Arithmetical Squares

An arithmetical magical square consists of numbers so disposed in
parallel and equal lines, that the sum of each, taken any way of the
square, amounts to the same.

Any five of these sums taken in a right line make 65. You will observe
that five numbers in the diagonals A to D, and B to C, of the magical
square, answer to the ranks E to F, and G to H, in the natural square,
and that 13 is the centre number of both squares.

A Natural Square. A Magical Square.
A G B A B
+--+--+--+--+--+ +--+--+--+--+--+
1 2 3 4 5 1124 720 3
+--+--+--+--+--+ +--+--+--+--+--+
6 7 8 910 41225 816
+--+--+--+--+--+ +--+--+--+--+--+
E 1112131415 F 17 51321 9
+--+--+--+--+--+ +--+--+--+--+--+
1617181920 1018 11422
+--+--+--+--+--+ +--+--+--+--+--+
2122232425 23 619 215
+--+--+--+--+--+ +--+--+--+--+--+
C H D C D

To form a magical square, first transpose the two ranks in the natural
square to the diagonals of the magical square; then place the number 1
under the central number 13, and the number 2 in the next diagonal
downward. The number 3 should be placed in the same diagonal line; but
as there is no room in the square, you are to place it in that part it
would occupy if another square were placed under this. For the same
reason, the number 4, by following the diagonal direction, falling out
of the square, it is to be put into the part it would hold in another
square, placed by the side of this. You then proceed to numbers 5 and
6, still descending; but as the place 6 should hold is already filled,
you then go back to the diagonal, and consequently place the 6 in the
second place under the 5, so that there may remain an empty space
between the two numbers. The same rule is to observed, whenever you
find a space already filled.

You proceed in this manner to fill all the empty cases in the angle
where the 15 is placed: and as there is no space for the 16 in the
same diagonal, descending, you must place it in the part it would hold
in another square, and continue the same plan till all the spaces are
filled. This method will serve equally for all sorts of arithmetical
progressions composed of odd numbers; even numbers being too
complicated to afford any amusement.

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