"MAGIC square, in arithmetic, a square figure made up of numbers in arithme tical proportion, so disposed in parallel and equal ranks, that the sums of each row, taken either perpendicularly, hori zontally, or diagonally, are equal : thus, Magic squares seem to have been so called, from their being used in the con struction of talismans.
Take another instance : where every row and diagonal in the magic square makes just the sum 65, be ing the same as the two diagonals of the ronerai square.
It is probable that these magic squares were so called, both because of this pro perty in them, viz. that the ranks in every direction make the same sum, appeared, extremely surprising, especially in the more ignorant ages, when mathematics passed for magic, and because also of the superstitious operations they were em ployed in, as the construction of talismans, &c.; tor, according to the childish philo sophy of those days, which ascribed vir. toes to numbers, what might not be ex. petted frorn numbers so seemingly wonder ful 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 inclosed by a regular polygon, inscribed in 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 angles of the planet, and the signs of the zodiac, written upon the void spaces between the polygon and the circumference ache circumscribed circle. Such a talisman or metal they vainly ima gined would, upon occasion, befriend the person who carried it about him. To Sa turn they attributed the square of 9 places or cells, the side being 3, and the sum of the numbers 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 269 : 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. To form a magic square of an odd num ber of terms in the arithmetic progression 1, 2, 3, 4, &c. place the least term 1 in the cell immediately under the middle or central one ; and the rest of the terms, in their natural order, in a descending diagonal direction, till they run off either at the bottom, or on the side: when the cumber runs off at the bottom, carry it to the uppermost cell, that is not occupied, of the same column that it would have fall en in below, and then proceed descending diagonalwise again as tar as you can, or till the numbers either run offal. bottom or side, or are interrupted by coming at a cell already filled: now when any number runs off at the right-hand side, then bring it to the furthest cell on the left-hand of the same row or line it would have fallen in towards the right-hand : and when the progress diagonalwise is interrupted by meeting with a cell already occupied by some other number, then descend di agonally to the left from this cell till an empty one is met with, where enter it; and thence proceed as before. Thus, First place the 1 next below the cen tre cell, and thence descend to the right till the 4 runs off at the bottom, which therefore carry to the top corner on the same column as it would have fallen in ; but as that runs off at the side, bring it to the beginning of the second line, and thence descend to the right till they ar rive at the cell occupied by 1; carry the 8 therefore to the next diagonal cell to Ale left, and so proceed till 10 runs off at the bottom, which carry therefore to the top of its column, and so proceed till 13 runs off at the side, which therefore bring to the beginning of the same line; and thence proceed till 15 arrives at the cell occupied by 8; from this therefore descend diagonally to the left; but as 16 runs off at the bottom, carry it to the top of its proper column, and thence descend till 21 runs off at the side, which is there fore brought to the beginning of its pro per line ; but as 22 arrives at the cell oc copied by 15, descend diagonally to the left, which brings it into the first column, but off at the bottom, and therefore it is carried to the top of that column ; thence descending till 29 runs off both at bottom and side, which therefore early to the highest unoccupied cell in the last co lumn ; and here, as 30 runs off at the side, bring it to the beginning of its pro per column, and thence descend till 35 runs off at the bottom, which therefore carry to the beginning or top of its own column ; and here, as 36 meets with the cell occupied by 29, it is brought from thence diagonally to the left ; thence de scending, 38 runs off at the side, and therefore it is brought to the beginning of its proper line ; thence descending, 41 runs off at the bottom, which therefore is carried to the beginning or top of its co lumn; from whence descending, 43 ar rives at the cell occupied by 36, and therefore it is brought down from thence to the left ; thence descending, 46 runs off at the side, which therefore is brought to the beginning of its line ; but here, as 47 runs off at the bottom, it is carried to the beginning or top of its column, from whence descending with 48 and 49, the square is completed, the sum of every row and column and diagonal making just 175. Dr. Franklin carried this cu
rious speculation further than any of his predecessors in the same way. He con structed both a magic square of squares, and a magic circle of circles, the descrip tion of which is as follows. The magic square of squares is formed by dividing the great square into 256 little squares, in which all the numbers from 1 to 256, or the square of 16, are placed, in 16 co lumns, which may be taken either hori zontally or vertically. Their chief pro perties are as follow. 1. The sum of the 16 numbers in each column or row, ver tical or horizontal, is 2056. 2. Every half column, vertical and horizontal, makes 1028, or just one half of the same sum 2056. 3. Half a diagonal ascending, added to half a diagonal descending, makes also the same sum 2056; taking these half diagonals from the ends of any side of the square to the middle of it ; and so reckoning them either upward or downward, or sideways from right to left, or from left to right. 4. The same with all the parallels to the half diagonals, as many as' can be drawn in the great square : for any two of them being di rected upward and downward, from the place where they begin, to that where they end, their sums still make the same 2056. Also the same holds true down. ward and upward ; as well as if taken sideways to the middle, and back to the same side again. Only one set of these half diagonals and their parallels, is drawn in the same square upward and down ward ; but another set may be drawn from any of the other three sides. 5. The four corner numbers in the great square, added to the four central numbers in it, make 1028, the half sum of any vertical or horizontal column, which contains 16 numbers ; and also equal to half a dia gonal or its parallel. 6. If a square hole, equal in breadth to four of the little squares or cells, be cut in a paper, through which any of the 16 little cells in the great square may be seen, and the paper be laid upon the great square ; the sum of all the 16 numbers, seen throukh the hole, is always equal to 2056, the sum of the 16 numbers in any horizontal or ver tical column.