Sphaerica

Sphaerica, also known as Sphaerica sive de sphaeris (On the Sphere), is a classical treatise on spherical geometry and astronomy from antiquity. Traditionally attributed to Claudius Ptolemy, though some manuscripts present it as part of his Almagest corpus, it was likely composed in the 2nd century CE and transmitted through Greek, Arabic, and Latin versions.

The treatise introduces the geometry of the sphere, including definitions of the sphere, great circles, and angles between arcs. It develops methods for solving problems posed on the celestial sphere, particularly spherical triangles formed by great circles. It employs chord-based trigonometry (the predecessor of modern trigonometric definitions) and derives relationships analogous to the plane laws of sines and cosines, adapted to spherical geometry. The work also addresses practical applications to astronomy, such as determining declinations, ascensions, and the coordinates of celestial bodies, as well as the effects of diurnal rotation.

Sphaerica has a long textual history: it circulated in Arabic translations during the medieval period and was later rendered into Latin, becoming an important source for Islamic astronomers and for European scholars studying celestial mechanics before the advent of modern trigonometry.

Overall, Sphaerica is regarded as one of the foundational texts in the history of spherical geometry and astronomy, providing essential mathematical tools that influenced later developments in both Eastern and Western traditions.