Aristarchos' half moon
Illustration of Gomez's VA theory on the origin of Aristarchos' 87° angle.
According to Aristarchos, ‘When the Moon appears to us halved, its distance from the Sun is then less than a quadrant by one-thirtieth of a quadrant.’ Heath (1913:352) interpreted these words as meaning that the angular distance between a half moon and the Sun ‘is less than 90° by 1/30 of 90° (or 3°) and is therefore equal to 87°.’ This interpretation is correct, since this is the angle that puts the Sun 18 to 20 times as far away as the Moon, in agreement with what is said of Aristarchos in On Sizes (Heath 1913:376) and in the Sandreckoner (1.9).
Why Aristarchos should choose this angle becomes apparent when noticing that one-thirtieth of the Moon's disc is one arc-minute (since the Aristarchan Moon is thirty arc-minutes wide), and one arc-minute is the typical resolution of the human eye. So bare-eyed humans can't tell a half moon to better than one-thirtieth of a quadrant, or 90° ± 3°.
In the present, simplified model, however, one arc-minute seems to translate into something closer to 2 degrees (instead of the expected 3) when converted into spherical degrees. (Notice that the dotted lines halving the Moon here represent a gap of one visual arc-minute and this gap doesn't match a lunar elongation of 87 degrees exactly; it rather seems to match an elongation of 88 degrees more closely.)
This is because the present model assumes the observer to be at the centre of the Earth. In reality, observers are on the surface of the Earth, and from here this tiny mismatch is cancelled out almost completely by the effect of the Moon’s daily wobble (which is caused by the Earth’s rotation). So Aristarchos, according to the VA theory, correctly derived a (3-D) elongation of 87 degrees from a (2-D) visual arc-minute gap on the lunar disc.
His half-moon experiment was designed to prove beyond doubt that the Sun is many times farther away than the Moon, rather than to give a definite figure for the Sun’s distance. As for the latter, his ultimate take was reported in the Sandreckoner (2.1) to be somewhat ‘less than ten thousand Earth radii.’