# Epacts

There are several different types of epacts, but we’ll limit our scope to the most interesting one of them all, and that is the epact that was introduced during the Gregorian calendar reform by Luigi Lili. This epact goes by many names, such as the New Year’s Epact, Lili's Epact, and the Gregorian Epact, and is a necessary prerequisite for calculating Easter Sunday. In the Gregorian calendar, the epact for any given year is equal to the age of the phase of the Moon (in days) since the new moon that occurred on 31 December of the previous year. Unfortunately there is a bit of confusion in the existing literature, with some counting the age of the Moon from 0 and others from 1. To help clarify:

• if the age of the moon on 31 December of the previous year is set to 0, then the Gregorian epact for that year is equal to the age of the moon on 1 January
• if the age of the moon on 31 December of the previous year is set to 1, then the Gregorian epact for that year is equal to the age of the moon on 31 December of the previous year (this option is used for martyrology)

This gives us the numbers 1 to 30, but as mentioned above, some calendars use 0 instead of the number 30, and therefore only count as high as 29. Now, from here we can clearly and unambiguously assign a new moon table for the whole year that matches up to each epact. While these “cyclic calculations” may not be as accurate as astronomical calculations, they are far and away much easier calculations to perform. Essentially, if a year has a Gregorian epact of 1, it means that the cyclical new moon was 31 December of the previous year and its next new moon would be on 30 January of that year, and so on and so forth, until the last new moon in the same year would happen on the 20th of December.

The following table compares the accuracy of the epact with the age of the Moon on 1 January at 00:00 GMT for the years 2001 to 2030. The accuracy of the epact is admirable, with the error rarely ever exceeding one day.

The accuracy of Gregorian epacts
yearepactage of the Moon
200156.28
20021617.13
20032727.69
200488.60
20051919.94
200600.87
20071111.42
20082222.26
200934.48
20101415.50
20112526.27
201267.25
20131718.64
20142928.99
2015109.93
20162120.56
201722.71
20181313.73
20192424.70
202055.78
20211617.32
20222727.68
202388.57
20241919.02
202501.07
20261111.93
20272222.96
202834.16
20291415.91
20302526.36