Preparation of the Natural Square.
6. Mark all the corner cells on the left of the a upper quarter - The corner cells on the right of do.
The middle cells of the upper quarter The cells on the left quarter opposite the centre c In the even belts alone, Mark a cell in each, in the upper2 on the left quarter, next the corner cells, S on the right r Mark the cells in the left quarter immediately under a, And the cells on the right quarter immediate ly above those which are opposite the centre, Rules for transferring the Minors from the Natural to the Magic Square.
7. General rules. These numbers when carried from any belt in the natural square, must be placed in a similar belt in the magic square.
8. They must never be placed opposite, either di agonally, or facing each other.
9. Particular rules. For the odd belts.
In the left corner cells of the upper quarter, place c In the right corner cells of do. place - In any cells out of the corners in the lower guar- Z a ter, place - S In any cells out of do. in the left quarter 10. For the even belts. In the left corner cells of the upper quarter, place a In the right corner cells of do. - - - m In any cells on the left quarter between the dial , gonals, place - - - In any cells on the right quarter do. and not ? facing b, place - - - . s and n In any cells in the lower quarter, between the di agonals place, s, c, and r. The minors which are let tered, being thus inserted in the magic square, the remainder must be transferred by the following gene ral rules: 11. In the upper quarter, the cells remaining un marked, are either in number 4 or its multiples, as 8, 12, 16, in each belt. Of the numbers in these then, place the extremes in the upper, and the means in the lower quarter of the magic square, in their ap propriate belts. Thus, suppose there are four unlet tered in any belt, the numbers in which are 2, 3, 5, and 6, place No. 2 and 6 in the upper, and 3 and 5 in the lower quarter and similar belt of the magic square. Or the extremes may be carried to the lower, and the means to the upper quarter, no matter which.
12. In the right and left quarters, the cells unmark ed in each belt arc always in pairs, as 2, 4, 6, 8, &c. Carry the numbers in one half, say the upper half of those on the left quarter to the left quarter of the ma gic square; and the lower half to the right quarter. Do the same with those which are unlettered on the right quarter, but in the reverse order, so that the amount of the numbers so transferred shall be the same in each of these quarters.
13. The minors being all transposed, will each of them, if properly placed, have a corresponding empty cell in its own belt, either diagonally opposite, if it is in a corner, or facing it, if in any cell between the di agonals.
14. These empty cells are now to be filled with the majors, which is done without any reference to the natural square. Each minor must have its proper major, which is that number which the minor wants of the amount of the first and last number of the pro gression. Thus, if the series runs from 1 to 25, the amount of these is 26; and if the minor in question be 7, its major of course must be 19, &c. Instrting the majors, therefore, diagonally opposite those numbers which arc in the corner cells, and facing those which are situated between the diagonals, each in the same belt with its minor, the magic square will be com pleted.
Magic Squares of Even Numbers.
These are generally subdivided into two kinds,- 1st, Oddly even squares, which arc those whose roots when halved produce odd numbers,as the squares of 6, 10, 14, 18, &c.
2d, Evenly even squares, are the squares of 4 and its muliples, as 8, 12, 16, &c.
The first kind, though possessing fewer properties, is more difficult of construction than the second. We have seen no method superior to the following one, which embraces both kinds, and at the same time the additional property of being bordered, so that the ex terior surrounding row or rows may be removed, and the square remaining still continue magic.