Source edition
Euclid. Euclidis Opera Omnia, Volume 8. Menge, Heinrich, editor. Leipzig: Teubner, 1916.
Source data
Open Greek and Latin · CC BY-SA 4.0
Cloned and adapted by Humanitext, with ongoing edits.
Summary
This work is a mathematical treatise on spherical astronomy, discussing the geometric proofs related to the motion of the ecliptic and parallel circles on the celestial sphere. The primary subject is the rigorous proof of how different arcs of the ecliptic rise and set over varying durations of time. The treatise begins by demonstrating through contradiction that when one point of a diameter on the ecliptic rises, the opposite point sets. It then establishes the relations between the horizon, the equator, and the tropics to compare the setting times of equal arcs along the ecliptic. By offering alternative, clearer proofs for existing astronomical propositions, it demonstrates that arcs closer to the summer solstice take longer to set than those closer to the equator. Finally, the work concludes with a geometric proof showing that for two equal and opposite arcs, the time one takes to leave the visible hemisphere equals the time the other takes to leave the invisible hemisphere.