EPAOT
482
EPACT
19 solar years averaging 365^ days equal 6939 J
But later computators found that the average luna- tion lasts 29 days, 12 hours, 44 minutes, 3 seconds, consequently: — 235 calendar lunations (one
Lunar Cycle) equal 6939 d. IS h. m. s.
235 astronomical lunations
equal 6939 d. 16 h. 31m. 45 s.
Difference 1 h. 28 m. 15 s.
We thus see that the average Lunar Cycle is about 1^ hour too long, and that, though the new moons occur on the same dates in successive cycles, they occur, on an average, lA hour earlier in the day. The astrono- mers entrusted with the reformation of the calendar calculated that after a period of 312J years (310 years is according to our figvires a closer approximation) the new moons occur on the day preceding that intlicated by the Lunar Cycle, that is, that the moon is one day older at the beginning of the year than the Jletonic 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 SX312A or 2500 years). This addition of one to the Epacts is known as the Lunar Equation, and occurs at the beginning of the years ISOC), 2100, 2400, 2700, 3000, 3300, 3600, 3900, 4300, 4600, etc. A second disturbance of the Epacts is caused by the oc- currence of the non-bissextile centurial years. We have seen above that the assigning of 6939} days to 19 lunar years leads to an error of one day every 3V2h years, and that witliin these limits the lunar calendar must not be disturbed; but the assigning of 0939 1 tlays 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 centiu'ial years than in- dicated by the Lunar Cycle (e. g. a new moon which under ordinary circumstances would have occurred on 29 February will occur on 1 March), 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 prop- erly indicated is immaterial in a system whose sole object is to indicate as nearly as practicable the fourteenth day of the moon after 21 Marcli, the sul> traction of one from the Epacts takes place at the begin- ning of all non-bissextile centurial years and is known as the Solar Equation. In the following table, -M is written after the years which have the Lunar Equa- tion, and — 1 after those which have the Solar: —
1600
2800
4000
1700
— 1
2900
— 1
4100
— 1
lSOO+1
— 1
3000 f 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. .\s 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 bo 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 . . . Ill, 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 . 5 Feb., 4 April,
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 nine-
teen years (the case is exceedingly rare even in the
purely astronomical calendar); consequently, when-
ever the two Epacts XXV and XXIV occur in the
same nineteen years, the new moons of the year whose
Epact is XX\' 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 Ep.\ct. — 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 characteris- tic of the Metonic Cycle were to be retained in chang- ing 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 funda- mentally 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 improb- able, 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 in
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 de- pends 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,
the Goklen Number of 1484 is 3, since — '^j^ — = 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 reform- ing 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 equa- tions to show the changes in the Epacts neces.';ary to preserve the relations between the ecclesiastical and astronomical calendars, proceeiled to calcvilate the
