HYPERBOLA

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 1⁄2(a2 + b2) and 1⁄2a2 respectively. (see geometry: _analytical_; _projective_.) hyperbole (from gr. [greek: hyperballein], 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 ([greek: hyperboreoi, hyperboreioi]), 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 perpherees, 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 perpherees are identical, is now widely accepted. in some of the dialects of northern greece (especially macedonia and delphi) [phi] had a tendency to become [beta]. the original form of [greek: perpherees] was [greek: hyperpheretai] or [greek: hyperphoroi] ("those who carry over"), which becoming [greek: hyperboroi] gave rise to the popular derivation from [greek: boreas] ("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. schroder that the form [greek: perpherees] 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 the mountains," that is, in heaven. under the influence of the derivation from [greek: boreas], the home of the hyperboreans was placed in a region beyond the north wind, a paradise like the elysian plains, inaccessible by land or sea, whither apollo could remove those mortals who had lived a life of piety. it was a land of perpetual sunshine and great fertility; its inhabitants were free from disease and war. the duration of their life was 1000 years, but if any desired to shorten it, he decked himself with garlands and threw himself from a rock into the sea. the close connexion of the hyperboreans with the cult of apollo may be seen by comparing the hyperborean myths, the characters of which by their names mostly recall apollo or artemis (agyieus, opis, hecaergos, loxo), with the ceremonial of the apolline worship. no meat was eaten at the pyanepsia; the hyperboreans were vegetarians. at the festival of apollo at leucas a victim flung himself from a rock into the sea, like the hyperborean who was tired of life. according to an athenian decree (380 b.c.) asses were sacrificed to apollo at delphi, and pindar (_pythia_, x. 33) speaks of "hecatombs of asses" being offered to him by the hyperboreans. as the latter conveyed sacrificial gifts to delos hidden in wheat-straw, so at the thargelia a sheaf of corn was carried round in procession, concealing a symbol of the god (for other resemblances see crusius's article). although the hyperborean legends are mainly connected with delphi and delos, traces of them are found in argos (the stories of heracles, perseus, io), attica, macedonia, thrace, sicily and italy (which niebuhr indeed considers their original home). in modern times the name has been applied to a group of races, which includes the chukchis, koryaks, yukaghirs, ainus, gilyaks and kamchadales, inhabiting the arctic regions of asia and america. but if ever ethnically one, the asiatic and american branches are now as far apart from each other as they both are from the mongolo-tatar stock. see o. crusius in roscher's _lexikon der mythologie_; o. schroder in _archiv fur religionswissenschaft_ (1904), viii. 69; w. mannhardt, _wald- und feldkulte_ (1905); l. r. farnell, _cults of the greek states_ (1907), iv. 100. hypereides (c. 390-322 b.c.), one of the ten attic orators, was the son of glaucippus, of the deme of collytus. having studied under isocrates, he began life as a writer of speeches for the courts, and in 360 he prosecuted autocles, a general charged with treason in thrace (frags. 55-65, blass). at the time of the so-called "social war" (358-355) he accused aristophon, then one of the most influential men at athens, of malpractices (frags. 40-44, blass), and impeached philocrates (343) for high treason. from the peace of 346 to 324 hypereides supported demosthenes in the struggle against macedon; but in the affair of harpalus he was one of the ten public prosecutors of demosthenes, and on the exile of his former leader he became the head of the patriotic party (324). after the death of alexander, he was the chief promoter of the lamian war against antipater and craterus. after the decisive defeat at crannon (322), hypereides and the other orators, whose surrender was demanded by antipater, were condemned to death by the athenian partisans of macedonia. hypereides fled to aegina, but antipater's emissaries dragged him from the temple of aeacus, where he had taken refuge, and put him to death; according to others, he was taken before antipater at athens or cleonae. his body was afterwards removed to athens for burial. hypereides was an ardent pursuer of "the beautiful," which in his time generally meant pleasure and luxury. his temper was easy-going and humorous; and hence, though in his development of the periodic sentence he followed isocrates, the essential tendencies of his style are those of lysias, whom he surpassed, however, in the richness of his vocabulary and in the variety of his powers. his diction was plain and forcible, though he occasionally indulged in long compound words probably borrowed from the middle comedy, with which, and with the everyday life of his time, he was in full sympathy. his composition was simple. he was specially distinguished for subtlety of expression, grace and wit, as well as for tact in approaching his case and handling his subject matter. sir r. c. jebb sums up the criticism of pseudo-longinus (_de sublimitate_, 34) in the phrase--"hypereides was the sheridan of athens." seventy-seven speeches were attributed to hypereides, of which twenty-five were regarded as spurious even by ancient critics. it is said that a ms. of most of the speeches was in existence in the 16th century in the library of matthias corvinus, king of hungary, at ofen, but was destroyed at the capture of the city by the turks in 1526. only a few fragments were known until comparatively recent times. in 1847 large fragments of his speeches against _demosthenes_ (see above) and _for lycophron_ (incidentally interesting as elucidating the order of marriage processions and other details of athenian life, and the athenian government of lemnos), and the whole of the _for euxenippus_ (c. 330, a _locus classicus_ on [greek: eisangeliai] or state prosecutions), were found in a tomb at thebes in egypt, and in 1856 a considerable portion of a [greek: logos epitaphios], a _funeral oration_ over leosthenes and his comrades who had fallen in the lamian war, the best extant specimen of epideictic oratory (see babington, churchill). towards the end of the century further discoveries were made of the conclusion of the speech _against philippides_ (dealing with a [greek: graphe paranomon], or indictment for the proposal of an unconstitutional measure, arising out of the disputes of the macedonian and anti-macedonian parties at athens), and of the whole of the _against athenogenes_ (a perfumer accused of fraud in the sale of his business). these have been edited by f. g. kenyon (1893). an important speech that is lost is the _deliacus_ (frags. 67-75, blass) on the presidency of the delian temple claimed by both athens and delos, which was adjudged by the amphictyons to athens. on hypereides generally see pseudo-plutarch, _decem oratorum vitae_; f. blass, _attische beredsamkeit_, iii.; r. c. jebb, _attic orators_, ii. 381. a full list of editions and articles is given in f. blass, _hyperidis orationes sex cum ceterarum fragmentis_ (1894, teubner series), to which may be added i. bassi, _le quattro orazioni di iperide_ (introduction and notes, 1888), and j. e. sandys in _classical review_ (january 1895) (a review of the editions of kenyon and blass). for the discourse against athenogenes see h. weil, _etudes sur l'antiquite grecque_ (1900).