Glossarium Technicum (Technical Glossary)

Lexicon of Intrinsic Spherical Geometry, Chord Trigonometry, and Projective Transversals

This lexicon defines 15 foundational technical terms across the Greek, Latin, Arabic, Hebrew, and modern projective traditions of Menelaus’s Sphaerica. Every inline :term: tag resolves directly to its #term-<slug> anchor below.

Lexicon Definitions

triangulum sphaericum (τρίπλευρον σφαιρικόν) — spherical triangle

  • Tradition: Latin (Halley 1758 Def. I) / Greek (Pappus VI)

  • English Rendering: spherical triangle (spherical trilateral)

Space on the surface of a sphere bounded by three arcs of great circles, each strictly less than a semicircle (Menelaus Sphaerica Book I, Def. I–II, Halley 1758 p. 1).

circulus magnus (μέγιστος κύκλος) — great circle

  • Tradition: Latin / Greek

  • English Rendering: great circle (intrinsic geodesic of the sphere)

Intersection of the spherical surface \(S^2\) with a plane passing through the sphere’s center; minor arcs of great circles serve as intrinsic geodesic segments.

angulus sphaericus — spherical (dihedral) angle

  • Tradition: Latin (Halley 1758 Def. III–VI)

  • English Rendering: spherical (dihedral) angle

Angle contained by two intersecting great-circle arcs on a sphere, measured by the inclination of their planes (or by the arc intercepted on the polar great circle, Halley p. 1).

polus (πόλος) — pole of a great circle

  • Tradition: Latin / Greek

  • English Rendering: pole of a great circle

Either of the two antipodal points on \(S^2\) at quadrant arc distance (\(90^\circ\)) from every point of a given great circle.

excessus sphaericus — spherical excess

  • Tradition: Latin / Modern Spherical Geometry

  • English Rendering: spherical excess (\(E = A + B + C - \pi\))

Positive excess by which the three interior angles of any spherical triangle exceed two right angles (Sphaerica Prop. I.11, Halley p. 11).

triangulum polare — polar (supplementary) triangle

  • Tradition: Latin

  • English Rendering: polar (supplementary) triangle

Dual spherical triangle \(\triangle A^*B^*C^*\) whose vertices are the poles of \(BC, CA, AB\); its sides and angles are exact supplements (\(\pi - A, \pi - a\)), grounding AAA spherical congruence (Sphaerica I.17).

subtensa dupli arcus (ἡ ὑπὸ τὴν διπλῆν τῆς περιφερείας / נכחות) — chord of the double arc

  • Tradition: Latin (Halley 1758 p. 82) / Greek (Almagest I.13) / Hebrew

  • English Rendering: chord of the double arc (\(\operatorname{crd}(2\theta) = 2R\sin\theta\))

Classical Greek trigonometric primitive used by Hipparchus, Menelaus, and Ptolemy; ratios of double-arc chords equal ratios of sines identically.

sinus (جيب / jayb) — sine

  • Tradition: Latin / Arabic

  • English Rendering: sine (half-chord of the double arc)

Half the chord of twice the arc, adopted in Abū Naṣr Manṣūr’s and Halley’s redactions of Menelaus Book III (Halley 1758 Scholion, p. 82).

Figura Catha / al-Shakl al-Qaṭṭāʿ (القطاع) — The Sector Figure

  • Tradition: Medieval Latin / Arabic

  • English Rendering: The Sector (or Transversal) Figure

Traditional medieval name for the six-quantity Menelaus configuration of four intersecting great-circle arcs (Halley 1758 p. 82 Scholion).

ratio composita (λόγος συνημμένος) — compound ratio

  • Tradition: Latin / Greek (Euclid VI Def. 5; Almagest I.13)

  • English Rendering: compound ratio

Product of two ratios (A:B compounded of C:D and E:F, i.e., \(\frac{A}{B} = \frac{C}{D}\cdot\frac{E}{F}\)).

componendo et dividendo (κατὰ σύνθεσιν καὶ κατὰ διαίρεσιν) — composition & division of ratios

  • Tradition: Latin / Greek (Euclid V.17–18; Almagest I.13)

  • English Rendering: by composition and by division of ratios

Euclidean proportion transformations governing Ptolemy’s internal and external Menelaus cases.

transversalis — transversal line or great circle

  • Tradition: Latin (Carnot 1806)

  • English Rendering: transversal

Line or great circle intersecting all three extended sides \(BC, CA, AB\) of a reference triangle at traces \(D, E, F\).

sectio harmonica — harmonic range / harmonic conjugate

  • Tradition: Latin

  • English Rendering: harmonic range (\((A, B; C, D) = -1\))

Collinear or great-circle quadruple whose internal and external division ratios are equal in magnitude and opposite in sign (Sphaerica III.6, III.8, III.10).

polaris trilinearis — trilinear polar of a point

  • Tradition: Modern Latin / Projective Geometry

  • English Rendering: trilinear polar

The Menelaus transversal joining the harmonic conjugates of the Ceva traces of a point \(P\) with respect to \(\triangle ABC\).

quadrance & spread (Q, s) — Rational Trigonometry invariants

  • Tradition: Rational Trigonometry (Wildberger)

  • English Rendering: squared distance Q and squared sine s between lines

Polynomial invariants over \(\mathbb{Q}\) certifying Planar Menelaus without square roots.