Astronomy is based on observed patterns. Observed patterns of the moon, observed patterns of the stars. Things are only predictable because they come in patterns. It is possible to create an equation to express those patterns, but they are only valuable in that they might produce a right answer.

read the literature i linked more carefully. here are the equations of motion:

this is pretty much exactly how eclipse tables are calculated. the nasa page you always link says this itself at the bottom.

The problem is a good deal more subtle than that. We know what the equations are - but we cannot solve them for more than two bodies. So we can say (for two bodies) - here are there masses, here is where they are in space - here is an equation into which you can plug any future time and the equation will tell you the location of the two bodies. THAT can be solved.

However, for three or more bodies - no such equation is possible (it can actually be proven to be impossible). Hence you cannot write an equation to tell you PRECISELY when the next eclipse will happen because the sun, moon and earth constitute a three body system.

BUT this doesn't mean that we use "patterns" to figure it out. We simply don't use a single equation - we can do one of two things:

1) We can calculate the Earth/Moon orbits precisely as a "two-body problem" - then we can calculate how "EarthMoon" orbits the Sun as a two-body problem. The solution isn't exact - but it's good enough to predict eclipses to within a fraction of a second over a century. If we continually correct the data after the time of each eclipse is measured - then the results will be essentially perfect.

2) We can use "numerical integration". So we calculate the two-body solution above over a time-step of (say) 1 second. Over such a tiny time interval, the error will be about the diameter of an atom. Definitely too small to matter. Then we take the new positions and do it again over 1 second...we repeat this (using a computer) over a hundred billion seconds - and we arrive at a series of eclipse predictions over the next 3,100 years. Then, we change the time step from 1 second to (say) a half second - and repeat the calculations. The result is a new series of predictions. If the difference between the first set of predictions and the second is "close enough" then we can be happy to say that this approach isn't introducing large errors.

In practice, it's worse than this because Jupiter and Saturn add significantly to the motion - but by breaking the problem down into separate 2-body steps, and integrating over tiny time intervals - we can produce answers as accurately as you'd like. With modern computers, we can narrow down the answers to the point where we're talking distances the size of an atom and times down to nanoseconds.

So just because there is no single equation - we can still predict eclipses without using the idea of seeing "patterns"...which in itself isn't ever going to produce a perfect result because the effects of the gravity of the very slow moving outer planets means that the pattern of eclipses isn't ever "perfect" and has never precisely repeated over all of human history.