But you are not observing at the same time, you are observing at the same local time. So, you trade off the longitudinal parallax error from (8) for the error in the rectascension of the sun (5). "It's the statue, man, The Statue."
Parallax wouldn't count if you compared local times at the differing absolute times when the Moon was in the same position in the sky. Then you also need the absolute time difference. That absolute time difference is (ignoring point 10) proportional to the longitude... *Lunatic*, n. One whose delusions are out of fashion.
I thought about the method's details. On one hand, it can be used to establish a first approximation of latitude without an initial guess: the apparent motion of the Moon between the moment of the observatory local midnight and the ship getting into the same position (relative to the stars) on the same day is roughly the same as the latitude difference per sidereal month in sidereal days. You don't even need to correct for the Sun's motion relative to the stars.
However, this time, we have another parallax, not because of differing observing positions, but because we aren't watching the Moon from Earth's center, but its surface. When the Moon is near the horizon at the equator, the distance from the Meridian exceeds the Moon's geocentric position angle from the same Meridian by those up to 61 arc seconds [somewhat less when the Moon is not in apogeon].
Once we have the more or less correct position angles (the Moon's changing distance from the Earth means an error of up to c. 10% in the parallax, up to 7'), we still have the bigger problem of the elliptic nature of the Moon's orbit. Near the perigeon and apogeon, the Moon's angular speed is more than 10% faster/slower, assuming constant speed can mean up to 1.5 degrees error [when the ship is at latitude 180°].
So, for precise position determination, a) the almanach-makers should be aware of the elliptic nature of the Moon's orbit, b) the navigator must be supplied with a data series or formula (epicycles?) giving the time differences corresponding to the Moon's location, c) the navigator must be able to calculate parallax.
Realistically, I wouldn't expect a predicted Moon position precision for the ancients better than 15'. That would convert to 6.85 degrees precision in latitude, or 760 km at the equator. *Lunatic*, n. One whose delusions are out of fashion.
