Celestial illusions and ancient astronomers: Aristarchus and Eratosthenes

When the moon is half, one would expect that a line starting from the moon’s center and being perpendicular to the “shadow diameter” would, if extended, go through the center of the light source, namely, the sun. It turns out that, when the sun is visible, this extended line appears to aim significantly above the sun, which is the essence of the “half-moon illusion”. The explanation advanced here is that this is not an optical illusion; instead, it can be explained by the relative sizes and distances of the earth, moon, and sun, and it hinges on the fact that the sunrays are nearly parallel with respect to the earth-moon system.

It turns out that the ancients knew and used this near-parallelism of the sunrays. Eratosthenes, for example, used a simple but ingenious scheme to obtain a good estimate of the earth’s circumference. An interesting question is: How did the ancients arrive at the conclusion that the sunrays are nearly parallel?

This was probably a corollary, based on the immense size of the sun and its huge distance from the earth, as estimated by, among others, Aristarchus of Samos by a brilliantly simple method.