MAGIC SQUARES. A magic square is a square divided into equal smaller squares, each containing a term of a series of integers, the sums of the numbers in any horizontal, vertical and diagonal line being the same.
be due to peculiarities in the conntosition of the metal in some of the mirrors, the pressure used in polishing the thicker portions contain ing the raised ornaments resulting in a differ ence on the reflecting surface too minute to The construction of such squares is an amusement of great antiquity. They were known in India and China before the Christian Era, and a knowledge of them was introduced into Europe by Moschopulus who flourished in Constantinople early in the 15th century. Talismanic virtues and occult properties were ascribed to them by the ancients. They were engraved on metal and stone and worn as amulets, as in India at the present day. A magic square of the fourth order is engraved on the gate of the fort at Gwalior in that country. Mediaeval astrologers and physicians were filled with superstitions in regard to magic squares. They associated the squares of the orders 3, 4, 5, 6, 7, 8 and 9 with the astrological planets Saturn, Jupiter, Mars, the Sun, Venus, Mercury and the Moon. A square containing one cell symbolized the unity of the Deity; one of the second order, not being possible, signified the imperfection of the elements air, earth, fire and water. Albert Diirer's well-known paint ing, 'Melancholy,' contains a magic square of the fourth order, doubtless because of its sup posed mystical significance. They have been made the subject of elaborate research by vari ous investigators but the world is indebted chiefly to the French mathematicians for the de velopment of the theory of magic squares.
In this article general rules for the construc tion of magic squares of any order will be given, illustrated by particular examples. The squares produced by these methods by no means exhaust all possible arrangements, but the rules furnish squares in great number and variety.
Magic squares are divided into two general classes according as the numbers of cells on a side is odd or even. Even squares are sub divided into doubly even, i.e., when the root is divisible by 4, and singly even, when the is divisible by 2 but not by 4. A horizontal line of cells is called a row, and a vertical line a column. Two cells in a row equidistant from
the ends are termed a horizontal pair, and two cells in a column equidistant from the ends are termed a vertical pair. In a series of natural numbers any two equidistant from the ends are said to be complementary.
Magic Squares of an Odd Order.—La Hire's method for constructing odd magic squares requires the formation of two auxiliary squares A and B. For a square of the fifth order diagram A is formed with the series of natural numbers 1, 2, 3, 4 and 5 as follows: First, put 3 (the mean of the numbers) in the top left-hand corner cell, and the numbers 1, 2, 4 and 5 in the cells of the top row in any order. Next, the number in each cell of the top row is repeated in the cells of a diagonal sloping downward to the right. The cells filled by the same number form a broken diagonal.
Form a new square by making the left-hand column of A (beginning with its bottom num ber) the first row in the new square, and so on. Next, instead of the numbers 1, 2, 3, 4 and 5, substitute respectively the numbers 0, 5, 10, 15 and 20, thus producing square B. In each cell of square C place the sum of the numbers in similarly situated cells of squares A and B. The result is a magic square of the fifth order. Any magic square of an odd order can be constructed in a similar manner.
La Loubere's order to con struct a magic square of an odd order by this method, place 1 in the middle cell of the upper row, and using the series of natural numbers (any arithmetical series will answer) proceed always diagonally upward to the right, except when the edge of the square or a cell already filled is reached. When a number would fall outside the square, carry it to the extreme cell in that row or column in which the cell outside would fall. When a cell is reached that is al ready filled or when the righthand upper corner cell is reached, place the number in the cell just below. The magic square D is formed by this rule. It may be remarked here that from any magic square, whether odd or even, a number of other magic squares can be formed by the mere interchange of the row and column which intersect in a diagonal with the row and column which intersect in some other cell in the same diagonal. In this way from each magic square of the fifth order 48 other magic squares can be formed.