Unit IV · Latin Diplomatic Lemmata (Heiberg 1907, pp. 227–234)

Claudii Ptolemaei Planisphaerium (trans. Hermann of Carinthia, Toulouse, 1143)

Structured diplomatic lemmata and geometric propositions keyed to J. L. Heiberg's public-domain Teubner edition (Opera Astronomica Minora, Vol. II, 1907, pp. 227–234).

South Pole North Pole e b d t (r_Can) m (r_Cap) I. Stereographic Section from South Pole (et · em = ed²) e a g b d l t m k r II. Astrolabe Plate: Equator abgd, Tropics tl & km, Ecliptic mbtd

§1 · Principium Repraesentationis Planae (Heiberg II, pp. 227–229)

  • Exordium & Aequator (`p. 227, ll. 1–12`): Ptolemy opens to Syrus (“Cum sit possibile, Jesure, et plerumque necessarium, ut in plano represententur circuli in speram corpoream incidentes…”) by establishing a single foundational circle abgd (unius circuli positione, qui rectam speram determinat) around center e representing the visible North Pole (manifesti poli), cut orthogonally by diameters ag and bd.

  • Circuli Paralleli (`p. 227, ll. 13–24`): Every parallel circle on the sphere is described concentrically around e by visual rays drawn from the hidden South Pole (ab occulto polo): northern parallels toward the visible pole fall inside abgd as circle tl (Cancer), whereas southern parallels toward the hidden pole fall outside abgd as circle km (Capricorn).

  • Circulus Declivis / Signifer (`p. 228, ll. 14–28`): Bisecting segment tm at r and describing the eccentric circle mbtd tangent to Cancer at t and Capricorn at m, Ptolemy proves from the folded meridian rays dm and dt that the angle subtending the solstitial diameter is right (“manifestum est angulum mdt rectum esse”, Eucl. III.31). Hence the circle on diameter tm passes strictly through the opposite equinoctial points b and d, bisecting the equator in the plane exactly as on the solid sphere.

§§2–3 · Circulus Finitor / Horizon (Heiberg II, pp. 229–232)

  • Sectio Aequatoris et Zodiaci (`pp. 229–232`): Every inclined great circle—and specifically the local horizon (finitor)—is represented by an eccentric circle whose diameter lies along the meridian and which bisects both the equatorial circle abgd and the oblique zodiacal circle mbtd into equal semicircles of six zodiacal signs each.

§4 · Proportio Semidiametrorum Tropicarum (Heiberg II, pp. 233–234)

  • Geometric Inversion Proportion (`p. 233, ll. 1–24`): Comparing the similar right triangles formed by the stereographic rays from the pole across the equatorial radius ed, Ptolemy establishes the continuous geometric proportion between the winter tropic radius (ez), the equatorial radius (ed), and the summer tropic radius (ek):

    $$ez : ed = ed : ek \qquad (\textit{linee ez ad lineam ed atque ed ad lineam ek}).$$

  • Numerical Evaluation from the *Almagest* (`p. 234, ll. 5–27`): Taking the tropical obliquity $\varepsilon = 23^\circ 51' 20''$ (partibus XXIII punctis LI secundis XX) and the Almagest chords (quemadmodum in Almagesti constitutum est) $\operatorname{crd}(113^\circ 51' 20'') = 100;33,28^p$ and $\operatorname{crd}(66^\circ 8' 40'') = 65;29,0^p$ for an equatorial radius $ed = 60^p$ (semidiametros circuli recti absolute LX partium):

    • Hyemalis Tropici Semidiametros (`ez`, Capricorn): XCII puncta VIII secunde XV ($92;8,15^p$).

    • Aestivi Tropici Semidiametros (`ek`, Cancer): partes XXXIX puncta IIII secunde XVIIII ($39;4,19^p$).

    • Totam Zodiaci Diametrum (`ez + ek`): partes CXXXI puncta XII [secunde XXXIIII] ($131;12,34^p$).