I must bring forth the fact that radio waves can't be sent from USA to China directly, without bouncing them off satellites or the ionosphere. This is due to the fact that the radio waves go off on a tangent line to the surface and out into space and do not circle the Earth. This wouldn't be so on a flat Earth. On a flat Earth, you could send a signal straight to China (assuming you had sufficient ground clearence).
Changes in Polaris' angle above the horizon with distance traveled north or south only make sense if the path is an arc of a circle, or very nearly so.
Land surveyers use plane geometry which doesn't work over large distances. The methods of spherical geometry, however, do fit the surface of the earth very closely. For example, if you survey a large triangle, the sum of the interior angles is greater than 180 deg.
If the Earth were flat, then no matter how far away the sun was, there would never be an Arctic Circle. Draw a horizontal line on a paper. Now mark a point above that line to represent the god Mercury in his chariot crossing the sky at noon. All points along that line would have a direct line of sight to the sun god. Not only that, but from two different points, you would be able to determine how far up Mercury's chariot was.
Get a peice of paper and try this...
Start with point A and point B at a distance d to the north. A makes an angle theta with the sun, and B makes an angle phi. The angle between the hypotenuse of A and the hypotenuse of B is phi - theta. By trigonometric Law of Sines, you should be able to compute the hypotenuses (hypoteni?) of both A and B. And from there, by basic trig, you can compute how high up Mercury's chariot is. If the assumption that the Earth is flat is correct, then you should be able to take another point C still further north of B and run the numbers and come up with the same distance. If the numbers don't add up, then the hypothesis is flawed.
DO YOU REMEMBER GEOMETRY IN HIGH SCHOOL? WHY DOES SPHERICAL GEOMERTY FIT THE EARTH SO WELL.......THINK!!!!