Technical Glossary of Apollonian Conics

Greek, Latin, and English geometrical terminology of Apollonius of Perga, verified against T. L. Heath’s Notes on the Terminology of Greek Geometry (1896, pp. clvii–clxx) and Heiberg’s Index Graecitatis (1893).

parabolē (παραβολή)

  • Greek: παραβολή

  • Latin: parabola

  • English Rendering: parabola (exact application)

  • Heath (1896) Loci: I.11 [Prop. 1], pp. lxxx–lxxxiv, 8–9

  • Definition: Coined by Apollonius in I.11 from Euclidean application of areas (VI.27–29): the square on the ordinate QV equals the rectangle applied exactly along the latus rectum PL with width equal to the abscissa PV (y² = px), neither exceeding nor falling short.

elleipsis (ἔλλειψις)

  • Greek: ἔλλειψις

  • Latin: ellipsis

  • English Rendering: ellipse (application falling short)

  • Heath (1896) Loci: I.13 [Prop. 3], pp. lxxx–lxxxiv, 11–13

  • Definition: Coined by Apollonius in I.13 from Euclid VI.28: the square on the ordinate QV equals a rectangle applied to the latus rectum p with width PV, falling short (ἐλλεῖπον) by a figure similar and similarly situated to the rectangle contained by the transverse diameter d and parameter p: y² = px − (p/d)x².

hyperbolē (ὑπερβολή)

  • Greek: ὑπερβολή

  • Latin: hyperbola

  • English Rendering: hyperbola (application exceeding)

  • Heath (1896) Loci: I.12 [Prop. 2], pp. lxxx–lxxxiv, 9–11

  • Definition: Coined by Apollonius in I.12 from Euclid VI.29: the square on the ordinate QV equals a rectangle applied to the latus rectum p with width PV, exceeding (ὑπερβάλλον) by a figure similar and similarly situated to the rectangle contained by the transverse diameter d and parameter p: y² = px + (p/d)x².

orthia / par’ hēn dynantai (ὀρθία / παρ’ ἣν δύνανται)

  • Greek: ἡ ὀρθία (πλευρά)

  • Latin: latus rectum

  • English Rendering: parameter / latus rectum

  • Heath (1896) Loci: I.11–13 [Props. 1–3], pp. clxi, 8–13

  • Definition: The ‘upright’ fixed segment PL erected perpendicular to the diameter at the vertex P against which the rectangle equal to the square of the ordinate is applied; determined by the cone’s axial triangle geometry (e.g., p = 2b²/a along the principal axis of a central conic).

plagia pleura (πλαγία πλευρά)

  • Greek: ἡ πλαγία πλευρά

  • Latin: latus transversum

  • English Rendering: transverse diameter

  • Heath (1896) Loci: I.12–13 [Props. 2–3], pp. clxi, 9–13

  • Definition: The segment PP’ intercepted on the cutting plane’s diameter between the two generators of the axial triangle (internal for the ellipse, external across the opposite cone napes for the hyperbola), forming with the orthia p the ‘figure’ (εἶδος) of the section.

eidos (τὸ τῆς τομῆς εἶδος)

  • Greek: τὸ εἶδος

  • Latin: figura sectionis

  • English Rendering: figure of the section

  • Heath (1896) Loci: I.12–14, III.45, VII.12–31, pp. 9–14, 113, 228

  • Definition: The rectangle d · p contained by a transverse diameter PP’ (= d) and its corresponding parameter PL (= p); for the principal axis AA’, one-fourth of the figure equals d·p / 4 = CB² = b², which governs both the asymptote slope and the focal application.

tetagmenōs katagomenē (τεταγμένως καταγομένη)

  • Greek: τεταγμένως καταγομένη

  • Latin: ordinatim applicata

  • English Rendering: ordinate

  • Heath (1896) Loci: Book I Defs. 4–5, pp. clix–clxi, 1–2

  • Definition: A straight line QV drawn from a point Q on the conic parallel to the tangent at the vertex P of a chosen diameter (hence bisecting parallel chords), meeting the diameter at V; not restricted to right angles except when the diameter is a principal axis.

diametros & syzygeis diametroi (διάμετρος / συζυγεῖς διάμετροι)

  • Greek: συζυγεῖς διάμετροι

  • Latin: diametri coniugatae

  • English Rendering: conjugate diameters

  • Heath (1896) Loci: Book I Defs. 4, 6; I.15–16, I.49–51, VII.12–31

  • Definition: Any straight line bisecting a family of parallel chords of a conic is a diameter; two diameters are conjugate when each bisects all chords parallel to the other (equivalent to mutual polarity with respect to the line at infinity).

hypenantia tomē (ὑπεναντία τομή)

  • Greek: ὑπεναντία τομή

  • Latin: sectio subcontraria

  • English Rendering: subcontrary circular section

  • Heath (1896) Loci: I.5, I.9, pp. 4–6

  • Definition: In an oblique scalene circular cone, the second family of circular plane sections (I.5 [Heath p. 4]) inclined antiparallel to the circular base across the axial triangle ABC so that ∠AKH = ∠ABC, proving that an oblique circular cone possesses two distinct parallel families of circular sections.

hai antikeimenai (αἱ ἀντικείμεναι)

  • Greek: αἱ ἀντικείμεναι (τομαί)

  • Latin: sectiones oppositae

  • English Rendering: opposite branches of the hyperbola

  • Heath (1896) Loci: I.14 [Prop. 4], I.16 [Prop. 6], pp. lxxxiv–lxxxv, 13–17

  • Definition: Apollonius’s revolutionary recognition (I.14, I.16) that when a plane cuts both napes of the double cone, the two resulting hyperbolic branches share a single transverse diameter, center, conjugate diameter, and common pair of asymptotes—forming one unified algebraic curve.

symptōma (σύμπτωμα / ἀρχικὸν σύμπτωμα)

  • Greek: σύμπτωμα

  • Latin: symptoma

  • English Rendering: fundamental symptom (defining relation)

  • Heath (1896) Loci: pp. lxxix, xcvii, cxv–cxviii

  • Definition: The invariant planimetric relation linking every point’s ordinate QV and abscissa PV on the curve once severed from the 3D generating cone; the direct ancient counterpart of the algebraic equation of an affine curve.

asymptōtoi (ἀσύμπτωτοι)

  • Greek: αἱ ἀσύμπτωτοι

  • Latin: asymptoti

  • English Rendering: asymptotes

  • Heath (1896) Loci: II.1–16 [Props. 28–36], pp. 53–63

  • Definition: Constructed in II.1 by setting off PL = PL’ on the vertex tangent with PL² = ¼ p·d and joining the center C to L, L’: straight lines that never meet the hyperbola at any finite distance yet approach nearer to it than any assigned distance (II.14), satisfying xy = const in asymptotic coordinates (II.12).

ta ek tēs parabolēs ginomena sēmeia (τὰ ἐκ τῆς παραβολῆς γινόμενα σημεῖα)

  • Greek: τὰ ἐκ τῆς παραβολῆς γινόμενα σημεῖα

  • Latin: puncta ex comparatione (foci)

  • English Rendering: foci of a central conic

  • Heath (1896) Loci: III.45–52 [Props. 69–73], pp. 113–118

  • Definition: Apollonius’s technical name in III.45–52 [Props. 69–73] for the two foci S, S’ on the major/transverse axis AA’ determined by applying a rectangle AS · SA’ = ¼ p_a · AA’ = b² (falling short by a square for the ellipse, c² = a² − b²; exceeding by a square for the hyperbola, c² = a² + b²). Because a parabola lacks a center C and second vertex A’, this 2-vertex application does not apply to it.

topos pros treis kai tessaras grammas (τόπος πρὸς τρεῖς καὶ τέσσαρας γραμμάς)

  • Greek: τόπος πρὸς τρεῖς καὶ τέσσαρας γραμμάς

  • Latin: locus ad tres et quattuor lineas

  • English Rendering: Three-Line and Four-Line Locus

  • Heath (1896) Loci: Preface I, Intro Ch. V, III.53–56 [Props. 74–76], pp. lxx, cxxxviii–cl, 119–125

  • Definition: The celebrated locus of a point P whose oblique distances d₁, d₂, d₃, d₄ to four given lines satisfy d₁·d₃ = λ d₂·d₄ (or d₁·d₃ = λ d₂² when two lines coincide); Apollonius states in Preface I that Book III (Props. 53–56) finally yields its complete synthesis after Euclid achieved only a partial special case.

diorismos (διορισμός)

  • Greek: διορισμός

  • Latin: determinatio

  • English Rendering: diorismos (discriminant / solvability condition)

  • Heath (1896) Loci: Prefaces V & VII, V.51–52 [Prop. 99], pp. lxxv–lxxvi, 168–179

  • Definition: The rigorous determination of the exact parameter bounds separating zero, one (double/tangent), or two real solutions—reaching its supreme ancient expression in Book V Props. 51–52 [Prop. 99] (the evolute discriminant separating 2 vs. 3 vs. 4 normals) and in Book VII’s inequality theorems created for Book VIII.

evolute / envelope of normals (V.51–63)

  • Greek: διορισμὸς τῶν ἐλαχίστων (Book V)

  • Latin: evoluta

  • English Rendering: evolute of a conic

  • Heath (1896) Loci: Intro pp. lxxv–lxxvi, cxxvii–cxxix; V.51–63 [Props. 99–106], pp. 168–186

  • Definition: Although Apollonius has no single noun for the curve as a locus, Book V Props. 51–52 [Prop. 99] construct for every abscissa x the exact critical ordinate y below which 4 normals pass through (x, y), on which 2 normals coalesce, and above which only 2 normals exist—yielding point-for-point the semicubical parabola 27py² = 4(x − p/2)³ and the astroidal Lamé curve (ax)^(2/3) + (by)^(2/3) = (a² − b²)^(2/3).

epaphai (Περὶ ἐπαφῶν / Tangencies)

  • Greek: Περὶ ἐπαφῶν

  • Latin: De Tactionibus

  • English Rendering: Problem of Apollonius (Tangencies)

  • Heath (1896) Loci: Intro Part II Ch. I; Pappus Synagoge VII

  • Definition: Apollonius’s two-book analytic treatise summarized in Pappus Synagoge VII: given any three elements each being a point (P), straight line (L), or circle (C), construct a circle tangent to all three (10 cases: PPP, PPL, PPC, PLL, PLC, PCC, LLL, LLC, LCC, and the crowning CCC case with up to 8 solutions).

neusis (Περὶ νεύσεων / Inclinations)

  • Greek: νεῦσις

  • Latin: inclinatio

  • English Rendering: neusis (verging insertion)

  • Heath (1896) Loci: Intro pp. cxxv–cxxix; Pappus Synagoge VII

  • Definition: Inserting a straight segment of given length between two given lines or circles so that its prolongation verges through a given pole; Apollonius restricted his two-book treatise On Inclinations strictly to those neuseis constructible by plane means (straightedge and compass).