There is a certain engraving “Melancholy”, owned by the German artist Albrecht Durer, which is better known to mathematicians and occultists than to those interested in painting.
At least - you can check this - very little has been written about it on the Internet. But this is a really cool thing. And the only more or less detailed source is Dan Brown’s book “The Lost Symbol”.
I read this book and neither the plot nor the square stuck in my head. And then it suddenly popped up from an unexpected direction.
Engraving “Melancholy” - pay attention to the square in the upper right corner:
Here it is larger:
The essence of all “magic squares” is generally clear: the sum of the columns and diagonals is equal to some number. So it is here. This number is 34. But the fact is that this number appears in absolutely ANY scenario. The sum of the upper left square is 34, the same is true for the upper right, lower right and lower left small squares. And also the central square - 10+11+6+7=34. And also, if you add the corner numbers 16,13, 4 and 1, you also get 34.
And also, if you start laying a line from 1 to 16, you will get this absolutely symmetrical (and in a mirror relation!) figure:
And at the very bottom, the numbers 15 and 14 indicate the date of creation of the engraving - 1514. And the numbers in the lower corners - 4 and 1 - are the digital designations of the artist’s initials: D A - Dürer Albrecht.
All this mathematical “palmistry”, according to some, indicates that Dürer created his square not by poking or picking, but by using other measurements. In the sense - going beyond 3 dimensions and.... somehow on the seventh-dimensional(????) level?…. Perhaps with the help of the so-called “conchoids” or “shells,” as Dürer called it (in his mathematical monograph “Guide to Measuring with Compass and Ruler,” published in 1525) and of which he was the author, he created his “magic square.”
"Conchoid":
And pay attention to the stone in the engraving - a parallelepiped truncated at two corners, the side faces of which are 2 regular triangles and 6 pentagons:
Robert Langdon, the symbolist detective in Dan Brown's The Lost Symbol, superimposes the 16-digit cipher from the base of the Masonic pyramid onto the Durer square and receives the decryption:
that is, JEOVA SANCTUS UNUS - the One True God.
Dürer in all likelihood belonged to a certain Secret Society. And perhaps he possessed some secret sacred knowledge...
Or maybe this is all a hoax?!..
Let's draw 16 cells and put numbers from 1 to 16 in them in order. Now just swap 1 and 16, 4 and 13 (these are the corners), 6 and 10 and 7 and 11 (the square in the middle). And also 2 and 3 and 14 and 15 standing next to each other.
VOILA! This is the magic square of the coolest degree. Just? Just! But guess what and how to change... On the other hand, the absolute symmetry of replacing numbers cannot but suggest the simplicity and universality of the solution. Or is it easy for us to reason now, but Dürer needed to use his conchoid (see above) to understand how and what to swap places?...
The correction in the engraving, which Dürer INTENTIONALLY left so obvious, can be seen with the naked eye:
When replacing the numbers in the square drawn for us from 1 to 16 in order, only the side 5 and 9 on the left and 8 and 12 on the right remain unchanged. Initially, Dürer wanted to swap them too, but this turned out to be unnecessary. Why did he leave his mistake for everyone to see? Show me how your thoughts work? Vanity? And the year 1514, which fits so well into the square, is also a merit or did the artist simply wait for the desired date for greater effect, having thought through all the mathematics earlier?))
Maybe so. Even the fields of higher mathematics can be explained by the vanity of an artist who considered himself handsome and regularly painted his self-portraits so that everyone could admire him.
Returning to Melancholia, magic squares and the occult. The engraving was written for Emperor Maximilian I (for those who know, the husband of Mary of Burgundy, son-in-law of Charles the Bold and grandfather of Emperor Charles V).
Here is his portrait, also by Durer:
Maximilian considered himself melancholic. In the Middle Ages (and even now) it was believed that melancholic people were influenced by the planet Saturn. The magic square was supposed to be a kind of talisman that would ward off the dark influence of Saturn, while simultaneously attracting the more positive energy of Jupiter.
In general, you can write a lot about this engraving. You can still consider all the attributes - but that’s for another time. In this case, mathematics seemed more interesting to me than painting.
A.V. Lantratov
ContentRice. 16. Position of the planets in the morning (two hours after sunrise)
September 1 Art. Art. (September 11 AD) 1624 AD Observation location: Nuremberg.
Based on StarCalc screen
Thus, we have three options, assigning the “zodiac” date recorded in the painting, depending on the possible calendar ideas of its customer, to the end of the eighth month of January 1624, the end of the first ten days of the first month, or exactly to the beginning of September 1625.
A natural question arises: which of these options best matches the image on? As we will now see, it is the latter, since it is with it that a number of other details of the picture under consideration are ideally consistent.
4. “Year of Saturn” and the symbolic meaning of the “lion” horoscope
First of all, let's look at the two figures depicted in the lower left and right corners of the portrait, Fig. 17, and let's try to understand what they mean.
Rice. 17. Figures at the bottom of the portrait of Johann Kleberger.
Enlarged fragments of Fig. 3
With the left one - the clover shamrock growing on the top of the mountain - there are no questions. This is an ordinary coat of arms with the symbols of the owner. Exactly the same symbol can be seen in another surviving image of Johann Kleberger (by the way, his very name comes from clover), fig. 18.
Rice. 18. Johann Kleberger on a medal from an unknown Nuremberg master,
dated, like the portrait of Dürer, to 1526. Visible on the reverse side
a helmet above which is depicted a mountain with a trefoil growing on its top
But what exactly does right mean? Of course, it is quite possible to say that this is “just a beautiful picture, paired with the shield,” and be satisfied with that. However, taking into account the above, in this picture it is easy to recognize an astronomical plot slightly veiled under heraldic style. In fact, we see here a long-bearded old man holding two shamrocks in his hands. The background of this composition suggests itself: six identical leaves (as well as six identical stars in the opposite corner of the same portrait) most likely represent six planets, Fig. 19-21, and the old man - some seventh planet.
Rice. 19. Planetary tree. Title page alchemical treatise:
Rice. 20. Planets (aka alchemical elements),
depicted as leaves on the branches of a tree.
Rice. 21. Sun, Moon and planets on the branches of the alchemical tree.
Illustration from the treatise: Johann Mylius, Philosophia Reformata, Frankfurt, 1622
The question is, which one specifically? Obviously, this is either Jupiter or Saturn, since these two planets were most often (and the latter almost always) depicted in this form, Fig. 22.
Rice. 22. Jupiter (left) and Saturn (right) in engravings by Hans Burgkmair.
Allegedly the end of the 15th - beginning of the 16th century
Strictly speaking, more or less similar images are sometimes found for Mars, Mercury and the Sun, but they always have signatures or characteristic attributes (the sword of Mars, the winged staff of Mercury, etc.) that make it possible to understand which planet is meant , . In the absence of such attributes, it is Jupiter and Saturn that remain, since the only sign for identification, in this case, turns out to be age itself, and the latter are the eldest among the “planetary” gods.
So, let's consider the first option. In this case, it turns out that the six planets are divided into two triplets, depicted as trefoils in the hands of the elder Jupiter. From an astronomical point of view, this means that three planets should be on one side of Jupiter, and three on the other. But this is exactly how things stood in the “New Year’s” resolution of 1624/25 obtained above: to the left of Jupiter, on the side of Virgo, were Mercury, the Sun and Venus, to the right were Mars, the Moon and Saturn, . That is, when the elder is identified with Jupiter, the entire composition takes on the meaning of an additional astronomical indication to the main horoscope.
In the second case, such a transparent correspondence, of course, is no longer observed, however, as it turns out, it does not in the least contradict the “New Year’s” version of the dating obtained above. And even more than that, it not only further confirms it, but also allows us to better understand the logic and way of thinking that guided the author and/or customer of the portrait in question.
Namely, let us ask ourselves the question: what else, besides dividing the planets into two groups, can the fact that they are all depicted the same, small, and, moreover, being in the hands of an old man, personifying (this time) Saturn? It is obvious that the latter holds them all in some kind of subordination (literally, “in his hands”). The question is, what kind of “subordination” can we talk about? The answer is given again. The fact is that the observer, looking at the starry sky in New Year's Eve September 1625, I saw how Saturn rose about two hours before dawn, half an hour later the Moon (in the form of a barely noticeable or even completely indistinguishable crescent), and another hour later - all the other planets. That is, figuratively speaking, in these pre-dawn hours, Saturn “reigned” undividedly in the sky, thereby announcing that the coming months would pass under his “control” (as well as all the other planets, equally “subordinate” to him, whose fate, in the near future , ended up “in his hands”, and, of course, with earthly affairs).
And, as is well known, this kind of correlation of the year with the planet “ruling” it was indeed a widespread practice in the era of Kleberger-Dürer, Fig. 23-24.
Rice. 23. Saturn is the ruler of the annual circle. Illustration from
medieval astrological almanac. Allegedly 1491
Rice. 24. Saturn. On the reverse side there is a vestal virgin at the altar and an inscription
“Good luck in the new year” (SPENDE NEUES GLUCK IM WECHSEL DES JAHRES).
Medal issued in Nuremberg around 1810
This tradition has been preserved to this day, fig. 25-29.
Rice. 25. “Saturn is the ruler of the year” (JAHRES REGENT SATURN).
Medal from the “calendar” series produced in Austria
from 1933 to present
Rice. 26. Obverse sides of two more Austrian calendar medals
(for 1937 and 1972), dedicated to Saturn
Rice. 27. Jupiter and Mars on Austrian calendar medals
Rice. 28. Venus and Mercury on Austrian calendar medals
Rice. 29. Sun and Moon on Austrian calendar medals
Thus, the identification of the old man with Saturn also perfectly corresponds to the solution found above. Except that the reading of the composition turns out to be a little more intricate, and the resulting meaning shifts from a purely astronomical to an allegorical plane.
The latter, however, can be objected to by the fact that Saturn, according to medieval ideas, was considered an ominous, extremely unfavorable planet associated with death and various kinds of bad influences. The publication [Saplin] summarizes these views as follows: “Saturn is the fifth astronomical planet... In individual astrology, the following concepts are subordinated to Saturn: partings, obstacles, difficulties, losses, confrontations, endurance, patience, perseverance, solidity, alienation, loneliness, cold , age, difficulty, cruelty, steadfastness, constancy, envy and greed. In world astrology... Saturn is responsible for national disasters, epidemics, famine, etc. ..." And also: “Great misfortune (lat. Infortuna major) is an epithet of the planet Saturn, which was considered the most unfavorable planet, often used in medieval astrology.”
In general, at first glance, it is difficult to imagine a reason that could prompt someone to order their portrait against such a background. And in most cases, this would be quite enough to reject the option of identifying the elder with Saturn (thus leaving Jupiter for him as the only candidate). However, in this particular case, such proximity can be very easily explained. The fact is that the picture described above of how the “sinister” Saturn was the first to rise on New Year’s Eve in September 1625 was not entirely complete. To be completely precise, then, as is again clearly visible in , “the very first” - according to calculated data, three minutes earlier than Saturn - one of the brightest stars in the sky, Regulus, appeared on the horizon. And after Regulus, it was the turn of the “reigning” Saturn (by the way, the name of this star is also associated with royal power and means, translated from Latin, “little king”).
Regarding Regulus, the publication [Saplin] says this: “Regulus, the Heart of Leo ... is the star α Leo, ... indicates happiness.” That is, from the point of view of the same medieval ideas, by the time of the rise of the “Great Misfortune” = Saturn, his malevolent hypostasis was neutralized by the “happy” Regulus, and, therefore, came to the fore positive features- “endurance, patience, perseverance, thoroughness, ... steadfastness, constancy.” Strengthened in addition by the “royal” essence of Regulus. Who wouldn't want a set like this?
By the way, it immediately becomes clear why Saturn could be depicted as a good-natured old man, without his usual attributes in the form of a scythe and a devoured baby. In this case, they were obviously no longer needed. On the other hand, the author’s train of thought could have been more sophisticated and consisted in the fact that, by depicting the named old man without any characteristic attributes that would clearly indicate Saturn or Jupiter, he thereby provided the viewer, who was quite experienced in such subtleties, with the opportunity correlate it with each of them, in both cases revealing an important part of the overall meaning embedded in the picture.
By the way, Saturn has another aspect, which could also be considered as one of the fragments of the multifaceted symbolism of the picture. Namely, Saturn-Kronos was also associated with the ageless Chronos, that is, Time. And, therefore, the placement of his figure in the portrait, when looking at it from such an angle, could promise a long life for the person depicted, fig. 30-31.
Rice. 30. Saturn-Chronos, wishing good luck in the new year
(VERTENTE ANNO - literally: “throughout the whole year”).
Medal issued in Augsburg and dating from 1635
Rice. 31. Leopold Habsburg with his son Joseph at the altar of Eternity, opposite them
— Chronos-Saturn with a broken scythe and an hourglass thrown to the ground
and Fortune with a cornucopia. On the reverse side there is a picture of someone sitting in the clouds
Chronos, holding in his hand a snake coiled around the number XVII, biting himself
by the tail (symbol of cyclicity, rebirth, etc.). Augsburg medal,
issued in 1700 to commemorate the coming new century
Thus, we see that even the standard interpretation of the symbol as denoting the constellation Leo leads us to a very interesting and symbolically rich result. However, as mentioned above, there is another reading option, according to which this symbol points to a specific star in the sky - Regulus. Let us now consider this possibility.
5. The second version of the horoscope is “with Regulus”.
When was Johann Kleberger born?
In this case, we get that the following horoscope is presented - all the planets in the vicinity of Regulus. At first glance, it may seem that there is no difference with the already discussed version - all the planets are in Leo - since, as stated just above, one of the names of Regulus used by medieval astronomers was “Heart of Leo” (Cor Leonis), and the phrases “ in Leo" and "in the vicinity of the heart of Leo" give the impression of being, on the whole, synonymous. However, from a purely astronomical point of view, there is a significant difference between them, caused by the fact that the exact position of the star Regulus does not coincide with the geometric center of the figure of the constellation Leo (and its projection onto the ecliptic). In fact, Regulus is noticeably closer to Cancer than to Virgo. Consequently, in order to correctly take into account this nuance, the area permissible for planets must be expanded somewhat by adding the half of Cancer bordering Leo.
Thus, the expanded “according to Regulus” horoscope takes the following form - all planets from the middle of Cancer to Virgo.
It is quite obvious that among his astronomically possible solutions there will still remain the “New Year’s” days found above, August 31 - September 1, 1624 (as well as the obviously out-of-date year 1007). This means that everything said earlier will remain fully valid. The question, in this case, is only whether new solutions will appear, and if so, which ones? Perhaps among them there will be one that would correspond to the Scaligerian dating of the painting in question?
- 1) August 30 - September 1, 1445 AD;
2) October 10-11, 1564 AD;
3) August 3-6 old style 1624 AD.
The first of these decisions, 1445 AD, however, immediately disappears, since it turns out to be forty years earlier than the Scaligerian date of birth of Johann Kleberger - 1485/86. The second, 1564 AD, also disappears, since the latter, according to the same Scaligerian chronology, died twenty years before this date, in 1546. Equally, both of these decisions do not correspond to the Scaligerian dating of the life of Albrecht Dürer, allegedly 1471-1528, whose monogram is signed on the painting in question. That is, as we see, its dating proposed by Scaligerian historians - supposedly 1526 - is not confirmed in this case either.
As a result, only one solution remains for further analysis - August 3-6 of the old style (August 13-16 of the new style) 1624 AD. Let's take care of it.
What immediately attracts attention is that this solution turns out to be “paired” with the previously found “New Year’s” resolution of the end of August - beginning of September of the same 1624, separated from the latter by only three weeks. This can obviously mean one of two things. Either the newly found solution is a side one, representing nothing more than a consequence of excessive expansion of the ecliptic region permissible for the location of planets, or there is some separate plot behind it, revealing an additional important fragment of the general plan embedded in the symbolism of the picture. And then you need to understand which one it is.
Let's try to find out. Above, we have already examined the details of the picture, located in all four corners of it. Only the central part remained unaffected, with the actual image of Johann Kleberger and the Latin inscription running along the rim of the portrait circle. We will now turn to it.
So, this inscription is E IOA[N]NI KLEBERGERS NORICI AN AETA SVAE XXXX. Translated into Russian: “portrait of Johann Kleberger from Norik (Nuremberg - A.), [depicted] at the age of 40." What attracts attention here is the “round” age of the person portrayed—exactly forty years. This is on the one hand. On the other hand, as reported in, his birthday was August 15 (which exact source contains this number is not said, but since, unlike the year, the day is much less susceptible to chronological distortions, both random and features, intentional - then, most likely, this instruction can be completely trusted).
This means that Kleberger’s fortieth birthday fell on August 15, 1624 (and he was born, therefore, on August 15, 1584). But it is precisely this date (or, more strictly, the interval from August 13 to 16, with the smallest deviation from the “best points” of August 15) that the “extended” solution points to!
Thus, the following completely natural picture opens before us. On August 15, 1624, Johann Kleberger turned forty years old (which, at that time, was a very respectable age). And literally three weeks later, the new year of September 1625 arrived, which promised, according to the ideas of the era, to be very favorable (and should pass, in particular, under the sign of solidity and steadfastness - qualities that are undoubtedly very important for a banker). All this, taken together, was reflected in the wonderful portrait executed by Albrecht Dürer.
Which of these two dates simultaneously recorded in the painting was considered to be the “main” one cannot be said with certainty. In the case of the first, the result was a fixation of the age and status already achieved by the depicted person, in the case of the second, a focus on further growth and prosperity in the future. But, most likely, the overall symbolic message of the portrait was formed - in the perception of the author, the customer and their contemporaries - as a balance of both. Or, to put it simply, the portrait in question was “read” by them in approximately the same way as modern anniversary cards (from this, by the way, it follows that it could have been commissioned not by the “hero of the occasion” himself, as is believed, but by one of his relatives or friends as a gift).
This is the result following from the purely historical background of the “anniversary” date of August 15 (new style) 1624. Let's now look at its astronomical aspect. Namely, to what extent the picture of the sky observed on this date corresponded to what we see on.
So, the table below shows the calculated positions of the planets for the specified date, and in Fig. 32 provides a visual “snapshot” starry sky, obtained using the planetarium program StarCalc.
JULIAN DAY (JD) = 2314441 |
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MERCURY |
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Rice. 32. Position of the planets on the morning of August 15 AD. Art. 1624 AD Comparison of these data with again reveals complete agreement. In fact, of the seven planets, five (Sun, Moon, Saturn, Jupiter and Mars) were strictly in Leo, the sixth (Venus) was on the border of Leo and Cancer, and only Mercury was directly in Cancer. At the same time, Regulus was located right in the middle of the planetary “line,” which ideally corresponds to the identification of the central symbol on it with him. Moreover, there - in the center - was the Sun, which again fits perfectly with the image of the latter not in the form of one of the faceless stars that are in no way different from each other, but in the form of a radiance that equally embraces each from the latest. This is what is inferred from the formal arrangement of the planets on the date in question. From an observational point of view, the picture of the pre-dawn Nuremberg sky was as follows. A little more than an hour and a half before dawn, Mercury rose, and Venus appeared forty minutes after it. Then, almost simultaneously - literally within seven minutes - Saturn, the Sun and Mars rose (and a few minutes later - Regulus). This procession was completed by Jupiter and the Moon, appearing only an hour and a half after dawn. Consequently, of all the planets in the night sky on the anniversary date for Johann Kleberger, only Mercury and Venus could be observed. It is possible that this circumstance was reflected in the fact that all the stars on it are divided into two groups in relation to Regulus (and the two lower star-planets, in a sense, are opposed to the four upper ones). If this is indeed the case, then the entire composition could be read as follows (from bottom to top): first, two planets appeared in the sky (represented by two stars at the bottom), and then Regulus (the symbol in the center), the Sun (radiance) and all the rest of the planets. And one more observation that well complements the overall picture. The fact that on the date found above, Mercury was the first to appear in the night sky - and, therefore, the day of Kleberger's anniversary was the “day of Mercury” - was probably also viewed by the people of that era in a symbolic way and was perceived by them as a good omen, since Mercury was considered ( and is considered) the patron of trade and wealth, often depicted with a money bag, fig. 33-35, and Kleberger, as already mentioned, was a banker and merchant.  Rice. 33. "Children of Mercury." Engraving by Hans Sebald Beham of a riding
Rice. 34. Mercury with a money bag on the relief of Artus Quellin
Rice. 35. Year of Mercury (JAHR DES MERKUR). Last of twelve released 6. “The Sign of Regulus” and the “magical” nature of Kleberger’s portraitThus, we see that on both dates we found - the “anniversary” and the “New Year’s” - the decisive role was played both in the immediate astronomical picture and, especially, in its symbolic interpretation, by the “royal” Regulus. That is, speaking figuratively, it is quite possible to say that the portrait of Johann Kleberger is dedicated to the events that happened “under the sign of Regulus.” And it is remarkable that it is precisely this premise that turns out to be expressed in a completely transparent way in the last remaining unexamined detail. Namely, at the very end of the inscription running along the rim of the portrait circle (the text of which was given in full above) there is a bizarre symbol representing the so-called “sign of the spirit” of Regulus, fig. 36.  Rice. 36. Fragment of the inscription on the portrait of Johann Kleberger Such signs, designed to accumulate the influences exerted by one or another astral (angelic, demonic, etc.) entity, go back (in most cases) to the famous work of Cornelius Agrippa “Occult Philosophy” (first published, it is believed, in 1531) , rice. 37, and are found on many objects of a “magical” nature created in the era in question, Fig. 38-39.  Rice. 37. Page from “Occult Philosophy” with images of signs
Rice. 38. “Astronomical Seal of Leo” (ASTRONOMICVM SIGILLVM LEONIS).
Rice. 39. “Magic sign” of Regulus. Enlarged fragment of Fig. 5 It is clear that the presence of this sign in Kleberger’s portrait was supposed, on the one hand, to “attract” the beneficial influence exerted by Regulus to its owner, and on the other, it emphasized that the picture itself was painted “under a lucky star.” The latter, by the way, should be understood almost literally, since Regulus, as mentioned above, according to the ideas of that time “indicated happiness.” Thus, in general, it turns out that the portrait of Johann Kleberger painted by Dürer is a kind of talisman, which, according to the author or customer, should bring happiness to the person depicted on it, sent by the “royal” Regulus. And also, apparently, “endurance, patience, perseverance, thoroughness, ... steadfastness, constancy,” [Saplin] bestowed by the “neutral positive” Saturn. By the way, the same “lucky” meaning could partly (along with the heraldic and astronomical components indicated above) be carried by the image of a clover, since it is a well-known symbol of good luck. 7. “The sun in the heart of Leo” and the birthday of Johann KlebergerSo, we are convinced that the portrait of Johann Kleberger is literally completely permeated with the symbolism of Regulus. Both in an abstract allegorical and, most importantly, in a concrete astronomical sense. Moreover, the first is only an appendage to the second, allowing a more complete revelation of the multi-layered content of the picture. And now it would be useful to see what Scaligerian historians say about this. We already know that the symbolism of Regulus was noticed by them. But how exactly is it interpreted by them? The answer to this question is found in the description of the medal given in: “Agrippa"s seals and characters frequently occur, with many other signs, on post-mediaeval medallions and seem usually to be taken from his works. … The medallion of the Sun shows the Sun in his astrological house, the Lion, as on the coat of arms of Persia. On the reverse there appear the character of the Intelligence of the Sun (Nachiel), the zodiacal sign of Leo, and the sign of the "heart of the lion" (Cor leonis), i.e., the bright fixed star Regulus. This sign appears also in Agrippa's book, and moreover, Durer depicted it in the portrait of Kleeberger in Vienna; for Kleeberger was born in this significant conjunction of the Sun and Regulus (Sol in Corde leonis)." Translated: “Seals and characters (“conventional icons for frequently used concepts found on astrological medals and talismans, magical symbols”, [Saplin] - A.) Agrippa is often found, along with many other characters, on post-medieval medallions and is usually taken from his work. ... The Sun Medallion shows the Sun in its astrological house, Leo, as in the coat of arms of Persia. The reverse side depicts the character of the spirit of the Sun (Nachiel), the zodiac sign of Leo and the sign of the “heart of the lion” (Cor leonis), that is, the bright fixed star Regulus. This sign is also present in the book of Agrippa and, moreover, Dürer depicted it in the portrait of Kleberger from Vienna, because Kleberger was born at a significant conjunction of the Sun with Regulus (Sol in Corde leonis)." Without going into the subtleties of interpretation of each of the listed symbols, it is proposed here that the reason that prompted Dürer to place the signs of Regulus on the portrait of Kleberger was simply that at the moment of his birth the Sun was in conjunction (“conjunction”) with the latter. Or, literally translated, it was “in the Heart of the Lion” (“Sol in Corde leonis”). That's it, simple and uncomplicated. With such a simple indication of Kleberger’s birth “under the sign of Regulus,” according to the author, the astronomical content of the portrait is exhausted. In principle, one could hardly expect anything else, since in the overwhelming majority of cases, when encountering one or another ancient zodiac, Scaligerian historians do their best to avoid the potentially dangerous topic of its astronomical dating as quickly as possible (since in this case it is rarely possible to obtain acceptable dates) into a convenient plane of safe reasoning “about something else.” For example, about the occult (however, in fairness it should be said that the work is devoted to a rather narrow issue of constructing “magical seals”, and in this regard there can be no complaints about it). However, the author of the above-quoted description quite correctly notes that on the birthday of Johann Kleberger the Sun was “in the heart of Leo.” In fact, this is exactly how everything was exactly on the date of the latter’s fortieth anniversary found above. The sun that day was in the closest (about two degrees of arc) approach to Regulus. That is, indeed, “in the Heart of the Lion,” . And it is clear that the same picture was observed forty years before this date - on the day Kleberger was born. From this point of view, the latter was indeed born “under the sign of Regulus.” After these words, it may seem that the same is true regarding the Scaligerian date of birth of Johann Kleberger. However, it is not. After all, the “anniversary” August 15, 1624 is the fifteenth day of August according to the new style. That is, according to the Gregorian calendar we are used to. Counting from him forty years ago, we find ourselves on the day Johann Kleberger was born - August 15, 1584. Again a new style, since the Gregorian reform was carried out two years earlier than this date. But if, as Scaligerian historians claim, Kleberger was born in 1485/86 and died in 1546, then August 15 was Julian throughout his life. And if you now launch any computer planetarium program and open a “picture” in it for August 15, 1485, it turns out that the Sun on that day was, although quite close - about 11 degrees of arc - but, nevertheless, not “in the Heart of a Lion.” And besides, in such a magnitude of approach - about ten degrees, that is, a third of the average length of the constellation - there is, in general, nothing particularly outstanding, and no matter what arbitrarily chosen position of the Sun one takes, there will certainly be a fairly bright star located at a comparable , or even noticeably smaller distance from the latter. This means that even the above-mentioned and extremely “reduced” explanation proposed by Scaligerian historians corresponds much better to the independent astronomical dating of Kleberger’s horoscope that we obtained than to the Scaligerian dating of the era of his life. By the way, speaking about the presence of the Sun “in the heart of Leo”, one cannot help but pay attention to the fact that on Kleberger’s anniversary date of August 15, 1624, by a happy coincidence, an even more striking event occurred - the conjunction of four “luminaries” at once: Regula , the Sun, Saturn and Mars, simultaneously appearing in a tiny area of the sky, . And although such a “double” connection, for obvious reasons, could not be directly observed, it is clear that, from the point of view of various kinds of mystical ideas of that era, such an event should have been regarded as important (and, due to the logic stated above, very favorable ) sign. This means that it could well have become one of the immediate reasons for painting the picture in question. 8. Is the star or sign zodiac depicted in Kleberger's portrait?Before we come to the final conclusion, let’s deal with a few expected objections. First of all, surely one of the defenders of Scaligerian chronology, who does not want to part with its proposed dating of the portrait in question in 1526, will object in the spirit that the stars symbolize, in fact, not planets at all, but “just stars” of the constellation Leo ( and, accordingly, there is no horoscope in Kleberger’s portrait,” he adds with a sigh of relief, “no). The following can be answered. Firstly, the very configuration of these stars does not even remotely resemble the pattern of the constellation Leo. Secondly, there are more than six stars in Leo that have their own names - assuming that they are the ones depicted. Thirdly, if the artist wanted to indicate the birth of Kleberger “under the sign of Regulus” and nothing more, then his “magic” sign alone, completing the inscription with his name and age, would have been enough. But the main thing, and this is fourth, lies elsewhere. Namely, the very fact that identifying the stars on with planets and not with some nameless stars immediately leads to two dates that ideally agree with each other, with the astronomical situation and all the details of the picture, directly indicates that such identification is correct, since It is quite obvious that such an impressive result could hardly have happened by pure chance. Another potential opportunity to “tie up loose ends,” that is, to obtain the “correct” date from the Scaligerian point of view for the horoscope in the portrait of Johann Kleberger, opens up if we assume that this horoscope is not an observational-stellar horoscope, but a calculation-sign one, that is, the positions of the planets in it are tied not to the constellation, but to the sign of the same name (but by no means coinciding with it) Leo. It is clear that such an interpretation, in this case, is extremely doubtful, but, nevertheless, for the sake of completeness, it was also taken into account. A verification calculation performed in the ZET 9 program showed that in the interval from 1400 to 1800 the iconic horoscope “all planets in Leo” has no solutions. If we weaken the required accuracy and conduct a second search in the most “gentle” version, allowing planets to “climb” by as much as 10 degrees into the signs of Cancer and Virgo adjacent to Leo, then two solutions appear - September 2-6, 1622 AD. and August 12-16, 1624 AD. (both in new style). What do we see here? Firstly, the fact that none of these solutions still fell into the 16th century required to confirm the Scaligerian dating of the painting. And secondly, the date August 15, 1624 was again among them! That is, the date of Kleberger's fortieth anniversary. According to the calculation, all the planets that day were in the sign of Leo, and only Jupiter was at 7 degrees Virgo. True, such a significant deviation practically excludes the possibility of making this decision, since Jupiter is not a fast-moving Mercury, and it takes about 12 days to travel just 1 degree along the ecliptic, and 7 degrees takes almost three whole months! But if you close your eyes to this circumstance (assuming that the calculation was carried out according to a theory that gave extremely poor accuracy, and the would-be astrologer who performed it had never looked at the real sky in his life), then it turns out that even with the “sign” interpretation of the horoscope depicted in Kleberger's portrait, the latter's decision - 1624 - remains unchanged. 9. About the September and January counting of yearsAnother possible objection that should be considered arises from the direct comparison of the August-September 1624 dates obtained above with each other. Namely, some doubt may be caused by the fact that one of these dates - the “anniversary” - turned out to be recorded in the new style, and the second - in the old style, and moreover, with the beginning of the year in September. But by that time four decades had already passed since the introduction of the Gregorian calendar, not to mention the fact that New Year in Western Europe in that era, it is believed that it began in March. The question is, how can all this be combined? More precisely, could the second of these dates really be perceived - by Kleberger and at least some of his contemporaries - as the date of the beginning of the new 1625 (September) year? And did they really live simultaneously according to two calendars, switching between them every now and then? As a matter of fact, with regard to the last question, it could well be so. But, in this case, the confusion of dates from different calendars can be explained in a more elegant way. In fact, as we have already seen, the painting in question is not so much an ordinary portrait as a talisman, as is directly indicated by the “magic” sign of Regulus depicted on it. This means that it is through this prism, first of all, that one should look at the dates recorded on this talisman portrait. And then everything is explained very simply. Indeed, it is obvious that the “magical” date (that is, the date in which some “magical” meaning is invested), namely, the “New Year’s” resolution of September 1 (old style) 1624, does not have to be written down at all. as everybody". On the contrary, the more unusual and, at first glance, more incomprehensible, the better. This is on the one hand. On the other hand, it is well known that it was fashionable to consider “Ancient” Egypt as the birthplace of the “magical” practices that were “revived” in the era of the 16th-17th centuries (“The Renaissance magician thought that he was returning to Egyptian wisdom”, and also: “... the content of the Hermetic texts nourished in the Renaissance magician the illusion that before him is a mysterious, precious story about the most ancient Egyptian wisdom, philosophy, magic” [Yeats]). Therefore, attention to what this or that “ordinary” date indicates, if you look at it from the “ancient” Egyptian point of view (and, of course, according to the old style), is quite understandable. By the way, it is worth noting here that the presence of the “magic” sign of Regulus in the portrait in question could carry, along with the above, another aspect. Namely, this sign, borrowed from “Occult Philosophy”, well known in certain circles - a book dedicated to “secret” knowledge hidden from the uninitiated - could not only accumulate the beneficial influence of Regulus, directing it to the person depicted (that is, fulfill its direct “ magical" function), but also, in itself, to signal to a sufficiently knowledgeable viewer that the portrait contains some kind of secret message hidden from ordinary eyes. That is, pointing to dates recorded in an astronomical way that is not obvious to the untrained eye, revealing, when interpreted in a mystical spirit, the meaning inherent in the picture to a much greater extent than a simple signature with numbers. A small nuance. Strictly speaking, when correlating the January year with the September year, two recalculation options are permissible. In the first, the September year is ahead of the January year by four months (and on September 1, 1624 of the January year, the September 1625 year begins), in the second, it lags behind by eight months (and then the beginning of the September 1624 year falls on the same date). From a formal point of view, both of these options are equivalent, but from a “magical” point of view, the first is undoubtedly preferable, since recalculating the date “ahead” exactly corresponds to the aspiration to the future, that is, the reason for which any talisman is created. Therefore, the above-accepted correlation of the date September 1, 1624 with the beginning of the 1625th year, and not the 1624th September year, in the case of the Kleberger portrait is fully justified. And one last thing. In light of everything stated above, it is possible that the “digital” date “1526” placed on the portrait, meaning, as we now understand, 1625, could, in fact, refer not to January, but to the same September year. And, therefore, this portrait itself could have been painted by Dürer as early as September 1624 (the usual January year for us), that is, literally “hot on the heels” of the events that had just passed. 10. Conclusion. When did Albrecht Durer live?So, let us briefly list the three main astronomical plots that follow from the obtained datings of the zodiac depicted in the portrait of Johann Kleberger: 1) on the first of the two dates recorded in the portrait—the date of Kleberger’s fortieth anniversary—Mercury, the patron saint of money and wealth, “reigned” in the night sky; 2) on the same date the “royal” Sun was in conjunction with the “happiness-bringing” Regulus; 3) finally, the same “happy” Regulus, together with Saturn (also identified with the “eternal” Chronos), “ruled” in the sky on the second – “New Year’s” – date. It turns out a whole bunch of events, each of which, according to medieval views, was interpreted as a favorable sign sent down by the heavens themselves and promising wealth, happiness and good luck to Kleberger who was awarded them. In the same vein - as a symbol of good luck - the heraldic attribute of Kleberger depicted in the picture - the clover leaf - could also be “read”. Finally, the intention of the customer of the painting is most clearly revealed by the “magic sign” of Regulus placed on it. All this, taken together, clearly indicates that the portrait of Johann Kleberger is not just an ordinary work of painting, intended to decorate the interior and delight the eyes of viewers, but, first of all, a talisman, at the heart of the creation (and correct understanding) which lay the ancient language of astronomical symbolism, Fig. 40.  Fig, 40. Love talisman of Catherine de Medici. On both sides of it In conclusion, we can say the following. We see that unbiased astronomical dating of the zodiac depicted in the portrait of Johann Kleberger allows us not only to determine the true time of his life - as it turns out, a century closer to us - but also to obtain another important proof that the numbers 5 and 6 once really had a meaning opposite to the modern one. Today's number 6 was originally perceived as a five, and vice versa, today's number 5 at first meant a six. As for Albrecht Dürer, whose author’s mark is signed on this wonderful “magical” portrait, the result obtained once again confirms the statement formulated and substantiated in detail in [KHRON5] that either he (like Kleberger himself) lived later than this It is generally believed that he “went” into the past as a result of a hundred-year chronological shift, or a certain artist with that name really existed in the 15th-16th centuries and was very famous, but practically nothing remains of his original works. And, looking today at the works attributed to Dürer, we see, in fact, the creations of a whole galaxy of remarkable masters, created mainly in the 17th century and retroactively declared works of the 15th-16th centuries. 11. Conclusions1. In the portrait of Johann Kleberger written by Albrecht Dürer (whoever was hiding, in this case, behind this famous name), the artist depicted not one, as it seems at a quick glance, but three related dates at once. 2. The first - August 15 (new style) 1624 - indicated exactly the fortieth birthday of the person depicted. 3. Second. - September 1 (old style) 1624 - marked the moment of the onset of the new year 1625 according to the “ancient” Egyptian account. 4. The third - presented in explicit form and indicating the time of the painting, the date “1526” - did not, in fact, indicate the end of the first quarter of the 16th century, as it is perceived today, but, taking into account the old meaning of the numbers 5 and 6, the same the year itself is 1625 (according to January or September calculations). 5. In general, the portrait of Johann Kleberger is a highly artistic talisman created on the occasion of the customer’s anniversary, which should contribute to a long life full of earthly joys, in which at least the “explicit” (portrait) and “secret” (hidden from the untrained eye) stand out. astronomical), and, in fact, “magical-functional” (following from the previous one and additionally emphasized by the “sign of Regulus”) components. Literature[KhRON1] Fomenko A.T. "The Foundations of History". - Moscow, Rimis, 2005.[KhRON2] Fomenko A.T. "Methods". - Moscow, Rimis, 2005. [KhRON4] Nosovsky G.V., Fomenko A.T. "New chronology of Rus'". - Moscow, Rimis, 2004. [KhRON5] Nosovsky G.V., Fomenko A.T. "Empire". - Moscow, Rimis, 2004. Nowotny K.A. "The Construction of Certain Seals and Characters in the Work of Agrippa of Nettesheim". — Journal of the Warburg and Courtauld Institutes, Vol. 12, 1949, pp. 46-57. Roos A.M. ""Magic Coins" and "Magic Squares": The Discovery of Astrological Sigils in the Oldenburg Letters." — Notes and Records of the Royal Society of London, Vol. 62, No. 3, 2008, pp. 271-288. [Yeats] Yeats F. "Giordano Bruno and the Hermetic Tradition." - Moscow, New Literary Review, 2000. [Saplin] Saplin A.Yu. “Astrology is for everyone. Encyclopedia". - Moscow, Geleos, 2007. | ||||||
Albrecht Durer's magic square. The 4x4 magic square depicted in Albrecht Durer's engraving "Melancholy I" is considered the earliest in European art. The two middle numbers in the bottom row indicate the date of creation of the painting (1514). The sum of the numbers on any horizontal, vertical and diagonal is 34. This sum is also found in all corner squares 2?2, in the central square (10+11+6+7), in the square of corner cells (16+13+4+1 ), in squares built by the “knight’s move” (2+8+9+15 and 3+5+12+14), in rectangles formed by pairs of middle cells on opposite sides (3+2+15+14 and 5+8 +9+12).
Slide 13 from the presentation "Square in life". The size of the archive with the presentation is 388 KB.Geometry 8th grade
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Based on a theoretical analysis of 4×4 pandiagonal squares, their “structure” features are shown: the invariants of the structure of 4×4 pandiagonal squares are pairs of numbers equal in total to one of the two Fibonacci numbers - 13 or 21. It is revealed that any variant of the set of six digits of this and that There are 51 similar pandiagonal squares 4x4 forming a continuous symmetrical configuration. A geometric figure “cube in a cube” has been constructed, which has the properties of “golden symmetry” of pandiagonal squares 4x4. All numbers of diagonals of a cube have the properties of “golden symmetry” (two numbers form in one case - the total number is 13, in another – 21), and all planes that have 4 angles (numbers) of both internal and external squares geometric figure form a total Fibonacci number of 34.
Introduction
Based on a theoretical analysis of Khajuraho, Dürer squares and similar 4x4 squares, the features of their “structure” have been identified: the invariants of the structure of pandiagonal 4x4 squares are pairs of numbers equal in sum to one of the two Fibonacci numbers - 13 or 21.
A magic square is an n×n square table filled with n 2 different numbers such that the sum of the numbers in each row, each column and on both diagonals is the same. The earliest unique 4x4 magic square was discovered in an 11th century inscription Indian city Khajuraho. The 4x4 square depicted in Albrecht Durer's engraving "Melancholy" is considered the earliest in European art (1514). The sum of the numbers of a Durer square on any horizontal, vertical and diagonal is 34. This sum is also found in all 2x2 corner squares, in the central square, in the square of corner cells, in squares built by the “knight’s move” (2+12+15 +5 and 3+8+14+9), at the vertices of rectangles parallel to the diagonals (2+8+15+9 and 3+12+14+5), in rectangles formed by pairs of middle cells on opposite sides (3+2 +15+14 and 5+8+9+12). Most additional symmetries arise from the fact that the sum of any two centrally symmetrically located numbers is 17.
There are 48 pandiagonal 4x4 squares with rotation and reflection precision. If we also take into account the symmetry with respect to toric parallel translations, then only 3 significantly different squares remain (Figure 2).
Main part
I analyzed the “structure” of 4x4 pandiagonal squares and identified the invariant parts of their structure (Figure 3). The invariants of the structure of 4x4 pandiagonal squares are pairs of numbers equal in sum to one of two Fibonacci numbers - 13 or 21. Various options for symmetrically combining these number pairs form a set of 4x4 pandiagonal squares.
The Dürer square (and similar 4x4 pandiagonal squares) have golden ratio symmetry. For example, in Figure 4, red and blue squares show variants of symmetries, in which the arithmetic mean of the sum of the red components of the squares in possible positions (4 or 2, when rotating in different directions) is 51. Thus, the sum of all numbers of the square is 136, of which 85 are blue, 51 are red. 136/85=1.6; 85/51=1.667.
Based on the Dürer square, we constructed a geometric figure “cube in a cube”, which has the symmetry properties of pandiagonal 4×4 squares (Figure 5). Such a “transformation” became possible by arranging the vertical columns of numbers in the Dürer square at a certain angle, thus forming a cube within a cube. At the same time, all the numbers of diagonals of the cube have the properties of “golden symmetry” (two numbers form in one case – the total number is 13, in the other – 21), and all planes that have 4 angles (numbers) of both the internal and external squares of the constructed figure form a total Fibonacci number of 34.
Conclusion
- Based on a theoretical analysis of 4x4 pandiagonal squares, their “structure” features are shown: the invariants of the structure of 4x4 pandiagonal squares are pairs of numbers equal in sum to one of two Fibonacci numbers - 13 or 21.
- It has been revealed that any version of the set of six digits of the Dürer square and similar pandiagonal 4x4 squares, forming a continuous symmetrical configuration, is equal to the total number - 51.
- A geometric figure “cube within a cube” has been constructed, which has the properties of “golden symmetry” of pandiagonal 4x4 squares. All numbers of diagonals of a cube have the properties of “golden symmetry” (two numbers form in one case – the total number is 13, in the other – 21), and all planes having 4 angles (numbers) of both the internal and external squares of a geometric figure form in The total Fibonacci number is 34.
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The magic square, reproduced by the German artist Albrecht Durer in the engraving “Melancholy”, is known to all researchers of magic squares.
Square in its usual form (Fig. 6.1):
Figure 6.1
Interestingly, the two middle numbers in the last line of the square (they are highlighted) make up the year the engraving was created - 1514.
It is believed that this square, which so fascinated Albrecht Durer, came to Western Europe from India at the beginning of the 16th century. In India, this square was known in the 1st century AD.
It is believed that magic squares were invented by the Chinese, since the earliest mention of them is found in a Chinese manuscript written 4000-5000 BC. That's how old magic squares are!
Let us now consider all the properties of this amazing square. But we will do this on another square, the group of which includes the Durer square.
This means that the Dürer square is obtained from the square that we will now consider by one of the seven main transformations of magic squares, namely a rotation of 180 degrees. All 8 squares that form this group have properties that will now be listed, only in property 8 for some squares the word “row” will be replaced by the word “column” and vice versa.
You can see the main square of this group in Fig. 6.2.
Figure 6.2
Properties of this square:.
Property 1. This square is associative, that is, any pair of numbers symmetrically located relative to the center of the square gives a sum 17=1+n2.
Property 2. The sum of the numbers located in the corner cells of the square is equal to the magic constant of the square - 34 .
Property 3. The sum of the numbers in each corner 2x2 square, as well as in the central 2x2 square, is equal to the magic constant of the square.
Property 4. The magic constant of a square is equal to the sum of the numbers on opposite sides of the two central 2x4 rectangles, namely: 14+15+2+3=34, 12+8+9+5=34.
Property 5. The magic constant of the square is equal to the sum of the numbers in the cells marked by the move of the chess knight, namely: 1+6+16+11=34, 14+9+3+8, 15+5+2+12=34 and 4+10+13 +7=34.
Property 6. The magic constant of a square is equal to the sum of the numbers in the corresponding diagonals of the 2x2 corner squares adjacent to the opposite vertices of the square.
For example, in the 2x2 corner squares, which are highlighted in Fig. 4, the sum of the numbers in the first pair of corresponding diagonals: 1+7+10+16=34 (this is understandable, since these numbers are located on the main diagonal of the square itself). The sum of the numbers in the other pair of corresponding diagonals: 14+12+5+3=34.
Property 7. The magic constant of the square is equal to the sum of the numbers in the cells marked by a move similar to the move of a chess knight, but with an elongated letter G. I show these numbers: 1+9+8+16=34, 4+12+5+13=34, 1+2 +15+16=34, 4+3+14+13=34.
Property 8. In each row of the square there is a pair of adjacent numbers, the sum of which is 15, and another pair of adjacent numbers, the sum of which is 19. In each column of the square there is a pair of adjacent numbers, the sum of which is 13, and another pair of also adjacent numbers , the sum of which is 21. brain cell square sudoku
Property 9. The sums of the squares of the numbers in the two outer rows are equal to each other. The same can be said about the sums of the squares of the numbers in the two middle rows. See:
12 + 142 + 152 + 42 = 132 + 22 + 32 + 162 = 438
122 + 72 + 62 + 92 = 82 + 112 + 102 + 52 = 310
Numbers in the columns of a square have a similar property.
Property 10. If we inscribe a square with vertices in the middle of the sides into the square under consideration (Fig. 6.3), then:
- · the sum of the numbers located along one pair of opposite sides of an inscribed square is equal to the sum of the numbers located along the other pair of opposite sides, and each of these sums is equal to the magic constant of the square;
- The sums of squares and sums of cubes of the indicated numbers are equal:
- 122 + 142 + 32 + 52 = 152 + 92 + 82 + 22 = 374
- 123 + 143 + 33 + 53 = 153 + 93 + 83 + 23 = 4624
Figure 6.3
These are the properties of the magic square in Fig. 5.2
It should be noted that in an associative square, which is the square in question, you can also perform such transformations as rearranging symmetrical rows and/or columns. For example, in Fig. 5.4 shows a square obtained from the square in Fig. 4 by rearranging the two middle columns.
Figure 6.4
In the new associative squares obtained by such transformations, not all of the properties listed above are satisfied, but many of the properties hold. Readers are invited to check the fulfillment of the properties in the square shown in Fig. 6.4.
