Mark Swaney on the History of Magic Squares
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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 that it's just ripped hot off the neurons, and mayhave a detail or two out of place. I'm reading all this stuff and then roaringoff an epistle. Later, I always think I should have done it differently, butwhat the hell? Also, I find that I like to write a lot of text when I'm feelingradioactive.
After Mark's review of the history of Magic Squares, I have listed some further information and references on the squares. Mark:I am still getting references andpicking up information, so the following is subject to revision andexpansion.The history of magic squares is murky, mysterious, and has not beenwell researched by academics. Consequently the claims arecontradictory, and in some cases exaggerated. Very little is knownabout the origin of magic squares. Next to nothing is known about themovement of the idea of a magic square before about 1300 AD. Threecultures are known to have created magic squares, the Chinese, theIndian, and the Arabic. In each culture they were viewed as havingsupernatural properties.ChinaThe first magic square in history was created in China by an unknownmathematician, probably sometime before the first century AD. Calledthe Lo Shu square, it is a magic square of 3 that was said to haveappeared on the back of a turtle that came up out of the river. Lo Shusupposedly means "river map" and the story of the appearance of theturtle had to do with a sacrifice to the river god. Right from the beginning we are seeing anessentially mathematical construction combined with the supernatural. Ihave not found an analysis of the story of the turtle and the Lo Shusquare from the point of view of folklore or mythology that would shedmore light on the story. The Lo Shu square is later associated with thefloor plan of a mythical palace, that of Ming'tang. Again, this is fragmentary, Ihave seen a diagram that shows the floor plan, but no explanation as towhat 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 isalso connected to the I-Ching, though there is no explicit plan ofcorrespondence that I know of. The oldest documents that refer to theLo Shu square are ambiguous, but one reference lists a Shu Ching in 650 BCE who makes areference to the "river map" which may be the magic square of 3. In 500BCE, and 300 BCE, the river map is mentioned, but no explicit magicsquare is given. In 80 AD Ta Tai Li Chi gives the first clear reference to amagic square. In 570 AD Shuzun gives an actual description of a magicsquare of 3. Not until 1275 do we hear of the Chinese making squares oforder larger than 3. Norman Biggs says that this is because the Chineseregarded the Lo Shu square as an object of the supernatural, rather thanas an object of human curiosity, and it was therefore not a subject for study.
IndiaWe find the first magic square of 4 in the first century in India bya mathematician named Nagarajuna. This is all that I know at the momentabout the early development of the magic square in India. However,India is the birthplace of much superior mathematics, and was advancedin other areas of combinatorics at an early date. I would be surprisedif it did not eventually turn out that India has an older traditioninvolving magic squares. Still, this approximate date is interestingfor other types of analysis. The next known date in the Indiandevelopment is an 11th or 12th century Jaina inscription that includes amagic square of 4. This particular magic square of 4 has unusualproperties not found in other magic squares before that time, and thewhole class of squares having these properties is called "Jainasquares", including squares of order larger than 4. I have noinformation on the document, why it includes the magic square, or whatconnection it has to the Jaina religion in medieval India. Much remainsto be explained.
IslamThe first magic squares of 5 and 6 appear in an encyclopedia inBaghdad about 983 AD by Ikhw'n al-Saf' Ras'il, though several earlierArab mathematicians also wrote about magic squares. How it came to passthat the Arabs acquired knowledge of magic squares is unknown. It isnot known if they invented them separately or if they were introduced tothem by another culture. Biggs assumes that the Arabs got the idea fromthe Chinese, though he doesn't know how the connection was made. Ithink it far more likely that the Arabs got magic squares from the samesource that they got decimal arithmetic, namely India. The Arab Jihad ofthe 7th century succeeded in conquering portions of India, and the Arabsabsorbed a great deal of Indian mathematics and astronomy. It is knownthat many other aspects of combinatorial mathematics passed from Indiato the Arabs in this way. Al-Buni was an Arab mathematician that workedon magic squares and also believed in the mystical properties of magicsquares, though no details on this number mysticism are available.Al-Buni did his work on the squares about 1200 AD. Sources have alsoreferred to the Arabs using magic squares in making astrologicalcalculations and predictions, again no details are given. Theassociation of the squares with astrology and the heavens appears to beoriginal with the Arabs, but again, much is unknown concerning theIndian tradition.
EuropeIt is from the Arabs that the West finally receives the idea of magicsquares. In 1300 Manual Moschopoulos, a Greek Byzantine scholar, writesa 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 thesquares in a purely mathematical way in contrast to the mystical ideasof the Arabs. Moschopoulos is generally considered to be the firstwesterner to know of the squares. A mistaken attribution of knowledgeto 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 anatural square.
After Moschopoulos, in the 1450's Luca Pacioli of Italyworked on magic squares and owned a large collection of examples ofmagic squares. With Pacioli we come to the doorstep of the knownWestern mystical tradition concerning magic squares. What Paciolihimself believed about the squares I don't know, but in the 1480's Italywas to see the birth of the Renaissance which revolutionized Europeanthinking. Marsilio Ficino wrote about and propounded a school of magicbased on his translations of the Hermeticdocuments that were at the time believed to be as ancient as Moses.Pico Della Mirandola wrote the "Nine Hundred Theses" - much of it basedon the translations of older Jewish Kabbalistic texts. Artists likeAlbrecht Durer eagerly absorbed the new perspective painting based onthe mathematical developments of Della Franscisca, who was popularizedby 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 themagic squares, and supplies examples of them in the orders 3- 9. Thisbook became famous throughout Europe and was very influential until thecounter-reformation and the witch-hunts that followed. Most what iscommonly thought of about Agrippa is the result of the witch-hunts andpropaganda, 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 thanthat, and in the explanation of Agrippa's book we get the first inklingof a detailed worked out system of mysticism concerning magic squares.However, though we find out some details about the squares in their role as supernaturaldevices, we are still left with conflicts and unanswered questions.In 1514 Albrecht Durer made his famous woodcut "Melancholia I," whichfeatures a magic square of 4 on the wall behind the "brooding genius"that became the archetype of all the "thinker" type sculptures in lateryears. The reason for the magic square of 4 being included in thewoodcut has been analyzed by the authors of "Saturn and Melancholy".Briefly, the square of 4 is the square of Jupiter. The planet jupiter was consideredbeneficial and was associated with the "sanguine" humor. Even today wespeak of someone's being "jovial" at a party. Durer's brooding geniussuffers from melancholia, which we call depression, and the square ofJupiter 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 byMarsilio Ficino. Ficino's magic is a kind of sympathetic magic whereobjects, colors, sounds, etc. are all categorized as to what"influences" they excite. Ficino's influences come primarily from theplanets, the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. Themagic is aimed at "drawing down" the influence or power of specificplanets in order to accomplish some end, such as protection from diseaseor a psychological cure. In this magic we see the role of the squaresas being the mathematical archetypes of the planets themselves. As eachsquare has a set of characteristic numbers, these numbers then alsocarry 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 ishow the particular planets come to be associated with particularsquares. More than one source has it that the correspondences betweenthe squares and the planets were the invention of Agrippa himself. The description of Agrippa and his bookby Francis Yates makes it appear that Agrippa made no originalcontributions to magical theory in his book, but merely collected thethought of others. Other sources simply say that the Arabs assigned thesquares to the planets. David Fidler in his book "Jesus Christ, Sun ofGod" says that the arrangements came from the Babylonians. The ancientsystem of cosmology had 7 planets, each in a concentric shell thatrotated around the earth. The Babylonians believed that the closestplanet was the moon, followed by Mercury, Venus, the Sun, Mars, Jupiter,and Saturn. They placed the order of the squares such that the smallestsquares 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 isthe fact that the system assigns the square of 6 to the sun. By makingthis assignment, the system is made to resonate with one of the mostancient of numerological systems, namely that of the Sumerians. It wasthe Sumerians with their Solar worship and their sexigesimal countingsystem that firmly fixed the hours of the day at 24, sun nominallyrising at 6:00AM and setting at 6:00PM, and who gave us the still used360 degree circle. The association of the number 6 with the sun is avery ancient western tradition. Pythagoras on account of numericaltheory called 6 the first "perfect" number. In view of these facts, themagic square of 6 with a sum total of 666 must have made quite animpression even in the 14th century, the earliest date that modernconventional scholarship will allow a western knowledge of magicsquares.Other points may be considered from the assignment of squares toplanets. First, consider that the ordering of the planets does notfollow the simple digits. That is, there is no planet associated withthe number 1 or the number 2. Does this mean that the correspondenceswere made based on the squares (there is no magic square of 2) and notsimply on the single digit numbers 1-7? If this is the case then we might infer that theancient Babylonians had knowledge of magic squares. The whole issue ofthe use by the author of the book of Revelations of the number 666 torepresent the Anti-Christ, in a passage that also includes the onlyunmistakable reference in the Bible to the practice of gematria has beenbut barely dealt with. Does this passage have to do with magic square numerology? Can othernames and phrases in the New Testament have had their values constructedin such a way as to yield gematria values that have numerologicalsignificance based on magic squares? Do various diagrams andgeometrical constructions such as the Tree of Life have a basis in thegeometry inherent in magic squares? These questions remain astantalizing possibilities, as yet not definitively answered byscholarship.Mark Swaney, January, 2000Dan W. writes:Below are the magic square numbers and their attributions to theplanets. If you recognize any of these numbers from astrology,astronomy, ancient texts, musical tuning theory, the proportions ofbuildings, sacred geometry, mathematics, physics, architecture, arthistory or anywhere at all, please let me know. Planetside of squareboxes in squaresum of any symmetrical group of four boxessum of any line - horizontal, vertical, diagonalsum of perimeter values of the squaresum of the entire squaresaturn 3 - 9 - xx - 15 - 40 - 45jupiter 4 - 16 - 34 - 34 - 102 - 136mars 5 - 25 - 52 - 65 - 208 - 325sun 6 - 36 - 74 - 111 - 370 - 666venus 7 - 49 - 100 - 175 - 600 - 1225mercury 8 - 64 - 130 - 260 - 910 - 2080moon 9 - 81 - 164 - 369 - 1312 - 3321
Mark Swaney writes:The squares of 5 and 8 have a direct to relationship to the numericalcosmology/religion of the Mayans. Read any text on their culture to findtheir sacred numbers which are 4, 13, 20, 52, & 260. If Dan had listed thenumbers for the additional squares of 10,11,12,&13, you could see thenatural mathematical affinity that exists between 4 and 13. As it is, checkout the relationship between the squares of 5 and 8. In Magic Square math,the "characteristic" number of the square is the sum of the first and lastnumbers in the series 1 - x^2. (Alpha and Omega?) The CN for square of 4 is16 + 1 = 17. This number for the square of 13 is 169 + 1 = 170.
Catherine Yronwode writes:This point has long puzzled me -- who assigned each ofthese magic number squares to a planet -- and by what logical reasoning?Has ANY logical reason EVER been given for the assignments? If so, didit make sense.?Mark Swaney replies:I am continuing to research the squares, and one of the many questionsthat I want to answer is just the one you have asked. Several sourcescredit Agrippa with the assignment of the squares to the planets, but thatis certainly not true, as Agrippa did not originate any of the Magic helists in his Occult Philosophy. At this time, I don't have any definitiveanswer, and I don't believe that the question has been tackled byacedemics. Fidler says that the Babylonians originated the system, that isthe ORDER of planets, Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon.This specific order (one of 7! = 5040 possibilities) was known long beforeAgrippa's time, because it is given in the Sepher Yetzirah, circa 300AD.The question is though, when and who and why did the magic squares getassigned to the planets? An interesting point is that there is no planetwith the number 1 or 2, (there is no magic square of two.) The bestspeculation is that the assignments were made by the Arabs, because it isknown that the Arabs incorporated Magic Squares with their Astrologicalcalculations. It is the classical position (such as has been researched -meaning NOT MUCH) that Magic Square esotericism passed to the West from theArabs in about the 13th century.Now there are a couple of points to make about the assignment of thesquares to the planets. First off, who ever came up with the arrangementcontrived it so that the Sun was assigned to the number 6. In doing so, theSumerian numerical/religious structure was preserved, because it was theSumerians who originally established the association of 6 with the Sun.They were Solar worshipers, and they also possessed a base 60 numberingsystem. They gave us 360 degree circles, and a time system that has the Sunrising (nominally) at 6 and setting at 6. So the famous or infamous Squareof the Sun is esoterically consistent with the oldest Western culture.For those who like to find a pattern common to modern knowledge of theSolar system, consider that the Sun and the Moon are "special cases". Theirapparent motion in the sky is due to the Earth's rotation and the Moon'sorbit about the Earth. So if we allow the ancient astronomer to be wrongabout the Sun and Moon and only consider the "real" planets, then the orderis (from top down), Mercury, Venus, Mars, Jupiter, Saturn. In just theproper order that we know them to be. Five objects ordered properly is arandom chance of 1 in 5! = 120. I consider it possible that the assignmentof the squares to the planets was made as the result of a serious (and nottoo far wrong) attempt to order the planets according to their actualarrangement in the sky. Such an effort would be doomed to inaccuracy by thefact that the planets actually move around the Sun and not the Earth as wasbelieved.Notice also that in the Tree of Life (a fairly modern geometry -probably from about the 17th century) the planets are shown almost in anorbital system about the SUN. Exceptions being Saturn and Mercury.However, also note that with the Earth as the 10th square/planet, the Sun,Moon, and Earth are shown in the position of a solar eclipse.
Dan W writes:Here is a quote from page 377 of William Eisen's The Cabalah of Astrology (1986)"Eventually Ptolemy’s Tetrabiblos was translated into Arabic in the 8th centuryby the Jewish astrologer Al Batrig Mashallah of Baghdad, and for the next 500years, up until the middle of the 13th century, [Ralph William] Holden [in hisbook The Elements of House Division, 1977,] traces the passing of theastrological torch into the hands of the great Muslim astrologers. A renownedschool of astronomy and astrology was established in Baghdad and flourished formany years. Among the most important literary works to be produced during thisperiod was the Elements of Astrology, written by Al Biruni in the 11th century.This book carried the Equal House system of Ptolemy even further. These menthoroughly understood the value of the Solar Houses (where the Sun is placed atthe ascendant, or at the East point in the chart), and they established a systemof Arabian Points, or Parts. The position of the Moon then became the “Point ofFortune,” Mercury the “Point of Commerce,” Venus the “Point of Love,” etc. Thehouses in which these sensitive points appeared, when compared with the actualhouses of the birth chart, thus enhanced the over-all interpretation of thehoroscope to a remarkable degree."Dan W comments: The translation of the Tetrabiblos was done by a Jewish astrologer and weknow that later the rabbi of Damascus was deep into magic square Kabbalahmysticism around 1500. Further the magic square references in Aggrippa's TheOccult Philosophy show associations with Hebrew, indicating a Jewish source. Wecan conclude that even if the Arabs originally assigned the squares to theplanets, there was an interchange of ideas going on between Arab and Jewishmystical astrologer-mathematicians. The place of this syncretism was probably in Baghdad. Now Baghdad is inMesopotamia, the site of the ancient civilizations of Sumeria and Babyloniawhere the order of the planets used in assigning the magic squares was firstworked out.Mark Swaney writes:The Brethern of Purity: I'm still working on the main leads, but thebackground information is very interesting in itself. The BP were agroup of Ismaili scholars. The Ismaili are a sub-sect of the Shia brandof Islam, and are and were, considered heretics by the other Moslems.The history on this sect is obscure and I am working on finding out somemore about the Ismaili, but the main points I know are these. TheIsmailis have/had their own Imam, like a Pope I think, except theposition is hereditary, Ismaili Imams are not the same as those of theother Shias. They were oppressed vigorously, and many Ismailis werekilled in wars of suppression waged by the orthodox Sunni who were inpower in Egypt, the Fatimids. At the time of the production of theRasail of Ikhwan as-Safa, or the Encyclopedia of the Brethren of Purity,the Ismailis were an underground organization in the neighborhood ofBaghdad, which was under the rule of the Fatimids in Egypt.The BP were said to have been organized a century or so before thetime of the Rasail, but no one is exactly sure when they started.Anyway, it seems that the Imam in about 989 AD decided to have the BPproduce an encyclopedia. This was because the BP represented theintelligentsia of the Ismailis at the time, and because they werealready an organized group. Under the direct supervision of the Imam,the scholars of the BP met secretly in a cave and began work on theRasail. After the completion of the Rasail, copies were made and onewas placed in every Mosque in the area in and around Baghdad.The purpose of the Rasail was to reconcile Greek philosophy with theprecepts of the Ismaili sect, to show that the beliefs of the Ismailiswere in accord with the scientific knowledge of the day! Now keep inmind that this book, the Rasail, contains the FIRST recorded example ofa magic square of 6. All the numbers in the New Testament that relateto the multiples of 37 are said by Fidler and Michell to be based onthe square of the sun. However, the straight academic world says"whoa!" the first magic square of 6 was invented by the ARABS 1000 yearsAFTER the NT was written. The Greeks are NOT supposed to have anyknowledge of magic squares at all. But, when we look further, it turnsout that we don't have the name of the first mathematician to create asquare of 6, and we don't know if he/she was Greek, Arab, or a Martian.We just have an encyclopedia where knowledge is collected, presumablyknowledge that was accepted and common to the Greeks in the 10thcentury, otherwise how could the Rasail have performed as it wasintended?So we have a new lease on the idea that MAYBE there was a knowledgeof magic squares in Greece in antiquity. Or still possible, and just asintriguing, the Babylonians. We are talking about Baghdad where thisheretical Islamic sect creates the Rasail, the sum of all knowledge thatcontains magic squares and proclaims that the Ismailis arescientifically correct in their beliefs.There is more. Ever hear of Hassan a-Sabah? Well, if not, look himup. He is considered the inventor of the modern 20th centuryintelligence service, 1000 years ahead of his time. The words Assassinand Hashish? They derive from the name of the warrior/terroristsorganized by Hassan a-Sabah, the Hashishim. Who were they? You guessedit, the Ismailis. I am now looking into the links between a-Sabah andthe BP. It gets very interesting, doesn't it?Mark Swaney writes further:
I have found that the BP and the Ismailis werespecifically influenced by Neo-Platonism in the 8th century. I havelocated a reference that should be of some help in understanding therole of the magic squares in the Ismaili beliefs. Neoplatonists: An Introduction Into the Thought of the Brethern ofPurity, Allen&Unwin, London 1982.another reference is;Early Philosophical Shiism by P.E. Walker, Cambridge, 1993.I am also finding information on the theory of the concentric spheresmentioned by Fidler, as well as information on Al-Biruni.Ancient CosmologiesAllen & Unwin London 1975.Dan W writes:The relation between the Assassins and the BP is weird inthe extreme. See if you can find out anything about thepsycho-spiritual/brainwashing disciplines the Assassins used, would you.Maybe the same methods were used in working with the squares, a la the Rabbiof Damascus and his Kabbalistic system for meditating on the squares.Mark Swaney writes: More on the BP, Sabah, etc. The BP are thought to have been formed aboutthe same time as the formation of the Ismaili sect, about three hundred yearsbefore the publication of the Rasail in the 10th century. The sects of Islamseem to all be named after the person whom that sect originally believes to bethe Imam. I have read so many names of Imams that I am a little confused as towho is who, but the Imam that preceded the split that created the Ismailis was avery educated man, and very interested in philosophy. It is he who is creditedwith infusing the Ismailis with a philosophically based belief system. I haveread a critique of the story of Adam and Eve by an medevial Ismaili writer thatis in my mind remarkable for it's very modern analysis of the story. TheIsmaili brand of Islam (which still has millions of followers) is described bymodern scholars of religion as "gnostic" and "neoplatonic". The Ismailicosmology is certainly concerned with the hierarchy of the universe. Theirbelief in the succession of the Imams as the living representatives of God onEarth was fused with the Platonic/Kabbalistic Theory of Emanation at a veryearly period. I am fascinated by the thought of these medieval Islamic Mystics,they certainly deserve attention and consideration for their contributions tothe thread of thoughts that we pursue.Sabah is another mystery and a famous character from the late 11th centuryin Iran. Hassan a-Sabah was also known by the moniker "The Old Man of theMountain". Almost everything that we know about Sabah has been written by hisenemies. So we should give the guy a break and try to look at him objectively.Actually, what is known about Sabah for sure is that he must be considered agenius. He founded an intelligence service that was unrivaled in the worlduntil the 20th century. He invented what his enemies call terrorism, his modernfriends would call it "guerilla warfare". He was a military genius and afteroccupying the "Eagle's Nest", his famous fortress in northern Iran, he neverlost it, and he conquered several other forts in the surrounding area. Hisbattles with the Seljuck Turks made history. Sabah converted to the Ismailifaith as a young man, and he is still considered by the modern Ismailis to be aHero of the Faith. True, he was supposed to have been cruel and bloodthirsty.Yeah, and your mother wears army boots.The really interesting thing about Sabah for our studies is that in additionto being a military genius, he was also known to be a scholar. The organizationhe created, the Hashishim, or Assassins, was a "Masonic" military organization.By the way, the words Assassin and Hashishim and Hashish are all thought to becorruptions of Sabah's first name, Hassan. The Assassins were in essenceKamikaze's. They were trained to strike an enemy and not escape, but stay andfight 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 hisfortress. He was known to have had an interest in mathematics, and to haveencouraged the study of mathematics and philosophy by his followers. TheAssassins practiced initiation rites, and had strict grades of hierarchy, sothat modern historians have described them as "Masonic" in nature. Sabah andthe Assassins also had intriguing contacts with the Crusaders that I am nowtrying to find out more about. All this is hugely interesting for all theobvious reasons.The initiation rites are the probable source of the story about Sabah's useof drugs to fool initiates into thinking they had gone to heaven when in realitythey were only in Sabah's garden. This story was written by Marco Polo whopassed through the area of the Eagle's Nest 150 years after Sabah and theAssassins. There is no other documentation to back it up, and so it must betaken with a grain of salt. Personally, I think that no matter howmuch hash someone ate, it is very unlikely that they would wake up after fallingasleep and think themselves to be in heaven. But the available evidence doesindicate that the Assassins practiced some form of discipline that may havebordered on modern theories of mind control. Another example of Sabah'sprescient inventions.After the Mongols conquered the Eagle's Nest in the late 13th century, theAssassins and the Ismailis in general declined from any power in the politicalsense. The Mongols burned the library at the Eagle's Nest, so no books by Sabahor the Assassins survive today. The whole essence of the organization built bySabah rested on obedience, faith, and above all else, secrecy. We should not besurprised that a great deal of the knowledge of the Assassins was lost. Weshould also keep in mind that secrecy was one of the hallmarks of the gnosticsand other early mystery cults.
Dan Washburn writes:
Hmmm. Since we know so little about them its hard to tell what kind ofdisciplines the Assassins practiced. As you say, the rumor is drugs. However, nowthat I think about it, if they were supposed to visit paradise, maybe they werepracticing some form of Ascension similar to Jewish merkabah mysticism, whichinvolves trance visits to the heaven world. Also if the Assassins were trulygnostic, then ascension is an even more likely methodology, since the gnostics andhermetics were practitioners of ascending through the seven spheres of the planets.Ah ha - another connection to the seven planets which are connected to the squares.You might take a look at Dan Merkur's book Gnosis to get more of an idea on allthis.
Mark Swaney writes:I am suspecting more and more that the Brethrenof Purity's book the Rasail may have assigned the magic squares to the planets, but Iwill have to wait until I can get my hands on some source material that finally getsdown to the dirty details of just WHAT the BP saw in the squares, and why. Thebackground data certainly suggests to me that the BP associated the squares with theplanets in accordance with the medieval cosmology.That puts us at about 990 AD for the publication of the Rasail that contains thefirst recorded example of a magic square of 6. The interesting thing about this fact isthe inherent nature of the number 6. Here is where the mathematics provides someclues. Has you ever tried to create a magic square? If you study the squaresmathematically you will see that different orders have different properties and some areeasy and some are difficult. In brief, there are even numbers and there are oddnumbers. Further, the even numbers come in two varieties, those that are evenlydivisible by 4 and those that are not. Each magic square has a particular pattern thatis made by connecting the numbers in order as they appear in the square. Odd ordermagic squares have patterns that are symmetrical about a principal diagonal. Evensquares that are divisible by 4 have patterns that either have symmetry about 1 or 2axis that are either horizontal or vertical or they may have no symmetry at all. Butsquares of 6, 10, 14, 18 etc. are strange. The patterns of the un-evenly even squareshave no symmetry. There are other aspects of the UE squares that make them unusual.Magic squares are related mathematically to another kind of square numberarrangement called a Latin square (also called Greco-Latin squares). Methods ofconstructing magic squares by using Latin squares were published by a latermathematician named De LaHire. Latin squares have been (and are) studied by numbertheorists and also have found applications in modern technology. The idea of a Latinsquare is to arrange items with attributes in rows and columns so that each row andcolumn has 1 and only 1 of each kind of attribute. For example, make a Latin square of4 by using 16 cards, 4 from each suit in the order Ace, King, Queen, Jack. Now arrangethe cards in a square so that each row and each column has 1 each of the suits and 1each of the ranks. When you have done it, you have a Latin square. Now in later historythe most famous of the mathematicians took up the problem of magic squares, among thembeing Euler, the greatest mathematician of all time. Euler studied magic squares andLatin squares. One of his famous unsolved problems was the Euler Conjecture - astatement to the effect that UE Latin Squares are impossible. Actually, if I rememberit right, the EC had been disproved by 1960, in as much as larger UE Latin squares (suchas a LS of 10) can be constructed, however they proved Euler right in the case of aLatin Square of 6. It's impossible.The paper I received from the University of New South Wales on the Magic Squares ofManuel Moschopoulos, written in 1306 AD, is the first above-ground western writing thattells how to construct a magic square. But it is deficient in that it only givesmethods for constructing odd and evenly-even squares. It says nothing about how to makesquares of 6,10, etc. Furthermore, the methods of constructing odd squares and EEsquares take advantage of the inherent symmetry of those numbers, so that once known,anyone can create multitudes of different squares of various sizes without understandingany of the real combinatorial principals that magic squares are ultimately based on.Except that you can't make a magic square of 6 by resorting to such simplistic methods.That one you have to do the hard way.From the mathematical point of view of the construction of a square of 6 is a moredifficult task, and the appearance of the first magic square of 6 is far moresignificant mathematically than the first appearance of a square of 7 or 8 or 9. Didthe medieval mystic/mathematicians understand these points? They must have. And yet wedo not see any publication of their methods. Al-Buni the Arab mathematician who wroteabout magic squares around 1200 AD may have known how to construct a square of 6, butthe references I have so far indicate that he only gave the methods later published byMoschopoulos. The Rasail Ikhwan as-Safa is therefore crucial to our quest because itback dates knowledge of the square of 6 to sometime before 990AD in the west.
Dan Washburn Writes (04/02/00)
Nigel Pennick's Magical Alphabets has a section on magic squares in the chapter on the hebrew alphabet and also a chapter titled 'magic squares, literary labyrinths, and modern uses.'Here is a quote from the hebrew alphabet chapter, p34:"In Hebrew magic each planet can be seen as a symbol of one of theSephirah on the tree of life. Saturn signifies Binah, the thirdSephirah, whilst Jupiter, corresponds with the fourth Sephirah. Chesed.Mars parallels the fifth Sephirah, Geburah, whilst Mercury is the eighthSephirah, Hod. This planetary scheme was the basis of Assyrian andBabylonian ziggurats, of which the ill-fated Tower of Babel was anexample. In their purest cosmologically defined form, each of theziggurats’ seven stages or platforms represented one of the sevenplanets. The magic squares are arranged in a sequence that starts withthe smallest grid at the outermost. For structural reasons, theoutermost square on a ziggurat was the largest. However, it was ruled bythe corresponding magic square, and painted in the corresponding colour.The engraving of the Khorsabad ziggurat, reproduced in Fig. 10, gives agood idea of the principle.Summary of ziggurat drawing:level 7 - moon - whitelevel 6 - mercury - bluelevel 5 - venus - greenlevel 4 - sun - yellowlevel 3 - mars - redlevel 2 - jupiter - orangebase level - saturn - blackThe simplest magic square is the square of three by three, ascribed toSaturn, in which each line adds up to 15 and the total of all thenumbers added together is 45. This is the square most commonly used byEuropean magicians. Its traditionally associated colour is black,signifying the outermost planet and the bottom tier of the ziggurat."Anyone know anything about ziggurats? Does this color scheme actuallygo back to Sumerian/Babylonian sources or is it a later occultist'sdream? When did the magic squares become color coordinated? This isthe first I've heard of it. Did Aggrippa mention this in his 'OccultPhilosophy' (I saw a translation of the book at Borders the other daybut haven't had time to look at it)? Or is this an invention of S. L.MacGregor Mathers or one of the other Golden Dawn boys?I've known that the tiers of the ziggurat represented the planets butalways assumed that it was an allegorical ascension thru the heavenlyspheres to the realm of the fixed stars in an order of the planets thatstarted with the nearest to the earth and progressed outward. The orderoutlined here seems to be a mirror image of the heavens. With saturn atthe base the ascension of the pyramid is a descent thru the planets.This may offer an explanation for something that has troubled scholarsof Merkabah mysticism, the form mysticism took in Judaism for thethousand years before the rise of the Kabbalah around 1200 AD. In atrance the mystic ascends through seven visionary palaces to the thronechariot of God, the Merkabah. The movement is upward but the existenttexts refer to it as the Decent to the Merkabah. If the seven palacesare based on the seven planets, then the visionary may indeed bedescending through the heavenly spheres.Mark Swaney Writes (04/17/00) Have received a fairly exhaustive list of references and information onmagic squares and their history from David Singmaster, the Englishmathematician and Rubick's Cube expert. Dr. Singmaster has sent to me a pileof references about a quarter inch thick. This material is exclusivelyREFERENCE material, so that the complete works cited would fill a truck, I'msure. It's taken the weekend just to read through and assimilate the material,and there is much in it that is new to me, and bears directly on problems weare interested in.The first news is that we have new information on the question of theassignment of the squares to the planets. The following is a list I made uplast night from the Singmaster chronology and reference.The (so far) earliest known explicit association of the Planets with theSquares occurs sometime before 1384 in a document titled QABS al-ANWAR byNadruni. I don't know anything more about this document except that Nadrunigives the same associations of the squares with the planets as that later givenby Agrippa, 3 - Saturn, 4 - Jupiter, 5 - Mars, 6 - Sun, 7 - Venus, 8 - Mercury,9 - Moon.Another Arab manuscript, untitled, author unknown, circa 1466, gives areverse mapping of squares to planets, i.e. 3 - Moon, 4 - Mercury, etc.The first known European set of Magic Squares associated with the planetsis in a 15 century Latin manuscript in Cracow, described as Jagiellonian MS#753. The reference does not say what the explicit ordering is, or who wroteit, or exactly when, or for what purpose.In 1498 Pacioli wrote DeViribus, which gives squares and planets in thesame relationship as Nadruni and later Agrippa. This is important as apossible source for Albrecht Durer's square shown in Melancholia I.In 1531 Agrippa publishes De Occulta Philosophia, the second book of whichgives magic squares associated with the planets in the well-known order 3 -Saturn, 4 - Jupiter, etc.In 1539 Cardan writes Practica Arithmetica and gives the squares andplanets in the reverse as that published by Nadruni, Pacioli and Agrippa.Several interesting points to made about this information even inadvance of receiving the works cited are;1. The association of planets to squares is first PUBLISHED in the Arab worldin sometime in the mid 14th century.2. The mapping of planets to squares is given in two orders, each the reverseof the other, and each mapping is used in both the European and Arabicpublications. Attention DAN WASBURN - you asked about the order of planets vs.squares in your post on magic squares and Ziggurats, indicating a possibleconfusion about the Merkabah mystics as to whether they are asending ordescending. It appears that there were TWO "traditional" orders of planets.3. Each of the orders results in the square of 6 being assigned to the Sun.This is due to the position of the Sun in the "center" of the list of planets,an at least allegorically helio-centric system. As I have said, this hasimplications for us in our interest in the numerology of the square of 6.Another interesting assertion in the Singmaster material is a reference toan author who claims that the 3rd Century neo-platonists had knowledge of themagic square of 3. This is the sole academic reference to a western knowledgeof magic squares before the 14th century. A very interesting development thatwe will be sure to investigate for obvious reasons. If supportable, this wouldput a new light on the order of planets given in the Sepher Yitzarah, writtenin about the 3rd century in Palestine, and push back knowledge of the squaresin the West 11 centuries.As a "bonus" for me especially, Singmaster has references to manymathematicians and puzzle-authors who have contributed to magic squareliterature over the years. Quite unexpectedly, I discover that Dr. Singmasterhas excerpts from the notebooks of CHARLES BABBAGE, the famous Englishmathematician inventor and "grandfather of the computer". It seems thatBabbage was interested in Latin Squares, Magic Squares, and Magic Cubes.Along this line, (though not concerned with my main interest in the squaresas magical devices) we can add some famous names to the list of people whosquandered their time wrestling with these devices. Among them are;Leonard Euler - often considered the world's greatest mathematician -studied Latin Squares and Magic Squares, author of the famous "EulerConjecture".Pierre Fermat - super-famous author of the finally-proved "Fermat's LastTheorm" - Fermat conceived and produced the world's first Magic Cube, alsodiscussed magic triangles with Frenicle.M. Mersenne - friend of Fermat's and fellow number theorist.Bernard Frenicle - friend and correspondent of Fermat and Mersenne - firstto list and produce all the 880 possible magic squares of 4.Benjamin Franklin - worked with large magic squares and magic circles.Charles Babbage - the 19th century genius who first described the conceptof the modern computer and who attempted to build one - called a "differenceengine"Additional news to pass along - the Singmaster material provides thestarting point for a LARGE research project, including as it does virtually allthe previous research along the lines we wish to continue. What is clear froma review of everything I have collected is that while some information isknown, a lifetime could be devoted to working on the history of magic squaresand the associated religious/magical ideas. The contributions of the Indiansin particular has not been very well understood, and the authors disagree amongthemselves as to the sources of the Arab magic squares, Indian or Chinese.Dr. Singmaster has not touched the esoteric side of the question. None ofthe mathematical historians (who nevertheless give by far the most detailedinformation on sources) treat the larger subject of the relationship betweenmathematics and mystical experiences and philosophy. I think the territory isjust waiting for someone to make a PhD dissertation on the subject. Anytakers?Finally, I have run out of time to write today, but there is more, such asthe several sources that recommend that the magic square of 3 be used as aremedy for a hard labor during childbirth, such as the 5 15th century cast ironplates found in central China in 1958 with the magic square of 6 inscribed onthem, such as a reference to the Chinese god of the POLE STAR (attentionBarry!) and a description of the path of this god through the "houses" of theLo Shu square. Information on Chinese beliefs about the Ming-Tang Palace andthe Emperor.In short, enough meat to chew on for quite a while. Anybody wanna dive inand help find and analyze this stuff?Mark Swaney********************************************************************More information and links
China
Because there are 64 slots in the 8x8 matrix of the Magic Square of Mercury and because magic squares were important in China, I wondered ifthere was any connection with the 64 hexagrams of the I Ching.I discovered an essay by Tayagi Nagasiva on Magic Squares and the I Ching
Quote from John Opsopaus 4 9 2 3 5 78 1 6Chinese knowledge of the Saturn Square is shown by the ground plan of theMing-T'ang temple, which was built in A.D. 56. However, as Stapleton (Antiq. Alch. 15) says, "a much greater antiquity for this form of temple is indicated, firstly, by a templeof this plan being essential for Imperial worship, and, secondly, that in the7th century B.C., during the time of the warring Lords, it was believed to have been used by Wu, the alleged founder of the Chou dynasty in 1025 B.C., when sacrificing to hisancestors. Moreover, if this tradition be correct, the Magic Square form oftemple may ultimately be of Scythian origin, introduced at this time from Bactria, or ancient Iran, with the foreign mercenaries from the West, to whose help Wu owed his success in establishing a new dynasty." (From Bactria it may be traceable backto Mesopotamia.)The Ming-T'ang had twelve stations for the monthly "Proclamation of Space andTime." There is one station for each line segment on the perimeter of the square, that is, two for each corner (even) square, one for each side (odd) square. The eight squares on the perimeter represent the eightfold year (3 = vernal equinox, 9 = summer solstice, 7 = autumnalequinox, 1 = winter solstice). The central square corresponds to the additionaldays of the year beyond the twelve lunar months represented by the twelve line segments of the outer squares. Thus the Son of Heaven visited the central room of the temple(numerically 5, the Emblem of the Center) at "the end of summer - a criticalperiod when the transition was made from the yang seasons to the yin seasons" (Granet, Rel. Ch. 67). Alternately, the twelve line segments of the perimeter can represent the solaryear and the zodiac. Thus the representation of Time; the temple also represented Space by assigning 8+3 = east, 4+9 = south, 2+7 = west, 6+1 = north (the same four numbers as the elements, though not the same pairs of squares); opposing directions balance to 20, as do opposing elements. (Granet, Rel. Ch. 66-8; Stapleton, "Antiq. Alch.")Blofeld (I Ching, 218) says that mankind once understood how the Lo-Shu Squareis connected with the (apparently illogical) Later Heaven Sequence of the I Ching, but that it has been forgotten and now only the gods know it. I certainly have not been able to find it. (The connection established by Hacker (41) seems to me to be contrived, although it is remarkable enough that any connection can be established at all.)
Christianity
To get a look at some work by Dan Gleason relating the 888 of Jesus tothe 666 of the magic square of the Sun click on the link below:http://www.jesus8880.com/gematria/666.htm
IslamKieth Critchlow has a chapter on Magic Squares in Islam in his bookIslamic Patterns.
The following link has a picture of a page from one of Al-Buni's books with the caption given below:
http://www.vh.org/Welcome/UIHC/MedMuseum/ArtThatHeals/10Knowledge.html
21. Diagrams from the Book of Buni, theGeez translation of a version of the "Sun ofKnowledge" (Shams al-Marif), a bookattributed to Al-Buni, an Egyptian authorof the thirteenth century. Each diagramcontaining figures or letters isaccompanied by its method of use. Bookof Buni, eighteenth century to nineteenthcentury, parchment, 27.5 x 24 cm. Privatecollection. Photo courtesy of Guy Vivien
A quotation from John Opsopaus:Magic Square of 34 9 2 3 5 78 1 6According to the Theory of Balance attributed to 8th century Muslim alchemistJabir ibn Hayyan (based on 3rd century works by Zosimos and others), the Cosmos and everything in it is made from the numbers 1, 3, 5, 8, 17 and 28; they are thefoundation of all matter, of every science, and even of any possible language.The first four numbers were assigned by the Jabirian alchemists to the elements, 1=fire, 3=earth, 5=water, 8=air. The sum of these is 17, which is the fifth number. TheGnomon, which gives the larger square, sums to 4+9+2+7+6 = 28, the sixth number,the second Perfect Number.
Quoted from the following site:JudaismI've been reading Aryeh Kaplan's Meditation andthe Kabbalah, He has a chapter on the Kabbalistic writingsof Rabbi Joseph Tzayach (1505-1573) the Rabbi of Damascus.Kaplan says that magic squares were well known in ancient Indiaand China and that they were introduced into the west in the1400s by Moschopulus of Constatinople. Kaplan believesthey were also known to the ancient Kabbalists. Tzayachprobably received a hidden tradition concerning these figures.His system attributes the traditional magic squares to theseven planets but goes on to attribute higher order magic squaresto the Sephira on the Tree of Life, a Jewish mystical diagram: The square of tento Kether, of 11 to Chokmah, etc. skipping the square of15, so that Malkuth is the 20x20 matrix.Tzayach apparently had a system that involved meditatingon the color, number, and letter forms in one room after anotherin a square. Each row was called a house and each box a room.What we might be seeing here is a form of Merkabah Ascension.Up through the spheres of the 7 planets and then through 10(usually 7 in other literature) palaces of the King to the throneof God. The position of meditation described is that of Elijah on Mt. Carmel,sitting cross legged with head between the knees, a position associated with MerkabahAscension.Magic square lore may have passed back and forth betweenJewish and Muslim mystics.General
David Singmaster gives a number of dates relating to the history of Magic Squares in his"Chronology of Recreational Mathematics."
Here are some book references (I have not looked at) from the bilbiography at
Edward Falkener, "Games ancient and oriental and how to play them : being the games of the ancient Egyptians, the hiera gramme of the Greeks, the ludus latrunculorum of the Romans and the oriental games of chess, draughts, backgammon and magic squares" :New Dover ed:New York : Dover Publications , (1961) Soror A.L. ,Compton, Madonna, "Western mandalas of transformation : magical squares, tattwas, qabalistic talismans", St. Paul, MN, U.S.A. : Llewellyn Publications , (1995) Richard Webster, "Talisman magic : yantra squares for tantric divination", St. Paul, Minn., U.S.A. : Llewellyn Publications , (1995)
www.dhyansanjivani.org
You gotta tell them that it's just ripped hot off the neurons, and mayhave a detail or two out of place. I'm reading all this stuff and then roaringoff an epistle. Later, I always think I should have done it differently, butwhat the hell? Also, I find that I like to write a lot of text when I'm feelingradioactive.
After Mark's review of the history of Magic Squares, I have listed some further information and references on the squares. Mark:I am still getting references andpicking up information, so the following is subject to revision andexpansion.The history of magic squares is murky, mysterious, and has not beenwell researched by academics. Consequently the claims arecontradictory, and in some cases exaggerated. Very little is knownabout the origin of magic squares. Next to nothing is known about themovement of the idea of a magic square before about 1300 AD. Threecultures are known to have created magic squares, the Chinese, theIndian, and the Arabic. In each culture they were viewed as havingsupernatural properties.ChinaThe first magic square in history was created in China by an unknownmathematician, probably sometime before the first century AD. Calledthe Lo Shu square, it is a magic square of 3 that was said to haveappeared on the back of a turtle that came up out of the river. Lo Shusupposedly means "river map" and the story of the appearance of theturtle had to do with a sacrifice to the river god. Right from the beginning we are seeing anessentially mathematical construction combined with the supernatural. Ihave not found an analysis of the story of the turtle and the Lo Shusquare from the point of view of folklore or mythology that would shedmore light on the story. The Lo Shu square is later associated with thefloor plan of a mythical palace, that of Ming'tang. Again, this is fragmentary, Ihave seen a diagram that shows the floor plan, but no explanation as towhat 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 isalso connected to the I-Ching, though there is no explicit plan ofcorrespondence that I know of. The oldest documents that refer to theLo Shu square are ambiguous, but one reference lists a Shu Ching in 650 BCE who makes areference to the "river map" which may be the magic square of 3. In 500BCE, and 300 BCE, the river map is mentioned, but no explicit magicsquare is given. In 80 AD Ta Tai Li Chi gives the first clear reference to amagic square. In 570 AD Shuzun gives an actual description of a magicsquare of 3. Not until 1275 do we hear of the Chinese making squares oforder larger than 3. Norman Biggs says that this is because the Chineseregarded the Lo Shu square as an object of the supernatural, rather thanas an object of human curiosity, and it was therefore not a subject for study.
IndiaWe find the first magic square of 4 in the first century in India bya mathematician named Nagarajuna. This is all that I know at the momentabout the early development of the magic square in India. However,India is the birthplace of much superior mathematics, and was advancedin other areas of combinatorics at an early date. I would be surprisedif it did not eventually turn out that India has an older traditioninvolving magic squares. Still, this approximate date is interestingfor other types of analysis. The next known date in the Indiandevelopment is an 11th or 12th century Jaina inscription that includes amagic square of 4. This particular magic square of 4 has unusualproperties not found in other magic squares before that time, and thewhole class of squares having these properties is called "Jainasquares", including squares of order larger than 4. I have noinformation on the document, why it includes the magic square, or whatconnection it has to the Jaina religion in medieval India. Much remainsto be explained.
IslamThe first magic squares of 5 and 6 appear in an encyclopedia inBaghdad about 983 AD by Ikhw'n al-Saf' Ras'il, though several earlierArab mathematicians also wrote about magic squares. How it came to passthat the Arabs acquired knowledge of magic squares is unknown. It isnot known if they invented them separately or if they were introduced tothem by another culture. Biggs assumes that the Arabs got the idea fromthe Chinese, though he doesn't know how the connection was made. Ithink it far more likely that the Arabs got magic squares from the samesource that they got decimal arithmetic, namely India. The Arab Jihad ofthe 7th century succeeded in conquering portions of India, and the Arabsabsorbed a great deal of Indian mathematics and astronomy. It is knownthat many other aspects of combinatorial mathematics passed from Indiato the Arabs in this way. Al-Buni was an Arab mathematician that workedon magic squares and also believed in the mystical properties of magicsquares, though no details on this number mysticism are available.Al-Buni did his work on the squares about 1200 AD. Sources have alsoreferred to the Arabs using magic squares in making astrologicalcalculations and predictions, again no details are given. Theassociation of the squares with astrology and the heavens appears to beoriginal with the Arabs, but again, much is unknown concerning theIndian tradition.
EuropeIt is from the Arabs that the West finally receives the idea of magicsquares. In 1300 Manual Moschopoulos, a Greek Byzantine scholar, writesa 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 thesquares in a purely mathematical way in contrast to the mystical ideasof the Arabs. Moschopoulos is generally considered to be the firstwesterner to know of the squares. A mistaken attribution of knowledgeto 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 anatural square.
After Moschopoulos, in the 1450's Luca Pacioli of Italyworked on magic squares and owned a large collection of examples ofmagic squares. With Pacioli we come to the doorstep of the knownWestern mystical tradition concerning magic squares. What Paciolihimself believed about the squares I don't know, but in the 1480's Italywas to see the birth of the Renaissance which revolutionized Europeanthinking. Marsilio Ficino wrote about and propounded a school of magicbased on his translations of the Hermeticdocuments that were at the time believed to be as ancient as Moses.Pico Della Mirandola wrote the "Nine Hundred Theses" - much of it basedon the translations of older Jewish Kabbalistic texts. Artists likeAlbrecht Durer eagerly absorbed the new perspective painting based onthe mathematical developments of Della Franscisca, who was popularizedby 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 themagic squares, and supplies examples of them in the orders 3- 9. Thisbook became famous throughout Europe and was very influential until thecounter-reformation and the witch-hunts that followed. Most what iscommonly thought of about Agrippa is the result of the witch-hunts andpropaganda, 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 thanthat, and in the explanation of Agrippa's book we get the first inklingof a detailed worked out system of mysticism concerning magic squares.However, though we find out some details about the squares in their role as supernaturaldevices, we are still left with conflicts and unanswered questions.In 1514 Albrecht Durer made his famous woodcut "Melancholia I," whichfeatures a magic square of 4 on the wall behind the "brooding genius"that became the archetype of all the "thinker" type sculptures in lateryears. The reason for the magic square of 4 being included in thewoodcut has been analyzed by the authors of "Saturn and Melancholy".Briefly, the square of 4 is the square of Jupiter. The planet jupiter was consideredbeneficial and was associated with the "sanguine" humor. Even today wespeak of someone's being "jovial" at a party. Durer's brooding geniussuffers from melancholia, which we call depression, and the square ofJupiter 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 byMarsilio Ficino. Ficino's magic is a kind of sympathetic magic whereobjects, colors, sounds, etc. are all categorized as to what"influences" they excite. Ficino's influences come primarily from theplanets, the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. Themagic is aimed at "drawing down" the influence or power of specificplanets in order to accomplish some end, such as protection from diseaseor a psychological cure. In this magic we see the role of the squaresas being the mathematical archetypes of the planets themselves. As eachsquare has a set of characteristic numbers, these numbers then alsocarry 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 ishow the particular planets come to be associated with particularsquares. More than one source has it that the correspondences betweenthe squares and the planets were the invention of Agrippa himself. The description of Agrippa and his bookby Francis Yates makes it appear that Agrippa made no originalcontributions to magical theory in his book, but merely collected thethought of others. Other sources simply say that the Arabs assigned thesquares to the planets. David Fidler in his book "Jesus Christ, Sun ofGod" says that the arrangements came from the Babylonians. The ancientsystem of cosmology had 7 planets, each in a concentric shell thatrotated around the earth. The Babylonians believed that the closestplanet was the moon, followed by Mercury, Venus, the Sun, Mars, Jupiter,and Saturn. They placed the order of the squares such that the smallestsquares 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 isthe fact that the system assigns the square of 6 to the sun. By makingthis assignment, the system is made to resonate with one of the mostancient of numerological systems, namely that of the Sumerians. It wasthe Sumerians with their Solar worship and their sexigesimal countingsystem that firmly fixed the hours of the day at 24, sun nominallyrising at 6:00AM and setting at 6:00PM, and who gave us the still used360 degree circle. The association of the number 6 with the sun is avery ancient western tradition. Pythagoras on account of numericaltheory called 6 the first "perfect" number. In view of these facts, themagic square of 6 with a sum total of 666 must have made quite animpression even in the 14th century, the earliest date that modernconventional scholarship will allow a western knowledge of magicsquares.Other points may be considered from the assignment of squares toplanets. First, consider that the ordering of the planets does notfollow the simple digits. That is, there is no planet associated withthe number 1 or the number 2. Does this mean that the correspondenceswere made based on the squares (there is no magic square of 2) and notsimply on the single digit numbers 1-7? If this is the case then we might infer that theancient Babylonians had knowledge of magic squares. The whole issue ofthe use by the author of the book of Revelations of the number 666 torepresent the Anti-Christ, in a passage that also includes the onlyunmistakable reference in the Bible to the practice of gematria has beenbut barely dealt with. Does this passage have to do with magic square numerology? Can othernames and phrases in the New Testament have had their values constructedin such a way as to yield gematria values that have numerologicalsignificance based on magic squares? Do various diagrams andgeometrical constructions such as the Tree of Life have a basis in thegeometry inherent in magic squares? These questions remain astantalizing possibilities, as yet not definitively answered byscholarship.Mark Swaney, January, 2000Dan W. writes:Below are the magic square numbers and their attributions to theplanets. If you recognize any of these numbers from astrology,astronomy, ancient texts, musical tuning theory, the proportions ofbuildings, sacred geometry, mathematics, physics, architecture, arthistory or anywhere at all, please let me know. Planetside of squareboxes in squaresum of any symmetrical group of four boxessum of any line - horizontal, vertical, diagonalsum of perimeter values of the squaresum of the entire squaresaturn 3 - 9 - xx - 15 - 40 - 45jupiter 4 - 16 - 34 - 34 - 102 - 136mars 5 - 25 - 52 - 65 - 208 - 325sun 6 - 36 - 74 - 111 - 370 - 666venus 7 - 49 - 100 - 175 - 600 - 1225mercury 8 - 64 - 130 - 260 - 910 - 2080moon 9 - 81 - 164 - 369 - 1312 - 3321
Mark Swaney writes:The squares of 5 and 8 have a direct to relationship to the numericalcosmology/religion of the Mayans. Read any text on their culture to findtheir sacred numbers which are 4, 13, 20, 52, & 260. If Dan had listed thenumbers for the additional squares of 10,11,12,&13, you could see thenatural mathematical affinity that exists between 4 and 13. As it is, checkout the relationship between the squares of 5 and 8. In Magic Square math,the "characteristic" number of the square is the sum of the first and lastnumbers in the series 1 - x^2. (Alpha and Omega?) The CN for square of 4 is16 + 1 = 17. This number for the square of 13 is 169 + 1 = 170.
Catherine Yronwode writes:This point has long puzzled me -- who assigned each ofthese magic number squares to a planet -- and by what logical reasoning?Has ANY logical reason EVER been given for the assignments? If so, didit make sense.?Mark Swaney replies:I am continuing to research the squares, and one of the many questionsthat I want to answer is just the one you have asked. Several sourcescredit Agrippa with the assignment of the squares to the planets, but thatis certainly not true, as Agrippa did not originate any of the Magic helists in his Occult Philosophy. At this time, I don't have any definitiveanswer, and I don't believe that the question has been tackled byacedemics. Fidler says that the Babylonians originated the system, that isthe ORDER of planets, Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon.This specific order (one of 7! = 5040 possibilities) was known long beforeAgrippa's time, because it is given in the Sepher Yetzirah, circa 300AD.The question is though, when and who and why did the magic squares getassigned to the planets? An interesting point is that there is no planetwith the number 1 or 2, (there is no magic square of two.) The bestspeculation is that the assignments were made by the Arabs, because it isknown that the Arabs incorporated Magic Squares with their Astrologicalcalculations. It is the classical position (such as has been researched -meaning NOT MUCH) that Magic Square esotericism passed to the West from theArabs in about the 13th century.Now there are a couple of points to make about the assignment of thesquares to the planets. First off, who ever came up with the arrangementcontrived it so that the Sun was assigned to the number 6. In doing so, theSumerian numerical/religious structure was preserved, because it was theSumerians who originally established the association of 6 with the Sun.They were Solar worshipers, and they also possessed a base 60 numberingsystem. They gave us 360 degree circles, and a time system that has the Sunrising (nominally) at 6 and setting at 6. So the famous or infamous Squareof the Sun is esoterically consistent with the oldest Western culture.For those who like to find a pattern common to modern knowledge of theSolar system, consider that the Sun and the Moon are "special cases". Theirapparent motion in the sky is due to the Earth's rotation and the Moon'sorbit about the Earth. So if we allow the ancient astronomer to be wrongabout the Sun and Moon and only consider the "real" planets, then the orderis (from top down), Mercury, Venus, Mars, Jupiter, Saturn. In just theproper order that we know them to be. Five objects ordered properly is arandom chance of 1 in 5! = 120. I consider it possible that the assignmentof the squares to the planets was made as the result of a serious (and nottoo far wrong) attempt to order the planets according to their actualarrangement in the sky. Such an effort would be doomed to inaccuracy by thefact that the planets actually move around the Sun and not the Earth as wasbelieved.Notice also that in the Tree of Life (a fairly modern geometry -probably from about the 17th century) the planets are shown almost in anorbital system about the SUN. Exceptions being Saturn and Mercury.However, also note that with the Earth as the 10th square/planet, the Sun,Moon, and Earth are shown in the position of a solar eclipse.
Dan W writes:Here is a quote from page 377 of William Eisen's The Cabalah of Astrology (1986)"Eventually Ptolemy’s Tetrabiblos was translated into Arabic in the 8th centuryby the Jewish astrologer Al Batrig Mashallah of Baghdad, and for the next 500years, up until the middle of the 13th century, [Ralph William] Holden [in hisbook The Elements of House Division, 1977,] traces the passing of theastrological torch into the hands of the great Muslim astrologers. A renownedschool of astronomy and astrology was established in Baghdad and flourished formany years. Among the most important literary works to be produced during thisperiod was the Elements of Astrology, written by Al Biruni in the 11th century.This book carried the Equal House system of Ptolemy even further. These menthoroughly understood the value of the Solar Houses (where the Sun is placed atthe ascendant, or at the East point in the chart), and they established a systemof Arabian Points, or Parts. The position of the Moon then became the “Point ofFortune,” Mercury the “Point of Commerce,” Venus the “Point of Love,” etc. Thehouses in which these sensitive points appeared, when compared with the actualhouses of the birth chart, thus enhanced the over-all interpretation of thehoroscope to a remarkable degree."Dan W comments: The translation of the Tetrabiblos was done by a Jewish astrologer and weknow that later the rabbi of Damascus was deep into magic square Kabbalahmysticism around 1500. Further the magic square references in Aggrippa's TheOccult Philosophy show associations with Hebrew, indicating a Jewish source. Wecan conclude that even if the Arabs originally assigned the squares to theplanets, there was an interchange of ideas going on between Arab and Jewishmystical astrologer-mathematicians. The place of this syncretism was probably in Baghdad. Now Baghdad is inMesopotamia, the site of the ancient civilizations of Sumeria and Babyloniawhere the order of the planets used in assigning the magic squares was firstworked out.Mark Swaney writes:The Brethern of Purity: I'm still working on the main leads, but thebackground information is very interesting in itself. The BP were agroup of Ismaili scholars. The Ismaili are a sub-sect of the Shia brandof Islam, and are and were, considered heretics by the other Moslems.The history on this sect is obscure and I am working on finding out somemore about the Ismaili, but the main points I know are these. TheIsmailis have/had their own Imam, like a Pope I think, except theposition is hereditary, Ismaili Imams are not the same as those of theother Shias. They were oppressed vigorously, and many Ismailis werekilled in wars of suppression waged by the orthodox Sunni who were inpower in Egypt, the Fatimids. At the time of the production of theRasail of Ikhwan as-Safa, or the Encyclopedia of the Brethren of Purity,the Ismailis were an underground organization in the neighborhood ofBaghdad, which was under the rule of the Fatimids in Egypt.The BP were said to have been organized a century or so before thetime of the Rasail, but no one is exactly sure when they started.Anyway, it seems that the Imam in about 989 AD decided to have the BPproduce an encyclopedia. This was because the BP represented theintelligentsia of the Ismailis at the time, and because they werealready an organized group. Under the direct supervision of the Imam,the scholars of the BP met secretly in a cave and began work on theRasail. After the completion of the Rasail, copies were made and onewas placed in every Mosque in the area in and around Baghdad.The purpose of the Rasail was to reconcile Greek philosophy with theprecepts of the Ismaili sect, to show that the beliefs of the Ismailiswere in accord with the scientific knowledge of the day! Now keep inmind that this book, the Rasail, contains the FIRST recorded example ofa magic square of 6. All the numbers in the New Testament that relateto the multiples of 37 are said by Fidler and Michell to be based onthe square of the sun. However, the straight academic world says"whoa!" the first magic square of 6 was invented by the ARABS 1000 yearsAFTER the NT was written. The Greeks are NOT supposed to have anyknowledge of magic squares at all. But, when we look further, it turnsout that we don't have the name of the first mathematician to create asquare of 6, and we don't know if he/she was Greek, Arab, or a Martian.We just have an encyclopedia where knowledge is collected, presumablyknowledge that was accepted and common to the Greeks in the 10thcentury, otherwise how could the Rasail have performed as it wasintended?So we have a new lease on the idea that MAYBE there was a knowledgeof magic squares in Greece in antiquity. Or still possible, and just asintriguing, the Babylonians. We are talking about Baghdad where thisheretical Islamic sect creates the Rasail, the sum of all knowledge thatcontains magic squares and proclaims that the Ismailis arescientifically correct in their beliefs.There is more. Ever hear of Hassan a-Sabah? Well, if not, look himup. He is considered the inventor of the modern 20th centuryintelligence service, 1000 years ahead of his time. The words Assassinand Hashish? They derive from the name of the warrior/terroristsorganized by Hassan a-Sabah, the Hashishim. Who were they? You guessedit, the Ismailis. I am now looking into the links between a-Sabah andthe BP. It gets very interesting, doesn't it?Mark Swaney writes further:
I have found that the BP and the Ismailis werespecifically influenced by Neo-Platonism in the 8th century. I havelocated a reference that should be of some help in understanding therole of the magic squares in the Ismaili beliefs. Neoplatonists: An Introduction Into the Thought of the Brethern ofPurity, Allen&Unwin, London 1982.another reference is;Early Philosophical Shiism by P.E. Walker, Cambridge, 1993.I am also finding information on the theory of the concentric spheresmentioned by Fidler, as well as information on Al-Biruni.Ancient CosmologiesAllen & Unwin London 1975.Dan W writes:The relation between the Assassins and the BP is weird inthe extreme. See if you can find out anything about thepsycho-spiritual/brainwashing disciplines the Assassins used, would you.Maybe the same methods were used in working with the squares, a la the Rabbiof Damascus and his Kabbalistic system for meditating on the squares.Mark Swaney writes: More on the BP, Sabah, etc. The BP are thought to have been formed aboutthe same time as the formation of the Ismaili sect, about three hundred yearsbefore the publication of the Rasail in the 10th century. The sects of Islamseem to all be named after the person whom that sect originally believes to bethe Imam. I have read so many names of Imams that I am a little confused as towho is who, but the Imam that preceded the split that created the Ismailis was avery educated man, and very interested in philosophy. It is he who is creditedwith infusing the Ismailis with a philosophically based belief system. I haveread a critique of the story of Adam and Eve by an medevial Ismaili writer thatis in my mind remarkable for it's very modern analysis of the story. TheIsmaili brand of Islam (which still has millions of followers) is described bymodern scholars of religion as "gnostic" and "neoplatonic". The Ismailicosmology is certainly concerned with the hierarchy of the universe. Theirbelief in the succession of the Imams as the living representatives of God onEarth was fused with the Platonic/Kabbalistic Theory of Emanation at a veryearly period. I am fascinated by the thought of these medieval Islamic Mystics,they certainly deserve attention and consideration for their contributions tothe thread of thoughts that we pursue.Sabah is another mystery and a famous character from the late 11th centuryin Iran. Hassan a-Sabah was also known by the moniker "The Old Man of theMountain". Almost everything that we know about Sabah has been written by hisenemies. So we should give the guy a break and try to look at him objectively.Actually, what is known about Sabah for sure is that he must be considered agenius. He founded an intelligence service that was unrivaled in the worlduntil the 20th century. He invented what his enemies call terrorism, his modernfriends would call it "guerilla warfare". He was a military genius and afteroccupying the "Eagle's Nest", his famous fortress in northern Iran, he neverlost it, and he conquered several other forts in the surrounding area. Hisbattles with the Seljuck Turks made history. Sabah converted to the Ismailifaith as a young man, and he is still considered by the modern Ismailis to be aHero of the Faith. True, he was supposed to have been cruel and bloodthirsty.Yeah, and your mother wears army boots.The really interesting thing about Sabah for our studies is that in additionto being a military genius, he was also known to be a scholar. The organizationhe created, the Hashishim, or Assassins, was a "Masonic" military organization.By the way, the words Assassin and Hashishim and Hashish are all thought to becorruptions of Sabah's first name, Hassan. The Assassins were in essenceKamikaze's. They were trained to strike an enemy and not escape, but stay andfight 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 hisfortress. He was known to have had an interest in mathematics, and to haveencouraged the study of mathematics and philosophy by his followers. TheAssassins practiced initiation rites, and had strict grades of hierarchy, sothat modern historians have described them as "Masonic" in nature. Sabah andthe Assassins also had intriguing contacts with the Crusaders that I am nowtrying to find out more about. All this is hugely interesting for all theobvious reasons.The initiation rites are the probable source of the story about Sabah's useof drugs to fool initiates into thinking they had gone to heaven when in realitythey were only in Sabah's garden. This story was written by Marco Polo whopassed through the area of the Eagle's Nest 150 years after Sabah and theAssassins. There is no other documentation to back it up, and so it must betaken with a grain of salt. Personally, I think that no matter howmuch hash someone ate, it is very unlikely that they would wake up after fallingasleep and think themselves to be in heaven. But the available evidence doesindicate that the Assassins practiced some form of discipline that may havebordered on modern theories of mind control. Another example of Sabah'sprescient inventions.After the Mongols conquered the Eagle's Nest in the late 13th century, theAssassins and the Ismailis in general declined from any power in the politicalsense. The Mongols burned the library at the Eagle's Nest, so no books by Sabahor the Assassins survive today. The whole essence of the organization built bySabah rested on obedience, faith, and above all else, secrecy. We should not besurprised that a great deal of the knowledge of the Assassins was lost. Weshould also keep in mind that secrecy was one of the hallmarks of the gnosticsand other early mystery cults.
Dan Washburn writes:
Hmmm. Since we know so little about them its hard to tell what kind ofdisciplines the Assassins practiced. As you say, the rumor is drugs. However, nowthat I think about it, if they were supposed to visit paradise, maybe they werepracticing some form of Ascension similar to Jewish merkabah mysticism, whichinvolves trance visits to the heaven world. Also if the Assassins were trulygnostic, then ascension is an even more likely methodology, since the gnostics andhermetics were practitioners of ascending through the seven spheres of the planets.Ah ha - another connection to the seven planets which are connected to the squares.You might take a look at Dan Merkur's book Gnosis to get more of an idea on allthis.
Mark Swaney writes:I am suspecting more and more that the Brethrenof Purity's book the Rasail may have assigned the magic squares to the planets, but Iwill have to wait until I can get my hands on some source material that finally getsdown to the dirty details of just WHAT the BP saw in the squares, and why. Thebackground data certainly suggests to me that the BP associated the squares with theplanets in accordance with the medieval cosmology.That puts us at about 990 AD for the publication of the Rasail that contains thefirst recorded example of a magic square of 6. The interesting thing about this fact isthe inherent nature of the number 6. Here is where the mathematics provides someclues. Has you ever tried to create a magic square? If you study the squaresmathematically you will see that different orders have different properties and some areeasy and some are difficult. In brief, there are even numbers and there are oddnumbers. Further, the even numbers come in two varieties, those that are evenlydivisible by 4 and those that are not. Each magic square has a particular pattern thatis made by connecting the numbers in order as they appear in the square. Odd ordermagic squares have patterns that are symmetrical about a principal diagonal. Evensquares that are divisible by 4 have patterns that either have symmetry about 1 or 2axis that are either horizontal or vertical or they may have no symmetry at all. Butsquares of 6, 10, 14, 18 etc. are strange. The patterns of the un-evenly even squareshave no symmetry. There are other aspects of the UE squares that make them unusual.Magic squares are related mathematically to another kind of square numberarrangement called a Latin square (also called Greco-Latin squares). Methods ofconstructing magic squares by using Latin squares were published by a latermathematician named De LaHire. Latin squares have been (and are) studied by numbertheorists and also have found applications in modern technology. The idea of a Latinsquare is to arrange items with attributes in rows and columns so that each row andcolumn has 1 and only 1 of each kind of attribute. For example, make a Latin square of4 by using 16 cards, 4 from each suit in the order Ace, King, Queen, Jack. Now arrangethe cards in a square so that each row and each column has 1 each of the suits and 1each of the ranks. When you have done it, you have a Latin square. Now in later historythe most famous of the mathematicians took up the problem of magic squares, among thembeing Euler, the greatest mathematician of all time. Euler studied magic squares andLatin squares. One of his famous unsolved problems was the Euler Conjecture - astatement to the effect that UE Latin Squares are impossible. Actually, if I rememberit right, the EC had been disproved by 1960, in as much as larger UE Latin squares (suchas a LS of 10) can be constructed, however they proved Euler right in the case of aLatin Square of 6. It's impossible.The paper I received from the University of New South Wales on the Magic Squares ofManuel Moschopoulos, written in 1306 AD, is the first above-ground western writing thattells how to construct a magic square. But it is deficient in that it only givesmethods for constructing odd and evenly-even squares. It says nothing about how to makesquares of 6,10, etc. Furthermore, the methods of constructing odd squares and EEsquares take advantage of the inherent symmetry of those numbers, so that once known,anyone can create multitudes of different squares of various sizes without understandingany of the real combinatorial principals that magic squares are ultimately based on.Except that you can't make a magic square of 6 by resorting to such simplistic methods.That one you have to do the hard way.From the mathematical point of view of the construction of a square of 6 is a moredifficult task, and the appearance of the first magic square of 6 is far moresignificant mathematically than the first appearance of a square of 7 or 8 or 9. Didthe medieval mystic/mathematicians understand these points? They must have. And yet wedo not see any publication of their methods. Al-Buni the Arab mathematician who wroteabout magic squares around 1200 AD may have known how to construct a square of 6, butthe references I have so far indicate that he only gave the methods later published byMoschopoulos. The Rasail Ikhwan as-Safa is therefore crucial to our quest because itback dates knowledge of the square of 6 to sometime before 990AD in the west.
Dan Washburn Writes (04/02/00)
Nigel Pennick's Magical Alphabets has a section on magic squares in the chapter on the hebrew alphabet and also a chapter titled 'magic squares, literary labyrinths, and modern uses.'Here is a quote from the hebrew alphabet chapter, p34:"In Hebrew magic each planet can be seen as a symbol of one of theSephirah on the tree of life. Saturn signifies Binah, the thirdSephirah, whilst Jupiter, corresponds with the fourth Sephirah. Chesed.Mars parallels the fifth Sephirah, Geburah, whilst Mercury is the eighthSephirah, Hod. This planetary scheme was the basis of Assyrian andBabylonian ziggurats, of which the ill-fated Tower of Babel was anexample. In their purest cosmologically defined form, each of theziggurats’ seven stages or platforms represented one of the sevenplanets. The magic squares are arranged in a sequence that starts withthe smallest grid at the outermost. For structural reasons, theoutermost square on a ziggurat was the largest. However, it was ruled bythe corresponding magic square, and painted in the corresponding colour.The engraving of the Khorsabad ziggurat, reproduced in Fig. 10, gives agood idea of the principle.Summary of ziggurat drawing:level 7 - moon - whitelevel 6 - mercury - bluelevel 5 - venus - greenlevel 4 - sun - yellowlevel 3 - mars - redlevel 2 - jupiter - orangebase level - saturn - blackThe simplest magic square is the square of three by three, ascribed toSaturn, in which each line adds up to 15 and the total of all thenumbers added together is 45. This is the square most commonly used byEuropean magicians. Its traditionally associated colour is black,signifying the outermost planet and the bottom tier of the ziggurat."Anyone know anything about ziggurats? Does this color scheme actuallygo back to Sumerian/Babylonian sources or is it a later occultist'sdream? When did the magic squares become color coordinated? This isthe first I've heard of it. Did Aggrippa mention this in his 'OccultPhilosophy' (I saw a translation of the book at Borders the other daybut haven't had time to look at it)? Or is this an invention of S. L.MacGregor Mathers or one of the other Golden Dawn boys?I've known that the tiers of the ziggurat represented the planets butalways assumed that it was an allegorical ascension thru the heavenlyspheres to the realm of the fixed stars in an order of the planets thatstarted with the nearest to the earth and progressed outward. The orderoutlined here seems to be a mirror image of the heavens. With saturn atthe base the ascension of the pyramid is a descent thru the planets.This may offer an explanation for something that has troubled scholarsof Merkabah mysticism, the form mysticism took in Judaism for thethousand years before the rise of the Kabbalah around 1200 AD. In atrance the mystic ascends through seven visionary palaces to the thronechariot of God, the Merkabah. The movement is upward but the existenttexts refer to it as the Decent to the Merkabah. If the seven palacesare based on the seven planets, then the visionary may indeed bedescending through the heavenly spheres.Mark Swaney Writes (04/17/00) Have received a fairly exhaustive list of references and information onmagic squares and their history from David Singmaster, the Englishmathematician and Rubick's Cube expert. Dr. Singmaster has sent to me a pileof references about a quarter inch thick. This material is exclusivelyREFERENCE material, so that the complete works cited would fill a truck, I'msure. It's taken the weekend just to read through and assimilate the material,and there is much in it that is new to me, and bears directly on problems weare interested in.The first news is that we have new information on the question of theassignment of the squares to the planets. The following is a list I made uplast night from the Singmaster chronology and reference.The (so far) earliest known explicit association of the Planets with theSquares occurs sometime before 1384 in a document titled QABS al-ANWAR byNadruni. I don't know anything more about this document except that Nadrunigives the same associations of the squares with the planets as that later givenby Agrippa, 3 - Saturn, 4 - Jupiter, 5 - Mars, 6 - Sun, 7 - Venus, 8 - Mercury,9 - Moon.Another Arab manuscript, untitled, author unknown, circa 1466, gives areverse mapping of squares to planets, i.e. 3 - Moon, 4 - Mercury, etc.The first known European set of Magic Squares associated with the planetsis in a 15 century Latin manuscript in Cracow, described as Jagiellonian MS#753. The reference does not say what the explicit ordering is, or who wroteit, or exactly when, or for what purpose.In 1498 Pacioli wrote DeViribus, which gives squares and planets in thesame relationship as Nadruni and later Agrippa. This is important as apossible source for Albrecht Durer's square shown in Melancholia I.In 1531 Agrippa publishes De Occulta Philosophia, the second book of whichgives magic squares associated with the planets in the well-known order 3 -Saturn, 4 - Jupiter, etc.In 1539 Cardan writes Practica Arithmetica and gives the squares andplanets in the reverse as that published by Nadruni, Pacioli and Agrippa.Several interesting points to made about this information even inadvance of receiving the works cited are;1. The association of planets to squares is first PUBLISHED in the Arab worldin sometime in the mid 14th century.2. The mapping of planets to squares is given in two orders, each the reverseof the other, and each mapping is used in both the European and Arabicpublications. Attention DAN WASBURN - you asked about the order of planets vs.squares in your post on magic squares and Ziggurats, indicating a possibleconfusion about the Merkabah mystics as to whether they are asending ordescending. It appears that there were TWO "traditional" orders of planets.3. Each of the orders results in the square of 6 being assigned to the Sun.This is due to the position of the Sun in the "center" of the list of planets,an at least allegorically helio-centric system. As I have said, this hasimplications for us in our interest in the numerology of the square of 6.Another interesting assertion in the Singmaster material is a reference toan author who claims that the 3rd Century neo-platonists had knowledge of themagic square of 3. This is the sole academic reference to a western knowledgeof magic squares before the 14th century. A very interesting development thatwe will be sure to investigate for obvious reasons. If supportable, this wouldput a new light on the order of planets given in the Sepher Yitzarah, writtenin about the 3rd century in Palestine, and push back knowledge of the squaresin the West 11 centuries.As a "bonus" for me especially, Singmaster has references to manymathematicians and puzzle-authors who have contributed to magic squareliterature over the years. Quite unexpectedly, I discover that Dr. Singmasterhas excerpts from the notebooks of CHARLES BABBAGE, the famous Englishmathematician inventor and "grandfather of the computer". It seems thatBabbage was interested in Latin Squares, Magic Squares, and Magic Cubes.Along this line, (though not concerned with my main interest in the squaresas magical devices) we can add some famous names to the list of people whosquandered their time wrestling with these devices. Among them are;Leonard Euler - often considered the world's greatest mathematician -studied Latin Squares and Magic Squares, author of the famous "EulerConjecture".Pierre Fermat - super-famous author of the finally-proved "Fermat's LastTheorm" - Fermat conceived and produced the world's first Magic Cube, alsodiscussed magic triangles with Frenicle.M. Mersenne - friend of Fermat's and fellow number theorist.Bernard Frenicle - friend and correspondent of Fermat and Mersenne - firstto list and produce all the 880 possible magic squares of 4.Benjamin Franklin - worked with large magic squares and magic circles.Charles Babbage - the 19th century genius who first described the conceptof the modern computer and who attempted to build one - called a "differenceengine"Additional news to pass along - the Singmaster material provides thestarting point for a LARGE research project, including as it does virtually allthe previous research along the lines we wish to continue. What is clear froma review of everything I have collected is that while some information isknown, a lifetime could be devoted to working on the history of magic squaresand the associated religious/magical ideas. The contributions of the Indiansin particular has not been very well understood, and the authors disagree amongthemselves as to the sources of the Arab magic squares, Indian or Chinese.Dr. Singmaster has not touched the esoteric side of the question. None ofthe mathematical historians (who nevertheless give by far the most detailedinformation on sources) treat the larger subject of the relationship betweenmathematics and mystical experiences and philosophy. I think the territory isjust waiting for someone to make a PhD dissertation on the subject. Anytakers?Finally, I have run out of time to write today, but there is more, such asthe several sources that recommend that the magic square of 3 be used as aremedy for a hard labor during childbirth, such as the 5 15th century cast ironplates found in central China in 1958 with the magic square of 6 inscribed onthem, such as a reference to the Chinese god of the POLE STAR (attentionBarry!) and a description of the path of this god through the "houses" of theLo Shu square. Information on Chinese beliefs about the Ming-Tang Palace andthe Emperor.In short, enough meat to chew on for quite a while. Anybody wanna dive inand help find and analyze this stuff?Mark Swaney********************************************************************More information and links
China
Because there are 64 slots in the 8x8 matrix of the Magic Square of Mercury and because magic squares were important in China, I wondered ifthere was any connection with the 64 hexagrams of the I Ching.I discovered an essay by Tayagi Nagasiva on Magic Squares and the I Ching
Quote from John Opsopaus 4 9 2 3 5 78 1 6Chinese knowledge of the Saturn Square is shown by the ground plan of theMing-T'ang temple, which was built in A.D. 56. However, as Stapleton (Antiq. Alch. 15) says, "a much greater antiquity for this form of temple is indicated, firstly, by a templeof this plan being essential for Imperial worship, and, secondly, that in the7th century B.C., during the time of the warring Lords, it was believed to have been used by Wu, the alleged founder of the Chou dynasty in 1025 B.C., when sacrificing to hisancestors. Moreover, if this tradition be correct, the Magic Square form oftemple may ultimately be of Scythian origin, introduced at this time from Bactria, or ancient Iran, with the foreign mercenaries from the West, to whose help Wu owed his success in establishing a new dynasty." (From Bactria it may be traceable backto Mesopotamia.)The Ming-T'ang had twelve stations for the monthly "Proclamation of Space andTime." There is one station for each line segment on the perimeter of the square, that is, two for each corner (even) square, one for each side (odd) square. The eight squares on the perimeter represent the eightfold year (3 = vernal equinox, 9 = summer solstice, 7 = autumnalequinox, 1 = winter solstice). The central square corresponds to the additionaldays of the year beyond the twelve lunar months represented by the twelve line segments of the outer squares. Thus the Son of Heaven visited the central room of the temple(numerically 5, the Emblem of the Center) at "the end of summer - a criticalperiod when the transition was made from the yang seasons to the yin seasons" (Granet, Rel. Ch. 67). Alternately, the twelve line segments of the perimeter can represent the solaryear and the zodiac. Thus the representation of Time; the temple also represented Space by assigning 8+3 = east, 4+9 = south, 2+7 = west, 6+1 = north (the same four numbers as the elements, though not the same pairs of squares); opposing directions balance to 20, as do opposing elements. (Granet, Rel. Ch. 66-8; Stapleton, "Antiq. Alch.")Blofeld (I Ching, 218) says that mankind once understood how the Lo-Shu Squareis connected with the (apparently illogical) Later Heaven Sequence of the I Ching, but that it has been forgotten and now only the gods know it. I certainly have not been able to find it. (The connection established by Hacker (41) seems to me to be contrived, although it is remarkable enough that any connection can be established at all.)
Christianity
To get a look at some work by Dan Gleason relating the 888 of Jesus tothe 666 of the magic square of the Sun click on the link below:http://www.jesus8880.com/gematria/666.htm
IslamKieth Critchlow has a chapter on Magic Squares in Islam in his bookIslamic Patterns.
The following link has a picture of a page from one of Al-Buni's books with the caption given below:
http://www.vh.org/Welcome/UIHC/MedMuseum/ArtThatHeals/10Knowledge.html
21. Diagrams from the Book of Buni, theGeez translation of a version of the "Sun ofKnowledge" (Shams al-Marif), a bookattributed to Al-Buni, an Egyptian authorof the thirteenth century. Each diagramcontaining figures or letters isaccompanied by its method of use. Bookof Buni, eighteenth century to nineteenthcentury, parchment, 27.5 x 24 cm. Privatecollection. Photo courtesy of Guy Vivien
A quotation from John Opsopaus:Magic Square of 34 9 2 3 5 78 1 6According to the Theory of Balance attributed to 8th century Muslim alchemistJabir ibn Hayyan (based on 3rd century works by Zosimos and others), the Cosmos and everything in it is made from the numbers 1, 3, 5, 8, 17 and 28; they are thefoundation of all matter, of every science, and even of any possible language.The first four numbers were assigned by the Jabirian alchemists to the elements, 1=fire, 3=earth, 5=water, 8=air. The sum of these is 17, which is the fifth number. TheGnomon, which gives the larger square, sums to 4+9+2+7+6 = 28, the sixth number,the second Perfect Number.
Quoted from the following site:JudaismI've been reading Aryeh Kaplan's Meditation andthe Kabbalah, He has a chapter on the Kabbalistic writingsof Rabbi Joseph Tzayach (1505-1573) the Rabbi of Damascus.Kaplan says that magic squares were well known in ancient Indiaand China and that they were introduced into the west in the1400s by Moschopulus of Constatinople. Kaplan believesthey were also known to the ancient Kabbalists. Tzayachprobably received a hidden tradition concerning these figures.His system attributes the traditional magic squares to theseven planets but goes on to attribute higher order magic squaresto the Sephira on the Tree of Life, a Jewish mystical diagram: The square of tento Kether, of 11 to Chokmah, etc. skipping the square of15, so that Malkuth is the 20x20 matrix.Tzayach apparently had a system that involved meditatingon the color, number, and letter forms in one room after anotherin a square. Each row was called a house and each box a room.What we might be seeing here is a form of Merkabah Ascension.Up through the spheres of the 7 planets and then through 10(usually 7 in other literature) palaces of the King to the throneof God. The position of meditation described is that of Elijah on Mt. Carmel,sitting cross legged with head between the knees, a position associated with MerkabahAscension.Magic square lore may have passed back and forth betweenJewish and Muslim mystics.General
David Singmaster gives a number of dates relating to the history of Magic Squares in his"Chronology of Recreational Mathematics."
Here are some book references (I have not looked at) from the bilbiography at
Edward Falkener, "Games ancient and oriental and how to play them : being the games of the ancient Egyptians, the hiera gramme of the Greeks, the ludus latrunculorum of the Romans and the oriental games of chess, draughts, backgammon and magic squares" :New Dover ed:New York : Dover Publications , (1961) Soror A.L. ,Compton, Madonna, "Western mandalas of transformation : magical squares, tattwas, qabalistic talismans", St. Paul, MN, U.S.A. : Llewellyn Publications , (1995) Richard Webster, "Talisman magic : yantra squares for tantric divination", St. Paul, Minn., U.S.A. : Llewellyn Publications , (1995)
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