re: The Earth is Flat

I'm going to assume you're familiar with them.

We have 2 theodolites greater than one mile apart. We set them plumb at each location. Then we go to one of them. From here we measure from straight up, down to the 2nd theodolite and record that angle. We then go to 2nd teodolite and do the same back to the first.

We sum those angles. Now this measurement does not assume a shape, so we cannot be begging the question of the shape of the surface. We just need to know basic geometry.

Which means that no matter what the altitude difference is between the 2 stations that a flat earth would make it to where the sum of these angles would be 180. If the Earth were concave, then it would be less than 180 degrees.

But the Earth always measures greater than 180 degrees using this measurement, at all locations and this is not direction dependent. Meaning that the shape of the earth can only be roughly spherical.

Addendum: don't forget the average condition of the atmosphere favors light to bend down on average, which would not create an illusion of downward curve but rather make a flat earth look concave.