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.
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.
Excerpt form the book:
ENDLESS AMUSEMENT: A COLLECTION OF NEARLY 400 ENTERTAINING EXPERIMENTS IN VARIOUS BRANCHES OF SCIENCE;
LEA AND BLANCHARD.