§50.7. Magic Squares
Ferdinand Maack, the original and indefatigable campaigner for a
“scientific xenology,” published a series of articles,
“Magisch-Quadratische Studien,” in the journal he founded, Wissenschaftliche Xenologie
(No. 2, July 1899). The following quote comes from the introduction to
the series: “[The magic square] contains an ancient mathematical
problem, which is continually attacked from new angles over the passage
of the centuries. Even the greatest brains of all nations have occupied
themselves with this problem, as the bibliography shows” (Maack gives
an extensive bibliography, ibid. pp. 33-38). “Academic journals opened
their columns to the magic square; university professors gave inaugural
lectures about it; it represents an established category in
mathematical dictionaries and encyclopedias. All this would surely not
be the case if the 'tetragram' were only a 'learned trumpery of
numbers,' an 'arithmetical diversion and amusement,' only a number
game, a graphic puzzle. Most authors do claim this. But one can read
between their lines that the 'divine square' was more to them than just
a simple arithmetical example, that along the way they have the feeling
that there might yet be something peculiar behind it.”
What is a magic square?
The discovery of a secret number-figure, the “Lo-shu,” containing the
numbers 1-9, is attributed to the legendary Fou-hi. Fou-hi supposedly
saw this figure in a wondrous vision on the shell of a tortoise
(Thimus, Harmonikale Symbolik
I, 101). The illustration that the Chinese writers drew from this
figure, which have been found on a four to five thousand-year-old
Chinese tablet (the Lo-shu), and which is probably the oldest example
of a magic square, is the number-square represented in knots:
8 3 4
1 5 9
6 7 2
If one writes out the progression of numbers 1-9 in the form shown in Fig. 457:

Figure 457

Figure 458
and
adds on the numbers standing outside the square thus, in these empty
fields, as is indicated by the numbers in parentheses, then Fig. 458
appears, i.e. precisely this ancient Chinese tetragram. This is one of
the simplest methods of constructing the magic square. The arithmetical
characteristics of the magic square, of which there are an infinite
number with various indexes, is in the following: the sums of all
vertical, horizontal, and diagonal rows are the same, likewise the sum
of each pair of numbers opposite one another; and the parallel main and
secondary diagonals form an equidistant series. Larger squares show
further mathematical curiosities. But these are only the arithmetical
attributes. Each magic square is also in equilibrium. If, for example,
we set up the Natural Square (N.S.) that belongs to every magic square,
i.e. Fig. 457, with weights according to the measures of the numbers,
for example 1 g on the topmost, 4 g and 2 g
on the next fields, etc., and then hang the N.S. by a thread in the
middle, then the system tips over; if we do the same with the magic
square (Fig. 458), it stays in balance. Thus, the problem of magic
squares is a problem of equilibrium. Furthermore, Maack discovered
(ibid.) that all magic squares can be developed from the natural
squares through “torsion of the magical system,” i.e. through rotations
of certain inner configurations. If one writes the Natural Square and
Magic Square of Figures 457 and 458 as follows (Figure 459 and 460):

Figure 459
Figure 460
then
the shaded fields are symmetrical around the middle field. The sum of
each pair of opposite shaded fields is twice the value of the middle
field in both squares (Figure 459 and 460). F. Maack calls these shaded
and non-shaded fields, which thus appear both in the magic square and
the natural square, “magic systems.” If one now imagines the middle
field as an axis, the magic square (Fig. 460) appears from the natural
square (Fig. 459) through torsion of the magic system. Namely, Fig. 460
appears from Fig. 459 in such a way that first the shaded system of
Fig. 459 must be rotated 225° to the right, then the non-shaded system
of Fig. 459 must be rotated 45° to the right.

Figure 461
If we now question the “harmonics” of the magic squares, an equal
intervallic quality in this simple example (here the octave would be
set at 15) catches our attention. As a comparative theorem, the
“complete P” would first come to mind. But there is an even closer
parallel. If we construct the corresponding magic square, analogous to
the above specifications, from the natural seven-square (Fig. 461), in
which we inscribe the diatonic seven-step scale from 1-7, 8-14, etc.,
in seven different octaves (Fig. 462):

Figure 462
then
we are immediately reminded of our diagram of fifths (Fig. 463), in
which we find the tone-values of the magic square (Fig. 462) drawn (in
a thick line) starting from 1/3 f. Admittedly, in this fifth diagram, the ratio progression is limited at h, due to the 6 fifth intervals f c g d a e h.

Figure 463
Furthermore, the lines of the equal octave tone-values are different.
But yet the ordering of the magic square appears undoubtedly to agree
more closely with that of the fifth diagram, and therefore also with
that of the “open P.” Mathematically oriented readers who are
interested in this problem will certainly find an exciting field of
operation on the basis of the above. In itself, the domain of the
square (see Bibliography) is already interesting enough, and although
it is still considered by many today to be a useless game, since no
machines, bombs, or grenades can be manufactured de facto
with its problems, I still believe that any serious mathematician would
dismiss such a superficial judgment. Otherwise he would have to think
of all of mathematics as a “useless” or “profitable” game, against
which admittedly nothing could be said.
From the many historical writings about the magic square, one more is
introduced here that is important for us. Around the year 1000, in
Arabic cultural circles, there existed a strange Neoplatonist sect
known as the “Brethren of Purity.” It was a secret society that was
founded in Basra around the year 960, and they opposed strict
Mohammedan faith with free philosophical thought. Within this order, an
encyclopedia of 51 treatises was created, which arranged and summarized
the knowledge of the times according to its elements. “The center of
their thought is the study of the soul, which for them tends towards a
doctrine of its development from the lower forms to human
individuality, and its regeneration in the perfection of its origin.
The worldview at the core of this speculation is the emanation doctrine
of the Plotinians, with their universal Intellect and their
all-pervading Universal Soul, whose formative activity is evident in
nature and in the creation of humans, to which Pythagorean
number-symbolism is added to make the emanation system more
understandable through number ratios. Also various symbolisms, such as
that of letters, tone-harmonies [!], etc., are used to demonstrate that
all in the material and spiritual world is an image of the stepwise
emanation from the world of pure spirituality” (J. Goldhizer, Allgemeine Geschichte der Philosophie, Kultur der Gegenwart, Vol. I, 5, 1909, p. 53). Unfortunately, I have not yet been able to obtain a copy of the 8-volume Philosophie der Araber nach den Schriften der Lauteren Brüder (1858-1886) by F. Dieterici. I have only been able to look at the Naturanschauung und Naturphilosophie der Araber im 10. Jahrhundert nach den Schriften der Lauteren Brüder (Berlin 1861) by the same author, but it was not very fruitful with regard to our harmonic interests. Moritz Cantor, in his Vorlesungen über Geschichte der Mathematik (4th
ed., 1922, p. 516), states that “magic squares play a mysterious role
in the Arabic philosophical sect of the so-called Brethren of Purity,
squares with 9, 16, 25, 36, 64, and 81 fields were especially familiar
to them, therefore there must surely have been a method available with
which to construct them.”