Greek & Latin Mathematical Glossary

Technical mathematical and astronomical lexicon for Almagest Book I and Planisphaerium.

Μαθηματικὴ Σύνταξις (Syntaxis Mathematica / Almagest)

Ptolemy’s original Greek title (“Mathematical Systematic Treatise”); later styled ἡ μεγίστη (“the greatest”), whence Arabic al-Majisṭī and Latin Almagestum.

τμῆμα, pl. τμήματα (part / unit of diameter)

One of the $120$ equal parts ($120^p$) into which Ptolemy divides the diameter of the reference circle (Almagest I.10), making the radius $R = 60^p$.

μοῖρα, pl. μοῖραι (degree of arc)

One of the $360$ equal divisions ($360^\circ$) of the circle’s circumference.

ἑξηκοστόν (πρῶτον, δεύτερον) (sexagesimal sixtieth)

Positional base-60 fractional place ($p; a, b = p + \frac{a}{60} + \frac{b}{3600}$) replacing awkward Greek unit fractions (μοναριασμοί).

εὐθεῖα ὑποτείνουσα περιφέρειαν (chord subtending an arc, $\operatorname{crd}\theta$)

The straight line joining the endpoints of an arc $\theta$ on a circle of diameter $D = 120^p$: $\operatorname{crd}(\theta) = 120\sin(\theta/2)$.

ἄκρον καὶ μέσον λόγον (extreme and mean ratio / golden section $\phi$)

Division of a segment $\Gamma Z$ at $\Delta$ such that $\Gamma Z : \Gamma\Delta = \Gamma\Delta : \Delta Z$ (Elements VI.Def.3); in Almagest I.10, $\Gamma\Delta = R = 60^p$ and $\Delta Z = R/\phi = s_{10}$.

τετράπλευρον ἐν κύκλῳ ἐγγεγραμμένον (cyclic quadrilateral)

A four-sided figure inscribed in a circle; governed by Ptolemy’s Theorem ($AC \cdot BD = AB \cdot CD + BC \cdot DA$).

διὰ τῶν γραμμῶν (by means of lines)

Exact geometric construction; Ptolemy notes (Almagest I.10, p. 42) that trisecting $1\frac{1}{2}^\circ$ to obtain $\operatorname{crd}(\frac{1}{2}^\circ)$ cannot be achieved διὰ τῶν γραμμῶν.

κατὰ σύνθεσιν (by composition)

First form of the Menelaus compound-ratio theorem (Almagest I.13, p. 69): $\frac{\Gamma A}{AE} = \frac{\Gamma\Delta}{\Delta Z}\cdot\frac{ZB}{BE}$.

κατὰ διαίρεσιν (by division / separation)

Second form of the Menelaus compound-ratio theorem (Almagest I.13, p. 69): $\frac{\Gamma E}{EA} = \frac{\Gamma Z}{Z\Delta}\cdot\frac{\Delta B}{BA}$.

λόξωσις τοῦ διὰ μέσων τῶν ζωδίων κύκλου (obliquity of the ecliptic, $\varepsilon$)

Angle between the celestial equator and ecliptic; measured in Almagest I.12 as $\varepsilon = \frac{11}{166} \times 360^\circ \approx 23^\circ 51'20''$.

Planisphaerium (Ἅπλωσις ἐπιφανείας σφαίρας)

Ptolemy’s treatise on stereographic projection of the celestial sphere onto the equatorial plane.

circulus declivis / signifer (oblique / zodiacal circle)

Latin term in Planisphaerium §1 for the projected ecliptic circle mbtd bisecting the equator at b and d.

ὑπεναντία τομή / sectio subcontraria (subcontrary section)

Anti-parallel plane section of an oblique circular cone (Apollonius Conics I.5), proving that circles on $S^2$ map to circles under stereographic projection.

almucantar (al-muqanṭarāt)

Small circle of constant altitude $h$ above the horizon, projected stereographically onto an eccentric circle on the astrolabe plate.