Mark Swaney on the History of Magic Squares

16 FEBRUARY 2004


Swaney on the History of Magic Squares

4 9 2

3 5 7

8 1 6

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

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

   My friend Mark
Swaney has been working on the history of Magic Squares and has said yes
to my passing on some of his preliminary results with the following warning:

“You gotta tell them 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.”


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


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. 


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. 


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. 


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. 

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

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

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

The Squares and the Planets

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

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


Mark Swaney, January, 2000

Note from someone else:

See if you can find out
anything about the psycho-spiritual/brainwashing disciplines [Hasan i Sabah’s]
Assassins used, would you. Maybe the same methods were used in working
with the squares, a la the Rabbi of Damascus and his Kabbalistic system
for meditating on the squares.

Mark Swaney writes: 

The really interesting thing
about Sabah for our studies is that in addition to being a military genius,
he was also known to be a scholar. The organization he created, the Hashishim,
or Assassins, was a “Masonic” military organization. By the way, the words
Assassin and Hashishim and Hashish are all thought to be corruptions of
Sabah’s first name, Hassan. The Assassins were in essence Kamikaze’s. They
were trained to strike an enemy and not escape, but stay and fight to the
death. So you can see why these people were so feared.

But the organization was not solely based on military/political adventures.
That’s the mystery. Sabah was known to have amassed a large library
in his fortress. He was known to have had an interest in mathematics, and
to have encouraged the study of mathematics and philosophy by his followers.
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.

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.