This, in arithmetic, is a square figure made up of numbers in arithmetical proportion, so disposed in parallel and equal ranks, that the sums of each row, taken either perpendicularly, horizontally, or diagonally, are equal: thus—
| Natural Square. | Magic Square. | |||||
| 1 | 2 | 3 | 2 | 7 | 6 | |
| 4 | 5 | 6 | 9 | 5 | 1 | |
| 7 | 8 | 9 | 4 | 3 | 8 | |
Magic squares seem to have been so called, from their being used in the construction of talismans.
Take another instance:—
| 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 | |
where every row and diagonal in the magic square, makes just the sum 65, being the same as the two diagonals of the natural square.
It is probable that these magic squares were so called, both because of this property in them, viz. that the ranks in every direction make the same sum, which appeared extremely surprising, especially in the more ignorant ages, when mathematics passed for magic; and because also of the superstitious operations they were employed in, as, the construction of talismans, &c.; for, according to the childish philosophy of those days, which ascribed virtues to numbers, what might not be expected from numbers so seemingly wonderful? The magic square was held in great veneration among the Egyptians, and the Pythagoreans their disciples, who, to add more efficacy and virtue to this square, dedicated it to the then known seven planets, divers ways, and engraved it upon a plate of the metal that was esteemed in sympathy with the planet. The square, thus dedicated, was enclosed by a regular polygon, inscribed into a circle, which was divided into as many equal parts as there were units in the side of the square; with the names of the angels of the planet, and the signs of the zodiac written upon the void spaces between the polygon and the circumference of the circumscribed circle. Such a talisman, or metal, they vainly imagined would, upon occasion, befriend the person who carried it about him. To Saturn, they attributed the square of 9 places, or cells, the side being 3, and the sum of the number in every row 15: to Jupiter, the square of 16 places, the side being 4, and the amount of each row 34: to Mars, the square of 25 places, the side being 5, and the amount of each row 65: to the Sun, the square with 36 places, the side being 6, and the sum of each row 111: to Venus, the square of 49 places, the side being 7, and the amount of each row 175: to Mercury, the square with 64 places, the side being 8, and the sum of each row 260: and to the Moon, the square of 81 places, the side being 9, and the amount of each row 369. Finally, they attributed to imperfect matter, the square with 4 divisions, having 2 for its side: and to God, the square of only one cell, the side of which is also an unit, which, multiplied by itself, undergoes no change.