1. explain why eratosthenes' method would not work if earth were flat.

Eratosthenes lived in the city of Alexandria (northern Egypt) and he knew that the Sun was never directly overhead at this location, even on the Summer Solstice. However, he knew that on Summer Solstice, in a town located in southern Egypt and named Syene, there was no shadow at the bottom of a well. This meant the Sun was directly overhead in the town of Syene at noon just on that day.

So, taking into account the sun is radiating to both places at the same time, but in Syene there is no shadow and in the second place,Alexandria, a shadow is observed, the logical explanation  Eratosthenes concluded is that this must be a curved surface. With this in mind, Eratosthenes calculated the circumference of the Earth.

Therefore, if the Earth were flat, at both places the shadow would have been the same, and the calculation of the curvature of the Earth would have not been possible.