# Extensive Definition

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.

## See also

epact in German: Epakte

epact in Spanish: Epacta

epact in Italian: Epatta

epact in Dutch: Epacta

epact in Japanese: エパクト

epact in Piemontese: Epata