Jean-Paul RIOPELLE, L'hommage à Rosa Luxemburg, 1992 (détail)[ * ]

Albrecht Dürer and Nicholas Cusanus:
the Real, the Ideal, and the Quantification of the Body

Allister Neher

Abstract:

For Albrecht Dürer the study of human proportions and the study of perspective were correlative fields of inquiry, and central to the quest for a naturalistic science of painting. Their conjunction and natural point of coincidence, it will be argued, is to be found in Dürer's stereometrical studies of the human figure. In these studies perspective and proportions are joined to create a system for manipulating representations of the human form. This paper will set out the central ideas that direct this approach to representation, and relate them to the philosophy of Nicholas Cusanus's De docta ingnorantia. Both Dürer's art theory and Cusanus's epistemology link the real and the ideal through the language of quantification. This paper explores a structural commonality in their different approaches to the question of truth and the development of knowledge.

                  

This paper arose from a structural intuition about truth as it is conceived in Albrecht Dürer’s art and in Nicholas Cusanus’s philosophy. The phrase “structural intuition” is borrowed from Martin Kemp, while severed from his empiricist epistemology. [1] I find it an apt phrase for what I am trying to articulate. In historical studies we are often struck by similarities that appear to connect disparate activities and inquiries, and make them seem fitting counterparts in their culture. This fittingness can show itself in many ways. With Dürer and Cusanus it is revealed through their use of the language of geometry. The role that was given to mathematics in Renaissance reflections on the nature of knowledge and reality is not a new subject. The aim of this paper is to set out a new aspect of it.

I

My discussion of Dürer will have as its focus his studies on human proportions. I would like to enter this discussion by considering an evaluation that has often been made of these studies. Frequently, they are treated as a curiosity, a kind of intellectual excess that must have been due to some private preoccupation or obsession that led Dürer out of the realm of art theory and into the purely descriptive science of anthropometry (illustration 1). [2] Federico Zuccaro, for instance, thought that Dürer’s efforts were simply a waste of time, artistically irrelevant, an amusement for those more inclined to explore Nature’s wonders than to represent them. [3]   Even Erwin Panofsky, perhaps Dürer’s most sympathetic commentator, offered an assessment of these studies that in one aspect agreed with Zuccaro’s conclusion. In “The History of the Theory of Proportions as a Reflection of the History of Styles,” Panofsky tells us

Dürer’s Vier Bücher von menschlicher Proportions marks a climax which the theory of proportions had never reached before nor was to reach ever after. It also marks, however, the beginning of its decline. Dürer himself succumbed, to a degree, to the temptation of pursuing the study of human proportions as an end in itself: by their very exactitude and complexity his investigations went more and more beyond the bounds of artistic usefulness, and finally lost almost all connection with artistic practice. . . .And if we remember that the smallest unit of his metrical system, the so-called “particle” (Trümlein), was equal to less than a millimeter, the chasm between theory and practice becomes obvious. [4]

Zuccaro, Panofsky, and others who would join them in expressing such reservations about Dürer’s project are of course quite right: one does not need the kind of detail that Dürer provides in order to be able to produce worthy and compelling naturalistic figures. I would like to propose though a different perspective from which to view the contribution made by these studies to the history of representation. For while they do appear to be early contributions to the science of anthropometry, they can also be seen as part of a project to further sophisticate the techniques of representation, and advance the visual arts.

Dürer fashioned a number of approaches to the study of human proportions. For my purposes they are not so different that one cannot stand in for the rest. As my example, then, let me choose the one referred to in the quotation from Panofsky.

The general framework within which Dürer made his measurements was derived from the Vitruvian canon (illustration 2). Leaving aside long-standing debates about the proper interpretation of Vitruvius, the general idea behind his approach is that one establishes human proportions by organic differentiation, not through the use of an absolute module. One measures the relation of fingers to hand, hand to forearm, forearm to arm, and all limbs in relation to the length of the body. All of these relations are expressed as fractions of total body length, giving one a harmonious, organic unity of the whole. Of course one needs a system of measurement in order to state fractional relations, and the system that Dürer proposed has a certain debt to Alberti:

I will, in this book, teach how to measure out the human figure with a rule, which I make long or short according as the figure is to be large or small. I make the rule always one sixth of the length of the figure. . . . Then I divide the rule into ten equal parts and each part I call a zall, each zall I divide into ten and call each tenth a teil, each teil into three and call each third a trümlein. [5]

As Panofsky suggested, this seems to be a rather refined system of measurement for an artist’s workshop of that time, where concerns rarely extended beyond the larger divisions of the body (illustration 3). This is true, if one is talking about the typical artist of Dürer’s era, but Dürer’s theoretical ambitions did not make him a typical artist. He took the Italian idea of a “science of painting” seriously, and he was trying to advance that science. Dürer was living in an era that would to bring about a complete transformation of humankind’s relation to the world. As is well known, between 1300 and 1600 Western European civilization gained control of the basic quantificational techniques that would allow it to dominate nature in ways that would be without parallel. [6] One hundred years after Dürer’s studies on human proportions the transformation was essentially complete, announced in this famous pronouncement by Galileo:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze, but the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth. [7]

No artist would have endorsed this more whole-heartedly than Dürer. In fact, he had already expressed his support for this vision, for the future of art, in the dedication of his Treatise on Measurement:

In our Germany, most excellent Wilibald, are to be found at the present day many young men of a happy talent for the Art Pictorial, who without any artistic training whatever, but taught only by their exercise of it, have run riot like an unpruned tree, so that unhesitatingly and without compunction they turn out their works, purely according to their own judgment. But when great and ingenious artists behold their so inept performances, not undeservingly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who themselves are ignorant of this art. Since this is in very truth the foundation of the whole graphic art, it seems to me a good thing to set down for studious beginners a few rudiments, in which I might, as it were, furnish them with a handle for using the compass and the rule, and thence, by seeing Truth itself before their eyes, they might become not only zealous of the arts, but even arrive at a great and true understanding of them. [8]

Dürer dedicated his treatise on measurement to Willibald Pirckheimer because Pirckheimer, and the circle of humanist scholars around him, saw Dürer as the person who could elevate German painting to the level of a truly serious intellectual endeavour; art would become the representational science of Nature. And, given the naturalistic aims of the era, this would also be an advance for art. “Depart not from Nature,” Dürer enjoins us, “for Art is rooted in Nature, and whoever can pull it out, has it.” [9]

So, what has Dürer pulled out in his studies on proportions, and how is it to be returned to representation? His meticulous, seemingly excessive, studies of the human body have a direct relation to its representation. The connection I want to establish can be made most clearly through another set of studies—stereometrical drawings—that originally had a somewhat different purpose (illustration 4). Dürer’s stereometrical drawings can be enlisted to graphically link his labours on proportions to his contributions to perspective, and provide an explanation of why the former are artistically relevant to an art underpinned by the latter.

Like Piero della Francesca and other Italian artists, Dürer made numerous studies of geometrical solids and how to represent them in correct perspective. The stereometrical man of illustration 4 is obviously an assemblage of geometrical solids, and he could easily be represented from any perspective. He does not of course much resemble a man; however, he could be worked to a closer approximation of the human form. With the appropriate measurements, the cube that serves as his head, for example, could be refined into a multi-planed solid with the basic configuration of a human head (illustration 5). There is no reason in principle why this process could not be continued, and the head refined to an ever-increasing number of planes, even to planes no bigger than one Trümlein, at which point the geometrical construction would be indistinguishable from the organic object. The cubic, grid-like structure that typically surrounds the head of the figure in one of Dürer’s complete proportion studies (illustration 1) would facilitate such a process of refinement, in that it would offer the framework for both the proportional measurements of a head and its formation in perspective. Evidently, this procedure could be extended to any other part of the body and to the body as a whole (illustration 6). Thus, one would have complete control of the object of representation through the means of representation, which, in the opinion of Dürer and his followers, would be a considerable advancement in the “science of painting.”

One would need the diligence and infinite patience of a Dürer to realise a painting in this way, but it is certainly an attainable goal, especially in our era. If many of Dürer’s images seem curiously modern that is not surprising, for the systems of perspective and proportions that are used in contemporary computer-imaging programmes are direct descendants of his techniques. Dürer did not of course have it in mind to create a kind of manual computer-imaging programme. There is no doubt though that he meant for his various studies to be united in the service of naturalistic depiction, as I think is indicated by illustration 6, and testified to by the following passage: “In order that this teaching of mine might be better understood I have already published a book about Measurements, that is to say of lines, planes, bodies, and the like, without which this my theory [on human proportions] can not be properly understood. It is therefore necessary for all who would try this art that they be well instructed in Measurements and know how to draw a plan and elevation of anything.” [10] If proportions and perspective are harmonised mathematically, then the essential structures are in place for the creation of a universal system for the representation of any body, real or ideal, at ever-increasing levels of complexity.

II

Let us now turn to the other figure I want to bring into these reflections. Even though my intention is not to tell a tale of influence that links Dürer to Nicholas Cusanus, it would not be difficult to do so. Although Cusanus (1401-1464) was dead before Dürer (1471-1528) was born, it is probable that he had a place in Dürer’s intellectual environment. After all, Cusanus was one of the few Germans to achieve a high position in Italian intellectual circles. Dürer associated with the humanist scholars of Nuremberg who of course admired Italian intellectual culture and were familiar with Cusanus’s writings. It has been documented that many of the members of this circle owned the works of Cusanus, for example, Hartmann Schedel and Conrad Celtis. Furthermore, Dürer’s closest friend was Willibald Pirckheimer, the most eminent of the Nuremberg humanists, and Pirckheimer’s grandfather knew Cusanus. Everything considered, it would be something of a surprise if Dürer knew nothing of Cusanus’s doctrines. But this not a question I want to pursue. [11] As I stated at the outset, my interest is only in revealing a structural affinity between Dürer’s approach to visual truth and Cusanus’s analysis of epistemological truth.

Some see Cusanus as the last medieval philosopher, some as the first philosopher of the Renaissance, and others as a Janus faced figure posed on the threshold between the two eras. The last description fits my understanding of him. Although Cusanus was an inventive and powerful renaissance thinker, his approach to philosophy was profoundly influenced by his scholastic heritage. Nevertheless, from within the framework of scholasticism Cusanus was able to introduce questions that pointed beyond it. This was affected, principally, through a rethinking of scholasticism’s debt to Platonism, a rethinking that was felt throughout Cusanus’s wide-ranging interests.

The central question for Cusanus was the relation between the sensible and the intelligible, between what Plato called the world of appearances and the realm of the forms. This was a question that he posed in relation to both empirical knowledge and theology. Many medieval thinkers had been eager to try to bridge the gap between the sensible and the intelligible, for human beings needed assurance that their knowledge of the imperfect particulars given to them in experience held something of the truth found in the perfect, intelligible forms that these particulars were mere instances of. Similarly, as imperfect creatures in God’s world, human beings needed to believe that there was some means by which they could immediately come to know their creator. The fundamental insight of Cusanus’s philosophy was that the gap between the sensible and the intelligible, between appearances and forms, could not be bridged. This was the founding idea of his work De docta ignorantia (On Learned Ignorance). [12]

Consider first what this implies for theology. God as the infinite Maximum cannot be comprehended through the means by which we comprehend the world of experience. For the Maximum is not the superlative in a graduated chain of comparisons that has led up to it: it is not like the largest mammal or tallest mountain. We cannot know God through the natural progression of knowledge in this world. In fact, the Maximum is the antithesis to every possible comparison. The difference is qualitative not quantitative: the most intelligent person in the world is still an infinite distance from God’s omniscience, and any increase in intelligence will not make that distance less than infinite. Something similar holds for our claims to empirical knowledge. Any actual existent known to us is inferior to the ideal, and this inferiority is a necessary inferiority. All real circles exist in the realm of “more or less,” whereas the ideal circle, what Plato would call the intelligible form of circularity, is by definition independent of questions of “more or less.” There is an unbridgeable gap between the real and the ideal.

This does not mean though that the human effort to understand the world and God must end in failure. There is a positive doctrine that comes from accepting this irremovable barrier: De docta ignorantia. It is because of its separation from the intelligible that the sensible gains its significance. No empirical knowledge is possible that is not related to the ideal. The character of the ideal is its delimitation, its determinateness. The character of empirical knowledge is its determinability, its capacity to refine itself as an ever-closer approximation of the ideal. The ideal, then, to use a mathematical concept, is the limit to which empirical knowledge strives, but can never reach. In Kantian terms, the ideal is a condition of possibility for our knowledge of our world.

How does all this relate to Dürer? Because humankind is God’s creation, and because we gain our knowledge of the ideal through sensibility, the world of sense in Cusanus’s philosophy receives a promotion: it is no longer the source of base knowledge and deception from which we must escape. Rather, it is the foundation of knowledge and the impetus for the movement of thought towards the ideal. Correspondingly, those who explore and articulate the world of sense also receive a promotion. In his set of dialogues Idiota, Cusanus has a layman instruct an orator and a philosopher about the nature of knowledge and wisdom. Knowledge, he tells them, is not to be found in the writings of others and the authority of the ancients. It is to be found in experience, in the everyday activities of everyday life: in the weighing, measuring and counting found in the market place, for example. Knowledge for Cusanus presupposes comparison, which is, more precisely, measurement. Measurement itself presupposes a common unit and a homogeneous quantitative order.

Merchants may know how to operate in the world of experience better than bookish scholars, but they are rarely inclined to explore it and articulate the means through which it is known. If we want to make the principles of our knowledge explicit, and if we want to draw out the ideal from what is given to us in experience, we will need more astute and careful observers. We will need scientists and artists, or, in an era in which they are not clearly distinguished, we will need artist/scientists such as Leonardo da Vinci and Dürer.

Dürer’s stereometrical studies of the human figure are the perfect counterpart to Cusanus’s doctrines. It is striking how they are almost literal instantiations of the mathematical images that Cusanus relies upon to explicate his ideas. In De docta ignorantia, when trying to explain the nature of the unbridgeable gap between the real and the ideal, Cusanus asks the reader to think of the relation between a polygon and a circle. We can increase indefinitely the number of sides that a polygon has but it will never become a circle, even when it has a million sides, or a billion, and is visually indistinguishable from a circle. Dürer’s stereometrical drawings, we will remember, can be understood in relation to just such an idea of continual geometrical refinement. Student of the sensible world and its quantificational determination, Dürer pursued the ideal in the sensible with ever-increasing precision, up to the limits of his system of measurement, which were also the limits of representation, and, in Cusanus’s philosophy, the limits of human knowledge. It would be difficult to think of a more fitting way to exemplify Cusanus’s doctrines on the nature of knowledge and how it is achieved.

It might be remarked though that the fit is not what it appears to be: whereas Cusanus was concerned with the process by which our knowledge moves towards an ideal (an abstract general concept) Dürer was concerned with how to better represent individual figures (instances of the abstract general concept ‘human being’). Such a remark captures only part of the truth for it overlooks the more universal aspect of Dürer’s investigations. Dürer did indeed want artists to be able to better represent individual figures, that was his most concrete artistic aim. But the more general goal of his studies was to find the proportions appropriate to the human form. Dürer famously gave up the normative ideal of one, true set of proportions, and turned instead to studying the proportions found in a variety of human types. The fact that he moved from a normative quest to a descriptive project does not entail though that he decided to confine himself to the realm of particulars. Dürer measured hundreds of individuals so that he could deduce from his measurements the proportions of different general types. What he could not do was find a descriptive model that would allow these to be resolved in turn at another, more general, conceptual level. He was nevertheless continually adding sides to Cusanus’s polygon, while providing further mathematical information for the depiction of individuals.

A question that remains to be addressed is how deep does this structural affinity between Dürer’s art theory and Cusanus’s epistemology run? Are the statements we find in Dürer’s writings that touch on questions of knowledge compatible with Cusanus’s doctrines? Given that Dürer was not a philosopher and his writings were typically aimed at artists, I do not think that this matter can be satisfactorily resolved. Dürer’s pronouncements are often too vague to betray an allegiance to a definite philosophical position. That said relevant passages do seem to run in the same direction as Cusanus’s doctrines. Consider this well-known one from the fourth book on human proportion: “The Creator fashioned men once for all as they must be, and I hold that the perfection of form and beauty is contained in the sum of all men.” [13] He then goes on to add: “We are considering about the most beautiful human figure conceivable, but the Maker of the world knows how that should be. Even if we succeed well we do but approach towards it somewhat from afar.” [14] All of this seems very much in line with Cusanus’s Christian Platonism. The ideal human form, of necessity, exists in the mind of God. We can only approach an understanding of that ideal through studying the totality of real human beings and their imperfect ways of participating in it. This of course will not give us God’s knowledge, but it will give us an understanding that can be a more or less successful approximation of the ideal.

 

 

Illustration 1. Albrecht Dürer, strong handsome man of eight head lengths, 1507.

 

Illustration 2. Leonardo da Vinci, study of human proportions according to Vitruvius, c.1485-1490.

 

Illustration 3. Albrecht Dürer, constructed foot, 1513.

 

Illustration 4. Albrecht Dürer, stereometic man, 1523.

 

Illustration 5. Albrecht Dürer, head as an assembly of geometrical planes, around 1523.

 

Illustration 6. Albrecht Dürer, projection of a section of a chest, 1528.

 


[1] Martin Kemp, Visualizations (Oxford: Oxford University Press, 2000).

[2] Albrecht Dürer, Four Books of Human Proportion (New York: Dover, 1972). We know that Dürer finished the manuscript in 1523. It appears that he delayed publication because he acquired, on the recommendation of his friend Willibald Pirckheimer , the noted humanist scholar, ten books considered to be of interest to painters from the library of Bernhard Walther, the Nuremberg astronomer and mathematician. Dürer died April 6, 1528. The manuscript was posthumously published in 1532.

[3] Federico Zuccaro, “L’Idea de’ pittori, scultori e architetti” in D. Heikamp, Scritti d’arte di Federico Zuccaro (Florence, 1967) p. 250.

[4] Erwin Panofsky, “The History of the Theory of Proportions as a Reflection of the History of Styles” in Meaning and the Visual Arts (Chicago: University of Chicago Press, 1955): pp. 103-104.

[5] Dürer, Four Books of Human Proportion, Second Book, first pages of manuscript.

[6] For a recent treatment of the subject see Alfred W. Crosby, The Measure of Reality: Quantification and Western Society, 1250-1600 (Cambridge: Cambridge University Press, 1999).

[7] Galileo Galilei, Discoveries and Opinions of Galileo, trans. Stillman Drake (Garden City, NY: Doubleday, 1957), pp. 237-238.

[8] I have chosen to use the translation of the dedication from Of the Just Shaping of Letters, (New York: Dover, 1965), pp. 1-2. The Dover edition is a reprint of a 1917 Grolier edition in which the translator is not recognized.  Originally Of the Just Shaping of Letters was part of the Treatise on Measurement, which was published in 1525. The Treatise on Measurement has been translated into English under the title The Painter’s Manual by Walter L. Strauss (New York: Abaris Books, 1977).

[9] Quoted in Jane Campbell Hutchison, Albrecht Dürer: A Biography (Princeton: Princeton University Press, 1990), p. 69.

[10] Albrecht Dürer, The Writings of Albrecht Dürer, ed. and trans. William Martin Conway (New York: The Philosophical Library, 1958), p. 231.

[11] For a brief discussion of this topic and further references see Joseph Leo Koerner, The Moment of Self-Portraiture in German Renaissance Art (Chicago: University of Chicago Press, 1996), pp. 129-130. See as well Campbell Hutchison, pp. 60-61.

[12] Nicholas of Cusa, Nicholas of Cusa: Selected Spiritual Writings, trans. and  intro. H. Lawrence Bond (New York: Paulist Press, 1997), pp. 85-206.

[13] Albrecht Dürer, The Writings of Albrecht Dürer, p.250.

[14] Ibid., p.251.


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