On the previous page, we encountered the Tritos, a period which matches a recurrence of half the draconic month with the synodic month, that the Maya made use of in the Dresden Codex, and it was noted that two other longer periods which match even closer recurrences, the Saros and the Inex, could be used to take partial sequences of eclipse semesters from a repeated Tritos that could be repeated with better accuracy.
After the Inex, the next very close approximation to the ratio between (half) the draconic month and the synodic month is reached at 4,161 lunations, and the next one after that is only a bit larger at 4,519 lunations.
With the Tritos cycle at 135 lunations, the Saros at 223, and the Inex at 358, the cycle of 4,519 lunations would consist of twelve Inex cycles and one Saros, which is equivalent to twelve Tritos cycles and thirteen Saros cycles, and the cycle of 4,161 lunations of eleven Inex cycles and one Saros. The longer cycle is about twenty times more accurate than the shorter, so the next good eclipse cycle will also be quite long: 94541, 99060, and 193,601 lunations are the next few attractive cycles.
The Saros, however, is also close to a multiple of the anomalistic month, and so while other eclipse cycles will relate eclipses of some sort, a Saros will tend to relate total eclipses to total eclipses and annular eclipses to annular eclipses. As well, the elliptical orbit of the Moon also affects when it reaches a particular phase, and when it reaches its nodes. Thus, an eclipse is almost certain to be followed by another eclipse one Saros cycle later, and this eclipse is likely to be of the same kind. A series of days separated by an Inex that have some eclipses on some of those days can also have intervening days without an eclipse, but most of the time there will be eclipses as well, although the character of the eclipses will vary. Thus, the Saros cycle is much more striking, although, since the Inex more accurately approximates the relation between the two cycles it reflects, the series of eclipses related by the Inex will continue on for a longer period.
The number of lunations in a Saros is 223, and that in an Inex is 358; these two numbers are relatively prime. Any lunation can be reached by a combination of those two numbers, so it's not surprising that any lunation associated with an eclipse can be reached that way. But because these are valid eclipse cycles, far-out combinations that reach numbers that aren't close to multiples of the Saros and Inex individually are not required. A graphic chart showing the sequence of solar eclipse types separated by the interval of a Saros vertically, and an Inex horizontally, is presented on this page, this type of graphic chart was justly called the Saros-Inex Panorama by George van der Bergh.
The synodic, draconic, and anomalistic months are not constant, but change in length over time, and the diagram on that page covers such a length of time that it reflects this, with the sequence of eclipses forming a curved arc.
Just as it was possible to repeat the Saros three times in order to obtain the Exeligmos, which is close to a multiple of the day, so as to find repeated eclipses in the same location, it might be possible to do the same thing with the other eclipse cycles to bring them into correspondence with the anomalistic month, in order to obtain an eclipse cycle that is genuinely beyond the Saros.
Given that the synodic month is 29.530588853 days long, and half the draconic month is 13.6061104085 days long, the ratio between these two periods is 2.17039168185. Thus, some successive approximations to coincidences between these cycles are as shown below:
Synodic Half Draconic Days Anomalistic months months months 1 2.17039168 29.5306 1.072 Month 6 13.02235009 177.1835 6.430 Eclipse semester 135 293.00288771 3986.6295 144.681 Tritos 223 483.99734505 6585.3213 238.992 Saros 358 777.00022210 10571.9508 383.674 Inex 4161 9030.99978820 122876.7802 4459.401 4519 9808.00001030 133448.7310 4843.074 Square Year 94541 205190.99999419 2791851.4008 101320.886 99060 214999.00000449 2925300.1318 106163.960 193601 420189.99999868 5717151.5325 207484.846
The 99060-lunation cycle comes close to being a multiple of the anomalistic month, but not as close as the Saros. Five repetitions of the 4161-lunation cycle, however, will provide a closer recurrence of the anomalistic month, as well as a recurrence of the half draconic month that is still more than twice as good as a single Saros cycle. This cycle, 20,805 lunations, is known as the Selenid I cycle. It was named by George van den Bergh (who also coined the name of the Inex), who, through study of the famous Canon of Eclipses by Oppolzer, wrote of how the interplay between the Saros and the Inex is reflected in the pattern of eclipses. A triple Inex also brings the Anomalistic month close to a recurrence, but not as close as a Saros.
It should be noted, too, that the lifetime of an Inex cycle is so much longer than that of a Saros cycle, that it is limited largely by tidal changes in the Earth's rotation and the Moon's orbit. Thus, longer cycles that are more precise than the Inex are not really that useful.
As noted above, there are 223 lunations in a Saros, and 358 lunations in an Inex.
The time from one eclipse to the next one is usually 6 lunations, and sometimes 5 lunations or 1 lunation.
An eclipse that is one Saros later than another eclipse belongs to the same Saros cycle as the other eclipse, because a Saros cycle is a series of eclipses that is one Saros apart, but to the next Inex cycle. Similarly, an eclipse one Inex later than another eclipse belongs to the same Inex cycle, but is in the next Saros cycle.
Eight Saros cycles of 223 lunations are 1784 lunations; five Inex cycles of 358 lunations are 1790 lunations. Thus, six lunations are equal to five Inex cycles minus eight Saros cycles. Thus, noting the statement in the previous paragraph, when two eclipses are six lunations apart, the next eclipse has a Saros number that is increased by five, and an Inex number that is decreased by eight.
The occasional pairs of eclipses separated by five lunations make up for the inaccuracy in approximating the Saros as 5/8 of an Inex, and eclipses, usually both partial, separated by one lunation result from there being an "extra" eclipse so that two groups of eclipses, each six lunations apart, are linked by more than one pair of eclipses five lunations apart.
When two eclipses are five lunations apart, the Saros number of the next eclipse is decreased by 33, and the Inex number of the next eclipse is increased by 53. Thus, the change when two eclipses are one lunation apart is an increase of 38 in the Saros number, and a decrease of 61 in the Inex number, because it amounts to backtracking five lunations and advancing six.