Epact

Encyclopedia

The

s). 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.

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

. The extra 209 days fill 7 embolismic months, for a total of 19×12 + 7 = 235 lunations.

reform was instituted in 1582, the lunar cycle previously used with the Julian calendar to complete the calculation of Easter dates was adjusted also, in accordance with a (modification of a) scheme devised by Aloysius Lilius

. There were two adjustments of the old lunar cycle:

The "solar equation" adjusts exactly for the Gregorian change in the solar calendar. The "lunar equation" adjusts approximately for what had (by 1582) become the experience of many centuries, that the Moon moves a little faster than the expectation of its rate embodied in the old lunar cycle. By 1582 it was noted (e.g. in the text of the bull Inter gravissimas

itself) that the new and full moons were occurring "four days and something more" sooner than the old lunar cycle had been indicating.

**epact**was originally defined as the age of the moon in days on January 1, and occurs primarily in connection with tabular methods for determining the date of Easter. It varies (usually by 11 days) from year to year, because of the difference between the solar year of 365 days and the 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 yearLeap year

A leap year is a year containing one extra day in order to keep the calendar year synchronized with the astronomical or seasonal year...

s). 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 yearTropical year

A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

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

Metonic cycle

In astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of very close to 19 years which is remarkable for being very nearly a common multiple of the solar year and the synodic month...

). 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

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

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*(leap of the moon). The sequence number of the year in the 19-year cycle is called the Golden NumberGolden numbers

A golden number is a number assigned to each year in sequence to indicate the year's position in a 19-year Metonic cycle. They are used in the computus and also in the Runic calendar. The golden number of any Julian or Gregorian calendar year can be calculated by dividing the year by 19, taking...

. The extra 209 days fill 7 embolismic months, for a total of 19×12 + 7 = 235 lunations.

### Lilian (Gregorian) epacts

When the Gregorian calendarGregorian calendar

The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...

reform was instituted in 1582, the lunar cycle previously used with the Julian calendar to complete the calculation of Easter dates was adjusted also, in accordance with a (modification of a) scheme devised by Aloysius Lilius

Aloysius Lilius

Aloysius Lilius , also variously referred to as Luigi Lilio, Luigi Giglio, or Aluise Baldassar Lilio, was an Italian doctor, astronomer, philosopher and chronologist, and also the "primary author" who provided the proposal that became the basis of the Gregorian Calendar reform of 1582.The crater...

. There were two adjustments of the old lunar cycle:

- a "solar equation", decrementing the epact by 1, whenever the Gregorian calendar drops a leap day (3 times in 400 calendar years), and
- a "lunar equation", incrementing the epact by 1, 8 times in 2500 calendar years (seven times after an interval of 300 years, and the eighth time after an interval of 400 years).

The "solar equation" adjusts exactly for the Gregorian change in the solar calendar. The "lunar equation" adjusts approximately for what had (by 1582) become the experience of many centuries, that the Moon moves a little faster than the expectation of its rate embodied in the old lunar cycle. By 1582 it was noted (e.g. in the text of the bull Inter gravissimas

Inter gravissimas

Inter gravissimas was a papal bull issued by Pope Gregory XIII on February 24, 1582. The document reformed the Julian calendar and created a new calendar which came to be called the Gregorian calendar, which is used in most countries today.-Description:...

itself) that the new and full moons were occurring "four days and something more" sooner than the old lunar cycle had been indicating.

## See also

- ComputusComputusComputus is the calculation of the date of Easter in the Christian calendar. The name has been used for this procedure since the early Middle Ages, as it was one of the most important computations of the age....
- Wikisource English translation of the (Latin) 1582 papal bull 'Inter gravissimasInter gravissimasInter gravissimas was a papal bull issued by Pope Gregory XIII on February 24, 1582. The document reformed the Julian calendar and created a new calendar which came to be called the Gregorian calendar, which is used in most countries today.-Description:...

' instituting Gregorian calendar reform

## External links

- Epacts from the Catholic EncyclopediaCatholic EncyclopediaThe Catholic Encyclopedia, also referred to as the Old Catholic Encyclopedia and the Original Catholic Encyclopedia, is an English-language encyclopedia published in the United States. The first volume appeared in March 1907 and the last three volumes appeared in 1912, followed by a master index...