Mark Swaney on the History of Magic Squares

 

4 9 2
3 5 7
8 1 6

This is a magic square of order 3 (three numbers to the side of the square).  If you add up any row, column, or diagonal, it sums to the same number, 15.  There are magic squares of order 4, 5, 6, etc.

See this link for a listing of magic squares of order 3 through 11:

http://www.pse.che.tohoku.ac.jp/~msuzuki/MagicSquare.byMATLAB.html

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 may
have a detail or two out of place. I'm reading all this stuff and then roaring
off an epistle. Later, I always think I should have done it differently, but
what the hell? Also, I find that I like to write a lot of text when I'm feeling
radioactive.

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 and
picking up information, so the following is subject to revision and
expansion.

The history of magic squares is murky, mysterious, and has not been
well researched by academics. Consequently the claims are
contradictory, and in some cases exaggerated. Very little is known
about the origin of magic squares. Next to nothing is known about the
movement of the idea of a magic square before about 1300 AD. Three
cultures are known to have created magic squares, the Chinese, the
Indian, and the Arabic. In each culture they were viewed as having
supernatural properties.

China

The first magic square in history was created in China by an unknown
mathematician, probably sometime before the first century AD. Called
the Lo Shu square, it is a magic square of 3 that was said to have
appeared on the back of a turtle that came up out of the river. Lo Shu
supposedly means "river map" and the story of the appearance of the
turtle had to do with a sacrifice to the river god. Right from the beginning we are seeing an
essentially mathematical construction combined with the supernatural. I
have not found an analysis of the story of the turtle and the Lo Shu
square from the point of view of folklore or mythology that would shed
more light on the story. The Lo Shu square is later associated with the
floor plan of a mythical palace, that of Ming'tang. Again, this is fragmentary, I
have seen a diagram that shows the floor plan, but no explanation as to
what the thinking about the square was, why it was used as a floor plan for a palace,
or other information to flesh out the picture. The Lo Shu square is
also connected to the I-Ching, though there is no explicit plan of
correspondence that I know of. The oldest documents that refer to the
Lo Shu square are ambiguous, but one reference lists a Shu Ching in 650 BCE who makes a
reference to the "river map" which may be the magic square of 3. In 500
BCE, and 300 BCE, the river map is mentioned, but no explicit magic
square is given. In 80 AD Ta Tai Li Chi gives the first clear reference to a
magic square. In 570 AD Shuzun gives an actual description of a magic
square of 3. Not until 1275 do we hear of the Chinese making squares of
order larger than 3. Norman Biggs says that this is because the Chinese
regarded the Lo Shu square as an object of the supernatural, rather than
as an object of human curiosity, and it was therefore not a subject for study.

India

We find the first magic square of 4 in the first century in India by
a mathematician named Nagarajuna. This is all that I know at the moment
about the early development of the magic square in India. However,
India is the birthplace of much superior mathematics, and was advanced
in other areas of combinatorics at an early date. I would be surprised
if it did not eventually turn out that India has an older tradition
involving magic squares. Still, this approximate date is interesting
for other types of analysis. The next known date in the Indian
development is an 11th or 12th century Jaina inscription that includes a
magic square of 4. This particular magic square of 4 has unusual
properties not found in other magic squares before that time, and the
whole class of squares having these properties is called "Jaina
squares", including squares of order larger than 4. I have no
information on the document, why it includes the magic square, or what
connection it has to the Jaina religion in medieval India. Much remains
to be explained.

Islam

The first magic squares of 5 and 6 appear in an encyclopedia in
Baghdad about 983 AD by Ikhw'n al-Saf' Ras'il, though several earlier
Arab mathematicians also wrote about magic squares. How it came to pass
that the Arabs acquired knowledge of magic squares is unknown. It is
not known if they invented them separately or if they were introduced to
them by another culture. Biggs assumes that the Arabs got the idea from
the Chinese, though he doesn't know how the connection was made. I
think it far more likely that the Arabs got magic squares from the same
source that they got decimal arithmetic, namely India. The Arab Jihad of
the 7th century succeeded in conquering portions of India, and the Arabs
absorbed a great deal of Indian mathematics and astronomy. It is known
that many other aspects of combinatorial mathematics passed from India
to the Arabs in this way. Al-Buni was an Arab mathematician that worked
on magic squares and also believed in the mystical properties of magic
squares, though no details on this number mysticism are available.
Al-Buni did his work on the squares about 1200 AD. Sources have also
referred to the Arabs using magic squares in making astrological
calculations and predictions, again no details are given. The
association of the squares with astrology and the heavens appears to be
original with the Arabs, but again, much is unknown concerning the
Indian tradition.

Europe

It is from the Arabs that the West finally receives the idea of magic
squares. In 1300 Manual Moschopoulos, a Greek Byzantine scholar, writes
a mathematical treatise on the subject of the magic squares.
Moschopoulos' book builds on the work of Al-Buni who preceded him.
Western authors are quick to point out that Moschopoulos treats the
squares in a purely mathematical way in contrast to the mystical ideas
of the Arabs. Moschopoulos is generally considered to be the first
westerner to know of the squares. A mistaken attribution of knowledge
to Theon of Smryna in about 130 AD has continued to be cited, but the
"square" in question is definitely not a magic square, being just a
natural square. 

After Moschopoulos, in the 1450's Luca Pacioli of Italy
worked on magic squares and owned a large collection of examples of
magic squares. With Pacioli we come to the doorstep of the known
Western mystical tradition concerning magic squares. What Pacioli
himself believed about the squares I don't know, but in the 1480's Italy
was to see the birth of the Renaissance which revolutionized European
thinking. Marsilio Ficino wrote about and propounded a school of magic
based on his translations of the Hermetic
documents that were at the time believed to be as ancient as Moses.
Pico Della Mirandola wrote the "Nine Hundred Theses" - much of it based
on the translations of older Jewish Kabbalistic texts. Artists like
Albrecht Durer eagerly absorbed the new perspective painting based on
the mathematical developments of Della Franscisca, who was popularized
by the later books of Pacioli.

In about 1510 Cornielius Agrippa, that problematical character,
wrote "De Occulta Philosophia" in which he expounds on the powers of the
magic squares, and supplies examples of them in the orders 3- 9. This
book became famous throughout Europe and was very influential until the
counter-reformation and the witch-hunts that followed. Most what is
commonly thought of about Agrippa is the result of the witch-hunts and
propaganda, i.e. he was a sorcerer, he was in league with the devil,
etc. The truth about Agrippa and his book is much more complex than
that, and in the explanation of Agrippa's book we get the first inkling
of a detailed worked out system of mysticism concerning magic squares.
However, though we find out some details about the squares in their role as supernatural
devices, we are still left with conflicts and unanswered questions.

In 1514 Albrecht Durer made his famous woodcut "Melancholia I," which
features a magic square of 4 on the wall behind the "brooding genius"
that became the archetype of all the "thinker" type sculptures in later
years. The reason for the magic square of 4 being included in the
woodcut has been analyzed by the authors of "Saturn and Melancholy".
Briefly, the square of 4 is the square of Jupiter. The planet jupiter was considered
beneficial and was associated with the "sanguine" humor. Even today we
speak of someone's being "jovial" at a party. Durer's brooding genius
suffers from melancholia, which we call depression, and the square of
Jupiter was thought to bring down the influence of the planet Jupiter,
thereby helping to cure the depression. 

The Squares and the Planets

This is an example of the theory of magic propounded by
Marsilio Ficino. Ficino's magic is a kind of sympathetic magic where
objects, colors, sounds, etc. are all categorized as to what
"influences" they excite. Ficino's influences come primarily from the
planets, the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. The
magic is aimed at "drawing down" the influence or power of specific
planets in order to accomplish some end, such as protection from disease
or a psychological cure. In this magic we see the role of the squares
as being the mathematical archetypes of the planets themselves. As each
square has a set of characteristic numbers, these numbers then also
carry the influences of the various planets. In this way certain numbers can be said to be
"Solar" or "Lunar" numbers.

In this system, for our study, the important issue to understand is
how the particular planets come to be associated with particular
squares. More than one source has it that the correspondences between
the squares and the planets were the invention of Agrippa himself. 
The description of Agrippa and his book
by Francis Yates makes it appear that Agrippa made no original
contributions to magical theory in his book, but merely collected the
thought of others. Other sources simply say that the Arabs assigned the
squares to the planets. David Fidler in his book "Jesus Christ, Sun of
God" says that the arrangements came from the Babylonians. The ancient
system of cosmology had 7 planets, each in a concentric shell that
rotated around the earth. The Babylonians believed that the closest
planet was the moon, followed by Mercury, Venus, the Sun, Mars, Jupiter,
and Saturn. They placed the order of the squares such that the smallest
squares were associated with the farthest planets, thus Saturn is 3,
Jupiter is 4, etc. This relationship is important for several reasons,
but the one reason that is most striking is
the fact that the system assigns the square of 6 to the sun. By making
this assignment, the system is made to resonate with one of the most
ancient of numerological systems, namely that of the Sumerians. It was
the Sumerians with their Solar worship and their sexigesimal counting
system that firmly fixed the hours of the day at 24, sun nominally
rising at 6:00AM and setting at 6:00PM, and who gave us the still used
360 degree circle. The association of the number 6 with the sun is a
very ancient western tradition. Pythagoras on account of numerical
theory called 6 the first "perfect" number. In view of these facts, the
magic square of 6 with a sum total of 666 must have made quite an
impression even in the 14th century, the earliest date that modern
conventional scholarship will allow a western knowledge of magic
squares.

Other points may be considered from the assignment of squares to
planets. First, consider that the ordering of the planets does not
follow the simple digits. That is, there is no planet associated with
the number 1 or the number 2. Does this mean that the correspondences
were made based on the squares (there is no magic square of 2) and not
simply on the single digit numbers 1-7? If this is the case then we might infer that the
ancient Babylonians had knowledge of magic squares. The whole issue of
the use by the author of the book of Revelations of the number 666 to
represent the Anti-Christ, in a passage that also includes the only
unmistakable reference in the Bible to the practice of gematria has been
but barely dealt with. Does this passage have to do with magic square numerology? Can other
names and phrases in the New Testament have had their values constructed
in such a way as to yield gematria values that have numerological
significance based on magic squares? Do various diagrams and
geometrical constructions such as the Tree of Life have a basis in the
geometry inherent in magic squares? These questions remain as
tantalizing possibilities, as yet not definitively answered by
scholarship.

Mark Swaney, January, 2000


Dan W. writes:

Below are the magic square numbers and their attributions to the
planets. If you recognize any of these numbers from astrology,
astronomy, ancient texts, musical tuning theory, the proportions of
buildings, sacred geometry, mathematics, physics, architecture, art
history or anywhere at all, please let me know.  

Planet
side of square
boxes in square
sum of any symmetrical group of four boxes
sum of any line - horizontal, vertical, diagonal
sum of perimeter values of the square
sum of the entire square

saturn    3 -  9 -  xx   - 15   -   40   - 45

jupiter    4 - 16 -  34  - 34   - 102   - 136

mars      5 - 25 -  52  - 65   - 208   - 325

sun        6 - 36 -  74  - 111 - 370   - 666

venus     7 - 49 - 100 - 175 - 600  - 1225

mercury 8 - 64 - 130 - 260 - 910  - 2080

moon     9 - 81 - 164 - 369 - 1312 - 3321

Mark Swaney writes:

The squares of 5 and 8 have a direct to relationship to the numerical
cosmology/religion of the Mayans. Read any text on their culture to find
their sacred numbers which are 4, 13, 20, 52, & 260. If Dan had listed the
numbers for the additional squares of 10,11,12,&13, you could see the
natural mathematical affinity that exists between 4 and 13. As it is, check
out 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 last
numbers in the series 1 - x^2. (Alpha and Omega?) The CN for square of 4 is
16 + 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 of
these magic number squares to a planet -- and by what logical reasoning?
Has ANY logical reason EVER been given for the assignments? If so, did
it make sense.?

Mark Swaney replies:

I am continuing to research the squares, and one of the many questions
that I want to answer is just the one you have asked. Several sources
credit Agrippa with the assignment of the squares to the planets, but that
is certainly not true, as Agrippa did not originate any of the Magic he
lists in his Occult Philosophy. At this time, I don't have any definitive
answer, and I don't believe that the question has been tackled by
acedemics. Fidler says that the Babylonians originated the system, that is
the ORDER of planets, Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon.
This specific order (one of 7! = 5040 possibilities) was known long before
Agrippa'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 get
assigned to the planets? An interesting point is that there is no planet
with the number 1 or 2, (there is no magic square of two.) The best
speculation is that the assignments were made by the Arabs, because it is
known that the Arabs incorporated Magic Squares with their Astrological
calculations. It is the classical position (such as has been researched -
meaning NOT MUCH) that Magic Square esotericism passed to the West from the
Arabs in about the 13th century.

Now there are a couple of points to make about the assignment of the
squares to the planets. First off, who ever came up with the arrangement
contrived it so that the Sun was assigned to the number 6. In doing so, the
Sumerian numerical/religious structure was preserved, because it was the
Sumerians who originally established the association of 6 with the Sun.
They were Solar worshipers, and they also possessed a base 60 numbering
system. They gave us 360 degree circles, and a time system that has the Sun
rising (nominally) at 6 and setting at 6. So the famous or infamous Square
of the Sun is esoterically consistent with the oldest Western culture.

For those who like to find a pattern common to modern knowledge of the
Solar system, consider that the Sun and the Moon are "special cases". Their
apparent motion in the sky is due to the Earth's rotation and the Moon's
orbit about the Earth. So if we allow the ancient astronomer to be wrong
about the Sun and Moon and only consider the "real" planets, then the order
is (from top down), Mercury, Venus, Mars, Jupiter, Saturn. In just the
proper order that we know them to be. Five objects ordered properly is a
random chance of 1 in 5! = 120. I consider it possible that the assignment
of the squares to the planets was made as the result of a serious (and not
too far wrong) attempt to order the planets according to their actual
arrangement in the sky. Such an effort would be doomed to inaccuracy by the
fact that the planets actually move around the Sun and not the Earth as was
believed.

Notice also that in the Tree of Life (a fairly modern geometry -
probably from about the 17th century) the planets are shown almost in an
orbital 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 century
by the Jewish astrologer Al Batrig Mashallah of Baghdad, and for the next 500
years, up until the middle of the 13th century, [Ralph William] Holden [in his
book The Elements of House Division, 1977,] traces the passing of the
astrological torch into the hands of the great Muslim astrologers. A renowned
school of astronomy and astrology was established in Baghdad and flourished for
many years. Among the most important literary works to be produced during this
period 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 men
thoroughly understood the value of the Solar Houses (where the Sun is placed at
the ascendant, or at the East point in the chart), and they established a system
of Arabian Points, or Parts. The position of the Moon then became the “Point of
Fortune,” Mercury the “Point of Commerce,” Venus the “Point of Love,” etc. The
houses in which these sensitive points appeared, when compared with the actual
houses of the birth chart, thus enhanced the over-all interpretation of the
horoscope to a remarkable degree."

Dan W comments: 

   The translation of the Tetrabiblos was done by a Jewish astrologer and we
know that later the rabbi of Damascus was deep into magic square Kabbalah
mysticism around 1500. Further the magic square references in Aggrippa's The
Occult Philosophy show associations with Hebrew, indicating a Jewish source. We
can conclude that even if the Arabs originally assigned the squares to the
planets, there was an interchange of ideas going on between Arab and Jewish
mystical astrologer-mathematicians.

   The place of this syncretism was probably in Baghdad. Now Baghdad is in
Mesopotamia, the site of the ancient civilizations of Sumeria and Babylonia
where the order of the planets used in assigning the magic squares was first
worked out.


Mark Swaney writes:

The Brethern of Purity: I'm still working on the main leads, but the
background information is very interesting in itself. The BP were a
group of Ismaili scholars. The Ismaili are a sub-sect of the Shia brand
of 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 some
more about the Ismaili, but the main points I know are these. The
Ismailis have/had their own Imam, like a Pope I think, except the
position is hereditary, Ismaili Imams are not the same as those of the
other Shias. They were oppressed vigorously, and many Ismailis were
killed in wars of suppression waged by the orthodox Sunni who were in
power in Egypt, the Fatimids. At the time of the production of the
Rasail of Ikhwan as-Safa, or the Encyclopedia of the Brethren of Purity,
the Ismailis were an underground organization in the neighborhood of
Baghdad, which was under the rule of the Fatimids in Egypt.

The BP were said to have been organized a century or so before the
time 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 BP
produce an encyclopedia. This was because the BP represented the
intelligentsia of the Ismailis at the time, and because they were
already an organized group. Under the direct supervision of the Imam,
the scholars of the BP met secretly in a cave and began work on the
Rasail. After the completion of the Rasail, copies were made and one
was placed in every Mosque in the area in and around Baghdad.

The purpose of the Rasail was to reconcile Greek philosophy with the
precepts of the Ismaili sect, to show that the beliefs of the Ismailis
were in accord with the scientific knowledge of the day! Now keep in
mind that this book, the Rasail, contains the FIRST recorded example of
a magic square of 6. All the numbers in the New Testament that relate
to the multiples of 37 are said by Fidler and Michell to be based on
the square of the sun. However, the straight academic world says
"whoa!" the first magic square of 6 was invented by the ARABS 1000 years
AFTER the NT was written. The Greeks are NOT supposed to have any
knowledge of magic squares at all. But, when we look further, it turns
out that we don't have the name of the first mathematician to create a
square 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, presumably
knowledge that was accepted and common to the Greeks in the 10th
century, otherwise how could the Rasail have performed as it was
intended?

So we have a new lease on the idea that MAYBE there was a knowledge
of magic squares in Greece in antiquity. Or still possible, and just as
intriguing, the Babylonians. We are talking about Baghdad where this
heretical Islamic sect creates the Rasail, the sum of all knowledge that
contains magic squares and proclaims that the Ismailis are
scientifically correct in their beliefs.

There is more. Ever hear of Hassan a-Sabah? Well, if not, look him
up. He is considered the inventor of the modern 20th century
intelligence service, 1000 years ahead of his time. The words Assassin
and Hashish? They derive from the name of the warrior/terrorists
organized by Hassan a-Sabah, the Hashishim. Who were they? You guessed
it, the Ismailis. I am now looking into the links between a-Sabah and
the BP. It gets very interesting, doesn't it?

Mark Swaney writes further:

I have found that the BP and the Ismailis were
specifically influenced by Neo-Platonism in the 8th century. I have
located a reference that should be of some help in understanding the
role of the magic squares in the Ismaili beliefs. 

Neoplatonists: An Introduction Into the Thought of the Brethern of
Purity, 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 spheres
mentioned by Fidler, as well as information on Al-Biruni.

Ancient Cosmologies
Allen & Unwin London 1975.

Dan W writes:

The relation between the Assassins and the BP is weird in
the extreme. See if you can find out anything about the
psycho-spiritual/brainwashing disciplines the Assassins used, would you.
Maybe the same methods were used in working with the squares, a la the Rabbi
of Damascus and his Kabbalistic system for meditating on the squares.


Mark Swaney writes: 

More on the BP, Sabah, etc. The BP are thought to have been formed about
the same time as the formation of the Ismaili sect, about three hundred years
before the publication of the Rasail in the 10th century. The sects of Islam
seem to all be named after the person whom that sect originally believes to be
the Imam. I have read so many names of Imams that I am a little confused as to
who is who, but the Imam that preceded the split that created the Ismailis was a
very educated man, and very interested in philosophy. It is he who is credited
with infusing the Ismailis with a philosophically based belief system. I have
read a critique of the story of Adam and Eve by an medevial Ismaili writer that
is in my mind remarkable for it's very modern analysis of the story. The
Ismaili brand of Islam (which still has millions of followers) is described by
modern scholars of religion as "gnostic" and "neoplatonic". The Ismaili
cosmology is certainly concerned with the hierarchy of the universe. Their
belief in the succession of the Imams as the living representatives of God on
Earth was fused with the Platonic/Kabbalistic Theory of Emanation at a very
early period. I am fascinated by the thought of these medieval Islamic Mystics,
they certainly deserve attention and consideration for their contributions to
the thread of thoughts that we pursue.

Sabah is another mystery and a famous character from the late 11th century
in Iran. Hassan a-Sabah was also known by the moniker "The Old Man of the
Mountain". Almost everything that we know about Sabah has been written by his
enemies. 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 a
genius. He founded an intelligence service that was unrivaled in the world
until the 20th century. He invented what his enemies call terrorism, his modern
friends would call it "guerilla warfare". He was a military genius and after
occupying the "Eagle's Nest", his famous fortress in northern Iran, he never
lost it, and he conquered several other forts in the surrounding area. His
battles with the Seljuck Turks made history. Sabah converted to the Ismaili
faith as a young man, and he is still considered by the modern Ismailis to be a
Hero 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 addition
to being a military genius, he was also known to be a scholar. The organization
he created, the Hashishim, or Assassins, was a "Masonic" military organization.
By the way, the words Assassin and Hashishim and Hashish are all thought to be
corruptions of Sabah's first name, Hassan. The Assassins were in essence
Kamikaze's. They were trained to strike an enemy and not escape, but stay and
fight to the death. So you can see why these people were so feared.

But the organization was not solely based on military/political adventures.
That's the mystery. Sabah was known to have amassed a large library in his
fortress. He was known to have had an interest in mathematics, and to have
encouraged the study of mathematics and philosophy by his followers. The
Assassins practiced initiation rites, and had strict grades of hierarchy, so
that modern historians have described them as "Masonic" in nature. Sabah and
the Assassins also had intriguing contacts with the Crusaders that I am now
trying to find out more about. All this is hugely interesting for all the
obvious reasons.

The initiation rites are the probable source of the story about Sabah's use
of drugs to fool initiates into thinking they had gone to heaven when in reality
they were only in Sabah's garden. This story was written by Marco Polo who
passed through the area of the Eagle's Nest 150 years after Sabah and the
Assassins. There is no other documentation to back it up, and so it must be
taken with a grain of salt. Personally, I think that no matter how
much hash someone ate, it is very unlikely that they would wake up after falling
asleep and think themselves to be in heaven. But the available evidence does
indicate that the Assassins practiced some form of discipline that may have
bordered on modern theories of mind control. Another example of Sabah's
prescient inventions.

After the Mongols conquered the Eagle's Nest in the late 13th century, the
Assassins and the Ismailis in general declined from any power in the political
sense. The Mongols burned the library at the Eagle's Nest, so no books by Sabah
or the Assassins survive today. The whole essence of the organization built by
Sabah rested on obedience, faith, and above all else, secrecy. We should not be
surprised that a great deal of the knowledge of the Assassins was lost. We
should also keep in mind that secrecy was one of the hallmarks of the gnostics
and other early mystery cults.

Dan Washburn writes:

Hmmm. Since we know so little about them its hard to tell what kind of
disciplines the Assassins practiced. As you say, the rumor is drugs. However, now
that I think about it, if they were supposed to visit paradise, maybe they were
practicing some form of Ascension similar to Jewish merkabah mysticism, which
involves trance visits to the heaven world. Also if the Assassins were truly
gnostic, then ascension is an even more likely methodology, since the gnostics and
hermetics 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 all
this.

Mark Swaney writes:

I am suspecting more and more that the Brethren
of Purity's book the Rasail may have assigned the magic squares to the planets, but I
will have to wait until I can get my hands on some source material that finally gets
down to the dirty details of just WHAT the BP saw in the squares, and why. The
background data certainly suggests to me that the BP associated the squares with the
planets in accordance with the medieval cosmology.

That puts us at about 990 AD for the publication of the Rasail that contains the
first recorded example of a magic square of 6. The interesting thing about this fact is
the inherent nature of the number 6. Here is where the mathematics provides some
clues. Has you ever tried to create a magic square? If you study the squares
mathematically you will see that different orders have different properties and some are
easy and some are difficult. In brief, there are even numbers and there are odd
numbers. Further, the even numbers come in two varieties, those that are evenly
divisible by 4 and those that are not. Each magic square has a particular pattern that
is made by connecting the numbers in order as they appear in the square. Odd order
magic squares have patterns that are symmetrical about a principal diagonal. Even
squares that are divisible by 4 have patterns that either have symmetry about 1 or 2
axis that are either horizontal or vertical or they may have no symmetry at all. But
squares of 6, 10, 14, 18 etc. are strange. The patterns of the un-evenly even squares
have no symmetry. There are other aspects of the UE squares that make them unusual.

Magic squares are related mathematically to another kind of square number
arrangement called a Latin square (also called Greco-Latin squares). Methods of
constructing magic squares by using Latin squares were published by a later
mathematician named De LaHire. Latin squares have been (and are) studied by number
theorists and also have found applications in modern technology. The idea of a Latin
square is to arrange items with attributes in rows and columns so that each row and
column has 1 and only 1 of each kind of attribute. For example, make a Latin square of
4 by using 16 cards, 4 from each suit in the order Ace, King, Queen, Jack. Now arrange
the cards in a square so that each row and each column has 1 each of the suits and 1
each of the ranks. When you have done it, you have a Latin square. Now in later history
the most famous of the mathematicians took up the problem of magic squares, among them
being Euler, the greatest mathematician of all time. Euler studied magic squares and
Latin squares. One of his famous unsolved problems was the Euler Conjecture - a
statement to the effect that UE Latin Squares are impossible. Actually, if I remember
it right, the EC had been disproved by 1960, in as much as larger UE Latin squares (such
as a LS of 10) can be constructed, however they proved Euler right in the case of a
Latin Square of 6. It's impossible.

The paper I received from the University of New South Wales on the Magic Squares of
Manuel Moschopoulos, written in 1306 AD, is the first above-ground western writing that
tells how to construct a magic square. But it is deficient in that it only gives
methods for constructing odd and evenly-even squares. It says nothing about how to make
squares of 6,10, etc. Furthermore, the methods of constructing odd squares and EE
squares take advantage of the inherent symmetry of those numbers, so that once known,
anyone can create multitudes of different squares of various sizes without understanding
any 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 more
difficult task, and the appearance of the first magic square of 6 is far more
significant mathematically than the first appearance of a square of 7 or 8 or 9. Did
the medieval mystic/mathematicians understand these points? They must have. And yet we
do not see any publication of their methods. Al-Buni the Arab mathematician who wrote
about magic squares around 1200 AD may have known how to construct a square of 6, but
the references I have so far indicate that he only gave the methods later published by
Moschopoulos. The Rasail Ikhwan as-Safa is therefore crucial to our quest because it
back 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 the
Sephirah on the tree of life. Saturn signifies Binah, the third
Sephirah, whilst Jupiter, corresponds with the fourth Sephirah. Chesed.
Mars parallels the fifth Sephirah, Geburah, whilst Mercury is the eighth
Sephirah, Hod. This planetary scheme was the basis of Assyrian and
Babylonian ziggurats, of which the ill-fated Tower of Babel was an
example. In their purest cosmologically defined form, each of the
ziggurats’ seven stages or platforms represented one of the seven
planets. The magic squares are arranged in a sequence that starts with
the smallest grid at the outermost. For structural reasons, the
outermost square on a ziggurat was the largest. However, it was ruled by
the corresponding magic square, and painted in the corresponding colour.
The engraving of the Khorsabad ziggurat, reproduced in Fig. 10, gives a
good idea of the principle.

Summary of ziggurat drawing:
level 7 - moon - white
level 6 - mercury - blue
level 5 - venus - green
level 4 - sun - yellow
level 3 - mars - red
level 2 - jupiter - orange
base level - saturn - black

The simplest magic square is the square of three by three, ascribed to
Saturn, in which each line adds up to 15 and the total of all the
numbers added together is 45. This is the square most commonly used by
European 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 actually
go back to Sumerian/Babylonian sources or is it a later occultist's
dream? When did the magic squares become color coordinated? This is
the first I've heard of it. Did Aggrippa mention this in his 'Occult
Philosophy' (I saw a translation of the book at Borders the other day
but 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 but
always assumed that it was an allegorical ascension thru the heavenly
spheres to the realm of the fixed stars in an order of the planets that
started with the nearest to the earth and progressed outward. The order
outlined here seems to be a mirror image of the heavens. With saturn at
the base the ascension of the pyramid is a descent thru the planets.

This may offer an explanation for something that has troubled scholars
of Merkabah mysticism, the form mysticism took in Judaism for the
thousand years before the rise of the Kabbalah around 1200 AD. In a
trance the mystic ascends through seven visionary palaces to the throne
chariot of God, the Merkabah. The movement is upward but the existent
texts refer to it as the Decent to the Merkabah. If the seven palaces
are based on the seven planets, then the visionary may indeed be
descending through the heavenly spheres.


Mark Swaney Writes (04/17/00)

Have received a fairly exhaustive list of references and information on
magic squares and their history from David Singmaster, the English
mathematician and Rubick's Cube expert. Dr. Singmaster has sent to me a pile
of references about a quarter inch thick. This material is exclusively
REFERENCE material, so that the complete works cited would fill a truck, I'm
sure. 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 we
are interested in.

The first news is that we have new information on the question of the
assignment of the squares to the planets. The following is a list I made up
last night from the Singmaster chronology and reference.

The (so far) earliest known explicit association of the Planets with the
Squares occurs sometime before 1384 in a document titled QABS al-ANWAR by
Nadruni. I don't know anything more about this document except that Nadruni
gives the same associations of the squares with the planets as that later given
by Agrippa, 3 - Saturn, 4 - Jupiter, 5 - Mars, 6 - Sun, 7 - Venus, 8 - Mercury,
9 - Moon.

Another Arab manuscript, untitled, author unknown, circa 1466, gives a
reverse mapping of squares to planets, i.e. 3 - Moon, 4 - Mercury, etc.

The first known European set of Magic Squares associated with the planets
is 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 wrote
it, or exactly when, or for what purpose.

In 1498 Pacioli wrote DeViribus, which gives squares and planets in the
same relationship as Nadruni and later Agrippa. This is important as a
possible source for Albrecht Durer's square shown in Melancholia I.

In 1531 Agrippa publishes De Occulta Philosophia, the second book of which
gives 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 and
planets in the reverse as that published by Nadruni, Pacioli and Agrippa.

Several interesting points to made about this information even in
advance of receiving the works cited are;

1. The association of planets to squares is first PUBLISHED in the Arab world
in sometime in the mid 14th century.

2. The mapping of planets to squares is given in two orders, each the reverse
of the other, and each mapping is used in both the European and Arabic
publications. Attention DAN WASBURN - you asked about the order of planets vs.
squares in your post on magic squares and Ziggurats, indicating a possible
confusion about the Merkabah mystics as to whether they are asending or
descending. 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 has
implications for us in our interest in the numerology of the square of 6.

Another interesting assertion in the Singmaster material is a reference to
an author who claims that the 3rd Century neo-platonists had knowledge of the
magic square of 3. This is the sole academic reference to a western knowledge
of magic squares before the 14th century. A very interesting development that
we will be sure to investigate for obvious reasons. If supportable, this would
put a new light on the order of planets given in the Sepher Yitzarah, written
in about the 3rd century in Palestine, and push back knowledge of the squares
in the West 11 centuries.

As a "bonus" for me especially, Singmaster has references to many
mathematicians and puzzle-authors who have contributed to magic square
literature over the years. Quite unexpectedly, I discover that Dr. Singmaster
has excerpts from the notebooks of CHARLES BABBAGE, the famous English
mathematician inventor and "grandfather of the computer". It seems that
Babbage was interested in Latin Squares, Magic Squares, and Magic Cubes.

Along this line, (though not concerned with my main interest in the squares
as magical devices) we can add some famous names to the list of people who
squandered 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 "Euler
Conjecture".

Pierre Fermat - super-famous author of the finally-proved "Fermat's Last
Theorm" - Fermat conceived and produced the world's first Magic Cube, also
discussed magic triangles with Frenicle.

M. Mersenne - friend of Fermat's and fellow number theorist.

Bernard Frenicle - friend and correspondent of Fermat and Mersenne - first
to 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 concept
of the modern computer and who attempted to build one - called a "difference
engine"

Additional news to pass along - the Singmaster material provides the
starting point for a LARGE research project, including as it does virtually all
the previous research along the lines we wish to continue. What is clear from
a review of everything I have collected is that while some information is
known, a lifetime could be devoted to working on the history of magic squares
and the associated religious/magical ideas. The contributions of the Indians
in particular has not been very well understood, and the authors disagree among
themselves as to the sources of the Arab magic squares, Indian or Chinese.

Dr. Singmaster has not touched the esoteric side of the question. None of
the mathematical historians (who nevertheless give by far the most detailed
information on sources) treat the larger subject of the relationship between
mathematics and mystical experiences and philosophy. I think the territory is
just waiting for someone to make a PhD dissertation on the subject. Any
takers?

Finally, I have run out of time to write today, but there is more, such as
the several sources that recommend that the magic square of 3 be used as a
remedy for a hard labor during childbirth, such as the 5 15th century cast iron
plates found in central China in 1958 with the magic square of 6 inscribed on
them, such as a reference to the Chinese god of the POLE STAR (attention
Barry!) and a description of the path of this god through the "houses" of the
Lo Shu square. Information on Chinese beliefs about the Ming-Tang Palace and
the Emperor.

In short, enough meat to chew on for quite a while. Anybody wanna dive in
and 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 if
there 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 at

http://www.hollyfeld.org/heaven/Avidyana/Dozen/cl.mgksqrs.fn

Quote from John Opsopaus (site http://www.cs.utk.edu/~mclennan/BA/PT/M19.html )

4 9 2 
3 5 7
8 1 6

Chinese knowledge of the Saturn Square is shown by the ground plan of the
Ming-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 temple
of this plan being essential for Imperial worship, and, secondly, that in the
7th 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 his
ancestors. Moreover, if this tradition be correct, the Magic Square form of
temple 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 back
to Mesopotamia.)

The Ming-T'ang had twelve stations for the monthly "Proclamation of Space and
Time." 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 = autumnal
equinox, 1 = winter solstice). The central square corresponds to the additional
days 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 critical
period 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 solar
year 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 Square
is 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 to
the 666 of the magic square of the Sun click on the link below:

http://www.jesus8880.com/gematria/666.htm

Islam

Kieth Critchlow has a chapter on Magic Squares in Islam in his book
Islamic 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, the
Geez translation of a version of the "Sun of
Knowledge" (Shams al-Marif), a book
attributed to Al-Buni, an Egyptian author
of the thirteenth century. Each diagram
containing figures or letters is
accompanied by its method of use. Book
of Buni, eighteenth century to nineteenth
century, parchment, 27.5 x 24 cm. Private
collection. Photo courtesy of Guy Vivien

A quotation from John Opsopaus:

Magic Square of 3

4 9 2 
3 5 7
8 1 6

According to the Theory of Balance attributed to 8th century Muslim alchemist
Jabir 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 the
foundation 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. The
Gnomon, 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:

http://www.cs.utk.edu/~mclennan/BA/PT/M19.html


Judaism

I've been reading Aryeh Kaplan's Meditation and
the Kabbalah
, He has a chapter on the Kabbalistic writings
of Rabbi Joseph Tzayach (1505-1573) the Rabbi of Damascus.

Kaplan says that magic squares were well known in ancient India
and China and that they were introduced into the west in the
1400s by Moschopulus of Constatinople. Kaplan believes
they were also known to the ancient Kabbalists. Tzayach
probably received a hidden tradition concerning these figures.

His system attributes the traditional magic squares to the
seven planets but goes on to attribute higher order magic squares
to the Sephira on the Tree of Life, a Jewish mystical diagram: The square of ten
to Kether, of 11 to Chokmah, etc. skipping the square of
15, so that Malkuth is the 20x20 matrix.

Tzayach apparently had a system that involved meditating
on the color, number, and letter forms in one room after another
in 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 throne
of 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 Merkabah
Ascension.

Magic square lore may have passed back and forth between
Jewish and Muslim mystics.

General

David Singmaster gives a number of dates relating to the history of Magic Squares in his
"Chronology of Recreational Mathematics."

http://www.geocities.com/SiliconValley/9174/recchron.html

Here are some book references (I have not looked at) from the bilbiography at

http://www.pse.che.tohoku.ac.jp/~msuzuki/refference.html

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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