Antipodes are points on a sphere that are exactly diametrically opposite. If we assign latitude and longitude in the usual RE manner, the co-ordinates of the antipodes are (λ, φ) and (–λ, φ ± 180º), where we take the sign that makes the result between -180º and 180º.
What is interesting for antipodes on a sphere is that the shortest path between them is along any great circle on which they lie. There are infinitely many great circles that fulfill this condition, among which, there is always the one that also passes through the North Pole.
The situation is quite different on a FE. While it is easy to estimate the longitude of a point, since the meridians are the rays passing through the North Pole, and the longitude is the angle between the Greenwich meridian and the meridian passing through the considered point, the situation is quite different for the latitude. Even though it is said that the Equator is a circle exactly half the distance between the North Pole and the Ice Wall and the “parallels” (lines of points with constant latitude) are concentric circle with the North Pole at their center, there is no described procedure for determining the latitude of different points. Nevertheless, this is sufficient for our further discussion. We shall simply impose the condition that the antipodes lie on the same “parallel”, or have the same latitude. But then, the following equation must hold:
λ = –λ => λ = 0
Points with zero latitude (RE) are on the Equator.
Now, the shortest distance between the antipodes on a FE is a straight line which always passes through the North Pole, and it is the only one. For antipodes on the Equator, this distance is equal to the FE radius (whatever its value). However, the distance along the Equator is a semicircle and is π/2 times greater than the previous one.
You can see the discrepancy between the two models. This would imply that, if the Earth had indeed been as described by the FE model, then the time that it would have taken to get from one equatorial point to its antipode along the Equator would have been about 1.57 times greater than along the polar route, assuming the same speed. On the RE, on the other hand, these times are equal.
So, the experiment is the following. Fly from one equatorial point along the polar route to its antipode and then head back along the Equator. If the two times are equal, the Earth is round. If the ratio of the second to the first is approximately 1.57, than the Earth is flat.