1911 Encyclopædia Britannica/Calendar/Ecclesiastical Calendar - Easter

Easter.—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:—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.

Table IV.—Gregorian Calendar.

Days.	Jan.		Feb.		March.		April.		May.		June.		July.		August.		Sept.		October.		Nov.		Dec.
	E	L	E	L	E	L	E	L	E	L	E	L	E	L	E	L	E	L	E	L	E	L	E	L
1	*	A	29	D	*	D	29	G	28	B	27	E	26	G	25 24	C	23	F	22	A	21	D	20	F
2	29	B	28	E	29	E	28	A	27	C	25 26	F	25′25	A	23	D	22	G	21	B	20	E	19	G
3	28	C	27	F	28	F	27	B	26	D	25 24	G	24	B	22	E	21	A	20	C	19	F	18	A
4	27	D	25′26	G	27	G	25′26	C	25′25	E	23	A	23	C	21	F	20	B	19	D	18	G	17	B
5	26	E	25 24	A	26	A	25 24	D	24	F	22	B	22	D	20	G	19	C	18	E	17	A	16	C
6	25′25	F	23	B	25′25	B	23	E	23	G	21	C	21	E	19	A	18	D	17	F	16	B	15	D
7	24	G	22	C	24	C	22	F	22	A	20	D	20	F	18	B	17	E	16	G	15	C	14	E
8	23	A	21	D	23	D	21	G	21	B	19	E	19	G	17	C	16	F	15	A	14	D	13	F
9	22	B	20	E	22	E	20	A	20	C	18	F	18	A	16	D	15	G	14	B	13	E	12	G
10	21	C	19	F	21	F	19	B	19	D	17	G	17	B	15	E	14	A	13	C	12	F	11	A
11	20	D	18	G	20	G	18	C	18	E	16	A	16	C	14	F	13	B	12	D	11	G	10	B
12	19	E	17	A	19	A	17	D	17	F	15	B	15	D	13	G	12	C	11	E	10	A	9	C
13	18	F	16	B	18	B	16	E	16	G	14	C	14	E	12	A	11	D	10	F	9	B	8	D
14	17	G	15	C	17	C	15	F	15	A	13	D	13	F	11	B	10	E	9	G	8	C	7	E
15	16	A	14	D	16	D	14	G	14	B	12	E	12	G	10	C	9	F	8	A	7	D	6	F
16	15	B	13	E	15	E	13	A	13	C	11	F	11	A	9	D	8	G	7	B	6	E	5	G
17	14	C	12	F	14	F	12	B	12	D	10	G	10	B	8	E	7	A	6	C	5	F	4	A
18	13	D	11	G	13	G	11	C	11	E	9	A	9	C	7	F	6	B	5	D	4	G	3	B
19	12	E	10	A	12	A	10	D	10	F	8	B	8	D	6	G	5	C	4	E	3	A	2	C
20	11	F	9	B	11	B	9	E	9	G	7	C	7	E	5	A	4	D	3	F	2	B	1	D
21	10	G	8	C	10	C	8	F	8	A	6	D	6	F	4	B	3	E	2	G	1	C	*	E
22	9	A	7	D	9	D	7	G	7	B	5	E	5	G	3	C	2	F	1	A	*	D	29	F
23	8	B	6	E	8	E	6	A	6	C	4	F	4	A	2	D	1	G	*	B	29	E	28	G
24	7	C	5	F	7	F	5	B	5	D	3	G	3	B	1	E	*	A	29	C	28	F	27	A
25	6	D	4	G	6	G	4	C	4	E	2	A	2	C	*	F	29	B	28	D	27	G	26	B
26	5	E	3	A	5	A	3	D	3	F	1	B	1	D	29	G	28	C	27	E	25′26	A	25′25	C
27	4	F	2	B	4	B	2	E	2	G	*	C	*	E	28	A	27	D	26	F	25 24	B	24	D
28	3	G	1	C	3	C	1	F	1	A	29	D	29	F	27	B	25′26	E	25′25	G	23	C	23	E
29	2	A			2	D	*	G	*	B	28	E	28	G	26	C	25 24	F	24	A	22	D	22	F
30	1	B			1	E	29	A	29	C	27	F	27	A	25′25	D	23	G	23	B	21	E	21	G
31	*	C			*	F			28	D			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. 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	”  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

L = 7m + 3 − x − (x/4)_w.

This equation gives the dominical letter of any year from the commencement of the era to the Reformation. In order to adapt it to the Gregorian calendar, we must first add the 10 days that were left out of the year 1582; in the second place we must add one day for every century that has elapsed since 1600, in consequence of the secular suppression of the intercalary day; and lastly we must deduct the units contained in a fourth of the same number, because every fourth centesimal year is still a leap year. Denoting, therefore, the number of the century (or the date after the two right-hand digits have been struck out) by c, the value of L must be increased by 10 + (c − 16) − (c − 16/4)_w. We have then

L = 7m + 3 − x − (x/4)_w + 10 + (c − 16) − (c − 16/4)_w;

that is, since 3 + 10 = 13 or 6 (the 7 days being rejected, as they do not affect the value of L),

L = 7m + 6 − x − (x/4)_w + (c − 16) − (c − 16/4)_w;

This formula is perfectly general, and easily calculated.

As an example, let us take the year 1839. this case, x = 1839, (x/4)_w = (1839/4)_w = 459, c = 18, c − 16 = 2, and (c − 16/4)_w = 0. Hence

L = 7m + 6 − 1839 − 459 + 2 − 0
L = 7m − 2290 = 7 × 328 − 2290.
L = 6 = letter F.

The year therefore begins with Tuesday. It will be remembered that in a leap year there are always two dominical letters, one of which is employed till the 29th of February, and the other till the end of the year. In this case, as the formula supposes the intercalation already made, the resulting letter is that which applies after the 29th of February. Before the intercalation the dominical letter had retrograded one place less. Thus for 1840 the formula gives D; during the first two months, therefore, the dominical letter is E.

In order to investigate a formula for the epact, let us make

E = the true epact of the given year;

J = the Julian epact, that is to say, the number the epact would have been if the Julian year had been still in use and the lunar cycle had been exact;

S = the correction depending on the solar year;

M = the correction depending on the lunar cycle;

then the equation of the epact will be

E = J + S + M;

so that E will be known when the numbers J, S, and M are determined.

The epact J depends on the golden number N, and must be determined from the fact that in 1582, the first year of the reformed calendar, N was 6, and J 26. For the following years, then, the golden numbers and epacts are as follows:

1583, N =  7, J = 26 + 11 − 30 =  7;
1584, N =  8, J =  7 + 11= 18;
1585, N =  9, J = 18 + 11= 29;
1586, N = 10, J = 29 + 11 − 30 = 10;

and, therefore, in general J = (26 + 11(N − 6)/30)_r. But the numerator of this fraction becomes by reduction 11 N − 40 or 11 N − 10 (the 30 being rejected, as the remainder only is sought) = N + 10(N − 1); therefore, ultimately,

J = (N + 10(N − 1)/30)_r.

On account of the solar equation S, the epact J must be diminished by unity every centesimal year, excepting always the fourth. After x centuries, therefore, it must be diminished by x − (x/4)_w. Now, as 1600 was a leap year, the first correction of the Julian intercalation took place in 1700; hence, taking c to denote the number of the century as before, the correction becomes (c − 16) − (c − 16/4)_w, which must be deducted from J. We have therefore

S = − (c − 16) + (c − 16/4)_w.

With regard to the lunar equation M, we have already stated that in the Gregorian calendar the epacts are increased by unity at the end of every period of 300 years seven times successively, and then the increase takes place once at the end of 400 years. This gives eight to be added in a period of twenty-five centuries, and 8x/25 in x centuries. But 8x/25 = 1/3(x − x/25). Now, from the manner in which the intercalation is directed to be made (namely, seven times successively at the end of 300 years, and once at the end of 400), it is evident that the fraction x/25 must amount to unity when the number of centuries amounts to twenty-four. In like manner, when the number of centuries is 24 + 25 = 49, we must have x/25 = 2; when the number of centuries is 24 + 2 × 25 = 74, then x/25 = 3; and, generally, when the number of centuries is 24 + n × 25, then x/25 = n + 1. Now this is a condition which will evidently be expressed in general by the formula n − (n + 1/25)_w. Hence the correction of the epact, or the number of days to be intercalated after x centuries reckoned from the commencement of one of the periods of twenty-five centuries, is $\left\lbrace {\frac {x-\left({\tfrac {x+1}{25}}\right)_{w}}{3}}\right\rbrace _{w}$. The last period of twenty-five centuries terminated with 1800; therefore, in any succeeding year, if c be the number of the century, we shall have x = c − 18 and x + 1 = c − 17. Let (c − 17/25)_w = a, then for all years after 1800 the value of M will be given by the formula (c − 18 − a/3)_w; therefore, counting from the beginning of the calendar in 1582,

M = {c − 15 − a/3}_w.

By the substitution of these values of J, S and M, the equation of the epact becomes

E = (N + 10(N − 1)/30)_r − (c − 16) + (c − 16/4)_w + (c − 15 − a/3)_w.

It may be remarked, that as a = (c − 17/25)_w, the value of a will be 0 till c − 17 = 25 or c = 42; therefore, till the year 4200, a may be neglected in the computation. Had the anticipation of the new moons been taken, as it ought to have been, at one day in 308 years instead of 3121/2, the lunar equation would have occurred only twelve times in 3700 years, or eleven times successively at the end of 300 years, and then at the end of 400. In strict accuracy, therefore, a ought to have no value till c − 17 = 37, or c = 54, that is to say, till the year 5400. The above formula for the epact is given by Delambre (Hist. de l’astronomie moderne, t. i. p. 9); it may be exhibited under a variety of forms, but the above is perhaps the best adapted for calculation. Another had previously been given by Gauss, but inaccurately, inasmuch as the correction depending on a was omitted.

Having determined the epact of the year, it only remains to find Easter Sunday from the conditions already laid down. Let
P = the number of days from the 21st of March to the 15th of the paschal moon, which is the first day on which Easter Sunday can fall;
p = the number of days from the 21st of March to Easter Sunday;
L = the number of the dominical letter of the year;
l = letter belonging to the day on which the 15th of the moon falls: then, since Easter is the Sunday following the 14th of the moon, we have

p = P + (L − l),

which is commonly called the number of direction.

The value of L is always given by the formula for the dominical letter, and P and l are easily deduced from the epact, as will appear from the following considerations.

When P = 1 the full moon is on the 21st of March, and the new moon on the 8th (21 − 13 = 8), therefore the moon’s age on the 1st of March (which is the same as on the 1st of January) is twenty-three days; the epact of the year is consequently twenty-three. When P = 2 the new moon falls on the ninth, and the epact is consequently twenty-two; and, in general, when P becomes 1 + x, E becomes 23 − x, therefore P + E = 1 + x + 23 − x = 24, and P = 24 − E. In like manner, when P = 1, l = D = 4; for D is the dominical letter of the calendar belonging to the 22nd of March. But it is evident that when l is increased by unity, that is to say, when the full moon falls a day later, the epact of the year is diminished by unity; therefore, in general, when l = 4 + x, E = 23 − x, whence, l + E = 27 and l = 27 − E. But P can never be less than 1 nor l less than 4, and in both cases E = 23. When, therefore, E is greater than 23, we must add 30 in order that P and l may have positive values in the formula P = 24 − E and l = 27 − E. Hence there are two cases.

When E < 24,$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	P = 24 − E.
	l = 27 − E, or (27 − E/7)r,
When E > 23,$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	P = 54 − E;
	l = 57 − E, or (57 − E/7)r.

By substituting one or other of these values of P and l, according as the case may be, in the formula p = P + (L − l), we shall have p, or the number of days from the 21st of March to Easter Sunday. It will be remarked, that as L − l cannot either be 0 or negative, we must add 7 to L as often as may be necessary, in order that L − l may be a positive whole number.

By means of the formulae which we have now given for the dominical letter, the golden number and the epact, Easter Sunday may be computed for any year after the Reformation, without the assistance of any tables whatever. As an example, suppose it were required to compute Easter for the year 1840. By substituting this number in the formula for the dominical letter, we have x = 1840, c − 16 = 2, (c − 16/4)_w = 0, therefore

L = 7m + 6 − 1840 − 460 + 2
  = 7m − 2292
  = 7 × 328 − 2292 = 2296 − 2292 = 4

L = 4 = letter D

(1).

For the golden number we have N = (1840 + 1/19)_r therefore

For the epact we have

(N + 10(N − 1)/30)_r = (17 + 160/30)_r = (177/30)_r = 27;

likewise c − 16 = 18 − 16 = 2, c − 15/3 = 1, a = 0; therefore

E = 27 − 2 + 1 = 26

(3).

Now since E > 23, we have for P and l,

P = 54 − E = 54 − 26 = 28,

l = (57 − E/7)_r = (57 − 26/7)_r = (31/7)_r = 3;

consequently, since p = P + (L − l),

p = 28 + (4 − 3) = 29;

that is to say, Easter happens twenty-nine days after the 21st of March, or on the 19th April, the same result as was before found from the tables.

The principal church feasts depending on Easter, and the times of their celebration are as follows:—

Septuagesima Sunday	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$is $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	9 weeks	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$	before Easter.
First Sunday in Lent		6 weeks
Ash Wednesday		46 days
Rogation Sunday	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$is $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}$	5 weeks	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$	after Easter.
Ascension day or Holy Thursday		39 days
Pentecost or Whitsunday		7 weeks
Trinity Sunday		8 weeks

The Gregorian calendar was introduced into Spain, Portugal and part of Italy the same day as at Rome. In France it was received in the same year in the month of December, and by the Catholic states of Germany the year following. In the Protestant states of Germany the Julian calendar was adhered to till the year 1700, when it was decreed by the diet of Regensburg that the new style and the Gregorian correction of the intercalation should be adopted. Instead, however, of employing the golden numbers and epacts for the determination of Easter and the movable feasts, it was resolved that the equinox and the paschal moon should be found by astronomical computation from the Rudolphine tables. But this method, though at first view it may appear more accurate, was soon found to be attended with numerous inconveniences, and was at length in 1774 abandoned at the instance of Frederick II., king of Prussia. In Denmark and Sweden the reformed calendar was received about the same time as in the Protestant states of Germany. It is remarkable that Russia still adheres to the Julian reckoning.

In Great Britain the alteration of the style was for a long time successfully opposed by popular prejudice. The inconvenience, however, of using a different date from that employed by the greater part of Europe in matters of history and chronology began to be generally felt; and at length the Calendar (New Style) Act 1750 was passed for the adoption of the new style in all public and legal transactions. The difference of the two styles, which then amounted to eleven days, was removed by ordering the day following the 2nd of September of the year 1752 to be accounted the 14th of that month; and in order to preserve uniformity in future, the Gregorian rule of intercalation respecting the secular years was adopted. At the same time, the commencement of the legal year was changed from the 25th of March to the 1st of January. In Scotland, January 1st was adopted for New Year’s Day from 1600, according to an act of the privy council in December 1599. This fact is of importance with reference to the date of legal deeds executed in Scotland between that period and 1751, when the change was effected in England. With respect to the movable feasts, Easter is determined by the rule laid down by the council of Nice; but instead of employing the new moons and epacts, the golden numbers are prefixed to the days of the full moons. In those years in which the line of epacts is changed in the Gregorian calendar, the golden numbers are removed to different days, and of course a new table is required whenever the solar or lunar equation occurs. The golden numbers have been placed so that Easter may fall on the same day as in the Gregorian calendar. The calendar of the church of England is therefore from century to century the same in form as the old Roman calendar, excepting that the golden numbers indicate the full moons instead of the new moons.