INVOLUTION AND EVOLUTION.. In mathematics. the raising to powers and the ex traction of roots, respectively. The result of taking a number twice as a factor is called the square of the ninnIter; the result of taking it three times as a factor, its cube; four limes, its fourth power, and so on; e.g. 3 . 3 = 3'. nr 9 is the square of 3; 3 :3 • 3 = 3', or 27 is the cube of 3; a•a•a•a=a' is the fourth power of u. This process is called involution. Evolution is the inverse of involution, nr it is the process which undoes involution. The square root of a number is one of the two equal factors of the number, the cube root tine of the three equal factors, and so on: e.g. the square root of 16, or ?16, is either 4 or —4, since 4 • 4 = and — 4 • — 4 = 16; the cube root of 27, or either 3 or —3 -4- 1/-3) since each of these cubed equals 27. The nth root of a perfect nth power is one of the n equal fas ters of that power. A number which is not a perfect nth power has not it equal factors. It is, however, said to have an nth root to any re quired degree of approximation. Thus the nth root. of m to 0.1 is that number of tenths whose nth power differs from m by less than the nth power of any other of tenths.
When the number ose root is sought is a perfect power, the process of factoring is one of the most practical methods except the use of tables. In the case of numbers which are not perfect. powers, and of certain algebraic expres sions, the binomial formula is usually employed where tables arc not available. Thus a' ±2ab It', the square of a b, may be applied to extract the square root of a number or of an alge braic expression, since the root ran always be expressed as a binomial whose square the power eontains. The older methods of square and cube root, depending upon the sections of a square and of a cube, were inferior, since they could not he extended to higher roots. The binomial formula is, however, of general application, and may be extended so as to extract the nth root. The detail
of these processes Ca11 best I obtained from text In Pr:tette:141y to obtain the square or (quite root of a number, reference is usually made to tables of roots or of logarithms. See 1,06A• arrnst.
In geometry two collinear ranges, three points each, are said to form an involution, when the anharmonie ratios (q.v.) of any four, not two pairs of vonjuoutc points, is equal to the anhar mimic ratio of their ftair conjugates. 'rims, in the figure A. 13, C', 13', A' form an involution, A, 13, C being conjugate to A', II', t" and tA B C = (A' Ii' C' B). This form of involution is IN(' to ]es: ( 639). Involutions of higher degrees have been developed by Poneelet (184:3) and :Mains ( IS55 ) .
IO, (Lat., from Ck. '10. In Creek legend, the daughter of Ina•hus or Iasus, and priestess of Hera at Argos. She was loved by Zeus, who, on account of llera's suspicions, changed her into a white cow. Ilera, having obtained of him the cow as a present, set the hundred-eyed Argus to watch her. Hermes, by command of Zeus, killed Argus and released her. liera then sent a gad fly, which pursued Jo until in her wanderings she 1-caviled Egypt, where she was restored to her original form, and became the mother of Elia plots. A somewhat different account of this myth is found in the Proelt(hruts of .•schylus. In the early Creek art Jo is represented as a cow; about n.•. 500 it became customary to represent her as a maiden with horns Oil her forehead; later still there is a return to the earlier method. Various attempts have been made to explain the myth by natural phenomena, but none have obtained any general acceptance. Consult. Engelman. De lone (Hall•, 1868) id., in Roseher, Lea-ikon der griechisrltcn and riimisehen ptholoqie (Leipzig, 1890.94) ; Near Jah•Liicber fur vol. vii. (Leipzig, 1870).