MAGIC SQUARE. A term applied to square arrays of numbers possessing the property that the sums of the various columns and rows, and of the two diagonals, are equal. In Fig. I this sum is 34. This square (the earliest known hi Europe) was represented in Diirer's copperplate entitled Melancholia. The origin of magic squares is generally ascribed to although with out any strict proof. A magic square is found on one of the gates of the fortifications, in the Sanskrit characters, in the East India n city of Owaloir, but its an tiquity is not such as to justify the assump tion of Hindu origin. Magic squares were, however, certainly known to the Arab astrologers, who claimed for them a peculiar supernatural power and recommend ed them as talismans and amulets. A similar power is attributed to them to-day among the Hindus and to some extent in Europe.
There are various methods for constructing magic squares of an odd number of cells. One of the oldest of which we have any knowledge is described by De la Loubere in his work Du, royatime de Siam (1691), and by him ascribed to the dus of Surat. The rule is as follows: Write 1 in the middle cell of the top row, 2 in the first cell to the right of the dle of the bottom row, then following the diagonal to the right until the right hand margin is reached; then go to the row above and take the left-hand cell, and again low the diagonal upward to the right, and when the upper margin is reached, go to the lowest, row and one cell to the right. If progress is barred by a filled cell, go one cell down from the last number written and proceed as before. (See
Fig. 2.) Another method for an odd number of cells, due to Bachet de 'AI6ziriac (see IlAcHET). is as follows: Take, for example, 25 cells. Arrange and number them as in Fig. 3. Then slide the outside cells to the sides opposite those on which they rest, thus filling all cells, as Fig. 4.
It is not so simple to construct magic squares with even numbers of cells. The following method is, however, not particularly difficult. Suppose the cells are filled, in the first place, with the numbers arranged in the natural order, as in Fig. 5.
Besides magic squares. magic polygons and solids of various forms have been studied.
BIBLIOGRAPHY. On the history of the subject, Bibliography. On the history of the subject, consult: Gunther, rermisehte Untersuchungen «fir Gcschichte der matkematischen 1T'isst"n schaften (Leipzig. 13711) : Lucas. Recreations mathematiques (Paris, 4 vols.. 2d ed.. 1891-94) ; McClintock. "On the Most Perfect Form of Magic Squares," in the American Journal of ]lathe maties (1397); Horner, "On the Algebra of Magic Squares," in the Quarterly Journal of Mathematics (1870); Schaller, Die niagischen Fiyar•n (Leipzig, 1882) ; Ball, "Even Magic :Nita res," in the M es.yeager of Mathe ta al es (London, 1894) ; Ball, Mathematical Recreations (London, 1892) ; Ahrens, Mathemotische Unter hallangen and 8picle, giving a very complete bibliography (Leipzig, 1901) ; Schubert, Motile matisehe .1lussestanden (Leipzig, 3 vuls., 2d ed., 1900).