from the series of short articles about calendars
Let’s consider the 3 basic calendar cycles: the Indiction cycle lasting 15 years, the Solar Calendar cycle lasting 28 years and Meton Lunar cycle lasting 19 years. The division of time in 15-year periods started in the year 312/313 AD according to Chronicon Pascale. The years did not start always on January 1st but on August 29th (Old-Style Julian) or Thoth 1st in Coptic Calendar, or on September 23rd (the birthdate of Emperor August) in the Roman Empire and on September 1st, after 462 AD in the Eastern Roman (Byzantine) Empire. Any year has its position in this 15-year cycle. Therefore any year Y in Julian/Gregorian Calendar corresponds to a couple Z/(Z+1) in a time measurement system, where the years begin in some arbitrary date different from Jan. 1st. To find the position (1st to 15th) of a year Y in the Indiction cycle, we use the formula: I = (Y+2) mod 15 +1, where A mod B means the remainder of A divided by B. Then a year Y is said that it is the I/(I+1) of Indiction, that is I of Indiction until Aug. 31st and (I+1) of Indiction after Sept. 1st. For example 2018 A.D. is the 11th of Indiction until Aug. 31 and the 12th of Indiction after Sep. 1st. The Meton Lunar cycle is the basis of lunisolar calendars, such as the ancient Attic (Athenian) Calendar and the used until now Hebrew Calendar. As we saw in a previous info item concerning the Metonic cycle, we connected each year of the Julian/ Gregorian calendar to a couple of successive Meton numbers: M/(M+1). On the other hand, the Meton number used by Christians for the computation of the date of the Easter, let call it Easter Meton number is considered in alignment with whole years (from Jan. 1st to Dec. 31st) and therefore a year does not correspond to a couple but to a single number from 1 to 19. To find the Easter Meton number of a year Y, we use the formula: M = Y mod 19 +1. Thus, the year 1 AD is considered of Easter Meton number 2 (as 1 mod 19 + 1) and the year 2018 A.D. is considered of number 5. The Solar Calendar cycle concerns the position in the year of the first Sunday (Solar day) of the year. Due to the leap years the period for the reappearance of the same sequence of days of week in the whole range of the year (measuring the dates in Old-Style Julian Calendar) is 4x7=28. To find the position of a year Y in the 28-year Solar Calendar cycle, under the requirement that the first of the 28 years sequence is a leap year starting in Monday, we use the formula: S = (Y+9) mod 28 +1. The Julian Day Number has been proposed by the scholar Joseph Scaliger, in 1583, one year after the Gregorian calendar reform, as the order of a day in a great cycle of days corresponding to a number of years, which is the least common multiple (LCM) of the above mentioned three calendar cycles (Indiction, Meton, Solar): 15 (Indiction cycle) × 19 (Christian-Meton cycle) × 28 (Solar cycle) = 7980 years or 2,914,695 days. Its epoch falls at the last time when the three cycles, (if they are extended backwards) were in their first year together. How can we compute this year, the first year of this artificial era, called by Scaliger, Julian Period? Through our knowledge that year 1 AD was the 10th year of the Solar Cycle, the 2nd of the Easter Meton Cycle, and the 4th of the Indiction Cycle. This is equivalent to the following problem: Which is the order of the year of this artificial sequence, which, when divided by 28 will leave a remainder 10, divided by 19 will leave a remainder 2, and divided by 15 will leave a remainder 4. It can be estimated through Mathematics that this is the year 4714. So we have the required correspondence: 4714 of Julian Period = 1 AD. Thus 1 BC = 4713 of J.P. Thus the required first year is 4713 BC, a year before any historical record. Jan. 1st of 4713 BC is the day 1 of this important calendar, appraised a lot by many astronomers, such as Hershel. Simple algorithms are available for the computation of the Julian-Day-Number (JDN) of any date either in the Julian or in the Gregorian Calendar. Since JDN is the number of days passed after a predefined date (Jan. 1st of 4713 BC), we can compute the number of days passed between any 2 dates as the subtraction of the 2 JDNs. Another important application of JDN is the computation of the day of week of any date either of the Old-Style Julian or of the New-Style Gregorian Calendar. The formula is the following: W = (JDN + 1) mod 7, where W=0 for Sunday, 1 for Monday etc. The JDN of the current date or of any desired date is given in our calendar site.
About the Metonic cycle