Hence we derive the following rule for finding Easter Sunday from the tables:—1st, Find the golden number, and, from Table III., the epact of the proposed year. 2nd, Find in the calendar (Table IV.) 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, 3rd, 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. 4th, Find from Table I. 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.
Table IV.—Gregorian Calendar.
Days. | Jan. | Feb. | March. | April. | May. | June. | ||||||
E | L | E | L | E | L | E | L | E | L | E | L | |
1 | * | A | 29 | D | * | D | 29 | G | 28 | B | 27 | E |
2 | 29 | B | 28 | E | 29 | E | 28 | A | 27 | C | 25 26 | F |
3 | 28 | C | 27 | F | 28 | F | 27 | B | 26 | D | 25 24 | G |
4 | 27 | D | 25 26 | G | 27 | G | 25′26 | C | 25′25 | E | 23 | A |
5 | 26 | E | 25 24 | A | 26 | A | 25 24 | D | 24 | F | 22 | B |
6 | 25′25 | F | 23 | B | 25′25 | B | 23 | E | 23 | G | 21 | C |
7 | 24 | G | 22 | C | 24 | C | 22 | F | 22 | A | 20 | D |
8 | 23 | A | 21 | D | 23 | D | 21 | G | 21 | B | 19 | E |
9 | 22 | B | 20 | E | 22 | E | 20 | A | 20 | C | 18 | F |
10 | 21 | C | 19 | F | 21 | F | 19 | B | 19 | D | 17 | G |
11 | 20 | D | 18 | G | 20 | G | 18 | C | 18 | E | 16 | A |
12 | 19 | E | 17 | A | 19 | A | 17 | D | 17 | F | 15 | B |
13 | 18 | F | 16 | B | 18 | B | 16 | E | 16 | G | 14 | C |
14 | 17 | G | 15 | C | 17 | C | 15 | F | 15 | A | 13 | D |
15 | 16 | A | 14 | D | 16 | D | 14 | G | 14 | B | 12 | E |
16 | 15 | B | 13 | E | 15 | E | 13 | A | 13 | C | 11 | F |
17 | 14 | C | 12 | F | 14 | F | 12 | B | 12 | D | 10 | G |
18 | 13 | D | 11 | G | 13 | G | 11 | C | 11 | E | 9 | A |
19 | 12 | E | 10 | A | 12 | A | 10 | D | 10 | F | 8 | B |
20 | 11 | F | 9 | B | 11 | B | 9 | E | 9 | G | 7 | C |
21 | 10 | G | 8 | C | 10 | C | 8 | F | 8 | A | 6 | D |
22 | 9 | A | 7 | D | 9 | D | 7 | G | 7 | B | 5 | E |
23 | 8 | B | 6 | E | 8 | E | 6 | A | 6 | C | 4 | F |
24 | 7 | C | 5 | F | 7 | F | 5 | B | 5 | D | 3 | G |
25 | 6 | D | 4 | G | 6 | G | 4 | C | 4 | E | 2 | A |
26 | 5 | E | 3 | A | 5 | A | 3 | D | 3 | F | 1 | B |
27 | 4 | F | 2 | B | 4 | B | 2 | E | 2 | G | * | C |
28 | 3 | G | 1 | C | 3 | C | 1 | F | 1 | A | 29 | D |
29 | 2 | A | 2 | D | * | G | * | B | 28 | E | ||
30 | 1 | B | 1 | E | 29 | A | 29 | C | 27 | F | ||
31 | * | C | * | F | 28 | D | ||||||
Days. | July. | August. | Sept. | October. | Nov. | Dec. | ||||||
E | L | E | L | E | L | E | L | E | L | E | L | |
1 | 26 | G | 25 24 | C | 23 | F | 22 | A | 21 | D | 20 | F |
2 | 25′25 | A | 23 | D | 22 | G | 21 | B | 20 | E | 19 | G |
3 | 24 | B | 22 | E | 21 | A | 20 | C | 19 | F | 18 | A |
4 | 23 | C | 21 | F | 20 | B | 19 | D | 18 | G | 17 | B |
5 | 22 | D | 20 | G | 19 | C | 18 | E | 17 | A | 16 | C |
6 | 21 | E | 19 | A | 18 | D | 17 | F | 16 | B | 15 | D |
7 | 20 | F | 18 | B | 17 | E | 16 | G | 15 | C | 14 | E |
8 | 19 | G | 17 | C | 16 | F | 15 | A | 14 | D | 13 | F |
9 | 18 | A | 16 | D | 15 | G | 14 | B | 13 | E | 12 | G |
10 | 17 | B | 15 | E | 14 | A | 13 | C | 12 | F | 11 | A |
11 | 16 | C | 14 | F | 13 | B | 12 | D | 11 | G | 10 | B |
12 | 15 | D | 13 | G | 12 | C | 11 | E | 10 | A | 9 | C |
13 | 14 | E | 12 | A | 11 | D | 10 | F | 9 | B | 8 | D |
14 | 13 | F | 11 | B | 10 | E | 9 | G | 8 | C | 7 | E |
15 | 12 | G | 10 | C | 9 | F | 8 | A | 7 | D | 6 | F |
16 | 11 | A | 9 | D | 8 | G | 7 | B | 6 | E | 5 | G |
17 | 10 | B | 8 | E | 7 | A | 6 | C | 5 | F | 4 | A |
18 | 9 | C | 7 | F | 6 | B | 5 | D | 4 | G | 3 | B |
19 | 8 | D | 6 | G | 5 | C | 4 | E | 3 | A | 2 | C |
20 | 7 | E | 5 | A | 4 | D | 3 | F | 2 | B | 1 | D |
21 | 6 | F | 4 | B | 3 | E | 2 | G | 1 | C | * | E |
22 | 5 | G | 3 | C | 2 | F | 1 | A | * | D | 29 | F |
23 | 4 | A | 2 | D | 1 | G | * | B | 29 | E | 28 | G |
24 | 3 | B | 1 | E | * | A | 29 | C | 28 | F | 27 | A |
25 | 2 | C | * | F | 29 | B | 28 | D | 27 | G | 26 | B |
26 | 1 | D | 29 | G | 28 | C | 27 | E | 25′26 | A | 25′25 | C |
27 | * | E | 28 | A | 27 | D | 26 | F | 25 24 | B | 24 | D |
28 | 29 | F | 27 | B | 25′26 | E | 25′25 | G | 23 | C | 23 | E |
29 | 28 | G | 26 | C | 25 24 | F | 24 | A | 22 | D | 22 | F |
30 | 27 | A | 25′25 | D | 23 | G | 23 | B | 21 | E | 21 | G |
31 | 25′26 | B | 24 | E | 22 | C | 19′20 | A | ||||
Example.—Required the day on which Easter Sunday falls in the year 1840? 1st, For this year the golden number is ((1840 + 1) / 19)r = 17, and the epact (Table III. line C) is 26. 2nd, After the 7th of March the epact 26 first occurs in Table III. at the 4th of April, which, therefore, is the day of the new moon. 3rd, Since the new moon falls on the 4th, the full moon is on the 17th (4 + 13 = 17). 4th, The dominical letters of 1840 are E, D (Table I.), of which D must be taken, as E belongs only to January and February. After the 17th of April D first occurs in the calendar (Table IV.) at the 19th. Therefore, in 1840, Easter Sunday falls on the 19th of April. The operation is in all cases much facilitated by means of the table on next page.
Such is the very complicated and artificial, though highly ingenious method, invented by Lilius, for the determination of Easter and the other movable feasts. Its principal, though perhaps least obvious advantage, consists in its being entirely independent of astronomical tables, or indeed of any celestial phenomena whatever; so that all chances of disagreement arising from the inevitable errors of tables, or the uncertainty of observation, are avoided, and Easter determined without the
possibility of mistake. But this advantage is only procured by the sacrifice of some accuracy; for notwithstanding the cumbersome apparatus employed, the conditions of the problem are not always exactly satisfied, nor is it possible that they can be always satisfied by any similar method of proceeding. The equinox is fixed on the 21st of March, though the sun enters Aries generally on the 20th of that month, sometimes even on the 19th. It is accordingly quite possible that a full moon may arrive after the true equinox, and yet precede the 21st of March. This, therefore, would not be the paschal moon of the calendar, though it undoubtedly ought to be so if the intention of the council of Nice were rigidly followed. The new moons indicated by the epacts also differ from the astronomical new moons, and even from the mean new moons, in general by one or two days. In imitation of the Jews, who counted the time of the new moon, not from the moment of the actual phase, but from the time the moon first became visible after the conjunction, the fourteenth day of the moon is regarded as the full moon: but the moon is in opposition generally on the 16th day; therefore, when the new moons of the calendar nearly concur with the true new moons, the full moons are considerably in error. The epacts are also placed so as to indicate the full moons generally one or two days after the true full moons; but this was done purposely, to avoid the chance of concurring with the Jewish passover, which the framers of the calendar seem to have considered a greater evil than that of celebrating Easter a week too late.
Table V.—Perpetual Table, showing Easter.
Epact. | Dominical Letter. | ||||||
A | B | C | D | E | F | G | |
* | Apr. 16 | Apr. 17 | Apr. 18 | Apr. 19 | Apr. 20 | Apr. 14 | Apr. 15 |
1 | " 16 | " 17 | " 18 | " 19 | " 13 | " 14 | " 15 |
2 | " 16 | " 17 | " 18 | " 12 | " 13 | " 14 | " 15 |
3 | " 16 | " 17 | " 11 | " 12 | " 13 | " 14 | " 15 |
4 | " 16 | " 10 | " 11 | " 12 | " 13 | " 14 | " 15 |
5 | " 9 | " 10 | " 11 | " 12 | " 13 | " 14 | " 15 |
6 | " 9 | " 10 | " 11 | " 12 | " 13 | " 14 | " 8 |
7 | " 9 | " 10 | " 11 | " 12 | " 13 | " 7 | " 8 |
8 | " 9 | " 10 | " 11 | " 12 | " 6 | " 7 | " 8 |
9 | " 9 | " 10 | " 11 | " 5 | " 6 | " 7 | " 8 |
10 | " 9 | " 10 | " 4 | " 5 | " 6 | " 7 | " 8 |
11 | " 9 | " 3 | " 4 | " 5 | " 6 | " 7 | " 8 |
12 | " 2 | " 3 | " 4 | " 5 | " 6 | " 7 | " 8 |
13 | " 2 | " 3 | " 4 | " 5 | " 6 | " 7 | " 1 |
14 | " 2 | " 3 | " 4 | " 5 | " 6 | Mar. 31 | " 1 |
15 | " 2 | " 3 | " 4 | " 5 | Mar. 30 | " 31 | " 1 |
16 | " 2 | " 3 | " 4 | Mar. 29 | " 30 | " 31 | " 1 |
17 | " 2 | " 3 | Mar. 28 | " 29 | " 30 | " 31 | " 1 |
18 | " 2 | Mar. 27 | " 28 | " 29 | " 30 | " 31 | " 1 |
19 | Mar. 26 | " 27 | " 28 | " 29 | " 30 | " 31 | " 1 |
20 | " 26 | " 27 | " 28 | " 29 | " 30 | " 31 | Mar. 25 |
21 | " 26 | " 27 | " 28 | " 29 | " 30 | " 24 | " 25 |
22 | " 26 | " 27 | " 28 | " 29 | " 23 | " 24 | " 25 |
23 | " 26 | " 27 | " 28 | " 22 | " 23 | " 24 | " 25 |
24 | Apr. 23 | Apr. 24 | Apr. 25 | Apr. 19 | Apr. 20 | Apr. 21 | Apr. 22 |
25 | " 23 | " 24 | " 25 | " 19 | " 20 | " 21 | " 22 |
26 | " 23 | " 24 | " 18 | " 19 | " 20 | " 21 | " 22 |
27 | " 23 | " 17 | " 18 | " 19 | " 20 | " 21 | " 22 |
28 | " 16 | " 17 | " 18 | " 19 | " 20 | " 21 | " 22 |
29 | " 16 | " 17 | " 18 | " 19 | " 20 | " 21 | " 15 |
We will now show in what manner this whole apparatus of methods and tables may be dispensed with, and the Gregorian calendar reduced to a few simple formulae of easy computation.
And, first, to find the dominical letter. Let L denote the number of the dominical letter of any given year of the era. Then, since every year which is not a leap year ends with the same day as that with which it began, the dominical letter of the following year must be L - 1, retrograding one letter every common year. After x years, therefore, the number of the letter will be L - x. But as L can never exceed 7, the number x will always exceed L after the first seven years of the era. In order, therefore, to render the subtraction possible, L must be increased by some multiple of 7, as 7m, and the formula then becomes 7m + L - x. In the year preceding the first of the era, the dominical letter was C; for that year, therefore, we have L = 3; consequently for any succeeding year x, L = 7m + 3 - x, the years being all supposed to consist of 365 days. But every fourth year is a leap year, and the effect of the intercalation is to throw the dominical letter one place farther back. The above expression must therefore be diminished by the number of units in x/4, or by (x/4)w (this notation being used to denote the quotient, in a whole number, that arises from dividing x by 4). Hence in the Julian calendar the dominical letter is given by the equation