square figure, formed of a series of numbers in mathematical proportion, so disposed in parallel and equal ranks, that the sums of each row, taken either perpendicularly, horizontally, or diagonally, are equal.
Let the several numbers which compose any square number (for instance, 1, 2, 3, 4, 5, &c., to 25, the square number, inclusive) be disposed in their natural order after each other in a square figure of twenty-five cells, each in its cell; if now you change the order of these numbers, and dispose them in the cells in such manner that the five numbers which fill a horizontal rank of cells, being added together, shall make the same sum with the five numbers in any other rank of cells, whether horizontal or vertical, and even the same number with the five in each of the two diagonal ranks; this disposition of numbers is called a magic square, in opposition to the former disposition, which is called a natural square. See the following figures.
| Natural Square | Magic Square | |----------------|--------------| | 1 2 3 4 5 | 16 14 8 2 25 | | 6 7 8 9 10 | 3 22 20 11 9 | | 11 12 13 14 15 | 15 6 4 23 17 | | 16 17 18 19 20 | 24 18 12 10 1 | | 21 22 23 24 25 | 7 5 21 19 13 |
The Hindus, Egyptians, and Chinese are said to have been acquainted with those arrangements at a very early period, and adhered to the belief that such squares were possessed of powers of divination.
Moschopulus, a Greek author of no great antiquity, is the first who appears to have spoken of magic squares; and, by the age in which he lived, there is reason to imagine he did not look on them merely as a mathematician. However, he has left us some rules for their construction. In the treatise of Cornelius Agrippa, so much accused of magic, we find the squares of seven numbers, viz., from three to nine inclusive, disposed magically. Bachet applied himself to the study of magic squares, on the hint which he had taken from the planetary squares of Agrippa.
After him came Frenicle, who took the same subject in hand. He gives, however, no general demonstration of his methods, and frequently seems to have no other guide than chance. It is true, his book was not published by himself, nor did it appear till after his death, viz., in 1693.
In 1703 Poignard, a canon of Brussels, published a treatise on sublime magic squares. Before him there had been no magic squares constructed but for series of natural numbers that formed a square. Instead of taking all the numbers that fill a square, he only takes as many successive numbers as there are units in the side of the square, which, in this case, are six; and these six numbers alone be disposed in such manner in the thirty-six cells, that none of them are repeated twice in the same rank, whether it be horizontal, vertical, or diagonal; whence it follows, that all the ranks, taken all the ways possible, must always make the same sum, which Poignard calls repeated progression. Instead of being confined to take these numbers according to the series and succession of the natural numbers, that is, in arithmetical progression, he takes them likewise in a geometrical progression, and even in an harmonical progression.
The book of Poignard gave occasion to M. de la Hire to turn his thoughts the same way, which he did with such success that he seems to have well nigh completed the theory of magic squares. He first considers uneven squares, all his predecessors on the subject having found the construction of even ones by much the most difficult; for which reason M. de la Hire reserves these for the last.
M. de la Hire proposes a general method for uneven squares, which has some similitude to the theory of compound motions, so useful and fertile in mechanics. As that consists in decomposing motions, and resolving them into others more simple, so does M. de la Hire's method consist in resolving the square that is to be constructed, into two simple and primitive squares.
As to the even squares, M. de la Hire constructs them, like the uneven ones, by two primitive squares; but the construction of primitives is different in general, and may be so a great number of ways; and those general differences admit of a great number of particular variations, which give as many different constructions of the same even square.
Dr Franklin seems to have carried this curious speculation farther than any of his predecessors. He constructed not only a magic square of squares, but likewise a magic circle of circles. The details are not, however, of such importance as to require particular notice in this place. In addition to the writers on this subject, already specified, we may add the names of Stifels, Leibnitz, Bachet, and Ozanam. (For a complete history of this subject, see Montucla, vol. I., p. 346; Hutton's Dictionary; and Ozanam and Montucla's Mathematical Recreations.)