I.

INTRODUCTORY.

Among the philosophies of modern times there is no other which emphasises so much the importance of form and formal thought as the monism of _The Monist_. An expression thereof is found in the following passages:

“The order that prevails among the facts of reality is due to the laws of form. Upon the order of the world depends its cognisability.

“... The laws of form are no less eternal than are matter and energy and ‘Verily I say unto you, till heaven and earth pass, one jot or one tittle shall in no wise pass from the law!’

“The laws of form and their origin have been a puzzle to all philosophers. ‘Ay, there’s the rub!’ The difficulties of Hume’s problem of causation, of Kant’s _a priori_, of Plato’s ideas, of Mill’s method of deduction, etc., etc., all arise from a one-sided view of form and the laws of form and formal thought.”

Considering the great results which engineering and other applied sciences accomplish through the assistance of mathematics, we must confess that the forms of thought are wonderful indeed, and it is not at all astonishing that the primitive thinkers of mankind when the importance of the laws of formal thought in some way or another first dawned on their minds, attributed magic powers to numbers and geometrical figures.

We shall devote the following pages to a brief review of magic squares, the consideration of which has made many a man believe in mysticism. And yet there is no mysticism about them unless we either consider everything mystical, even that twice two is four, or join the sceptic in his exclamation that we can truly not know whether twice two might not be five in other spheres of the universe.

[Illustration: ALBERT DÜRER’S ENGRAVING

MELANCHOLY OR THE GENIUS OF THE INDUSTRIAL SCIENCE OF MECHANICS]

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.