MELANCHOLIA & MATHEMATICS

In the early 19th century, Wackenroder and the Nazarenes re-appropriated Albrecht Dürer by characterizing him as a nationalist figure and a “painter of Christian subjects". Such prescriptive views of Dürer will be challenged here. In doing so Dürer’s understanding and exploration of mathematics will be utilised to establish that Dürer’s primary focus was the achievement of a greater understanding of its application to beauty. Yet, this focus will be explored to uncover Dürer’s awareness of Platonic transcendence and its potential for a trans-historical beauty. With this focus on mathematics, Dürer’s own conception of the role of an artist and the role of a mathematician will show how these are united by the same goal, the goal of attaining beauty. It will be discussed how particular elements in Dürer’s renowned engraving ‘Melencolia I’ [Figure 1], can be used to suggest that Dürer’s knowledge of mathematical beauty not only plays into the general themes of the engraving but illustrates Dürer’s capacity to distinguish between different hierarchies of beauty, as well as demonstrating Dürer’s aim in pursuing such a quest. Such distinguished thought will then be further discussed to address Dürer’s philosophical involvement with classical metaphysics by looking at Dürer’s writing on Plato; I will draw conclusions that Dürer was cognisant of the potential transcendental quality of art and sought to achieve in his work. Finally, it will be suggested it is possible to unite the two apparently polarised positions on Dürer’s self-consciousness as a artist through an understanding of how Dürer placed Platonic transcendental philosophy in relation to his religious beliefs and how this impacted upon the legacy that Dürer sought to establish. THE NAZARENE’S DÜRER Wackenroder was a writer acclaimed with being the co-founder of the German Romantic movement. He sought to glorify the age where “daily life was supposedly based on art, craft and religion”, the very elements that appear to oppose the Renaissance fervour for science, learning and logic in the elevation of the status of the ‘artist’ from craftsman. Wackenroder’s writings had two central aims, to emphasise the religiosity of Dürer and his work and consequently, the connection between art and religion in general and to posit Dürer as a figurehead for nationalistic German art. As Sanford indicates, in the Dürer chapters, “Wackenroder upheld the superiority of medieval art and the Catholic Church and precipitated the militantly nationalistic spirit of the late nineteenth century”.  Wackenroder formed the basis for the Nazarenes, a group of artists founded by Overbeck and Pforr, calling themselves the ‘Brotherhood of St. Luke’ who put into practice Wackenroder’s aesthetics. The Nazarenes elevated Dürer’s character to an “almost mythical status” and assigned a quasi-religious status to his character. Overbeck himself describes the ceremonial honoring of Dürer at an altar-like table with collections of his best work in a letter to Joseph Sutton. Furthermore, in “Dürer and Raphael before the Throne of Art” [Figure 2], Pforr displayed a symbolic unification of Germany and Italy in the Renaissance and the veneration and connection with the Catholic Church. It was through this re-characterisation that “Dürer was seen no longer as a representative of the secular Renaissance but as an artist with a Christian approach to art and life”. Placing these works and writings in historical context, we can see that the German Romantics re-appropriated Dürer for their own purposes. The preceding era of the enlightenment had portrayed Dürer as a talented painter but one that had “squandered his genius by remaining in Germany amidst the superstitious masses of rough-hewn peasants”. This comment is reminiscent of Vasari who spoke of Dürer: “If this man, so able, so diligent and so very versatile, had had Tuscany instead of Flanders for his country, and had been able to study the treasures of Rome, as we ourselves have done, he would have been the best painter in our land” Wackenroder, sought to recalibrate thinking about Dürer in relation to Vasari’s regret that Dürer was not an Italian: “I find nothing to regret, but I am glad that the fate of the German soil has indulged in this man-a really patriotic painter.” Sanford astutely assessed the way in which Wackenroder redressed the specific criticisms leveled at Dürer in the 18th Century: “[Wackenroder] offered a positive evaluation of precisely the same aspects in which superstition became piety; provincialism was transformed into laudable national pride” Wackenroder looked at the art of his time disparagingly, claiming that his contemporary art was a “frivolous game of the senses” and placed Dürer as an idol for German art to inspire young artists of the time to alter their aesthetic. Thus, this characterisation is a historical re-appropriation of Dürer utilised in such a way to redress an imbalanced view of Dürer’s work and to reassert a “haughty moral posture” and the connection between art and religion. Such a characterisation of Dürer is a limited view of Dürer’s work, for it is undeniable that Dürer was a religious man who did work from biblical narratives but this idealization of his nature is simplified. As it will now be established, Dürer did not found his work on lowly craftsmanship but sought to elevate his skill and art through a profound understanding and practice of geometry, mathematics and proportion.

DÜRER AND MATHEMATICS In order to deconstruct these romanticised notions of Dürer’s role as an artist, it is important to consider the role of mathematics and geometry in the formation of Dürer’s images. Dürer felt that the lack of proportions was the resounding problem with the contemporary German artists. As he wrote in his ‘Treaty on Human Proportions’: “It is evident that, though the German painters are not a little skillful with the hand and in the use of colours, they have as yet been wanting in the arts of measurement, perspective and other like matters…without proportion no figure can ever be perfect.” Indeed there has been a long tradition of the link between mathematics and beauty even Aristotle in ‘The Metaphysics’ recognised “the chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree”. Dürer’s mathematical enquiry began with his journey to Italy in 1500 where he was to discover the work of Jacopo de’ Barbari which altered his perspective on art for the rest of his life’s work. Dürer was altogether taken with the excellence in proportion that Barbari demonstrated. Dürer himself writes: “Jacopo (de’ Barbari), a charming painter, he showed me the figures of a man and woman, which he had drawn according to a canon of proportions; and now I would rather be shown what he meant (i.e. upon what principles the proportions were constructed) than behold a new kingdom…so I set myself to discover how such a canon might be wrought out.” This declaration of intention was then the focus of Dürer mathematical study, centreing his research on reading Vitruvius’ ‘De Architectura’ and Euclid’s ‘Elements’, Dürer meticulously calculated the correct proportions of the human form [Figure 3]. Barbari’s unwillingness to share in his mathematical secret to beauty spurred Dürer on and he “devoted the remaining three decades of his life to the search”. ‘Melencolia I’ has been subject to much discussion as Dodgson states: “The literature on Melancholia is more extensive than that on any other engraving by Dürer: that statement would probably remain true if the last two words were omitted” Consequently, to exemplify Dürer’s approach to mathematical beauty, only the magic square will be discussed here. A magic square is one where all rows, both vertically and horizontally, sum to the same number, in the case of Dürer’s magic square [Figure 4] (hereafter DA) this number is 34. DA is not however, just an ‘ordinary’ magic square but one that possesses ‘extra-magic qualities’. Dürer has also inscribed the year of the engraving 1514 in the bottom row of the magic square. All magic squares are part of a family of reflection/rotation symmetries; DA’s family of magic squares is as follows:

There are two hierarchical assessments of ‘magic-ness’ for magic squares, The following table provides statistical analysis of those magic squares containing these properties:

In the early 19th century, Wackenroder and the Nazarenes re-appropriated Albrecht Dürer by characterizing him as a nationalist figure and a “painter of Christian subjects". Such proscriptive views of Dürer will be challenged here. In doing so Dürer’s understanding and exploration of mathematics will be utilised to establish that Dürer’s primary focus was the achievement of a greater understanding of its application to beauty. Yet, this focus will be explored to uncover Dürer’s awareness of Platonic transcendence and its potential for a trans-historical beauty. With this focus on mathematics, Dürer’s own conception of the role of an artist and the role of a mathematician will show how these are united by the same goal, the goal of attaining beauty. It will be discussed how particular elements in Dürer’s renowned engraving ‘Melencolia I’ [Figure 1], can be used to suggest that Dürer’s knowledge of mathematical beauty not only plays into the general themes of the engraving but illustrates Dürer’s capacity to distinguish between different hierarchies of beauty, as well as demonstrating Dürer’s aim in pursuing such a quest. Such distinguished thought will then be further discussed to address Dürer’s philosophical involvement with classical metaphysics by looking at Dürer’s writing on Plato; I will draw conclusions that Dürer was cognisant of the potential transcendental quality of art and sought to achieve in his work. Finally, it will be suggested it is possible to unite the two apparently polarised positions on Dürer’s self-consciousness as a artist through an understanding of how Dürer placed Platonic transcendental philosophy in relation to his religious beliefs and how this impacted upon the legacy that Dürer sought to establish.

THE ‘FINAL FORMS’ AND TRANSCENDENCE The reference to ‘final forms’ connects Dürer with Plato, to whom he refers directly: “For a good painter is full of inward figures and if it were possible for him to live forever he would always have to pour forth something new from the inner ideas of which Plato writes” These Forms come from the story of Plato’s cave in which expresses that the ever-changing aesthetic and sensation-based world that surrounds us is subservient to the transcendental truth of the Forms which are non-material, abstract ideas, beyond historical condition or temporality. Dürer indicates above that these Forms are within the artist. Kuspit assesses Dürer’s nature as one which is a “Procrustes bed of forms”, a standard that is enforced uniformly without regard to individuality. His investigations into mathematics were all to work out “a single idea” . Plato agreed in this respect, knowledge must be geared towards the essence of objectivity, which can be uncovered by searching for commonality between objects. For both Plato and Dürer this commonality was to be found in mathematics, namely proportion. “The fairest bond is that which makes the most complete fusion of itself and the things which it combines; and proportion is best adapted to effect such a union…the body of the world was created, and it was harmonized by proportion”  Transcendental knowledge has both a religious and philosophical meaning. Kant spoke at length about transcendental knowledge as being “knowledge which is not so much occupied with objects as with the mode of our cognition of these objects, so far as this mode of cognition is possible a priori”. By such definition, knowing the Forms would constitute a type of transcendental knowledge as it would gain access to a world beyond that of the aesthetic. It is thus suggested here that Dürer’s mathematical quest for beauty may serve as the evidence for the link between his work and the goals of accessing transcendental knowledge. Art has long been considered prime means of accessing truth. Having denounced the very idea of morality, Nietzsche provided a counter-doctrine that “art, and not morality, is…the truly metaphysical activity of man…the existence of the world is justified only as an aesthetic phenomenon”. Going so far as to claim that the “world is a work of art that gives birth to itself”, in Nietzsche’s system all nihilism is sublimated to art.  Burckhardt makes reference to the trans-historicity of the artwork and how it links to the platonic forms. The artwork becomes a “means of communicating knowledge about the truth of the whole.”. Burckhardt then discusses the relationship between logic and reason (easily substituted with mathematics) and transcendence: “Action of reason…makes possible the transcendence of man’s historical self because all acts of knowledge and of spiritual insight attain an objective stand beyond the flux of time” The combination of such knowledge through reason and spiritual insight describes the very approach that Dürer took. Such knowledge, according to Burckhardt achieves objectivity, something that is beyond temporality. Dürer seeks to perfect human proportion in his ‘Theory of Human Proportions’ he describes that “the Creator fashioned men once and for all as they must be, and I hold that the perfection of form and beauty is contained in the sum of all men”. This perfection rests on conquering the form of an “earthly thing”, an action, Burckhardt describes, as “the deepest ground of all religion and knowledge”. Barbari’s secret evidences Dürer’s belief in the idea of obtainable truth, the idea that earthly things can be conquered through the ascertaining of their perfect depiction in art. Kuspit remarks: “[Dürer’s art] is constituted by precision in the depiction of nature, symptomatic of a desire for objectivity; on the other hand it is characterized by intense religiosity” It is in this respect that we must seek to unite the truth that lay at the base of the Nazarene characterisation and that which has been established here – Dürer’s conception of the role of the artist as one to uncover the mathematical science of proportion to achieve the perfection of ‘earthly things’ and bring the artist closer to the divine. DURER AND LEGACY Before these positions are united, it is necessary to consider how Dürer correlates with Barbari and how this establishes his conception of the artist as a legacy. Barbari’s refusal to divulge his learning, seemed to vex Dürer whose attitude to learning appeared to one that recognised a collaboratively trans-historical progress towards artistic perfection as he states “If I had [Barbari’s canon], I would put it into print in his honour, for the use of all men”. Dürer saw his role as an artist as pedagogical, one that had a role in passing on a legacy to new artists. Dürer writes: “Some hide their art in great secrecy…I would gladly give everything I know to the light for the good of cunning students who prize art more highly than silver or gold. I further admonish all those who have knowledge in these matters that they write it down…to the great honour of God and your own praise” Dürer saw the quest for mathematical beauty as a common endeavour, relating the achievement of this knowledge to his praise of God, evidencing his belief that perfection in art was affiliated with religious truth. Something that one may classify as a quest for transcendental knowledge. Conway, assessed Dürer’s character in relation to Barbari as “free from all meanness” and that his efforts in divulging information to the masses was “not because of the advantage it gave him over his contemporaries but because it enabled him to be of use to those with whom his sympathies were most closely bound up” . Burckhardt reiterates collaborative, generational achievement through art: “Works of Art are interpreted as the ideal expression of a historical situation, as the unending effort of the human generations in the sequence of time to interpret their being in time and space in symbols and images” UNITING THE POSITIONS Returning to ‘Melencolia I’, we can see the metaphor for a unification of the positions taken by the Nazarenes and Dürer’s mathematical project. The area concealed above the frame of the engraving represents the ‘super-celestial’ world, as something beyond our access. Though, if we look to the ladder in the right of the piece, we see that this is cut-off from the ‘super-celestial’ world above. Pingree discuss this ladder in relation to the tower on which it rests: “[The tower] is, as Heckscher shows, a House of Wisdom; more specifically, it symbolizes the intellectual mode of ascent to the supercelestial, as the ladder refers to ascent by faith.” Kuspit sees Dürer’s mathematical and religious aims as “conflicting”, but it is shown that they need not be. For Kuspit, Dürer is not a man of modern science because of his piety but that he sits within the classical tradition of science one that “had not abandoned absolute truths…a science in search of generality”. As discussed earlier, this generality was one believed to be accessed by Plato through proportion. Therefore, using our discussion of the ‘ladder of faith’ resting against the ‘tower of intellect’, we may propose that Dürer saw these two fields as unified, that the intellectual pursuit of maths and geometry, brings one closer to the divine. It is thus asserted here that Dürer’s own self consciousness as an artist was concerned with temporality. In recognising the legacy of an artist, Dürer recognised a collaborative aesthetic project, one that was directed toward beauty. It was through his studied application of mathematics that Dürer sought to uncover the Form of beauty. Reasserting the religious nature of Dürer’s work, he carried out the Platonic search for transcendental knowledge of beauty in order to get closer to divinity through mathematical perfection of ‘earthly things’. For Dürer, maths was a means of access to religion not a defiance or neglect of faith. Any espousal of Dürer’s self-consciousness, as seen from the German Romantics, of his status as an artist cannot be explored without an understanding of how Dürer’s aesthetics sought to answer questions of truth through a devout application of mathematics. The ladder and tower in ‘Melencolia I’ for Dürer were united in their transcendence to the ‘super-celestial’ but having abandoned the tools of geometry, the dejected angel is unable to uncover this secret before time runs out, but perhaps the quest to unlock beauty’s secret itself will be her, or Dürer's, legacy.

THE ‘FINAL FORMS’ AND TRANSCENDENCE The reference to ‘final forms’ connects Dürer with Plato, to whom he refers directly: “For a good painter is full of inward figures and if it were possible for him to live forever he would always have to pour forth something new from the inner ideas of which Plato writes” These Forms come from the story of Plato’s cave in which expresses that the ever-changing aesthetic and sensation-based world that surrounds us is subservient to the transcendental truth of the Forms which are non-material, abstract ideas, beyond historical condition or temporality. Dürer indicates above that these Forms are within the artist. Kuspit assesses Dürer’s nature as one which is a “Procrustes bed of forms”, a standard that is enforced uniformly without regard to individuality. His investigations into mathematics were all to work out “a single idea” . Plato agreed in this respect, knowledge must be geared towards the essence of objectivity, which can be uncovered by searching for commonality between objects. For both Plato and Dürer this commonality was to be found in mathematics, namely proportion. “The fairest bond is that which makes the most complete fusion of itself and the things which it combines; and proportion is best adapted to effect such a union…the body of the world was created, and it was harmonized by proportion”  Transcendental knowledge has both a religious and philosophical meaning. Kant spoke at length about transcendental knowledge as being “knowledge which is not so much occupied with objects as with the mode of our cognition of these objects, so far as this mode of cognition is possible a priori”. By such definition, knowing the Forms would constitute a type of transcendental knowledge as it would gain access to a world beyond that of the aesthetic. It is thus suggested here that Dürer’s mathematical quest for beauty may serve as the evidence for the link between his work and the goals of accessing transcendental knowledge. Art has long been considered prime means of accessing truth. Having denounced the very idea of morality, Nietzsche provided a counter-doctrine that “art, and not morality, is…the truly metaphysical activity of man…the existence of the world is justified only as an aesthetic phenomenon”. Going so far as to claim that the “world is a work of art that gives birth to itself”, in Nietzsche’s system all nihilism is sublimated to art.  Burckhardt makes reference to the trans-historicity of the artwork and how it links to the platonic forms. The artwork becomes a “means of communicating knowledge about the truth of the whole.”. Burckhardt then discusses the relationship between logic and reason (easily substituted with mathematics) and transcendence: “Action of reason…makes possible the transcendence of man’s historical self because all acts of knowledge and of spiritual insight attain an objective stand beyond the flux of time” The combination of such knowledge through reason and spiritual insight describes the very approach that Dürer took. Such knowledge, according to Burckhardt achieves objectivity, something that is beyond temporality. Dürer seeks to perfect human proportion in his ‘Theory of Human Proportions’ he describes that “the Creator fashioned men once and for all as they must be, and I hold that the perfection of form and beauty is contained in the sum of all men”. This perfection rests on conquering the form of an “earthly thing”, an action, Burckhardt describes, as “the deepest ground of all religion and knowledge”. Barbari’s secret evidences Dürer’s belief in the idea of obtainable truth, the idea that earthly things can be conquered through the ascertaining of their perfect depiction in art. Kuspit remarks: “[Dürer’s art] is constituted by precision in the depiction of nature, symptomatic of a desire for objectivity; on the other hand it is characterized by intense religiosity” It is in this respect that we must seek to unite the truth that lay at the base of the Nazarene characterisation and that which has been established here – Dürer’s conception of the role of the artist as one to uncover the mathematical science of proportion to achieve the perfection of ‘earthly things’ and bring the artist closer to the divine. DURER AND LEGACY Before these positions are united, it is necessary to consider how Dürer correlates with Barbari and how this establishes his conception of the artist as a legacy. Barbari’s refusal to divulge his learning, seemed to vex Dürer whose attitude to learning appeared to one that recognised a collaboratively trans-historical progress towards artistic perfection as he states “If I had [Barbari’s canon], I would put it into print in his honour, for the use of all men”. Dürer saw his role as an artist as pedagogical, one that had a role in passing on a legacy to new artists. Dürer writes: “Some hide their art in great secrecy…I would gladly give everything I know to the light for the good of cunning students who prize art more highly than silver or gold. I further admonish all those who have knowledge in these matters that they write it down…to the great honour of God and your own praise” Dürer saw the quest for mathematical beauty as a common endeavour, relating the achievement of this knowledge to his praise of God, evidencing his belief that perfection in art was affiliated with religious truth. Something that one may classify as a quest for transcendental knowledge. Conway, assessed Dürer’s character in relation to Barbari as “free from all meanness” and that his efforts in divulging information to the masses was “not because of the advantage it gave him over his contemporaries but because it enabled him to be of use to those with whom his sympathies were most closely bound up”. Burckhardt reiterates collaborative, generational achievement through art: “Works of Art are interpreted as the ideal expression of a historical situation, as the unending effort of the human generations in the sequence of time to interpret their being in time and space in symbols and images”

UNITING THE POSITIONS Returning to ‘Melencolia I’, we can see the metaphor for a unification of the positions taken by the Nazarenes and Dürer’s mathematical project. The area concealed above the frame of the engraving represents the ‘super-celestial’ world, as something beyond our access. Though, if we look to the ladder in the right of the piece, we see that this is cut-off from the ‘super-celestial’ world above. Pingree discuss this ladder in relation to the tower on which it rests: “[The tower] is, as Heckscher shows, a House of Wisdom; more specifically, it symbolizes the intellectual mode of ascent to the supercelestial, as the ladder refers to ascent by faith.” Kuspit sees Dürer’s mathematical and religious aims as “conflicting”, but it is shown that they need not be. For Kuspit, Dürer is not a man of modern science because of his piety but that he sits within the classical tradition of science one that “had not abandoned absolute truths…a science in search of generality”. As discussed earlier, this generality was one believed to be accessed by Plato through proportion. Therefore, using our discussion of the ‘ladder of faith’ resting against the ‘tower of intellect’, we may propose that Dürer saw these two fields as unified, that the intellectual pursuit of maths and geometry, brings one closer to the divine. It is thus asserted here that Dürer’s own self consciousness as an artist was concerned with temporality. In recognising the legacy of an artist, Dürer recognised a collaborative aesthetic project, one that was directed toward beauty. It was through his studied application of mathematics that Dürer sought to uncover the Form of beauty. Reasserting the religious nature of Dürer’s work, he carried out the Platonic search for transcendental knowledge of beauty in order to get closer to divinity through mathematical perfection of ‘earthly things’. For Dürer, maths was a means of access to religion not a defiance or neglect of faith. Any espousal of Dürer’s self-consciousness, as seen from the German Romantics, of his status as an artist cannot be explored without an understanding of how Dürer’s aesthetics sought to answer questions of truth through a devout application of mathematics. The ladder and tower in ‘Melencolia I’ for Dürer were united in their transcendence to the ‘super-celestial’ but having abandoned the tools of geometry, the dejected angel is unable to uncover this secret before time runs out, but perhaps the quest to unlock beauty’s secret itself will be her, or Dürer's, legacy.