# Epact

### Surplus days of the solar over the lunar year

*Epac* t (Gr. Epaktai hemerai; Lat. dies adjecti), the surplus days of the solar over the lunar year; hence, more freely, the number of days in the age of the moon on January 1 of any given year. The whole system of epacts is based on the Metonic Lunar Cycle (otherwise known as the Cycle of Golden Numbers), and serves to indicate the *days* of the year on which the new moons occur.

THE CHURCH LUNAR CALENDAR.—It is generally held that the Last Supper took place on the Jewish Feast of the Passover, which was always kept on the fourteenth day of the first month of the old Jewish calendar. Consequently, since this month always began with that new moon of which the fourteenth day occurred on or next after the vernal equinox, Christ arose from the dead on Sunday, the seventeenth day of the so-called paschal moon. It is evident, then, that an exact anniversary of Easter is impossible except in years in which the seventeenth day of the paschal moon falls on Sunday. In the early days of Christianity there existed a difference of opinion between the Eastern and Western Churches as to the day on which Easter ought to be kept, the former keeping it on the fourteenth day and the latter on the Sunday following. To secure uniformity of practice, the Council of Nicaea (325) decreed that the Western method of keeping Easter on the Sunday after the fourteenth day of the moon should be adopted throughout the Church, believing no doubt that this mode fitted in better with the historical facts and wishing to give a lasting proof that the Jewish Passover was not, as the Quartodeciman heretics believed, an ordinance of Christianity.

As in the Julian calendar the months had lost all their original reference to the moon, the early Christians were compelled to use the Metonic Lunar Cycle of the Greeks to find the fourteenth day of the paschal moon. This cycle in its original form continued to be used until 1582, when it was revised and embodied in the Gregorian calendar. The Church claims no astronomical exactness for her lunar calendar; we shall show presently the confusion which would necessarily result from an extreme adherence to precise astronomical data in determining the date of Easter. She wishes merely to ensure that the fourteenth day of the calendar moon shall fall on or shortly after the real fourteenth day but never before it, since it would be chronologically absurd to keep Easter on or before the Passover. Otherwise, as Clavius plainly states (Romani Calendarii a Gregorio XIII P.M. restituti explicatio, cap. V, § 13, p. 85), she regards with indifference the occurrence of the moons on the day *before or after their* proper seats and cares much more for peace and uniformity than for the equinox and the new moon. It may be mentioned here that Clavius's estimate of the accuracy of the calendar, in the compilation of which he took such a leading part, is extremely modest, and the seats assigned by him to the new moons tally with strict astronomical findings in a degree which he seems never to have anticipated. The impossibility of taking the astronomical moons as our sole guide in finding the date of Easter will be best understood from an example: Let us suppose that Easter is to be kept (as is at least implied by the British Act of Parliament regulating its date) on the Sunday after the astronomical full moon, and that this full moon, as sometimes happens, occurs just before midnight on Saturday evening in the western districts of London or New York. The full moon will therefore happen a little after midnight in the eastern districts, so that Easter, if regulated strictly by the paschal full moon, must be kept on one Sunday in the western and on the following Sunday in the eastern districts of the same city. Lest it be thought that this is carrying astronomical exactness to extremes, we may say that, if Easter were dependent on the astronomical moons, the feast could not always be kept on the same Sunday in England and America. Seeing, therefore, that astronomical accuracy must at some point give way to convenience and that an arbitrary decision on this point is necessary, the Church has drawn up a lunar calendar which maintains as close a relation with the astronomical moons as is practicable, and has decreed that Easter is to be kept on the Sunday after the fourteenth day of the paschal moon as indicated by this calendar.

METONIC LUNAR CYCLE OR CYCLE OF GOLDEN NUMBERS.—In the year now known as 432 B.C., Meton, an Athenian astronomer, discovered that 235 lunations (i.e. lunar months) correspond with 19 solar years, or, as we might express it, that after a period of 19 solar years the new moons occur again on the same days of the solar year. He therefore divided the calendar into periods of 19 years, which he numbered 1, 2, 3, etc. to 19, and assumed that the new moons would always fall on the same days in the years indicated by the same number. This discovery found such favor among the Athenians that the number assigned to the current year in the Metonic Cycle was henceforth written in golden characters on a pillar in the temple, and, whether owing to this circumstance or to the importance of the discovery itself, was known as the Golden Number of the year. As the 19 years of the Metonic Cycle were purely lunar (i.e. each contained an exact number of lunar months) and contained in the aggregate 235 lunations, it was clearly impossible that all the years should be of equal length. To twelve of the 19 years 12 lunations were assigned, and to the other seven 13 lunations, the thirteenth lunation being known as the embolismic or intercalary month.

Length of the Lunation.—The latest calculations have shown that the average duration of the lunar month is 29 days, 12 hours, 44 mins., 3 secs. To avoid the difficulty of reckoning fractions of a day in the calendar, all computators, ancient and modern, have assigned 30 and 29 days alternately to the lunations of the year, and regarded the ordinary lunar year of 12 lunations as lasting 354 days, whereas it really lasts some 8 hours and 48 mins. longer. This underestimation of the year is compensated for in two ways: (I) by the insertion of one extra day in the lunar (as in the solar) calendar every fourth year, and (2) by assigning 30 days to six of the seven embolismic lunations, although the average lunation lasts only about 291 days. A comparison of the solar and lunar calendars for 76 years (one cycle of 19 years is unsuitable in this case, since it contains sometimes 4, sometimes 5, leap years) will make this clearer:—76 solar years=76 X 365+19, i.e. 27,759 days. Therefore 940 calendar lunations (since 19 years equal 235 lunations) contain 27,759 days (29 d., 12 hrs., 44 mins., 3 secs. X940= 2. 7,758 d., 18 hrs., 7 mins.). But 940 lunations averaging 291 days equal only 27,730 days. Consequently, if we assign 30 and 29 days uninterruptedly to alternate lunations, the lunar calendar will, after 76 years, anticipate the solar by 29 days. The intercalation of the extra day every fourth year in the lunar calendar reduces the divergence to 10 days in 76 years, i.e. 2+1 days in 19 years. The divergence is removed by assigning to the seven embolismic months (which would otherwise have contained 7 X 291, or 2061, days) 209 days, 30 days being assigned to each of the first six and 29 to the seventh.

MANNER OF INSERTION OF THE EMBOLISMIC MONTHS.—As the Gregorian and Metonic calendars differ in the manner of inserting the embolismic months, only the former is spoken of here. It has just been said that seven of the 19 years of the lunar cycle contain a thirteenth, or embolismic, month, consisting in six cases of 30 days and in the seventh of 29 days. Granted that the first solar and lunar years begin on the same day (i.e. that the new moon occurs on January 1), it is evident that, as the ordinary lunar year of 12 lunations is 11 days shorter than the solar, the lunar calendar will, after three years, anticipate the solar by 33 days. To the third lunar year, then, is added the first embolismic month of 30 days, reducing the divergence between the calendars to three days. After three further years, i.e. at the end of the sixth year, the divergence will have mounted to 36 (3 X 11+ 3) days, but, by the insertion of the second embolismic lunation, will be reduced to six days. Whenever, then, the divergence between the calendars amounts to more than 30 days, an embolismic month is added to the lunar year; at the end of the nineteenth lunar year, the divergence will be 29 days, and, as the last embolismic month consists of 29 days, it is clear that after the insertion of this month the nineteenth solar and lunar years will end on the same day and that the first new moon of the twentieth (as of the first) year will occur on January 1. The divergence, therefore, at the end of the 19 successive years of the lunar cycle is: 11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, and 0 days.

CYCLE OF EPACTS.—We have defined an epact as the age of the moon on January 1, i.e. at the beginning of the year. If, then, the new moon occurs on January 1 in the first year of the Lunar Cycle, the Epact of the year is 0 or, as it is more usually expressed, *; and, since the lunar year always begins with the new moon, it is clear that the divergence between the solar and lunar calendars, of which we have just been speaking, gives the Epacts of the succeeding years. Thus, after the first year, the divergence between the calendars amounts to 11 days; therefore, the new moon occurs 11 days before January 1 of the second solar year, which is expressed by saying that the Epact of the second solar year is XI. Granted, then, that the new moon occurs on January 1 in the first year of the Lunar Cycle, the epacts of the 19 years are as follows:

Golden Numbers 1 2 3 4

Epacts * XI XXII III

5 6 7 8 9 10

XIV XXV VI XVII XXVIII IX

Golden Numbers 11 12 13 14

Epacts XX I XII XXIII

15 16 17 18 19

IV XV XXVI VII XVIII

INACCURACY OF THE METONIC CYCLE.—Meton's theory, as adopted by the Church until the year 1582, might be briefly expressed as follows:

The average Lunar Cycle consists of:

19 lunar years averaging 354¬º days i.e.......... ......6730¬æ Days

6 extra, or embolismic, months of 30 days, i.e....... 180 Days

1 embolismic month of 29 days......................... 29 Days

Total..............................................6939¬æ Days

19 solar years averaging 365, days equal................6939¬æ. But later computators found that the average lunation lasts 29 days, 12 hours, 44 minutes, 3 seconds, consequently:

235 calendar lunations (one Lunar Cycle) equal....... 6939 d. 18 h. 0 m. 0 s.

235 astronomical lunations equal......................... 6939 d. 16 h. 31 m. 45 s.

Difference:................................1 h. 28 m. 15 s.

We thus see that the average Lunar Cycle is about 14 hour too long, and that, though the new moons occur on the same dates in successive cycles, they occur, on an average, 1¬Ω hour earlier in the day. The astronomers entrusted with the reformation of the calendar calculated that after a period of 312¬Ω years (310 years is according to our figures a closer approximation) the new moons occur on the day preceding that indicated by the Lunar Cycle, that is, that the moon is one day older at the beginning of the year than the Metonic Cycle, if left unaltered, would show, and they removed this inaccuracy by adding one day to the age of the moon (i.e. to the Epacts) every 300 years seven times in succession and then one day after 400 years (i.e. eight days in 8 X 312¬Ω or 2500 years). This addition of one to the Epacts is known as the Lunar Equation, and occurs at the beginning of the years 1800, 2100, 2400, 2700, 3000, 3300, 3600, 3900, 4300, 4600, etc. A second disturbance of the Epacts is caused by the occurrence of the non-bissextile centurial years. We have seen above that the assigning of 6939¬æ days to 19 lunar years leads to an error of one day every 312¬Ω years, and that within these limits the lunar calendar must not be disturbed; but the assigning of 6939¬æ days to every 19 solar years amounts to an error of 3 days every 400 years, and it is therefore necessary to omit one day from the solar calendar in every centurial year not divisible by 400. Consequently, since this extra day in February every fourth year is an essential part of the lunar calendar, the new moons will occur one day later in the non-bissextile centurial years than indicated by the Lunar Cycle (e.g. a new moon which under ordinary circumstances would have occurred on February 29 will occur on March 1), and the age of the moon will, after the omission of the day, be one day less on all succeeding days of the solar year. As the fact that the January and February moons are not properly indicated is immaterial in a system whose sole object is to indicate as nearly as practicable the fourteenth day of the moon after March 21, the subtraction of one from the Epacts takes place at the beginning of all non-bissextile centurial years and is known as the Solar Equation. In the following table, +1 is written after the years which have the Lunar Equation, and -1 after those which have the Solar:

1600 2800 4000

1700 -1 2900 -1 4100 -1

1800 +1 -1 3000 +1 -1 4200 -1

1900 -1 3100 -1 4300 +1 -1

2000 3200 4400 2100 +1 -1 3300 +1 -1 4500 -1 2200 -1 3400 -1 4600 + 1 -1 2300 -1 3500 -1 4700 -1

2400 +1 3600 + 1 4800

2500 -1 3700 -1 4900 + 1 -1

2600 -1 3800 -1 5000 -1 2700 +1 -1 3900 + 1 -1 5100 -1

Clavius continued this table as far as the year 300,000, inserting the Lunar Equation eight times every 2500 years and the Solar three times every 400 years. As he thus treats the year 5200 as a leap year his table is untrustworthy after 5199.

INDICATION OF NEW MOONS.—Before proceeding further, it will be convenient to consider the method devised by Lilius of indicating the new moons of the year in the Gregorian calendar. As the first lunation of the year consists of 30 days, he wrote the Epacts *, XXIX, XXVIII... III, II, I opposite the first thirty days of January; then continuing, he wrote * opposite the thirty-first, XXIX opposite the first of February and so on to the end of the year, except that in the case of the lunations of 29 days he wrote the two Epacts XXV, XXIV opposite the same day (cf. February 5, April 4, etc. in the Church calendar). From this arrangement it is evident that if, for example, the Epact of a year is X, the new moons will occur in that year on the days before which the Epact X is placed in the calendar. One qualification must be made to this statement. According to the Metonic Cycle, the new moon can never occur twice on the same date in the same nineteen years (the case is exceedingly rare even in the purely astronomical calendar); consequently, whenever the two Epacts XXV and XXIV occur in the same nineteen years, the new moons of the year whose Epact is XXV are indicated in the months of 29 days by Epact XXVI with which the number 25 is for this object associated in the Church calendar.

HOW TO FIND THE EPACT.—We have already seen that the Church used the Metonic Cycle until the year 1582 as the only practical means devised of finding the fourteenth day of the paschal moon. Now, this cycle has always been regarded as starting from the year 1 B.C., and not from the year of its introduction (432 B.C.), probably (although all the authors we have seen appear to have overlooked the point) because such change was found necessary if the leading characteristic of the Metonic Cycle were to be retained in changing from a lunar to a solar calendar, viz., that the first lunar and solar years of the cycle should begin on the same day. That two nations with calendars so fundamentally different as those of the Greeks and the Romans should regard the solar year as beginning with the same phases of the sun would be highly improbable, even if there were no direct evidence that such was not the case. But we have shown that when the solar and lunar years begin on the same day, the Epacts of the successive years of the cycle are:

Golden Numbers 1 2 3 4

Epacts * XI XXII III

5 6 7 8 9 10

XIV XXV VI XVII XXVIII IX

Golden Numbers 11 12 13 14

Epacts XX I XII XXIII

15 16 17 18 19

IV XV XXVI VII XVIII

Consequently, if we divide the calendar into cycles of 19 years from 1 B.C., the first year of each cycle will have the Epact *, the second the Epact XI and so on, or, in other words, the Epact of any year before 1582 depends solely on its Golden Number. The Golden Number of any year may be found by adding 1 to the year and dividing by 19, the quotient showing the number of complete cycles elapsed since 1 B.C. and the remainder (or, if there be no remainder, 19) being the Golden Number of the year. Thus, for example,

*1481 + 1*

the Golden Number of 1484 is 3, since 19 = 78,

with 3 as remainder; therefore the Epact of the year 1484 is XXII.

In the course of time it was found that the paschal moon of the Metonic Cycle was losing all relation to the real paschal moon, and in the sixteenth century (c. 1576) Gregory XIII entrusted the task of reforming the calendar to a small body of astronomers, of whom Lilius and Clavius are the most renowned. These astronomers having drawn up the table of equations to show the changes in the Epacts necessary to preserve the relations between the ecclesiastical and astronomical calendars, proceeded to calculate the proper Epacts for the years of the Lunar Cycle after 1582. These they found to be as follows:

Golden Numbers 1 2 3 4

Epacts I XII XXIII IV

5 6 7 8 9 10

XV XXVI VII XVIII XXIX X

Golden Numbers 11 12 13 14

Epacts XXI II XIII XXIV

15 16 17 18 19

V XVI XXVII VIII XIX

Now the essential difference between the Metonic Cycle and the Gregorian system of Epacts lies in this, that, whereas the sphere of application of the former was held to be unlimited, that of the latter is bounded by the Lunar and Solar Equations. Since, then, a Solar Equation occurs in 1700, the Cycle of Epacts just given holds only for the period 1582-1699, after which a new cycle must be formed. To understand the reason of the changes we must remember (I) that by treating 365 days as equivalent to one solar year and to 12 lunations plus 11 days, we underestimate the solar year by about 5 4/5 hours and the lunations by 8 4/5 hours; (2) that in consequence of this underestimation of the solar year, one day must be inserted in every fourth solar year except in the case of the centurial years not divisible by 400; and (3) that the underestimation of the lunations by 6 hours every year (the additional 2 4/5 hours are compensated for in the embolismic months and by the Lunar Equation) necessitates the insertion of one extra day in the lunar calendar every fourth year without exception. To take an example: the Epact of 1696 (its Golden Number being 6) is XXVI, and since this Epact is found opposite February 4 in the Church calendar we know that in 1696 the new moon happened on that date and that consequently February 23 was the twentieth day of the calendar moon. But, since the underestimation of the lunations amounts to one day in every four years, the following day (our February 24) was only nominally the twenty-first day of the moon and the proper twenty-first was our February 25. The Church therefore inserted an extra day after February 23 and treated this and the real February 24 (our 24 and 25) as one continuous day in both the solar and lunar calendars, and consequently February 25 (our 26) was again legitimately regarded as the twenty-second day of the moon and the fifty-sixth day of the astronomical solar year. Coming now to the year 1700, we find its Epact to be X, consequently the new moon occurred on February 19 and February 23 was the fifth day of the calendar moon. But, since no extra day could be inserted in February, 1700, the twenty-fourth and twenty-fifth of this month had to be treated as the sixth day of the moon, and the age of the moon on every subsequent day of the year 1700 was one day less than indicated by the Epact X. As the moons of January and February are of very secondary importance in the Church calendar, we may say that the age of the moon in 1700 and all subsequent years was one day less than indicated by the above Cycle of Epacts, and thus the Epacts for the years of the Lunar Cycle after 1700 are:

Golden Numbers 1 2 3 4

Epacts * XI XXII III

5 6 7 8 9 10

XIV XXV VI XVII XXVIII IX

Golden Numbers 11 12 13 14

Epacts XX I XII XXIII

15 16 17 18 19

IV XV XXVI VII XVIII

In the year 1800, both the Lunar and Solar Equations (i.e. the addition and subtraction of 1) occur and no change of Epacts takes place. In 1900 the Solar Equation occurs and we must again subtract 1 from the Epacts. No change takes place in 2000 or in 2100, the former being a leap year and the latter having both equations. In 2200 and in 2300, we must again subtract 1, while in 2400, in which the Lunar Equation occurs and is not neutralized as usual by the Solar Equation, we add 1 to all the Epacts. The accompanying table gives the Epact of every year from 1 B.C. to A.D. 3099.

Examples.—(I) To find the Epact of the year 3097.

3097 + 1

Golden Number is 1, since 19 = 163, with 1 as remainder. Epact corresponding to Golden Number 1 after 2900 is XXV; therefore the Epact of 3097 is XXV.

(2) On what Sunday will Easter fall in the year 2459? Golden Number of 2459 is 9, and Epact of ninth year of Lunar Cycle after 2400 is XXVI. Since the Epact of 2459 is XXVI, the new moons of this year will occur on the days before which XXVI is placed in the Church calendar (e.g. in the Breviary). Now, since the paschal moon is that whose fourteenth day falls on or next after March 21, the paschal new moon can never happen before March 8. The first day after March 8 to which the Epact XXVI is prefixed in the Church calendar is April 4; consequently the paschal new moon in the year 2459 will occur on April 4. Counting 14 days from April 4, which we include in our reckoning, we find the fourteenth day of the paschal moon to be April 17. In 2459, therefore, Easter will be kept on the Sunday after April 17, which with the help of the Dominical Letters is found to be April 20. (See Dominical Letter.)

THOMAS KENNEDY