As far as the Lunar Eclipse goes, there is no evidence that the shadow which manifests on the moon during a Lunar Eclipse originates from the earth at all. That shadow could come from any celestial body intersecting the light between the sun and moon.
The only reason the Greeks and ancients (and modern astronomers) are able to predict the Lunar Eclipses was because the predictions are based on recurring charts and tables of past eclipses. It had nothing to do with the shape of the earth or the actual geometry of the cosmos. The Lunar Eclipse is a phenomenon which comes in patterns. By studying these patterns it is possible to predict when the next transit or eclipse will occur. The astronomer can use historic charts and tables with a few equations to predict the time, magnitude, and duration of a future eclipse.
This does not apply only to the eclipse, either: All recurring phenomena such as the transits of planets, occultations of bodies, and precision of paths across the sky are predicable only because they are phenomenons which come in patterns. Astronomers predict celestial events by studying the patterns and predicting when the next occurrence will occur.
Modern Eclipse Models
Q. But what about NASA's yearly lunar eclipse predictions? They must be based on a geometric model, and not simply cycles in the sky, surely?
A. Actually, NASA freely admits that they use ancient cycle charts for their eclipse predictions. The Saros Cycle and those cobby old ancient methods which simply look at past patterns in the sky to predict the next one is precisely how "modern theorists" predict the lunar eclipse today.