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MELENCOLIA I :

The physics of Albrecht Durer

David.Finkelstein ∗

February 28, 2006

Abstract

Besides its conspicuous early Renaissance scientific instruments and tools, Durer’s engraving ME-LENCOLIA I also contains numerous apparently unreported concealments. Their decryption re-veals heresies concealed in the work. The main one is encoded in the motto that has named thepiece: Natural Philosophy is superior to Mathematical Philosophy and Theological Philosophy asa way to knowledge. Numerous secondary images demonstrate the relativity and ambiguity ofperception and indicate that Durer like Newton was a secret unitarian.

1 Background

Many details of the MELENCOLIA I (1514) of Albrecht Durer (1471-1528) seem to have beenoverlooked in recent centuries, and the meaning is in the details [6]. I report some of them here.My interpretations of some of these led to the discovery of others, giving me enough confidence toalso make a stab at interpreting what I found. Mistakes in my interpretation undoubtedly existbut need not cast doubt on the observations themselves, which I bullet for reference. A betterinterpretation ought still to take these data into account. I have made an effort not to projectmy own views across half a millennium onto Durer. The point is not what I see in the engravingbut what Durer expected his viewers to see. The philosophical views I read in the engraving andpresent below are not necessarily my own. They are selected from or extend those expressed inwriting by Durer and his acquaintances.

Anyone can see the impressive angel sitting in the right foreground and the small winged serpentflying in the left background, like virtue and evil personified. A winged boy, or putto, a famil-iar convention of Renaissance art, perches between them on an up-turned millstone and intentlyinscribes a block. Paraphernalia of the arts and sciences clutter the scene in a significant pattern.

The art historian Erwin Panofsky [19] saw the engraving as intensely autobiographical and readits melancholy as Durer’s own frustration at the gap between artistic and divine creation. FrancesYates, historian of the Hermetic tradition, took its melancholia to be one that inspired creativity,

∗School of Physics, Georgia Institute of Technology. df4@mail.gatech.edu

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not mere depression, and read the engraving as a declaration of the harmony between microcosmand macrocosm: as above, so below [24]. The art historian Patrick Doorly sees it as an illustrationfor Plato’s Greater Hippias, a dialogue on beauty [4], with the angel melancholy because the artistcannot define absolute beauty. They pass over more than a score of individual concealments andparadoxes in the engraving: a key anagram, rebuses, Cabalistic gematria, puns, a Star of David,over a dozen hidden faces, two Jews blowing the ram’s horn, the entire figure of a kneeling warrior inArab dress, and even the downfall of Babylon. There are references to Genesis, Revelation, Cabala,the Nicean Creed, and the Durer family history. There are numerous demonstrations of the limitsto human perception, and capital crimes that explains much of the concealment. These illuminateDurer’s views on art, science, philosophy, and religion, and support some of the interpretations ofothers that I have just cited, but not all. The interpretations I cite do not mention the main pointof the engraving.

One element of this work is relativity in the broad sense, the study of how what we see depends onour viewpoint. Durer uses perspective to construct several examples of relativity and ambiguity.It is not anachronistic to attribute this idea to Durer, since he is one of the creators of descriptivegeometry, which is precisely a theory of how the same object looks from different points of view,and is probably the first mathematical theory of relativity in the broad sense. He is significant inthe history of the perspective technique, which the Romans and Greeks practiced and the Churchsuppressed as deceptive. Durer carried “the secret of perspective,” as he called it, from Italy tonorthern Europe, and some say that his book on the subject founded the mathematical theory ofdescriptive or projective geometry, though this developed over centuries with growing rigor.

Perspective, special relativity, and general relativity all rest on an understanding of visual percep-tion and light, and so there are family resemblances among them. The apparent foreshortening oflimbs 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 trajectories in space-time. Theyheld that in perception one such pyramid diverges from the point observed, like a future light cone,and another converges to the the observing eye, like a past light cone. The perspective transfor-mations of the graphic plane that we carry out when we shift our viewpoint are special cases of thetransformations of space-time used later in modern relativity. Both are represented by matricesor tables of two rows and columns, with real or complex number entries respectively. Projectivegeometry is a playground for relativists and made Durer an eminent mathematician of his time.

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

1. In the 2nd-4th century CE, a certain Horapollo wrote Hieroglyphica, a spurious rendering ofthe Egyptian hieiroglyphic symbols into Greek, centuries before the Rosetta Stone made a properreading possible. This figures in MELENCOLAI I too.

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

3. Durer illustrated Pirckheimer’s Hieroglyphica.

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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 intoan allegorical fantasy of dogs, goats, harpies, cocks, snakes, scepters and the like, making up thecenterpiece of the Triumphal Arch [19].

The code-book and the clear message were also published, so the Triumphal Arch was an attemptto communicate, not conceal. Neither a plain text nor a code-book for MELENCOLIA I has beenpassed down to us, however, and we must resort to a kind of cryptography to deduce them. Onemight think that Durer had something to hide. The message I find confirms this suspicion.

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

There are indeed well-known limits to Durer’s knowledge. Most famously, the astronomer Keplerpointed out that in Durer’s work on measurement [6] he drew the ellipse produced by cutting a conewith a plane as egg-shaped, with its small end near the tip of the cone. I found a striking error inoptics in this engraving. Nevertheless, to catch his meanings we must not underestimate his powerwith the burin, his skill in perspective, his devotion to precise language, and his acute observationsof nature. We should remember that he was a humanist, knowledgable in mathematics, poetry,and antiquity 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 of noneof these when I began this study, but they are indispensable for reading the engraving so I willstate them briefly before we set out into the world of MELENCOLIA I .

Like Leonardo, Durer began his training under a goldsmith, namely his father, a Hungarian immi-grant in Nuremberg. By the year on the engraving, he had made two historic visits to Venice, wherehe studied perspective and human proportion and developed a passion for geometry. Eventuallyhe extended Leonardo’s work on beauty in a relativistic direction, using projective transformationsto allow for different standards of beauty.

Durer brings us back to many beginnings. He did not have to please art critics to succeed becausethey had not yet been invented. The first published art criticism in history was actually the praiseof this very engraving by Joachim Camerarius (1500-1574), a young friend of Durer who watchedhim at work, translated his book on human proportions from German into Latin, and at the endwrote his eulogy, in which he declared that although Durer’s art was great, it was the least of hisaccomplishments.

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 and thefirst known specific landscape. He invented etching and some perspective drawing machines. Heseems to have invented the modern artist, being the first to charge for his special talent instead of

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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.

Number lifted the medieval quadrivium of arithmetic, geometry, astronomy, and music out of themundane and into the divine. So Leonardo, Durer, and others sought to found art and beauty tooon number. The concept of the Golden Rectangle, Leonardo’s famous drawing of a man inscribedin a circle, and Durer’s books on descriptive geometry and body proportions are mementos of thatsearch.

2 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 posted a copy with a thin black outline of a face drawn onthe polyhedron in the engraving sometimes called “Durer’s Solid.” I chanced on this once and havenever found it again. I went to my own copy and found at least the implication of a face that I hadoverlooked for decades. If we step back several paces from a good print and look at the shading ofthe front face of the Solid for a minute or so, we can see a woman’s face in profile. Or perhaps weconstruct it.

There are three faces in the Solid and which you see depends on how you hold your head relativeto the print. I will call these people A., B., and C. Solid.

• 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 noseof C. Solid when we turn the picture on its left edge.

I found C. Solid, smaller and more caricaturish than his two companions, some time after A. andB., but not by some careful search. I predicted his existence as described in what follows, and thenfound him at once. The existence of this triplet may have been known in the time of Durer, but Ido 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 subtleand draw much on our own perceptive process. We are not certain whether we see them or createthem. I thought that the internet outline of A. Solid was an accidenal product of scanning ordigitization until I realized that most of the left edge of her face has almost no physical correlatein the engraving. Durer compels us to see it, when we look for it, by supplying chin and foreheadlines and eye and nostril shadows, but we ourselves create the rest of the face as no point-by-pointscanning process could. Part of the art behind these hidden faces is that they are ambiguous,deniable, and lie at the edge of perception. They are joint productions of the hand of the artistand the mind of the viewer, demonstrating Durer’s mastery of both media. When we see one ofthe faces in this triplet we do not see the other two, because of elements they share. For example,A. and B. Solid share an eye-pit.

Ambiguous faces are frequently reproduced, but I have never seen a triply ambiguous one beforethese, 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 of

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sight. 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 distracting low-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 anonymous donor saw one ofthe faces before me, and who knows how many of the others. I cannot find hidden faces in mostof Durer’s works but only in some. I verify the faces I see with other viewers. Ultimately I knowthese faces are not accidental random patterns in much the same way that you know that the wordsyou are looking at now are not random selections of letters: They make more sense than randompatterns do.

On the other hand, in the end I cannot count the faces of MELENCOLIA I exactly. I have named10 faces that are clearly intended, others are clearly unintended. nut there are a few intermediatecases that cannot be settled. Any noise, such as that produced by digitization, distorts or eliminatesthe intended faces and creates accidental ones. Our propensity to find faces is so strong that nowI wonder why Schiaperelli saw canals on Mars and not a face as in the Moon.

• Viewers of Durer’s time may have seen most of these faces at once, alerted by the many hiddenfaces in earlier works of Durer. His watercolor View of Arco has a cliff that clearly is also a famousscowling 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 byDeath (page 201 of [19]) has a remarkable secondary image of a male face formed of and loomingover both the lady and the gentleman in question. Their heads are his eyes, her arm is his mouth.In this picture Durer does not use the watermark technique but a kind of sleight-of-hand. The linesof the hidden face are all drawn in high contrast but other more expected forms divert us from thecovert one.

• In the 1494 drawing Orfeus der erst Pusaner (Orpheus the First Pederast) or The Death ofOrpheus, 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, and theDevil . Durer portrays the Knight as spectacularly oblivious to Death and the Devil, though in allother such encounters in the work of Durer, the knight engages evil in mortal combat. It is one thingto overlook an angel and a small demon as the Boy in MELENCOLIA I does; another to ignoreDeath and the Devil themselves as this Knight does. The Knight is generally said to representthe ideal Christian Soldier described by Erasmus (1466?-1536), who was a revered acquaintance ofDurer. So virtuous was the Christian Soldier that he was blind to the evil about him. On the otherhand, Durer wrote a fervent plea to Erasmus, still extant though perhaps never sent, to join withhim and Luther in battle against the “Cave of Hell” and die a martyr’s death. Erasmus remainedwithin the Catholic church and worked for reform instead of applying for martyrdom. For Durerand his viewers, the unseeing Knight may also have represented Erasmus himself. In any case Durerpays tribute to the Knight in a marvellous way that others have noted:

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

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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.

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

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 mother andfather and so perhaps they fit well into Panofsky’s theory that this engraving is autobiographical.If so, then C. Solid might be Durer the son. In his last portrait of his mother, Durer shows herdivergent strabismus, a family trait that he inherited known as the Durereblick . The resolution ofDurer’s images may be too low for certainty but A. and C. Solid can be seen as strabismic.

3 Rebuses

There are two rebuses in the engraving, and two in the Durer coat-of-arms. These are clues to thecentral mystery.

The famous Durer monogram AD appears under the date 1514 on the edge of the stone slab underthe angel. It seems to pun on Anno Domini. The capital A is also a rebus. To read it we needsome family background.

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 Ajts in Hungary to finally resettle in Nuremberg. Ajto is Hungarian for door, or Thur in theGerman of the time, so Albrecht of Ajts 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. His family coat-of-arms tells us much about his self-conception andopens the doors of this work.

The coat-of-arms is topped with conventional symbols found in any dictionary of heraldry. Eaglewings mean fame and glory. A closed helmet in full profile declares Durer an esquire, the lowestestate. A blackamoor at the top stands for heroic action by an ancestor in the Crusades. Its racismwas common in the time and place, and recurs in MELENCOLIA I too. Below the cliches is ashield that carries specific family symbols. I point out two that are obvious.

• 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.

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It is easy to read this too. Coats of arms usually show off the pride of the family. The cloud shouldstand for some great personal accomplishment. To proclaim what he was most famous for, Durercould simply have drawn his burin across the shield. But the burin is also a mere artisan’s tool,and Durer’s life mission was to sanctify the artist above the artisan. He presumably knew thatcaelum, the Latin for burin, is also an ordinary word for “the sky, ” “heaven,” and “the heavens,”with the same root as our “celestial.” Durer seized this heaven-sent opportunity to sanctify himselfand his art.

• The cloud in the Durer coat-of-arms is a rebus that puts his family gates “at Heaven” (caelo)and at the same time says “I engrave” (caelo).

The dative case of the noun is the first person present tense of the verb. The cloud is our secondrebus.

”Gate to Heaven,” another reading of the coat-of-arms, became a common metaphor for the RomanChurch itself. An educated person of the time might take this claim of divinity not as a sacrilegiousdisplay of ego but as a pious reaffirmation that we are made in the image of God. We will meetthe Durer gateway in the sky again in MELENCOLIA I .

With Durer’s coat-of-arms in mind, one now sees the oddly-formed A in his monogram as a gatewayas well as a letter. The legs of the gateway are the uprights of the A, the lintel connects them atthe top, and the stiffener just beneath the lintel is the cross-bar of the A. The cross-bar of Durer’sA is peculiarly high for an A but for proper for a gateway. The polymorphic A-gate is our thirdrebus for the Durer family, the first of several gates in the picture important for its interpretation.

4 Gematria

The matrix of numbers set into the masonry wall like a window lattice is a magic square, meaningthat it is made of consecutive numbers starting from 1 without omission or repetition, and everyrow, every column, and both main diagonals add up to the same number. Here that number is 34.This magic square is called a gnomon magic square because in addition its four quadrants, its fourcorners, and its central tetrad add up to the same number. In addition, the sum of any pair ofnumbers symmetric about the center is 17.

The idea of the magic square and much of the cosmology in this engraving may have come to Durerfrom Heinrich Cornelius Agrippa von Nettesheim (1486-1535), a prolific and controversial figure inthe Florentine Neo-Platonist school, the original of Dr. Faust. Agrippa wandered Europe seekingroyal support, never settling in one city as much as four years. He visited Nuremberg in 1510with the manuscript of his De Occulta Philosophia, a compendium of astrology, alchemy, Cabala,and absurd propositions of biology, physics, and medicine, for example, that diamond destroys themagnetism of lodestone. He said that he studied the occult philosophy because it traced back toa secret divine revelation given at the same time as the open revelation of the Torah of Moses. Init Agrippa assigns a magic square to each of the seven “planets” then known, in both Arabic andHebrew numerals, in the ancient order Saturn, Jupiter, Mars, Sun, Venus, Mercury, and Moon.This is the order of their periods, since the geocentric system gave to the Sun the heliocentricperiod of the Earth. These magic 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 Saturn hasa 3-by-3 table, Jupiter a 4-by-4 table and so on to the 9-by-9 table of the Moon. The number of

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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 prescribedwearing 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 number andthat therefore every Hebrew word has a number, the sum of its letters. He found reassuring wordsin Jupiter’s Table by gematria. Some say Durer borrowed Jupiter’s Table from Agrippa [19, 24].

• Here are the two magic squares:

4 14 15 1

9 7 6 12

5 11 10 8

16 2 3 13

16 3 2 13

5 10 11 12

9 6 7 12

4 15 14 1

Agrippa Durer

They have no row or column in common and Drer’s table lacks Agrippa’s protective words.

Durer borrowed the idea of the Table, perhaps, but he performed non-trivial permutations —not mere reflections of the whole table — that left it both magic and gnomon but changed thebottom line to the date 1514 of the engraving flanked by the Latin gematria for D A, matching themonogram and date at the bottom of the engraving. Durer used the Table to perpetuate his nameand the date, not to ward off melancholia. That he saw and used this combinatorial possibility tellsus more about his vision and alerted me to other uses of the permutation principle and gematriain MELENCOLIA I .

I found both. Pursuing the autobiography hypothesis, I computed the Latin gematria for “AlbrechtDurer,” 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 the sum, since “v” would also replace“u” as letter 21. The summation of the whole table is 136 and is not under Durer’s control. Hewould have separated the 1 from the rest of the table to make the sum equal 135 or “AlbrechtDurer.” This prediction we can verify.

• One wing of the angel has brushed the 1 and no other numeral in Durer’s Table, blessing it,enlarging it, and raising it in a way that violates the conventions of realism. I did not see theenlarged 1 until I did the gematria and looked for it. The difference looks small to me, but Durerhad extraordinary vision for detail.

It is a possibility that this holy 1 represents the Holy One. It is the 1×1 magic square that Agrippaleft out. Its size further distinguishes Durer’s Table from Jupiter’s. The magic square thus carriesDurer 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 egotism after we see other instances. We have noticed oneself-deification in his monogram and another in his coat-of-arms. For another example, see Durer’spainting in the Alte Pinakothek of the Mourning over the Body of Jesus. The mourners form atriangle. The base of the triangle is the body of Jesus, and the body of the triangle is made up ofmourners, looking down as symbolizes mournful melancholy. But the apex is Durer himself, lookingup.

One moral of the magic square is so obvious that it is important to point it out again: Things arenot what they seem in this engraving. What we see depends on our point of view.

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5 Bat

The chimera with head of mouse, wings of bat, and tail of serpent flies at us out of the engraving.In earlier works Durer gives bat-wings and snake-tail to minions of the devil such as the dragonof St. George, suggesting that this is a demon, but Agrippa assigns the color black, serpents, andbats to the planet Saturn, so this could merely be a most Saturnian messenger, and the ambiguitycould well be intentional. There are so many Agrippan elements in the engraving that I adopt theSaturnian interpretation and call this messenger the Bat. It carries the banner “MELENCOLIA I.”This clearly refers to the ancient Greek humor of melancholia, and likely to Agrippa’s revision ofthe teaching of the Florentine Neo-Platonist Marsilio Ficino (1433-1499). I summarize the languageof the Spheres and Humors that seems to be used in this engraving [19, 24], capitalizing terms usedin a now archaic sense. [1]. For a millenium and more, the West divided the world into threeconcentric Worlds of Spheres, the Terrestrial or Elemental, the Celestial, and the Supercelestial,Intellectual, or Spiritual. They were studied in Natural, Mathematical, and Theological Philosophyrespectively [1].

Part of the Renaissance was the return to the subtler understanding of the Greeks, many of whomknew that the Earth moved. In this time of transition, I will suppose, Da Vinci and Durer used amore physical cosmology in their science, while in their art they still used the naive Neo-Platoniccosmology of geocentric worlds because that was still the language that they shared with theirviewers.

The Terrestrial Sphere is our normal abode, angels and Platonic Ideas inhabit the IntellectualSphere, and between the two lies the Celestial Sphere, made up of one sphere for the stars andmany others to work the planets. Ptolemy used over 90 spheres in all, which are all lumped intothe Celestial Sphere. Neo-Platonists analogously divided the human psyche into three Mentalitiesor Faculties specially suited to the corresponding Spheres [24]:

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 know whenstars rise and set and thus through astrology it allows statesmen to know when kingdoms rise andfall.

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

Their names vary considerably. Agrippa called them idolus, ratio, and mens respectively [18]. Ficinoand Agrippa also kept the ancient Greek psychology of the four humors, one for each Element: bloodfor Air and the sanguinary temperament, yellow bile or choler for Fire and the bilious temperament,phlegm for Water and the phlegmatic temperament, and black bile or melancholia for Earth andacute depression. 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 was a revolutionary astrologist while he practiced astrology and even after he rejected it

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[18]. He promoted Saturn from the lowest status among the planets to the highest, empowering himto inspire all three Faculties to their peak creativity, and transferred the demi-godlike geometers —“earth measurers,” after all — to the Saturnians. According to Agrippa, the truly inspired artisthas a creative melancholia acting on the Imaginative Faculty [24]. Durer was a revolutionary artist,aiming to promote art and artists from the lowest to the highest rank. Since artists are assignedto the Terrestrial Sphere of Saturn, this must have created a natural alliance between Agrippa andDurer.

Moreover both Agrippa and Durer were humanists and admired Erasmus . Agrippa was a Hebraist,advocated simplicity and Bible-reading. These became part of both the Reformation and theUnitarian movement, but Agrippa himself remained in the Catholic Church. In the last phase ofhis life he hated astrology and was skeptical about all access to truth accept faith in the Bible. Heoffers us three increasingly reliable gateways to knowledge of God: the Book of Nature, the Bookof the Law, and the Book of the Gospel. In the Book of Nature we see merely the creatures ofGod and use reason to deduce the existence and nature of God, but original sin clouds our soulsand makes our reason unreliable. Reason even led to the original sin when Eve reasoned with theserpent. The Book of the Law and the Book of the Gospel, the two books of the Christian Bible,are the result of revelation and require only faith. His pessimistic work on the uncertainty andvanity of the sciences, and the superiority of faith in Scriptures and divine revelation to all the artsand sciences, including astrology, was written after his optimistic Occult Philosophy but appearedin print two years earlier. His self-contradiction is also built into MELENCOLIA I .

Panofsky and Yates saw many Saturnian melancholic elements in the engraving, taking its melan-cholia in either the more depressive sense of Ficino or the more creative sense of Agrippa. Theytake it that Durer accepted Agrippa’s philosophy at least for the purpose of this engraving. In-deed, much in the engraving supports this view or is at least consistent with it. Nevertheless whileErasmus, Agrippa and Durer all took the Christian Bible as divine revelation, they took signifi-cantly different positions with respect to the Reformation. Erasmus remained loyal to the Churchand distanced himself from Agrippa when Agrippa’s protest against the corruption of the monkswho ran the Inquisition became radical [10]. Durer left the Church to follow Luther, but if I readMELENCOLIA I correctly, it is more radical than anything we have from Agrippa or Luther.

Also 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 follower atonce. Luther specifically linked Agrippa’s Occult Philosophy with witchcraft, and Agrippa arousedire when he defended a woman accused of witchcraft from her corrupt accusers. Agrippa eventuallywithdrew to a pious faith in God and the Bible.

In view of the motto, the darkness of the engraving can be taken to indicate Saturnian melancholia.Some say that this motto names the angel diagonally opposite it. Durer labels things unambiguously

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in his other works. He puts his labels on or near the things labeled. The label is on the Bat orthe whole scene, not the Angel. Durer fills his engraving with Agrippan elements. I supposed thatby giving the message “MELENCOLIA I, ” to the Bat Durer identified the messenger as Agrippadelivering his theory of melancholy. Agrippa was such a charismatic figure that Durer’s audiencemay have known at once who he meant by the Bat carrying Agrippa’s famous catch-word. Let ussee how far this takes us.

6 Anagram

It seemed strange to me that Durer would lower the tone of his classic Master Engraving with thelugubrious figure of the Bat,. There is also something off-key about the motto on the banderoleof the Bat. Melancholia is transliterated Greek and also Latin for melancholy. The German isMelancholie. In no language is it melencolia. The spelling of the time was variable and phonetic,but this spelling is not quite phonetic either, for both the Greek and the German word have theguttural ch, different from any c sound. The encodings we have already found, the misspelling, andthe cryptic extra letter suggested the possibility of an anagram to me. The odd choice of lettersmay have been constrained by a message that Durer wished both to record and conceal.

Agrippa could have put anagrams as well as gematria and magic squares into Durer’s mind. Theyare all in his book on the occult philosophy. Anagrams, gematria, acrostics, and acronyms areamong 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, theanagram. This can conceal even the existence of a hidden message. “MELENCOLIA I” has donethis well. Only the cover message has been read in recent centuries as far as I know. I decoded itas follows.

Since so many Durer’s already populate the engraving, it is a reasonable first guess that the mottomight hide another signature, so I first went to Durer’s family coat-of-arms to see how he mightsign himself and found the caelo rebus already described by which Durer represented his heavenlyart. CAELO indeed fits into MELENCOLIA. The leftover letters promptly spell out LIMEN,commonly meaning gateway, a near-synonym for Durer, as well as doorway, threshold, lintel, walls,house, home, boundary path, and limit, according to context.

• MELENCOLIA is thus an anagram of LIMEN CAELO, literally “gateway at Heaven.”

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The hidden motto thus combines the family name-rebus, Durer’s art, and the Durer coat-of-armsitself. This verifies the guess: The anagram is indeed another signature. Indirectly, it supports theinterpretations that Durer intended the clouds in the coat-of-arms as a symbol of Heaven, and didnot intend to demonize Agrippa. The hidden phrase also applies to the dim archway in the heavensthat frames the motto, and to other elements that we take up later. There are so many points ofcontact with the anagram and the engraving that I am pretty sure I have read it correctly. Theovert message and image, both dark, cover a positive one. “Limen caelo” is as bright in spirit as“Melencolia” is depressive. It seems to tell us that Agrippa is or has a new gateway to Heaven. Wemust still figure out what that is.

Possibly Durer drew Agrippa with diabolical attributes to distance himself from Agrippa as Erasmusdid. There are four living things in the engraving and only three Faculties. Someone has to be leftout, and the Bat is the best candidate. I suppose that the role of the Bat is to carry the anagraminto the picture, provide a cover story for it, and as Agrippa, alert viewers to the many Agrippanelements hidden in the engraving. It is not one of the Faculties.

7 Boy

The Boy — the root meaning of putto — is the central figure in several senses. The geometric centralaxis of the engraving is the vertical defined by the leftmost edge of the house. It passes almostexactly through the nearest eye of the Boy, a common way to indicate centrality in Renaissanceart. The line of the comet, suitably extended, also strikes the Boy in the eye.

One way to incorporate the I in the anagram is “Caelo limine” which can mean “I engrave at theedge” or “at the wall.” The Boy does all of these.

• His instrument has a cross-bar at its top, and so is not a stylus as Panofsky said but a graver,the celestial burin again. His block is not a slate but a copper sheet or a block of wood.

And he sits against a wall and on the edge of the stone disk as he engraves, as the anagram says.This could be a multiple coincidence, but more likely Durer is making the Boy fit the anagram sowe will know that the Boy is Durer too. This supports Panofsky’s conclusion that Durer meant thiswork to show many aspects of himself, but not that the Boy is the generic starving artist scribblingmeaninglessly on a slate. It is the famous holy Engraver himself, at work with his heavenly Burin,perhaps with the creative frenzy that melancholia can inspire in the artist, as Yates said. Panofskywas misled by the paradoxical gloom of the work, which we must still understand.

Yates, on the other hand, suggested that the Boy is Durer engraving “MELENCOLIA I” itself, ina self-referential way. This is not consistent with projective geometry. It ignores the line of sightand the scale of the engraving. 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.

• Since Angel had her fools, I returned to the Boy to look for his wise men and found the face Icall Mr. A. Boy in his blouse, between two straps.

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8 Mystery

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 magicians and naturalphilosophers alike. Agrippa was linked with the devil by Martin Luther (1483-1546). Agrippa mayhave influenced Servetus, who was burned in 1535, and David, who was imprisoned and died in1579 [10], both for the charge of “innovation” in religion. Eventually Agrippa renounced his majoropus and adopted an uncritical fideism. There was ample reason for Durer and his heirs to fearexecution and preserve silence.

This theory of a hidden heresy runs into a certain obstacle. In about 1498 Durer executed andpublished 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?

The answer to this question dawned on me slowly over a period of months but I could accept itonly after answering easier ones.

9 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.

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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 ofthe God of Israel. Early on, therefore, the porta caeli of the Vulgate also became a metaphor forthe Catholic church itself. We have found the limen caelo we were expecting, the predecessor ofthe limen caelo I , yet another hidden gateway. It suggests that the limen caelo I is a successor tothe Church. This is at least consistent with the conversion of Durer to Lutheranism, and hints atDurer’s dread heresy and new gateway to Heaven. But if we jump to conclusions we will fall short.

10 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 a sidewindow, 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, ordemons from vermin, was an optical illusion often practiced by artists of the era. Here there isallusion as well as illusion. If the house is what the sanctified instruments on its walls open into,it is the humanly perceptible universe, the entire physical cosmos. The ladder and the Angel havealready 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.

Then there is the curious matter of the door to the house. The curious matter, of course, is thatthere is no door to the house. The house that we see as we step back has several windows. The belland the magic square become one window for arithmetic and music, an hourglass and a sundialturn into another window for time, and the scales turn into a third for weight. The Angel blesses allthree windows with her wings. There is a distorted image in the hour-glass that by all the laws ofoptics should be the reflection of Durer himself, standing before the Globe as he sketches: anothersignature. But nowhere do we see a door. When Mr. Door draws the House of God without a doorhe is likely making 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 to absolutereality. This was a common idea by Durer’s time, for example, from writings of Nicholas of Cusaand Erasmus already cited. Durer could expect some of his viewers to make out this message atfirst sight. It recurs in modern physics in the limits to perception imposed so far by light-speed c

and action-quantum ~, with more surely to come. Before Durer did this engraving he had alreadywritten that the human mind cannot know absolute beauty, and in the picture he may be sayingthe same about absolute truth in several ways. We must not assume that he separated truth and

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beauty, or intellect and emotion, as cleanly as some people claim to do today, especially if heexpected a mathematical theory of both.

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

11 Angel

The burin identifies the Boy as Durer, but the Angel is probably Durer too, sitting on and touchingher name-tags, the Durer monogram engraved on the edge of her stony seat and echoed in themagic square at her wing-tip. In the tripartite psychology of the times, the artist’s special Facultyis the Imaginative one, so the Boy is probably the Imaginative Faculty.

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]. The Angel’s expression is alert and focused, not soft and sad. Above all her gaze isnot downcast, as was absolutely mandatory for portraits of melancholia, but elevated.

Her pose more likely shows contemplation, as in Rodin’s Thinker, than melancholy. This and herwings say rather clearly that she is the Contemplative Faculty proper to the Intellectual Sphere.

12 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, not the Bat, the Faculty of mathematical reason? To be sure, in a Durer woodcut of 1504, TheAstronomer measures a globe with compasses under a full moon, so compasses are also allowed inthe Celestial Sphere, but this does not mean that 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.

The Angel grasps only one arm of the compass and that near the point. She holds the compassesin a way unsuitable for ordinary use, which requires control of both points. She has no drawingboard or table beneath it but seems to be sticking herself in the thigh. Since this is absurd we mustlook deeper. And lower.

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13 Arab and Jews

• The angel’s right knee, bright in the moonlight, is also the top of a hood worn by a bearded manwhose head is cocked at about 30◦ counterclockwise, shown in profile with only one eye visible.

Call him A. Angel. He has an aquiline nose and wears a black headband across his brow. Slowly weconnect his face to a powerful shoulder, almost a hump back, on which Angel rests her elbow andcompasses, and to his left arm, outstretched toward the saw-sword on the ground. The left footof Angel, where it extends beyond her robe, is also his left hand, her toes are his fingers. In theother direction his shoulder leads us to his torso, waist, and knees. He is kneeling as he reaches forthe sword blade. The Angel steps on the hilt with her right foot and pricks him with the point ofthe compasses in her left hand. His right hand is reaching up past his face to fend off the compasspoint that is evidently causing him discomfort. He kneels there covered from head to feet in anArab burnoose.

• He too is one of a triplet, the first that I found in the engraving, simply because it is the leasthidden. To see the other two hidden faces, find a moonlit isosceles triangle next to the face ofA. 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 spiral whitehorn leading to the dark line of a mouth in the face of a man in a skull-cap blowing a shofar, aram’s horn, as is done on Rosh Hashonah, the Jewish New Year, in a full front view. I call himB. Angel.

• There is a second older and hairier face blowing the same horn, whom I call C. Angel, shownin partial profile looking to the left with a hooked nose, a nasty expression, and a skull-cap. C.is looks like an anti-Semitic sterotype of a Jew. Young B. and old C. Angel share an eye. Thistriplet has higher contrast than the triplet of the Solid. They are hidden by sleight-of-hand, notthe watermark technique. Their elements belong to a larger gestalt which compels us to give thema totally different interpretation. Again the people in the triplet can be seen only one at a time.

We already read the Angel as the Contemplative Faculty of the theologians. Durer seems to showby this triplet that her entourage includes Jews and Muslims as well as Christians. The swordfor which the Muslim is reaching suggests that Durer associated Islam with militancy. This is notpalusible since Hungary was mauled by both Christian and Islamic combatants in the 15th century.Muhammed II, founder of the Ottoman Empire, took Constantinople in 1453 and launched apowerful counterattack on Hungary in 1455, the year in which Albert Senior left Ajtos. The attackwas stopped by the forces of John Hunyadi, who became a Hungarian national hero. Since thefall of Constantinople was so epochal for Christianity, the viewers of Durer’s day might have readA. Angel as Muhammad II. But then we would have no clue to whom Durer might have meant byB. and C. Angel.

The configuration makes more sense if the triplet represents the founders of the three monotheisticfaiths of the time. This reading makes sense for the Faculty of the theologians and fits the scripturalreports that two of the founders were Jews of whom one lived to old age and one died young, andthat the third was Arabian and a great warrior. The probability that these agreements are chanceis not overwhelming so I pursue this possibility only.

When Durer unites these three figures in one triplet so that we can see only one at a time dependingon our point of view, he represents them to be close in many respects. Since Moses and Muhammadwere Prophets, Durer would seem to assume the same for Jesus. I considered this unthinkable on

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the basis of Durer’s other works until I learned that Agrippa probably links Durer to two foundersof Unitarianism, Michael Servetus and Francis David [10].

We have found the capital crime of Albrecht Durer: innovation. Both Servetus and David werecharged with innovation and were sent to their deaths for this crime by Reformation courts, notthe Roman Church. Being more explicit might have led Durer to the same fate. If I read himcorrectly on this point, then most likely the downfall he shows would represent the overthrow ofthe hierarchy of the Roman Church, whom he had already slain in effigy.

• In any case, now we see what the Angel is doing with her compasses. She is sticking the Arab.

This violates realism, but Durer left realism behind when his Angel raised the painted One in themagic 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 invisible attack isprominent in Durer’s repertory. His Knight too seems oblivious to Death and the Devil but crushesone of them between his fists without visible feeling. The sneering cliffs around the Arco cathedraland much of MELENCOLIA I itself are acts of concealed aggression by Durer himself against histargets. At least we can cast the Angel as the Contemplative Faculty with more confidence, nowthat we understand her anomalous compasses and her hidden faces.

14 Fools

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 hides non-Christiansin her garment. Only a fool would worship her, the artist seems to say. And there the fool is, a crudelow-browed caricature hidden near her feet in the hem of her gown, next to Durer’s monogram.Call him D. Angel.

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

With no forehead, the caricature is so cruel that I suspected it was unintended. A powerful wingedhousewife with a low-browed fool in her hem does not seem to be a standardized symbol.

• Actually it is. In a wood-cut Von Buolschafft (On Amours) attributed to Durer. in the Narren-schyff [3], the winged woman of On Amours is Venus, the goddess of the fools of love, the mostcommon variety of fool according to Brant. Her wings are those of an eagle signifying fame, notthose of an angel signifying divinity. We know she too is a housewife because Brant calls her sar-castically “Frau Venus.” In MELENCOLIA I similarly we see Frau Angel, her housewife statusconveyed now not by words but by purse and keys, hanging from her waist to the ground. (TheMadonna by the Wall (Durer 1514) also wears keys and purse, on short leashes.) “Frau” is evidentlyno term of respect for Durer or Brant. These fools-in-winged-wives’-hems are clearly not accidentsbut part of Durer’s artistic language.

Durer seems to say that the philosopher/theologian of the Contemplative Faculty is foolish, likeone in love. This echoes and amplifies doctrines already expressed by Nicholas of Cusa in his OnLearned Ignorance (1440) and Erasmus in his In Praise of Folly (1511), and later written down byAgrippa in his De uncertitudine et vanitate scientiarum (1526-1530).

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15 Globe

Durer’s goddesses Fortuna and Nemesis both stand on globes, liable to roll in any direction at theleast provocation, because Luck and Fate are unpredictable. Chance governs the Terrestrial Sphere,mathematical law governs the Celestial Sphere, and God alone knows what governs the IntellectualSphere. Therefore Durer’s Globe seems to represent the Terrestrial Sphere. It is indeed the lowestof the three stone objects in this picture, as if to confirm that they are Durer’s version of thetripartite cosmology. Durer, like Nicolas of Cusa and Leonardo, has replaced the naive tri-sphericalgeocentric Neo-Platonist cosmology by a tri-partite philosophy with three Faculties and Spheresthat are no longer geocentric, concentric, or places at all.

16 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 closed surfaceformed of six congruent rhombuses. Most people see the Solid as a truncated rhombohedron withits longer axis vertical [26]. If it represents any of the neo-Platonic Spheres, then its mathematicalnature and its altitude in the picture suggest that it represents the Intellectual Sphere, the thirdrealm, the abode of Platonic Ideas and angels. The ghosts in the Solid also support this conclusion,for this puts them in a philosophical Heaven. I tentatively adopt this hypothesis in what 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. Onecan 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, on anearly rectangular slab with two diagonally opposite corners trimmed, cocked and cantilevered backtoward the horizon.

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

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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 declares the acute angle ofthe rhombus to be a fifth of a circle, 72◦. [26]. 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 frommeasurements 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, thoughit has been recognized as that only recently [23]. The sketch shows an irregular pentahedroninscribed 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 compass intoseven 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 heptagon. Thetwo long edges of the pentagon are chords of the heptagon that connect the apical vertex to itsnext-nearest neighbors on each side, The remaining three edges of the pentagon are edges of theheptagon 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 occupymuch 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 as atruncated rhombohedron when seen from one orientation, and as a truncated slab from another.

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The Solid is a puzzle that is in principle unsolvable without another view. I suspect that is whythe Angel is smiling.

17 Millstone

The most cryptic of the three stone things may be the wheel holding the Boy. Panofsky and Yatescall it a grindstone and Doorly calls it a millstone. Its perimeter is broken while its face is smoothso it would make a passable millstone but a bone-rattling grindstone and likely Doorly is right.This is also the only stone in the engraving that could serve Jacob both as pillow and pillar.

MORE Since we have argued that the stone Globe stands for the Terrestrial Sphere and the stoneSolid 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.

This turns out to be an easy one. For centuries cultures as diverse as Babylon, Greece, Arabia,Scandinavia, and Rome had represented the Celestial Sphere by a millstone. The celestial millstonecaused the precession of the equinoxes when it flew off its axis into the sea [20]. It makes a cameoappearance in the Icelandic legend of Amleth, the original Hamlet, and so Santillana and vonDechend called it Hamlet’s Millstone. In the Golden Age it ground out gold, in later, lesser timesit ground out cheaper stuff such as sea salt and in our decadent age mere sand. The ScandinavianAmleth is as melancholy as Shakespeare’s Dane and Durer’s Boy. I will point out an indication ofthe melancholy celestial millstone in Scripture too.

The Boy-artist is the Imaginative Faculty and the Angel the Contemplative one, so by eliminationthe Dog is the Rational Faculty proper to the Celestial Sphere.

Since a revolution is going on in this engraving, or perhaps two, some confusion in the seating isunderstandable, but the Faculties are out of place. The seated Angel, the Contemplative Faculty,is nominally of the Intellectual Sphere, the highest, but here she is grounded and lowest. The Boy,the Imaginative Faculty, is traditionally of the Terrestrial Sphere, but he is here esconced on theCelestial, fulfilling Durer’s life dream. Perhaps the Dog, the Rational Faculty, is in his proper place,resting with open eyes between the two spheres, but he too is on the ground. Durer has clearlypromoted Natural Philosophy and demoted Theological Philosophy. We can read this as the Agrip-pan revolution, but it is also the scientific revolution. The Natural Philosopher observes matterin space and time, traditionally the lowest Sphere, yet thereby attains truth. 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, not tothe Intellectual Sphere represented by the Angel. “As above, so below” may well be Agrippa’scredo but it presumably is not Durer’s. Like Da Vinci, Durer had already mocked the idea thatevents in the sky foretold events on Earth. He had drawn and published maps of the stars aboveand the planet below that showed no similarity or correspondence.

20

18 I

Some suggest that the “I” of the banerole 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 . They make a plausible triptych of thethree Faculties, the artist, the statesman, and the contemplative. We know from Durer’s recordsthat when he made a present of MELENCOLIA I he often accompanied it with Jerome, never withthe Knight . But this does not explain why one engraving bore its own caption and not the othertwo, and so does not explain the I .

Another reading is natural for a mathematician like Durer. Even in 1514, an “x primus” in amathematical writing more often referred back to an earlier x than ahead to a later x secundus orx tertius. This would mean that Durer was superceding the old gateway to heaven with a new one,limen caleo primus, and so was claiming innovation. I had to abandon this interpretation, whichI held at first, not when I realized that innovation was a capital crime, but when I realized thatDurer and Agrippa did not consider themselves innovators but as renovators stripping away theinnovations made by the Church.

I propose that the ”I” indeed refers to a triple based on the three Neo-Platonic Spheres, namelythe three Agrippan gateways to God of Natural, Mathematical, and Theological Philosophy, butwe do not need to postulate lost engravings. This very engraving shows all three gateways andshows Durer choosing gateway I, the Natural Philosophy that evolved into Physics. This is the Iof MELENCOLIA I.

19 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 Durer wrote ofseeing a comet himself in 1503. The physical natures of meteors and comets were not yet known in1514. Galileo would still believe in the next century that comets and meteors were meteorological,weather of the highest atmosphere.

Since the Bible has already explained many elements of the picture, let us search it for this onetoo. The answer is immediate and unique. The only blinding light in the biblical sky is the originallight of the divine creation. Then the sky recalls the time in the Bible story after the creation oflight but before the creation of the stars. This is a joyous occasion: Happy birthday, universe! TheLight may represent Divinity itself, and divine revelation. Then the fact that the comet is headingexactly for the Boy suggests that he, not the Angel, the Dog, or the Bat, is chosen by God. Yetthe tone is somber. The tension grows.

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

21

divine.

It took me a long time to understand this engraving even with all these clues, because I couldnot figure out whether Durer’s gateway to Heaven was mathematical or religious. This is a goodexample of the kind of projection to avoid in such a study. Durer does not separate science and faith.On the 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.

20 Dog

The Dog is the last piece of the puzzle to fall into place. It had several possible interpretations.Durer’s animals are among his finest works of art and science, and he had drawn at least threedogs in the year before that are still with us. A similar hunting dog in another work of Durer isequally gaunt, perhaps as part of the training of the breed. Dog and hourglass are the only twocomponents common to all three Master Engravings. Presumably hourglasses represent time andmortality. To me, dogs represent fidelity, to Agrippa love or flattery. There must be a better fit.

This Dog lies rather too close to the line of Comet, Solid, and Globe for accident. He curls up justwhere the Celestial Sphere would lie in the old cosmology. The battered millstone has been liftedup on one edge and leaned on the House, and the Dog is where it used to be. The people associatedwith the Celestial Sphere were Mathematical Philosophers, astrologers, and kings. The connectionof the dog to the Celestial Sphere is no more obvious than that of the Millstone.

We know that when Durer illustrated the Hieroglyphica, he learned to represent the ruler by adog wearing the royal mantle on its naked body [14]. That is why in his Triumphal Arch Durer sorepresented the Emperor Maximilian I. At the same time he also learned that the way to represent aprophet — one whose Rational Faculty is excited by Saturn, in the theory of Ficino and Agrippa —is by a dog. So the Dog — not the Bat, say — represents the Rational Faculty. This interpretationof the Dog fits all three Master Engravings at least loosely.

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 tellus whether Noah saw a rainbow by day or a moonbow by night. God’s memorandum to himselfto send no more genocidal floods could have been either, but Durer had no choice, once he set thescene in the dark of night. In Genesis, the light in the heavens and the bow in the clouds representtwo of the greatest 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 plain of the bow passing throughthe bow center. Durer’s watching eye should lie directly before the point of the horizon beneaththe highest point of the moonbow. This is not exactly where he is located by the vanishing pointof the building lines, which is under the eye of the Bat, but it is only several centimeters off.

22

The moon must lie on the same line, behind the viewer. It is therefore well to the left of center,like the viewer and the moonbow. But according to the shadows on the House and the face of theAngel 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 light.

22 Babylon

My Bible-searches for ladders, sky lights, and sky bows all bore fruit so I looked for Durer’s Millstonein the Bible too. I found 11 millstones in the Christian Bible. The first ten serve as metaphorsfor livelihood or industry, and refer to no other element of the engraving so I put them aside. Theeleventh and last is in Revelation. In the King James version:

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 three elements of the engraving appear here: the mighty angel, the millstone, and the sea.The millstone of Revelation is therefore a provisional candidate for Durer’s millstone, the best inthe Bible. But it is itself a puzzle. Tossing an arbitrary millstone into the sea is an oddly lameimage for the downfall of Babylon, one of the great empires of Bible times. A bath would not harma millstone and might do it good. But if the millstone of Revelation is again the Celestial Sphere,throwing it into the sea is the same astronomical catastrophe as Hamlet’s millstone rolling into thesea. It ends one world age and begins a new one [20], a simile fit for a a king even of Babylon. SoI looked for Babylon and its downfall in the engraving.

The engraving shows a city by the sea with ships in port. Over the left-most pan of the scalesin the engraving, through a small triangle framed by ladder, house, and pan, we see a massif offlow-lines pointing toward the city, well-resolved in a full-size print if not in [19]. How could I haveoverloooked it?

• There is a great wave looming high over the city, about to destroy it as prophesied in Revelation.

Or the massif could be far up the coast beyond the city. The flow can be read in two ways due tothe intrinsic ambiguity of perspective, which we at last begin to realize is always deliberate, thevery point of the engraving, hammered home again and again until we could not miss it. The flowlines are then geological, not hydrodynamical. We cannot read the depth of the flow in the pictureuniquely from one perspective view. The ladder hides connecting clues that might force one viewor the other upon us.

Now the engraving and the Biblical verse share Millstone, Angel, sea, Babylon, destruction, andeven melancholy, for the verse I quoted is imbedded in a score that lament the destruction to come.If authors as far apart as those of Revelation and the Hamlet legend used the Millstone for theCelestial Sphere, then this usage may have still been current in Durer’s time too. The connectionwe sketched in lightly between Durer’s millstone and Hamlet’s is now less tenuous. Probably themillstones of Revelation, MELENCOLIA I, and the original Hamlet, separated by centuries, areone and all the Celestial Sphere, and this scene in the engraving probably refers to Revelation. TheBiblical verse would then refer to the end of the Age of Aries and the beginning of Pisces, the next

23

after the writing of Revelation. If Durer meant the metaphor seriously, he referred to the end ofthe Age of Pisces, and his Babylon might be the hierarchy of the Roman Church that he alreadyso mistreated in his Apocalypse. In 1514 he was not yet disillusioned with the Reformation clergyas well, but Pirckheimer attested that both clergies eventually disgusted him beyond his tolerance.

23 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 Shield ofDavid framed in a hexagon.

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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 infera certain philo-Semitism. Centuries earlier the Khazars of the Crimea had adopted the hexagramfor their flag when they adopted Judaism, but in 1514 Nuremberg the hexagram was still mainlya magical device, an amulet, possibly referring to the Hebrews of the Bible but probably not tocontemporary Jews. Durer’s use was likely a Hebraism, not philo-Semitism. The Shield of Davidwas called that because it was supposed to shield its bearer from evil spirits. It was more oftenfound in churches than synagogues. A two-footed hexagram like Durer’s ground plan is seen ona German altar of Durer’s time. Some said the hexagram was the Seal of Solomon, and othersthe pentagram. Traditions firmed later that gave Solomon’s star five points to David’s six, andstood Solomon’s on two points to David’s one. Not long after Durer’s death the Jewish communityof Prague was granted the privilege of a flag and chose the hexagram, possibly because of itsassociation with the Khazars, and this continued its evolution from magic charm to Judaic symbol.

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 truncated atrue cube, we would have seen it as such in any view. Stretching the cube made the figure triply

24

ambiguous, and demonstrated our inability to reach truth through our senses. Circumcising itHebraicized it, connected the new gateway to 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.

24 Space and time

Durer uses the vertical dimension on his plate in a traditional way. Higher is holier. Jacob’s angelsget to heaven by climbing a ladder. Angel and Boy have equal divinity because they have equalaltitude. The tools of artisans litter the ground, they are profane, while the instruments of scienceand mathematics hang carefully on the wall nearer to the sky, they are more sacred.

The horizontal dimension is time. It is cut neatly in half by one edge of the House. The cut separatesthe new and old gateways to Heaven, which use the Rational and Contemplative Faculties. Yet allthree Spheres of the cosmos and all four actors in the drama are to the left of the dividing wall ofthe House, though perspective puts the Boy and the Angel on the right-hand half of the paper.

All the instruments of science are on the right-hand side. The crucible is no exception. If it werean instrument of alchemical science, as I thought at first, it would be elevated, but it is on thestone floor. Only perspective puts it high on the paper, above all other tools; perspective andfilial feelings. This is further indication that Doorly read this correctly, This crucible is not aninstrument of science but a remembrance of Durer’s goldsamith father.

The near eye of the Boy lies just on the center line. Durer the artist-scientist withdraws from theschism about him and does his inspired work.

Perhaps the vanishing points tell us how to hang all three Meisterstiche as a triptych on three wallsof one room: Jerome on the wall to our right, Knight on the wall before us, and MELENCOLIA onthe wall to our left. Then the horizontal axis can be time passing from right to left as in Hebrewwriting.

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

25 Melancholy

The light and the bow and even the destruction of the city of evil are good news, and yet the pictureis set in the gloom of night. Durer has even transformed the rainbow of the common understandinginto a moonbow. The comet illuminates nothing and hides the stars. 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 in theiroutlook. 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 is

25

I know not. Nobody knows it but God.” The darkness of the setting and the multiple ambiguitiesof the Solid suggest that after all Durer did not expect humanity to attain the supreme mathesisthat optimists like Ramon Llull had expected and even claimed to have attained. He expressedthis pessimism in the engraving atmospherically, by omitting the door of the House of God, andby numerous skillful demonstrations of ambiguity and relativity.

The most unambiguous aspect of this engraving is its ambiguity. Durer wants us to know clearlythat we cannot know. The Solid is a truncated rhombohedron with vertical axis or a slab tiltedtoward the horizon. There are three faces hidden in the Solid or none. The city is to be destroyedor not. The Angel is good or evil, the Bat too. The house has windows or not. The melancholy isinspired by the inaccessibility of the absolute, or by the influence of Saturn, or because the CelestialSphere is off it axis. What we see depends on how we look at the scene and what we expect. Mostof the elements of this engraving that we have discussed here are intentionally ambiguous, eitherin meaning or form.

If we cannot believe what we see, what can we believe? This questioning occurs famously in theFirst Meditation of Descartes (1596-1650), who cites optical illusions as a reason for his methodof universal doubt. It seems that MELENCOLIA I anticipates Cartesian doubt. And its responsehas a mathematical element that anticipates the Mathesis Universalis of Descartes. The ideal ofa complete mathematical theory of beauty lies on the same long line of distinguished fantasiesof mathematical wisdom as the number mysticism of Pythagoras and Plato, the Ars Magna ofRamon Llull (whom Agrippa studied) and Giordano Bruno (who studied Llull and Agrippa), thevision of Mathesis Universalis that Descartes and Leibniz shared, and the Ars Combinatorix ofLeibniz. 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 and beautyas 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. The engraving seems to come froma later more despairing phase of Durer’s life. This suggests to me that the date on the engravingis an afterthought, chosen to agree with the bottom line of the magic square and the death yearof his mother. One can also suspect the date because in 1514 Luther had not yet linked Agrippato witchcraft and the demon image might have been obscure. This reasoning is somewhat circular,and only further historical data will break the circle.

The engraving should indeed be read as an illustration, as Doorly says, but of Durer’s own lamentand credo. Its night is our benighted ignorance. Its darkness is the darkness “firmly intrenchedin our mind.” The mystery of the Solid, crafted to be insoluble, is one of many metaphors in theengraving for our inability to see the Absolute Truth from our limited perspective. In acknowledgingthe limits of human knowledge, Durer went a step beyond the more optimistic Leonardo towardmodernity. As Panofsky and Doorly said, the melancholy is Durer’s over the limit of human artand understanding. And as Yates said, it is also the Saturnian melancholy that inspires artisticcreation.

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 rather

26

than abstract speculation, but it became relativistic. He included a variable horizontal scale-factortransformation in order to represent human beauty of the lean, average, and plump variety as thereference frame of the artist requires.

26 Trinities

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 he is a triplet, Durerdoes not intend triplets to be divine as opposed to human. Following this reasoning I returned tothe engraving and looked at the fool from another angle.

• It was then easy to see faces E. and F. Angel superimposed on D. D. in profile looks to our right,E. in full face looks slightly to our left; and F. in profile looks upward and to our left. I originallyfound C. Solid in the same way.

I looked in vain for such triplets in the other figures of this engraving or in other works of Durer.The fool of Frau Venus too is single, not three-faced. This supports two inferences at once. Thetriplets refer to the Doctrine of the Trinity, and the Angel and Solid alone belong to the SpiritualSphere. It seems probable that Drer introduced these three relativistic triplets in order to reformthe Doctrine of the Trinity. He shows one individual with three aspects, not three entities and oneat the same time.

This puts Durer closer to Unitarianism than Trinitarianism. This is not completely absurd, itseems, due to at least three circumstances.

First, since he agrees with Da Vinci that knowledge comes through the senses, and since theDoctrine of the Trinity comes more from the Council of Nicea (325) and faith than from directobservation, it would not be surprising if an independent thinker like Durer had doubts about theTrinity by 1514.

Second, the Hebraism of Agrippa might also stimulate a revision of the Trinitarian doctrine. Gellerdproposes that Agrippa provided the Transylvanian Unitarian David with some of his epistemology[10]. Panofsky and Yates make it clear that Agrippa influenced Durer’s thinking. It seems possiblethat the charismatic Agrippa inspired Durer too to express the unitarian tendencies we find inMELENCOLIA I.

Michael Servetus, traveling in Germany, published his historic unitarian treatise De trinitatis er-roribus, perhaps the first, in 1531 at Hagenau, only 100 km from Nuremberg. The publication andthis engraving close enough in space and time to suggest that unitarian thoughts were part of alocal Zeitgeist, perhaps inspired by Agrippa, though in 1531 he was publishing his occult philosophyat last in Antwerp.

Once we look for unitarianism in the engraving, it is easy enough to find. The Deity is shown hereas the single point of light that is also the comet. In his Landauer altarpiece, on the contrary, Durerpresents each member of the Trinity as a fully conceived independent entity in itself, representedby a dove, an old man, and a young man on a cross, and indicates their unity by the conventionof arranging them in a vertical straight line. In Melencolia I instead he presents a relativistic kindof trinity more natural to a descriptive geometer. Each of the members of such a triplet is not a

27

different and separate individual but a different aspect of one unity. This representation capturesthe impossibility of seeing three separate things where there is only one in a way that the Landaueraltarpiece 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 Boy hedeclines the traditional ways to knowledge of the Contemplative and Rational Faculties for the wayof the Imaginative Faculty, the way of artists and artisans and natural philosophers. “All that isin the mind comes through the senses.” Finally, he suggests that the three entities of the Trinityare 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 era when the closest thing to physics was natural magic, the natural wasstill emerging from the magic, and the Bible was more trusted than our senses.

Durer shows a single divinity connected to the human world through the realm of pure mathematics.He demonstrates multiply that relativity keeps us from the absolute. He shows how this fills himwith melancholy, and how this melancholy inspires the frenzy of artistic creation and naturalphilosophy that we see before us.

27 Disclaimer

The views expressed by this engraving are not necessarily mine.

Durer seems to find that relativity is a melancholy science. Many poets, philosophers, artists,and even some scientists, express sadness, pain, or nausea over losing the absolute, and many stillcontinue to seek it. I do not.

28 Acknowledgments

I thank an anonymous donor (§2) for starting me on this study. It would have been impossiblewithout the studies on MELENCOLIA I by Erwin Panofsky and by Francis Yates and works onAgrippa by Judit Gellerd and by Charles Nauert. I have also drawn on the study of Hamlet’smillstone by Giorgio de Santillana and Hertha von Dechend, on a study of MELENCOLIA I byPatrick Doorly, on a study of Hamlet by Stephen Greenblatt.

The heraldry dictionary of Cecil Wade helped me to read Durer’s coat-of-arms. I benefited fromgematria on the website of the Aiwaz Gallery, though our arithmetic and interpretation differ. Mythanks go to Roy Skodnick for introducing me to the work of Yates; to Prof. Heinrich Saller forthe print of MELENCOLIA I that triggered this process; to Rabbi Mario Karpuj for remindingme that Jacob’s ladder is the original 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 the

28

reference 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 §11.

[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

[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.

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

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

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

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

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

[27] 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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