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The epact (from Greek: epaktai hèmerai = added days) is, as the second Canon of the Gregorian calendar reform puts it, "nothing else than the number of days by which the common solar year of 365 days surpasses the common lunar year of 354 days" (Latin: Epacta nihil aliud est quam numerus dierum quibus annus solaris communis dierum 365 annum communem lunarem dierum 354 superat).

## Lunar calendarEdit

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

### Solar and lunar yearsEdit

A (solar) calendar year usually has 365 days (366 days in leap years). A lunar year usually has 12 synodic months, that last about 29½ days (and a bit) on average. So the lunar year has months that start with the New Moon and alternate between 30 and 29 days in length, and the lunar year is counted to have 12 × 29.5 = 354 days. So the solar year is 11 days longer than the lunar year.

Suppose a solar and lunar year start at the same day. At the start of the next solar year, already 11 days of the new lunar year have passed. After two years the difference has accumulated to 22: the start of the lunar months fall 11 days earlier into the solar calendar each year. These days in excess of the solar year over the lunar year are called epacts. You have to add them to the day of the solar year to know the day in the lunar year. Whenever the epact reaches or exceeds 30, an extra (so-called embolismic or intercalary) month has to be inserted into the lunar calendar; then 30 has to be subtracted from the epact.

What about leap days? These are ignored in the computation. Leap days get inserted into the lunar calendar month in which they fall, extending them from 29 to 30 days or even from 30 to 31 days. So the next lunar month starts at the same solar calendar date as without a leap day and can be ignored.

### 19-year cycleEdit

The tropical year is about 1/4 day longer than 365 days, but the synodic month is also longer than 29½ days. This gets corrected in the following way. 19 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 so-called 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) epactsEdit

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 1/30 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 computusfor 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,24 and xxvi,25; one of these is used depending on the Golden Number). Therefore the correction of the epact by 1 unit does not always result in a shift of all dates of the New Moon (and Full Moon) by 1 day: for epacts 25 in short lunar months there is no difference. So the epact corrections are less than 1 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.

## Table of extended epactsEdit

Epacts.—Epact is a word of Greek origin, employed in the calendar to signify the moon's age at the beginning of the year. The common solar year containing 365 days, and the lunar year only 354 days, the difference is eleven; whence, if a new moon fall on the 1st of January in any year, the moon will be eleven days old on the first day of the following year, and twenty-two days on the first of the third year. The numbers eleven and twenty-two are therefore the epacts of those years respectively. Another addition of eleven gives thirty-three for the epact of the fourth year; but in consequence of the insertion of the intercalary month in each third year of the lunar cycle, this epact is reduced to three. In like manner the epacts of all the following years of the cycle are obtained by successively adding eleven to the epact of the former year, and rejecting thirty as often as the sum exceeds that number. They are therefore connected with the golden numbers by the formula $11n \over 30$ in which n is any whole number; and for a whole lunar cycle (supposing the first epact to be 11), they are as follows:—11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29. But the order is interrupted at the end of the cycle; for the epact of the following year, found in the same manner, would be 29 + 11 = 40 or 10, whereas it ought again to be 11 to correspond with the moon's age and the golden number 1. The reason of this is, that the intercalary month, inserted at the end of the cycle, contains only twenty-nine days instead of thirty; whence, after 11 has been added to the epact of the year corresponding to the golden number 19, we must reject twenty-nine instead of thirty, in order to have the epact of the succeeding year; or, which comes to the same thing, we must add twelve to the epact of the last year of the cycle, and then reject thirty as before.

This method of forming the epacts might have been continued indefinitely if the Julian intercalation had been followed without correction, and the cycle been perfectly exact; but as neither of these suppositions is true, two equations or corrections must be applied, one depending on the error of the Julian year, which is called the solar equation; the other on the error of the lunar cycle, which is called the lunar equation. The solar equation occurs three times in 400 years, namely, in every secular year which is not a leap year; for in this case the omission of the intercalary day causes the new moons to arrive one day later in all the following months, so that the moon's age at the end of the month is one day less than it would have been if the intercalation had been made, and the epacts must accordingly be all diminished by unity. Thus the epacts 11, 22, 3, 14, &c., become 10, 21, 2, 13, &c. On the other hand, when the time by which the new moons anticipate the lunar cycle amounts to a whole day, which, as we have seen, it does in 308 years, the new moons will arrive one day earlier, and the epacts must consequently be increased by unity. Thus the epacts 11, 22, 3, 14, &c., in consequence of the lunar equation, become 12, 23, 4, 15, &c. In order to preserve the uniformity of the calendar, the epacts are changed only at the commencement of a century; the correction of the error of the lunar cycle is therefore made at the end of 300 years. In the Gregorian calendar this error is assumed to amount to one day in 312½ years or eight days in 2500 years, an assumption which requires the line of epacts to be changed seven times successively at the end of each period of 300 years, and once at the end of 400 years; and, from the manner in which the epacts were disposed at the Reformation, it was found most correct to suppose one of the periods of 2500 years to terminate with the year 1800.

The years in which the solar equation occurs, counting from the Reformation, are 1700, 1800, 1900, 2100, 2200, 2300, 2500, &c. Those in which the lunar equation occurs are 1800, 2100, 2400, 2700, 3000, 3300, 3600, 3900, after which, 4300, 4600 and so on. When the solar equation occurs, the epacts are diminished by unity; when the lunar equation occurs, the epacts are augmented by unity; and when both equations occur together, as in 1800, 2100, 2700, &c., they compensate each other, and the epacts are not changed.

In consequence of the solar and lunar equations, it is evident that the epact or moon's age at the beginning of the year, must, in the course of centuries, have all different values from one to thirty inclusive, corresponding to the days in a full lunar month. Hence, for the construction of a perpetual calendar, there must be thirty different sets or lines of epacts. These are exhibited in the subjoined table (Table III.) called the Extended Table of Epacts, which is constructed in the following manner. The series of golden numbers is written in a line at the top of the table, and under each golden number is a column of thirty epacts, arranged in the order of the natural numbers, beginning at the bottom and proceeding to the top of the column. The first column, under the golden number 1, contains the epacts, 1, 2, 3, 4, &c., to 30 or 0. The second column, corresponding to the following year in the lunar cycle, must have all its epacts augmented by 11; the lowest number, therefore, in the column is 12, then 13, 14, 15 and so on. The third column corresponding to the golden number 3, has for its first epact 12 + 11 = 23; and in the same manner all the nineteen columns of the table are formed. Each of the thirty lines of epacts is designated by a letter of the alphabet, which serves as its index or argument. The order of the letters, like that of the numbers, is from the bottom of the column upwards.

In the tables of the church calendar the epacts are usually printed in Roman numerals, excepting the last, which is designated by an asterisk (*), used as an indefinite symbol to denote 30 or 0, and 25, which in the last eight columns is expressed in Arabic characters, for a reason that will immediately be explained. In the table here given, this distinction is made by means of an accent placed over the last figure.

At the Reformation the epacts were given by the line D. The year 1600 was a leap year; the intercalation accordingly took place as usual, and there was no interruption in the order of the epacts; the line D was employed till 1700. In that year the omission of the intercalary day rendered it necessary to diminish the epacts by unity, or to pass to the line C. In 1800 the solar equation again occurred, in consequence of which it was necessary to descend one line to have the epacts diminished by unity; but in this year the lunar equation also occurred, the anticipation of the new moons having amounted to a day; the new moons accordingly happened a day earlier, which rendered it necessary to take the epacts in the next higher line. There was, consequently, no alteration; the two equations destroyed each other. The line of epacts belonging to the present century is therefore C. In 1900 the solar equation occurs, after which the line is B. The year 2000 is a leap year, and there is no alteration. In 2100 the equations again occur together and destroy each other, so that the line B will serve three centuries, from 1900 to 2200. From that year to 2300 the line will be A. In this manner the line of epacts belonging to any given century is easily found, and the method of proceeding is obvious. When the solar equation occurs alone, the line of epacts is changed to the next lower in the table; when the lunar equation occurs alone, the line is changed to the next higher; when both equations occur together, no change takes place. In order that it may be perceived at once to what centuries the different lines of epacts respectively belong, they have been placed in a column on the left hand side of the table on next page.

The use of the epacts is to show the days of the new moons, and consequently the moon's age on any day of the year. For this purpose they are placed in the calendar (Table IV.) along with the days of the month and dominical letters, in a retrograde order, so that the asterisk stands beside the 1st of January, 29 beside the 2nd, 28 beside the 3rd and so on to 1, which corresponds to the 30th. After this comes the asterisk, which corresponds to the 31st of January, then 29, which belongs to the 1st of February, and so on to the end of the year. The reason of this distribution is evident. If the last lunation of any year ends, for example, on the 2nd of December, the new moon falls on the 3rd; and the moon's age on the 31st, or at the end of the year, is twenty-nine days. The epact of the following year is therefore twenty-nine. Now that lunation having commenced on the 3rd of December, and consisting of thirty days, will end on the 1st of January. The 2nd of January is therefore the day of the new moon, which is indicated by the epact twenty-nine. In like manner, if the new moon fell on the 4th of December, the epact of the following year would be twenty-eight, which, to indicate the day of next new moon, must correspond to the 3rd of January.

When the epact of the year is known, the days on which the new moons occur throughout the whole year are shown by Table IV., which is called the Gregorian Calendar of Epacts. For example, the golden number of the year 1832 is $(1832 + 1 \over 19) \sub{r}$ = 9, and the epact, as found in Table III., is twenty-eight. This epact occurs at the 3rd of January, the 2nd of February, the 3rd of March, the 2nd of April, the 1st of May, &c., and these days are consequently the days of the ecclesiastical new moons in 1832. The astronomical new moons generally take place one or two days, sometimes even three days, earlier than those of the calendar.

There are some artifices employed in the construction of this table, to which it is necessary to pay attention. The thirty epacts correspond to the thirty days of a full lunar month; but the lunar months consist of twenty-nine and thirty days alternately, therefore in six months of the year the thirty epacts must correspond only to twenty-nine days. For this reason the epacts twenty-five and twenty-four are placed together, so as to belong only to one day in the months of February, April, June, August, September and November, and in the same months another 25′, distinguished by an accent, or by being printed in a different character, is placed beside 26, and belongs to the same day. The reason for doubling the 25 was to prevent the new moons from being indicated in the calendar as happening twice on the same day in the course of the lunar cycle, a thing which actually cannot take place. For example, if we observe the line B in Table III., we shall see that it contains both the epacts twenty-four and twenty-five, so that if these correspond to the same day of the month, two new moons would be indicated as happening on that day within nineteen years. Now the three epacts 24, 25, 26, can never occur in the same line; therefore in those lines in which 24 and 25 occur, the 25 is accented, and placed in the calendar beside 26. When 25 and 26 occur in the same line of epacts, the 25 is not accented, and in the calendar stands beside 24. The lines of epacts in which 24 and 25 both occur, are those which are marked by one of the eight letters beknr, B, E, N, in all of which 25′ stands in a column corresponding to a golden number higher than 11. There are also eight lines in which 25 and 26 occur, namely, cflps, C, F, P. In the other 14 lines, 25 either does not occur at all, or it occurs in a line in which neither 24 nor 26 is found. From this it appears that if the golden number of the year exceeds 11, the epact 25, in six months of the year, must correspond to the same day in the calendar as 26; but if the golden number does not exceed 11, that epact must correspond to the same day as 24. Hence the reason for distinguishing 25 and 25′. In using the calendar, if the epact of the year is 25, and the golden number not above 11, take 25; but if the golden number exceeds 11, take 25′.

Another peculiarity requires explanation. The epact 19′ (also distinguished by an accent or different character) is placed in the same line with 20 at the 31st of December. It is, however, only used in those years in which the epact 19 concurs with the golden number 19. When the golden number is 19, that is to say, in the last year of the lunar cycle, the supplementary month contains only 29 days. Hence, if in that year the epact should be 19, a new moon would fall on the 2nd of December, and the lunation would terminate on the 30th, so that the next new moon would arrive on the 31st. The epact of the year, therefore, or 19, must stand beside that day, whereas, according to the regular order, the epact corresponding to the 31st of December is 20; and this is the reason for the distinction.

 Years. Index. Golden Numbers. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1700 1800 8700 C * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 1900 2000 2100 B 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25′ 6 17 2200 2400 A 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 2300 2500 u 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 2600 2700 2800 t 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 2900 3000 s 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 3100 3200 3300 r 24 5 16 27 8 19 * 11 22 3 14 25′ 6 17 28 9 20 1 12 3400 3600 q 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 3500 3700 p 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 3800 3900 4000 n 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25′ 6 17 28 9 4100 m 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 4200 4300 4400 l 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 4500 4600 k 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25′ 6 4700 4800 4900 i 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 5000 5200 h 16 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 5100 5300 g 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 5400 5500 5600 f 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 5700 5800 e 13 24 5 16 27 8 19 * 11 22 3 14 25′ 6 17 28 9 20 1 5900 6000 6100 d 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 6200 6400 c 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 6300 6500 b 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25′ 6 17 28 6600 6800 a 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 6700 6900 P 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7000 7100 7200 N 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 3 14 25′ 7300 7400 M 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 2 13 24 7500 7600 7700 H 5 16 27 8 19 * 11 22 3 14 25 6 17 28 9 20 1 12 23 7800 8000 G 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 * 11 22 7900 8100 F 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 10 21 8200 8300 8400 E 2 13 24 5 16 27 8 19 * 11 22 3 14 25′ 6 17 28 9 20 1500 1600 8500 D 1 12 23 4 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19

As an example of the use of the preceding tables, suppose it were required to determine the moon's age on the 10th of April 1832. In 1832 the golden number is $({1832+1} \over {19} ) _{r}$ = 9 and the line of epacts belonging to the century is C. In Table III, under 9, and in the line C, we find the epact 28. In the calendar, Table IV., look for April, and the epact 28 is found opposite the second day. The 2nd of April is therefore the first day of the moon, and the 10th is consequently the ninth day of the moon. Again, suppose it were required to find the moon's age on the 2nd of December in the year 1916. In this case the golden number is $({1832+1} \over {19} ) _{r}$ = 17, and in Table III., opposite to 1900, the line of epacts is B. Under 17, in line B, the epact is 25′. In the calendar this epact first occurs before the 2nd of December at the 26th of November. The 26th of November is consequently the first day of the moon, and the 2nd of December is therefore the seventh day.

Table of centuries
Century digit
0 1 2 3 4 5 6 7 8 9
Millennium digit 0 D D D D D D D D D D
1 D D D D D D D C C B
2 B B A u A u t t t s
3 s r r r q p q p n n
4 n m l l l k k i i i
5 h g h g f f f e e d
6 d d c b c b a P a P
7 N N N M M H H H G F
8 G F E E E D D C C B
9 B B A u A u t t t s

## Gregorian calendar to find an epactEdit

The next, and indeed the principal use of the calendar, is to find Easter, which, according to the traditional regulation of the council of Nice, must be determined from the following conditions:

1. Easter must be celebrated on a Sunday;
2. this Sunday must follow the 14th day of the paschal moon, so that if the 14th of the paschal moon falls on a Sunday then Easter must be celebrated on the Sunday following;
3. the paschal moon is that of which the 14th day falls on or next follows the day of the vernal equinox;
4. the equinox is fixed invariably in the calendar on the 21st of March.

Sometimes a misunderstanding has arisen from not observing that this regulation is to be construed according to the tabular full moon as determined from the epact, and not by the true full moon, which, in general, occurs one or two days earlier.

From these conditions it follows that the paschal full moon, or the 14th of the paschal moon, cannot happen before the 21st of March, and that Easter in consequence cannot happen before the 22nd of March. If the 14th of the moon falls on the 21st, the new moon must fall on the 8th; for 21 - 13 = 8; and the paschal new moon cannot happen before the 8th; for suppose the new moon to fall on the 7th, then the full moon would arrive on the 20th, or the day before the equinox. The following moon would be the paschal moon. But the fourteenth of this moon falls at the latest on the 18th of April, or 29 days after the 20th of March; for by reason of the double epact that occurs at the 4th and 5th of April, this lunation has only 29 days. Now, if in this case the 18th of April is Sunday, then Easter must be celebrated on the following Sunday, or the 25th of April. Hence Easter Sunday cannot happen earlier than the 22nd of March, or later than the 25th of April.

Hence we derive the following rule for finding Easter Sunday from the tables:

1. Find the golden number, and, from Table III., the epact of the proposed year.
2. Find in the calendar (see Easter) the first day after the 7th of March which corresponds to the epact of the year; this will be the first day of the paschal moon.
3. Reckon thirteen days after that of the first of the moon, the following will be the 14th of the moon or the day of the full paschal moon.
4. Find from Dominical letter. the dominical letter of the year, and observe in the calendar the first day, after the fourteenth of the moon, which corresponds to the dominical letter; this will be Easter Sunday.
Days Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
ELELELELELELELELELELELELEL
1 *A29D*D29G28B27E26G25 24C23F22A21D20F
2 29B28E29E28A27C25 26F25′25A23D22G21B20E19G
3 28C27F28F27B26D25 24G24B22E21A20C19F18A
4 27D25′26G27G25′26C25′25E23A23C21F20B19D18G17B
5 26E25 24A26A25 24D24F22B22D20G19C18E17A16C
6 25′25F23B25′25B23E23G21C21E19A18D17F16B15D
725*D24G22C24C22F22A20D20F18B17E16G15C14E
824E23A21D23D21G21B19E19G17C16F15A14D13F
923F22B20E22E20A20C18F18A16D15G14B13E12G
1022G21C19F21F19B19D17G17B15E14A13C12F11A
1121A20D18G20G18C18E16A16C14F13B12D11G10B
1220B19E17A19A17D17F15B15D13G12C11E10A9C
1319C18F16B18B16E16G14C14E12A11D10F9B8D
1418D17G15C17C15F15A13D13F11B10E9G8C7E
1517E16A14D16D14G14B12E12G10C9F8A7D6F
1616F15B13E15E13A13C11F11A9D8G7B6E5G
1715G14C12F14F12B12D10G10B8E7A6C5F4A
1814A13D11G13G11C11E9A9C7F6B5D4G3B
1913B12E10A12A10D10F8B8D6G5C4E3A2C
2012C11F9B11B9E9G7C7E5A4D3F2B1D
2111D10G8C10C8F8A6D6F4B3E2G1C*E
2210E9A7D9D7G7B5E5G3C2F1A*D29F
239F8B6E8E6A6C4F4A2D1G*B29E28G
248G7C5F7F5B5D3G3B1E*A29C28F27A
257A6D4G6G4C4E2A2C*F29B28D27G26B
266B5E3A5A3D3F1B1D29G28C27E25′26A25′25C
275C4F2B4B2E2G*C*E28A27D26F25 24B24D
284D3G1C3C1F1A29D29F27B25′26E25′25G23C23E
293E2A 2D*G*B28E28G26C25 24F24A22D22F
302F1B 1E29A29C27F27A25′25D23G23B21E21G
311G*C *F 28D 25′26B24E 22C 19′20

* = golden number