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.)

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

**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).

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 PN^{2}: AN·NA′: :
SX^{2}: AX·A′X, *i.e.* PN^{2} 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 = BC^{2}; similarly if g and G be the corresponding intersections
of the normal, PG : Pg : : BC^{2} : AC^{2}. 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′ = CS^{2}. The geometry of the rectangular hyperbola
is simplified by the fact that its principal axes are equal.

Analytically the hyperbola is given by ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0
wherein ab > h^{2}. Referred to the centre this becomes
Ax^{2} + 2Hxy + By^{2} + C = 0; and if the axes of coordinates be the
principal axes of the curve, the equation is further simplified to
Ax^{2} − By^{2} = C, or if the semi-transverse axis be a, and the semi-conjugate
b, x^{2}/a^{2} − y^{2}/b^{2} = 1. This is the most commonly used form.
In the rectangular hyperbola a = b; hence its equation is x^{2} − y^{2} = 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 = k^{2}; obviously the axes are oblique in the general
hyperbola and rectangular in the rectangular hyperbola. The values
of the constant k^{2} are ½(a^{2} + b^{2}) and ½a^{2} respectively. (See

*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