My text says that Aristarchus (310 BC – ~230 BC) measured the "angle subtended by the Earth-Moon distance at the Sun" ($\theta$ in the figure below) to establish the relative Earth-Moon and Earth-Sun distances.
I understand that he must, in fact have used the Moon-Earth-Sun angle, and then subtracted that from 90° to arrive at $\theta$; but how did he establish the Moon-Earth-Sun angle? The reference points for all three objects is their centers, yet what Aristarchus must have in fact measured was the angle between the Moon and the Sun at the surface of the Earth.
Did Aristarchus take this discrepancy into account in his calculations? If so, how?