MELENCOLIA I:The postmodern art of Albrecht Durer

David.Finkelstein@Physics.GaTech.edu

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Abstract

Besides its conspicuous early Renaissance scientific instruments and tools, Durer’s engraving ME-LENCOLIA I also contains numerous apparently unreported concealments whose decryption revealsambiguously heretic messages about physics and religion. The motto that has named the piece isan anagram. Decoded, it refers to Natural Philosophy as opposed to Mathematical Philosophyand Theological Philosophy as a way to knowledge. Numerous secondary images demonstrate therelativity and ambiguity of perception and indicate that Durer expressed unitarian ideas.

1 Disclaimer I

Views expressed by this engraving are not necessarily mine.

2 Background

Many meaningful details of the MELENCOLIA I (1514) of Albrecht Durer (1471-1528) seem tohave been overlooked in recent centuries [6]. I report some of them here. My interpretations of someled to the discovery of others, giving me enough confidence to present my tentative interpretationof the lot. Mistakes in my interpretation of such a rich work undoubtedly exist, and certainly thereare gaps in the story, but this need not reflect on the observations themselves, which I bullet forany other interpretation to take into account. I have made an effort not to project my own viewsacross half a millennium onto Durer. The point is not what I see in the engraving but what Durercould have expected his viewers to see. The philosophical views I read in the engraving and presentbelow are not necessarily my own. They are expressed in writing by Durer or current in his timeand place.

Shannon pointed out that improbable messages carry more information than probable ones.The unusual elements of this engraving therefore most likely carry the key to its meaning. Unfor-tunately Shannon’s principle is less helpful in this picture than most because we encounter so muchthat is unusual in it.

Anyone can see the impressive angel sitting in the right foreground and the small wingedserpent flying in the left background, on first sight the personifications of virtue and evil. A putto,a winged boy, a familiar convention of Renaissance art, perches between them on an up-turnedmillstone and intently inscribes a block. Paraphernalia of the arts and sciences clutter the scene ina significant pattern.

Schuster provides a grand overview of how MELENCOLIA I has been interpreted in the lasthalf millennium [22]. The art historian Erwin Panofsky [19] saw the engraving as intensely autobio-graphical and read its melancholy as Durer’s own frustration at the gap between artistic and divinecreation. Frances Yates, historian of the Hermetic tradition, took its melancholia to be one thatinspired creativity, not mere depression, and read the engraving as a declaration of the harmonybetween microcosm and macrocosm: as above, so below [25]. The art historian Patrick Doorly seesit as an illustration for Plato’s Greater Hippias, a dialogue on beauty [4], with the angel melan-choly because the artist cannot define absolute beauty. The museum-director Peter-Klaus Schustersees it as Durer’s virtues-book. These judgments seem based on incomplete evidence. Amazingly,none of the interpretations compiled by Schuster touch on a score of individual concealments andintentional ambiguities in the engraving. There is a key anagram, rebuses, Cabalistic gematria,puns, a Star of David, a dozen hidden faces including two Jews and an Arab warrior, and eventhe downfall of Babylon. There are plausible references, mostly unnoticed, to Genesis, Revelation,

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the Nicean Creed, and the Durer family history. There are numerous missed demonstrations of thelimits to human perception and comprehension. There are also hidden capital crimes that explaintheir own concealment.

Many of my interpretations fit Durer’s known views on art, science, philosophy, and religionsupport but some go further and add new insights on Durer if they are correct. The interpretationsI have cited do not mention the main point of the engraving, hidden in the title and depicted inthe work.

A main element of this work is relativity in the broad sense, the study of how what we seedepends on our viewpoint. Durer uses perspective to construct several examples of relativity andambiguity. It is not in the least anachronistic to attribute relativistic ideas to Durer, since he isone of the creators of descriptive geometry. This is precisely a theory of how the same objectlooks from different points of view, and is probably the first mathematical theory of relativity inthe broad sense. He is a significant figure in the history of the perspective technique, which theRomans and Greeks practiced and the Church suppressed as deceptive. Durer carried “the secretof perspective,” as he called it, from Italy to northern Europe, and some say that his book on thesubject founded the mathematical theory of descriptive or projective geometry, though clearly thisdeveloped over centuries with growing rigor.

Perspective, special relativity, and general relativity all rest on an understanding of visual per-ception and light, and so there are family resemblances among them. The apparent foreshorteningof limbs in perspective and the apparent shortening of rulers and slowing of clocks in relative motionor in gravitational valleys are all effects of relativity in the broad sense. Where relativists todayspeak of light-cones, Leonardo and Durer spoke of “pyramids of vision,” which are light-cones withtime omitted, being made of paths of light in space rather than in space-time. They held that inperception one such pyramid diverges from the point observed, like a future light cone, and anotherconverges to the the observing eye, like a past light cone. The perspective transformations of thegraphic plane that we carry out when we shift our viewpoint are special cases of the transformationsof space-time used later in modern relativity. Both are represented by matrices or tables of tworows and columns, with real or complex number entries respectively. Projective geometry madeDurer an eminent mathematician of his time and is a playground for relativists today.

That Durer’s art can be programmatic is indisputable, for he literally encoded messages intoanother work of this period, leaving us message, code-book, and work. I needed his code-book tointerpret MELENCOLIA I. In 1514 — the year inscribed on this engraving — he had just finishedhis Triumphal Arch of Maximilian I , an elaborate eulogy of Maximilian assembled in several stages:

1. In the 2nd-4th century CE, a certain Horapollo Niliaci wrote Hieroglyphica, a wild andspurious rendering of Egyptian hieroglyphics into Greek, centuries before the Rosetta Stone madea proper reading possible.

2. In 1512/1513 Willibald Pirckheimer (1470-1530), Durer’s learned best friend, Humanist,state councillor, and translator of Greek and Hebrew classics, translated the fictional Hieroglyphicafrom Greek into Latin.

3. Durer illustrated Pirckheimer’s Hieroglyphica.4. Pirckheimer wrote out a eulogy of Maximilian I in plain Latin, still extant.5. With the Hieroglyphica as his code-book Durer encoded Pirckheimer’s eulogy word by word

into an allegorical fantasy of dogs, goats, harpies, cocks, snakes, scepters and the like that becamethe centerpiece of the Triumphal Arch [19].

The code-book and the clear message were also published, so the Triumphal Arch was anattempt to communicate, not conceal. Neither a plain text nor a code-book for MELENCOLIA Ihas been passed down to us, however, and we must resort to a kind of cryptography to deduce them.One might think that Durer had something to hide. The message I find supports this suspicion.

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MELENCOLIA I conceals more deeply than the Triumphal Arch. Some of its elements havebeen interpreted as slips of the artist and so seen but not read. Some viewers have suggested thatDurer did not know how to spell, since he misspelled the motto MELENCOLIA I; or how to drawa cube in perspective, since the polyhedron is not quite a truncated cube.

There are indeed well-known limits to Durer’s knowledge. Most famously, the astronomerKepler pointed out that in Durer’s work on measurement [6] he drew the ellipse produced by cuttinga cone with a plane as egg-shaped, with its small end near the tip of the cone. I found a strikingerror in optics in this engraving. Nevertheless, to catch his meanings we must not underestimatehis power with the burin, his skill in perspective, his devotion to precise language, and his acuteobservations of nature.

We must also know the following about his background. To read it we need some familybackground.

In 1455 Gutenberg printed his first Bible, Mohammed II, founder of the Ottoman Empire andconqueror of Constantinople, invaded Hungary, and a goldsmith named Albrecht, 23, left his villageof Ajtos in Hungary to finally resettle in Nuremberg. Ajto is Hungarian for door, or Thur in theGerman of the time, so Albrecht of Ajtos became Albrecht Thurer. Had he moved to Englandhe could have become Albert Gates. He soon simplified the spelling to Durer, a homonym inone dialect. He married his master’s daughter, Barbara Holper, and she had eighteen children.Three lived, inheriting his Hungarian nationality. One, Albrecht Junior, became the famous artist-philosopher-mathematician.

We should remember that he was a humanist, knowledgable in mathematics, poetry, andantiquity as well as art, and that he argued philosophy hotly and on equal terms with a famousscholar like Pirckheimer and conversed with Erasmus. We should be aware of his impassionedcriticism of corrupt ecclesiastical practices and his support of the Reformation. I was aware ofnone of these when I began this study, but they are indispensable for reading the engraving. LikeLeonardo, Durer began his training under a goldsmith, namely his father, a Hungarian immigrantin Nuremberg. By the year on the engraving, he had made two historic visits to Venice, where hestudied perspective and human proportion and developed a passion for geometry. Eventually heextended Leonardo’s work on beauty in a relativistic direction, using projective transformations toallow for different standards of beauty.

Durer brings us back to many beginnings. He did not have to please art critics to succeedbecause they had not yet been invented. The first published art criticism in history was actuallythe praise of this very engraving by Joachim Camerarius (1500-1574), a young friend of Durer whowatched him at work, translated his book on human proportions from German into Latin, and atthe end wrote his eulogy, in which he declared that although Durer’s art was great, it was the leastof his accomplishments.

Durer was the first to publish in German a mathematical proof or a book on pure mathematics.He produced the first printed star-chart and the first printed map of the world as a sphere viewedfrom outer space. He shaped the German scientific language much as Leonardo did the Italian: bydescribing the processes of art and nature in ordinary language instead of flowery Latin, painstak-ingly picking out the prose ornaments and classical references that infested scholarly writing in histime.

He was a brilliant and recognized originator in art too. He did the first known self-portrait andthe first known specific landscape. He invented etching and some perspective drawing machines.He seems to have invented the modern artist, being the first to charge for his special talent insteadof just billing for time and materials like Raphael or any house-painter of the time. His life missionseems to have been to sanctify art above artisanry and even above philosophy; to elevate graphicart to the status of music.

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Figure 1: A., B., and C. Solid, rotated and enlarged details

The medieval quadrivium consisted of arithmetic, geometry, astronomy, and music. Numberlifted three of these out of the mundane and into the divine; perhaps geometry did not merge witharithmetic until Descartes. So Leonardo, Durer, and others sought to found art and beauty too onnumber. The concept of the Golden Rectangle, Leonardo’s famous drawing of a man inscribed ina circle, and Durer’s books on descriptive geometry and body proportions are mementos of thatsearch.

3 Hidden faces

I had known MELENCOLIA I for many years when it caught my attention again in April 2004.On the internet, an anonymous donor (a. d.) posted a copy with a thin black outline of a facedrawn on the polyhedron in the engraving sometimes called “Durer’s Solid.” I chanced on this onceand have never found it again. I went to my own copy and found at least the implication of a facethat I had overlooked for decades. If we step back several paces from a good print and look at theshading of the front face of the Solid for a minute or so, we can see a woman’s face in profile. Orperhaps we construct it. Schuster mentions this face and sees it as frightening [22]

There are actually three faces in the Solid and which you see depends on how you hold yourhead relative to the print. I will call these people A., B., and C. Solid. I have rotated and rescaledthe details from the engraving that show their three faces.

• A. Solid is a woman with her head erect.• B. Solid is a man with his head cocked 30◦ clockwise.• At the left edge of the face of B. Solid is a vee-shaped dimple in the stone that becomes the

nose of C. Solid when we stand the picture on its left edge.I found C. Solid, smaller and more caricaturish than his two companions, some time after A.

and B., but not by some careful search. I predicted his existence as described in what follows, andthen found him at once. The existence of this triplet may have been known in the time of Durer,but I do not find them mentioned by any modern writer on Durer.

We have to scan these hidden faces with some persistence before they emerge. They are subtle,ambiguous, and strain our perception. Do we see them or create them? I thought that the outlineof A. Solid that I saw on the internet was an accidenal product of scanning or digitization untilI realized that most of the left edge of her face has almost no physical correlate in the engraving.Durer compels us to see it, when we look for it, by supplying chin and forehead lines and eye andnostril shadows, but we ourselves create the rest of the face as no point-by-point scanning processcould. Part of the art behind these hidden faces is that they are ambiguous, deniable, and lie atthe edge of perception. They are joint productions of the hand of the artist and the mind of theviewer, demonstrating Durer’s mastery of both media. When we see one of the faces in this tripletwe do not see the other two, because of elements they share. For example, A. and B. Solid share

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an eye-pit.Ambiguous faces are frequently reproduced, but I have never seen a triply ambiguous one before

these, so these may not be well known today. They go beyond the familiar concept of relativity,in that each face in turn appears and disappears as we merely rotate our heads about the line ofsight. They show that what we see depends on our reference frame more than we ordinarily realize.Our innate face-detector is most sensitive by far to upright faces.

These three faces also hide in the way watermarks do. We apparently filter out distractinglow-contrast images within a high-contrast one unless we consciously search for them.

I have doubted the reality of some of these hidden faces but several circumstances reassure me:Durer had extraordinary vision; what I see he saw more clearly. The a. d. saw one of the facesbefore me, and who knows how many of the others. Schuster [22] attests to A. Solid, which hedescribes as frightening (erschreckended). I cannot find hidden faces in most of Durer’s works. Isee only two in The Knight, Death, and the Devil and none in St. Jerome in his Study, works donewithin a year of this engraving and in similar styles. I verify the faces I see with other viewers.Ultimately I know these faces are not accidental random patterns in much the same way that youknow that the words you are looking at now are not random selections of letters: They make moresense than random patterns do.

On the other hand, in the end I cannot count the faces of MELENCOLIA I exactly. BelowI name 10 faces that are clearly intended. Others are clearly unintended. But there are a fewintermediate cases that I cannot settle. Any noise, such as that produced by digitization or silkscreen, distorts or eliminates the intended faces and creates accidental ones.

• Viewers of Durer’s time may well have seen and enjoyed these faces at once, alerted by themany hidden faces in earlier works of Durer. His watercolor View of Arco has a cliff that clearly isalso a famous scowling face.

• His drawing of Arco hides three such unhappy faces in the landscape.• His engraving Der Spaziergang , or The Lady and the Gentleman, or Young Couple Threatened

by Death (page 201 of [19]) has a remarkable and incontrovertible secondary image of a male faceformed of and looming over both the lady and the gentleman in question. Their heads form hiseyes, her arm his mouth. In this picture Durer does not use the watermark technique but a kindof sleight-of-hand. The elements of the hidden face are in high contrast but the overt form divertsus from the covert one.

• In the 1494 drawing Orfeus der erst Pusaner (Orpheus the First Pederast) or The Deathof Orpheus, a hidden face in the tree-top over Orpheus watches in astonishment as two Thracianwomen club Orpheus to death and a boy flees.

Others have pointed out a significant hidden face in Durer’s engraving The Knight, Death, andthe Devil . Durer portrays the Knight as spectacularly oblivious to Death and the Devil, thoughin all other such encounters in the work of Durer, the knight engages evil in mortal combat. It isone thing to overlook an angel and a little demon as the Boy — this is the root meaning of putto— in MELENCOLIA I does; another to ignore Death and the Devil themselves as this Knightdoes. The Knight is generally said to represent the ideal Christian Soldier described by Erasmus(1466?-1536), who was a revered acquaintance of Durer. So virtuous was the Christian Soldier thathe was blind to the evil about him. On the other hand, Durer wrote a fervent plea to Erasmus, stillextant though perhaps never sent, to join with him and Luther in battle against the “Cave of Hell”and die a martyr’s death. Erasmus remained within the Catholic church and worked for reforminstead of applying for martyrdom. For Durer and his viewers, the unseeing Knight may also haverepresented Erasmus himself. In any case Durer pays tribute to the Knight in a marvellous waythat others have noted:

• Part of the garment of Death forms an agonized face smashed between the Knight’s two fists.

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Figure 2: Durer’s coat-of-arms (1523): A gateway in heaven

The Knight may seem blind to Death but still punishes him. Eventually, Pirckheimer tells us,the high-minded Durer would lament that the behavior of the Protestant clerics made even theCatholic clergy look respectable by comparison, and there is indication that Durer never lost hishigh regard for Erasmus:

• Durer’s 1526 portrait Erasmus of Rotterdam has a gleeful hidden face high on the left sleeveof Erasmus.

These examples indicate that the hidden faces are not all just puzzles and games. Some advancethe story like a Greek chorus. When Durer’s hidden faces look on and express emotion, they tell ushow Durer felt about the subject. For example, I suppose that the cliffs of Arco scowled because,as we now know, Durer disapproved of some aspect of the Arco cathedral. We will find still moresurprising and informative hidden faces in MELENCOLIA I .

As for these three, the poses of A. and B. Solid are those of Durer’s last portraits of his motherand father and so perhaps they fit well into Panofsky’s theory that this engraving is autobiograph-ical. If so, then C. Solid might be Durer the son.

• In the Boy’s blouse, between two straps, is an erect face I call A. Boy, its meaning unclear.

4 Coat-of-arms

Durer’s family coat-of-arms tells us about his self-conception and opens the doors of this work. Theversion of 1523, a copy of an earlier drawing that antedates MELENCOLIA I, has his monogramand date at top center. Beneath the monogram come conventional symbols found in any dictionaryof heraldry. Eagle wings mean fame and glory. A blackamoor at the top stands for heroic action byan ancestor in the Crusades. A closed helmet in full profile declares Durer an esquire, the lowestestate. Below the cliches, finally, stands a shield bearing two symbols specific to the family. Theseare easily read.

• A gateway with open doors can be read as a rebus for both “Durer” and “Ajtos,” the familyname and origin.

This sent me looking for other rebuses. I did not have to look far.The gateway stands on a cloud, making it a gateway in the sky. It is easy to read this too. It

is the key to MELENCOLIA I, and it does not seem to have been turned in earlier studies.Coats of arms usually show off the pride of the family. The cloud should stand for some great

personal accomplishment. To proclaim what he was most famous for, Durer could simply havedrawn his burin across the shield or included the word. I looked up the Latin for burin and foundthat it is caelum. This is most suggestive, for caelum is also the ordinary word for “the sky, ”“heaven,” and “the heavens,” with the same root as our “celestial” and the alternate form coelum.Durer could not miss this heaven-sent opportunity to sanctify himself and his art. The form caelo

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can mean both in Heaven and “I engrave.• The cloud in the Durer coat-of-arms is a rebus that points his family gates “in Heaven”

(caelo) and at the same time says “I engrave” (caelo).The cloud is our second rebus. Durer could assume that it was easily read; more easily, certainly,

than the familiar fish rebus for Jesus.“Gate of Heaven,” close in meaning and form to the Durer coat-of-arms, had long been a

common metaphor for the Roman Church itself. An educated person of the time might takeDurer’s claim of divinity not as a sacrilegious display of ego but as a pious reaffirmation that weare made in the image of God. We will meet the Durer gateway in the sky again in MELENCOLIAI .

There is another rebus in the famous Durer monogram AD at the top of the 1523 coat-of-armsand at the bottom right corner of the engraving. It seems to pun on Anno Domini. With Durer’scoat-of-arms in mind, one now sees the oddly-formed A in his monogram as a gateway as well as aletter. The legs of the gateway are the uprights of the A, the lintel sits on them and connects them,and the stiffener just beneath the lintel crosses the A. The cross-bar of Durer’s A is peculiarly highfor an A but proper for a gateway. The polymorphic A-gate is our third rebus for the Durer family,the first of several gates in the picture important for its interpretation. It makes the monogramat the top of the coat-of-arms an echo of the coat of arms itself. The A-gate of the monogramrepeats the open gateway of the coat-0f-arms. The D of the monogram, however, is in the positionof someone passing through the gateway, which is empty in the coat-of-arms.

5 Numerology

The matrix of numbers set into the masonry wall in MELENCOLIA I like a window lattice is amagic square, meaning that it is filled with consecutive numbers starting from 1 and every row,every column, and both main diagonals add up to the same number. In a 4× 4 magic square thatnumber must be 34: one quarter the sum of the integers from 1 to 16. This magic square is specialand called a gnomon magic square because in addition its four quadrants, its four corners, and itscentral tetrad add up to the same number. In addition, the sum of any pair of numbers symmetricabout the center is 17.

The idea of the magic square and much of the cosmology in this engraving may have come toDurer from Heinrich Cornelius Agrippa von Nettesheim (1486-1535), a prolific and controversialfigure in the Florentine Neo-Platonist school, the original of Christopher Marlowe’s Dr. Faust.Agrippa never settled in one city as much as four years, but wandered Europe fleeing the enemieshe made and seeking royal support. He visited Nuremberg in 1510 with the manuscript of his DeOcculta Philosophia, a compendium of astrology, alchemy, Cabala, and false propositions of biology,physics, and medicine, for example, that diamond destroys the magnetism of lodestone. He saidthat he studied the occult philosophy because it traced back to a secret divine revelation given atthe same time as the open revelation of the Torah of Moses. In his writing Agrippa assigns a magicsquare to each of the seven “planets” then known, in both Arabic and Hebrew numerals, in theancient order Saturn, Jupiter, Mars, Sun, Venus, Mercury, and Moon. This is the order of theirperiods, since the geocentric system gave to the Sun the heliocentric period of the Earth. Thesemagic squares have ninth century Arab sources [19] but may be older.

Agrippa omitted the 1-by-1 magic square [1] and there is no 2-by-2 magic square, so Saturnhas a 3-by-3 table, Jupiter a 4-by-4 table and so on to the 9-by-9 table of the Moon. The numberof Jupiter is then 1+2+ . . .+16 = 136, as the number of the Sun is 1+2+ . . .+36 = 666. Agrippawarned that unshielded Saturn caused acute melancholia, a clinical mental illness, and prescribed

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4 14 15 1

9 7 6 12

5 11 10 8

16 2 3 13

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

Figure 3: The Magic Squares of Agrippa and Durer: with rescaled detail

wearing Jupiter’s Table as shield. Agrippa proved the virtue of Jupiter’s Table by gematria, aHebrew numerology based on the fact that any letter of the Hebrew alphabet is also a numberand that therefore every Hebrew word has a number, the sum of its letters. He found reassuringHebrew words in Jupiter’s Table by gematria. Some say Durer borrowed Jupiter’s Table fromAgrippa [19, 25].

• The two magic squares are shown in Figure 5. They have no row or column in common andDrer’s table lacks Agrippa’s protective words. Durer borrowed the idea of the Table, perhaps, buthe performed non-trivial permutations — not mere reflections of the whole table — that left itboth magic and gnomon but changed the bottom line to the date 1514 of the engraving flankedby the Latin gematria for D A, matching the monogram and date at the bottom of the engraving.Thus monogram signature echoes the body of the work, as in the coat-of-arms.

The number 5 is engraved over a false start that is still visible, as though Durer lapsed brieflyback to Agrippa’s magic square. Durer used the Table to incorporate his name and the date, notto ward off melancholia. If he saw and used this peculiar combinatorial possibility, this tells ussomething about his way of thinking. It led me to seek other uses of the permutation principle andgematria in MELENCOLIA I .

I found both at once. Pursuing Panofsky’s autobiography hypothesis, I computed the Latingematria for “Albrecht Durer,” ignoring the non-Latin umlaut. The sum is 135. Since there is no“u” in the Latin alphabet, the spelling can be “Albrecht Dvrer, ” but this does not change thesum, since “v” would also replace “u” as letter 21. The summation of the whole table is 136 and isnot under Durer’s control. He would have had to separate the 1 from the rest of the table to makethe sum equal 135 or “Albrecht Durer.” This prediction we can verify.

• One wing of the angel has brushed the 1 and no other numeral in Durer’s Table, presumablymaking it holy, and enlarging and raising it in a way that violates the conventions of realism. I didnot see the magnified 1 until I did the gematria and looked for it. As far as I can see, no one hasmentioned it for five centuries, but it is indubitably there and quite out of keeping with the realismof the piece. The difference looks small to me, but Durer had an extraordinary attention for detail.

Possibly this holy 1 represents the Holy One. It is the 1×1 magic square that Agrippa left out.Its size further distinguishes Durer’s Table from Jupiter’s. The magic square thus carries Durer

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signature twice, once in the bottom line and once in the total, next to the Holy One.It is easier to credit Durer with such self-deification after we see other instances. We have noted

others in his monogram and his coat-of-arms. For another important example, see Durer’s paintingin the Alte Pinakothek of the Mourning over the Body of Jesus. The mourners form a triangle.The base of the triangle is the body of Jesus, and the body of the triangle is made up of mourners,looking down as symbolizes mournful melancholy. But the apex is Durer himself, looking up.

One moral of the magic square is so obvious that it is important to point it out again: Thingsare not what they seem in this engraving. What we see depends strongly on our point of view.

6 Bat

The chimera with head of mouse, wings of bat, and tail of serpent flies at us out of the engraving.Perhaps it flees the light, perhaps it carries us a message from it.

It carries a banderole on which is printed “MELENCOLIA I.” I accept the view that this refersto the ancient Greek humor of melancholia, and specifically in Agrippa’s revolutionary form ofthe theory, derived from the teaching of the Florentine Neo-Platonist Marsilio Ficino (1433-1499)[19, 25, 1].

Part of the Renaissance was the return to the subtler understanding of the Greeks, many ofwhom knew that the Earth moved. In this time of transition, I will suppose, Leonardo and Durerused a more physical cosmology in their science, at the same time that they still used the naiveNeo-Platonic cosmology of geocentric worlds in their art because that was still the language thatthey shared with their audiences.

Sphere I is the Terrestrial Sphere, our normal abode, made of the four elements distributedin space and time. Angels and Platonic Ideas inhabit the Intellectual Sphere III. Between thesetwo Spheres lies the Celestial Sphere II, made up of one sphere for the stars and many others forthe planets. Ptolemy used over 90 spheres in all, all part of the Celestial Sphere. Neo-Platonistsanalogously divided the human psyche into three Mentalities or Faculties specially suited to thecorresponding Spheres [25]:

Mens imaginatio, the Imaginative Faculty, empowers the artisan, the artist, and the NaturalPhilosopher to view arrangements of the four Elements in Space and Time, but nothing more.

Mens ratio, the Rational Faculty, allows astronomers and Mathematical Philosophers to knowwhen stars rise and set and thus through astrology it allows statesmen to know when kingdomsrise and fall.

Mens contemplatrix , the Contemplative Faculty, enables Theological Philosophers to knowangels and Plato’s Forms or Ideas.

The three names vary. Agrippa called them idolus or imaginatio, ratio, and mens respectively[18]. Ficino and Agrippa also kept the ancient Greek psychology of the four humors, one foreach Element: blood for Air and the sanguinary temperament, yellow bile or choler for Fire andthe bilious temperament, phlegm for Water and the phlegmatic temperament, and black bile ormelancholia for Earth and acute depression.

Melancholia, the black bile, is a Greek myth invented to fit the mental state of acute depressioninto humor theory. There is no such stuff. Some said that the appendix secreted it, others thespleen; they secrete nothing. Astrologers associated Saturn, the slowest and darkest planet, with theleast favorable humor, the black bile or melancholia; the worst mental state, depression, thereforecalled melancholia; the lowest Element, Earth; and the lowest people, understood to be those whowork with or in the earth.

Agrippa promoted Saturn from the lowest status among the planets to the highest, arguing

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that Saturn is highest from the Sun and slowest of all. He gave Saturn the power to inspire all threeFaculties to their peak creativity, and transferred the demi-godlike geometers — “earth measurers,”after all — to the control of Saturn, besides the farmers. According to Agrippa, Saturn could inspirethe Imaginative Faculty [25] of the artist with a melancholia that takes the form of a creative frenzyor furor. Thus Agrippa was a revolutionary astrologist while he practiced astrology and even afterhe rejected it [18]. Durer was a revolutionary artist, aiming to promote art and artists from thelowest to the highest rank. Since artists are assigned to the Terrestrial Sphere of Saturn, this musthave created a natural alliance between Agrippa and Durer.

Moreover both Agrippa and Durer were humanists and admired Erasmus. Agrippa was aHebraist, advocating simplicity and Bible-reading. These became part of both the Reformationand the Unitarian movement, but Agrippa himself remained in the Catholic Church. In the lastphase of his life he hated astrology and was skeptical about all access to truth accept faith in theBible. He offers us three increasingly reliable ways to know God, and three Books: Gateway I,and the Book of Nature; Gateway II, and the Book of the Law, the Hebrew Bible; and GatewayIII, the Book of the Gospel. In Book I we see merely the creatures of God, not God. We may useReason to deduce the existence and nature of God, but original sin clouds our souls and makes ourreason unreliable. Reason even led to the original sin when Eve reasoned with the serpent. TheBook of the Law and the Book of the Gospel, the two books of the Christian Bible, are the resultof revelation and require only faith. His pessimistic work on the uncertainty and vanity of thesciences, and the superiority of faith in Scriptures and divine revelation to all the arts and sciences,including astrology, was written after his optimistic Occult Philosophy but appeared in print twoyears earlier. His self-contradiction is also built into MELENCOLIA I .

Panofsky and Yates saw many Saturnian melancholic elements in the engraving, in either themore depressive sense of Ficino or the more creative sense of Agrippa. They take it that Dureraccepted Agrippa’s philosophy at least for the purpose of this engraving. Much in the engravingsupports this view or is at least consistent with it, and I accept it in the following interpretation. Atthe same time, while Erasmus, Agrippa and Durer all took the Christian Bible as divine revelation,they took significantly different positions with respect to the Reformation. Erasmus remained loyalto the Church and distanced himself from Agrippa when Agrippa’s protest against the corruptionof the monks who ran the Inquisition became radical [10]. Durer left the Church to follow Luther,but if I read MELENCOLIA I correctly, it expresses something more radical than anything we havefrom Agrippa or Luther.

Durer probably rejected astrology before Agrippa. In 1514 Copernicus (1473-1543) had notyet published his heliocentric cosmology, but the cardinal, mathematician, and philosopher Nicolasof Cusa (1401-1464) had quite demolished the Ptolemaic spheres. Leonardo had already writtenthat the Earth moves, not the Sun, and that “those who have chosen to worship men as gods – asJove, Saturn, Mars and the like – have fallen into the gravest error.” He had derided necromancersand excluded every trace of astrology from his intensely rational works on astronomy. There is apicture of the folly of Attention to the Stars in the Narrenschyff (Ship of Fools) of Brant (1494) [3],plausibly attributed to Durer. It shows the astrologer-fool immersed in a motley flock of misshapenfowl flying in random directions, representing the products of the astrologer’s disordered mind.Birds were a common metaphor for thoughts.

Martin Luther tacked his theses to a church door in 1517, and Durer became an ardent fol-lower at once. Luther specifically linked Agrippa’s Occult Philosophy with witchcraft, and Agrippaaroused ire when he defended a woman accused of witchcraft from her corrupt accusers. Agrippaeventually withdrew to a pious faith in God and Bible.

The motto labels the Bat as Mr. Melancholia, hence Agrippa, who had recently passed throughNuremberg with his Occult Philosophy in manuscript form. In view of the motto, the darkness of

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the engraving can be taken to indicate Agrippan, hence Saturnian, melancholia.Some say that this motto names the angel diagonally opposite it. In his other works Durer

labels things unambiguously. He puts his labels on or near the things labeled. There is no reasonto deny him this practice here, except possible preconceptions. The label is on the Bat, not theAngel.

Durer fills his engraving with Agrippan elements. I supposed that by giving the message“MELENCOLIA I, ” to the Bat Durer identified the messenger as Agrippa delivering his theory ofmelancholy. Agrippa was such a charismatic figure that Durer’s audience may have known at oncewho he meant by the Bat carrying Agrippa’s famous catch-word.

7 Anagram

There is something off-key about the motto on the banderole of the Bat. Melancholia is thetransliterated Greek for melancholy. One Latin form is melancolicus; The German is Melancholie.The Italian was Malinconia. Durer himself uses the form Melancolia in a title-page woodcut of1503. In no language is the word melencolia; the spelling seems unnatural even for Durer. Thespelling of the time was so variable that I could not be sure, but the encodings we have alreadyfound, the misspelling, and the cryptic extra letter suggested the possibility of an anagram to me.We may be sensing the constraint imposed on the overt message by a covert one that Durer wishedboth to record and conceal.

Agrippa could have put anagrams as well as gematria and magic squares into Durer’s mind.They are all in his book on the occult philosophy. Anagrams, gematria, acrostics, and acronymsare among the techniques used to extract messages from biblical texts in the Baraita of Thirty-TwoRules of Rabbi Eliezer ben Jose the Galilean, who flourished in the second half of the second centuryC.E. and was cited importantly in later years. Such lore became more accessible to the FlorentineNeo-Platonists, including Agrippa, an active Hebraist, when some of the Jews expelled from Spainin 1492 went to Florence.

In any case, by the 16th century letter-permutation was a standard way to protect intellectualproperty. Writers of the time who solved an important problem could not guard their rights bypublishing or patenting. The Royal Society and the system of scientific archives were still over acentury in the future. Some protected their priority by permuting the letters of their solution andpublishing the result together with the problem. Earlier Roger Bacon had scrambled the formula forgunpowder in this way in order to prevent the proliferation of this terrible weapon. Later Galileowould scramble his discoveries of the phases of Venus and the rings of Saturn. Mathematiciansscrambled their theorems. The resulting meaningless jumble of letters declared on its face that itwas a cipher. Sometimes it was misconstrued by a false rearrangement.

In the next degree of concealment, the scrambled letters themselves spell a cover message,the anagram. This can conceal not only the content but even the existence of a hidden message.“MELENCOLIA I” has done this well. Only the cover message has been read in recent centuriesas far as I know. I decoded it as follows.

Since so many Durer’s already populate the engraving, it is a reasonable first guess that themotto might hide another signature, so I first went to Durer’s family coat-of-arms to see how hemight sign himself. There I found the caelo rebus already described by which Durer representedhis heavenly art. CAELO indeed fits into MELENCOLIA. The leftover letters promptly spellout LIMEN, commonly meaning gateway, a near-synonym for “Durer,” as well as gate, doorway,threshold, lintel, walls, house, home, boundary path, and limit, according to context. We are backto Mr. Gates.

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• MELENCOLIA is an anagram of LIMEN CAELO, which I venture to render as “Gatewayin Heaven.”

This unscrambling took no longer than it takes to describe, and made me confident that I wasfollowing the path that Durer intended us to take. The hidden motto combines the family name-rebus, Durer’s art, and the Durer coat-of-arms itself. This verifies the initial guess: The anagram isindeed another signature. Indirectly, it supports the interpretations that Durer intended the cloudsin the coat-of-arms as a symbol of Heaven, and that he did not intend to demonize Agrippa. Thehidden phrase also applies to the dim archway in the heavens that frames the motto, and to otherelements that we take up later. These points of contact between the anagram and the engravingsuggest that I have read it correctly.

The overt message and image, both dark, cover a positive one. “Limen caelo” is as bright inspirit as “Melencolia” is depressive, and yet the picture is dark. It seems to offer a gateway toHeaven, but we must still figure out what that is.

8 I

Some suggest that the “I” of the banderole refers to one third of the medieval tripartite universe,and expect that MELENCOLIA II and III were engraved and lost, or were at least planned.MELENCOLIA I is indeed one of three, but they all survive famously. They are the MasterEngravings (Meisterstiche) of Durer, all dated within a year of each other. The other two are TheKnight, Death, and the Devil , and St. Jerome in His Study . Jerome and MELENCOLIA exemplifyFaculties I and III so clearly that one must consider the possibility that the Knight exemplifies II.Durer records that when he made a present of MELENCOLIA I he often accompanied it withJerome, never with the Knight . But this does not explain why one engraving bore its own captionand not the other two, and so does not explain the I .

Another reading would have been natural for a mathematician like Durer. Even in 1514, an “xprimus” in a mathematical writing more often referred back to an earlier x than ahead to a later xsecundus or x tertius. This would mean that Durer was superceding the old gateway in heaven witha new one, limen caleo primus, and so was claiming innovation. I abandoned this interpretation notwhen I realized that innovation was a capital crime, but when I realized that Durer and Agrippadid not consider themselves innovators but as renovators stripping away the innovations made bythe Church.

I suppose that the “I” as numeral indeed refers to a triple based on the three Neo-PlatonicSpheres. Agrippa described three pathways to God of Natural, Mathematical, and TheologicalPhilosophy, using the three Faculties, Imaginative, Rational and Contemplative respectively [1].We do not need to postulate lost engravings. This very engraving shows all three Faculties, orGateways to Heaven, and shows the Boy Durer representing the Imaginative Faculty choosingAgrippa’s Gateway I, the Natural Philosophy that evolved into Physics, in preference to Gateway II,the way of the Mathematical Philosopher, and Gateway III, the way of the Theological Philosopher.For Durer Gateway I meant making precise and beautiful records of nature and knowing and obeyingthe laws of perspective and structure.

• MELENCOLIA I is an anagram of limen caelo I, Gateway I in Heaven.This cannot be all. Presumably Durer wished the sense of caelo as “I engrave” to manifest

itself as well as the celestial sense, in the engraving as in the coat-of-arms. This rebus is too fittingto omit. To preserve it Durer would have to form the dative case, limine, for which he need onlyadd an i. And the i is waiting as the second element of the motto.

• MELENCOLIA I is an anagram of “CAELO LIMINE .”

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This can mean, “I engrave at the edge,” or, “I engrave at the wall.” It can also mean, “At theGateway in Heaven.”

We have already seen that Durer is able to find and use such multiple possibilities, but weprobably will never know which of these he intended, since they are all relevant and appropriate.

9 Boy

The Boy engraves at the edge and engraves at a wall. He sits against a wall of the House as heengraves, and also on the edge of the stone disk, as the anagram says. This could be a multiplecoincidence, but it is also possible that Durer is making the Boy fit the anagram so that we willknow that the Boy is Durer too. Considering the number of things a boy can do, this agreementseems well outside the normal range of accident.

• The Boy’s writing instrument has a cross-bar at its top, and so is not a stylus or a piece ofchalk but a graver, the celestial burin again. His block is therefore not a slate but a copper sheetor a block of wood. He is engraving, not writing.

This supports Panofsky’s conclusion that Durer meant this work to show many aspects ofhimself, but the Boy is no generic starving artist scribbling meaninglessly on a slate, he is theMaster Engraver himself, at work with his heavenly Burin with the creative frenzy that melancholiacan inspire in the artist; quite as Yates said. The Boy is the sole melancholy figure in the work.

The Boy is the central figure in several senses. The geometric central axis of the engraving isthe vertical defined by the leftmost edge of the house. It passes almost exactly through the nearesteye of the Boy, a common way to indicate centrality in Renaissance art. The line of the cometsuitably extended also strikes the Boy’s eye.

Yates suggested that the Boy is Durer, engraving “MELENCOLIA I” itself in a self-referentialway. This is not consistent with the Boy’s line of sight and the scale of the engraving. We mustrespect geometry as Durer did. The Boy looks over his tablet directly at the Dog, and his tablet istoo small for “MELENCOLIA I.” The Boy is clearly engraving the Dog on his tablet; Durer hadin 1513; more autobiography.

The burin identifies the Boy as Durer. In the tripartite psychology of the times, the artist’sspecial Faculty is the Imaginative one, so the Boy probably represents the Imaginative Faculty too.

10 Crime

One mystery that faces us is why all the mystery. Durer could have explained his engraving forus. Leonardo made up many riddles, but he wrote the solutions next to them, protected only byhis mirror-writing. Durer and Pirckheimer left the key to the Triumphal Arch for us; why notthe key to MELENCOLIA I ? In his essay on this picture, furthermore, Camerarius described themagic square full of numbers as a spider-web full of dead flies, ignored the many concealments, andomitted the religious and the scientific contents [13]. Why was he disingenuous?

I hypothesized that Durer and his friends were less than frank in order to evade the stake. Themessages of the Triumphal Arch could be published in 1514, but the ideas in MELENCOLIA Imust have been more dangerous, this theory goes; and yet they must already have had a market.

In the times of this engraving, accusations of black magic and witchcraft beset magiciansand natural philosophers alike. Agrippa was linked with the devil by Martin Luther (1483-1546).Agrippa may have influenced Servetus, who was burned in 1535, and David, who was imprisonedand soon died in 1579 [10], both for the charge of “innovation” in religion. Eventually Agripparenounced his major opus, even before its publication, and adopted an uncritical fideism. There

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was ample reason for Durer and his heirs to fear execution and preserve silence.This theory of a hidden heresy runs into a certain obstacle. In about 1498 Durer executed and

published a volume of fourteen woodblock prints of the Apocalypse. One of them, The Openingof the Fifth and Sixth Seals,, shows the wicked hiding from the vengeance that God takes for theblood of the saints. The wicked are the Pope, Cardinal, Bishop, Kaiser, and Kaiserin, the top ofthe whole hierarchy of the Catholic Church and the Holy Roman Empire. In another, The Battleof the Angels, he shows the imminent execution of the Pope by an avenging angel. So by the ageof 27, 16 years before the date of MELENCOLIA I, Durer — and his master Wohlgemut — hadalready stood against the hierarchy of the Church and killed the Pope and the Emperor in effigy.The heresy of this engraving must be even more criminal since he had to hide it. It must also havea market large enough to wear out the copper plate in making copies of this engraving. It mustbe a crime that the Protestants too would punish, since Durer had already dismissed the Catholicthreat. What was the crime of Albrecht Durer?

I could answer this question only after I answered some easier ones.

11 Ladder

Any artist who put an endless ladder next to an angel, a house, and an up-turned stone in 1514could be sure that the educated viewer of the time would see Jacob’s ladder, angel, house, and stoneof Genesis 28. The unseen end of the ladder is in heaven. Ladders as gates of heaven occurredcommonly in earlier art, including Durer’s own. The most relevant verses are

Genesis 28:12 And he dreamed, and behold a ladder set up on the earth, and the topof it reached to heaven: and behold the angels of God ascending and descending on it.

Genesis 28:17 And he was afraid, and said, How dreadful is this place! this is noneother but the house of God, and this is the gate of heaven.

Genesis 28:18 And Jacob rose up early in the morning, and took the stone that he hadput for his pillows, and set it up for a pillar, and poured oil upon the top of it.

The house shows no door, so the ladder is the Gate of Heaven of the anagram and Genesis: anothersignature. Beth El, the House of God, is also the legendary site of the first house of worship of theGod of Israel. Early on, therefore, the porta caeli of the Vulgate also became a metaphor for theCatholic Church itself.

12 House

The house too is meaningfully polymorphic. From up close we see two blank walls supporting ahodge-podge of instruments. The side wall does not seem much wider than the Boy leaning againstit. It seems about the right thickness for an out-house or a chimney, not a residence. Its frontwall extends indefinitely to the right and holds a bell, Durer’s Table, an hour-glass, and a sun-dial.Its left edge bisects the picture to the millimeter. Its side wall holds a chemical balance. Thebell-rope trails off the edge to our right, probably to clear the magic square. Step back severalmeters however and the picture changes.

• Magic square and bell become a lattice window, the hour-glass a bay window, the scales aside window, and the out-house a full-sized house showing us three of its windows.

Assembling familiar objects from unrelated parts in this way, for example, faces from fruits,or demons from vermin, was an optical illusion often practiced by artists of the era. Here there

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is allusion as well as illusion. If the house is what the sanctified instruments on its walls openinto, it is the humanly perceptible universe, the entire physical cosmos. The ladder and the Angelhave already told us that this is also the House of God, and the instrumental windows tell us thatmeasurement and mathematics are a way of seeing into it. This is the declaration of Leonardo thatall knowledge comes from experience, but Durer clearly accepts the truth of the Scriptures as well.

13 Door

Then there is the curious matter of the door to the House. The curious matter, of course, is thatthere is none. The House that we see as we step back has several windows. The Bell and the MagicSquare become one window for arithmetic and music, the Hourglass and Sundial turn into anotherwindow for time, and the Scales turn into a third for weight. The Angel blesses all three windowswith her wings. There is a distorted image in the Hour-glass that by all the laws of optics should bethe reflection of Durer himself, standing before the Globe as he sketches: another signature. Butnowhere do we see a door. When Mr. Door draws the House of God without a door he is likelymaking a statement. It is one of the major statements of the piece.

If the house is indeed Reality, its doorlessness would mean that we have no direct access toabsolute reality. This was a common idea by Durer’s time, for example, from writings of Nicholasof Cusa and Erasmus already cited. Durer could expect some of his viewers to make out thismessage at first sight. It recurs in modern physics in the limits to perception imposed so far bylight-speed c and action-quantum ~, with more surely to come. Before Durer did this engraving hehad already written that the human mind cannot know absolute beauty, and in the picture he maybe saying the same about absolute truth in several ways. We must not assume that he separatedtruth and beauty, or intellect and emotion, as cleanly as some people claim to do today, especiallyif he expected a mathematical theory of both.

From the Terrestrial Sphere we can look at divine reality but we cannot touch it. If it canbe entered by the ladder, that is for the angels, we have seen. But Durer makes no cut betweenthe truth that we glean from measurements and that which we read from Scriptures. They form aseamless whole, except that the ladder reaches above the house.

14 Angel

Angel-wings indicate sanctity as eagle-wings represent fame. These sacred wings brush and bless theholy One in Durer’s Table, the hour-glass, the scales, and nothing else in the picture. Apparently theAngel blesses the scientific instruments above the workman’s tools that litter the ground. She holdsa sealed book and an idle pair of compasses. Most commentators also say that she is melancholy,and is even the spirit Melancholia herself. The motto on the banderole, the shadow on the Angel’sface, and the fist on her cheek all are indeed consistent with melancholia. But the hidden mottois not negative in mood but positive and in any case is attached to the Bat, not the Angel. Herwreath of water-cress and water-ranculus protect her from Earthy melancholia by their Waterynature [19], so she cannot be melancholy. The Angel’s expression is alert and focused, not soft andsad. Above all her gaze is not downcast, as was absolutely mandatory for portraits of melancholia,but elevated. She is smiling more than frowning.

It is remarkable how ambiguous her expression is. There is a definite upward line at the nearedge of her lips that can be read as a bright smile. But is can also be read as the edge of her sleeve,which is placed at precisely the right position by her posture. It seems absurd to imagine that thisambiguity is not completely under Durer’s remarkable control. He drew the Angel’s expression so

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that it could be taken as serious or smiling, depending on the expectation of the viewer. The faredge of her lips is almost hidden by the position of her head, so that only a trace of the smile isvisible there. The smile is there quite definitely if one looks for it, but Durer makes us work for it.

At the end Durer resolves this ambiguity, unlike others in the engraving. Even if he had not,since the smile is unexpected in the context of the motto and the nigt sky, it carries more meaning,by Shannon’s principle. I will explore the smile only in what follows, the assumed frown havingbeen studied for five centuries.

Then the pose shows contemplation, as in Rodin’s Thinker, not melancholy. It and her wingssay rather clearly that she is the Contemplative Faculty proper to the Intellectual Sphere andGateway III. Let us see how this interpretation fits the rest of the engraving.

15 Compasses

The compasses of the Angel test my assignment of Spheres and Faculties. Whenever the quadriviaare personified, Geometria is the one with the compasses. Since the Angel has the compasses, isn’tshe the Faculty II of mathematical reason?

To be sure, in a Durer woodcut of 1504, The Astronomer measures a globe with compassesunder a full moon, so compasses are also allowed in the Celestial Sphere, but this does not meanthat the Contemplative Faculty can have them.

Some say the Angel has measured the stone Globe that lies before the dog. But the Astronomerstudies his globe intently while the Angel looks right past this globe, as she should if she is Contem-plation and the globe is the the Terrestrial Sphere. Lines of sight matter in the work of a projectivegeometer.

To use compasses one must control both points at once, either holding the apex or both points.The Angel grasps only one arm of the compass and that near the point, so she cannot use them ascompasses.

She has no drawing board or table beneath the point. She seems to be sticking herself in thethigh. Since this is absurd we must look deeper; a few centimeters will suffice.

16 Arab and Jews

• The angel’s right knee, a bright patch in the moonlight, is also the top of a hood worn by abearded man whose head is cocked at about 30◦ counterclockwise, shown in profile with only oneeye visible. Call him A. Angel. He has an aquiline nose and wears a black headband across hisbrow. Slowly his face leads us to a powerful shoulder, almost a hump back, on which Angel restsher elbow and compasses. This connects to his left arm, outstretched toward the saw-sword on theground. The left foot of the Angel, where it extends beyond her robe, is also his left hand, her toesindicate his fingers. In the other direction his shoulder leads us to his torso, waist, and knees. Heis kneeling as he reaches for the sword blade. The Angel steps on the hilt to foil his effort with herright foot and pricks him with the point of the compasses in her left hand. His right hand, whichdoubles as a domestic tool, reaches up past his face to fend off the compass point, indicating thathe is in some discomfort. He kneels there covered from head to feet in an Arab burnoose.

• A. Angel is one of a triplet, the first that I found in the engraving, simply because it is theleast hidden. To see the other two hidden faces, find a moonlit isosceles triangle next to the faceof A. Angel, a lighter patch in the shading of the skirt directly below the Angel’s right knee. If weturn our head through 45◦ clockwise this time, we can see this lighter triangle as a flask leading tothe dark line of a mouth in the face of a man in a skull-cap in a full front view. I call him B. Angel.

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Figure 4: Angel: Smiling or not. Enlarged detail.

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Figure 5: A., B., and C. Angel, rotated and scaled details

A worm extruding from his right eye tells us with shocking clarity that B. Angel is dead.• A second older and hairier face whom I call C. Angel shows in partial profile looking to the

left with a hooked nose, a nasty expression, and a skull-cap. Young B. Angel and old C. Angelshare an eye and form a merely ambiguous face, like the well-known duck-rabbit. C. blows a shofar,a ram’s horn, as is done on Rosh Hashonah, the Jewish New Year. B.’s worm becomes C.’s nose,B.’s flask becomes C.’s horn.

This triplet of the Angel has higher contrast than the triplet of the Solid. It is hidden by thesleight-of-hand technique, not the watermark technique. Their elements belong to a larger Gestaltwhich compels us to give them a totally different interpretation. Again the people in the tripletcan be seen only one at a time.

We already read the Angel as the Contemplative Faculty of the theologians. Durer seems toshow by this triplet that her entourage includes Jews and Muslims as well as Christians. Thesword for which the Muslim is reaching suggests that Durer associated Islam with militancy. Thisis plausible since Hungary had been mauled by both Christian and Islamic combatants in the15th century. Muhammed II, founder of the Ottoman Empire, took Constantinople in 1453 andlaunched a powerful counterattack on Hungary in 1455, the year in which Albrecht’s father leftAjtos. I expect to learn that Ajtos, which I have not been able to locate, was in eastern Hungary,most exposed to the army of Muhammad II. The attack was stopped by the forces of John Hunyadi,who became a Hungarian national hero.

Since the fall of Constantinople was so epochal for Christianity, the viewers of Durer’s daymight have read A. Angel as Muhammad II. But this would not tell us who Durer might havemeant by B. and C. Angel. And since Muhammad II was a warrior, not a theologian, Durer wouldassociate him not with the Intellectual Sphere but the Celestial.

The configuration makes better sense if the figure in Arabian dress is the Prophet Muhammadhimself. Then the triplet represents the founders of the three monotheistic faiths of the time. Thisreading associates them with Faculty III, that of of the theologians, and fits the scriptural reportsthat two of the founders were Jews, that one Jew lived to old age, that the second Jew died youngand that his death was doubtful enough to merit explicit indication, and that the third founderwas Arabian and a great warrior. The three are dawn in the correct chronological order from leftto right. The probability that these agreements are chance is not overwhelming so I concentrateon this possibility in what follows.

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When Durer unites these three figures in one triplet so that we can see only one at a timedepending on our point of view, he represents them to be close in many respects. Since Moses andMuhammad were Prophets, Durer would seem to indicate the same for Jesus. This is consistentwith his emphasis on the death of the younger Jew. I rejected this possibility for many months onthe basis of Durer’s many Trinitarian works, but reconsidered when I learned recently that Agrippaprobably links Durer to two founders of Unitarianism, Michael Servetus and Francis David [10].

We have surely found the capital crime of Albrecht Durer: innovation. Both Servetus andDavid were charged with innovation and convicted by Reformation courts, not the Roman Church,so being more explicit might have earned Durer the same fate. If I read him correctly on thispoint, then most likely the downfall he shows would represent the overthrow of the hierarchy of theRoman Church, whom he had already slain in effigy.

• In any case, now we see what the Angel is doing with her apparently idle compasses. She issticking the Arab, not doing geometry. She may therefore be Faculty III.

This violates realism, but Durer left realism behind when his Angel raised the painted One inthe magic square, so the violation has lost its shock value. I regret having to propose that Durer sather there with an innocent expression on her face as she pricks the Arab, but Durer often attacksfrom hiding. His Knight too seems oblivious to Death and the Devil but crushes one of thembetween his fists without visible feeling. The sneering cliffs around the Arco cathedral and muchof MELENCOLIA I itself are acts of concealed aggression by Durer himself against his targets.At least we can cast the Angel as the Contemplative Faculty with more confidence, now that weunderstand her anomalous compasses and her hidden faces.

17 Fool

This is a curiously domestic Angel. She is dressed as a housewife, complete with keys and pursehanging from her belt which Durer says represent power and wealth, and she attracts Christiansand non-Christians alike. Only a fool would worship her, the artist seems to say. And there thefool is, a crude low-browed caricature hidden near her feet in the hem of her gown, next to Durer’smonogram. Call him D. Angel.

• We must turn the engraving about 60◦ counterclockwise to see D. best and then he looks toour right. His nose forms part of the bottom edge of the Angel’s robe. He has a mustache.

A powerful winged housewife with a fool in her hem does not seem to be a standard symbol,but apparently it was for Durer. In a wood-cut Von Buolschafft (On Amours) attributed to Durer.in the Narrenschyff [3], the winged housewife of On Amours is Venus, the goddess of the fools oflove, the most common variety of fool according to Brant. Her wings are those of an eagle signifyingfame, not those of an angel signifying divinity. We know she too is a housewife because Brant callsher sarcastically “Frau Venus.” In MELENCOLIA I similarly we see Frau Angel, her housewifestatus conveyed now not by words but by purse and keys, hanging from her waist to the ground.(The Madonna by the Wall (Durer 1514) also wears keys and purse, on short leashes.) “Frau” isevidently no term of respect for Durer or Brant. With no forehead, the caricature in her hem is socruel that I hoped it was unintended, but since they recur, these fools-in-winged-wives’-hems areclearly not accidents but part of Durer’s graphic language.

Durer seems to say that the philosopher/theologian of the Contemplative Faculty is foolish,like one in love. This echoes and amplifies doctrines already expressed by Nicholas of Cusa in hisOn Learned Ignorance (1440) and Erasmus in his In Praise of Folly (1511), and later written downby Agrippa in his De incertitudine et vanitate scientiarum [2].

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18 Globe

Durer’s goddesses Fortuna and Nemesis both stand on globes, liable to roll in any direction atany impulse, because Luck and Fate are unpredictable. Chance governs the Terrestrial Sphere,mathematical law governs the Celestial Sphere. Therefore Durer’s Globe seems to represent theTerrestrial Sphere. It is indeed the lowest of the three stone objects in this picture, as if to confirmthat they are Durer’s version of the tripartite cosmology. Durer, like Nicolas of Cusa and Leonardo,has replaced the naive tri-spherical geocentric Neo-Platonist cosmology by a tri-partite philosophywith three Faculties and Spheres that are no longer geocentric, concentric, or places at all.

19 Solid

Durer seems to put his Solid in a place of honor. The artist stands before it, the ghosts haunt it,the motto might seem to name it, it balances the Angel in the composition. Yet it is bleak andsterile against the Angel, and its shape is subtly ambiguous. One looks for a deeper significance toenrich it and justify it artistically, or at least to understand its shape.

A rhombus is a planar quadrilateral whose sides are all equal. A rhombohedron is a closedsurface formed of six congruent rhombuses. Most people see the Solid as a truncated rhombohedronwith its longer axis vertical [27]. If it represents any of the neo-Platonic Spheres, then its math-ematical nature and its altitude in the picture suggest that it represents the Intellectual Sphere,the third realm, the abode of Platonic Ideas and angels. The ghosts in the Solid also support thisconclusion, for this puts them in a philosophical Heaven. I tentatively adopt this hypothesis inwhat follows.

If the two missing vertices were restored, they would seem to lie in a vertical line in space. Thecompulsion to see the figure as a truncated rhombohedron is irresistible. Only when a model orprojection of such a solid is juxtaposed with the engraving does its untenability spring to the eye.

In principle, however, one cannot determine the form of a solid from one perspective view.One can shift any surface point along the line of sight from the eye without changing its apparentlocation in the perspective view. Therefore there are many solids all having the same perspectiveview. The Reversing Staircase, the Necker Cube and the Cameo Intaglio exhibit this ambiguity.Sometimes other cues to the interpretation are so strong and unconscious that we cannot helpinferring a particular solid. For example, it is hard not to see a perspective of a cube as a cube, ora perspective of a person as a person, even though intellectually we know the picture is ambiguous.The Durer Solid is deeper than the illusions I mentioned, in that they are not like anything we seein nature, so it is not surprising that our visual system is not able to cope with them, while theDurer Solid has two interpretations and looks real in both.

If one views the Solid with head erect, one sees the hidden face of the woman on a truncatedrhombohedron.

If one views the Solid with head cocked to the right, one sees the hidden face of the man, ona nearly rectangular slab with two diagonally opposite corners trimmed, cocked and cantileveredback toward the horizon.

I owe this second observation to Dr. Basimah Khulusi [15], who has constructed three-dimensional models to show that the rhombohedron theories do not fit in either view, and thetruncated slab does.

Some speak of measuring the angles of the Solid from the engraving. These angles depend onsecondary assumptions that are not forced on us by the engraving.

One apparently invalid interpretation currently presented on the Web seems the acute angle of

21

the rhombus to be a fifth of a circle, 72◦ [27]. If one puts an actual model of the proposed solid, ora perspective view of such a model, next to the engraving, the mismatch is clear. If one compressesthe engraving vertically by a factor of

√ϕ, its outline becomes square, and the Solid looks like a

truncated cube, until one actually juxtaposes a true cube, or a perspective of one, for comparison.Another apparently invalid interpretation sees the same angle as close to 80◦ [17]. This results

from measurements on the engraving assuming that certain lengths are show in true proportion. Butthere is no way to tell for sure from the engraving if a given length is shown in true proportion. Thisrequires additional assumptions, usually made unconsciously and therefore irresistible. Moreover,if one juxtaposes a model of such a rhomobohedron with the engraving, the discrepancy is clearlyvisible.

Before the engraving Durer apparently made a rough sketch of the Solid that has survived,though it has been recognized as that only recently [24]. The sketch shows an irregular pentagoninscribed in a circle. The apex angle in the drawing is indeed 79.5◦ ± .5◦. The agreement with oneinterpretation of the engraving seems noteworthy. The question remains: How did Durer arrive atthat angle?

One notices that the circle in the drawing put forward by Weitzel was divided by a compassinto seven equal arcs as if to inscribe a regular heptagon. There is no Euclidean construction for theregular heptagon but Durer gives an excellent approximation to it in his course on measurement,without telling us that it is approximate.

The vertices of the irregular pentagon seem to be five of the seven vertices of the regularheptagon. The two long edges of the pentagon are chords of the heptagon that connect the apicalvertex to its next-nearest neighbors on each side, The remaining three edges of the pentagon areedges of the heptagon joining the remaining vertices. This would lead to an apical angle of 77.2◦

theoretically. The angle of 80◦ fits the engraving better.Leonardo had already undertaken a mathematical theory of beauty. That goal would occupy

much of Durer’s later years, and result in works on human proportions and on constructions ofprojections and perspectives with compass and ruler. Durer’s study of human proportions is anamazingly dry gallery of stark outline drawings of standing human nudes and tables of anatomicaldimensions to three decimal places. It may never have been of much use to artists, but it fulfilledother functions for him. Human measurements were to be the foundation of a geometry of beauty,as stellar measurements were the foundations of Ptolemy’s geometry of the heavens. Music wasalready divine and part of the quadrivium because it was considered mathematical. By providinga mathematical basis for art, Durer hoped to sanctify the graphic arts as well. Possibly thisbelief in a mathematical art was only one aspect of a belief in a general mathematical wisdom, amathesis. Then the Solid does not simply show off Durer’s mathematical muscle, it represents themathematical nature of the Intellectual Sphere, the divinely inspired mathesis that many sought,and its inaccessibility to human cognition. We will give it deeper meaning later.

I propose that Durer deliberately designed the Solid so that it was irresistibly misconstrued asa truncated rhombohedron when seen from one orientation, and as a truncated slab from another.The Solid is a puzzle that is in principle unsolvable without another view. I suspect that is whythe Angel is smiling.

20 Millstone

The most cryptic of the three stones may be the wheel holding the Boy. Panofsky and Yates callit a grindstone and Doorly calls it a millstone. Its perimeter is broken while its face is smooth soit would make a passable millstone but a bone-rattling grindstone and likely Doorly is right. This

22

is also the only stone in the engraving that could serve Jacob both as pillow and pillar.Since we have argued that the stone Globe stands for the Terrestrial Sphere and the stone

Solid for the Intellectual Sphere, by elimination it seems that the Millstone stands for the CelestialSphere, the realm of Reason. Why Durer would show the Celestial Sphere as a millstone is thenext puzzle.

One possible answer is closer than it seems at first. For centuries cultures as diverse as Babylon,Greece, Arabia, Scandinavia, and Rome had represented the Celestial Sphere by a millstone. Thecelestial millstone caused the precession of the equinoxes when it flew off its axis into the sea,presumably the Milky Way [20]. Santillana and von Dechend called it Hamlet’s Millstone becauseit appears as a prop in the Icelandic legend of Ambleth, the original Hamlet, as retold by SaxoGrammaticus in his Historia Danica in the late 13th century and first printed in Paris in — by asuggestive coincidence — 1514.

In the Golden Age the legendary celestial millstone ground out gold, in later, lesser times itground out cheaper stuff such as sea salt, and in our decadent age mere sand. Saxo’s Amleth,Shakespeare’s Dane, and Durer’s Boy share melancholia, parental ghosts, millstones, and dynasticoverthrow, suggesting at least a certain spirit of the times. I will point out a curious coupling ofan ocean-going millstone, melancholia, and dynastic overthrow in Scripture too.

Since a revolution is going on in this engraving, or perhaps two, some confusion in the seating isunderstandable, but the Faculties are out of their proper places, and in a significant way. The seatedAngel, the Contemplative Faculty, is nominally of the Intellectual Sphere III, highest, but here she isgrounded and lowest. The Boy, as the Imaginative Faculty, is traditionally of the Terrestrial SphereI, but he is here esconced on the Celestial II, as though fulfilling Durer’s life dream. Durer hasclearly promoted Natural Philosophy and demoted Theological Philosophy. We can read this bothas the Agrippan revolution and the scientific revolution. The Natural Philosopher observes matterin space and time, traditionally the lowest Sphere, yet thereby attains God. The ExperimentalPhilosophy is in the seating plan.

Pace Yates, the balance between the Angel and the Boy, therefore, is not the Hermetic equalitybetween the worlds above and below. That related the Terrestrial Sphere to the Celestial, notto the Intellectual Sphere represented by the Angel. “As above, so below” may well have beenAgrippa’s credo at the time but it presumably is not Durer’s. Like Leonardo, Durer had alreadymocked the idea that events in the sky foretold events on Earth. He had drawn and publishedmaps of the stars above and the planet below that showed no similarity or correspondence.

21 Moonbow

I have never seen a moonbow but again NASA provides the picture and the Bible provides thecaption. The only bow in the Bible sky is the “bow in the clouds” that Noah saw after the flood,a supreme example of divine revelation, direct communication with divinity. Genesis does not saywhether Noah saw his bow by day or by night. Durer had no choice once he set the scene in thedark of night. In Genesis, the light in the heavens and the bow in the clouds represent two of thegreat gifts of God, so the picture should be joyous, and yet it is dark. And ambiguous. Again.

The viewer of any moonbow or rainbow lies on the normal to the bow plane passing throughthe bow center. Durer’s watching eye should lie directly before the horizon point beneath thehighest point of the moonbow. This is not exactly where he is located by the vanishing point ofthe building lines, which is under the eye of the Bat, but it is only about a centimeter off.

The moon must lie on the same line, and behind the viewer, well to the left of center, likethe viewer and the moonbow. But according to the shadows on the House and the face of the

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Angel the moonlight comes from well to the right of center, over the viewer’s right shoulder. Thisinconsistency in Durer’s optics is of no import for the content of the engraving but sets an upperlimit on Durer’s knowledge of rainbows.

22 Comet

The bright light in the sky is a comet, not a nova, though it lacks the curved tail of the mosttraditional renderings. The NASA online collection of astronomical photographs shows cometsquite like it. A comet is straight when it heads for the sun and sunlight pushes its tail out outbehind it. It may be an illusion of perspective, but this comet seems to be pointing straight at theBoy, and at his near eye.

The great comet 1471Y1 was first seen on Christmas Day in Durer’s birth year, and Durerwrote of seeing a comet himself in 1503. The physical natures of meteors and comets were notyet known in 1514. Galileo would still believe in the next century that comets and meteors weremeteorological, weather of the highest atmosphere.

Since the Bible has already explained other elements of the picture, let us search it for thisone too. The answer is immediate and unique. The only blinding light in the biblical sky is theoriginal light of the divine creation. Then the sky would recall the time in the Bible story after thecreation of light but before the creation of the stars; and the sky is indeed starless. This is a joyousoccasion: Happy birthday, universe! The Light may represent Divinity itself, and divine revelation.Then the fact that the comet is pointing exactly at the Boy suggests that he, not the Angel, theDog, or the Bat, is chosen by God. The news is good but the tone is somber. The tension grows.

But so does the ambiguity. This is a realistic comet. Nothing in the picture compels us to seeit as divine.

It took me a long time to understand this engraving even with all these clues, because I could notfigure out whether Durer’s Gate in Heaven was mathematical or religious. This is a good exampleof the kind of projection to avoid in such a study. Durer does not separate science and faith. Onthe right they entwine in the building and ladder complex. On the left, there is an unnaturallystraight line from the comet or the light of God through the Intellectual Sphere of MathematicalPhilosophy to the heart of the Terrestrial Sphere of Natural Philosophy. The Celestial Sphere isnot in that line. It is old and damaged and out of service. The Dog is there instead.

23 Dog

The Dog is the next-to-last piece of the puzzle to fall into place. To me, dogs represent fidelity, toAgrippa love or flattery. I inferred that the Dog of MELENCOLIA I is the Rational Faculty byelimination and guesswork and then had to understand how this can be.

This Dog lies rather too close to the straight line of Comet, Solid, and Globe for accident. Hecurls up just where the Celestial Sphere would lie in the old cosmology. The battered millstone hasbeen lifted up on one edge and leaned on the House, and the Dog is where it used to be. The peopleassociated with the Celestial Sphere were Mathematical Philosophers, astrologers, and kings; notdogs. The connection of the dog to the Celestial Sphere was no more obvious to me than that ofthe Millstone.

To read the Dog I looked at Durer’s other dogs. He had drawn at least three dogs in the yearbefore that are still with us. In his Triumphal Arch Durer had represented the Emperor MaximilianI as a dog with a stole exactly as the Hieroglyphica required. The Hieroglyphica also requires usto represent a prophet — one whose Rational Faculty is excited by Saturnian melancholia, in the

24

theory of Agrippa — by a dog, “because the dog looks intently beyond all other beasts upon theimages of the gods, like a prophet” [14]. Horapollo has one represent a judge by a dog next tothe royal stole, and a king by a dog wearing the royal stole, for equally convincing reasons. Thedog drawn by Durer when he illustrated the Hieroglyphica even seems to be of the same breed asthe one in MELENCOLIA I, judging by the unusual muzzle-shape and the gaunt belly. Since herepresents a prophet, judge, or king by that breed of dog in theTriumphal Arch I shall assume hedid so again in MELENCOLIA I, within a yesr.

I find reading the language of Hieroglyphica trickier than writing it. Besides a prophet, Ho-rapollo’s naked dog also represents an embalmer, the spleen, odour, laughter, sneezing, and rule.Apparently one decides by context. I think Durer meant the prophet this time.

The main people associated with the Celestial Sphere and with inspired Rational Faculty,according to Ficino, are the prophets and the kings. The Hieroglyphica represents both by dogs,naked or stoled. So Durer obediently represented the king by a dog with stole in the TriumphalArch and the Rational Faculty of prophets by a dog in MELENCOLIA I.

The interpretation of the Dog gives us a hermeneutic rule for reading Melencolia I : First consultthe Bible. If that fails, go to the Occult Philosophy. When all else fails, open the Hieroglyphica.All three of these were prominent in the Humanist community of the early 16th century. All arein print today.

The main puzzle outstanding is the Bat/Serpent. As it becomes increasingly clear that Dureris adopting Agrippan teachings, it becomes correspondingly puzzling that he represented Agrippaby such a demeaning symbol as a bat, let alone one with a serpent’s tail. Perhaps I have erred.

It seems, however, that Durer made Agrippa a bat for the same reason he made his emperorMaximilian I a dog: The Hieroglyphica told him to:

“When they wish to indicate a weak and rash man, they draw a bat. For the bat, though ithas no feathers, flies.” [14]

Agrippa might well seem rash. In 1531 when a plague broke out in Antwerp and all the doctorsfled, Agrippa stayed to tend the sick. When the doctors returned they filed charges against Agrippafor practicing medicine without a license, to recoup their wealthy patients. Agrippa generally hadto move to a new town every few years. He might indeed seem weak. As for the serpent’s tail:In Hieroglyphica a serpent represents the king (again), and “When they wish to symbolize a kingruling not the whole world but a part of it, they draw a serpent cut in half.”

So the Bat/Serpent symbol does no demean Agrippa, despite initial appearances. Durer canuse Agrippa’s theory of melancholia and nevertheless describe Agrippa as weak, rash, and of limitedrule. More important, some viewers of the time knew their Horapollo and may have recognizedthis representation as the very image of Agrippa himself.

The interpretation of the Dog as the Rational Faculty fits all three Master Engravings ratherwell. The dog of Jerome sleeps soundly because Jerome uses Faculty III, not II. The Boy’s Dogrests with half-closed eyes for a like reason. The dog of the Knight is wide awake and runs besidethe Knight because the Knight concerns himself with affairs of state, which require Faculty II.

24 Babylon

My Bible-searches for ladders, sky lights, and sky bows all bore fruit so following my hermeneuticrule I looked for Durer’s Millstone in the Bible too. My concordance found 11 millstones in theChristian Bible. The first ten serve as metaphors for livelihood or industry and refer to no otherpoint of the engraving. Drawing a straight line through one point is too easy, so I put them aside.The eleventh and last is in Revelation. In the King James version:

25

And a mighty angel took up a stone like a great millstone, and cast it into the sea,saying, Thus with violence shall that great city Babylon be thrown down, and shall befound no more at all. (Revelation 18:21.)

At least four points of the engraving appear here: the mighty angel, the millstone, the sea, andeven a kind of melancholia, for Revelation imbeds its millstone in a score of verses lamenting thedestruction to come. The millstone of Revelation is therefore a provisional candidate for Durer’smillstone, and the only one in the Bible.

But the Biblical millstone is as puzzling as Durer’s. A more cosmic image than sinking amillstone — say, sinking a continent — would show how great a force would destroy Babylon; amore domestic image — say, stepping on a wine-glass — would show how easy it would be for Godto destroy Babylon. But a millstone is the wrong size for Babylon, neither here nor there. Evenwe people can sink a millstone in the sea, and yet it takes some effort.

If on the other hand the millstone of Revelation again represents the Celestial Sphere, throwingit into the sea is the same astronomical catastrophe as Hamlet’s millstone rolling into the sea. Itis the wobble of the celestial sphere, and it ends the old world age and begins a new one [20], asimile fit for a king even of Babylon. I tentatively adopted this interpretation for both millstonesand looked for Babylon and its downfall in the engraving.

I did not have to look far. The engraving shows a city by the sea with ships in port.• Over the left-most pan of the scales in the engraving, through a small triangle framed by

ladder, house, and pan, we see a closely spaced system of fine parallel wavy flow-lines leadingcoming from behind the House and flowing leftward into a great Wave that looms high over thecity, about to crash down on it and wipe it out. How could I have overloooked the Wave?

Or it could merely be a massif far up the coast beyond the city, not over the city at all. Thenthe flow lines are geological strata, not hydrodynamical streamlines. We cannot tell whether theyare near the city from one perspective view, and the ladder carefully hides connecting clues thatmight force us to one view or the other. If the shore had a smooth C-shaped curve we would tendto see it as horizontal, so Durer has hidden part of the shoreline with the ladder and made a visiblepart of the shoreline rather straight and vertical on the canvas to enlist our propensity to recognizevertical lines. We can see this line either as a vertical wave or a horizontal shoreline due to theintrinsic ambiguity of perspective, which by this point I at last realized is always deliberate, thevery point of the engraving, hammered home again and again, until even I could not miss it. I donot insist on the face hiding in the vertical Wave/shoreline.

Now the engraving and the Biblical verse share Millstone, Angel, sea, Babylon, destruction,and melancholy. The connection we sketched in lightly between Durer’s millstone and Hamlet’s isnow less tenuous since it is reinforced by Revelation. I provisionally assume that the millstones ofRevelation, MELENCOLIA I, and the original Hamlet, separated by centuries and a continent, allrepresent the Celestial Sphere, and that Durer intended the engraving to refer to Revelation, theend of his Bible, as well as Genesis, the beginning. The Biblical verse would then refer to the endof the Age of Aries and the beginning of Pisces, the next after the writing of Revelation. If Durermeant the reference seriously, he referred to the end of the Age of Pisces the Fish. The fish is thestandard rebus of an acronym for the Jesus cultus. Since Durer painted himself not joining in theMourning over the Body of Jesus, he might have intended the Babylon of Revelation to refer tothe hierarchy of the Roman Church that he had already so punished in his Apocalypse. In 1514 hewas not yet disillusioned with the Reformation clergy as well, but Pirckheimer later attested thatboth clergies eventually disgusted him beyond his tolerance.

26

Figure 6: The Wave: enlarged detail

27

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25 Hexagram

Durer was a descriptive geometer, practiced in constructing views of bodies from all sides and inreconstructing a body from its views. I therefore looked at the Solid from all sides, as he likely did,while I was searching for his meaning.

• Viewed from the sky as a truncated rhombohedron, the ground plan of the Solid is a Shieldof David framed in a hexagon.

The top and bottom triangles of the Solid project into the two crossed triangles of the hexagram.Perhaps Durer truncated the cube, rather than some other regular solid, in order to create thishexagram. It would be anachronistic, however, to call this hexagram a Jewish star and to infer anyphilo-Semitism. Centuries earlier the Khazars of the Crimea had adopted the hexagram for theirflag when they adopted Judaism, but in 1514 Nuremberg the hexagram was still mainly a magicaldevice, an amulet, possibly referring to the Hebrews of the Bible but not to contemporary Jews.The Shield of David was called that because it was supposed to shield its bearer from evil spirits,and it was more often found in churches than synagogues. A two-footed hexagram like Durer’sground plan is seen on a German altar of Durer’s time. Some said the hexagram was the Seal ofSolomon, and others the pentagram. Traditions firmed later that gave Solomon’s star five pointsto David’s six, and stood Solomon’s on two points to David’s one. Not long after Durer’s death theJewish community of Prague was granted the privilege of a flag and chose the hexagram, possiblybecause of its association with the Khazars, and this continued its evolution from magic charm toJudaic symbol. Durer’s use was likely a Hebraism, not philo-Semitism.

The Hebraism of both Durer and Agrippa is conspicuous in their work. Durer admired Jeromeabove the other founding fathers for Jerome’s greater familiarity with the Hebrew Bible [19]. Inhis several drawings of Jerome studying Latin, Greek, and Hebrew Bibles, Durer invariably placedthe Hebrew Bible above the others, and higher is holier. The Shield of David is a further Hebraismconcealed in the Durer Solid.

Because we are so practiced at finding right angles in the world about us, had Durer truncateda true cube, we would have seen it as such in any view. Stretching the cube made the figure triplyambiguous, and demonstrated our inability to reach truth through our senses. Circumcising thecube Hebraicized it, connected Gateway III in Heaven with the Hebrew Bible. His stereotypes ofthe Jew and the Arab make it clear this hidden Shield of David is a Hebraism, not philo-Semitism,and this again aligns him with Agrippa.

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26 Space and time

Durer uses the vertical dimension on his plate in a traditional way. Higher is holier. Jacob’s angelsget to heaven by ladder. Angel and Boy have equal divinity because they have equal altitude. Thetools of artisans litter the ground because they are profane, while the instruments of science andmathematics hang carefully on the wall nearer to the sky because they are holier.

The horizontal dimension is cut neatly in half by the left-most wall of the House. The cutseparates the Gateway II in Heaven on the left from Gateway III on the right, and the Rationalfrom the Contemplative Faculties. Despite first appearances, all three Spheres of the cosmos andall four actors in the drama are to the left of the median wall, and the House, the Ladder, theComet, and most of the Moonbow are on the right with the Hourglass, Sundial, Bell, and MagicSquare.

The weighing scales are ambiguously located, hanging on the median wall itself. This is re-markably appropriate. Space and time were immaterial and therefore considered spiritual. Mass— “body” is the root meaning — was not defined in terms of them until four centuries later, byAlbert Einstein. Even Newton had to face the criticism of the Cartesians for taking mass to befundamental.

The division by the median plane is so meaningful and non-random that I will infer that itis Durer’s intention. The extreme point of view of the artist masks this clean cut by distributingelements of both halves of space on both halves of the paper.

All the Biblical and scientific elements are on the right-hand side of the median wall or hangfrom it. The crucible is no exception. If it were an instrument of alchemical science, as I thoughtat first, it would be elevated, but it is on the stone floor. Only perspective puts it high on thepaper, above all other tools; perspective and filial feelings. This is further indication that Doorlyread this correctly. This crucible is not an alchemical instrument but a remembrance of Durer’sgoldsmith father, who died in 1502. Its position to the far right gives us one point on the timeaxis. The House of God puts Biblical times on the right. This leaves little doubt in my mindabout Who is patiently holding the bell-rope off-stage right. Personally I find this non-depictionmore moving than the more famous anthopomorphic representation of God by Michelangelo in theCistine Chapel.

To reconcile the division by time and the division by Gateways, I must suppose that Durertook the occult philosophy to be later than Scripture. This certainly fits his personal experience ofthem, and the separation of the waters below from the waters above in Genesis might be the onlytrace of the three-sphere cosmology in the Bible.

The near eye of the Boy lies just on the center line of the paper, making him the centter ofattention, but he is to the left of the median plane too. Durer the artist-scientist is part of thecontemporary cosmology but takes Gateway in Heaven I, the subtext of the banderole, ignoring IIand III.

Perhaps the vanishing points tell us how to hang all three Meisterstiche as a triptych on threewalls of one room: Jerome of Gateway III on the wall to our right, the Knight of Gateway II on thewall before us, and MELENCOLIA, Gateway I, on the wall to our left. Then the time axis passesfrom right to left as in Hebrew writing.

Perspective provides a third dimension of depth into the scene along the line of sight. Durerseems to use this expressively too. On the right side our view ends in the House and on the leftthere is an infinite vista. It seems that perspective depth in the picture is the familiar metaphorfor depth of vision.

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27 Melancholy

The light and the bow and even the destruction of the city of evil are good news, cause for joy,and yet the picture is set in the gloom of night. Durer has even transformed the rainbow of thecommon understanding into a moonbow. The comet illuminates nothing. Only the moon behindus lights the scene and creates the shadows and the moonbow. What might have been a sunnyprophesy if Leonardo were speaking is dark as night.

While Leonardo and Durer were both mathematically ambitious, they differed importantly intheir outlook. Leonardo wrote confidently of “a complete knowledge of all the parts, which, whencombined, compose the totality of the thing which ought to be loved.” Leonardo was the rationaloptimist. Durer on the contrary said shortly before this engraving, “But what absolute beauty isI know not. Nobody knows it but God.” The darkness and the multiple ambiguities suggest thatafter all Durer did not expect humanity to attain the supreme mathesis that optimists like RamonLlull had expected and even claimed to have attained; and what is more, that he felt this as a greatbereavement. He expressed this pessimism in the engraving atmospherically, by omitting the doorof the House of God, and by numerous skillful demonstrations of ambivalence and relativity.

The most unambiguous aspect of this engraving is its ambiguity. Durer wants us to knowclearly that we cannot know anything clearly. The Solid is a truncated rhombohedron with verticalaxis or a slab tilted toward the horizon. There are three faces hidden in the Solid or none. Thecity is to be destroyed or not. The Angel is good or evil, and the Bat too. The house has windowsor not. The melancholy is inspired by the inaccessibility of the absolute, or by the influence ofSaturn, or because the Celestial Sphere is off its axis. What we see depends on us as well as whatwe look at. Most of the elements of this engraving that we have discussed here are intentionallyambiguous, either in meaning or form.

If we cannot believe what we see, what can we believe? This questioning occurs famouslyin the First Meditation of Descartes (1596-1650), who cites optical illusions as a reason for hismethod of universal doubt. MELENCOLIA I anticipates Cartesian doubt. And its response has amathematical element that anticipates the Mathesis Universalis of Descartes, but already correctsit by accepting the limitations to human knowledge.

The ideal of a complete mathematical theory of beauty lies on the same long line of distinguishedfantasies of mathematical wisdom as the number mysticism of Pythagoras and Plato, the Ars Magnaof Ramon Llull (whom Agrippa studied) and Giordano Bruno (who studied Llull and Agrippa),the vision of Mathesis Universalis that Descartes and Leibniz shared, and the Ars Combinatorixof Leibniz. Durer does not doubt the existence of absolute beauty but despairs of knowing it. TheBoy withdraws from both philosophy and astrology, ignores theory, and observes Nature.

Years after 1514, Durer like Agrippa explicitly abandoned the search for absolute truth andbeauty as futile and hopeless, crying out,

“The lie is in our understanding, and darkness is so firmly entrenched in our mind thateven our groping will fail” [Panofsky 1971].

In 1517 Durer was still optimistic enough to join with Luther, but the engraving already expressesdespair in 1514. The date 1514 for the engraving seems reliable, since several studies for thisengraving also bear that date. I conclude that Durer expressed his pessimsim about attainingabsolute truth in his engraving years before he put it in writing. First he drew it, then he wrote it.Because I write things first, it took me some time to realize that probably Durer drew things first.

The night of the engraving is our benighted ignorance. Its darkness is the darkness “firmlyintrenched in our mind.” The mystery of the Solid, crafted to be insoluble, is one of many metaphorsin the engraving for our inability to see the Absolute Truth from our limited perspective. His chosen

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Gateway in Heaven is Natural Philosophy, in spite of its limitations. In acknowledging the limits ofhuman knowledge, Durer went a step beyond the more optimistic Leonardo toward modernity. AsPanofsky and Doorly said, the melancholy is Durer’s over the limit of human art and understanding.And as Yates said, it is also the Saturnian melancholy that inspires artistic creation.

The posthumous publications of Durer on measurement and on human proportions show that henever quit searching for a mathematical theory of beauty and truth based on measurement ratherthan abstract speculation, but it became more relativistic, to the degree of including a variablehorizontal scale-factor transformation in order to represent human beauty of the lean, average, andplump variety as the reference frame of the artist requires.

28 Skull

Now at last we can cope with the most puzzling problem of all, that of the Skull. The Knight andJerome have skulls. Even the melancholy Dane has poor Yorick. Yet Durer conspicuously omitsthe Skull from this companion work, despite the title MELENCOLIA I . This is amazing, so wemust understand what Durer meant by it.

I propose that the Skull is absent for the same reason that the Angel is smiling. As the god ofthis engraving, Durer is subtle but not malicious. It seems that the melancholia of this engravingis not the classical one but the Agrippan as interpreted by Durer, all creative frenzy and pain,with no element of sadness. It does not lead to death but to its transcendent opposite, a form ofimmortality possessed by the engraving itself. Even later, when Durer later lamented the darknessin our minds, it must be the Agrippan melancholia, not depression, that he records. On this point,Yates was not merely right but very right.

Agrippa reversed the primary meaning of Melancholia just as Horapollo reversed Dog, Bat,and Serpent. By using the ephemeral and counterintuitive languages of Agrippa and Horapollo,Durer breaks our ordinary lines of interpretation, making the work surreal but also contributingto its remarkable endurance. The Skull would have misled the viewer and buried the anagram byvalidating its cover story, as the angelic smile and the missing Skull lead the viewer to attack theanagram.

Actually no one in the engraving is merely sad. The overall gloom of the engraving is theAgrippan melancholia, not depression but creative madness; probably of the Boy, and if I mayventure an interpretation that I cannot quite confidently attribute to Durer but which is consistentwith his self-glorification, perhaps of the Creator.

29 Three Fools

The triplets of the Angel and of the Solid, both in the theological realm, would likely be read byviewers as references to the Doctrine of the Trinity. It is not clear at first whether Durer meanstriplication to imply divinity. The fool provides a useful indicator for this. If the fool, D. Angel,is a triplet, Durer surely does not intend triplets to be more divine than other people. To controlthis theory I returned to the engraving and looked at the fool from another angle.

• It was then easy to see faces E. and F. Angel superimposed on D. Angel. D. in profile looksto our right, while E. in full face looks slightly to our left.

• F. Angel in profile looks upward and to our left. F., unlike D. and E. Angel, has a bodyfacing us, a complete arm with a mittenlike hand at his side, and much of a leg, and seems to havethe purse of the Angel.

All three faces can be seen one at a time in the triply ambiguous E. of Figure 29.

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Figure 8: D., E., and F. Angel, rotated and scaled details

This set me to searching for ambiguous triplets and so led me at once to C. Solid, alreadymentioned.

30 Purse and Keys

Durer’s Madonna by the Wall and his Frau Venus in the Ship of Fools of Brant both wear keys andpurse, so by Shannnon’s principle these carrry little information. What is unusual about them isthat they are apparently the only elements in the engraving whose meaning Durer made public,and while the Madonna and Venus wear their purses and keys at the waist, the Angel drags herson the ground.

If one did not know about the existence of fish it would be hard to understand why a manshould dangle a string in the water. It seems a poor way to wash ones thread, get a drink, oranchor a boat.

Similarly it is hard to understand the long purse-strings and key-string if one has not seen thefools. But we have, so there is an obvious explanation: She is fishing for fools. And she seems tohave caught F. Angel with her purse. This also explains Durer’s explanation: He wished to makeit clear that the fools were lured by power and wealth.

31 Unitarianism

I looked in vain for such triple ambiguities in other figures of this engraving or in other works ofDurer. The fool of Frau Venus too is single, not three-faced. The triplets are restricted to theIntellectual Sphere and the Rational Faculty of the theologians of this engraving. I infer tentativelythat the triplets refer to the Doctrine of the Trinity. If so, Durer modifies the doctrine. He showsone individual with three aspects, not three entities and one at the same time.

This makes Durer express something more unitarian than trinitarian. This is not completelyabsurd. Unitarianism was in the air. Michael Servetus, traveling in Germany, published his historicunitarian treatise De trinitatis erroribus, perhaps the first, in 1531 at Hagenau, 100 km fromNuremberg. Gellerd proposes that Agrippa provided the Transylvanian Unitarian Francis David

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with some of his epistemology [10]. Panofsky and Yates make it clear that Agrippa influencedDurer’s thinking. Agrippa’s Hebraism might have led him to reconsider the Trinitarian doctrine. Itseems possible that the charismatic Agrippa inspired Durer too to express the unitarian tendencieswe find in MELENCOLIA I.

Once we look for unitarianism in the engraving, it is easy enough to find. The Deity is shownhere as the single point of light that is also the comet. In his Landauer altarpiece, on the con-trary, Durer presents each member of the Trinity as a fully conceived independent entity in itself,represented by a dove, an old man, and a young man on a cross, and indicates their unity bythe convention of arranging them in an exactly vertical straight line. In MELENCOLIA I insteadhe presents a relativistic trinity more natural to a descriptive geometer. Each of the members ofsuch a triplet is not a different and separate individual but a different aspect of one unity. Thisrepresentation captures the impossibility of seeing three separate things where there is only one ina way that the Landauer altarpiece cannot.

Durer’s heresy is at least threefold if I see it correctly. First, he shows the same ContemplativeFaculty that serves Christian theologians as serving Jews and Muslims too. Secondly, as Boyhe declines the traditional Gateways II and III to knowledge, the Contemplative and RationalFaculties, for the way of the Imaginative Faculty, the Gateway I of artists and artisans and naturalphilosophers. “All that is in the mind comes through the senses.” Finally, he suggests that thethree entities of the Trinity are One seen from three viewpoints.

This engraving instantiates what Yates said about this transitional moment in the evolution ofscience. No sharp line yet separated the cultures of the old magic, the new natural philosophy, andreligion. This is the brief epoch when the closest thing to physics was natural magic, the naturalwas still emerging from the magic, and many trusted the Bible more than their senses.

Durer shows a single divinity connected to the human world through the realm of pure mathe-matics. He demonstrates repeatedly that relativity deprives us of the absolute, that interpretationis always multiply ambiguous. He shows how this inspires a melancholy that takes the form of afrenzy of artistic creation, such as the one that we see before us.

It is, however, no longer clear that the hiddenness in this engraving is entirely self-protection.The multiple ambiguity does not mask the message, as I thought at the start, it is the message.“The lie is in our understanding, and darkness is so firmly entrenched in our mind that even ourgroping will fail.” This approaches a postmodern position. Durer does not give up his belief thatthe world has a unique meaning, but he accepts that it is known only to God and cannot ever beknown to humanity.

Certainly this engraving does not have a unique meaning, and that is a central part of itsintention. This is a thoroughly postmodern work of art done in the 16th century. It would beunderstandable if it has had to wait until the postmodern era to be read; or to be re-invented, asthe case may be.

32 Disclaimer II

Views expressed by this engraving are not necessarily those of Albrecht Durer.

33 Acknowledgments

I thank the a. d. for starting me on this study as I mentioned in §3. It would have been impossiblewithout the studies on MELENCOLIA I by Erwin Panofsky and by Francis Yates and workson Agrippa by Judit Gellerd and by Charles Nauert. I have also drawn on the study of Hamlet’s

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millstone by Giorgio de Santillana and Hertha von Dechend, of MELENCOLIA I by Patrick Doorly,and of Hamlet by Stephen Greenblatt.

The heraldry dictionary of Cecil Wade helped me to read Durer’s coat-of-arms. I benefitedfrom gematria on the website of the Aiwaz Gallery, though our arithmetic and interpretation differ.My thanks go to Roy Skodnick for introducing me to the work of Yates; to Prof. Heinrich Sallerfor the print of MELENCOLIA I that triggered this process; to Rabbi Mario Karpuj for remindingme that Jacob’s ladder is the Biblical gate of heaven; to Prof. Shalom Goldman for stimulating,informative, and encouraging discussions, the reference to Yates on Durer, and the meaning of theMagen David in the Renaissance; to Dr. Basimah Khulusi for generous helpful discussions, modelsof the Durer Solid, the asymmetric interpretation of the Solid, and the reference to Von Buolschafft;to Danny Lunsford for refining the Schreiber construction of Durer’s Solid and for the reference tohttp://meteoros.de/halo/halo1.htm; to Carla Singer for her expertise in art history, including thereference to Mantegna; and to Shlomit Ritz Finkelstein, for lively discussions, help with Torah andDescartes, and useful expository suggestions. An earlier stage of this study was published in TheSt. Ann’s Review, whose reviewer improved the paper.

References

[1] Agrippa von Nettesheim, Heinrich Cornelius: 1531. Of Occult Philosophy, digital edition, ed.Joseph H. Peterson, http://www.esotericarchives.com/agrippa/agrippa1.html. Based on thetranslation of ”J.F.”, printed by Moule, London (1651).

[2] Agrippa von Nettesheim, Heinrich Cornelius: 1530. De incertitudine & vanitate scientiarum& artium atque excellentia verbi Dei declamatio. Antverpiae : Grapheus, 1530. OCLC:22080834.

[3] Brant, Sebastian. Narrenschiff (Ship of Fools). Illustrated by A. Durer. Basel, 1494 (firstedition). Translation by Alexander Barclay: Ship of Fools, Pynson 1509. Rhyming trans-lation by E. H. Zeydel: Ship of Fools. Columbia University Press, 1944, reprinted byDover Publishing Co., 1962. Espeically Chapter 13, On Amours. As of June 1, 2005, thesite http://www.loc.gov/exhibits/treasures/images/wb0013.1s.jpg shows the woodcut VonBuolschafft mentioned in §14.

[4] Doorly, Patrick. Durer’s Melencolia I : Plato’s Abandoned Search for the Beautiful. The ArtBulletin88, 255 (2004)

[5] Durer, Albrecht. The works of Durer mentioned in this paper are reproduced in [19] and arefound online in internet galleries. Such presentations are good enough to verify the featuresmentioned in this paper, but when we are verifying the secondary images, the beat betweenthe raster of the presentation and the shading of Durer combines with our propensity tosee faces and creates artifacts that should not be taken seriously. Commercially available artprints create no such problems.

[6] Durer, Albrecht. Vnderweysung der Messung mit dem Zirckel un Richtscheyt in Linien, Ebnenund Gantzen Corporen, Vol. 1-4. Nuremberg 1525. Republished with English translation ofWalter L. Strauss, revised, on compact disk. Editor, anonymous. Commentator, David Price.Octavo Editions, Oakland CA, 2003.

[7] Erasmus, Desiderius, of Rotterdam. In Praise of Folly. Translated by John Wilson. DoverThrift Editions. 1991

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[8] Finkelstein, David Ritz. Melencolia I: Durer Decoded, The St. Ann’s Review , 2004.

[9] Galileo Galilei, Dialogue on the Great World Systems. The Salisbury translation. Revised,annnotated, and with an Introduction by G. de Santillana. University of Chicago Press (1953)

[10] Gellerd, J. Francis David’s Epistemological Borrowings from Henry Cornelius Agrippa ofNettesheim. w3.enternet.hu/sandor64/cffr/papers/agrippa.htm

[11] Goldman, Shalom. God’s Sacred Tongue: Hebrew and The American Imagination; alsohttp://uncpress.unc.edu/booksT-6957.html. For the concept of Hebraicism.

[12] Greenblatt, Stephen. Will in the World: How Shakespeare Became Shakespeare. Norton(2004). On Hamnet and Hamlet.

[13] Heckscher, William S. Melancholia (1541). An Essay in the Rhetoric of Description by JoachimCamerarius. In: Frank Baron, ed., Joachim Camerarius (1500-1574). Beitrge zur Geschichtedes Humanismus im Zeitalter der Reformation. Essays on the History of Humanism Duringthe Reformation. Humanistische Bibliothek, Abhandlungen 24 (Munich, 1978): 32-103

[14] Horapollo Niliaci. The Hieroglyphics of Horapollo. Boas, George, translation and introduction.Princeton University Press 1993.

[15] Khulusi, Basimah. Personal communication.

[16] Lunsford, Daniel. Personal communications concerning the Durer Solid and Ratio (2004).

[17] MacGillavry, C. The polyhedron in A. Durer’s engraving Melencolia I. Nederl. Akad. Weten-schap. Proc. Series B 84, 287-232 (1981).

[18] Nauert, Jr., C. G. Agrippa and the Crisis of Renaissance Thought University of Illinois Press,Urbana, 1965.

[19] Panofsky, Erwin. The Life and Art of Albrecht Durer . Princeton University Press, 1971. Thestandard work. It reproduces the Durer works mentioned here.

[20] Santillana, Giorgio de, and Hertha von Dechend. Hamlet’s Mill: An Essay on Myth and theFrame of Time. Gambit, 1969. Argumentation for the astronomical interpretation of all myth.

[21] Peter Schreiber. “A New Hypothesis on Durer’s Enigmatic Polyhedron in His Copper En-graving Melancholia I.” Historia Mathematica 26, 369-377, 1999.

[22] Schuster, Peter-Klaus. MELENCOLIA I: Durers Denkbild. Berlin: Gebr. Mann Verlag, 1991.

[23] W. Cecil Wade. The Symbolisms of Heraldry or A Treatise on the Meanings and Derivationsof Armorial Bearings. London: George Redway, 1898.

[24] Weitzel. H. A further hypothesis on the polyhedron of A. Durer’s engraving Melencolia I.Historia Mathematica 31 (2004)

[25] Frances A. Yates. The Occult Philosophy in the Elizabethan Age. London ; Boston : Routledge& K. Paul, 1979. Especially chapter 6.

[26] http://gallery.euroweb.hu/html/d/durer/ shows all the Durer works I mention.

[27] http://mathworld.wolfram.com/DuerersSolid.html describes and names the Durer Solid.

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[28] http://meteoros.de/halo/halo1.htm is a remarkable gallery of portrayals of heavenly phenom-ena like moonbows and moon halos, including several from Durer’s time.

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Contents

1 Disclaimer I 2

2 Background 2

3 Hidden faces 5

4 Coat-of-arms 7

5 Numerology 8

6 Bat 10

7 Anagram 12

8 I 13

9 Boy 14

10 Crime 14

11 Ladder 15

12 House 15

13 Door 16

14 Angel 16

15 Compasses 17

16 Arab and Jews 17

17 Fool 20

18 Globe 21

19 Solid 21

20 Millstone 22

21 Moonbow 23

22 Comet 24

23 Dog 24

24 Babylon 25

25 Hexagram 28

26 Space and time 29

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27 Melancholy 30

28 Skull 31

29 Three Fools 31

30 Purse and Keys 32

31 Unitarianism 32

32 Disclaimer II 33

33 Acknowledgments 33

38