Ancients and Unobservable Lunar Eclipses

I have recently been wondering if ancient (pre-telescop)
astronomers/astrologers would have been able to predict upcoming lunar
eclipses which they would not have been able to see because the eclipse
would have happened while it was daytime for them.

I am leaning towards thinking that it would have been possible (as in
allowable) for the more sophisticated a/a's. It would have required only
four things.

1. Motive. They would have had the impetous to keep track of it because of
it's great importance in their belief systems.

2. The concept of a lunar eclipse being the moon passing through the
earth's sun-shadow. Would they have been aware of this phenomon, even if
they assumed an earth-centric system? I am thinking, yes.

3. The ability to track the invisible earth's shadow on the ecliptic. This
ought not be too difficult.

4. The practice of tracking the moon from evening to morning and projecting
it through the sky on a short-term basis.

Are there more items in the list that I have not included?

Am I being too gracious in supposing what their capabilities could have
been. Recall that I am asking about any society at any time before the
invention and spread of the telescope.

Any comments about the plausibility or weakness of my suppostion are
invited.

Thanks.

- Paul