The earliest recorded traces of geometry among the ancient Babylonian and Egyptian cultures place its origin in the practical problems of the construction of buildings (temples and tombs), and the administration of taxes on the land (Katz, 2008). Such problems were mastered by the scribes, an educated elite in these cultures. The word geometry is of Greek origin, γ∊ωμ∊τρíα, to “measure the earth.” Geometric ideas were collected, transformed by rigorous reasoning, and eventually developed by Euclid (ca. 300 b.c.e.) in his great work The Elements, which begins with the geometry of the plane, the abstract field of the farmer.

Another source of ancient geometric ideas is astronomy. The motions of the heavens determined the calendar and hence times for planting and for religious observances. The geometry at play in astronomy is spherical geometry, the study of the relations between figures on an idealized celestial sphere. Before discussing Euclid's work and its later generalizations, let us take a stroll in the garden of spherical geometry where many of the ideas that will later concern us arise naturally.