An argument against a flat earth

The flat Earth model can be refuted using a couple of simple observations that are easily checked.

First observation: the sun is exactly the same apparent size in the sky everywhere in the world and at all times of day. Where ever you are, as the sun crosses the sky from horizon to horizon it stays the same size.

Second observation: The angle of incoming sunlight changes very slowly as you move across the Earth. If you are at a place where the sun is directly overhead then the angle of incidence of the sunlight is 0°; angle of incidence is measured against the vertical. At a point on the Earth 500 miles away the angle of incidence will be 7°. Between those two points on the Earth the angle of incidence will have changed gradually.

If the Earth is flat then the sun would have to be about 4000 miles directly above the point on the Earth where the angle of incidence is 0°, that is the place where the sun is directly overhead. However, if the sun stays exactly the same size over the course of a day that means its distance can’t change significantly. So if the sun was 4000 miles overhead in the middle of the day it would have to be 4000 miles directly across land at sunset. Apart from the fact that this is absurd the sun would also only stay the same size in exactly one place on the Earth. Everywhere else its size would change hugely, in fact it would be landing on the ground in a couple of locations.

Suppose the sun was 4000 miles directly above a point on a flat Earth and followed a circular path centred at the North pole; its size would change significantly over the course of a day. Assuming the Earth is a flat disc with a diameter of 25 000 miles, the sun would move in a circle with a radius of about 6000 miles or so. If the sun happened to be directly overhead, then 12 hours later it would be 4 000 miles directly above a point on the Earth 12 000 miles away.  The sun would be about 3 times further away than it was when it was directly overhead which means it would appear to be 3 times smaller in the sky.

The only feasible explanation for the sun being the same size everywhere on Earth is that the sun is very big and is a long way away. However, if the sun is a long way away then all the light from it must be parallel. The angle of incidence of sunlight would be exactly the same all over a flat Earth at any given moment; it would be the same time of day everywhere.

There is only one model that fits the observed gradual change in the angle of incidence of sunlight and the unchanging size of the sun; that is a round Earth with a large sun a long way away.

That pretty well sums it up.

If any flat-earth proponents answer, it will probably be to suggest that the behavior of light can't be trusted; it bends (in unknown ways for unknown reasons) and causes the apparent size of the sum to remain constant. While it may be possible to explain some given phenomenon like rising and setting celestial objects with one particular behavior for light, it's hard to imagine the same principle applying consistently to other easily-observed phenomena (like the sun not changing apparent size and shape as it moves across the sky).

A flat earth may have some appeal as a simpler model than a spherical one, but it's not really simpler even though it may seem so on first glance. Once you start looking into details, everyday phenomena become devilishly difficult to explain systematically.