Epact
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The epact (from Greek: epaktai hèmerai = added days) is, as the second Canon of the Gregorian calendar reform puts it, "nothing else than the number of days which the common solar year of 365 days surpasses the common lunar year of 354 days" (Latin: Epacta nihil aliud est quam numerus dierum quibus annus solaris communis dierum 365 annum communem lunarem dierum 354 superat).
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[edit] Lunar calendar
Epacts are used to find the date in the lunar calendar from the date in the common solar calendar.
[edit] Solar and lunar years
A (solar) calendar year usually has 365 days (366 days in leap years). A lunar year usually has 12 synodic months, that last about 29½ days (and a bit) on average. So the lunar year has months that start with the New Moon and alternate between 30 and 29 days in length, and the lunar year is counted to have 12 × 29.5 = 354 days. So the solar year is 11 days longer than the lunar year.
Suppose a solar and lunar year start at the same day. At the start of the next solar year, already 11 days of the new lunar year have passed. After 2 years the difference has accumulated to 22: the start of the lunar months fall 11 days earlier into the solar calendar each year. These days in excess of the solar year over the lunar year are called epacts. You have to add them to the day of the solar year to know the day in the lunar year. Whenever the epact reaches or exceeds 30, an extra (so-called embolismic or intercalary) month has to be inserted into the lunar calendar; then 30 has to be subtracted from the epact.
What about leap days? These are ignored in the computation. Leap days get inserted into the lunar calendar month in which they fall, extending them from 29 to 30 days or even from 30 to 31 days. So the next lunar month starts at the same solar calendar date as without a leap day. This is why we can ignore them.
[edit] 19-year cycle
The tropical year is about 1/4 day longer than 365 days, but the synodic month is also longer than 29½ days. This gets corrected in the following way. 19 tropical years are as long as 235 synodic months (Metonic cycle). A cycle can last 6939 or 6940 full days, depending on whether there are 4 or 5 leap days in this 19-year period.
After 19 years the lunations should fall the same way in the solar years, so the epact should repeat after 19 years. However, 19 × 11 = 209 modulo 30 = 29, not 0. So after 19 years the epact must be corrected by +1 in order for the cycle to repeat over 19 years. This is the so-called saltus lunae. The sequence number of the year in the 19-year cycle is called the Golden Number. The extra 209 days fill 7 embolismic months, for a total of 19×12 + 7 = 235 lunations.
[edit] Lilian (Gregorian) epacts
Despite the statement in the second canon of the Gregorian reform quoted above, the epacts in this calendar can no longer be interpreted as days. The designer (Aloysius Lilius) broke the pure Metonic relation when allowing centennial corrections of the epacts by one unit:
- a "solar equation" by decrementing the epact for the years when the Gregorian calendar drops a leap day (3 times in 400 Gregorian years)
- a "lunar equation" by incrementing the epact 8 times in 2500 Gregorian years.
In the Gregorian calendar, there are 30 possible values for the epact. Epacts always are computed modulo 30, and always indicate the New Moon. Therefore the epacts are in units of 1/30 of a lunation (also called a tithi). However a lunation is less than 30 days, so the epact unit is less than a full day.
This can also be understood from the following fact (please read computus for an explanation of the terms and procedures referred to here): Almost half of the lunations last only 29 days. In the Calendarium 12 days in the year have a double epact label (xxiv,25 and xxvi,25; one of these is used depending on the Golden Number). Therefore the correction of the epact by 1 unit does not always result in a shift of all dates of the New Moon (and Full Moon) by 1 day: for epacts 25 in short lunar months there is no difference. So the epact corrections are less than 1 day on average, and therefore the epact itself is not measured in calendar days.
It may be argued that Lilius applied the "solar equations" in order to bring the lunar calendar back in sync with the original Julian calendar; the "lunar equations" would then make a long-term correction to the approximate Metonic relation between the Julian year and the mean lunation. However, the "lunar equations" are applied at the begin of Gregorian years, not Julian years. The Gregorian epact tables have a period of 5 700 000 years. When counting epacts as days, the lunar calendar does not repeat however with this period, neither in this many Gregorian nor in Julian years.
