The work of Archimedes was not lost. It continued to exist in the Eastern Empire and throughout the muslim world; it was simply less well known in the West because of a general deterioration of knowledge there. Just because a few well educated people -- many of them religious -- still knew the knowledge of antiquity that doesn't mean they invented it, any more than the muslims [who've also tried to claim credit for it.]
Heliocentricity was perfectly provable with the observations available to anyone. The Ptolemaic system was untenable; only ignormausses who wanted to believe humans were the center of the universe were determined to hang onto it.
The claim that Galileo got himself into trouble because he mocked the pope is both false and silly. Galileo got himself into trouble because when he was put to the question, he refused to back down. He took the correct position against an ignorant and evil group of men.
Even if his mockery of the pope had been the cause of his troubles, so what? The pope is a man and NOTHING MORE. He isn't entitled to any special respect by anyone, least of all someone who had the truth on his side while the pope and his flunkies were trying to suppress it.
Most of the knowledge for which Islam is given credit for preserving was actually kept alive by the Jews in their midst.
This pope had been his patron, and despite his knowledge, Galileo was in many respects a blustering fool. Bellarmine, who had protected him, tried to warn him not assert what few educated men of his time believed, which was that mathematics was a true measure of things in general.
Publicly mocking the powerful is seldom a good idea, even if they're only men.