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# Epact

[ee-pakt]
The epact (from Greek: epaktai hèmerai = added days) is a quantification of the difference between the solar and lunar calendars. It was defined by the second canon of the Gregorian calendar reform as "the number of days by which the common solar year of 365 days surpasses the common lunar year of 354 days".

## Lunar calendar

Epacts are used to find the date in the lunar calendar from the date in the common solar calendar.

### Solar and lunar years

A (solar) calendar year has 365 days (366 days in leap years). A lunar year has 12 lunar months which alternate between 30 and 29 days (in leap years, one of the lunar months has a day added).

If a solar and lunar year start on the same day, then after one year, the start of the solar year is 11 days after the start of the lunar year; after two years, it is 22 days after. These excess days are epacts, and are added to the day of the solar year to determine the day of the lunar year.

Whenever the epact reaches or exceeds 30, an extra (embolismic or intercalary) month is inserted into the lunar calendar, and the epact is reduced by 30.

Leap days extend both the solar and lunar year, so they do not affect epact calculations for any other dates.

### 19-year cycle

The tropical year is about 365¼ days, while the synodic month is also slightly longer than 29½ days on average. This gets corrected in the following way. Nineteen 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 , and this is not an integer multiple of the full cycle of 30 epact numbers (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 saltus lunae (jump of the moon). 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.

### 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 exactly 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 whenever 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 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 one unit does not always result in a shift of all dates of the New Moon (and Full Moon) by one day: for epacts 25 in short lunar months there is no difference. So the epact corrections are less than one 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.