from http://www.netmastersinc.com/secrets/magic_squares.htm Mark Swaney on the History of Magic Squares 4 9 2
3 5 7
8 1 6 This is a magic square of order 3 (three numbers to the side of the square). If you add up any row, column, or diagonal, it sums to the same number, 15. There are magic squares of order 4, 5, 6, etc.
See this link for a listing of magic squares of order 3 through 11: My friend Mark Swaney has been working on the history of Magic Squares and has said yes to my passing on some of his preliminary results with the following warning: “You gotta tell them tha it’s just ripped hot off the neurons, and may have a detail or two outof place. I’m reading all this stuff and then roaring off an epistle. Later, I always think I should have done it differently, but what the hell? Also, I find that I like to write a lot of text when I’m feeling radioactive.”
The history of magic squares is murky, mysterious, and has not been well researched by academics. Consequently the claims are contradictory, and in some cases exaggerated. Very little is known about the origin of magic squares. Next to nothing is known about the movement of the idea of a magic square before about 1300 AD. Three cultures are known to have created magic squares, the Chinese, the Indian, and the Arabic. In each culture they were viewed as having supernatural properties…. China
The first magic square in history was created in China by an unknown mathematician, probably sometime before the first century AD. Called the Lo Shu square, it is a magic square of 3 that was said to have appeared on the back of a turtle that came up out of the river. Lo Shu supposedly means “river map” and the story of the appearance of the turtle had to do with a sacrifice to the river god. Right from the beginning we are seeing an essentially mathematical construction combined with the supernatural. I have not found an analysis of the story of the turtle and the Lo Shu square from the point of view of folklore or mythology that would shed more light on the story. The Lo Shu square is later associated with the floor plan of a mythical palace, that of Ming’tang. Again, this is fragmentary, I have seen a diagram that shows the floor plan, but no explanation as to what the thinking about the square was, why it was used as a floor plan for a palace, or other information to flesh out the picture. The Lo Shu square is also connected to the I-Ching, though there is no explicit plan of correspondence that I know of. The oldest documents that refer to the Lo Shu square are ambiguous, but one reference lists a Shu Ching in 650 BCE who makes a reference to the “river map” which may be the magic square of 3. In 500 BCE, and 300 BCE, the river map is mentioned, but no explicit magic square is given. In 80 AD Ta Tai Li Chi gives the first clear reference to a magic square. In 570 AD Shuzun gives an actual description of a magic square of 3. Not until 1275 do we hear of the Chinese making squares of order larger than 3. Norman Biggs says that this is because the Chinese regarded the Lo Shu square as an object of the supernatural, rather than as an object of human curiosity, and it was therefore not a subject for study. India
We find the first magic square of 4 in the first century in India by a mathematician named Nagarajuna. This is all that I know at the moment about the early development of the magic square in India. However, India is the birthplace of much superior mathematics, and was advanced in other areas of combinatorics at an early date. I would be surprised if it did not eventually turn out that India has an older tradition involving magic squares. Still, this approximate date is interesting for other types of analysis. The next known date in the Indian development is an 11th or 12th century Jaina inscription that includes a magic square of 4. This particular magic square of 4 has unusual properties not found in other magic squares before that time, and the whole class of squares having these properties is called “Jaina squares”, including squares of order larger than 4. I have no information on the document, why it includes the magic square, or what connection it has to the Jaina religion in medieval India. Much remains to be explained. Islam
The first magic squares of 5 and 6 appear in an encyclopedia in Baghdad about 983 AD by Ikhw’n al-Saf’ Ras’il, though several earlier Arab mathematicians also wrote about magic squares. How it came to pass that the Arabs acquired knowledge of magic squares is unknown. It is not known if they invented them separately or if they were introduced to them by another culture. Biggs assumes that the Arabs got the idea from the Chinese, though he doesn’t know how the connection was made. I think it far more likely that the Arabs got magic squares from the same source that they got decimal arithmetic, namely India. The Arab Jihad of the 7th century succeeded in conquering portions of India, and the Arabs absorbed a great deal of Indian mathematics and astronomy. It is known that many other aspects of combinatorial mathematics passed from India to the Arabs in this way. Al-Buni was an Arab mathematician that worked on magic squares and also believed in the mystical properties of magic squares, though no details on this number mysticism are available. Al-Buni did his work on the squares about 1200 AD. Sources have also referred to the Arabs using magic squares in making astrological calculations and predictions, again no details are given. The association of the squares with astrology and the heavens appears to be original with the Arabs, but again, much is unknown concerning the Indian tradition. Europe
It is from the Arabs that the West finally receives the idea of magic squares. In 1300 Manual Moschopoulos, a Greek Byzantine scholar, writes a mathematical treatise on the subject of the magic squares. Moschopoulos’ book builds on the work of Al-Buni who preceded him. Western authors are quick to point out that Moschopoulos treats the squares in a purely mathematical way in contrast to the mystical ideas of the Arabs. Moschopoulos is generally considered to be the first westerner to know of the squares. A mistaken attribution of knowledge to Theon of Smryna in about 130 AD has continued to be cited, but the “square” in question is definitely not a magic square, being just a natural square.
After Moschopoulos, in the 1450’s Luca Pacioli of Italy worked on magic squares and owned a large collection of examples of magic squares. With Pacioli we come to the doorstep of the known Western mystical tradition concerning magic squares. What Pacioli himself believed about the squares I don’t know, but in the 1480’s Italy was to see the birth of the Renaissance which revolutionized European thinking. Marsilio Ficino wrote about and propounded a school of magic based on his translations of thd Hermetic documents that were at the time believed to be as ancient as Moses. Pico Della Mirandola wrote the “Nine Hundred Theses” – much of it based on the translations of older Jewish Kabbalistic texts. Artists like Albrecht Durer eagerly absorbed the new perspective painting based on the mathematical developments of Della Franscisca, who was popularized by the later books of Pacioli.
In about 1510 Cornielius Agrippa, that problematical character, wrote “De Occulta Philosophia” in which he expounds on the powers of the magic squares, and supplies examples of them in the orders 3- 9. This book became famous throughout Europe and was very influential until the counter-reformation and the witch-hunts that followed. Most what is commonly thought of about Agrippa is the result of the witch-hunts and propaganda, i.e. he was a sorcerer, he was in league with the devil, etc. The truth about Agrippa and his book is much more complex than that, and in the explanation of Agrippa’s book we get the first inkling of a detailed worked out system of mysticism concerning magic squares. However, though we find out some details about the squares in their role as supernatural devices, we are still left with conflicts and unanswered questions.
In 1514 Albrecht Durer made his famous woodcut “Melancholia I,” which features a magic square of 4 on the wall behind the “brooding genius” that became the archetype of all the “thinker” type sculptures in later years. The reason for the magic square of 4 being included in the woodcut has been analyzed by the authors of “Saturn and Melancholy”. Briefly, the square of 4 is the square of Jupiter. The planet jupiter was considered beneficial and was associated with the “sanguine” humor. Even today we speak of someone’s being “jovial” at a party. Durer’s brooding genius suffers from melancholia, which we call depression, and the square of Jupiter was thought to bring down the influence of the planet Jupiter, thereby helping to cure the depression. The Squares and the Planets
This is an example of the theory of magic propounded by Marsilio Ficino. Ficino’s magic is a kind of sympathetic magic where objects, colors, sounds, etc. are all categorized as to what “influences” they excite. Ficino’s influences come primarily from the planets, the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. The magic is aimed at “drawing down” the influence or power of specific planets in order to accomplish some end, such as protection from disease or a psychological cure. In this magic we see the role of the squares as being the mathematical archetypes of the planets themselves. As each square has a set of characteristic numbers, these numbers then also carry the influences of the various planets. In this way certain numbers can be said to be “Solar” or “Lunar” numbers.
In this system, for our study, the important issue to understand is how the particular planets come to be associated with particular squares. More than one source has it that the correspondences between the squares and the planets were the invention of Agrippa himself. The description of Agrippa and his book by Francis Yates makes it appear that Agrippa made no original contributions to magical theory in his book, but merely collected the thought of others. Other sources simply say that the Arabs assigned the squares to the planets. David Fidler in his book “Jesus Christ, Sun of God” says that the arrangements came from the Babylonians. The ancient system of cosmology had 7 planets, each in a concentric shell that rotated around the earth. The Babylonians believed that the closest planet was the moon, followed by Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. They placed the order of the squares such that the smallest squares were associated with the farthest planets, thus Saturn is 3, Jupiter is 4, etc. This relationship is important for several reasons, but the one reason that is most striking is the fact that the system assigns the square of 6 to the sun. By making this assignment, the system is made to resonate with one of the most ancient of numerological systems, namely that of the Sumerians. It was the Sumerians with their Solar worship and their sexigesimal counting system that firmly fixed the hours of the day at 24, sun nominally rising at 6:00AM and setting at 6:00PM, and who gave us the still used 360 degree circle. The association of the number 6 with the sun is a very ancient western tradition. Pythagoras on account of numerical theory called 6 the first “perfect” number. In view of these facts, the magic square of 6 with a sum total of 666 must have made quite an impression even in the 14th century, the earliest date that modern conventional scholarship will allow a western knowledge of magic squares.
– Mark Swaney, January, 2000 Note from someone else:
See if you can find out anything about the psycho-spiritual/brainwashing disciplines [Hasan i Sabah’s] Assassins used, would you. Maybe the same methods were used in working with the squares, a la the Rabbi of Damascus and his Kabbalistic system for meditating on the squares. Mark Swaney writes:
The really interesting thing about Sabah for our studies is that in addition to being a military genius, he was also known to be a scholar. The organization he created, the Hashishim, or Assassins, was a “Masonic” military organization. By the way, the words Assassin and Hashishim and Hashish are all thought to be corruptions of Sabah’s first name, Hassan. The Assassins were in essence Kamikaze’s. They were trained to strike an enemy and not escape, but stay and fight to the death. So you can see why these people were so feared.
But the organization was not solely based on military/political adventures. That’s the mystery. Sabah was known to have amassed a large library in his fortress. He was known to have had an interest in mathematics, and to have encouraged the study of mathematics and philosophy by his followers. The Assassins practiced initiation rites, and had strict grades of hierarchy, so that modern historians have described them as “Masonic” in nature. Sabah and the Assassins also had intriguing contacts with the Crusaders that I am now trying to find out more about. All this is hugely interesting for all the obvious reasons.
The initiation rites are the probable source of the story about Sabah’s use of drugs to fool initiates into thinking they had gone to heaven when in reality they were only in Sabah’s garden. This story was written by Marco Polo who passed through the area of the Eagle’s Nest 150 years after Sabah and the Assassins. There is no other documentation to back it up, and so it must be taken with a grain of salt. Personally, I think that no matter how much hash someone ate, it is very unlikely that they would wake up after falling asleep and think themselves to be in heaven. But the available evidence does indicate that the Assassins practiced some form of discipline that may have bordered on modern theories of mind control. Another example of Sabah’s prescient inventions.
After the Mongols conquered the Eagle’s Nest in the late 13th century, the Assassins and the Ismailis in general declined from any power in the political sense. The Mongols burned the library at the Eagle’s Nest, so no books by Sabah or the Assassins survive today. The whole essence of the organization built by Sabah rested on obedience, faith, and above all else, secrecy. We should not be surprised that a great deal of the knowledge of the Assassins was lost. We should also keep in mind that secrecy was one of the hallmarks of the gnostics and other early mystery cults.
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