Among the many ingenious squares given by various writers, this article may justly close with two by L. Euler, in the Histoire de l’académie royale des sciences (Berlin, 1759). In fig. 27 the natural numbers show the path of a knight that moves within an odd square in such a manner that the sum of pairs of numbers opposite to and equidistant from the middle figure is its double. In fig. 28 the knight returns to its starting cell in a square of 6, and the difference between the pairs of numbers opposite to and equidistant from the middle point is 18.
A model consisting of seven Nasik cubes, constructed by A. H. Frost, is in the South Kensington Museum. The centres of the cubes are placed at equal distances in a straight line, the similar faces looking the same way in a plane parallel to that line. Each of the cubes has seven parallel glass plates, to which, on one side, the seven numbers in the septenary scale are fixed, and behind each, on the other side, its value in the common scale. 1201, the middle number from 1 to 74 occupies the central cubelet of the middle cube. Besides each cube having separately the same Nasical summation, this is also obtained by adding the numbers in any seven similarly situated cubelets, one in each cube. Also, the sum of all pairs of numbers, in a straight line, through the central cube of the system, equidistant from it, in whatever cubes they are, is twice 1201.
(A. H. F.)
Fennell’s Magic Ring.—It has been noticed that the numbers of magic squares, of which the extension by repeating the rows and columns of n numbers so as to form a square of 2n − 1 sides yields n² magic squares of n sides, are arranged as if they were all inscribed round a cylinder and also all inscribed on another cylinder at right angles to the first. C. A. M. Fennell explains this apparent anomaly by describing such magic squares as Mercator’s projections, so to say, of “magic rings.”
The surface of these magic rings is symmetrically divided into n² quadrangular compartments or cells by n equidistant zonal circles parallel to the circular axis of the ring and by n transverse circles which divide each of the n zones between any two neighbouring zonal circles into n equal quadrangular cells, while the zonal circles divide the sections between two neighbouring transverse circles into n unequal quadrangular cells. The diagonals of cells which follow each other passing once only through each zone and section, form similar and equal closed curves passing once quite round the circular axis of the ring and once quite round the centre of the ring. The position of each number is regarded as the intersection of two diagonals of its cell. The numbers are most easily seen if the smallest circle on the surface of the ring, which circle is concentric with the axis, be one of the zonal circles. In a perfect magic ring the sum of the numbers of the cells whose diagonals form any one of the 2n diagonal curves aforesaid is ½n (n² + 1) with or without increment, i.e. is the same sum as that of the numbers in each zone and each transverse section. But if n be 3 or a multiple of 3, only from 2 to n of the diagonal curves carry the sum in question, so that the magic rings are imperfect; and any set of numbers which can be arranged to make a perfect magic ring or magic square can also make an imperfect magic ring, e.g. the set 1 to 16 if the numbers 1, 6, 11, 16 lie thus on a diagonal curve instead of in the order 1, 6, 16, 11. From a perfect magic ring of n² cells containing one number each, n² distinct magic squares can be read off; as the four numbers round each intersection of a zonal circle and a transverse circle constitute corner numbers of a magic square. The shape of a magic ring gives it the function of an indefinite extension in all directions of each of the aforesaid n² magic squares.
(C. A. M. F.)
See F. E. A. Lucas, Récréations mathématiques (1891-1894); W. W. R. Ball, Mathematical Recreations (1892); W. E. M. G. Ahrens, Mathematische Unterhaltungen und Spiele (1901); H. C. H. Schubert, Mathematische Mussestunden (1900). A very detailed work is B. Violle, Traité complet des carrés magiques (3 vols., 1837-1838). The theory of “path nasiks” is dealt with in a pamphlet by C. Planck (1906).
MAGINN, WILLIAM (1793-1842), Irish poet and journalist, was born at Cork on the 10th of July 1793. The son of a schoolmaster, he graduated at Trinity College, Dublin, in 1811, and after his father’s death in 1813 succeeded him in the school. In 1819 he began to contribute to the Literary Gazette and to Blackwood’s Magazine, writing as “R. T. Scott” and “Morgan O’Doherty.” He first made his mark as a parodist and a writer of humorous Latin verse. In 1821 he visited Edinburgh, where he made acquaintance with the Blackwood circle. He is credited with having originated the idea of the Noctes ambrosianae, of which some of the most brilliant chapters were his. His connexion with Blackwood lasted, with a short interval, almost to the end of his life. His best story was “Bob Burke’s Duel with Ensign Brady.” In 1823 he removed to London. He was employed by John Murray on the short-lived Representative, and was for a short time joint-editor of the Standard. But his intemperate habits and his imperfect journalistic morality prevented any permanent success. In connexion with Hugh Fraser he established Fraser’s Magazine (1830), in which appeared his “Homeric Ballads.” Maginn was the original of Captain Shandon in Pendennis. In spite of his inexhaustible wit and brilliant scholarship, most of his friends were eventually alienated by his obvious failings, and his persistent insolvency. He died at Walton-on-Thames on the 21st of August 1842.
His Miscellanies were edited (5 vols., New York, 1855-1857) by R. Shelton Mackenzie and (2 vols., London, 1885) by R. W. Montagu [Johnson].