Personally - I say this debate is won...the Flat Earthers no longer have a leg to stand on, and they should have the guts to admit it.

3DGeek,

Thank you for this thread. I read this forum occasionally mostly for entertainment, but this thread seemed to be different than a lot of others because it promises a mathematical proof of a round earth, rather than one that relies on observations and evidence that can be ignored by FE’ers. The only evidence that needs to be accepted is that flight times are not regularly wrong by 2 or 3 times the expected values, and that aircraft cruising speed is not wildly off of the expected speeds. Using these two nearly indisputable assumptions, you have given a calculation of the distances between cities on earth. Finally, you used these distances to show that no flat map can produce the correct distances.

While I do appreciate your argument, I’m not sure that it is the airtight argument you claim. It rests on one more assumption, which is that speed = distance * time is a good representation for the speeds, distances, and times at hand. For a plane, the ‘time’ variable is flight time and the ‘speed’ variable is average aircraft speed, so the ‘distance’ variable is the path length of the flight. You have made the assumption that flights travel in straight lines along their path. You must make this assumption in order to construct your mathematical proof, as your proof rests on an analogy between the sides of a quadrilateral and the paths of flights.

Instead of finding the “honest to goodness” distances between cities, I believe you have found upper bounds on the distances between cities. These cities can be separated by at most the distances you claim, assuming that flights travel in straight lines, or minimum distance paths. However, flight paths can always be devised that are longer than this minimum. Flights could circle cities for hours in order to extend flight time.

You may ask, “Why would airlines do this? This would be horribly inefficient and waste passengers’ and crews’ time.” I agree. However, this no longer constitutes an elegant mathematical proof free of assumptions, which is the reason I believe this thread is interesting.

One easy explanation that FE’ers might use is that the “Aether Wind” is blowing aircraft more significantly in the northern hemisphere, so when planes fly between these cities, they follow a nearly circular path between them, extending your expected distances by a factor of pi/2, or about 157%. Southern hemisphere flights, on the other hand, are less affected by the wind when in the southern hemisphere, and so have flight paths that are much closer to straight lines. This gets us half way to your 300% correction. The other half can be explained by the “Aether Wind” also causing altitude fluctuations of the plane in the northern hemisphere. These altitude fluctuations could cause planes to fly upward and downward sinusoidally, which would in fact not require a large amplitude to get the remaining correction that you have stated. This is just one of many examples of how planes could fly other than in straight lines that would allow for the flight times we observe on daily flights.

While I’m not trying to encourage FE’ers, I’m still curious to see an airtight mathematical proof, and I am not convinced that this qualifies.