Proving that the Earth is flat has to be done theoretically. That's what all proofs are, in reality. In any of the seven disciplines. The Earth can be proved flat in a fallacious geometry. It's the one Euclid defined and expounded on, minus the fifth postulate, and adding (which can be derived in absentia of postulate five) one triangle similarity proof. It was Rimann who first subtracted the parallel postulate and Lobachevsky who first added Angle Side Side. At least most people think so. It's "modern math" or in the lyrics of the profound (and maybe a bit profane) Tom Lehrer, New Math. Lehrer also thinks that Lobachevsky plagiarized, and I've heard attempted proofs before, but they all rely on grammar, and are not specifically geometrical.