So I heard the Greeks established the distance to the moon and the roundness of earth through parallax and the different of angles the measure when the sun was in its zenith due to the shade it made on a stick a two different places. Yet this also happens on flat land, or if the earth where flat, so how was it? Was it that it the shade provoked on a stick on flat surface is actually a little bit different than that one provoked on a curved surface, and did they find this doing this or how did they find about the curvature of the earth exactly?
I'm not sure about the distance to the moon, but I can explain the rest...
Lots of questions here...
First, they found out about the curvature of the earth mainly by observing the earth's shadow on the moon during a lunar eclipse. You can easily see the Earth's shadow as curved and not straight.
The experiment with the sticks, performed by Eratosthenes, was to determine the circumference of the earth, and thus, how big it is.
You're absolutely right with the argument that the shade produced by a stick on a flat surface would be different than that on a curved surface.
Eratosthenes went with the [correct] assumption that the sun was so far away, that its light rays striking the earth would be parallel to each other.
In truth, this only occurs for an object that is infinitely far away, but the sun is SO far away, that this difference, from two different places on Earth, is negligable.
So if you have a flat surface, then no matter where you are on the surface, the shadow cast by a stick should be the same. If its noon, and the light rights are parallel, and the Earth is flat, then there should be no shadow no matter where you are on the surface.
Now, Eratosthenes saw that at noon, in two different cities, that in one city, there was no shadow cast by the stick, meaning the sun was directly overhead, and at the same time in the other city, there was a small shadow, and measured the angle the stick made with the shadow.
He noticed the angle was about 7 degrees, which means its 1/50th of a full circle (360 degrees). Therefore, the two cities were 1/50th the way around the globe. Knowing how far the cities were, he multiplied their distance by 50 and got the distance around the globe. c = 2*pi*r, so he could then calculate the radius of the earth as well!
That's pretty much it, hoped you understood it. Its really amazing how they could get the size of the earth simply by using two old sticks and a little thought! :)