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ὑπερβολ-ή, ἡ, (ὑπερβάλλω) a throwing beyond others, δίσκων ὑπερβολαῖς Philostr.Im.2.19: in intr. sense, altitude of a star, Arist. Mete.342b32.

2. overshooting, superiority, χερῶν ὑπερβολαῖς E.Fr. 434; στρατιᾶς Th.6.31.

3. excess, τοῦ μεγέθεος Archyt.1; opp. ἔλλειψις or ἔνδεια, Pl.Prt.356a, 357a, 357b; θερμασίης Hp.Vict.2.65; ὑ. δισσὴ .., τῷ ποσῷ καὶ τῷ ποιῷ Arist.PA668b14; ὑπερβολὴν τῆς ἐπιθυμίας ἔχειν c. acc. et inf., And.3.33, etc.: hence in various phrases, χρημάτων ὑπερβολῇ ..πρίασθαι at an extravagant price, E.Med.232; ἐπέφερον τὴν ὑ. τοῦ καινοῦσθαι pushed on their extravagance in revolutionizing, Th.3.82; οὐκ ἔχον ἐστὶν ὑπερβολήν it can go no further, D.21.119, cf. 25.54; ἃ μηδὲ πιθανὰς τὰς ὑ. ἔχει Men.Her.62; so οὐδεμίαν or μηδεμίαν ὑ. λείπειν, Isoc.4.5,110, cf. D.3.25; οὐδεμίαν ὑ. καταλείπων φιλοτιμίας SIG545.13 (Delph., iii B. C.); εἴ τις ὑ. τούτου if there is aught beyond (worse than) this, D.19.66, cf. Isoc.5.42; ταῦτ' οὐχ ὑ.; is not this the extreme, the last degree? D.27.38; ὑπερβολὴν ποιήσομαι I will put an extreme case, Id.19.332; τοσαύτην ὑ. ποιεῖσθαι ὥστε ..to go so far that .., Id.18.190: folld. by a gen., ὑ. ποιεῖσθαι ἐκείνων τῆς αὑτοῦ βδελυρίας to carry his own rascality beyond theirs, Id.22.52, cf. 23.201, And.4.22, Lys.14.38; ὑ. ποιεῖν τῆς τιμῆς to raise the price, Arist.Pol.1259a26; εἰς ὑ. εὐδαιμονίας ἥκειν Isoc.11.14; τοσαύτας ὑ. δωρειῶν παρές χηται D.20.141; ὑ. ἀνοίας ἔχειν Polystr.p.27 W.

4. with Prep. in Adverbial phrases, = ὑπερβαλλόντως, εἰς ὑπερβολήν in excess, exceedingly; εἰς ὑ. ἄμεινον E.Fr.494; ἀγαθὸς εἰς ὑ. Antiph.80.11; ἐς ὑ. ἐκθερμαίνεσθαι Hp.Vict.2.65: c. gen., κτήσαιτ' ἂν ὄλβον εἰς ὑ. πατρός E.Fr.282.6 (v.l. εἰς ὑπεκτροφὴν πάτρας); far beyond, τοῦ πρόσθεν εἰς ὑ. πανοῦργος, i. e. far more wicked, Id.Hipp.939, cf. D.61.33; ἀναλίσκειν πάντα εἰς ὑπερβολάς Pl.Ep. 326d:—
ἐξ ὑπερβολῆς Plb.8.15.8:—
καθ' ὑπερβολὰν τοξεύσας with surpassing aim, S.OT1196 (lyr.); καθ' ὑ. ἐπαινεῖν extravagantly, Isoc. 5.11; οἱ καθ' ὑ. ἐν ἐνδείᾳ ὄντες in extremity of need, Arist.Pol.1295b18; αἱ καθ' ὑ. ἡδοναί Id.EN1151a12; τὸ καθ' ὑ. τραχύ Phld.Po.Herc. 994.35; καθ' ὑ. ἁμαρτωλός Ep.Rom.7.13: c. gen., καθ' ὑ. φιλοδοξίας OGI472.9 (Didyma, i A. D.):—
so in dat., εὐτελὴς ὑπερβολῇ Men.615; παχεῖ' ὑ. Philem.41; ὑ. ἀγαθός Arist.HA625a29, etc.

5. preeminence, perfection, without any notion of excess, δι' ἀρετῆς ὑπερβολήν Id.EN1145a24, cf. Rh.1367b1, Pol.1284a4; ἡ ὑ. τῆς φιλίας the best and noblest kind of friendship, Id.EN1166b1; but ἡ καθ' ὑ. φιλία, = ἡ καθ' ὑπεροχήν, Id.EE1238b18.

6. overstrained phrase, hyperbole, ὑπερβολὰς εἰπεῖν Isoc.4.88; οἱ πρὸς ὑπερβολὴν πεπονημένοι λόγοι ib.11; ὑπερβολὰς εἰπεῖν make strong statements, Id.3.35, D. 27.64; as a figure of speech, Arist.Rh.1413a29, Demetr.Eloc.52, Str.3.2.9; πρὸς -ὴν εἰρημένος Id.1.2.33.

7. τὸ καθ' ὑπερβολήν the superlative degree, in Adjectives, Arist.Top.134b24; τιθέναι ὑπερβολῇ ib.139a9; καθ' ὑ. εἰπεῖν Id.Cael.281a16.

8. overbid at auction, PPetr.2p.44 (iii B. C.).

II. crossing over, passage of mountains, etc., X.An.1.2.25, Plb.3.34.6, etc.

2. in sg. or pl., place of passage, mountain-pass, with or without τοῦ ὄρους, τῶν ὀρέων, X. An.3.5.18, 4.1.21, 4.4.18; ὑ. τοῦ Ταύρου Wilcken Chr.1 ii 14 (iii B. C.); αἱ τῶν Ἄλπεων ὑ. Plb.3.39.10; αἱ Ἄλπιαι ὑ. Str.7.1.5; ἡ κατὰ τὸν Αἷμον ὑ. D.S.19.73.

III. (from Med.) delay, τοῦ κακοῦ Hdt.8.112, cf. Decr. ap. D.18.29, Plb.14.9.8; [τῆς κρίσεως] ὑ. λαβούσης PEnteux.65.3 (iii B. C.).

IV. the conic section called hyperbola, because the square of the ordinate is equal to a rectangle with height equal to the abscissa applied to the parameter (as base) but exceeding (ὑπερβάλλον), i. e. overlapping, that base, Apollon. Perg.Con.1.12, Procl. in Euc.p.419F.