Alright! Over to the bending light issue.
The problem is with the suggestion that as we move away from the earth the "flatness" becomes round by bending light. This makes no sense what-so-ever. The light from background stars "behind" the Earth are not bending or warping around the earth. The Earth doesn't nearly have enough of a gravitational imprint to bend light in the same degree over such a short distance as required by this "flat earth seen-as-a-sphere" theory.
So, if the light from the stars isn't warping, then the light from the Earth's surface isn't warping either. In other words, if the Earth was a disk, it would STILL APPEAR AS A DISK!!!
Since the Earth appears to be a sphere, then it is, in fact, a sphere.
From a historical perspective on this subject, as stated before, the world was first proven to be round by measuring the angles of shadows that sunlight cast on a staff at different latitudes. The fact that the angles got shorter at lower latitudes indicated that the angle of the surface of the earth was changing, or curving.
Now that being said, if you subscribed to the "flat earth" theory, the way you could argue against those conclusions is by challenging the premise: that the sun is very far away. The premise states that if the sun is very far away then the rays of sunlight are parallel everywhere across earth. If they are parallel then the only explanation for different shadow lengths is curvature of the Earth's surface.
On the other hand, if you argued that the sun was very close, then you could explain the different shadow lengths by the close position of the sun. For example, if there is a light in the center of a room (on the ceiling), if a stick is in the center of the room, the shadow is short, as you move out to the edge of the room across the flat floor, the shadow lengthens. That would be your "flat earth" argument.
The problem with this is argument is that through triangulation, we can prove that the sun is actually VERY far away. It's an indisputable mathematically proven, repeatable observation. So, with even this simple and solid test, the Earth is proven to be round.
