ARITHMETIC.
As we do not mean to occupy our pages with the numerous arithmetical tricks which are now to be found in every popular work, we propose to confine our attention to the subject of magic squares and cir cles. The following treatise on this subject prepared for this work by an able correspondent, contains many new original views and constructions which cannot but prove interesting to the curious reader.
1. Magic Squares.
Magic squares are of two kinds; the roots of the one being even numbers, of the other odd numbers.
The rules for their construction are peculiar to each kind, and we shall begin with giving those for odd numbers.
Nag* Squares of odd numbers.
The lowest square of this kind has 3 for its root, but as it is incapable of any variation in its arrange ment, we shall elucidate the rules we give chiefly from the square of 5.
Having divided the square A, B, C, D, into 25 cells, fill them up with the numbers 1 to 25 in their natural order, as in Plate CCCCLXXXIV, Fig. 4.
In this square inscribe another square, E, F, G, II, and divide it likewise into 25 cells; 13 of which will now appear filled with numbers. The remaining 12, which are crossed by the subdivisions of the exterior square, being empty. To fill them up proceed as fol lows: Transfer the numbers in the upper triangle E, A, F, viz. 1, 6, 2, to the three empty cells immediately below the centre (13), and in the same order. Trans fer the numbers 24, 20, 25, in the lower triangle G, D, H, to the empty cells above the centre; the num bers 21, 16, 22, in the triangle C, E, G, on the left to the empty cells on the right of the centre, and the numbers 4, 10, 5, in the triangle F, H, B, on the right to the empty cells on the left of the centre.
The figures in the interior square being now made permanent with ink, and the pencil marks rubbed out, the magic square E, F, G, H, will remain. The amount of each column, horizontal or vertical, and also of each of the diagonals, being all the same or 65.
This is a very simple and easy method of making a magic square of odd numbers, and is applicable to every one of the kind, whatever may be its dimen sions. It is said to be the invention of AI. Bachet,
and some of the rides commonly given to make these squares are evidently derived from it. It would ap pear that it was thought incapable of being varied in the arrangement; as no mention is made of this pro perty in any treatise on the subject we have seen, we shall therefore show how this can be done with little trouble.
The natural arrangement of the num bers in the exterior square A, B, C, D, may be varied in two ways; 1st, In the vertical columns, any one of which may be shifted from its situation except the middle column, which contains the central number 13; 2d, In the hori zontal columns, which may be shift ed in the same way except the middle column, which contains the same num ber 13.
In this way, no less than 576 differ ent arrangements may be given to the square of 5. The square of 7 may be varied 518,400 different ways, and that of 9 upwards of twenty millions of ways.
If a still greater variety is wanted, the following very ingenious method, invented by Poiguard, and improved by De la Hire, will, we have no doubt, give ample satisfaction.
cells, place in the first horizontal column at top, the five first numbers of the natural progression in any order at pleasure, which we shall here suppose to be 1, 3, 5, 2., 4. Then make choice of a number which is prime to the root 5, and which, when diminished by unity, does not measure it. Let this number be 3, and for that reason begin with the third figure of the series, and count from it to fill up the second horizontal column 5, 2, 4, 1, 3. Then begin again with the next third figure, including the 5, that is to say by 4, which will give for the third column 4, 1, 3, 5, 2. By fol lowing the same process, we shall have the series of numbers 3, 5, 2, 4, 1, to fill up the fourth column, and 2, 4, 1, 3, 5, to fill up the fifth and last column. This square will be one of the com ponent parts of the required square, and will be magic, for the sum of each column, whether horizontal, vertical or diagonal, is the same, as the five figures of the progression are contained in each without the same figure being repeat ed.