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?
Best Answer
He ignored the radius of the Earth as negligible. His estimates for the angle were from the shape of the shadow the sun casts on the moon, and the difference between this and a straight line when the moon is halfway between full and new is too small to percieve precisely. He fooled himself into thinking he measured a different angle, so his estimate was really only giving a lower bound on the distance to the sun. As a lower bound, it was enough to establish that the sun is larger than the Earth, and this was important, in that it lent strong support to heliocentric models. But it was not an accurate method.