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Anamorphosis through Descartes or perspective gone awry.

The systems of perspective First produced during the fifteenth century and subsequently treated by mathematicians and artists in later centuries have been exhaustively discussed and analyzed by art historians. While many have focused on the technique's geometric and scientific history,(1) beginning with Panofsky and his remarkable treatment of perspective as symbolic form in 1927, a number of art historians have attempted to analyze the metaphorical and allegorical aspects of perspective.(2) Although varying greatly in tone and focus, there is one assumption that links all of these accounts. It is generally taken for granted that perspective somehow stands paradigmatically for Descartes's rationalism - for his search for ontological and epistemological certainty.

Many of the accounts that link Cartesian rationalism to perspective presume that both are systems that articulate a specifically modern, spatial ontology. This is true of Panofsky who connects the seemingly unified visual field of a vanishing point system to a historically modern, mathematical conception of infinity and argues that the ensuing understanding of space, "even with its still mystical coloring, is the same view that will later be rationalized by Cartesianism and formalized by Kant."(3) Thus for Panofsky perspective figuratively represents the spatial paradigm of Cartesian rationalism.

Norman Bryson, on the other hand, has argued that Alberti's perspective grid and vanishing point, based on the notion of the centric ray, construct a viewing subject that imagines him/herself as the objective focus of representation. In a tripartite system that Bryson calls the "logic" of representation, perspective embodies the viewer as both the origin of and object in the gaze and at the same time posits a third metaphysical or divine viewpoint (the disembodied view) that stands outside the closed system of references implied by the grid. This system sets up untenable mind/body contradictions similar to those found in Descartes's work, and it reveals that "Alberti's conception of the subject is already Cartesian in its reduction of the space of painting to dimensionless punctuality."(4) Bryson thus argues that the perspective grid corresponds to Descartes's rational subject in such a way that both posit a self-knowing, self-certain viewpoint that is somehow removed from the material, spatial conditions of embodied subjectivity.

In spite of their somewhat diverging interpretations, what Panofsky and Bryson share is the assumption that perspective has played an important role in the development of Cartesian rationalism. But this assumed link between rational philosophy and perspective remains confusing and ambiguous, especially when the actual techniques, practices, and historical theories of perspective are taken into consideration. Did perspective make Cartesian rationalism possible as both Panofsky and Bryson indicate? Or is perspective's conflation with Cartesianism better understood as a distinctly modern enterprise in which the perspective grid has become associated (subsequent to Descartes) with the activity of thinking itself, the "rational space" of Descartes's thought, as James Elkins has recently argued?(5) Is it possible and even necessary to rethink this particular conflation of Cartesian ontology and perspective and to begin problemizing what has become an overly comfortable and compromised philosophical cliche?

In what follows I will argue that anamorphic perspective challenges both the supposedly rational construction of vision associated with perspective and the assumed rationality of the Cartesian subject. Through anamorphosis we will see that the Cartesian cogito might be better understood not as a philosophical fulfillment of the spatially situated, rational subject seemingly promised by the linear grid but rather as a defense against perspective's disseminating threat to subjective self-certainty.

To begin this inquiry it is necessary to examine how and why Descartes's thought has been so inexorably and analogously associated with perspective. The association itself is of vital interest to any project that seeks to explain how and why perspective undergoes a historical shift from practice to metaphor from the fifteenth to the twentieth centuries. Elkins has argued that as the practice of perspective (increasingly relegated to mathematical treatises and less oriented to the practice of painting) became associated in the late sixteenth and early seventeenth centuries with mathematical arcana, a metaphoric use and conception of perspective gained greater currency.(6) The period in which this shift seems to occur most clearly is the seventeenth century, a century that has been historically characterized as the period in which the forms and methods of modern science and philosophy, via Galilean physics and Cartesian rationalism, were first realized.

However, there is no clear indication in Descartes's work that the practice of perspective in either mathematics or painting can be transcribed easily into a spatial metaphor for either subjective or objective understandings of human interactions with the world.(7) It takes, in effect, a giant associative leap to conflate the historical specificity of perspective practice and theory with Descartes's project of radical doubt, which starts with a rejection of sensuous apprehension of the world and then retreats into the "mind's eye" of subjective rationalism. Descartes's retreat itself is difficult enough to follow, but it is complicated exponentially by the subsequent philosophic and metaphoric transposition, which has firmly correlated perspective with the subject/object, spatial opposition long associated with Cartesianism.

For instance, twentieth-century philosophical critiques of Cartesian thought, such as Bergson's and Heidegger's, have focused on the static, timeless spatiality that the subject/object opposition seems to presume, arguing that Descartes privileged epistemological questions before ontological ones and in so doing subsumed the temporality and contingency of being to a spatially conceived ontology.(8) Thus, Bergson and Heidegger, whose radical reformulations of ontological questions have been among the most influential in the twentieth century, might be said to have begun with a "perspectivalist" reading or interpretation of Cartesian thought. Martin Jay has argued that twentieth-century continental philosophy can even be characterized as a radical rejection of the visual, spatial (perspectival) model associated with Cartesianism and its presumed relation to positivist construals of modernism.(9) The question is why this correlation has been made and whether this kind of interpretation of Descartes's thought and reasoning is fully justified. Where, in effect, is the perspective in Descartes?

This "perspectivalist" interpretation appears to be based largely on Descartes's explanation of the relation between the res cogitans and the res extensa, between a transcendentally self-sufficient, thinking being that itself is the only guarantee of certainty and the world that is posited as extended matter "outside" this being. Thus, assumptions about Cartesian "perspectivalism" seem to rest on Descartes's link between knowing and being, between identifying the foundations for accruing knowable things and establishing the autonomy of the subject who knows them. That is, modern notions of a "Cartesian world view" are usually correlated to this problematic movement between an "outside" and an "inside."

Descartes's initial articulation of this problem is generally traced to his first major public work, Discourse on the Method, which was published as the preface to three treatises, the Optics, Meteorology, and Geometry, in 1637.(10) In the preface he formulated for the first time the famous method of indubitability on which he founded the cogito argument. Beginning with the premise that it is possible to doubt everything about existence and the perceived world, Descartes was intent on finding some point of certainty, or something about which he could claim he had direct knowledge, and then from this point starting a larger project of accruing knowable things.(11) Engaging in an autobiographical narrative, Descartes describes in part 2 of the Discourse, the four rules (the method) that he follows in his investigation. He determines first never to accept anything as true except what presents itself to his mind "so clearly and distinctly" that he cannot doubt it; second, he will break all problems into separate parts, the better to examine them; third, he will order his thoughts by starting with the simplest and most basic knowable things and build on them to accrue greater knowledge of more complex things; and last, he will make this process so comprehensive and complete that nothing shall be left out.(12)

In part 4 of the Discourse he proceeds to the basic formulation of what has come to be called by its Latin rather than original French term, the cogito. Radical doubt serves as the necessary pretense of his investigation and becomes the foundation of his formulation. On the one hand, our senses can deceive us and thus lead us into logical fallacies and errors. On the other, when we dream, the thoughts we have may well be the same as those we have when we are awake, making it impossible to ascertain which thoughts are real and which are illusory. In neither state, awake or asleep, do we have any guarantee that our thoughts are true. Thus he will "pretend that all things that had ever entered my mind were no more true than the illusions of my dreams."(13) At this point he realizes that in this state of self-imposed radical doubt, the one thing he cannot doubt is the necessary truth that he himself exists. This is because the very act and process of doubting guarantees him surety of his existence. The only thing he cannot doubt when he is in a state of doubt is that he is doubting. And for Descartes, doubting is a form of thinking. Thus he determines that he is a thinking thing, or res cogitans:

I saw that while I could pretend that I had no body and that there was no world and no place for me to be in, I could not for all that pretend that I did not exist. I saw on the contrary that from the mere fact that I thought of doubting the truth of other things, it followed quite evidently and certainly that I existed; whereas if I had merely ceased thinking, even if everything else I had ever imagined had been true, I should have no reason to believe that I existed. From this I knew I was a substance whose whole essence or nature is simply to think, and which does not require any place, or depend on any material thing, in order to exist. Accordingly this "I" - that is, the soul by which I am what I am - is entirely distinct from the body, and indeed is easier to know than the body, and would not fail to be whatever it is, even if the body did not exist.(14)

This is the point at which he begins to establish a distinction between an internal "mind" and external "material world." Once the criterion of indubitability has been met for the first principle "I exist," then Descartes is left with the problem of connecting this seemingly transcendental, immaterial thinking "substance" back to the body and the world it inhabits.

He does this by insisting that there are only two essential attributes of and in the world, thought and extension. Because he has made his system or method entirely dependent on his own certainty of self, any proof of the existence of an external world is going to have to come from the same source. But in order to do this, he has to appeal to a transcendental source (God) for the innate ideas that emanate from the mind. He argues that because he has an idea of perfection against which he measures himself, he can only conceive of himself as imperfect or finite in the face of this perfection or infinitude. This puts him in a paradoxical position in which certitude of God's infinite nature and perfection emanates from the mind, which itself is imperfect and finite. As Bernard Williams points out, the tension produced by this seeming contradiction, which is present not only in Descartes but in much other seventeenth-century thought, is the result of a "genuine sense of contingency and limitation begin in conflict with a tendency to regard the power of the rational mind as virtually limitless."(15) The effect of this argument is that proof of the existence of God as a perfect, immaterial infinitude affords the grounds on which Descartes then argues that the world outside the mind, all matter, has the pure mathematical quality of extension. The world, he argues, is ordered and rational. His innate understanding of it must come from God and cannot derive from the body's sensual apprehensions.

In the transition from self-verifying transcendent subject back to a conception of the outside world as pure extension, Descartes doubles back on himself. He posits a self-knowing consciousness as the source and center of all knowledge of the world from which the world is then taken to extend. He then argues that the mind's grasp of the underlying structure of extended matter must come not from the world but from another external, perfect source. Thus Descartes's ontological argument produces a self-evident and self-verifying subject, the subject that "knows" and identifies itself first and then rests all subsequent knowledge on the basis of that first identification that in turn can only be guaranteed metaphysically.

There is no denying that it is tempting to draw an analogy between perspective and the Cartesian res cogitans. The notion of an all-seeing, all-knowing subject situated at the center of quantifiable matter to which the subject brings an innate and metaphysically conceived understanding of the rational order of the universe does indeed seem to have an affinity with perspective, especially since Descartes's arguments often draw from visual paradigms, i.e. he will only accept those ideas that appear "clearly and distinctly" to him. But since Descartes himself categorically rejected vision as a point of proof for existence, this doubling back to visual analogies remains problematic.

The close philosophical scrutiny given to Descartes's work more recently has produced a less reductive set of arguments concerning Descartes's methods, criteria, and conclusions. There have been numerous critiques within philosophy concerning the logic of Descartes's argument and whether the cogito has validity as an inference, with thinking as the proof of being. That is not of primary concern here since the point that I am pursuing is to work out whether or not Descartes's thought can be interpreted as "perspectival." But whether the cogito demonstrates a subject that can be imagined as geometrically and spatially situated in rational relation to the world is somewhat less assured than is generally assumed.

One obvious problem with the "perspectival" interpretation of Descartes is that he formulates both his epistemological and ontological arguments by rejecting the idea that certainty can be derived from sensual apprehension of the world. Doubt of the senses is a fundamental part of his construction of the cogito. While it is clear that the cogito seems based on a metaphoric transcription of visuality ("clear and distinct ideas") into a model of the mind and on making subjective self-certainty dependent on a model of internally clarified "vision," he does this by cutting the subject off from sensual apprehension of the world. Thus he states in the Discourse that "the sense of sight gives no less assurance of the reality of its objects than do the senses of smell and hearing, while neither our imagination nor our senses could ever assure us of anything without the intervention of our intellect."(16) Understanding of the world's rational order is not an affect of the senses and, in fact, the senses often work in opposition to the mind's rationality. The statement points out that for Descartes vision is the locus of neither certainty nor clarity; rather it is the locus of sensuous doubt.

Descartes's ambiguity regarding both vision and representation is apparent in his treatise on optics that follows the Discourse. Descartes's optical theory helped transform the terms with which visual perception was commonly represented. He acknowledged Kepler's discovery of the image cast at the back of the eye, but then rejected the notion that this image could conceivably be the basis for sight. There would have to be yet another pair of eyes within the mind that could then "see" the image.(17) Descartes alternatively adopted a pointillist theory of retinal apprehension in which patterns of stimulus pass across the retina causing movement or excitement in the mind. More interestingly, he remarks that philosophers have been too tempted to imagine that our conception of the world rests on a resemblance between objects in it that are perceived by the senses and ideas that reside in the mind. This "mistake" is the result of the influence that pictures have had on philosophers' notions of mental representation: "Their sole reason for positing such images [images that are conveyed intact from the senses to the brain] was that they saw how easily a picture can stimulate our mind to conceive the objects depicted in it, and so it seemed to them that, in the same way, the mind must be stimulated, by little pictures formed in our head, to conceive objects that affect our senses."(18)

But mental apprehension of the world does not have to be based on a notion of correspondence or "resemblance" between images in the world and the mind's understanding of the world. In fact, he says, the mind can understand certain aspects of the world by way of language and signs that have a purely arbitrary character and do not operate according to rules of resemblance: "We should, however, recall that our mind can be stimulated by many things other than images - by signs and words, for example, which in no way resemble the things they signify."(19)

He then suggests that perspective itself is no more based on resemblance than is the mind's apprehension of what the senses convey, and, in fact, it is perspective that reveals how irrelevant resemblance is to understanding:

We must at least observe that in no case does an image have to resemble the object it represents in all respects, for otherwise there would be no distinction between the object and its image. It is enough that the image resembles its object in a few respects. Indeed the perfection of an image often depends on its not resembling its object as much as it might.

. . . Moreover, in accordance with the rules of perspective they [engravers] often represent circles by ovals better than by other circles, squares by rhombuses better than by other squares, and similarly for other shapes. Thus it often happens that in order to be more perfect as an image and to represent an object better, an engraving ought not to resemble it.(20)

Thus for Descartes resemblance between object, image, and mental apprehension is not necessary for the mind's understanding of the world. In fact, perspective can in no way represent the multiple possibilities of extension that only the mind can apprehend. As Williams points out, the "pure intellectual conception of matter as extension is not derived from the senses, nor (and this for Descartes is a closely related point) can it be adequately represented in images: a favorite argument is that we have a rational comprehension of infinite variations in extension, and this comprehension cannot be adequately expressed in images.(21) For instance, in one famous example in the Meditations, Descartes refers to the changeable properties of wax. In his imagination, he pictures the wax changing from one shape to another, but with his intellect he grasps the seemingly infinite possible changes of which the wax is capable.(22) Later, in the Sixth Meditation, in which he distinguishes between imagination and true understanding, he uses another, more salient example:

When I imagine a triangle, for example, I do not merely understand that it is a figure bounded by three lines, but at the same time I also see the three lines with my mind's eye as if they were present before me; and this is what I call imagining. But if I want to think of a chiliagon, although I understand that it is a figure consisting of a thousand sides just as well as I understand the triangle to be a three-sided figure, I do not in the same way imagine the thousand sides or see them as if they were present before me.(23)

Thus for Descartes perspective inevitably misrepresents "reality" because reality is only and ever in the subject's rational grasp of the infinite possibilities of the world's two essential attributes, thought and extension. Perspective is incapable of presenting to the mind through visual sensation an adequate account of either thought or extension. Therefore there is a marked disjunction for him between what perspective can do and what the mind can conceive.

What this discussion shows is that for Descartes perspective could never serve as an adequate metaphor for the way in which the res cogitans and the res extensa are related. While the Cartesian res cogitans eventually became associated metaphorically with a "perspectival" conception of the subject as "viewing point," Descartes himself never thought of the res cogitans in terms of the fixed, perspectival point, and, in fact, did not subscribe to the notion that perspective presents a "resemblance" of the world that clearly mirrors the mind's understanding of extension. For Descartes perspective neither rationalized vision nor did it stand in metaphorically for the mind's rational view of the world. Instead, perspective was for him firmly in the material world, a visual and not a mental phenomenon, incapable even of presenting a description of mental activity. It should be clear that whenever the association is made between the perspectival viewpoint and the Cartesian subject, it is not made on the basis of a close reading of Descartes's work.

There are other problems with the notion that the relation between the res cogitans and res extensa is constituted "perspectivally." As we have seen, much of this connection seems to rest on the presumption that the cogito produces a mind-centered, stationary subject that is self-conceived in opposition to an equally stationary, and therefore spatially conceived, "objective" world. But as the above discussion indicates, Descartes does not give a perspectival description of the cogito in relation to extension. Recent reexaminations of Descartes's thought suggest that there is in it less of a subject/object dichotomy, or rather a dualistic, spatially conceived ontology, than is often assumed. The cogito is neither conceived solely in terms of a dualistic, spatial configuration, nor as a static, immutable "perspective" that this dualism would seem to imply. Rather it has both a meta-perspectival character and a synchronic, temporal element that emerges in the contingency of the enunciation "I am, I exist" that must necessarily determine self-certainty.

Karsten Harries, for example, has argued that the cogito presumes not simply a subject/object dualism but is contingent on an understanding of a third viewpoint, what he calls the "angelic eye." Descartes's self-defined project is to find an Archimedean point from which to comprehend reality in its entirety and to gain a mental toehold from which he can then grasp the world.(24) But the very nature of the project contradicts the possibility of finding such a point, which by his own definition cannot be conceived as a point at all. In fact, Descartes's search is conducted against the limiting and distorting power of the monoscopic point. If doubt becomes the testing ground of self-certainty for Descartes, it is because through doubt the mind escapes the closure and confinement of the single, fixed viewpoint. As Harries explains: "Doubt is tied to possibility. In order to doubt we must be able to conceive of the possibility that something may be different from the way it presents itself to us. Essential to doubt is the contrast between what is and what appears to be."(25)

In order to have assurance that he has his own viewpoint (to establish the grounds of self-verification) as such, Descartes must necessarily acknowledge that there is the possibility of other viewpoints different from his own. As Harries points out, however, even to recognize differences between viewpoints, one's own and those of others, presumes a kind of transcendental, mediating meta-view, not tied to any perspectivally circumscribed, embodied subject. This transcendental view, says Harries, is a disembodied, "angelic 'I'" that has no perspective but rather is objective in the sense that it is limited neither by sensory apprehension nor by the spatial orientation of perspective. For Descartes, it is this view that constitutes the "truth" of the mind's eye.

Thus the problem for Descartes is how to escape entirely from embodied subjects with contrasting perspectives and to reside only in the realm of the "angelic I" since it is in this a-temporal, a-perspectival realm that the mind grasps "reality" in its totality. However, as Harries demonstrates, there is a contradiction between Descartes's seeming desire for a clear and distinct mental representation of reality on the one hand (this is, after all, the stated goal of the method) and his rejection of a pictorial or perspectival understanding of that representation. Descartes's "demand for fully adequate representations of reality cannot be fulfilled" because reality "reveals itself to us as such precisely where it reveals itself to us as surpassing all our forms of representation."(26) In attempting to represent itself to itself, the rational subject can never entirely escape the strictures of the body. Descartes can catch sidelong glimpses of reality but never "see" it directly. "Reality" or rather the totality of thought and extension can never be represented or looked at directly but must be grasped intuitively, somehow glimpsed by looking off to one side rather than trying to stare directly at it.

Dalia Judovitz has similarly suggested that Descartes relies paradoxically on a pictorial conception of the mind's apprehension of the world while at the same time rejecting a mimetic or resemblance theory of sensuous apprehension. She argues that once the visible is reduced to the order of the sign, as when Descartes compares visuality/pictoriality to language, the senses are no longer the site of experiential reality. Reality is instead the product of the mind's schematic, abstract ordering of the objects of the senses. In Descartes's thought the mind's schematic representation of the world is made independent of perception while simultaneously relying on a model of perception to represent what is basically unrepresentable. In distinguishing between pure intellection and imagination, Descartes uses the example of the chiliagon to illustrate how the mind conceives of those things that are unrepresentable to the imagination. And yet he must fall back on pictorial modes of representation, even in his effort to define and distinguish the unrepresentable capacity of the intellect. Or as Judovitz puts it, "In the effort to validate itself, the Cartesian mind becomes an instrument of its own projection: an unimaginable point that determines the domain of the imagination."(27)

These arguments complicate the presumed relation between perspective, visuality, and the cogito. Harries's three terms - the subject, the other, and the transcendental "I" - do not diagram spatially in terms of a straight subject/object configuration. In fact, it comes much closer to Nicolas of Cusa's description of the multiple-part interaction of viewers with a painted image in the preface to De Visione Dei. A well-painted image of a face placed on a wall will perform the curious act of seeming to watch the viewer (or witness) who moves from side to side in the room. Each viewer becomes aware that the icon's eyes must follow all other viewers as well:

Owing to the revelation made by the witness (revelatio relatoris), he succeeds in realizing that the face abandons none of the walkers, even when their movements are contrary . . . . If he observes (attendere) that the gaze leaves none of the persons present, he will see (videre) that this gaze is concerned with each one with as much care as if it were the only one to have the experience of being followed, to the extent that the one who is being looked at cannot conceive that another might be the object of the same attention. He will see that this gaze watches with extreme care over the smallest creature (minima) as over the largest (maxima) and over the totality of the universe.(28)

As Michel de Certeau remarks, the gaze in the painting seems to mark a point, but for Nicolas of Cusa the point is mystical, at once "quasi-nothingness" (prope nihil) and infinite, both all-encompassing and immaterial.(29) The Cusan understanding of space is tied up in a similar conception of a universe "whose center is everywhere and whose circumference is nowhere."(30) The viewing subject's position in that space is ambiguously de-centered, the certainty of self coming only from an assumed but never perceived relation of the painting's gaze to the other. For de Certeau this marks a moment in philosophy when self-certainty (rationality) becomes based on a disorientation in which the philosopher must anchor self-knowledge to an impossibility (irrationality). The philosopher must "believe of the multitude what he does not see, in order to get out of his own uncertainty and to comprehend that the coincidence of all and each in 'one' (a gaze or faith) is 'possible'."(31) The spatiality of Descartes's "world view" is similarly ambiguous and in fact closer in spirit to the kind of decentered model presented by Nicolas of Cusa, in which subjectivity becomes a product of a kind of misperception, a dislocation rather than an emplacement.

Similarly the transcendent rationality of the cogito is being constantly interrupted and displaced by the temporality of the argument itself. The cogito moves between the immaterial, timeless, and a-perspectival "I" and the embodied, temporal, perspectival "I." The issue of temporality is integral to Descartes's conception of how doubt illuminates self-certainty. One distinctive feature of the cogito is that the statement "I am thinking" is verified for him simply by virtue of being thought or uttered. His purpose is to find a belief that is guaranteed by the fact that it is believed. In this sense the temporality of the thought or utterance of the cogito is of utmost importance to self-verification. Descartes himself says that only when he is thinking can he genuinely intuit that he exists. When or if he stops thinking, he may no longer exist, or at the very least may not know if he exists.

In the Discourse on the Method, Descartes presents the cogito as though it were a syllogistic inference: "But immediately I noticed that while I was trying thus to think everything false, it was necessary that I, who was thinking this, was something. And observing that this truth 'I am thinking; therefore I exist' (je pense, donc je suis) was so firm and sure that all the most extravagant suppositions of the sceptics were incapable of shaking it, I decided that I could accept it without scruple as the first principle of the philosophy I was seeking."(32) But later, in the Meditations, published in 1641, Descartes amends this to make it clear that this is not an inferential proposition. Self-verification is only conceivable in the moment that "I am, I exist" is uttered or thought: "So after considering everything very thoroughly, I must finally conclude that this proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in the mind."(33)

This emphasis on literal enunciation has led some to identify a performative aspect of the cogito.(34) The above quote from the Meditations reveals the extent to which doubt has less to do with verifying the statement "I am, I exist" than it does with constituting the conditional performance of the statement. For Descartes doubting is not simply a test case of thinking but is constitutive of thinking. What is distinctive about his argument is that doubting and thinking are indistinguishable and constitute a purely conditional and contingent intuition of being. In other word, it is in the moment when skeptical doubt is at its strongest and most overwhelming that a glimpse of the truth or certainty of being is possible, an idea that again calls up de Certeau's assertion that rationality must be founded on a moment of irrationality. But this moment of illumination is fragmentary and contingent and does not ultimately provide the grounds for certain understanding or knowledge of being. This is why Descartes must, in a sense, re-ground certainty in God.

Williams is more judicious about representing the cogito in these rather radical terms. He does however point to the special conditions of the cogito and suggests that they indeed hinge on Descartes's use of the first person.(35) One particular feature of the cogito is that it depends on the use of the pronoun "I," on the particular necessity of the first person for enunciation to guarantee subjective self-verification. The "I" of the cogito guarantees that it is the enunciator that is speaking/thinking. But if this is the case, then why can the formula not just as easily be "I am breathing, therefore I exist"? The problem here is that this would constitute a proposition about the self that must then be determined to be true or false (how do I know that I am breathing?). However, the point of the performativity argument is that thought itself does not have to be true or even a proposition about its own truth because even false thought requires thinking, and therefore the "I" is confirmed whether the thought is true or false. Descartes's ontology consequently rests on what he sees as the irrefutable intuition that the performance provides. In either case, whether the subject thinks or says, "I am thinking, therefore I exist," or its antithesis, "I am not thinking, therefore I do not exist," doubt remains the fundamental condition of thought.

Thus doubting is not simply one form of thinking for Descartes. Doubt must serve the difficult role of providing the conditions both for self-verification and thinking. The only kind of thinking that is indubitablity guaranteed by enunciation is doubting itself. In fact, the performance of the cogito is bound up in the presumed statement "I cannot intuit that I am not doubting, whenever I am doubting." The epistemological difficulty of doubting doubt is what resolves Descartes's total state of deception and leads him to subjective self-certainty.(36)

Herein lies the most problematic aspect of radical doubt. Much as he desires to remove certainty from corporeal sensations, it is precisely those sensations that cause him to doubt in the first place. In part 4 of the Discourse, he begins his argument for the cogito by saying that because our senses sometimes deceive us, he will begin by not trusting anything conveyed by them. In the Sixth Meditation he makes it clear that it is the sense of sight that signifies the deceivability of senses for him, for it is sight that illustrates best how the senses are deceived: "I had many experiences which gradually undermined all the faith I had in the senses. Sometimes towers which had looked round from a distance appeared square from close up; and enormous statues standing on their pediments did not seem large when observed from the ground."(37) He has a "pictorial" view of sensuous deception that comes up repeatedly in his arguments. Thus he comes full circle in his attempt to escape from the confines of the body and the cogito must necessarily rely on the senses, especially vision, as the locus of doubt. How, in effect, would we even know enough to doubt if the senses did not suggest to us already that things are not always as they appear?

It is clear that Descartes's rationalism is opposed to any "perspectival" interpretation of the relation between the res cogitans and the res extensa. But perspective and visual deception do figure in his elaboration of doubt as the proving ground of certainty. As Harries argues, perspective is limiting, it is corporeal, it is spatial, and for Descartes these are its liabilities and the reasons why perspective cannot serve as an adequate metaphor for the power and scope of the res cogitans. On the other hand, visual deception, such as that signified by the painter's art, provides the grounds for sensory deception and contradiction, and as such "doubt" would be impossible without it. The difficulty now is to examine in greater detail how perspective and doubt interact and in so doing to expand our understanding of the relation between vision and the cogito.

In his seminal history of anamorphic perspective, Jurgis Baltrusaitis suggests that anamorphosis was the visual counterpart of Descartes's doubt. Working his way through fifteenth- and sixteenth-century treatises on painting and geometry that mention anamorphosis, he traces a history of the theory and practice of anamorphic perspective beginning with Leonardo da Vinci who produced the first known images that conform to anamorphic perspective, a distorted eye and a child's face that appear on a page from the Codex Atlanticus [ILLUSTRATION FOR FIGURE 1 OMITTED].(38) But Baltrusaitis's main thesis concerns the relation between members of the Minim Order - Jean Francois Niceron, Emanuel Maignan, and Marin Mersenne - and Descartes, who maintained a close affiliation with the Parisian Minim chapter house near Place Royale by way of his friendship and correspondence with Mersenne.

Baltrusaitis's argument rests in part on the historical connection between Descartes and Mersenne, the Minim monk who was a consistently significant apologist within the Counter-Reformation Church for Galilean physics and who also served as a conduit for the exchange of scientific and philosophical ideas during the mid-seventeenth century via his prodigious correspondence.(39) Mersenne's protege within the order was Jean-Francois Niceron, a young mathematician who in 1638 published a widely-read treatise on optics and artificial perspective in which he gave a detailed discussion and geometric elaboration of anamorphosis, or what he called la perspective curieuse. Both Niceron, who lived in the Paris "convent," and Emanuel Maignan, another monk who resided for a short time at the Minim chapter house in Rome at Santa Trinita dei Monti before moving to Toulouse, not only wrote about perspective but also produced a number of different types of anamorphic images, some of immense dimensions, which took up entire walls of both the Roman and Parisian chapter houses.

Descartes knew of Niceron's work; in fact, the two exchanged books without ever having met - Niceron sent Descartes a copy of La perspective curieuse in 1639 and Descartes sent his Principles of Philosophy to Niceron in 1644.(40) But the question of influence or causal relations goes no further than this thin evidence of mutual admiration. Nevertheless, Baltrusaitis tries to connect Descartes's doubt of the senses specifically to anamorphosis. Finding little historical evidence that Descartes himself was particularly concerned with the techniques of anamorphic perspective, Baltrusaitis instead argues that Descartes's interest in anamorphic illusions can be traced to his descriptions of automata in the Discourse on the Method and more significantly in an earlier work, Treatise on Man, which was meant to be published in 1633 as a compendium to The World (neither of which was published until much later due to Descartes's reaction to the church's condemnation of Galileo in 1633). According to Baltrusaitis, Descartes's descriptions of "royal gardens" and their machines and grottos in the Treatise on Man suggest that he was familiar with the great royal garden works in Germany constructed by Salomon de Caus as well as with contemporary and ancient descriptions of mechanical wonders that change shape and reveal hidden secrets.(41) Baltrusaitis also points out that Descartes compares man himself to a wondrous automaton, constructed of pipes, devices, and springs, an image of man as a kind of machine similar to the garden works.(42)

The connection here is historically and theoretically weak. Even given Descartes's familiarity with contemporary automata and mechanical garden works, these machines are simply different from the anamorphic images with which the Minims were mostly concerned. Anamorphic images are not just images that transform to reveal hidden secrets or change from one expected view to another. They do both these things, but they do it in particular visual ways according to specific rules or guidelines based on distance point and catoptric perspective practices. In other words, although the automata described by Descartes indeed change shape and transform from one view to another, they do not do so according to the same structural rules as more strictly conceived, flat, anamorphic images constructed according to the rules of perspective. Another difference is that the garden works were constructed to transform in front of the viewer's eyes, like a shifting stage set, but two-dimensional anamorphoses require movement of the viewer - a literal transformation in the viewer's position in relation to the viewpoint. The automata are related to anamorphosis by analogy but do not conform to the same structure, method, or viewing conditions as, for instance, the anamorphic skull in Holbein's French Ambassadors.

Baltrusaitis carries more conviction when he connects Descartes's doubt of the senses specifically with anamorphic perspective. He argues that anamorphosis reveals how the apparent rationalization of sight proposed by perspective contains within it the possibility of its own de-rationalization. Thus, Baltrusaitis argues that Descartes wavers between the idea that the senses are completely deceivable and therefore not reliable bases for epistemological certainty and the idea that sight is the sense that serves as the vehicle of reason and judgement. For Baltrusaitis the theory of anamorphic perspective, proposed for instance by Niceron, demonstrates these opposing notions, contradictorily embodying and negating the rationalization of sight promised by perspective. This is both the most interesting and most confusingly argued part of Baltrusaitis's narrative, and it leads him to call the Minim Order a "Cartesian centre."(43) By calling attention to the seeming oppositions between Descartes's ontological and epistemological resolutions and at the same time pointing to similar oppositions in the perspective theory of Descartes's contemporaries, Baltrusaitis suggests, in effect, that the now-conventionally posited relation between perspective and the Cartesian "space of thought" requires careful reexamination. Taking up Baltrusaitis's suggestive proposition, the remainder of this article will ask how anamorphosis de-rationalizes perspective - if this is indeed what it does - and will explore what the technique could possibly mean for the critique of Cartesian ontology.

As mentioned earlier, the history of anamorphosis as a technique can be dated to Leonardo da Vinci's treatment of it in two drawings in the Codex Atlanticus and to references that he made to painterly problems of lateral distortion on flat surfaces and the painting of curved surfaces on walls and ceilings. He was concerned with problems of foreshortening, or accounting for the visual distortion that appears at the periphery of vision and in the frontmost plane of painting, and of painting domes and curved surfaces, where the angle of vision would be in some way oblique to the painted surface. In his treatise on painting Lomazzo describes anamorphic perspective, which he refers to as "prospettiva inversa" and in passing mentions an anamorphic profile of Christ by Gaudenzio Ferrari, a follower of Leonardo, which he himself saw, and two anamorphic pictures by Leonardo, neither of which he saw himself but were described to him by Francesco Melzi, one of a battle between a lion and a dragon and another of horses produced for Francis I.(44) However, the only extant examples of "prospettiva inversa" by Leonardo are the above-mentioned child's face and an eye from the Codex Atlanticus, which are meant to be viewed from the right side, close to the page.

Leonardo's experiments with anamorphosis seem to have derived from his investigation of lateral foreshortening and the relation between perspectiva naturalis (optics) and perspectiva artificialis (costruzione legittima). Leonardo became increasingly aware of how vulnerable the painter's perspective is to visual distortion because optical apprehension of the two-dimensional painted surface conflicts with apprehension of the surfaces through the painting if the viewpoint posited by perspective is not fixed or controlled. One problem that concerned him was the disjunction between lateral apprehension in "natural" perspective and lateral representation in "artificial" perspective. In "natural" perspective, when the eye apprehends a perpendicular plane along which are placed a series of objects of equal size and of equal distance from each other and from the eye - a row of unilateral columns for instance - the objects that appear to the side of the central axis of vision will be seen at narrower angles of vision and therefore will appear smaller and farther away than those near the center.(45) This presents a problem for painting however, because objects that are portrayed in the picture plane remain flat and adhere to only one viewpoint and one angle of vision. Thus, if a picture conforms too well to the rules of artificial perspective it can only be seen from a single fixed or immovable viewpoint. From any other viewpoint objects to the side in the frontmost picture plane will appear distorted to "natural" perspective.

According to Leonardo any effort to combine naturalis with artificialis will result in something he calls prospettiva composta, which he recommends that painters avoid.(46) This composite perspective appears to be anamorphosis. Composite perspective capitalizes on the tendency of artificial perspective to severely distort those images that appear closest to the viewer when the visual angle is at its widest. In order to compensate for the distortion that occurs in the frontal plane of the picture, a painter might create lateral images that when seen from a central viewpoint will appear normal. These images then would in actuality be seen from an "oblique" viewpoint, even though that same viewpoint is actually central for the painting as a whole. Thus these lateral images would be anamorphic and when seen from any angle other than the central one they would appear severely distorted. However, Leonardo emphasizes that the painter's task is to create a picture that can be seen from many different angles with no undue distortions.(47)

While Leonardo was opposed to the use of anamorphic or severely foreshortened images in painting he was also clearly interested in the possibilities of distortion, as witnessed by his anamorphic drawings and Lomazzo's references to his anamorphic paintings. The idea that perspective could be manipulated in order to produce a special kind of picture that could only be viewed from a severely oblique angle was obviously of interest to him.

One important aspect of the relation between linear perspective and anamorphosis is that both require the manipulation of the "distance point." In his investigation of Alberti's perspective Leonardo was perhaps the first to propose that Alberti's construction of the vanishing point presumes the use of a "distance point."(48) That is, the distance or depth implied in Alberti's grid - which extends back toward the vanishing point of the horizon - depends on a theoretically posited measure of distance between the viewer, the picture surface, and the vanishing point. An illusion of depth is fabricated by placing a distance point outside the frame of the intended image at the same height as the intended viewing point. This distance point is then used to determine the rate of diminution or measurement of the intervals that extend back into the picture toward the horizon. Lines are drawn from the distance point to the marked intervals that divided the picture's frontmost plane and as each line intersects the side of the frame, the point of intersection determines the measurement of intervals extending back toward the horizon [ILLUSTRATION FOR FIGURE 2 OMITTED].(49)

Depending on how far or how close the distance point is placed to the frame's edge and how high or low the horizon line is placed, the picture will appear to have either a great distance between viewpoint and horizon or a sharply raked effect, like a raked stage on which objects appear to be falling down toward the viewer, or a very low horizon, such that the viewer appears to be looking at a ledge that then drops down behind the frontmost picture plane. For Leonardo the distance point construction was mathematically precise in the sense that he claimed by knowing only a limited amount of information, i.e. the length in braccia of the base of the image, he could tell how far the viewer was supposed to stand from the work and how far, theoretically, any marked spot in the painting was supposed to be from the viewer? It is, however, precisely the painter's ability to manipulate the distance point so dramatically that caused Leonardo to warn against the possibilities of too much distortion.

Between Leonardo and Jean-Francois Niceron there were many treatises that dealt with anamorphic perspective and many artists who put the technique into practice. However, Baltrusaitis suggests that Niceron himself must have relied on Daniele Barbaro's "automatic method" to construct a model for anamorphic perspective.(51) In his Practica della Perspettiva, Barbaro sets up a system for figuring the foreshortening of objects.(52) One begins with a square called the perfetto, the top edge of which serves as the bottom edge of the proposed foreshortened image (in [ILLUSTRATION FOR FIGURE 3 OMITTED], the perfetto is the square bcfg).

The foreshortened trapezoid extending up above the square is called the digradato. In this diagram the proposed foreshortened image is a trapezoid, bcfg, which extends up toward a theoretical principal point on the right that would become apparent if you took a ruler and extended lines cf and bg until they met at a point. In order to perform the foreshortening of the digradato, Barbaro says to divide the perfetto into an equal number of squares, to create a grid. In the diagram of fig. 4 the perfetto is divided into a grid and the digradato rises to a triangular apex or principal point (point a). All vertical lines of the grid are extended up in the digradato toward this principal point a, which Barbaro calls the eye (occhio) meaning that it marks the height at which the eye is theoretically placed. The horizon of the foreshortened digradato is marked by the line fe. Both the perfetto and the digradato are bisected by a single diagonal. The diagonal on the digradato intersects with each vertical line drawn up from the perfetto toward the principal point a. At each point of intersection between the diagonal and vertical lines, a straight horizontal is drawn to mark the planes of diminution in the foreshortened image. The grid is now systematically reduced into a foreshortened trapezoid.

This method was particularly effective for transferring objects into perspective, a classic problem in perspective treatises that were often largely concerned with the problems of foreshortening mazzochi and other many-sided, complex objects. Niceron himself disparaged the complexity and abstraction of many previous commentators, including Barbaro, of whom he says in his preface: "How many others who have written on this, have had methods so abstract and speculative, such as Guidobaldo; or are muddled like that of Daniele Barbaro, whose [method] is very difficult to reduce in practice, if one is not already very learned."(53) Niceron preferred the systems of Vignola and Cousin (Cousin codified a double, distance-point system called tiers points, first proposed by Pelerin, and Vignola combined the tiers points with a plane intersection system)(54) and appears to have combined a digradato system with a distance point system for his representation of "perspective commune" or common perspective [ILLUSTRATION FOR FIGURE 5 OMITTED]. Here he shows a square perfetto ABCD divided into twenty-five smaller squares. Above it the digradato foreshortens the grid by extending the vertical lines back toward the principal point E. The diminution rate at which the squares recede is determined by the diagonal line drawn from the proposed distance point F. This diagonal line intersects each of the vertical lines and once again, at each one of these intersections, a horizontal line is drawn. Each horizontal then determines the depth of each successive bar of the foreshortened grid.

The distance point is a proposed point that corresponds in theory with the proposed viewing point of the actual observer in relation to the image. Thus it is often represented, as it is in Niceron's drawing, by a human figure whose eye meets the end point of the diagonal line. The figure's eye is also aligned, in accordance with Alberti's principles, with the intended principal point. It is a convenient way of literalizing the actual viewpoint, off to the side of the image. But in so doing, it effectively posits a third point outside of the principal point/vanishing point opposition, one that coincides directly with the picture plane. In anamorphosis the distance point becomes, at the limits of possibility, the actual viewpoint. In book 2 of his treatise Niceron begins his discussion of anamorphosis by noting in his first proposition that regular or "common" perspective posits a plane situated between the eye and the object. This is one kind of perspective, he says, but there is another kind "such that the object would have its place between the eye and the plane, or behind the plane."(55) This kind of perspective, in which the object projects out toward the viewer rather than back into the picture, is developed out of the projective possibilities of "common" perspective.

In the next proposition, 12, Faire une chaire en Perspective si difforme, qu'estant veue hors de son poinct, elle n'en air nulle aparence, Niceron shows how it is possible to use techniques of foreshortening to unnaturally elongate a chair and a bench.(56) Thus the same method that he used in the earlier book to foreshorten a chair [ILLUSTRATION FOR FIGURE 6 OMITTED] is employed to make an anamorphic image of a chair [ILLUSTRATION FOR FIGURE 7 OMITTED]. In fig. 6, there is a perfetto of sorts, square abcd (in the lower third of the table). The digradato extends up to the left toward principal point Q. The distance point is to the right at the upper corner above the chair and is marked R. In fig. 7 there is no longer a perfetto, but the digradato still follows the same rule, only now the horizon has been lowered and the principal point Q has been pulled much farther to the left. At the same time, the distance point R is now over next to Q, no longer in the opposite upper right corner. The chair is now severely elongated although it is still recognizable from a central vantage point (Niceron did this on purpose so as to keep the two diagrams correlated). But the ideal viewpoint is now not the presumed central view that corresponds in opposition to the principal or vanishing point. Rather it is now located at the distance point, so the chair is now properly seen by turning the page around, tilting it, and looking at it from the direction if RQ.

In proposition 3, Donner la methode de descrire toutes sortes de figures, images, & tableaux en la mesme facon, que les chaires de la precedents proposition, c'est a dire, qui semblant confuses en aparence, & d'un certain point represent parfaitement un objet propose, Niceron then proposes a specific method for producing this kind of elongated image.(57) In the accompanying illustration [ILLUSTRATION FOR FIGURE 8 OMITTED] he begins with a perfetto, a perfect square ABCD (illus. no. XXXIII), divided into thirty-six smaller squares. This is the grid that will become the trapezoid abcd in illustration no. XXXIV. The perfetto's side AD becomes the trapezoid's (digradato's) baseline ad. The diagram is now reduced to bare essentials, with P being the principal point and to the left of it R being the distance point. The principal point is now raised very far from the baseline ad, and the distance point is also raised and brought very close to the principal point. Unlike the distance point construction illustrated in figs. 2 and 5, there is no longer both a horizon line and a principal point that marks the viewing height, but rather they have been collapsed into one level. The rate of diminution is still determined by the distance point R, by drawing down straight lines to meet the points that divide the baseline ad. As each line intersects line aP, it marks a point that will determine the horizontal lines marking out the rest of the grid. In illustration no. XXXV [ILLUSTRATION FOR FIGURE 8 OMITTED] this process is simplified by drawing a single line dR, which bisects the trapezoid exactly, and as it crosses each vertical line of the grid, determines the horizontals at points onmlk. As the diagram shows, the baseline no longer marks the plane of intersection between the image and the eye. Any figure or object portrayed in this grid, with its highly elongated and precipitous rate of diminution, would be unrecognizably distorted if seen from a central vantage point. The image would have to be seen from above, from the vantage of the principal point P, in order for it to be legible. Because it is difficult to present an upside down viewpoint, the entire construction is turned on its side, so that now whatever appears in the grid will be seen from the right side at a severely oblique angle, which is precisely the case with the heads rendered by Niceron in fig. 9. As we saw in fig. 5, the distance point coincides with the flat plane of the paper, canvas, etc. So the anamorphic vantage point for this type of construction, which closes the gap between the principal, viewing, and distance point, will be to the side of and physically very close to the picture surface [ILLUSTRATION FOR FIGURE 9 OMITTED].

As noted earlier, Niceron suggests that while in "common" perspective the picture plane intersects between the eye and the object represented, in "curious" perspective the object comes between the eye and the picture plane. He meant that the diminution of the image now extends in reverse toward the viewer. Anamorphic perspective creates a strange boomerang effect in which the image originates and ends at the same point; i.e. the principal or "vanishing" point occupies almost the same space as the viewing or "distance" point. The image also now adheres rather tightly to the picture surface, and whatever depth that is supposedly indicated by the rate of diminution seems to extend out toward the viewer and not back beyond the picture plane. In fact, the distance point now functions not to produce the illusion of distance at all but rather to draw the viewer into a tighter, more restricted viewpoint which is so close to the picture surface that the image itself almost disappears from sight.

Employing all the same principles used for what is commonly called Albertian perspective or costruzione legittima, anamorphosis in one sense represents a kind of culmination of what perspective makes visually possible. It presents a picture that can only be seen from a single, fixed viewpoint, constructing a rigidly circumscribed viewing space, which presumes a single viewer. However, as we have also seen, it is a type of perspective that completely collapses the illusion of distance between viewer, picture plane, and horizon.(58) If Albertian perspective has been accused of fostering the illusion of the viewer as objective surveyor, positing the viewing subject as the spatial antithesis of the principal point, then anamorphosis, using exactly the same principles, completely resists this kind of spatializing metaphor. The subject (viewpoint) is nearly collapsed with the object (principal or vanishing point), and each is now aligned with the surface of the intersecting plane.

In conclusion, we are left with the question as to how anamorphosis interacts with Cartesian thought. From the previous discussion it is apparent that Cartesian subjectivity is not strictly definable according to the stable, spatially conceived model with which it is often associated. In fact, there are many fissures in Descartes's arguments that make it difficult to simply equate it with a rigid subject/object ontological opposition. As we have seen, the temporal, performative aspect of the cogito also negates any absolute assurance of ontological certainty. In a seemingly contradictory manner, certainty can be understood as contingent, embodied, and almost anamorphic for Descartes (glimpsed via the sidelong view). The Cartesian subject's certainty is only in and of the moment of enunciation and cannot be arrested or codified without an appeal to a transcendental source (God).

Anamorphosis, on the other hand, posits an embodied, seeing subject, but one that is still equally adherent to contingency, temporality, and performativity. While two-dimensional anamorphosis requires the viewer's movement from visual confusion to visual clarity, this does not simply reaffirm the connection between vision and rationality as some have argued. For instance, Catherine Chevalley has claimed that although anamorphosis first engenders a moment of radical doubt, visual uncertainty and opacity give way in a cathartic game that reinscribes the viewer into an apparent position of visual mastery.(59) The manipulation of the visual field effectively seems to restore "reason" to its rightful place. But, as we have seen, the movement to "clarity" involves the collapse of distance between subject and object. Sliding the distance point toward the principal point, the picture "looks" back at the viewer, making him/her both subject and object of the configuration, conjoining the viewpoint with the two-dimensional surface of the representation. The viewer is strangely but almost literally absorbed into the picture. There is nothing rational in the reversal, i.e. it can hardly be said to posit a "clear and distinct" perception of self-verification. "Clarity" in this case involves the near collapse of the subject/object distinction, and a condensed space of vision. In a sense then, anamorphosis, like Cartesian ontology, constructs a model viewpoint that depends on a de-spatialized vision. As Judovitz has argued, anamorphosis thus entails the displacement of the subjective eye, revealing the arbitrary and constructed relation between vision and the visible. In so doing, it corresponds to the "aperspectivism" inherent in Descartes's work and is a technique of illusion that "reduces ordinary vision to a kind of blindness."(60)

Anamorphosis thus presents a rejection of the visual as such, revealing the distortion and falsity inherent in the embodied viewpoint. However, rather than reconfirming the viewer's subjective certainty, anamorphosis repetitively replays radical doubt without recourse to rational resolution. Anamorphosis is, after all, still a visual and therefore embodied phenomenon. It cannot retreat along with Descartes into the mind. That the field of vision is indeterminate rather than determinate in the constitution of subjectivity, that far from isolating and confirming the subject/object relation, perspective visually enacts the tensions, contradictions, and reversals of subjectivity, is what anamorphosis makes apparent. What distinguishes anamorphosis from the cogito is that the anamorphic point resists the Cartesian recuperation to self-knowledge and instead reaffirms a divided subject who is split off from epistemological and ontological certainty.(61)

UNIVERSITY OF SOUTHERN CALIFORNIA

This article would not be possible without the close readings, suggestions, critiques, and general support offered by Mario Biagioli, Paula Findlen, Carlo Pedretti, Donald Preziosi, Kenneth Reinhard, Keith Topper, and Cecile Whiting.

1 The most comprehensive of these accounts is Kemp's recently published survey (1990) of the history of optics and pictorial representation over a period of some five hundred years.

2 Others who have examined perspective's symbolic dimensions include Edgerton, 1976, who employs Thomas Kuhn's notion of the paradigm shift in his analysis of perspective's origins arguing that the rediscovery of Ptolemy and three-dimensional mapping transformed the way in which the world was seen during the Renaissance and argues in Edgerton, 1991, that the relationship between geometry and perspective was crucial for the Renaissance rationalization of vision. Damisch takes issue with Edgerton's use of the paradigm notion, arguing that what is interesting about perspective is its "demonstrative character" that makes it paradigmatic in the literal or "technical" rather than the metaphoric sense of the word. Thus, Damisch argues, it is important to understand what it is that perspective "shows" or "demonstrates" to the viewing subject (an idea with which the present article is concerned) rather than how it represents a shift in world view.

3 Panofsky, 66.

4 Burson, 103.

5 Elkins, 29-30.

6 Elkins, 166-80. Elkins argues that late in the sixteenth century perspective became increasingly employed as a statement of its own arcane and irrational character and that it was used to represent melancholia or as a figure of vanitas (such as in Durer's famous print, Melencolia I, or in Holbein's French Ambassadors).

7 Subjective in the sense that Descartes's search for a rational foundation for knowledge was based on an argument designed to ensure ontological certainty and objective in the sense that the Cartesian empiricist tradition asserts that judgments about the systematic and coherent structures underlying the world can be made through observation of it.

8 For instance, Bergson, whose entire project in Matter and Memory is to overturn what he sees as a traditional hierarchical relationship between spatial and temporal ontology, says at the end of the third argument (in which he has reformulated perception as a relation between affection and duration) that "questions relating to subject and object, to their distinction and their union, should be put in terms of time rather than space" (71). Heidegger's concerns were similar, at least in one sense, because in Being and Time he develops a concept of Dasein (being-in and being-with the world) that contradicts a Cartesian notion of ontology as a self-sufficient, transcendental consciousness, the res cogitans, replacing it with a concept of being-in-the-world as constant temporal possibility. Cartesian space is occurrent space for Heidegger, measurable in physical terms. Against this he poses a notion of phenomenological space, the ready-to-hand, i.e. "the aroundness of the environment, the specific spatiality of beings encountered in the environment, is founded upon the worldliness of the world, while contrariwise the world, on its part, is not occurrent in space" (135). This passage and others related to Heidegger's critique of Cartesian spatiality are lucidly discussed by Dreyfus, chap. 7. My thanks to Keith Topper for drawing my attention to this point.

9 Jay uses the term "Cartesian perspectivalism" as a "way to characterize the dominant scopic regime of the modern era" (69-70).

10 It was first published in French with the title: Discours de la Methode pour bien conduire sa raison, et chercher la verite dans les sciences. Plus la Dioptrique, les Meteores et la Geometrie qui sont des essais de cette Methode.

11 Descartes's radical doubt was both a feature of contemporary skepticism and an attempt to counter that skepticism. In this sense his Discourse seems related in spirit to Fr. Marin Mersenne's great counter-attack on the then-current strain of skepticism inspired by a revival of pyrrhonism in the sixteenth century upon the rediscovery of and republication in Latin of Sextus Empiricus's Hypotyposes in 1562. Mersenne's work, La Verite des Sciences, contre les Septiques ou Pyrrhoniens, was published in 1625 and represents an attempt at what Popkin has called "constructive or mitigated scepticism" or rather the "formulation of a theory which could accept the full force of the sceptical attack on the possibility of human knowledge, in the sense of necessary truths about the nature of reality, and yet allow for the possibility of knowledge in a lesser sense, as convincing or probable truths about appearances" (129).

12 Descartes, 1985, 1:120. Henceforth, all translations of Descartes will come from this source.

13 Ibid., 1:127.

14 Ibid., 1:127.

15 Williams, 1967, 1:349.

16 Descartes, 1985, 1:129.

17 In fact, Descartes's insistence that neither visual nor mental apprehension of the world can be reduced to a "little man" model of the brain substantially reduces the force of the "homunculus" argument which has been used to critique Descartes's attempt to located a single source in the brain, the pineal gland, as the seat of the soul. See Cottingham, 121-22.

18 Descartes, 1985, 1:165. Here Descartes is attacking the Scholastic adherence to an Aristotelian notion of the "species" image.

19 Ibid., 1:65. Judovitz, 64, has similarly noted that Descartes established the arbitrary character of vision by drawing an analogy between pictorialism and language.

20 Descartes, 1985, 1:165-66.

21 Williams, 1967, 1-2:351.

22 Descartes, 1985, 2:20-21: "But what is meant here by 'flexible' and 'changeable'? Is it what I picture in my imagination: that this piece of wax is capable of changing from a round shape to a square shape, or from a square shape to a triangular shape? Not at all; for I can grasp that the wax is capable of countless changes of this kind, yet I am unable to run through this immeasurable number of changes in my imagination, from which it follows that it is not the faculty of imagination that gives me my grasp of wax as flexible and changeable. . . . I must therefore admit that the nature of this piece of wax is in no way revealed by my imagination, but is perceived by the mind alone."

23 Ibid., 2:50.

24 For instance, he states this explicitly at the beginning of the Second Meditation: "Archimedes used to demand just one firm and immovable point in order to shift the entire earth; so I too can hope for great things if I manage to find just one thing, however slight, that is certain and unshakeable" (ibid., 2:16).

25 Harries, 30.

26 Ibid., 39.

27 Judovitz, 81.

28 De Certeau, 12. De Vlsione Dei dates from 1453. For a more recent translation, see Hopkins, 117.

29 Ibid., 14.

30 Ibid., 18.

31 Ibid., 34.

32 Descartes, 1985, 1:127.

33 Ibid., 2:17.

34 See Hintikka. Williams, 1978, 76, acknowledges Hintikka's argument and expands on the possible meaning of "performativity" in reference to the cogito. He argues that Hintikka's contribution is to show that the relation between "I think" and "I exist" is not one between two propositions, but rather that what is essential for Descartes is that he should be thinking, "and it will be that thinking, and not a reflexive proposition recording it, which will somehow bring the indubitablity of 'sum' before him." Cottingham, 36, argues that the original formulation of the cogito in French must be read as employing not the simple present "I think" but the continuous present "I am thinking." Descartes makes it apparent that this is what he meant by latex modifying the cogito in the Meditations, pointing out that existence is only guaranteed in the moment of the cogito's enunciation. Thus "what makes me certain of my existence is not some static or timeless fact about me - that I am one who thinks; rather, it is the fact that I am at this moment engaged in thinking."

35 Williams, 1978, 92-97.

36 At the beginning of the Second Meditation, Descartes, 1985, 2:16, makes it clear that the state of doubt is the only thing he cannot doubt: "Anything which admits of the slightest doubt I will set aside just as if I had found it to be wholly false; and I will proceed in this way until I recognize something certain, or, if nothing else, until I at least recognize for certain that there is no certainty."

37 Ibid., 2:53.

38 The two drawings appear in the Codex Atlanticus, fol. [35.sup.v-a], mentioned in Richter, 1977, 2:384. The Codex Atlanticus has been dated to ca. 1478-1518, and the drawings are tentatively dated by Richter to ca. 1510.

39 There seems to be some confusion as to when and where Mersenne and Descartes met. Both were educated at the Jesuit College at La Fleche and their tenure there seems to have overlapped, but given the eight-year difference in age Mersenne would have been in a much higher class than Descartes and therefore they may not have come into contact with one another. According to the chronology given by Dear, 12-14, Mersenne entered in 1604, shortly after the college was founded, and probably left sometime in 1609 to attend the Sorbonne and College de France in Paris before entering the Minim Order in 1611. Dear relies on the information given by Mersenne's contemporary and biographer, Hilarion de Coste, in a biography written shortly after Mersenne's death. Descartes entered the college in 1606 and probably left sometime in 1614. See Cottingham, 8-9. Descartes's earliest biographer, Adrian Baillet, maintains that the two met first at La Fleche (although he wrongly says that Descartes entered the college in its founding year, 1604) and then continued their friendship beyond the confines of the college, with Descartes staying at the Parisian Minim "couvent" at Place Royale in the winter of 1622-23 and again from 1625 to late 1628 or early 1629. Charles Adam was convinced that Mersenne must have finished his studies before Descartes entered the school and thus suggests that the two may have met later when Descartes was in Paris in 1623 through a mutual acquaintance, Claude Mydorge. He is not convinced that Descartes actually stayed at the Minim Claude Mydorge. He is not convinced that Descartes actually stayed at the Minim convent in 1622 and in 1625, but will allow only that Descartes was indeed in Paris during this period. See Descartes, 1964-76, 12:34-74. In either case, Descartes probably met Mersenne upon his return from Holland in 1622 or a little later in 1623 when he spent the winter in Paris, and perhaps was in correspondence with him during the trip he subsequently took to Italy from 1623 to 1625. The earliest fragment of a letter from Descartes to Mersenne (although there is some doubt as to its authenticity) dates from the summer of 1625. See Mersenne, 1:256-59.

40 In a letter dated 30 April 1639, Descartes mentions to Mersenne that he received a packet of books, Niceron's Perspective curieuse among them: "Mon Reverend Pete, I'ay receu 4 pacquets de votre part depuis 8 ou 10 jours, sans avoir toutesfois receu qu'une de vos lettres. Car le [1.sup.er] ne contenoit que les livres de Monsieur Morin, de Monsieur Hardy, & les Theses du Pere Bourdin; le [2.sup.eme] que la Perspective curieuse" (Descartes, 1964-76, 2:529-30). According to Baillet, 2:300-01, Descartes also sent Niceron his Principia in 1644.

41 Descartes, 1985, 1:100.

42 Ibid., 1:100-01.

43 Baltrusaitis, 60.

44 Pedretti surmises that of these two lost paintings, the lion and the dragon probably dates to before 1480, and the other one, of horses, to sometime after Leonardo went to France. The term anamorphosis, from the Greek ana meaning "again" and morphe meaning "form," was coined by the Jesuit Gaspar Schott in 1657, a good one hundred fifty years after the technique's invention. Prior to Schott it was called by several names such as "prospettiva inversa" (Lomazzo, 335-36, cap. XIX: "Modo di fare la prospettiva inversa che paia vera, essendo veduta per un solo forame") and "perspective curieuse" (by Niceron). Elkins, 251, rejects the notion that anamorphosis is a kind of "inverse" or reverse perspective. He says that "reverse perspective" was the term applied to a medieval practice of "splaying" orthogonals instead of making them converge. It is clear, however, that Lomazzo had no difficulty describing anamorphosis as a reversed direction or inversion of linear perspective, and in fact was the first to describe in detail a method for producing anamorphic images which combines a grid with a projection system.

45 Richter, 1977, 1:127-28. The fragments referring to perspective come from Manuscript A and date to ca. 1490-91. The drawing and explanations of lateral diminution of columns come from an earlier sheet of studies from ca. 1483-85, Codex Atlanticus, [353.sup.r-b, v-b]

46 Richter, 1977, 1:147. Leonardo states, "It is well therefore to avoid such complex perspective ('prospettiva conposta [sic]') and hold to simple perspective which does not regard planes as foreshortened, but as much as possible in their proper form" (Manuscript E, 16b-16a, ca. 1513-14). See Richter, 1970, 1:64.

47 Although this has been used to claim that Leonardo developed a theory of "curvilinear" perspective to compensate for the "natural" curvature of vision, most writers on perspective as well as on Leonardo have since come to reject the notion that perspective theory incorporated the curve in order to compensate for "actual" visual experience. For a defense of the curvilinear theory, see White, 202-18; and Panofsky, 31-36. For arguments against, see Richter, 1977, 1:147; Damisch, 5-6; and Elkins, 181-216. While White defends the idea that painters like Leonardo were working toward constructing a curvilinear perspective, Panofsky identifies the theory of curvilinear vision as proof of how linear perspective is thoroughly symbolic. Elkins devotes a critical chapter to the philosophical problem presented by curvilinear perspective when it is used as a justification for either optical or perspectival "realism."

48 Richter, 1977, 1:128.

49 Ibid., 1:129.

50 Ibid., 1:129: "If you make a plan and show it to me with a mark or a dot made on it at random, and you tell me only how many braccia long is the foreside, I shall be able to tell you how many braccia you are standing away from it in observing it, and how many braccia removed is the point placed on the said square at random."

51 Baltrusaitis, 41.

52 Elkins, 89-101, discusses Barbaro's model and the related construction he calls the "circumscribed rectangle" method at length.

53 First published in 1638, Niceron's treatise was republished posthumously with a treatise by Mersenne called Optique et captoptrique in Paris in 1651. It is the 1651 edition to which I will refer throughout the remainder of this article. "Quant aux autres qui en ont escrit, ils ont des methodes des si abstraites & speculatives, comme Guide Ubalde; ou si embrouillees, comme Daniel Barbaro, qu'il est tres difficile de les reduire en pratique, si l'on n'a d'autres connoissances" (Niceron, 4). He is probably referring here to Guidobaldo del Monte whose treatise Perspective libri sex was published in 1600 in Pesaro.

54 Kemp discusses both Cousin, 64-68, and Vignola, 79-83. Tiers points employs two distance points on either side of the principal point. The intersection method is more difficult to comprehend as it employs an intersection of optical rays, those imagined extending from the eye with the picture plane. As Elkins, 84-89, points out, there is no sense of agreement among treatise writers about which method was best, and the tendency of each writer was to introduce yet more methods, each specified as particularly useful for certain kinds of representation.

55 Niceron, 90-91: "Or nous avons seulement considere jusques a present le plan situe entre l'oeil & l'objet, mais nous le considerons deformais in different, soit que l'objet ait sa place entre l'oeil & le plan, ou derriere le plan." This is not simply an ingenious and radical way of representing anamorphosis, but is another interesting caveat to Elkins's rejection of the idea that anamorphosis represents a "reverse" perspective.

56 Ibid., 92-93: "To make a chair in perspective so deformed, that seen from outside its point, it would have the appearance of nothing."

57 Ibid., 93: "To give a method of describing all sorts of figures, images and pictures in the same way, as the chain in the preceding proposition, that is to say, which seemed confused in appearance, and from a certain point represent a proposed object perfectly."

58 I will not be dealing with catoptric or reflective anamorphoses here, which figure prominently in book 2 of Niceron's treatise. But they are of special interest because the images that are produced raise a whole set of important questions concerning the relationship between perspective, reflection, and mimesis. The images that appear in the mirrors of these devices can literally only be seen or rather only exist in reflection, making the viewer's relation to the image one in which the only possible point of view is outside the closed system of reference imposed by the mirror.

59 Chevalley, 289-96.

60 Judovitz, 69.

61 The suggestiveness of anamorphosis has figured prominently in at least one twentieth-century construal of the subject. The reversals of anamorphosis clearly interested Jacques Lacan, who employed them to illustrate the disturbing subject/object displacements he himself attributed to the "gaze." Although it is not the purpose of this article to discuss Lacan's use of anamorphosis, it was the starting point for the present argument. In seminar 11, The Four Fundamental Concepts of Psychoanalysis Lacan capitalizes on the instability inherent in anamorphosis and uses it to undergird his critique of the Cartesian cogito, which he argues posits an impossible figuration exemplified in the phrase "I see myself seeing myself." Counter to Descartes's search for certainty, Lacan effectively argues that subjectivity as such must necessarily be understood as the negation of self-certainty and the impossibility of self-verification in the visual field, a negation given a figural shape in anamorphosis. My purpose here has been to argue that at least in the seventeenth century, the implications of anamorphosis were understood and that not only in Descartes's work but in the work of his contemporaries, the perspective model was by no means a universally accepted analogy for the philosopher's self-certainty, and in fact was perhaps understood to stand in opposition to Cartesian rationalism.

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