Caption: Edmond Halley (1656--1742) has several claims to fame: he discovered the periodic nature of the great comet we call Halley's comet, he encouraged Isaac Newton (1643--1727) to write the Principia (1687), and he misnamed the Saros cycle the Saros cycle.

Note right off the Saros cycle = 6,585.321347 days (J2000?) = 18 yr + 10.321347 days (5 leap years) or 11.321347 days (4 leap years) (west shift ∼ 120° longitude) and the triple Saros cycle (AKA exeligmos) = 19755.964041 days (J2000?) = 54 yr + 31.964041 days (14 leap years) 32.964041 days (13 leap years) (west shift ∼ 0° longitude). Note Wikipedia: Saros is NOT clear on whether the standard metric day = 24 h = 86400 s or the solar day = current mean value 86400.002 s is used in their description. Since the solar day evolves with time, it is a bit indefinite to use for high accuracy/precision description, and so I assume Wikipedia: Saros means standard metric day as an excellent approximation to solar day, but with more accuracy/precision.

To explicate the last claim to fame specified above, Halley in 1691 mistakenly concluded the cycle of eclipses known to Babylonian astronomy (centuries earlier than 1200 BCE--c.60 BCE) (Wikipedia: History of astronomy: Mesopotamia; Wikipedia: Babylonian astronomy; Wikipedia: Babylonian star catalogues) was called the Saros or something like that. Actually, the word "saru" means 3600 in some version of the Babylonian language (see Wikipedia: Saros: History).

Features of the Saros cycle:

1. There is NO exactly repeating cycle of eclipse phenomena because the various periods involved are effectively incommensurable because they are NOT exact integer multiples of some base period because they are empirical, and so have some uncertainty and because they vary a bit randomly and also slowly evolve with time due to astronomical perturbations.
   Note if the periods were exact integer multiples (e.g., I,J,K,...) of a base period p, then whatever state now would be repeated a time (I*J*K* ...)*p later. So there would be an exact cycle in which all eclipse phenomena with period (I*J*K* ...)*p.

2. Howsoever, the Saros cycle is an approximate cycle of eclipses.

   Using modern values (J2000?) its length is

     6585.321347 days = 13 common years + 5 leap years + 10.321347 days
                      = 14 common years + 4 leap years + 11.321347 days
                      = 18 Julian years + 10.821347 days  , 

   with common year = 365 days, leap years =366 days, and Julian year = 3652.25 days. See Wikipedia: Saros: Description.

   Note the period of the Saros cycle is empirical, and so has some uncertainty and it varies bit randomly and also slowly evolves with time due to astronomical perturbations.

3. The approximate 1/3 of a days that ends the value 6585.321347 days means that on a repeat of the Saros cycle the eclipse phenomena will happen at roughly the same local solar time, but the locality will be about 120° farther west in longitude.

   For a repetition in approximately the same locality with the same approximate solar time, you need to wait a triple Saros cycle (AKA exeligmos) with period 19755.964041 days ≅ 54 years + 32 or 33 days.

   So if a total solar eclipse happened right here at noon today, about 54 years + 33 days from now a total solar eclipse would happen somewhere near here near noon.

4. The Saros cycle was discovered by the Babylonian astronomers in the 2nd half of the 1st millennium BCE, but it was NOT called that by them---it was called that by Halley as aforesaid.

   The Saros cycle was later known to the ancient Greek astronomers, who probably got it from the Babylonian astronomers by some kind of trans-cultural diffusion.

5. Using the Saros cycle and probably the triple Saros cycle, the Babylonian astronomers were able to make approximate eclipse predictions.

   However, they were flat-Earthers and probably did NOT understand that solar time depends on longitude which they also did NOT know about. So they could only predict possibilities of eclipses. The predictions could be unrealized: e.g., a predicted lunar eclipse was unobserved because it was entirely below the horizon; a predicted total/annular solar eclipse was outside their geographical area, and so was unreported.

   The ancient Greek astronomers who did have the spherical Earth theory and eventually understood solar time probably did better.

   But high accuracy/precision eclipse predictions (eclipse ephemerides) were probably NOT available before the 17th century when more accurate procedures were available than the Saros cycle and triple Saros cycle.

   In fact, one of the first high accuracy/precision total solar eclipse predictions was made by none other than Edmond Halley (1656--1742) himself: the prediction of total solar eclipse of 1715 May03.