Unit IV · Historical & Mathematical Notes

1. Textual Survival: Alexandria $\to$ Baghdad $\to$ Córdoba $\to$ Toulouse (1143)

Unlike the Almagest, no Greek manuscript of Ptolemy's Planisphaerium (Ἅπλωσις ἐπιφανείας σφαίρας) has survived. Translated into Arabic in ninth-century Baghdad and revised with astrolabe chapters around 1000 CE in Córdoba by Maslama al-Majrītī, the work entered Latin Europe when Hermann of Carinthia (Hermannus Secundus) completed his Latin version at Toulouse on 1 June 1143 (dedicated to Thierry of Chartres). Notice the charming philological fossil in the very first sentence: the vocative ὦ Σύρε (“O Syrus”, matching Almagest I.1!) passed through Arabic transcription (Yā Sūrus $\to$ Iyāsūrus) into Hermann’s Latin as `Jesure`!

2. Apollonius Conics I.5 & Stereographic Circle Preservation

Why does an oblique circle on the sphere—whose visual rays from the South Pole form an oblique circular cone—cut the equatorial plane in a true circle rather than an ellipse? As Federico Commandino demonstrated synthetically in his 1558 commentary (Commandino 1558), the axial triangle of that oblique visual cone intersects the sphere in a meridian circle; by the inscribed-angle theorem (Eucl. III.21), the equatorial plane cuts the two extreme generators of the cone at anti-parallel (subcontrary) angles equal to those of the circular base on the sphere! By Apollonius's Subcontrary Section Theorem (Conics I.5), every subcontrary section of an oblique circular cone is an exact circle.

3. Exact Symbolic Proofs in geometor.ptolemy.stereographic

In geometor.ptolemy.stereographic:

  • prove_circle_preservation_symbolically() proves algebraically over $\mathbb{Q}(a, b, c, d, x, y)$ that $aX + bY + cZ + d = 0$ on $S^2$ maps to $(x + \frac{a}{c-d})^2 + (y + \frac{b}{c-d})^2 = \frac{a^2 + b^2 + c^2 - d^2}{(c - d)^2}$.

  • prove_conformal_metric_symbolically() computes the exact Jacobian Gram matrix $J^T J = \frac{4}{(1 + x^2 + y^2)^2} I_2$ ($g_{xx} - g_{yy} = 0$, $g_{xy} = 0$).

  • construct_astrolabe_planisphere() executes Ptolemy’s folded-meridian compass-and-straightedge construction in geometor.model, proving $\angle MDT = 90^\circ$, incidence of $B$ and $D$ on cMBTD, and $EL \cdot EK = ED^2$ with zero Float atoms.