Now in another square of 25 cells, Fig. 2. inscribe in the first column, the root 5 and its multiples, beginning with a cypher, viz. 0, 5, 10, 15, 20, and in any order at pleasure, such for example, 5, 0, 15, 10, 20,. Then fill up the square according to the same principles as before, taking care not to assume the same number in the se ries always to begin with. Thus for example,,as in the former square, the third figure in the series was taken, in the present one the fourth must be assumed, and thus we shall have a square of multiples as seen in Fig. 2. This is the second com ponent of the required magic square, and is itself magic, since the sum of each column is always the same.
Now to obtain the magic square required, nothing more is necessary but to inscribe in a third square of 25 cells, Fig. 3. the sum of the numbers found in the corresponding cells of the preceding two. For ex ample 5 and 1, or 6, on the first of the left at the top of the required square, 0 and 3, or 3 in the second, and so on. By these means we shall have the square Fig. 3. which will necessarily be magic.
By this method any of the numbers may be made to fall on any of the cells et pleasure; for example, 1 on the cen tral cell. We have only to fill up the middle band with the series of numbers in such a manner that 1 may be in the centre, as seen in Fig. 4. and then to fill up the rest of the square, according to the above principles, beginning at the highest column when the lowest has been filled up. To form the second square, place a cypher in the centre as seen in Fig. 5. and fill up the remain ing cells in the same manner as before, taking care as in the former, not to as sume the same quantities for beginning the columns.
In the last place, form a third square, by adding together the numbers in the similar cells, and you will have the an nexed square Fig. 6. where 1 will ne cessarily occupy the centre.
Remarks.
1st, We must here observe, that when the number of the root is not prime, that is, if it be 9, 15, 21, &c. it is impossible to avoid a repetition of some of the numbers, at least in one of the diagonals; but in that case it must be arranged in such a manner, that the number repeated in that diag onal, shall be the middle one of the progression; for example 5, if the root of the square be 9; 8, if it be 15: and as the square formed by the mul tiples will be liable to the same acci dent, care must be taken in filling them up, that the opposite diagonal shall contain the mean multiple between o and the greatest ; for example 36, if the root be 9, 105, if it be 15, Stc.
2d, The same thing may be done also in squares which have a prime number for their root. By way of ex ample, we shall form a magic square of the two first of the annexed ones, in the first of which, Fig. 7. the number 3 is repeated in the diagonal descending from right to left, and in the second, Fig. 8. 10 is repeated in the diagonal from left to right. This however, does not prevent the third square, Fig. 9. formed by their addition from being magic.
Magic Squares of Odd Numbers with Borders.
There is an additional property which it has been found can be given to these squares, viz. that whatever may be the dimensions, any one or two or more of the exterior rows may be removed all round the square, and the remaining square still continue magic. They are constructed by the following rules.
• Preliminary Remarks on the Natural Square.
1. In the middle the,re is a cell, which we shall call the centre..
2. One half of all the other numbers in the square are less, and the other half greater than the centre. The former we shall call Minors, the latter Majors.
Tlae cell in the centre is now to have a strong line drawn round it, the cells next to this are likewise to be bounded by a strong line, and so on with each sur rounding row to the extremity of the square. These lines will appear as so many eccentric squares, and the spaces bounded by them containing the numbers we shall call belts.
3: The belt next the centre we shall call the 1st belt, and continue numbering them outwards 2d, 3d, 4th belt, &c.
4. Those belts having the odd numbers we shall call the odd belts, those having the even numbers we shall call the even belts.
5. Supposing now that the square is divided diago nally into four parts, we shall distinguish them by the names of the upper, the lower, the left, and the right quarters; and we may here observe, that the minors occupy all the upper quarters, the left quarter from the top to the cells opposite the centre inclusive, and the right quarter from the top to the cells opposite the centre exclusive.