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 repertories of magical rituals from any part of the world. For certain departments of ancient magic, however, like the Pythagorean philosophy, there is no lack of illustrative material; it depended on mystical speculations based on numbers or analogous principles. The importance of numbers is recognized in the magic of America and other areas, but the science of the Mediterranean area, combined with the art of writing, was needed to develop such mystical ideas to their full extent. Among the neo-Platonists there was a strong tendency to magical speculation, and they sought to impress into their service the demons with which they peopled the universe. Alexandria was the home of many systems of theurgic magic, and gnostic gems afford evidence of the nature of their symbols. In the middle ages the respectable branches of magic, such as astrology and alchemy, included much of the real science of the period; the rise of Christianity introduced a new element, for the Church regarded all the religions of the heathen as dealings with demons and therefore magical (see ). In our own day the occult sciences still find devotees among the educated; certain elements have acquired a new interest, in so far as they are the subject matter of (q.v.) and  (q.v.). But it is only among what are regarded as the lower classes, and in England especially the rural population, that belief in its efficacy still prevails to any large extent.  Psychology of Magic.—The same causes which operated to produce a belief in (q.v.) aided the creed of magic in general. Fortuitous coincidences attract attention; the failures are disregarded or explained away. Probably the magician is never wholly an impostor, and frequently has a whole-hearted belief in himself; in this connexion may be noted the fact that juggling tricks have in all ages been passed off as magical; the name of “” (q.v.) survives in our own day, though the conjurer no longer claims that his mysterious results are produced by demons. It is interesting to note that magical leechcraft depended for its success on the power of (q.v.), which is to-day a recognized element in medicine; perhaps other elements may have been instrumental in producing a cure, for there are cases on record in which European patients have been cured by the apparently meaningless performances of medicine-men, but an adequate study of savage medicine is still a desideratum.   —For a general discussion of magic with a list of selected works see Hubert and Mauss in Année sociologique, vii. 1–146; also A. Lehmann, Aberglaube und Zauberei; the article “Religion” in La Grande encyclopédie; K. T. Preuss in Globus, vols. 86, 87; Mauss, L’Origine des pouvoirs magiques, and Hubert, La Réprésentation du temps (Reports of École pratique des hautes études, Paris). For general bibliographies see Hauck, Realencyklopädie, s.v. “Magie”; A. C. Haddon, Magic and Fetishism. J. G. T. Graesse’s Bibliotheca magica is an exhaustive list of early works dealing with magic and superstition. For Australia see Spencer and Gillen’s works, and A. W. Howitt, Native Tribes. For America see Reports of Bureau of Ethnology, vii. xvii. For India see W. Caland, Altindisches Zauber-ritual; and W. Crooke, Popular Religion; also V. Henry, La Magie. For the Malays see W. W. Skeat, Malay Magic. For Babylonia and Assyria see L. W. King’s works. For magic in Greece and Rome see Daremberg and Saglio, s.v. “Magia,” “Amuletum,” &c. For medieval magic see A. Maury, La Magie. For illustrations of magic see J. G. Frazer, The Golden Bough; E. S. Hartland, Legend of Perseus; E. B. Tylor, Primitive Culture; W. G. Black, Folkmedicine. For negative magic see the works of Frazer and Skeat cited above; also ''Journ. Anthrop.'' Inst. xxxvi. 92–103; Zeitschrift für Ethnologie (Verhandlungen) (1905), 153–162; Bulletin trimestriel de l’académie malgache, iii. 105–159. See also bibliography to and.

 MAGIC SQUARE, a square divided into equal squares, like a chess-board, in each of which is placed one of a series of consecutive numbers from 1 up to the square of the number of cells in a side, in such a manner that the sum of the numbers in each row or column and in each diagonal is constant. From a very early period these squares engaged the attention of mathematicians, especially such as possessed a love of the marvellous, or sought to win for themselves a superstitious regard. They were then supposed to possess magical properties, and were worn, as in India at the present day, engraven in metal or stone, as amulets or talismans. According to the old astrologers, relations subsisted between these squares and the planets. In later times such squares ranked only as mathematical curiosities; till at last their mode of construction was systematically investigated. The earliest known writer on the subject was Emanuel Moscopulus, a Greek (4th or 5th century). Bernard Frenicle de Bessy constructed magic squares such that if one or more of the encircling bands of numbers be taken away the remaining central squares are still magical. Subsequently Poignard constructed squares with numbers in arithmetical progression, having the magical summations. The later researches of Phillipe de la Hire, recorded in the Mémoires de l’Académie Royale in 1705, are interesting as giving general methods of construction. He has there collected the results of the labours of earlier pioneers; but the subject has now been fully systematized, and extended to cubes.

Two interesting magical arrangements are said to have been given by Benjamin Franklin; these have been termed the “magic square of squares” and the “magic circle of circles.” The first (fig. 1) is a square divided into 256 squares, i.e. 16 squares along a side, in which are placed the numbers from 1 to 256. The chief properties of this square are (1) the sum of the 16 numbers in any row or column is 2056; (2) the sum of the 8 numbers in half of any row or column is 1028, i.e. one half of 2056; (3) the sum of the numbers in two half-diagonals equals 2056; (4) the sum of the four corner numbers of the great square and the four central numbers equals 1028; (5) the sum of the numbers in any 16 cells of the large square which themselves are disposed in a square is 2056. This square has other curious 