Friday, November 20, 2009

Agrippa's Magic Squares - Part 2.

The Formation of the Magic Squares

In the previous part, we learned The Sequence of the Planets.

Now, we will apply that knowledge to form the Magic Squares (sometimes called Kameas) and attribute them to the Planets.

A Magic Square is an array of numbers where the sum of each of the columns, rows, and the two diagonals are all the same.  This sum is called the magic constant.

The type of squares we are dealing with are considered "Normal Magic Squares."  This means that each such square of an order N will contain integer values starting from 1 up to N-squared.

This means our Saturn square, being of the order 3, will contain integers from 1 to 9.

Magic Squares were known to Chinese mathematicians as early as 650 B.C.  There is essentially only one possibility for the normal order-3 square, although it can be reflected or rotated in various ways.  To the Chinese this was known as the Lo Shu square.

SATURN (3x3 ... Constant=15)

492
357
816

The Seal of Saturn (at the right, above) is made by connecting the cells with numbers 1-2-3, then 4-5-6 (the diagonal), then 7-8-9, thereby touching every cell in the Magic Square in the order it was created.  The diagonal is drawn a bit too long in the image.  The circles at the end of the line segments are most likely placed there for aesthetic purposes only, and differ between various depictions of the seal.

Agrippa's Magic Squares follow a certain pattern.  All of the squares of an "odd order" are made by placing the "1" on the middle column, one cell down from the center, then walking "southeast" and placing each subsequent number, allowing to wrap from the bottom to the top, and from the right to the left, whenever needed.  If an occupied cell is found, go "south" two squares instead of going southeast, and continue until the square is filled.  The Saturn square shown above agrees with this construction method.  (The diagonal formed by numbers 4, 5, and 6 are the easiest to see, since they neither wrap from side to side nor hit an occupied cell.)

Squares of an "double even order" (in other words, whose order is a multiple of 4) are made by counting from right-to-left, top-to-bottom starting from 1, but only filling in the diagonals (for order-4) of each 4x4 section. Then, repeating the process for the remaining squares, but counting down instead of up.  In other words, taking the black numbers from the following two squares, and combining them:

Order-4 Diagonals:

4321
8765
1211109
16151413

Order-4 Remaining:

13141516
9101112
5678
1234

The seal of Jupiter (above right) shows the method of constructing the 4x4 Square, by indicating the diagonals, and tracing a circle over the remaining cells, thereby touching every cell in the Magic Square.

Now, take the black numbers from both of the above squares, and combine them to get the following Magic Square:

JUPITER (4x4 ... Constant=34)

414151
97612
511108
162313

The following square is generated using the odd-order southeast walking method:

MARS (5x5 ... Constant=65)

11247203
41225816
17513219
101811422
23619215

I am currently uncertain what method was used to construct the following square:

THE SUN (6x6 ... Constant=111)

632334351
7112728830
191416152324
182022211713
25291092612
365334231

The following square is generated using the odd-order southeast walking method:

VENUS (7x7 ... Constant=175)

2247164110354
5234817421129
3062449183612
1331725431937
3814321264420
213983322745
461540934328

The following square is generated using the double-even-order method (I have indicated the lattice-work in bold):

MERCURY (8x8 ... Constant=260)

858595462631
4915145253111056
4123224445191848
3234352928383925
4026273736303133
1747462021434224
955541213515016
642361606757

The following square is generated using the odd-order southeast walking method:

THE MOON (9x9 ... Constant=369)

37782970216213545
63879307122631446
47739803172235515
16488408132642456
57174994173336525
26581850142743466
67275910512437535
36681960115234476
77286920611253445

There you have it!  All seven of our Planetary Magic Squares.

Next up...

The Names and Meanings of the Spirits and Intelligences

3 comments:

lanoo said...

hey i like the way you explain things. personally im trying to figure out how the sigils or intelligence symbols are made. especially related to mars. and what are thier effects. can u email me at thetwelveknights@yahoo.com thanks. cause im in south america and everytime i try to work on my blog i get stuck in the spanish language which i dont speak. Lanoo

lanoo said...

the way you illustrated out the order of the planets and the connection to thier number on the squares is cool.

Joseph Guerra said...

I would love to read more as I too am currently researching Agrippa's sigil making techniques. If you could message me back about the rest of this series I'd greatly appreciate it.