from the series of short articles about calendars
The Metonic Lunar cycle is the basis of lunisolar calendars, such as the ancient Attic (Athenian) Calendar and the used until now Hebrew calendar. Lunisolar are those lunar calendars, which, through the intercalation process, retain their alignment to the duration of the solar year. In this way they integrate both the movement of the Sun, resulting to the conclusion of the succession of the seasons within a tropical year and the movement of the Moon, resulting to the monthly cycle of the phases of the Moon (new moon - 1st quarter - full moon - 3rd quarter).
The cycle of the seasons was all along very important in the human life, since it defines the various phases of planting and breeding, as well as the feast days related to the countdown between light and darkness: Summer Solstice with the greatest day and start of the light weakening (Stonehedge great feast, start of Attic year), Fall Equinox with equal duration between light and darkness followed by the dominion of darkness (Eleusinian mysteries, start of Hebrew year), Winter Solstice with the greatest night and start of the light strengthening (birth of Mithra and Christ) and the Spring Equinox (sacrifice of Adonis and Christ) with equal duration between light and darkness followed by the dominion of light.
The cycle of the Moon was equally important since the absence of the main light of the night at new moon and the intense presence of it at full moon. The new moon meant symbolically the dominion of darkness, as an opportunity for immersing in the interiors of oneself. The full moon meant the dominion of light, as an opportunity to declare the never ending presence of the light within the darkness and express the gratitude to the gods as the hidden powers of the Nature.
But how did they succeed technically to hold a calendar which on the one hand follows the phases of the moon and on the other hand does not deviate after some months from the solar cycle of the seasons integrated within a tropical year (nearly 365,24 days). The synodic month of the moon phases is nearly 29.5 (precisely 29.530588) days. Therefore 12 months with alternating 30 and 29 days (in order to approach 29.5) amount to a sum of 354 days. This means that after 12 months an exclusively lunar calendar (as the Hijri Muslim Caledar, which we'll present in another article) loses nearly 11 days in respect to the solat cycle of nearly 365 days. The solution was an intercalation scheme followed by all the lunisolar calendars (Babylonian, Hebrew, Attic). Nearly every 3 years, an extra 30-day month (intercalated or embolismic) is added in some position of the calendar: in the Hebrew Calendar after the 6th month Adar another Adar (Adar II) is added and in the Attic Calendar after the 6th month Poseideon another Poseideon (Poseideon II) is added.
The intercalation process was studied in detail in the classical period of Athens by the astronomer Meton, who presented the 19-year cycle, after which dates and moon phases align again. Attic Calendar year starts on the Summer Solstice and Hebrew Calendar starts on the Fall Equinox. Thus, none of these widespread calendars start on a fixed date, as the fixed Julian and Gregorian calendars, that start always on January 1st. The ancient astronomers and mathematicians corresponded each year to a number showing the position of the year within the Metonic cycle, let call it Meton number of the year. This is the called Meton number and takes the values: 1 through 19 (replacing 0 with 19). The formula for the Meton number is:
Meton_number = year_number modulo 19, if (Meton_number=0) then Meton_number=19
The above formula contains the year_number, that is the number of the year in a year sequence, starting on a year, which has been defined as year 1 (the concept of 0 had not yet appeared). In the Hebrew calendar year 1 is the year starting on the Fall Equinox New Moon of Oct. 7th of 3761 BC (the supposed date of the creation of the world). In the Attic (Athenian) calendar, Meton defined as year 1 the year starting on the Summer Solstice New Moon of July 13th of 432 BC.
Therefore each year on the fixed Julian Calendar or Gregorian Calendar or proleptic Julian Calendar (covering the years before the Julian Caesar’s reform of 45 BC) corresponds to a couple of consecutive years of any of the lunisolar calendars, either Hebrew or Attic. Since each lunisolar year corresponds to a Meton number of the Metonic cycle, we can relate each Julian/Gregorian year to a couple of consecutive Meton numbers, such as 1/2 or 2/3 or … 18/19 or 19/1.
On the other hand, the Meton number used by Christians for the computation of the date of the Easter, is considered in alignment with whole years (from Jan. 1st to Dec. 31st) and therefore a year does not correspond to a couple of Meton numbers but to a single Meton number from 1 to 19, as we’ll see in detailed in a following info item, concerning the computation of the Easter date.
Year 2018 AD, for example, corresponds to the couple of Attic Meton numbers 17/18, because the attic year starting in the middle of 2018 AD is 2018+432=2450 and 2450 modulo 19 = 18. Similarly, year 2018 AD corresponds to the couple of Hebrew Meton numbers 2/3, because the hebrew year starting in the middle of 2018 AD is 2018+3761= the Hebrew year 5779 (from the creation of the world) and 5779 modulo 19 = 3.
Meton cycle consists of 6940 days distributed in the ancient Attic (Athenian) calendar as follows: 8 normal years of 354 days (Hakatombaion:30, Metageitnion: 29, Boedromion:30, Pyanepsion:29, Maimakterion:30, Poseideon:29, Gamelion:30, Anthesterion:29, Elaphebolion:30, Mounichion:29, Thargelion:30 and Skirophorion:29), 4 normal years of 355 days (by extending one 29-days month to 30 days) and 7 long (embolismic) years of 384 days (through the intercalation – embolism of one more 30-day month, named Poseideon II, after Poseideon). The order of a year is found as year modulo 19 (the remainder after integer division), with remainder 0 set as 19. Every attic year is characterized by its Meton number (1 to 19). The long years are the years: 3, 6, 8, 11, 14, 17, 19. We consider as normal years: 2, 4, 5, 9, 10, 13, 16, 18 and as extended years: 1, 7, 12, 15.
Though the actual beginning of attic years in repeated Meton cycles is 432 BC, we decide in our reconstruction of the Attic Calendar to extend it 20 cycles to the past in order to have the opportunity to use the Attic calendar for the definition of some certain dates recorded in the ancient texts (as the eclipse foretold by Thales, which has been identified with 28 May, 585 BC in proleptic Julian Calendar). The dates given before the summer solstice of 432 BC consist the proleptic Attic Calendar.