For calendars and for predictions it is important to be able to approximate an arbitrary number with a ratio of two whole numbers. For example, a solar calendar approximates the length of the year of seasons (for example the tropical year), and a lunar calendar approximates the length of the month of the phases of the Moon (the synodical month). For predicting solar eclipses and lunar eclipses it is important to be able to approximate the ratio of the length of the synodical month and the draconitic month.
If you use an approximation that isn't very good, then a calendar or prediction based on that approximation will quickly get out of step with the thing that the approximation was for. You want an approximation that is as good as possible. How you find such good approximations is explained on the page about the Extended Euclidean Algorithm.
The year that runs in step with the average of all seasons is the tropical year, which at the beginning of the 21st century had an average length of 365.2421898 dagen (365 days, 5 hours, 48 minutes, and 45.20 seconds). We seek good approximations of that value for making calendars with whole numbers of days.
Table 1: Calendars: Tropical Year in Days
The "offset" shows by how much that many days are offset from that many tropical years. The "difference" is the difference between the approximation and the true year. For comparison, the offset for the average Gregorian calendar year (146097 days equals 400 years) is 2 hours and 59 minutes, and the difference is 27 seconds. A calendar with 12053 days in 33 years follows the seasons better than the Gregorian calendar does.
The year that runs in step with the beginning of spring in the northern hemisphere now has a length of on average 365.24237404 days (365 days, 5 hours, 49 minutes, and 1.12 seconds). Good approximations for that year are:
Table 2: Calendars: March Equinox Years in Days
The Gregorian year tries to follow the March equinox, but a calendar based on 12053 days for 33 years follows the March equinoxes more closely.
The month of the lunar phases, the synodical month, is now 29.530588853 days (29 days, 12 hours, 44 minutes, and 2.88 seconds) long. Good approximations of that month with calendars with whole days only are:
Table 3: Calendars: Synodical Month in Days
The month of the stars, the sidereal month, is now 27.321661548 days (27 days, 7 hours, 43 minutes, and 11.56 seconds) long. Good approximations for this are:
Table 4: Calendars: Sidereal Month in Days
If we want to combine with synodical month and the tropical year, then the approximations from the following table are good. For convenience I've also included the average number of days: the average of the numbers of days that correspond to the given numbers of months and years. That number does not get ever closer to a whole number, which indicates that it is in general not possible to find a period that very accurately corresponds to a whole number of days, months, and years.
Table 5: Calendars: Tropical Year in Synodical Months
The first couple of very good approximations that we find for eclipses are listed in the following table. The very good approximations are \(a/b\). The corresponding period of prediction and great period (to be explained later) are \(y\) (in years) en \(c\) (in years). The number of successful predictions in a row to be expected is between \(n_1\) and \(n_2\). The fraction of successful predictions of further eclipses based on earlier eclipses and the prediction period is equal to \(P\). Very good approximation number 12 has the unusably large prediction period of 12393.4 years.
Table 6: Eclipse Periods