On how Aristarchos might have measured the Moon's phase angle (or Sun-Earth-Moon angle).
This file illustrates how the Sun-Earth-Moon (SEM) angle could be measured with naked-eye technology and without hurting your eyes. The method uses a polished surface (or a mirror) in order to reflect the Sun’s light towards the Moon. Viewing the Moon through a hole in a high wall or in the ceiling might help to point the Sun’s reflection in the right direction. A simple formula is needed (and provided in the attached file) to translate the angle formed by the polished surface and the reflected beam into the SEM angle we are after. If the former angle can be measured, the latter follows from the formula.