Solar eclipses, such as the most recent on July 2nd 2019 - viewable from South America - are simply due to the Moon's shadow casting onto the surface of the Earth. But this requires a very particular geometric alignment and timing that is decidedly non-trivial to predict - even if you understand orbital mechanics.

Yet for much of the past 5,000 years or thereabouts (and possibly earlier) humans have been able to make halfway decent predictions (although by no means highly accurate) for both lunar and solar eclipses. Even without a grasp of the nature of orbits and the three-dimensional state of our local cosmos. 

The reasons are both complex and subtle. The lunar orbit is inclined at about 5.1 degrees to the orbital plane of the Earth around the Sun. That means that the Moon crosses the Earth-Sun plane twice in each lunar orbit. That crossing point is called a node. When we have a New Moon - the general alignment of the Sun, Moon, and Earth with the Sun and Earth sandwiching the Moon - within about 17 degrees on the sky of a crossing (really from about 15.4 to 18.6 degrees because of the ellipticity of both the lunar orbit and Earth's) then there will be a solar eclipse (partial or total) visible somewhere from the surface of the Earth.

Because the lunar orbit has a period of 29.53 days with respect to this alignment - what's called the synodic period, which is a bit longer than the 27.3 day actual orbital period, because of the Earth-Moon motion around the Sun (phew) - and it takes about 34.5 days for the Sun to appear to cross the 17 degree radius eclipse zone (still with me?), there can be at most 2 solar eclipses during this time. The key is to figure out when these so-called 'eclipse seasons' are going to take place.

The first piece of that puzzle is solved by noticing that after roughly half a year the Moon has a chance at crossing the Earth-Sun plane again (the other node) within the magic 17 degrees of Earth-Sun alignment. So a second eclipse season can be anticipated at about 173.3 days after the first. That's not quite half a year because the location of the lunar crossing nodes also shifts in longitude by about 19 degrees a year. 

The upshot of all this is that about 11% of successive solar eclipses happen roughly a synodic month apart, some 23% happen 5 synodic months apart, and some 66% happen 6 synodic months apart. 

So far so good, but there's another set of complications on longer timescales. This is the so-called Saros cycle. Brace yourselves. Because the Synodic Month (29.53 days) is very, very similar to the Anomalistic Month (27.55 days and the period it takes the moon to go between closest approaches to Earth in its mildly elliptical orbit), and to the Draconic Month (27.21 days and the time between passages through the same orbital node), there is a harmonic behavior. In other words, much like musical harmony, there is a kind of beat frequency or period when things get into a particular alignment. 

That Saros period is about 18 years, 11 days, and 8 hours and corresponds to the repetition of eclipses (both solar and lunar) of very similar geometries - with shadow paths across the Earth that look very much alike, except shifted in longitude each time by about 120 degrees. That means that after 54 years and 34 days you'll be back to the same eclipse geometry over the same part of Earth. These eclipses will also occur at the same time of year.

But over longer timescales a Saros series like this will also drift across the Earth's latitude, and change from partial to full and back again, and eventually will end as all of the small differences in periods and nodes and orbits shift things out of harmony. What this means is that at any given time there are dozens of Saros series in play - with any given eclipse belonging to a particularly phased series. A single series will last for somewhere between 69 and 87 eclipses, or about 1,226 to 1,550 years. The recent July 2nd 2019 eclipse was part of the Saros cycle 127.

At this point you'd be forgiven for hoping that this is it, the sum total of things to worry about. However there is more. For example, the 'Inex' is a period of 358 synodic months (a bit under 29 years) and if a solar eclipse is seen in the southern hemisphere (for example) then exactly one Inex later there will be one visible in the northern hemisphere. There are others too: the Tritos (11 years, 1 month), the Metonic Cycle (19 years), the Exeligmos (triple Saros). Then there are periods that are combinations of Saros and Inex periods spanning a couple of thousand years that allow for prediction of the geographic latitude and longitude of the central path of the eclipse shadow, called the Accuratissima and Heliotrope respectively.

The bottom line to all of this is that today, with a grasp of orbital mechanics and computation we don't really need to know all of these things, we can simply model the Earth-Moon-Sun system with our digital orrerys and churn out tables of eclipse predictions and details. But for human across the centuries and the millennia these periods and patterns were key to both recognizing that solar (and lunar) eclipses were not just random and for attempting to predict their recurrence. Indeed, from these characteristics we know that there have been 11,898 solar eclipses in the past 5,000 years.

One can only marvel at the meticulous observation and bookkeeping that encouraged past astronomers to even take a shot at making these predictions.

Acknowledgment: many of the details I've written about here come from a remarkably complete set of information at this NASA site, derived from work on eclipse predictions by Fred Espenak (NASA's GSFC) and Jean Meeus.