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stevendaryl said:If you didn't know that the world was round, and all you knew was the above map, then you might be puzzled as to why airplanes naturally take the curved path, instead of the straight path. The two possible explanations are: (1) the world is round, and the Great Circle route is the shortest path, or (2) there is some mysterious force that causes the paths of airplanes to tend to curve to the south when they travel east to west.
It's possible locally (just considering the region shown on the above map, not the whole world--if you consider the whole world, the fact that you can sail completely around the world proves that it's not flat) to make the second explanation work.
I don't think I can agree with this as written, depending on one's interpretation of "make it work".
One can detect the curvature of a sphere by measuring the distance between as few as four points (a total of 6 possible pairs, giving six distance measurements).
There was a thread on this a while back, the original idea can be traced back to Synge. One way of describing the procedure is to use plane geometry and the law of cosines to calculate the interior angles of all 4 triangles.
Code:
....A
......
...B...D
.......C
The 4 points are A,B,C,D, by omitting each of the points in turn, one creates 4 triangles, namely ABC, ABD, ACD, BCD. There are a total of six sides, AB, AC, AD, BC, BD, CD.
Using the law of cosines, one can solve for the interior angles of each of the 4 triangles: https://en.wikipedia.org/wiki/Law_of_cosines
$$\cos \gamma = \frac{a^2 + b^2 - c^2}{2 a b}$$
If the sum of the interior angles one calculates by assuming the figure is planar in this manner is not 180 degrees, one knows the figure can't possibly be planar.
If one repeats the process using spherical trig, one needs to modify the law of cosines - and one has the well known relationship that the sum of the angles of a triangle increases with the area of the triangle. For instance, one can draw a triangle on a sphere with three right angles whose area is 1/4 the area of the sphere the triangle is on.