Illustration of Hypothesis 4 of the book On Sizes.
Hypothesis 4 of On Sizes states that when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant [that is, it is less than 90° by 1/30th of 90° or 3°, and is therefore equal to 87°].
Aristarchos' 87° value occurs when the sun and moon are placed each at a distance of 1146 and 60 earth-radii away from the earth.