Epacts can also be used to relate dates in the lunar calendar to dates in the common solar calendar.
Solar and lunar years
A calendar year has 365 days. A lunar year has 12 lunar months which alternate between 30 and 29 days. 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 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 slightly longer than 29½ days, on average; both are non-integers. This gets corrected in the following way. Nineteen tropical years are as long as 235 synodic months. 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. 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. 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
When the Gregorian calendar 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 scheme devised by Aloysius Lilius. 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, and
a "lunar equation", incrementing the epact by 1, 8 times in 2500 calendar years.
The "solar equation" would adjust for the Gregorian change in the solar calendar, if they were applied at 1 January of the Julian calendar instead of the Gregorian calendar as the reformers implemented it; moreover the corrections to the solar calendar are leap days, whereas there are 30 epact values for a mean lunar month of 29.5 days and a bit: therefore changing the epact by one does not exactly compensate for a dropped leap day. The "lunar equation" adjusts approximately for what had 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 that the new and full moons were occurring "four days and something more" sooner than the old lunar cycle had been indicating.
History
The discovery of the epact for computing the date of Easter has been attributed to Patriarch Demetrius I of Alexandria, who held office from 189 to 232. In the year 214 he used the epact to produce an Easter calendar, which has not survived, which used an eight-year luni-solar cycle. A subsequent application of the epact to an Easter calendar, using a sixteen-year cycle, is found in the Paschal Table of Hippolytus, a 112-year list of Easter dates beginning in the year 222 which is inscribed on the side of a statue found in Rome. Augustalis, whose dates have been disputed from the third to the fifth century, computed a laterculus of Easter dates. As reconstructed, it uses epacts and an 84-year luni-solar cycle to compute the dates of Easter using a base date of A.D. 213. If we accept Augustalis's earlier dates, his laterculus extends from 213 to 312 and Augustalis originated the use of epacts to compute the date of Easter. As early as the fourth century. we see Easter computus using the epact and the nineteen-year Metonic cycle in Alexandria, and subsequent computistical tables were influenced by the structure of the Alexandrian calendar. The epact was taken as the age of the Moon on 26 Phamenoth but that value of the epact also corresponded to the age of the Moon on the last epagomenal day of the preceding year. Thus the epact can be seen as having been established at the beginning of the current year. Subsequent Easter tables, such as those of Bishop Theophilus or Alexandria, which covered 100 years beginning in A.D. 380, and of his successor Bishop Cyril, which covered 95 years beginning in A.D. 437 discussed the computation of the epact in their introductory texts. Under the influence of Dionysius Exiguus and later, of Bede, the Alexandrian Easter Tables were adopted throughout Europe where they established the tradition that the epact was the age of the Moon on 22 March. This Dionysian epact fell into disuse after the introduction of a perpetual calendar based on the golden number, which made the calculation of epacts unnecessary for ordinary computistical calculations. Two factors led to the creation of three new forms of the epact in the fifteenth and sixteenth centuries. The first was the increasing error of computistical techniques, which led to the introduction of a new Julian epact around 1478, to be used for practical computations of the phase of the Moon for medical or astrological purposes. With the Gregorian reform of the calendar in 1582, two additional epacts came into use. The first was the Lilian epact, developed by Aloisius Lilius as an element of the ecclesiastical computations using the Gregorian calendar. The Lilian epact included corrections for the motions of the Sun and the Moon that broke the fixed relationship between the epact and the golden number. The second new epact was a simple adjustment of the practical Julian epact to account for the ten-day change produced by the Gregorian Calendar.
