That's exactly what I mean. The important phrases here are:
The detailed prediction of eclipses requires complex calculations of the celestial mechanics of the Moon's orbit, but amazingly good approximate solutions to the problem of predicting eclipses were known to the ancient Greeks and Babylonians.
Just what I said, pattern-fitting can only go so far.
and
the resulting solar eclipse would be visible about 1/3 of the way around the world
Modern methods can provide an exact path for the lunar eclipse, rather than "about 1/3".
and
...another solar eclipse would occur at the same approximate range of longitudes on Earth. This time, however, the next eclipse in the 54 year cycle would be some distance further north or further south of the previous track, depending on whether or not the eclipses happen on the descending node (Moon going south as it crosses the ecliptic) or ascending node.
As you can see, the Saros cycle can only deal in approximates. Even knowing whether it is going to appear North or South of last time requires orbital mechanics.
From the link you provided:
"Let A, B, R, (in the following diagram) be a section of the earth's shadow at the distance of the moon; S, n, the path described by its centre, S, on the ecliptic; M, n, the relative orbit of the moon; M, n, S, n, being considered straight lines. Draw S, o, perpendicular to S, n, and S, m, to M, n; then o, and m, are in the places, with respect to S, of the moon in opposition, and at the middle of the eclipse.
"Earth's shadow at the distance of the moon?" "ecliptic"? "orbit of the moon"? "moon in oposition"? These are all RE ideas based on the moon orbiting the Earth.
In FET the Earth does not cast a shadow towards the moon, there is no ecliptic plane, the moon does not orbit the Earth, and thus cannot be in opposition [to the sun].
Furthemore, variables are ill-defined, for example:
h = the moon's horary [hourly] motion in the relative orbit.
This does not tell me what
h is, or what units it is measured in. I can only guess it means the distance the moon moves in an hour, but this assumes that the moon's orbit is circular, and doesn't say what units
h is measured in.
It is bad maths and bad physics.