Eratothenes heard a report that at noon on the longest day of the year, shadows of columns disappeared in Syene, Egypt, indicating that the Sun was directly overhead.
Realizing that this did NOT occur in Alexandria where he lived (about 800 km away), he postulated that a curved (rather than a flat) Earth was required, and even computed the diameter of the Earth to within a few percent!
Eratothenes reasoning went like this.
If the Sun is really far away, the light from it arrives almost parallel to itself at Earth. So if the sun's rays produce no column shadows in Syene, but column shadows at Alexandria, the columns at Alexandria must be tilted with respect to those at Syene.
Eratosthenes' evidence was NOT enough to prove the Earth was a sphere (though he made that correct deduction), just not flat.
Actually, Eratosthenes' argument depends on the Sun being very far away from Earth, which was not known at the time. If the Sun was closer, the columns at the two locations could have different shadows and the Earth could still be flat.
So Eratosthenes' argument, brilliant as it is, relied on unverified assumptions, and was fundamentally flawed. Something we'll have to be careful of ourselves...
In 1611, Galileo met Maffeo Barberini, a prominent cardinal who would become Pope Urban VIII in 1623
In 1616, the Church bans as heretical teachings of the heliocentric theory as being in fact true, "which appears to contradict Scriptures": hypothetical discussions of it are still permitted.
In 1623, Galileo publishes "The Assayer" whose title page shows the crest of his patron, Barberini.
In 1632, Galileo comes to Rome and has 6 interviews with the new Pope: the Pope subcribes to the Earth-centred theory but Galileo is given permission to discuss the Sun-centred theory (but still only as a hypothesis.)
In 1632, Galileo publishes his famous book Dialogues Concerning Two Chief World Systems, written as a conversation between three characters, Salviati, Sagredo and Simplicio. The Earth-centred system is defended by Simplicio, an obvious fool who loses every point of the debate, using arguments Barberini himself had presented to Galileo during their discussions.
Soon thereafter, Galileo was questioned by the Inquisition, forced to recant his views, and placed under house arrest for the rest of his life. He died in 1642.
Though there was effort on the part of some in the Church to pardon him, Barberini steadfastly rejected all pleas on his account.