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.
Starting with number 7), tell me where I go wrong.
The moon is 0.5 of a degree wide (give or take) when seen from earth. This means that if you drew a moon next to a moon next to a moon all the way around its orbital path you could draw 720 moons.
You say that the navigator necessarily has an error when finding the centre of the moon. Could you expand on that? If I had, say, a circle cut in a piece of metal that (give or take) represented the moon, then by positioning the moon within that circle, the plumb line would cut straight through the centre and down...so one could see the stars in another cut out, maybe one with degree markings--even a grid of some kind (fine hairs stretched tight?)
So you are saying that the human eye-moon relationship has a necessary innacuracy--error-of 1/27.4th of the width of the moon?
I'm thinking of a network of astronomers, spreading out from central points, so the mariner could arrive in a port and know "the time" at that port and how it related to their home port. Don't fight forces, use them R. Buckminster Fuller.
(For other lattitudes, almanac dots can be routinely adjusted.)
If you measure position of the moon at fixed solar moments, you are more lucky, since synodic month is 29.53 days long, so daily moon positions from a reference point (London) differ by 360/29.53 = ~12.2 degrees. You can be 12 times more precise. Better still, measure Moons position at a lunar moment (moonset or so) - sidereal month is 27.32 days, so you have ~13.2 degrees interpolation spans. Isn't this what Crichton tries to do?
Can the difference between sidereal and synodic months employed? I doubt it. It seems to me that you have to employ the fastest movement (discarding Earth's rotation) across the skies. Apart from Sun and Moon, we have Venus moving in 95.6 degrees span around the Sun in 224.7 days twice, Mercury moving in 56.6 degrees span in 88 days, and much slower planets. The average angular speed for Venus and Mercury (relative to the Sun) is 0.9 or 1.3 degrees per day, with fastest movement across Sun's disk unobservable - that's hopeless.
If our civilisation is about to collapse, with technology setback for long centuries, it would be smart to launch a couple of "bright" satelites with the lapse period of exactly 12 hours (or maybe better, half of sidereal day, 11 hours 58 minutes, 2 seconds), with the most conveninet orbit circular equatorial (or ecliptic, respectively).
