The author of the short article on “Magic Squares” in the English
Cyclopædia (Vol. III, p. 415), presumably Prof. DeMorgan, says:
“Though the question of magic squares be in itself of no use,
yet it belongs to a class of problems which call into action
a beneficial species of investigation. Without laying down
any rules for their construction, we shall content ourselves
with destroying their magic quality, and showing that the
non-existence of such squares would be much more surprising
than their existence.”
This is the point. There obtains a symphonic harmony in mathematics which
is the more startling the more obvious and self-evident it appears to him
who understands the laws that produce this symphonic harmony.
* * * * *
On the wood-cut named “Melancholia”[68] of the famous Nuremberg painter,
Albrecht Dürer, is found among a number of other emblems, which the
reader will notice in our reproduction of the cut, the subjoined square.
This arrangement of the sixteen natural numbers from 1 to 16 possesses
the remarkable property that the same sum 34 will always be obtained
whether we add together the four figures of any of the horizontal rows
or the four of any vertical row or the four which lie in either of the
two diagonals. Such an arrangement of numbers is termed a magic square,
and the square which we have reproduced above is _the first magic square
which is met with in the Christian Occident_.
[Illustration: Fig. 1.
+--+--+--+--+
| 1|14|15| 4|
+--+--+--+--+
|12| 7| 6| 9|
+--+--+--+--+
| 8|11|10| 5|
+--+--+--+--+
|13| 2| 3|16|
+--+--+--+--+
]
Like chess and many of the problems founded on the figure of the
chess-board, the problem of constructing a magic square also probably
traces its origin to Indian soil. From there the problem found its way
among the Arabs, and by them it was brought to the Roman Orient. Finally,
since Albrecht Dürer’s time, the scholars of Western Europe also have
occupied themselves with methods for the construction of squares of this
character.
The oldest and the simplest magic square consists of the quadratic
arrangement of the nine numbers from 1 to 9 in such a manner that the sum
of each horizontal, vertical, or diagonal row, always remains the same,
namely 15. This square is the adjoined.
[Illustration: Fig. 2.
+-+-+-+
|2|7|6|
+-+-+-+
|9|5|1|
+-+-+-+
|4|3|8|
+-+-+-+
]
Here, we will find, 15 always comes out whether we add 2 and 7 and 6, or
9 and 5 and 1, or 4 and 3 and 8, or 2 and 9 and 4, or 7 and 5 and 3, or 6
and 1 and 8, or 2 and 5 and 8, or 6 and 5 and 4.
The question naturally presents itself, whether this condition of the
constant equality of the added sum also remains fulfilled when the
numbers are assigned different places. It may be easily shown however
that 5 necessarily must occupy the middle place, and that the even
numbers must stand in the corners. This being so, there are but 7
additional arrangements possible, which differ from the arrangement
above given and from one another only in the respect that the rows at
the top, at the left, at the bottom, and at the right, exchange places
with one another and that in addition a mirror be imagined present with
each arrangement. So too from Dürer’s square of 4 times 4 places, by
transpositions, a whole set of new correct squares may be formed. A magic
square of the 4 times 4 numbers from 1 to 16 is formed in the simplest
manner as follows. We inscribe the numbers from 1 to 16 in their natural
order in the squares, thus:
[Illustration: Fig. 3.
+--+--+--+--+
| 1| 2| 3| 4|
+--+--+--+--+
| 5| 6| 7| 8|
+--+--+--+--+
| 9|10|11|12|
+--+--+--+--+
|13|14|15|16|
+--+--+--+--+
]
We then leave the numbers in the four corner-squares, viz. 1, 4, 13, 16,
as well also as the numbers in the four middle-squares, viz. 6, 7, 10,
11, in their original places; and in the place of the remaining eight
numbers, we write the complements of the same with respect to 17: thus 15
instead of 2, 14 instead of 3, 12 instead of 5, 9 instead of 8, 8 instead
of 9, 5 instead of 12, 3 instead of 14, and 2 instead of 15. We obtain
thus the magic square
[Illustration: Fig. 4.
=34 =34
\ /
+--+--+--+--+
| 1|15|14| 4|=34
+--+--+--+--+
|12| 6| 7| 9|=34
+--+--+--+--+
| 8|10|11| 5|=34
+--+--+--+--+
|13| 3| 2|16|=34
+--+--+--+--+
34 34 34 34
]
from which the same sum 34 always results. It is an interesting property
of this square that any four numbers which form a rectangle or square
about the centre also always give the same sum 34; for example, 1, 4,
13, 16, or 6, 7, 10, 11, or 15, 14, 3, 2, or 12, 9, 5, 8, or 15, 8, 2,
9, or 14, 12, 3, 5. We may easily convince ourselves that this square is
obtainable from the square of Dürer by interchanging with one another the
two middle vertical rows.