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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