The Saros series of an eclipse, numbered as NASA's eclipse catalogues number them.

Eclipses recur after a saros of 223 lunations, 18 years 11 days and about 8 hours: the Sun, the Moon and the Moon's node return to nearly the same arrangement, so an eclipse is followed by a similar one 223 lunations later. The eclipses a saros apart form a Saros series, which begins with small partial eclipses near one pole, runs through central eclipses that move across the Earth and ends with partial eclipses near the other pole: 69 to 87 eclipses over 1,226 to 1,550 years.

An inex of 358 lunations, about 29 years, links neighbouring series: the eclipse an inex after an eclipse of series s belongs to series s + 1. So every eclipse of series s falls at a lunation n (Astro.Lunar.lunation_at/1) with n ≡ 358 s + c (mod 223) for a constant c, one for solar eclipses and another for lunar, and the series follows from the lunation to within a multiple of 223: s ≡ 38 (n − c) (mod 223), 38 being the inverse of 358 modulo 223.

The series in progress at a time form a band about 50 numbers wide, which moves on by about one number every 30 years, so the multiple of 223 is the one that puts the series nearest a straight-line estimate of the band from the lunation. The numbering is G. van den Bergh's, from his Saros–Inex panorama, as NASA's catalogues use it. The constants and the band are fitted to the catalogues and reproduce the series of every eclipse in six of their centuries between −999 and 3000; far beyond them the numbering continues by the same rule.

Reference

• G. van den Bergh, Periodicity and Variation of Solar (and Lunar) Eclipses (1955).

• Fred Espenak, Eclipses and the Saros, NASA Goddard Space Flight Center.