amatter of discretion as Poincareacute argued so too is our choiceof the interpretive frameworks that we use in making ouraesthetic judgments As an artist Duchamp was engaged inself-referential contemplative activities He tried to look

at himself seeing and by so doing to dislocate himself fromthe center of his own perspective

1 Interview with Francis Roberts ldquoI Propose toStrain the Laws of PhysicsrdquoArt News 67 (December 1968) 62

2Marcel Duchamp Salt Seller The Writings ofMarcel Duchamp (Marchand du Sel) ed Michel Sanouillet andElmer Peterson (New YorkOxford University Press 1973) 33

3In a note included in the Box of 1914 Duchampsays that ldquothe Three Standard Stoppages are the meterdiminishedrdquoIbid 22

4Interview with Katherine Kuh The ArtistrsquosVoice Talks with Seventeen Artists (New York Harper amp Row1960) 81

5The Network of Stoppages and its relationship tothe Large Glass is explained by Richard Hamilton The AlmostComplete Works of Marcel Duchamp (London Arts Council ofGreat Britain1966) 49 ldquoThe curved lines are drawn usingeach template of the Standard Stoppages three times once ineach of the three groups It was Duchamprsquos intention tophotograph the canvas from an angle in order to put the linesinto the perspective required for the Large Glassndasha means ofovercoming the difficulty of transferring the amorphous curves

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

A similar language could be used to describe the intersectionof the strings with the glass plates of the Three StandardStoppages They trail off at right-angles as it were alonglines that are orthogonal to the canvas strips as if they hadbeen rotated out of the virtual space of the ldquoPrussian bluerdquointo the actual space of the canvases If the strings areanalogous to ldquolines of sightrdquo they are like threads lyingldquoinrdquo the surface of the perspectival plane as we have seen inDesarguesrsquo perspective renderings (Figs 13 and 14) or inNiceacuteronrsquos illustration (Fig 23) In this sense the stringscan be taken as anamorphic lines crossing the representationalspace of the sheets of glass Recall what Duchamprsquos space wasintended to show his glass has ldquoneither front nor backneither top nor bottomrdquo and it can be used as a ldquothree-dimensional physical mediumrdquo in the construction of a ldquofour-dimensional perspectiverdquo In the Large Glass and the ThreeStandard Stoppages Duchamp was both literally andfiguratively boxing and encasing the geometrical elements ofhis iconographyndashinside glass and inside an n-dimensionalprojective system With Tu mrsquo he was also enclosing the basicelements of his own working method and indeed the basicelements of painting as a general practice inside a complexpictorial space one with unusual curvatures

Duchamprsquos works such as the ones I have discussed in thispaper with their various projections and intersections eachin their turn folding up into the next suggest that he wasthinking about different kinds of geometries Henri Poincareacuteamong the artistrsquos most likely mathematical sources often

discusses the interrelationships of geometries(72)

Projective geometry which was prefigured in Renaissanceperspective and initially elaborated in the work of suchseventeenth-century mathematicians as Desargues and Blaise

Pascal(73)

was later during the nineteenth century recognized as beingcentral to mathematics in general By the end of the century

both Euclidean and non-Euclidean geometry had been subsumedunder the principles of projective geometry(74)

Projective geometry deals with properties of geometricalfigures that remain invariant under transformation It studiesmappings of one figure onto another brought about byprojection and section and it tries to find qualities thatremain fixed during these procedures (Desarguesrsquo Theorem andPascalrsquos Theorem describe famous examples) Twentieth-centurymathematicians have invented methods of transformation thatare even more general than projection and section One of themost important of these approaches topology considersgeometrical properties of figures that are unchanged whilethese figures undergo deformations such as stretching andbending Especially in the context of the present discussionPoincareacute can be thought of as the ldquofather

of modern topologyrdquo (75) a subject that he referred to asanalysis situs (Latin for ldquoanalysis of the siterdquo ldquotopologyrdquocoming from the Greek equivalent for ldquostudy of the placerdquo) Hepoints out that this geometry ldquogives rise to a series oftheorems just as closely interconnected as those of Euclidrdquo(76)

Duchamprsquos Tu mrsquo can very nearly serve as an illustration forPoincareacutersquos arguments As pointed out earlier the elongatedshadows can be taken as anamorphic deformations and thus asreferences to topological transformations with four-dimensional or more generally n-dimensional ramifications(branchings) particularly insofar as anamorphic projectionsseem to intersect normal space at oblique angles In ways thatare like Holbeinrsquos famous skull the cast shadows in Tu mrsquoseem to traverse the space of the picture and in this sensethey are orthogonal to it (shadows are literally orthogonal tothe surfaces on which they are cast) From the perspective ofthe fourth dimension the strings in Three Standard Stoppagescan also be interpreted as falling away from normal space

along perpendicular lines at least insofar as they plummettoward the horizon of the Bride Duchamprsquos cast shadows andperhaps his cast segments of strings are projective analogiesfor higher-dimensional spaces His general approach can beseen in the following note

For an ordinary eye a point in a three-dimensional spacehides conceals the fourth direction of the continuumndashwhich isto say that this eye can try to perceive physically thisfourth direction by going around the said point From whateverangle it looks at the point this point will always be theborder line of the fourth directionndashjust as an ordinary eyegoing around a mirror will never be able to perceive anythingbut the reflected three-dimensional image and nothing from

behind(77)

Looked at ldquoedge-onrdquo in the sense of being seen undergoing ann-dimensional rotation the individual ldquostoppagesrdquo can betaken as trailing off into the fourth direction of whatDuchamp

calls the ldquoeacutetenduerdquo(78)From such a perspective they would beperceived as points The viewer equipped with a four-dimensional visual system to use Duchamprsquos words would beable to ascertain that a ldquopointrdquo is always a ldquoborder linerdquo ofthis ldquofourth directionrdquo At the center of the Bridersquosgarments the Stoppages recede anamorphically into thelabyrinth of the fourth dimension a space that isorthogonal to normal space Duchamp was probably aware that indescriptions of n-dimensional geometry when n is greater than3 the convention is to say that planes intersect at pointsunlike what happens in three-dimensional space where of

course they intersect along lines(79) The curvature of thestring does not really affect this n-dimensional argumentsince curvature depends upon whether or not the space

is Euclidean non-Euclidean or whatever(80) We can in a sense

choose the space to have any curvature we want(81)

In Tu mrsquo readymades cast shadows onto the surface of thepainting but these shadows do more than ride on the surfaceAs we have seen they are interlocked in curious ways with theentities depicted in the space of the picture convolutionsthat indicate Duchamp was interested in the readymades andtheir shadows as geometrical objects The shadows themselveshave perspectival implications and topological associationsand they are obviously seen differently under changing anglesof view As we walk ldquoaroundrdquo the picture it presents shiftingaspects In Tu mrsquo and indeed in most of his works Duchampwas interested in exploring both actual viewpoint andphilosophical point of view as well as the effects of the twoacting together

Such consequences were apparently on Duchamprsquos mind when hechose readymades bicycle wheels corkscrews and hat rackswere works of art depending upon how they were perceived Hewas involved with a discourse of surface (and reflectivesurface) in many of his works (often using glass and mirror intheir construction) Because projective analogies such asshadows and falling pieces of string can be related to severaldifferent geometries not just to n-dimensional Euclidean orfor that matter n-dimensional non-Euclidean geometry Duchampcan entail other regimes of meaning into his system Withinany given framework one which might say be used tointerpret theThree Standard Stoppages Network of StoppagesTu mrsquo the Large Glass Nine Malic Molds or the readymadesDuchamp understood that the implications of choosing onestandpoint over another were manifold (and the etymological

associations of this last term are germane here)(82)

Duchamp believed that just as how we use a particulargeometry to interpret the shape of the world is largely amatter of discretion as Poincareacute argued so too is our choiceof the interpretive frameworks that we use in making ouraesthetic judgments As an artist Duchamp was engaged inself-referential contemplative activities He tried to look

at himself seeing and by so doing to dislocate himself fromthe center of his own perspective

1 Interview with Francis Roberts ldquoI Propose toStrain the Laws of PhysicsrdquoArt News 67 (December 1968) 62

2Marcel Duchamp Salt Seller The Writings ofMarcel Duchamp (Marchand du Sel) ed Michel Sanouillet andElmer Peterson (New YorkOxford University Press 1973) 33

3In a note included in the Box of 1914 Duchampsays that ldquothe Three Standard Stoppages are the meterdiminishedrdquoIbid 22

4Interview with Katherine Kuh The ArtistrsquosVoice Talks with Seventeen Artists (New York Harper amp Row1960) 81

5The Network of Stoppages and its relationship tothe Large Glass is explained by Richard Hamilton The AlmostComplete Works of Marcel Duchamp (London Arts Council ofGreat Britain1966) 49 ldquoThe curved lines are drawn usingeach template of the Standard Stoppages three times once ineach of the three groups It was Duchamprsquos intention tophotograph the canvas from an angle in order to put the linesinto the perspective required for the Large Glassndasha means ofovercoming the difficulty of transferring the amorphous curves

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

both Euclidean and non-Euclidean geometry had been subsumedunder the principles of projective geometry(74)

Projective geometry deals with properties of geometricalfigures that remain invariant under transformation It studiesmappings of one figure onto another brought about byprojection and section and it tries to find qualities thatremain fixed during these procedures (Desarguesrsquo Theorem andPascalrsquos Theorem describe famous examples) Twentieth-centurymathematicians have invented methods of transformation thatare even more general than projection and section One of themost important of these approaches topology considersgeometrical properties of figures that are unchanged whilethese figures undergo deformations such as stretching andbending Especially in the context of the present discussionPoincareacute can be thought of as the ldquofather

of modern topologyrdquo (75) a subject that he referred to asanalysis situs (Latin for ldquoanalysis of the siterdquo ldquotopologyrdquocoming from the Greek equivalent for ldquostudy of the placerdquo) Hepoints out that this geometry ldquogives rise to a series oftheorems just as closely interconnected as those of Euclidrdquo(76)

Duchamprsquos Tu mrsquo can very nearly serve as an illustration forPoincareacutersquos arguments As pointed out earlier the elongatedshadows can be taken as anamorphic deformations and thus asreferences to topological transformations with four-dimensional or more generally n-dimensional ramifications(branchings) particularly insofar as anamorphic projectionsseem to intersect normal space at oblique angles In ways thatare like Holbeinrsquos famous skull the cast shadows in Tu mrsquoseem to traverse the space of the picture and in this sensethey are orthogonal to it (shadows are literally orthogonal tothe surfaces on which they are cast) From the perspective ofthe fourth dimension the strings in Three Standard Stoppagescan also be interpreted as falling away from normal space

along perpendicular lines at least insofar as they plummettoward the horizon of the Bride Duchamprsquos cast shadows andperhaps his cast segments of strings are projective analogiesfor higher-dimensional spaces His general approach can beseen in the following note

For an ordinary eye a point in a three-dimensional spacehides conceals the fourth direction of the continuumndashwhich isto say that this eye can try to perceive physically thisfourth direction by going around the said point From whateverangle it looks at the point this point will always be theborder line of the fourth directionndashjust as an ordinary eyegoing around a mirror will never be able to perceive anythingbut the reflected three-dimensional image and nothing from

behind(77)

Looked at ldquoedge-onrdquo in the sense of being seen undergoing ann-dimensional rotation the individual ldquostoppagesrdquo can betaken as trailing off into the fourth direction of whatDuchamp

calls the ldquoeacutetenduerdquo(78)From such a perspective they would beperceived as points The viewer equipped with a four-dimensional visual system to use Duchamprsquos words would beable to ascertain that a ldquopointrdquo is always a ldquoborder linerdquo ofthis ldquofourth directionrdquo At the center of the Bridersquosgarments the Stoppages recede anamorphically into thelabyrinth of the fourth dimension a space that isorthogonal to normal space Duchamp was probably aware that indescriptions of n-dimensional geometry when n is greater than3 the convention is to say that planes intersect at pointsunlike what happens in three-dimensional space where of

course they intersect along lines(79) The curvature of thestring does not really affect this n-dimensional argumentsince curvature depends upon whether or not the space

is Euclidean non-Euclidean or whatever(80) We can in a sense

choose the space to have any curvature we want(81)

In Tu mrsquo readymades cast shadows onto the surface of thepainting but these shadows do more than ride on the surfaceAs we have seen they are interlocked in curious ways with theentities depicted in the space of the picture convolutionsthat indicate Duchamp was interested in the readymades andtheir shadows as geometrical objects The shadows themselveshave perspectival implications and topological associationsand they are obviously seen differently under changing anglesof view As we walk ldquoaroundrdquo the picture it presents shiftingaspects In Tu mrsquo and indeed in most of his works Duchampwas interested in exploring both actual viewpoint andphilosophical point of view as well as the effects of the twoacting together

Such consequences were apparently on Duchamprsquos mind when hechose readymades bicycle wheels corkscrews and hat rackswere works of art depending upon how they were perceived Hewas involved with a discourse of surface (and reflectivesurface) in many of his works (often using glass and mirror intheir construction) Because projective analogies such asshadows and falling pieces of string can be related to severaldifferent geometries not just to n-dimensional Euclidean orfor that matter n-dimensional non-Euclidean geometry Duchampcan entail other regimes of meaning into his system Withinany given framework one which might say be used tointerpret theThree Standard Stoppages Network of StoppagesTu mrsquo the Large Glass Nine Malic Molds or the readymadesDuchamp understood that the implications of choosing onestandpoint over another were manifold (and the etymological

associations of this last term are germane here)(82)

Duchamp believed that just as how we use a particulargeometry to interpret the shape of the world is largely amatter of discretion as Poincareacute argued so too is our choiceof the interpretive frameworks that we use in making ouraesthetic judgments As an artist Duchamp was engaged inself-referential contemplative activities He tried to look

at himself seeing and by so doing to dislocate himself fromthe center of his own perspective

1 Interview with Francis Roberts ldquoI Propose toStrain the Laws of PhysicsrdquoArt News 67 (December 1968) 62

2Marcel Duchamp Salt Seller The Writings ofMarcel Duchamp (Marchand du Sel) ed Michel Sanouillet andElmer Peterson (New YorkOxford University Press 1973) 33

3In a note included in the Box of 1914 Duchampsays that ldquothe Three Standard Stoppages are the meterdiminishedrdquoIbid 22

4Interview with Katherine Kuh The ArtistrsquosVoice Talks with Seventeen Artists (New York Harper amp Row1960) 81

5The Network of Stoppages and its relationship tothe Large Glass is explained by Richard Hamilton The AlmostComplete Works of Marcel Duchamp (London Arts Council ofGreat Britain1966) 49 ldquoThe curved lines are drawn usingeach template of the Standard Stoppages three times once ineach of the three groups It was Duchamprsquos intention tophotograph the canvas from an angle in order to put the linesinto the perspective required for the Large Glassndasha means ofovercoming the difficulty of transferring the amorphous curves

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

along perpendicular lines at least insofar as they plummettoward the horizon of the Bride Duchamprsquos cast shadows andperhaps his cast segments of strings are projective analogiesfor higher-dimensional spaces His general approach can beseen in the following note

For an ordinary eye a point in a three-dimensional spacehides conceals the fourth direction of the continuumndashwhich isto say that this eye can try to perceive physically thisfourth direction by going around the said point From whateverangle it looks at the point this point will always be theborder line of the fourth directionndashjust as an ordinary eyegoing around a mirror will never be able to perceive anythingbut the reflected three-dimensional image and nothing from

behind(77)

Looked at ldquoedge-onrdquo in the sense of being seen undergoing ann-dimensional rotation the individual ldquostoppagesrdquo can betaken as trailing off into the fourth direction of whatDuchamp

calls the ldquoeacutetenduerdquo(78)From such a perspective they would beperceived as points The viewer equipped with a four-dimensional visual system to use Duchamprsquos words would beable to ascertain that a ldquopointrdquo is always a ldquoborder linerdquo ofthis ldquofourth directionrdquo At the center of the Bridersquosgarments the Stoppages recede anamorphically into thelabyrinth of the fourth dimension a space that isorthogonal to normal space Duchamp was probably aware that indescriptions of n-dimensional geometry when n is greater than3 the convention is to say that planes intersect at pointsunlike what happens in three-dimensional space where of

course they intersect along lines(79) The curvature of thestring does not really affect this n-dimensional argumentsince curvature depends upon whether or not the space

is Euclidean non-Euclidean or whatever(80) We can in a sense

choose the space to have any curvature we want(81)

In Tu mrsquo readymades cast shadows onto the surface of thepainting but these shadows do more than ride on the surfaceAs we have seen they are interlocked in curious ways with theentities depicted in the space of the picture convolutionsthat indicate Duchamp was interested in the readymades andtheir shadows as geometrical objects The shadows themselveshave perspectival implications and topological associationsand they are obviously seen differently under changing anglesof view As we walk ldquoaroundrdquo the picture it presents shiftingaspects In Tu mrsquo and indeed in most of his works Duchampwas interested in exploring both actual viewpoint andphilosophical point of view as well as the effects of the twoacting together

Such consequences were apparently on Duchamprsquos mind when hechose readymades bicycle wheels corkscrews and hat rackswere works of art depending upon how they were perceived Hewas involved with a discourse of surface (and reflectivesurface) in many of his works (often using glass and mirror intheir construction) Because projective analogies such asshadows and falling pieces of string can be related to severaldifferent geometries not just to n-dimensional Euclidean orfor that matter n-dimensional non-Euclidean geometry Duchampcan entail other regimes of meaning into his system Withinany given framework one which might say be used tointerpret theThree Standard Stoppages Network of StoppagesTu mrsquo the Large Glass Nine Malic Molds or the readymadesDuchamp understood that the implications of choosing onestandpoint over another were manifold (and the etymological

associations of this last term are germane here)(82)

Duchamp believed that just as how we use a particulargeometry to interpret the shape of the world is largely amatter of discretion as Poincareacute argued so too is our choiceof the interpretive frameworks that we use in making ouraesthetic judgments As an artist Duchamp was engaged inself-referential contemplative activities He tried to look

at himself seeing and by so doing to dislocate himself fromthe center of his own perspective

1 Interview with Francis Roberts ldquoI Propose toStrain the Laws of PhysicsrdquoArt News 67 (December 1968) 62

2Marcel Duchamp Salt Seller The Writings ofMarcel Duchamp (Marchand du Sel) ed Michel Sanouillet andElmer Peterson (New YorkOxford University Press 1973) 33

3In a note included in the Box of 1914 Duchampsays that ldquothe Three Standard Stoppages are the meterdiminishedrdquoIbid 22

4Interview with Katherine Kuh The ArtistrsquosVoice Talks with Seventeen Artists (New York Harper amp Row1960) 81

5The Network of Stoppages and its relationship tothe Large Glass is explained by Richard Hamilton The AlmostComplete Works of Marcel Duchamp (London Arts Council ofGreat Britain1966) 49 ldquoThe curved lines are drawn usingeach template of the Standard Stoppages three times once ineach of the three groups It was Duchamprsquos intention tophotograph the canvas from an angle in order to put the linesinto the perspective required for the Large Glassndasha means ofovercoming the difficulty of transferring the amorphous curves

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

In Tu mrsquo readymades cast shadows onto the surface of thepainting but these shadows do more than ride on the surfaceAs we have seen they are interlocked in curious ways with theentities depicted in the space of the picture convolutionsthat indicate Duchamp was interested in the readymades andtheir shadows as geometrical objects The shadows themselveshave perspectival implications and topological associationsand they are obviously seen differently under changing anglesof view As we walk ldquoaroundrdquo the picture it presents shiftingaspects In Tu mrsquo and indeed in most of his works Duchampwas interested in exploring both actual viewpoint andphilosophical point of view as well as the effects of the twoacting together

Such consequences were apparently on Duchamprsquos mind when hechose readymades bicycle wheels corkscrews and hat rackswere works of art depending upon how they were perceived Hewas involved with a discourse of surface (and reflectivesurface) in many of his works (often using glass and mirror intheir construction) Because projective analogies such asshadows and falling pieces of string can be related to severaldifferent geometries not just to n-dimensional Euclidean orfor that matter n-dimensional non-Euclidean geometry Duchampcan entail other regimes of meaning into his system Withinany given framework one which might say be used tointerpret theThree Standard Stoppages Network of StoppagesTu mrsquo the Large Glass Nine Malic Molds or the readymadesDuchamp understood that the implications of choosing onestandpoint over another were manifold (and the etymological

associations of this last term are germane here)(82)

Duchamp believed that just as how we use a particulargeometry to interpret the shape of the world is largely amatter of discretion as Poincareacute argued so too is our choiceof the interpretive frameworks that we use in making ouraesthetic judgments As an artist Duchamp was engaged inself-referential contemplative activities He tried to look

at himself seeing and by so doing to dislocate himself fromthe center of his own perspective

1 Interview with Francis Roberts ldquoI Propose toStrain the Laws of PhysicsrdquoArt News 67 (December 1968) 62

2Marcel Duchamp Salt Seller The Writings ofMarcel Duchamp (Marchand du Sel) ed Michel Sanouillet andElmer Peterson (New YorkOxford University Press 1973) 33

3In a note included in the Box of 1914 Duchampsays that ldquothe Three Standard Stoppages are the meterdiminishedrdquoIbid 22

4Interview with Katherine Kuh The ArtistrsquosVoice Talks with Seventeen Artists (New York Harper amp Row1960) 81

5The Network of Stoppages and its relationship tothe Large Glass is explained by Richard Hamilton The AlmostComplete Works of Marcel Duchamp (London Arts Council ofGreat Britain1966) 49 ldquoThe curved lines are drawn usingeach template of the Standard Stoppages three times once ineach of the three groups It was Duchamprsquos intention tophotograph the canvas from an angle in order to put the linesinto the perspective required for the Large Glassndasha means ofovercoming the difficulty of transferring the amorphous curves

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

at himself seeing and by so doing to dislocate himself fromthe center of his own perspective

1 Interview with Francis Roberts ldquoI Propose toStrain the Laws of PhysicsrdquoArt News 67 (December 1968) 62

2Marcel Duchamp Salt Seller The Writings ofMarcel Duchamp (Marchand du Sel) ed Michel Sanouillet andElmer Peterson (New YorkOxford University Press 1973) 33

3In a note included in the Box of 1914 Duchampsays that ldquothe Three Standard Stoppages are the meterdiminishedrdquoIbid 22

4Interview with Katherine Kuh The ArtistrsquosVoice Talks with Seventeen Artists (New York Harper amp Row1960) 81

5The Network of Stoppages and its relationship tothe Large Glass is explained by Richard Hamilton The AlmostComplete Works of Marcel Duchamp (London Arts Council ofGreat Britain1966) 49 ldquoThe curved lines are drawn usingeach template of the Standard Stoppages three times once ineach of the three groups It was Duchamprsquos intention tophotograph the canvas from an angle in order to put the linesinto the perspective required for the Large Glassndasha means ofovercoming the difficulty of transferring the amorphous curves

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

through normal perspective projection Photography did notprove up to the assignment and a perspective drawing had to bemaderdquo

6 Linda Dalrymple Henderson Duchamp in ContextScience and Technology in the ldquoLarge Glassrdquo and Related Works(Princeton Princeton University Press 1998) 63 105 shecredits Ulf Linde with drawing her attention to the differentcolors of the glass plates see his Marcel Duchamp (StockholmRabeacuten and Sjoumlgren 1986) 138

7 Ulf Linde ldquoMARieacutee CELibatairerdquo in WalterHopps Ulf Linde and Arturo Schwarz Marcel Duchamp Ready-Mades etc (1913-1964) (Paris Le Terrain Vague 1964) 48see also Arturo Schwarz The Complete Works of Marcel Duchamp(New York Abrams 1970) 463 Henderson (cited n 6) 105quotes this passage from Linde in her interpretation of theBridersquos ldquoclothingrdquo as a condenser

8Duchamp Salt Seller (cited n 2) 22 33

9This important discovery was made recently byRhonda Roland Shearerand Stephen Jay Gould see their essayldquoHidden in Plain SightDuchamprsquos Three Standard StoppagesMore Truly a `Stoppage(An Invisible Mending) Than We EverRealizedrdquo Tout-FaitThe Marcel Duchamp Studies Online Journal1 no 1 (December1999) Newslthttpwwwtoutfaitcomduchampjsppostid=677ampkeyword=

10See Craig Adcock Marcel Duchamprsquos Notes fromthe ldquoLarge Glassrdquo An N-Dimensional Analysis (Ann ArborMichUMI Research Press 1983) esp 135-46 189-90 see alsoidemrdquoMarcel Duchamprsquos `Instantaneacutesrsquo Photography and theEventStructure of the Ready-Madesrdquo in ldquoEventrdquo Arts and ArtEvents ed Stephen C Foster (Ann Arbor Mich UMI ResearchPress 1988) 239-66

11Duchamprsquos Three Standard Stoppages and Mareyrsquoschronophotographs are discussed by Jean Clair Duchamp et laphotographie Essai drsquoanalyse drsquoun primat technique sur ledeacuteveloppement drsquoune oeuvre (Paris Eacuteditions du Checircne 1977)26-28 52 For statements by Duchamp about chronophotographysee his interviews with James Johnson Sweeney ldquoElevenEuropeans in Americardquo Museum of Modern Art Bulletin 13(1946) 19-21 reprinted in Duchamp Salt Seller 123-26 andwith Pierre Cabanne Dialogues with Marcel Duchamp trans RonPadgett (New York Viking Press 1971) 34 For Mareyrsquos worksee Eacutetienne-Jules Marey Le Mouvement (Paris G MassonEacutediteur 1894)

12Schwarz (cited n 7) 444 says that Duchamprsquoschose his title after seeing a sign on a Parisian shopadvertizing ldquostoppagerdquo see also Francis Naumann The Mary andWilliam Sisler Collection (New York Museum of Modern Art1984) 168-71 Jennifer Gough-Cooper and Jacques CaumontldquoEphemerides on and about Marcel Duchamp and Rrose Seacutelavy1887-1968rdquo in Marcel Duchamp Work and Life ed PontusHulten (Cambridge MIT Press 1993) in their entry for May19 1914 have suggested that the sign read ldquostoppages et

talonsrdquo which would imply fixing holes in the heels (talons)of socks and stockings

13Robert Lebel Marcel Duchamp with texts byAndreacute Breton and H-P Rocheacute trans George Heard Hamilton(New York Grove Press 1959) 54

14In an interview with James Johnson Sweeneyfilmed at the Philadelphia Museum of Art and broadcast as partof the ldquoWisdomrdquo series on NBC television in January 1956Duchamp himself put forward a similar argument ldquoI like thecracks the way they fall You remember how it happened in1926 in Brooklyn They put the two panes on top of oneanother on a truck flat not knowing what they were carryingand bounced for sixty miles into Connecticut and thatrsquos theresult But the more I look at it the more I like the cracksthey are not like shattered glass They have a shape There isa symmetry in the cracking the two crackings aresymmetrically arranged and there is more almost an intentionthere an extrandasha curious intention that I am not responsiblefor a ready-made intention in other words that I respectand loverdquo ldquoA Conversation with Marcel Duchamprdquo reprinted inDuchampSalt Seller (cited n 2) 127-37 the quote is from p127 The Large Glass was on view at the ldquoInternationalExhibition of Modern Artrdquo at the Brooklyn Museum betweenNovember 17 1926 and January 9 1927 It thus must have beenbroken on its way back to Katherine S Dreierrsquos home in WestRedding Connecticut in early 1927 rather than in 1926 asDuchamp says

15Interview with Cabanne (cited n 11) 75 ldquoItrsquosa lot better with the breaks a hundred times better Itrsquos thedestiny of thingsrdquo See also Mark B Pohlad ldquo`MacaroniRepaired is Ready for Thursday rsquo Marcel Duchamp asConservatorrdquo Tout-Fait The Marcel Duchamp Studies OnlineJournal 1 no 3 (December 2002) Articleslthttpwwwtoutfaitcomduchampjsppostid=910ampkeyword=gt

16Henderson (cited n 6) discusses the Bridersquosldquogarmentsrdquo and their relationship with the Three StandardStoppages in terms of ldquotelegraphyrdquo comparing the glass platesin these works to such devices as condensers and insulatorssee especially her chap 8 ldquoThe Large Glass as a Painting ofElectromagnetic Frequencyrdquo

17Duchamp Salt Seller (cited n 2) 39

18Marcel Duchamp Notes ed and trans PaulMatisse (Paris Centre Georges Pompidou 1980) no 154

19Marcel Duchamp Notes ed and trans PaulMatisse (ParisCentre Georges Pompidou 1980) no 154

20For a more complete discussion of these ideassee Craig Adcock ldquoConventionalism in Henri Poincareacute andMarcel Duchamprdquo Art Journal 44 (fall 1984) 249-58 see alsoidem Marcel Duchamprsquos Notes (cited n 10) 149-54

talonsrdquo which would imply fixing holes in the heels (talons)of socks and stockings

13Robert Lebel Marcel Duchamp with texts byAndreacute Breton and H-P Rocheacute trans George Heard Hamilton(New York Grove Press 1959) 54

14In an interview with James Johnson Sweeneyfilmed at the Philadelphia Museum of Art and broadcast as partof the ldquoWisdomrdquo series on NBC television in January 1956Duchamp himself put forward a similar argument ldquoI like thecracks the way they fall You remember how it happened in1926 in Brooklyn They put the two panes on top of oneanother on a truck flat not knowing what they were carryingand bounced for sixty miles into Connecticut and thatrsquos theresult But the more I look at it the more I like the cracksthey are not like shattered glass They have a shape There isa symmetry in the cracking the two crackings aresymmetrically arranged and there is more almost an intentionthere an extrandasha curious intention that I am not responsiblefor a ready-made intention in other words that I respectand loverdquo ldquoA Conversation with Marcel Duchamprdquo reprinted inDuchampSalt Seller (cited n 2) 127-37 the quote is from p127 The Large Glass was on view at the ldquoInternationalExhibition of Modern Artrdquo at the Brooklyn Museum betweenNovember 17 1926 and January 9 1927 It thus must have beenbroken on its way back to Katherine S Dreierrsquos home in WestRedding Connecticut in early 1927 rather than in 1926 asDuchamp says

15Interview with Cabanne (cited n 11) 75 ldquoItrsquosa lot better with the breaks a hundred times better Itrsquos thedestiny of thingsrdquo See also Mark B Pohlad ldquo`MacaroniRepaired is Ready for Thursday rsquo Marcel Duchamp asConservatorrdquo Tout-Fait The Marcel Duchamp Studies OnlineJournal 1 no 3 (December 2002) Articleslthttpwwwtoutfaitcomduchampjsppostid=910ampkeyword=gt

16Henderson (cited n 6) discusses the Bridersquosldquogarmentsrdquo and their relationship with the Three StandardStoppages in terms of ldquotelegraphyrdquo comparing the glass platesin these works to such devices as condensers and insulatorssee especially her chap 8 ldquoThe Large Glass as a Painting ofElectromagnetic Frequencyrdquo

17Duchamp Salt Seller (cited n 2) 39

18Marcel Duchamp Notes ed and trans PaulMatisse (Paris Centre Georges Pompidou 1980) no 154

19Marcel Duchamp Notes ed and trans PaulMatisse (ParisCentre Georges Pompidou 1980) no 154

20For a more complete discussion of these ideassee Craig Adcock ldquoConventionalism in Henri Poincareacute andMarcel Duchamprdquo Art Journal 44 (fall 1984) 249-58 see alsoidem Marcel Duchamprsquos Notes (cited n 10) 149-54

15Interview with Cabanne (cited n 11) 75 ldquoItrsquosa lot better with the breaks a hundred times better Itrsquos thedestiny of thingsrdquo See also Mark B Pohlad ldquo`MacaroniRepaired is Ready for Thursday rsquo Marcel Duchamp asConservatorrdquo Tout-Fait The Marcel Duchamp Studies OnlineJournal 1 no 3 (December 2002) Articleslthttpwwwtoutfaitcomduchampjsppostid=910ampkeyword=gt

16Henderson (cited n 6) discusses the Bridersquosldquogarmentsrdquo and their relationship with the Three StandardStoppages in terms of ldquotelegraphyrdquo comparing the glass platesin these works to such devices as condensers and insulatorssee especially her chap 8 ldquoThe Large Glass as a Painting ofElectromagnetic Frequencyrdquo

17Duchamp Salt Seller (cited n 2) 39

18Marcel Duchamp Notes ed and trans PaulMatisse (Paris Centre Georges Pompidou 1980) no 154

19Marcel Duchamp Notes ed and trans PaulMatisse (ParisCentre Georges Pompidou 1980) no 154

20For a more complete discussion of these ideassee Craig Adcock ldquoConventionalism in Henri Poincareacute andMarcel Duchamprdquo Art Journal 44 (fall 1984) 249-58 see alsoidem Marcel Duchamprsquos Notes (cited n 10) 149-54

21Ecke Bonk Marcel Duchamp The Box in a Valisede ou par Marcel Duchamp ou Rrose Seacutelavy trans David Britt(New York Rizzoli 1989) 216-20 See also the letters Duchampsent to Dreier during late 1935 and early 1936 inAffectionately Marcel The Selected Correspondence of MarcelDuchamp ed Francis M Naumann and Hector Obalk (Ghent andAmsterdam Ludion Press 2000) 199-207

22For a discussion of Duchamprsquos approach alongsomewhat different lines see Craig Adcock ldquoDuchamprsquos WayTwisting Our Memory of the Past `For the Fun of Itrdquo in TheDefinitivelyUnfinished Marcel Duchamp ed Thierry de Duve (Halifax NovaScotia College of Art and Design Cambridge and London MITPress 1991) 311-34

23Interview Kuh (cited n 4) 92

24Interview with Cabanne (cited 11) 75

25Duchamp Duchamp du Signe (cited n 18) 50

26Esprit Pascal Jouffret Traiteacute eacuteleacutementaire degeacuteomeacutetrie agrave quatre dimensions et introduction agrave la geacuteomeacutetrie agrave

n dimensions (Paris Gauthier-Villars 1903) xxviii For amore detailed discussion of Jouffretrsquos usage and itsimportance for Duchamprsquos concept of inframince see AdcockMarcel Duchamprsquos Notes (cited n 10) 48-55

27 Duchamp Salt Seller (cited n 2) 88 Formore detailed analyses of Duchamprsquos use of glass and mirror asmetaphors for four-dimensional perspective see Adcock MarcelDuchamprsquos Notes (cited n 10) esp 75-79 146-49 also idemldquoGeometrical Complication in the Art of Marcel Duchamprdquo ArtsMagazine 58 (January 1984) 105-09

28Interview with Cabanne (cited n 11) 47

29Ibid 38

30Duchamp Notes (cited n 19) no 139 see alsono153

31See Henderson (cited n 6) 63 ldquoThe Stoppageslsquoarrangement of one clear and two greenish glass platesparallels exactly that of the glass strips mounted on theLarge Glass the top strip is clear and the two below aregreenish in hue Because Duchamp located the BridersquosldquoClothingrdquo at the midsection of the Glass the gravity-drawnthread lines of the Stoppages may have become for him a

metonymical sign for the fallen garment of the Briderdquo

32Linde ldquoMARieacutee CELibatairerdquo (cited n 7) 60Arturo Schwarz (cited n 7 p 463) says that Duchamp relatedTravelerrsquos Folding Item to a ldquofeminine skirtrdquo See also MollyNesbit and Naomi Sawelson-Gorse ldquoConcept of Nothing NewNotes by Marcel Duchamp and Walter Arensbergrdquo The DuchampEffect Essays Interviews Round Table ed Martha Buskirkand Mignon Nixon (Cambridge Mass and London MIT Press1996) 131-75 For a number of fascinating connections betweenDuchamprsquos Travelerrsquos Folding Item and the world at large seeRhonda Roland Shearer ldquoMarcel Duchamp A Readymade Case forCollecting Objects of Our Cultural Heritage along with Worksof Artrdquo Tout-Fait The Marcel Duchamp Studies Online Journal1 no 3 (December 2000) Collectionslthttpwwwtoutfaitcomduchampjsppostid=1090ampkeyword=gt

33Interview with Roberts (cited n 1) 62

34Hilary Putnam for example has said that ldquotheoverthrow of Euclidean geometry is the most important event inthe history of science for the epistemologistrdquo See hisMathematics Matter and Method 2d ed (Cambridge CambridgeUniversity Press 1979) x

35For one of the most complete discussions ofDesarguesrsquo work and for the most reliable translations of histexts see J V Field and J J Gray The Geometrical Work of

n dimensions (Paris Gauthier-Villars 1903) xxviii For amore detailed discussion of Jouffretrsquos usage and itsimportance for Duchamprsquos concept of inframince see AdcockMarcel Duchamprsquos Notes (cited n 10) 48-55

27 Duchamp Salt Seller (cited n 2) 88 Formore detailed analyses of Duchamprsquos use of glass and mirror asmetaphors for four-dimensional perspective see Adcock MarcelDuchamprsquos Notes (cited n 10) esp 75-79 146-49 also idemldquoGeometrical Complication in the Art of Marcel Duchamprdquo ArtsMagazine 58 (January 1984) 105-09

28Interview with Cabanne (cited n 11) 47

29Ibid 38

30Duchamp Notes (cited n 19) no 139 see alsono153

31See Henderson (cited n 6) 63 ldquoThe Stoppageslsquoarrangement of one clear and two greenish glass platesparallels exactly that of the glass strips mounted on theLarge Glass the top strip is clear and the two below aregreenish in hue Because Duchamp located the BridersquosldquoClothingrdquo at the midsection of the Glass the gravity-drawnthread lines of the Stoppages may have become for him a

metonymical sign for the fallen garment of the Briderdquo

32Linde ldquoMARieacutee CELibatairerdquo (cited n 7) 60Arturo Schwarz (cited n 7 p 463) says that Duchamp relatedTravelerrsquos Folding Item to a ldquofeminine skirtrdquo See also MollyNesbit and Naomi Sawelson-Gorse ldquoConcept of Nothing NewNotes by Marcel Duchamp and Walter Arensbergrdquo The DuchampEffect Essays Interviews Round Table ed Martha Buskirkand Mignon Nixon (Cambridge Mass and London MIT Press1996) 131-75 For a number of fascinating connections betweenDuchamprsquos Travelerrsquos Folding Item and the world at large seeRhonda Roland Shearer ldquoMarcel Duchamp A Readymade Case forCollecting Objects of Our Cultural Heritage along with Worksof Artrdquo Tout-Fait The Marcel Duchamp Studies Online Journal1 no 3 (December 2000) Collectionslthttpwwwtoutfaitcomduchampjsppostid=1090ampkeyword=gt

33Interview with Roberts (cited n 1) 62

34Hilary Putnam for example has said that ldquotheoverthrow of Euclidean geometry is the most important event inthe history of science for the epistemologistrdquo See hisMathematics Matter and Method 2d ed (Cambridge CambridgeUniversity Press 1979) x

35For one of the most complete discussions ofDesarguesrsquo work and for the most reliable translations of histexts see J V Field and J J Gray The Geometrical Work of

metonymical sign for the fallen garment of the Briderdquo

32Linde ldquoMARieacutee CELibatairerdquo (cited n 7) 60Arturo Schwarz (cited n 7 p 463) says that Duchamp relatedTravelerrsquos Folding Item to a ldquofeminine skirtrdquo See also MollyNesbit and Naomi Sawelson-Gorse ldquoConcept of Nothing NewNotes by Marcel Duchamp and Walter Arensbergrdquo The DuchampEffect Essays Interviews Round Table ed Martha Buskirkand Mignon Nixon (Cambridge Mass and London MIT Press1996) 131-75 For a number of fascinating connections betweenDuchamprsquos Travelerrsquos Folding Item and the world at large seeRhonda Roland Shearer ldquoMarcel Duchamp A Readymade Case forCollecting Objects of Our Cultural Heritage along with Worksof Artrdquo Tout-Fait The Marcel Duchamp Studies Online Journal1 no 3 (December 2000) Collectionslthttpwwwtoutfaitcomduchampjsppostid=1090ampkeyword=gt

33Interview with Roberts (cited n 1) 62

34Hilary Putnam for example has said that ldquotheoverthrow of Euclidean geometry is the most important event inthe history of science for the epistemologistrdquo See hisMathematics Matter and Method 2d ed (Cambridge CambridgeUniversity Press 1979) x

35For one of the most complete discussions ofDesarguesrsquo work and for the most reliable translations of histexts see J V Field and J J Gray The Geometrical Work of

Girard Desargues (New York Springer-Verlag 1987) Desarguesrsquoprincipal essay on projective geometry is Brouillon proiectdrsquoune atteinte aux evenemens des rencontres du Cone avec unPlan (Paris 1639) his earlier work on perspective isentitled Exemple de lrsquoune des manieres universelles duSGDL touchant la pratique de la perspective sans emploieraucun tiers point de distance ny drsquoautre nature qui foithors du champ de lrsquoouvrage (Paris 1636) ldquoSGDLrdquo is anabbreviation for ldquoSieur Girard Desargues Lyonnaisrdquo Thistwelve page brochure included the two high-quality engravedillustrations reproduced here which are almost certainly byAbraham Bosse (1602-1676) see J V Field The Invention ofInfinity Mathematics and Art in the Renaissance (Oxford andNew York Oxford University Press 1997) 192 Desarquesrsquoperspective treatise was included as an appendix in BossersquosManiere universelle de Mr Desargues pour pratiquer laperspective par petit-pied comme le Geometral (Paris 1648)

36For a discussion of this trend see MartinKemp ldquoGeometrical Perspective from Brunelleschi to DesarguesA Pictorial Means or an Intellectual Endrdquo Proceedings of theBritish Academy 70 (1984) 89-132

37Field (cited n 35) 192-95

38Erwin Panofsky Perspective as Symbolic Formtrans Christopher S Wood (New York Zone Books 1991)originally published as ldquoDie Perspektive als `symbolischeFormrdquo in Vortraumlge der Bibliothek Warburg 1924-1925 (Leipzigand Berlin 1927) 258-330 For a discussion of Panofskyrsquos

contributions to perspective studies particularly strong inits analysis of sources see Kim Veltman ldquoPanofskyrsquosPerspective A Half Century Laterrdquo in La Prospettivarinascimentale Codificazione e trasgressioni vol 1 edMarisa Dalai Emiliani (Florence Centro Di 1980) 565-84

39Duchamp Salt Seller (cited n 2) 42 ldquoThiscinematic blossoming which expresses the moment of thestripping should be grafted onto an arbor-type of the brideThis arbor-type has its roots in the desire-gears but thecinematic effects of the electrical stripping transmitted tothe motor with quite feeble cylinders leave (plasticnecessity) the arbor-type at rest (Graphically in Munich Ihad already made two studies of this arbor type) Do not touchthe desire-gears which by giving birth to the arbor-typefind within this arbor-type the transmission of the desire tothe blossoming into stripping voluntarily imagined by thebride desiringrdquo

40J V Field ldquoLinear Perspective and theProjectiveGeometry of Girard Desarguesrdquo Nuncius 2no 2(1987) 3-40

41Henderson (cited n 6) does not refer toDesargues in her discussion of the Bride as an ldquoarbor-typerdquoShe argues that because an ldquoarborrdquo is an ldquoaxlerdquo Duchamprsquosusage should be interpreted as a reference to such devices asthe shafts in automobile transmissions or electricalgenerators I completely agree that Duchamp could have hadthese kinds of associations in mind along with his taking an

ldquoarbrerdquo to refer to a geometrical axis of rotation

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

43Duchamp Salt Seller (cited n 2) 27 seealso idem Duchamp du Signe (cited n 18) 42

44Field ldquoLinear Perspective and the ProjectiveGeometry of Girard Desarguesrdquo (cited n 40) 21

45Duchamp Notes (cited n 19) no 57

46Ibid no 155

47There are two new sections in the upper rightcorner of the Large Glass with holes drilled through them tocreate the ldquonine shotsrdquo In photographs of the Large Glasstaken at the Brooklyn Museum in 1926-27 the ldquonine shotsrdquo arenot visible Duchamp may have incorporated them into the Glasswhen he was repairing it in 1936

48 ldquoArguesianrdquo would be the adjectivalcounterpart of ldquoCartesianrdquo Reneacute Descartes (1596-1650) andDesargues (1593-1662) were almost exact contemporaries andcommunicated with one another about mathematical matters seeField Invention of Infinity (cited n 35) 190-97 see alsoReneacute Taton LrsquoOeuvre matheacutematique de G Desargues Textespublieacutes et commenteacutes avec une introduction biographique ethistorique 2d rev ed (Lyon Institut InterdisciplinairedrsquoEtudes Episteacutemologiques distributed by the LibrairiePhilosophique J Vrin Paris 1988)

49I am indebted to Hector Obalk for drawing thisconnection (or reconnection) to my attention in his talk ldquoWhatIs an Object The Belated Career of the Readymaderdquo at theinterdisciplinary colloquium ldquoMethods of Understanding in Artand Science the Case of Duchamp and Poincareacuterdquo HarvardUniversity Cambridge Massachusetts November 7 1999

50Duchamp Notes (cited n 19) no 67

51Duchamp Salt Seller (cited n 2) 35

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

53Duchamp Duchamp du Signe (cited n 18) 54

54See Adcock Marcel Duchamprsquos Notes (cited n10)130-32

55Lebel (cited n 13) 132-33

56Robert Lebel Sur Marcel Duchamp with textsbyAndreacute Breton and H-P Rocheacute (Paris and LondonEacuteditionsTrianon 1958) a facsimile edition ofthis book was publishedby the Centre Georges PompidouParis in 1996

57This photograph appears in Robert LebelldquoDerniegravere soireacutee avec Marcel Duchamprdquo LrsquoOeil (Paris) no 167(November 1968) 18-21 also reproduced in the supplementldquoMarcel Duchamp et Robert Lebelrdquo in the facsimile edition ofSur Marcel Duchamp (cited n 56) see also Gough-Cooper andCaumont (cited n 12) under their entry for April 29 1942The photograph was taken just before Duchamp left France forthe United States Mirroring the famous movie script hesailed from Marseilles to Casablanca and from there to Lisbonand then to New York arriving on June 25 See Jennifer Gough-Cooper and Jacques Caumont Plan pour ecrire une vie de MarcelDuchamp vol 1 Marcel Duchamp catalogue (Paris CentreNational drsquoArt et de Culture Georges Pompidou Museacutee NationaldrsquoArt Moderne 1977) 23

58Bonk (cited n 21) 218 argues that Duchampfashioned the templates for the Three Standard Stoppages in1918 when he was working on Tu mrsquo and needed to draw theircurvatures several times This chronology would mean that heused something other than the templates perhaps tracing paperor some other means to draw the lines in Network of Stoppagesin 1914 See also Duchamprsquos correspondence with Katherine SDreier in Affectionately Marcel (cited n 21) 199-207

59For a more detailed discussion of Duchamprsquos useof shadows on Tu mrsquo see Adcock Marcel Duchamprsquos Notes (citedn 10) 41-49 For a more traditional approach but nonethelessinteresting for Duchamprsquos work see Thomas Da Costa KaufmannldquoThe Perspective of Shadows The History of the Theory ofShadow ProjectionrdquoJournal of the Warburg and CourtauldInstitutes 38 (1975) 258-87

60For a more detailed discussion of Tu mrsquo inrelation to non-Euclidean geometry and topology see AdcockMarcel Duchamprsquos Notes (cited n 10) 55-58 101-02

61Interview with Cabanne (cited n 12) 40

62Kemp (cited n 36) 123-24 points out thatDesarguesrsquo discussion of conic sections ldquohelped sow the seeds

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

contributions to perspective studies particularly strong inits analysis of sources see Kim Veltman ldquoPanofskyrsquosPerspective A Half Century Laterrdquo in La Prospettivarinascimentale Codificazione e trasgressioni vol 1 edMarisa Dalai Emiliani (Florence Centro Di 1980) 565-84

39Duchamp Salt Seller (cited n 2) 42 ldquoThiscinematic blossoming which expresses the moment of thestripping should be grafted onto an arbor-type of the brideThis arbor-type has its roots in the desire-gears but thecinematic effects of the electrical stripping transmitted tothe motor with quite feeble cylinders leave (plasticnecessity) the arbor-type at rest (Graphically in Munich Ihad already made two studies of this arbor type) Do not touchthe desire-gears which by giving birth to the arbor-typefind within this arbor-type the transmission of the desire tothe blossoming into stripping voluntarily imagined by thebride desiringrdquo

40J V Field ldquoLinear Perspective and theProjectiveGeometry of Girard Desarguesrdquo Nuncius 2no 2(1987) 3-40

41Henderson (cited n 6) does not refer toDesargues in her discussion of the Bride as an ldquoarbor-typerdquoShe argues that because an ldquoarborrdquo is an ldquoaxlerdquo Duchamprsquosusage should be interpreted as a reference to such devices asthe shafts in automobile transmissions or electricalgenerators I completely agree that Duchamp could have hadthese kinds of associations in mind along with his taking an

ldquoarbrerdquo to refer to a geometrical axis of rotation

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

43Duchamp Salt Seller (cited n 2) 27 seealso idem Duchamp du Signe (cited n 18) 42

44Field ldquoLinear Perspective and the ProjectiveGeometry of Girard Desarguesrdquo (cited n 40) 21

45Duchamp Notes (cited n 19) no 57

46Ibid no 155

47There are two new sections in the upper rightcorner of the Large Glass with holes drilled through them tocreate the ldquonine shotsrdquo In photographs of the Large Glasstaken at the Brooklyn Museum in 1926-27 the ldquonine shotsrdquo arenot visible Duchamp may have incorporated them into the Glasswhen he was repairing it in 1936

48 ldquoArguesianrdquo would be the adjectivalcounterpart of ldquoCartesianrdquo Reneacute Descartes (1596-1650) andDesargues (1593-1662) were almost exact contemporaries andcommunicated with one another about mathematical matters seeField Invention of Infinity (cited n 35) 190-97 see alsoReneacute Taton LrsquoOeuvre matheacutematique de G Desargues Textespublieacutes et commenteacutes avec une introduction biographique ethistorique 2d rev ed (Lyon Institut InterdisciplinairedrsquoEtudes Episteacutemologiques distributed by the LibrairiePhilosophique J Vrin Paris 1988)

49I am indebted to Hector Obalk for drawing thisconnection (or reconnection) to my attention in his talk ldquoWhatIs an Object The Belated Career of the Readymaderdquo at theinterdisciplinary colloquium ldquoMethods of Understanding in Artand Science the Case of Duchamp and Poincareacuterdquo HarvardUniversity Cambridge Massachusetts November 7 1999

50Duchamp Notes (cited n 19) no 67

51Duchamp Salt Seller (cited n 2) 35

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

53Duchamp Duchamp du Signe (cited n 18) 54

54See Adcock Marcel Duchamprsquos Notes (cited n10)130-32

55Lebel (cited n 13) 132-33

56Robert Lebel Sur Marcel Duchamp with textsbyAndreacute Breton and H-P Rocheacute (Paris and LondonEacuteditionsTrianon 1958) a facsimile edition ofthis book was publishedby the Centre Georges PompidouParis in 1996

57This photograph appears in Robert LebelldquoDerniegravere soireacutee avec Marcel Duchamprdquo LrsquoOeil (Paris) no 167(November 1968) 18-21 also reproduced in the supplementldquoMarcel Duchamp et Robert Lebelrdquo in the facsimile edition ofSur Marcel Duchamp (cited n 56) see also Gough-Cooper andCaumont (cited n 12) under their entry for April 29 1942The photograph was taken just before Duchamp left France forthe United States Mirroring the famous movie script hesailed from Marseilles to Casablanca and from there to Lisbonand then to New York arriving on June 25 See Jennifer Gough-Cooper and Jacques Caumont Plan pour ecrire une vie de MarcelDuchamp vol 1 Marcel Duchamp catalogue (Paris CentreNational drsquoArt et de Culture Georges Pompidou Museacutee NationaldrsquoArt Moderne 1977) 23

58Bonk (cited n 21) 218 argues that Duchampfashioned the templates for the Three Standard Stoppages in1918 when he was working on Tu mrsquo and needed to draw theircurvatures several times This chronology would mean that heused something other than the templates perhaps tracing paperor some other means to draw the lines in Network of Stoppagesin 1914 See also Duchamprsquos correspondence with Katherine SDreier in Affectionately Marcel (cited n 21) 199-207

59For a more detailed discussion of Duchamprsquos useof shadows on Tu mrsquo see Adcock Marcel Duchamprsquos Notes (citedn 10) 41-49 For a more traditional approach but nonethelessinteresting for Duchamprsquos work see Thomas Da Costa KaufmannldquoThe Perspective of Shadows The History of the Theory ofShadow ProjectionrdquoJournal of the Warburg and CourtauldInstitutes 38 (1975) 258-87

60For a more detailed discussion of Tu mrsquo inrelation to non-Euclidean geometry and topology see AdcockMarcel Duchamprsquos Notes (cited n 10) 55-58 101-02

61Interview with Cabanne (cited n 12) 40

62Kemp (cited n 36) 123-24 points out thatDesarguesrsquo discussion of conic sections ldquohelped sow the seeds

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

ldquoarbrerdquo to refer to a geometrical axis of rotation

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

43Duchamp Salt Seller (cited n 2) 27 seealso idem Duchamp du Signe (cited n 18) 42

44Field ldquoLinear Perspective and the ProjectiveGeometry of Girard Desarguesrdquo (cited n 40) 21

45Duchamp Notes (cited n 19) no 57

46Ibid no 155

47There are two new sections in the upper rightcorner of the Large Glass with holes drilled through them tocreate the ldquonine shotsrdquo In photographs of the Large Glasstaken at the Brooklyn Museum in 1926-27 the ldquonine shotsrdquo arenot visible Duchamp may have incorporated them into the Glasswhen he was repairing it in 1936

48 ldquoArguesianrdquo would be the adjectivalcounterpart of ldquoCartesianrdquo Reneacute Descartes (1596-1650) andDesargues (1593-1662) were almost exact contemporaries andcommunicated with one another about mathematical matters seeField Invention of Infinity (cited n 35) 190-97 see alsoReneacute Taton LrsquoOeuvre matheacutematique de G Desargues Textespublieacutes et commenteacutes avec une introduction biographique ethistorique 2d rev ed (Lyon Institut InterdisciplinairedrsquoEtudes Episteacutemologiques distributed by the LibrairiePhilosophique J Vrin Paris 1988)

49I am indebted to Hector Obalk for drawing thisconnection (or reconnection) to my attention in his talk ldquoWhatIs an Object The Belated Career of the Readymaderdquo at theinterdisciplinary colloquium ldquoMethods of Understanding in Artand Science the Case of Duchamp and Poincareacuterdquo HarvardUniversity Cambridge Massachusetts November 7 1999

50Duchamp Notes (cited n 19) no 67

51Duchamp Salt Seller (cited n 2) 35

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

53Duchamp Duchamp du Signe (cited n 18) 54

54See Adcock Marcel Duchamprsquos Notes (cited n10)130-32

55Lebel (cited n 13) 132-33

56Robert Lebel Sur Marcel Duchamp with textsbyAndreacute Breton and H-P Rocheacute (Paris and LondonEacuteditionsTrianon 1958) a facsimile edition ofthis book was publishedby the Centre Georges PompidouParis in 1996

57This photograph appears in Robert LebelldquoDerniegravere soireacutee avec Marcel Duchamprdquo LrsquoOeil (Paris) no 167(November 1968) 18-21 also reproduced in the supplementldquoMarcel Duchamp et Robert Lebelrdquo in the facsimile edition ofSur Marcel Duchamp (cited n 56) see also Gough-Cooper andCaumont (cited n 12) under their entry for April 29 1942The photograph was taken just before Duchamp left France forthe United States Mirroring the famous movie script hesailed from Marseilles to Casablanca and from there to Lisbonand then to New York arriving on June 25 See Jennifer Gough-Cooper and Jacques Caumont Plan pour ecrire une vie de MarcelDuchamp vol 1 Marcel Duchamp catalogue (Paris CentreNational drsquoArt et de Culture Georges Pompidou Museacutee NationaldrsquoArt Moderne 1977) 23

58Bonk (cited n 21) 218 argues that Duchampfashioned the templates for the Three Standard Stoppages in1918 when he was working on Tu mrsquo and needed to draw theircurvatures several times This chronology would mean that heused something other than the templates perhaps tracing paperor some other means to draw the lines in Network of Stoppagesin 1914 See also Duchamprsquos correspondence with Katherine SDreier in Affectionately Marcel (cited n 21) 199-207

59For a more detailed discussion of Duchamprsquos useof shadows on Tu mrsquo see Adcock Marcel Duchamprsquos Notes (citedn 10) 41-49 For a more traditional approach but nonethelessinteresting for Duchamprsquos work see Thomas Da Costa KaufmannldquoThe Perspective of Shadows The History of the Theory ofShadow ProjectionrdquoJournal of the Warburg and CourtauldInstitutes 38 (1975) 258-87

60For a more detailed discussion of Tu mrsquo inrelation to non-Euclidean geometry and topology see AdcockMarcel Duchamprsquos Notes (cited n 10) 55-58 101-02

61Interview with Cabanne (cited n 12) 40

62Kemp (cited n 36) 123-24 points out thatDesarguesrsquo discussion of conic sections ldquohelped sow the seeds

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

48 ldquoArguesianrdquo would be the adjectivalcounterpart of ldquoCartesianrdquo Reneacute Descartes (1596-1650) andDesargues (1593-1662) were almost exact contemporaries andcommunicated with one another about mathematical matters seeField Invention of Infinity (cited n 35) 190-97 see alsoReneacute Taton LrsquoOeuvre matheacutematique de G Desargues Textespublieacutes et commenteacutes avec une introduction biographique ethistorique 2d rev ed (Lyon Institut InterdisciplinairedrsquoEtudes Episteacutemologiques distributed by the LibrairiePhilosophique J Vrin Paris 1988)

49I am indebted to Hector Obalk for drawing thisconnection (or reconnection) to my attention in his talk ldquoWhatIs an Object The Belated Career of the Readymaderdquo at theinterdisciplinary colloquium ldquoMethods of Understanding in Artand Science the Case of Duchamp and Poincareacuterdquo HarvardUniversity Cambridge Massachusetts November 7 1999

50Duchamp Notes (cited n 19) no 67

51Duchamp Salt Seller (cited n 2) 35

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

53Duchamp Duchamp du Signe (cited n 18) 54

54See Adcock Marcel Duchamprsquos Notes (cited n10)130-32

55Lebel (cited n 13) 132-33

56Robert Lebel Sur Marcel Duchamp with textsbyAndreacute Breton and H-P Rocheacute (Paris and LondonEacuteditionsTrianon 1958) a facsimile edition ofthis book was publishedby the Centre Georges PompidouParis in 1996

57This photograph appears in Robert LebelldquoDerniegravere soireacutee avec Marcel Duchamprdquo LrsquoOeil (Paris) no 167(November 1968) 18-21 also reproduced in the supplementldquoMarcel Duchamp et Robert Lebelrdquo in the facsimile edition ofSur Marcel Duchamp (cited n 56) see also Gough-Cooper andCaumont (cited n 12) under their entry for April 29 1942The photograph was taken just before Duchamp left France forthe United States Mirroring the famous movie script hesailed from Marseilles to Casablanca and from there to Lisbonand then to New York arriving on June 25 See Jennifer Gough-Cooper and Jacques Caumont Plan pour ecrire une vie de MarcelDuchamp vol 1 Marcel Duchamp catalogue (Paris CentreNational drsquoArt et de Culture Georges Pompidou Museacutee NationaldrsquoArt Moderne 1977) 23

58Bonk (cited n 21) 218 argues that Duchampfashioned the templates for the Three Standard Stoppages in1918 when he was working on Tu mrsquo and needed to draw theircurvatures several times This chronology would mean that heused something other than the templates perhaps tracing paperor some other means to draw the lines in Network of Stoppagesin 1914 See also Duchamprsquos correspondence with Katherine SDreier in Affectionately Marcel (cited n 21) 199-207

59For a more detailed discussion of Duchamprsquos useof shadows on Tu mrsquo see Adcock Marcel Duchamprsquos Notes (citedn 10) 41-49 For a more traditional approach but nonethelessinteresting for Duchamprsquos work see Thomas Da Costa KaufmannldquoThe Perspective of Shadows The History of the Theory ofShadow ProjectionrdquoJournal of the Warburg and CourtauldInstitutes 38 (1975) 258-87

60For a more detailed discussion of Tu mrsquo inrelation to non-Euclidean geometry and topology see AdcockMarcel Duchamprsquos Notes (cited n 10) 55-58 101-02

61Interview with Cabanne (cited n 12) 40

62Kemp (cited n 36) 123-24 points out thatDesarguesrsquo discussion of conic sections ldquohelped sow the seeds

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

53Duchamp Duchamp du Signe (cited n 18) 54

54See Adcock Marcel Duchamprsquos Notes (cited n10)130-32

55Lebel (cited n 13) 132-33

56Robert Lebel Sur Marcel Duchamp with textsbyAndreacute Breton and H-P Rocheacute (Paris and LondonEacuteditionsTrianon 1958) a facsimile edition ofthis book was publishedby the Centre Georges PompidouParis in 1996

57This photograph appears in Robert LebelldquoDerniegravere soireacutee avec Marcel Duchamprdquo LrsquoOeil (Paris) no 167(November 1968) 18-21 also reproduced in the supplementldquoMarcel Duchamp et Robert Lebelrdquo in the facsimile edition ofSur Marcel Duchamp (cited n 56) see also Gough-Cooper andCaumont (cited n 12) under their entry for April 29 1942The photograph was taken just before Duchamp left France forthe United States Mirroring the famous movie script hesailed from Marseilles to Casablanca and from there to Lisbonand then to New York arriving on June 25 See Jennifer Gough-Cooper and Jacques Caumont Plan pour ecrire une vie de MarcelDuchamp vol 1 Marcel Duchamp catalogue (Paris CentreNational drsquoArt et de Culture Georges Pompidou Museacutee NationaldrsquoArt Moderne 1977) 23

58Bonk (cited n 21) 218 argues that Duchampfashioned the templates for the Three Standard Stoppages in1918 when he was working on Tu mrsquo and needed to draw theircurvatures several times This chronology would mean that heused something other than the templates perhaps tracing paperor some other means to draw the lines in Network of Stoppagesin 1914 See also Duchamprsquos correspondence with Katherine SDreier in Affectionately Marcel (cited n 21) 199-207

59For a more detailed discussion of Duchamprsquos useof shadows on Tu mrsquo see Adcock Marcel Duchamprsquos Notes (citedn 10) 41-49 For a more traditional approach but nonethelessinteresting for Duchamprsquos work see Thomas Da Costa KaufmannldquoThe Perspective of Shadows The History of the Theory ofShadow ProjectionrdquoJournal of the Warburg and CourtauldInstitutes 38 (1975) 258-87

60For a more detailed discussion of Tu mrsquo inrelation to non-Euclidean geometry and topology see AdcockMarcel Duchamprsquos Notes (cited n 10) 55-58 101-02

61Interview with Cabanne (cited n 12) 40

62Kemp (cited n 36) 123-24 points out thatDesarguesrsquo discussion of conic sections ldquohelped sow the seeds

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

58Bonk (cited n 21) 218 argues that Duchampfashioned the templates for the Three Standard Stoppages in1918 when he was working on Tu mrsquo and needed to draw theircurvatures several times This chronology would mean that heused something other than the templates perhaps tracing paperor some other means to draw the lines in Network of Stoppagesin 1914 See also Duchamprsquos correspondence with Katherine SDreier in Affectionately Marcel (cited n 21) 199-207

59For a more detailed discussion of Duchamprsquos useof shadows on Tu mrsquo see Adcock Marcel Duchamprsquos Notes (citedn 10) 41-49 For a more traditional approach but nonethelessinteresting for Duchamprsquos work see Thomas Da Costa KaufmannldquoThe Perspective of Shadows The History of the Theory ofShadow ProjectionrdquoJournal of the Warburg and CourtauldInstitutes 38 (1975) 258-87

60For a more detailed discussion of Tu mrsquo inrelation to non-Euclidean geometry and topology see AdcockMarcel Duchamprsquos Notes (cited n 10) 55-58 101-02

61Interview with Cabanne (cited n 12) 40

62Kemp (cited n 36) 123-24 points out thatDesarguesrsquo discussion of conic sections ldquohelped sow the seeds

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

of non-Euclidian geometry but was only to be fully taken upby Poncelet in the nineteenth century Vital steps in thedevelopment of new postulates appear to have been takenindependently by Kepler and Desargues The new geometrychallenged central assumptions of Euclidian theory Straightlines came to be interpreted as equivalent to circles whichpossess radiuses of infinite length and parallel linesregarded as meeting at infinityrdquo For the contributions ofPoncelet and Kepler alluded to here by Kemp see Jean-VictorPoncelet Traiteacute des proprieacuteteacutes projectives des figures(Paris 1822) Johannes Kepler Ad Vitellionem paralipomenaquibus astronomiae pars optica traditur (1604) a translationof this last work is included in an appendix ldquoKeplerrsquosInvention of Points at Infinityrdquo in Field and Gray (cited n35) 185-88

63See Field and Gray (cited n 35) 60-72

64Duchamp Salt Seller (cited n 2) 86

65Jean-Franccedilois Niceacuteron Thaumaturgus opticus(Paris 1646) Stephen Jay Gould and Rhonda Roland ShearerldquoDrawing the Maxim from the Minim The Unrecognized Source ofNiceacuteronrsquos Influence upon DuchamprdquoTout-Fait The MarcelDuchamp Studies Online Journal 1 no 3 (December 2000) arguethat Duchamp is very likely to have also used Niceacuteronrsquosearlier French edition which contains material not includedin the Latin edition see Jean-Franccedilois Niceacuteron LaPerspective curieuse ou magie artificielle des effetsmerveilleux (Paris 1638) News

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

lthttpwwwtoutfaitcomduchampjsppostid=896ampkeyword=gt Foran interesting discussion of the ways in which epistemologicalperspective can affect the interpretation of data see DavidMagnus ldquoDown the Primrose Path Competing Epistemologies inEarly Twentieth-Century Biologyrdquo in Biology and Epistemologyed Richard Creath and Jane Maienschein (Cambridge CambridgeUniversity Press 2000) 91-121

66 For a discussion of Niceacuteronrsquos image seeMartin Kemp The Science of Art Optical Themes in Western Artfrom Brunelleschi to Seurat (New Haven and London YaleUniversity Press 1990) 210-11 Kemp does not mention Duchampor Tu mrsquo Niceacuteronrsquos illustration was also included in Laperspective curieuse pl 33 see Kim H Veltman incollaboration with Kenneth D Keele Linear Perspective andthe Visual Dimensions of Science and Art Studies on Leonardoda Vinci I (Munich Deutscher Kunstverlag 1986) 164-65

67For a discussion of this painting see JurgisBaltrušaitis Anamorphic Art trans W J Strachan (New YorkAbrams 1977) 91-114 for interesting analyses ofanamorphosis see Fred Leeman Hidden Images Games ofPerception Anamorphic Art and Illusion from the Renaissanceto the Present trans Ellyn Childs Allison and Margaret LKaplan (New York Abrams 1976) see also Kim H VeltmanldquoPerspective Anamorphosis and Visionrdquo Marburger Jahrbuchfuumlr Kunstwissenshaft 21 (1986) 93-117

68Holbeinrsquos painting is discussed by JacquesLacan The Seminar of Jacques Lacan Book XI The Four

Fundamental Concepts of Psychoanalysis ed Jacques-AlainMiller trans Alan Sheridan (New York and London W WNorton 1981) 88 see also Martin Jay Downcast Eyes TheDenigration of Vision in Twentieth-Century French Thought(Berkeley Los Angeles and London University of CaliforniaPress 1994) 48 362-64 and Tom Conley ldquoThe Wit of theLetter Holbeinrsquos Lacanrdquo in Vision in Context Historical andContemporary Perspectives on Sight ed Teresa Brennan andMartin Jay (New York and London Routledge 1996) 45-60

69Dalia Judovitz in a discussion of ReneacuteDescartesrsquos interests in both ldquonormalrdquo and ldquoaberrantrdquoperspective systems makes a similar point about Holbeinrsquosimage see her essay ldquoVision Representation and Technologyin Descartesrdquo in Modernity and the Hegemony of Vision edDavid Michael Levin (Berkeley Los Angeles and LondonUniversity of California Press 1993) 66-67 Judovitzdiscusses Tu mrsquo in her book Unpacking Duchamp Art in Transit(Berkeley Los Angeles and London University of CaliforniaPress 1995) 221-26 but does not discuss the paintingrsquosanamorphic characteristics

70Jean Clair ldquoDuchamp and the ClassicalPerspectivistsrdquoArtforum 16 (March 1978) 40-49 the quote isfrom p 47

71Ibid emphasis in the original see alsoClairrsquos essay ldquoMarcel Duchamp et la tradition desperspecteursrdquo in Abeacuteceacutedaire vol 3 Marcel Duchamp catalogue(Paris Centre National drsquoArt et de Culture Georges Pompidou

Museacutee National drsquoArt Moderne 1977) 52-59

72See Adcock ldquoConventionalism in Henri Poincareacuteand Marcel Duchamprdquo (cited n 20) 257

73 For an early discussion of thesemathematicians in the context of art history see William MIvins Jr ldquoDesargues and Pascalrdquo chap 8 in Art amp GeometryA Study in Space Intuitions (Cambridge Harvard UniversityPress 1946)

74Morris Kline Mathematical Thought from Ancientto Modern Times (New York Oxford University Press 1972)285-301 834-60 see also idem ldquoProjective Geometryrdquo inMathematics in the Modern World Readings fromScientificAmerican ed Morris Kline (San Francisco and London W HFreeman 1968) 122-27

75For an accessible source that refers toPoincareacute in these terms see Albert W Tucker and Herbert SBailey Jr ldquoTopologyrdquo in Mathematics in the Modern WorldReadings from Scientific American ed Morris Kline (SanFrancisco and London W H Freeman 1968) 134-40

76Henri Poincareacute Mathematics and Science LastEssaystrans John W Bolduc (New York Dover 1963) 58-59

77Duchamp Salt Seller (cited n 2) 91

78The complexities of the four-dimensionalcontinuum are suggested by the following passage from the onlynote in the Green Box with a specific reference to a higherspace (Duchamprsquos term is ldquoeacutetendue 4rdquo in the original French)ldquoAs there is gradually less differentiation from axis to axisie as all the axes gradually disappear in a fadingverticality the front and the back the reverse and theobverse acquire a circular significance the right and theleft which are the four arms of the front and the back meltalong the verticals The interior and exterior (in a four-dimensional continuum) can receive a similar identificationrdquoSee Duchamp Salt Seller (cited n 2) 29 idem Duchamp duSigne (cited n 18) 45

79A modern way of putting this matter would be tosay ldquoTwo planes having a common point have at least one morecommon point If this is satisfied the space must be three-dimensional if it is not satisfied so that there are twoplanes with a unique common point then the space is at leastfour-dimensionalrdquo Encyclopaedia of Mathematics sv ldquoHigher-Dimensional Geometryrdquo by A D Aleksandrov For a moresophisticated definition see H S M Coxeter Introductionto Geometry (New York and London John Wiley amp Sons 1961)185-86

80There are a large number of possibilities Oneof the textbooks that I have on my shelves begins with thefollowing statement ldquoFrom the beginnings of geometry untilwell into the nineteenth century it was almost universallyaccepted that the geometry of the space we live in is the onlygeometry conceivable by man This point of view was mosteloquently formulated by the German philosopher Immanuel Kant(1724-1804) Ironically shortly after Kantrsquos death thediscovery of non-Euclidean geometry by Gauss Lobachevski andBolyai made his position untenable Today we study inmathematics not just one geometry or two geometries but aninfinity of geometriesrdquo Albrecht Beutelspacher and UteRosenbaum Projective Geometry From Foundations toApplications(Cambridge Cambridge University Press 1998) 1

81For one of the best discussions of the kinds ofissues this statement raises see Graham Nerlich The Shape ofSpace 2d ed (Cambridge Cambridge University Press 1994)

82

A generalized mathematical ldquosurfacerdquo is a ldquomanifoldrdquo and canhave any number of dimensions It can also have any number ofcurvatures This important way of thinking about geometricalconfigurations is due to Bernhard Riemann (1826-1866) and iscustomarily referred to as ldquoRiemannian Geometryrdquo This senseof ldquoRiemannian Geometryrdquo can be distinguished from the senseused to refer to his prior invention of a specific(ungeneralized) non-Euclidean geometry with constant positivecurvature customarily referred to as ldquoRiemann Geometryrdquo orelliptical geometry see Peter Petersen Riemannian Geometry(New York Springer-Verlag 1998) In a questionaire about the

Three Standard Stoppages in the archives of the Museum ofModern Art in New York dated 1953 (the year the work enteredtheir collection) Duchamp said that the assemblage was ldquoahumorous application of Riemannrsquos post-Euclidean geometrywhich was devoid of straight linesrdquo (see Naumann cited n 12p 170) That Duchamp used the term ldquopost-Euclideanrdquo ratherthan simply ldquonon-Euclideanrdquo indicates that he may very wellhave been sophisticated enough to have understood thedistinctions under discussion here

Figs 20-22copy2003 Succession Marcel Duchamp ARS NYADAGP Paris Allrights reserved

Museacutee National drsquoArt Moderne 1977) 52-59

72See Adcock ldquoConventionalism in Henri Poincareacuteand Marcel Duchamprdquo (cited n 20) 257

73 For an early discussion of thesemathematicians in the context of art history see William MIvins Jr ldquoDesargues and Pascalrdquo chap 8 in Art amp GeometryA Study in Space Intuitions (Cambridge Harvard UniversityPress 1946)

74Morris Kline Mathematical Thought from Ancientto Modern Times (New York Oxford University Press 1972)285-301 834-60 see also idem ldquoProjective Geometryrdquo inMathematics in the Modern World Readings fromScientificAmerican ed Morris Kline (San Francisco and London W HFreeman 1968) 122-27

75For an accessible source that refers toPoincareacute in these terms see Albert W Tucker and Herbert SBailey Jr ldquoTopologyrdquo in Mathematics in the Modern WorldReadings from Scientific American ed Morris Kline (SanFrancisco and London W H Freeman 1968) 134-40

76Henri Poincareacute Mathematics and Science LastEssaystrans John W Bolduc (New York Dover 1963) 58-59

77Duchamp Salt Seller (cited n 2) 91

78The complexities of the four-dimensionalcontinuum are suggested by the following passage from the onlynote in the Green Box with a specific reference to a higherspace (Duchamprsquos term is ldquoeacutetendue 4rdquo in the original French)ldquoAs there is gradually less differentiation from axis to axisie as all the axes gradually disappear in a fadingverticality the front and the back the reverse and theobverse acquire a circular significance the right and theleft which are the four arms of the front and the back meltalong the verticals The interior and exterior (in a four-dimensional continuum) can receive a similar identificationrdquoSee Duchamp Salt Seller (cited n 2) 29 idem Duchamp duSigne (cited n 18) 45

79A modern way of putting this matter would be tosay ldquoTwo planes having a common point have at least one morecommon point If this is satisfied the space must be three-dimensional if it is not satisfied so that there are twoplanes with a unique common point then the space is at leastfour-dimensionalrdquo Encyclopaedia of Mathematics sv ldquoHigher-Dimensional Geometryrdquo by A D Aleksandrov For a moresophisticated definition see H S M Coxeter Introductionto Geometry (New York and London John Wiley amp Sons 1961)185-86

80There are a large number of possibilities Oneof the textbooks that I have on my shelves begins with thefollowing statement ldquoFrom the beginnings of geometry untilwell into the nineteenth century it was almost universallyaccepted that the geometry of the space we live in is the onlygeometry conceivable by man This point of view was mosteloquently formulated by the German philosopher Immanuel Kant(1724-1804) Ironically shortly after Kantrsquos death thediscovery of non-Euclidean geometry by Gauss Lobachevski andBolyai made his position untenable Today we study inmathematics not just one geometry or two geometries but aninfinity of geometriesrdquo Albrecht Beutelspacher and UteRosenbaum Projective Geometry From Foundations toApplications(Cambridge Cambridge University Press 1998) 1

81For one of the best discussions of the kinds ofissues this statement raises see Graham Nerlich The Shape ofSpace 2d ed (Cambridge Cambridge University Press 1994)

82

A generalized mathematical ldquosurfacerdquo is a ldquomanifoldrdquo and canhave any number of dimensions It can also have any number ofcurvatures This important way of thinking about geometricalconfigurations is due to Bernhard Riemann (1826-1866) and iscustomarily referred to as ldquoRiemannian Geometryrdquo This senseof ldquoRiemannian Geometryrdquo can be distinguished from the senseused to refer to his prior invention of a specific(ungeneralized) non-Euclidean geometry with constant positivecurvature customarily referred to as ldquoRiemann Geometryrdquo orelliptical geometry see Peter Petersen Riemannian Geometry(New York Springer-Verlag 1998) In a questionaire about the

Three Standard Stoppages in the archives of the Museum ofModern Art in New York dated 1953 (the year the work enteredtheir collection) Duchamp said that the assemblage was ldquoahumorous application of Riemannrsquos post-Euclidean geometrywhich was devoid of straight linesrdquo (see Naumann cited n 12p 170) That Duchamp used the term ldquopost-Euclideanrdquo ratherthan simply ldquonon-Euclideanrdquo indicates that he may very wellhave been sophisticated enough to have understood thedistinctions under discussion here

Figs 20-22copy2003 Succession Marcel Duchamp ARS NYADAGP Paris Allrights reserved

77Duchamp Salt Seller (cited n 2) 91

78The complexities of the four-dimensionalcontinuum are suggested by the following passage from the onlynote in the Green Box with a specific reference to a higherspace (Duchamprsquos term is ldquoeacutetendue 4rdquo in the original French)ldquoAs there is gradually less differentiation from axis to axisie as all the axes gradually disappear in a fadingverticality the front and the back the reverse and theobverse acquire a circular significance the right and theleft which are the four arms of the front and the back meltalong the verticals The interior and exterior (in a four-dimensional continuum) can receive a similar identificationrdquoSee Duchamp Salt Seller (cited n 2) 29 idem Duchamp duSigne (cited n 18) 45

79A modern way of putting this matter would be tosay ldquoTwo planes having a common point have at least one morecommon point If this is satisfied the space must be three-dimensional if it is not satisfied so that there are twoplanes with a unique common point then the space is at leastfour-dimensionalrdquo Encyclopaedia of Mathematics sv ldquoHigher-Dimensional Geometryrdquo by A D Aleksandrov For a moresophisticated definition see H S M Coxeter Introductionto Geometry (New York and London John Wiley amp Sons 1961)185-86

80There are a large number of possibilities Oneof the textbooks that I have on my shelves begins with thefollowing statement ldquoFrom the beginnings of geometry untilwell into the nineteenth century it was almost universallyaccepted that the geometry of the space we live in is the onlygeometry conceivable by man This point of view was mosteloquently formulated by the German philosopher Immanuel Kant(1724-1804) Ironically shortly after Kantrsquos death thediscovery of non-Euclidean geometry by Gauss Lobachevski andBolyai made his position untenable Today we study inmathematics not just one geometry or two geometries but aninfinity of geometriesrdquo Albrecht Beutelspacher and UteRosenbaum Projective Geometry From Foundations toApplications(Cambridge Cambridge University Press 1998) 1

81For one of the best discussions of the kinds ofissues this statement raises see Graham Nerlich The Shape ofSpace 2d ed (Cambridge Cambridge University Press 1994)

82

A generalized mathematical ldquosurfacerdquo is a ldquomanifoldrdquo and canhave any number of dimensions It can also have any number ofcurvatures This important way of thinking about geometricalconfigurations is due to Bernhard Riemann (1826-1866) and iscustomarily referred to as ldquoRiemannian Geometryrdquo This senseof ldquoRiemannian Geometryrdquo can be distinguished from the senseused to refer to his prior invention of a specific(ungeneralized) non-Euclidean geometry with constant positivecurvature customarily referred to as ldquoRiemann Geometryrdquo orelliptical geometry see Peter Petersen Riemannian Geometry(New York Springer-Verlag 1998) In a questionaire about the

Three Standard Stoppages in the archives of the Museum ofModern Art in New York dated 1953 (the year the work enteredtheir collection) Duchamp said that the assemblage was ldquoahumorous application of Riemannrsquos post-Euclidean geometrywhich was devoid of straight linesrdquo (see Naumann cited n 12p 170) That Duchamp used the term ldquopost-Euclideanrdquo ratherthan simply ldquonon-Euclideanrdquo indicates that he may very wellhave been sophisticated enough to have understood thedistinctions under discussion here

Figs 20-22copy2003 Succession Marcel Duchamp ARS NYADAGP Paris Allrights reserved

80There are a large number of possibilities Oneof the textbooks that I have on my shelves begins with thefollowing statement ldquoFrom the beginnings of geometry untilwell into the nineteenth century it was almost universallyaccepted that the geometry of the space we live in is the onlygeometry conceivable by man This point of view was mosteloquently formulated by the German philosopher Immanuel Kant(1724-1804) Ironically shortly after Kantrsquos death thediscovery of non-Euclidean geometry by Gauss Lobachevski andBolyai made his position untenable Today we study inmathematics not just one geometry or two geometries but aninfinity of geometriesrdquo Albrecht Beutelspacher and UteRosenbaum Projective Geometry From Foundations toApplications(Cambridge Cambridge University Press 1998) 1

81For one of the best discussions of the kinds ofissues this statement raises see Graham Nerlich The Shape ofSpace 2d ed (Cambridge Cambridge University Press 1994)

82

A generalized mathematical ldquosurfacerdquo is a ldquomanifoldrdquo and canhave any number of dimensions It can also have any number ofcurvatures This important way of thinking about geometricalconfigurations is due to Bernhard Riemann (1826-1866) and iscustomarily referred to as ldquoRiemannian Geometryrdquo This senseof ldquoRiemannian Geometryrdquo can be distinguished from the senseused to refer to his prior invention of a specific(ungeneralized) non-Euclidean geometry with constant positivecurvature customarily referred to as ldquoRiemann Geometryrdquo orelliptical geometry see Peter Petersen Riemannian Geometry(New York Springer-Verlag 1998) In a questionaire about the

Three Standard Stoppages in the archives of the Museum ofModern Art in New York dated 1953 (the year the work enteredtheir collection) Duchamp said that the assemblage was ldquoahumorous application of Riemannrsquos post-Euclidean geometrywhich was devoid of straight linesrdquo (see Naumann cited n 12p 170) That Duchamp used the term ldquopost-Euclideanrdquo ratherthan simply ldquonon-Euclideanrdquo indicates that he may very wellhave been sophisticated enough to have understood thedistinctions under discussion here

Figs 20-22copy2003 Succession Marcel Duchamp ARS NYADAGP Paris Allrights reserved

Three Standard Stoppages in the archives of the Museum ofModern Art in New York dated 1953 (the year the work enteredtheir collection) Duchamp said that the assemblage was ldquoahumorous application of Riemannrsquos post-Euclidean geometrywhich was devoid of straight linesrdquo (see Naumann cited n 12p 170) That Duchamp used the term ldquopost-Euclideanrdquo ratherthan simply ldquonon-Euclideanrdquo indicates that he may very wellhave been sophisticated enough to have understood thedistinctions under discussion here

Figs 20-22copy2003 Succession Marcel Duchamp ARS NYADAGP Paris Allrights reserved