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HYPERBATON—HYPERBOREANS

was the wife of Isidorus; but this is chronologically impossible, since Isidorus could not have been born before 434 (see Hoche in Philologus). Shortly after the accession of Cyril to the patriarchate of Alexandria in 412, owing to her intimacy with Orestes, the pagan prefect of the city, Hypatia was barbarously murdered by the Nitrian monks and the fanatical Christian mob (March 415). Socrates has related how she was torn from her chariot, dragged to the Caesareum (then a Christian church), stripped naked, done to death with oyster-shells (ὀστράκοις ἀνεῖλον perhaps “cut her throat”) and finally burnt piecemeal. Most prominent among the actual perpetrators of the crime was one Peter, a reader; but there seems little reason to doubt Cyril’s complicity (see Cyril of Alexandria).

Hypatia, according to Suidas, was the author of commentaries on the Arithmetica of Diophantus of Alexandria, on the Conics of Apollonius of Perga and on the astronomical canon (of Ptolemy). These works are lost; but their titles, combined with expressions in the letters of Synesius, who consulted her about the construction of an astrolabe and a hydroscope, indicate that she devoted herself specially to astronomy and mathematics. Little is known of her philosophical opinions, but she appears to have embraced the intellectual rather than the mystical side of Neoplatonism, and to have been a follower of Plotinus rather than of Porphyry and Iamblichus. Zeller, however, in his Outlines of Greek Philosophy (1886, Eng. trans. p. 347), states that “she appears to have taught the Neoplatonic doctrine in the form in which Iamblichus had stated it.” A Latin letter to Cyril on behalf of Nestorius, printed in the Collectio nova conciliorum, i. (1623), by Stephanus Baluzius (Étienne Baluze, q.v.), and sometimes attributed to her, is undoubtedly spurious. The story of Hypatia appears in a considerably disguised yet still recognizable form in the legend of St Catherine as recorded in the Roman Breviary (November 25), and still more fully in the Martyrologies (see A. B. Jameson, Sacred and Legendary Art (1867) ii. 467.)

The chief source for the little we know about Hypatia is the account

given by Socrates (Hist. ecclesiastica, vii. 15). She is the subject of an epigram by Palladas in the Greek Anthology (ix. 400). See Fabricius, Bibliotheca Graeca (ed. Harles), ix. 187; John Toland, Tetradymus (1720); R. Hoche in Philologus (1860), xv. 435; monographs by Stephan Wolf (Czernowitz, 1879), H. Ligier (Dijon, 1880) and W. A. Meyer (Heidelberg, 1885), who devotes attention to the relation of Hypatia to the chief representatives of Neoplatonism; J. B. Bury, Hist. of the Later Roman Empire (1889), i. 208,317; A. Güldenpenning, Geschichte des oströmischen Reiches unter Arcadius und Theodosius II. (Halle, 1885), p. 230; Wetzer and Welte, Kirchenlexikon, vi. (1889), from a Catholic standpoint. The story of Hypatia also forms the basis of the well-known historical romance by Charles Kingsley

(1853).

HYPERBATON (Gr. ὑπέρβατον, a stepping over), the name of a figure of speech, consisting of a transposition of words from their natural order, such as the placing of the object before instead of after the verb. It is a common method of securing emphasis.

HYPERBOLA, a conic section, consisting of two open branches, each extending to infinity. It may be defined in several ways. The in solido definition as the section of a cone by a plane at a less inclination to the axis than the generator brings out the existence of the two infinite branches if we imagine the cone to be double and to extend to infinity. The in plano definition, i.e. as the conic having an eccentricity greater than unity, is a convenient starting-point for the Euclidian investigation. In projective geometry it may be defined as the conic which intersects the line at infinity in two real points, or to which it is possible to draw two real tangents from the centre. Analytically, it is defined by an equation of the second degree, of which the highest terms have real roots (see Conic Section).

While resembling the parabola in extending to infinity, the curve

has closest affinities to the ellipse. Thus it has a real centre, two foci, two directrices and two vertices; the transverse axis, joining the vertices, corresponds to the major axis of the ellipse, and the line through the centre and perpendicular to this axis is called the conjugate axis, and corresponds to the minor axis of the ellipse; about these axes the curve is symmetrical. The curve does not appear to intersect the conjugate axis, but the introduction of imaginaries permits us to regard it as cutting this axis in two unreal points. Calling the foci S, S′, the real vertices A, A′, the extremities of the conjugate axis B, B’ and the centre C, the positions of B, B′ are given by AB = AB′ = CS. If a rectangle be constructed about AA′ and BB′, the diagonals of this figure are the “asymptotes” of the curve; they are the tangents from the centre, and hence touch the curve at infinity. These two lines may be pictured in the in solido definition as the section of a cone by a plane through its vertex and parallel to the plane generating the hyperbola. If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the curve is called the rectangular hyperbola. The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola.

Some properties of the curve will be briefly stated: If PN be the ordinate of the point P on the curve, AA’ the vertices, X the meet of the directrix and axis and C the centre, then PN2: AN·NA′: : SX2: AX·A′X, i.e. PN2 is to AN·NA′ in a constant ratio. The circle on AA’ as diameter is called the auxiliarly circle; obviously AN·NA’ equals the square of the tangent to this circle from N, and hence the ratio of PN to the tangent to the auxiliarly circle from N equals the ratio of the conjugate axis to the transverse. We may observe that the asymptotes intersect this circle in the same points as the directrices. An important property is: the difference of the focal distances of any point on the curve equals the transverse axis. The tangent at any point bisects the angle between the focal distances of the point, and the normal is equally inclined to the focal distances. Also the auxiliarly circle is the locus of the feet of the perpendiculars from the foci on any tangent. Two tangents from any point are equally inclined to the focal distance of the point. If the tangent at P meet the conjugate axis in t, and the transverse in N, then Ct. PN = BC2; similarly if g and G be the corresponding intersections of the normal, PG : Pg : : BC2 : AC2. A diameter is a line through the centre and terminated by the curve: it bisects all chords parallel to the tangents at its extremities; the diameter parallel to these chords is its conjugate diameter. Any diameter is a mean proportional between the transverse axis and the focal chord parallel to the diameter. Any line cuts off equal distances between the curve and the asymptotes. If the tangent at P meets the asymptotes in R, R′, then CR·CR′ = CS2. The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.

Analytically the hyperbola is given by ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 wherein ab > h2. Referred to the centre this becomes Ax2 + 2Hxy + By2 + C = 0; and if the axes of coordinates be the principal axes of the curve, the equation is further simplified to Ax2 − By2 = C, or if the semi-transverse axis be a, and the semi-conjugate b, x2/a2 − y2/b2 = 1. This is the most commonly used form. In the rectangular hyperbola a = b; hence its equation is x2 − y2 = 0. The equations to the asymptotes are x/a = ±y/b and x = ±y respectively. Referred to the asymptotes as axes the general equation becomes xy = k2; obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola. The values of the constant k2 are ½(a2 + b2) and ½a2 respectively. (See

Geometry: Analytical; Projective.)

HYPERBOLE (from Gr. ὑπερβάλλειν, to throw beyond), a figure of rhetoric whereby the speaker expresses more than the truth, in order to produce a vivid impression; hence, an exaggeration.

HYPERBOREANS (Ὑπερβόρεοι, Ὑπερβόρειοι), a mythical people intimately connected with the worship of Apollo. Their name does not occur in the Iliad or the Odyssey, but Herodotus (iv. 32) states that they were mentioned in Hesiod and in the Epigoni, an epic of the Theban cycle. According to Herodotus, two maidens, Opis and Arge, and later two others, Hyperoche and Laodice, escorted by five men, called by the Delians Perphereës, were sent by the Hyperboreans with certain offerings to Delos. Finding that their messengers did not return, the Hyperboreans adopted the plan of wrapping the offerings in wheat-straw and requested their neighbours to hand them on to the next nation, and so on, till they finally reached Delos. The theory of H. L. Ahrens, that Hyperboreans and Perphereës are identical, is now widely accepted. In some of the dialects of northern Greece (especially Macedonia and Delphi) φ had a tendency to become β. The original form of Περφερέες was ὑπερφερέται or ὑπέρφοροι (“those who carry over”), which becoming ὑπέρβοροι gave rise to the popular derivation from βορέας (“dwellers beyond the north wind”). The Hyperboreans were thus the bearers of the sacrificial gifts to Apollo over land and sea, irrespective of their home, the name being given to Delphians, Thessalians, Athenians and Delians. It is objected by O. Schröder that the form Περφερέες requires a passive meaning, “those who are carried round the altar,” perhaps dancers like the whirling dervishes; distinguishing them from the Hyperboreans, he explains the latter as those who live “above