I've discussed this for the middle ages here, although Martianus Capella wrote before the canonical fall of Rome and Bede heavily cites Augustine to make his point. Also there is a link to an earlier thread where /u/jschooltiger discusses it briefly for the classical greeks as well. If I remember, I can have a look for some earlier references tomorrow.

Also, you are right that it is not easy to prove or rather measure, but their solution was to record the time of eclipses and note the difference in time, and indeed occurrence, east and west.