INVOLUTION (Lat. involvere, to roll up), a rolling up or complication. In arithmetic, involution is the operation of raising a quantity to any power; it is the converse of evolution, which is the operation of extracting any root of a quantity (see Arithmetic; Algebra). In geometry, an involution is a one-to-one correspondence between two ranges of points or between two pencils (see Geometry: Projective). The "involute" of a curve may be regarded as the locus of the extremity of a string when it is unwrapped from the curve (see Infinitesimal Calculus).
10, in Greek mythology, daughter of Inachus, the river-god of Argos and its first king. As associated with the oldest worship of Hera she is called the daughter of Peiren, who made the first image of that goddess out of a pear-tree at Tiryns; and under the name of Callithyia Io was regarded as the first priestess of Hera. Zeus fell in love with her, and, to protect her from the wrath of Hera, changed her into a white heifer (Apollodorus. ii. 1; Hyginus, Fab. 145; Ovid, Metam. i. 568-733); according to Aeschylus (Supplices, 299) the metamorphosis was the work of Hera herself. Hera, having persuaded Zeus to give her the heifer, set Argus Panoptes to watch her. Zeus thereupon sent Hermes, who lulled Argus to sleep and cut off his head with the sword with which Perseus afterwards slew the Gorgon. In another account Argus is killed by a stone thrown by Hermes. But the wrath of Hera still pursued Io. Maddened by a gadfly sent by the goddess she wandered all over the earth, swam the strait known on this account as the Bosporus (Ox-ford), and crossed the Ionian sea (traditionally called after her) until at last she reached Egypt, where she was restored to her original form and became the mother of Epaphus. Accounts of her wanderings (differing considerably in detail) are given in the Supplices and Prometheus Vinctus of Aeschylus. Various interpretations are given of the latter part of her story, which dates from the 7th century B.C., when intercourse was frequent between Greece and Egypt, and when much influence was exerted on Greek thought by Egyptian religion. According to the rationalistic explanation of Herodotus (i. 1) Io was an Argive princess who was carried off to Egypt by the Phoenicians. Epaphus, the son of Io, the supposed founder of Memphis, was identified with Apis. He was said to have been carried off by order of Hera to Byblus in Syria, where he was found again by Io. On returning to Egypt, Io, afterwards identified with Isis, married Telegonus and founded the royal families of Egypt, Phoenicia, Argos and Thebes. The journey to Syria in search of Epaphus was invented to explain the fact that the Phoenician goddess Astarte, who was sometimes represented as horned, was confounded with Io.
Io herself is variously interpreted. She is usually understood to be the moon in the midst of the mighty heaven, studded with stars, represented by Argus. According to others, she is the annual rising of the Nile; the personification of the Ionian race; the mist; the earth. It seems probable that she was a duplicate of Hera (Io i 30UKEpces is Hera (30c'ires), or a deity in primitive times worshipped under the symbol of a cow, whose worship was superseded by that of Hera; the recollection of this early identity would account for Io being regarded as the priestess of the goddess in later times. Amongst the Romans she was sometimes identified with Anna Perenna. The legend of Io spread beyond Argos, especially in Byzantium and Euboea, where it was associated with the town of Argura. It was a favourite subject among Greek painters, and many representations of it are preserved on vases and wall paintings; Io herself appears as a horned maiden or as the heifer watched by Argus.
See R. Engelmann, De Ione (1868), with notes containing references to authorities, and his article in Roscher's Lexikon der Mythologic; J. Overbeck, De Ione, telluris, non lunae, Dea (1872); P. W. Forchhammer, Die Wanderungen der Inachostochter Io (1881), with map and special reference to Aeschylus's account of Io's wanderings; F. Durrbach in Daremberg and Saglio's Dictionnaire des antiquites; G. Mellen, De Ius fabula (1901); Wernicke s.v. " Argos" in Pauly-Wissowa's Realencyclopadie, ii. pt. i. (1896); J. E. Harrison in Classical Review (18 93, p. 76); Bacchylides xviii. (xix.), with Jebb's notes.
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