Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/747

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 $\left({\frac {\scriptstyle 1916+1}{\scriptstyle {\text{19}}}}\right)_{r}$ = 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.

Easter.—The next, and indeed the principal use of the calendar, is to find Easter, which, according to the regulation of the Council of Nice, must be determined from the following conditions:—1st, Easter must be celebrated on a Sunday; 2nd, 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; 3rd, the paschal moon is that of which the 14th day falls on or next follows the day of the vernal equinox; 4th, 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:—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.

Example.—Required the day on which Easter Sunday falls in the year 1840? 1st, For this year the golden number is $\left({\frac {\scriptstyle 1840+1}{\scriptstyle {\text{19}}}}\right)_{r}$ = 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 following table:—

Table V.—Perpetual Table, showing Easter.

Epact.	Dominical Letter.
	For Leap Years use the second 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	,,	20	,,	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

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 Moon|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.

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 formulæ 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