By means of the lunar cycle the new moons of the calendar were indicated before the Reformation. As the cycle restores these phenomena to the same days of the civil month, they will fall on the same days in any two years which occupy the same place in the cycle; consequently a table of the moon's phases for 19 years will serve for any year whatever when we know its number in the cycle. This number is called theGolden Number, either because it was so termed by the Greeks, or because it was usual to mark it with red letters in the calendar. The Golden Numbers were introduced into the calendar about the year 530, but disposed as they would have been if they had been inserted at the time of the council of Nicaea. The cycle is supposed to commence with the year in which the new moon falls on the 1st of January, which took place the year preceding the commencement of our era. Hence, to find the Golden Number N, for any yearx, we have N = ((x+ 1) / 19)r, which gives the following rule:Add 1 to the date, divide the sum by 19; the quotient is the number of cycles elapsed, and the remainder is the Golden Number.When the remainder is 0, the proposed year is of course the last or 19th of the cycle. It ought to be remarked that the new moons, determined in this manner, may differ from the astronomical new moons sometimes as much as two days. The reason is that the sum of the solar and lunar inequalities, which are compensated in the whole period, may amount in certain cases to 10°, and thereby cause the new moon to arrive on the second day before or after its mean time.
Dionysian Period.—The cycle of the sun brings back the days of the month to the same day of the week; the lunar cycle restores the new moons to the same day of the month; therefore 28 × 19 = 532 years, includes all the variations in respect of the new moons and the dominical letters, and is consequently a period after which the new moons again occur on the same day of the month and the same day of the week. This is called theDionysianor GreatPaschal Period, from its having been employed by Dionysius Exiguus, familiarly styled "Denys the Little," in determining Easter Sunday. It was, however, first proposed by Victorius of Aquitain, who had been appointed by Pope Hilary to revise and correct the church calendar. Hence it is also called theVictorian Period. It continued in use till the Gregorian reformation.
Cycle of Indiction.—Besides the solar and lunar cycles, there is a third of 15 years, called the cycle of indiction, frequently employed in the computations of chronologists. This period is not astronomical, like the two former, but has reference to certain judicial acts which took place at stated epochs under the Greek emperors. Its commencement is referred to the 1st of January of the year 313 of the common era. By extending it backwards, it will be found that the first of the era was the fourth of the cycle of indiction. The number of any year in this cycle will therefore be given by the formula ((x+ 3) / 15)r, that is to say,add 3 to the date, divide the sum by 15, and the remainder is the year of the indiction. When the remainder is 0, the proposed year is the fifteenth of the cycle.
Julian Period.—The Julian period, proposed by the celebrated Joseph Scaliger as an universal measure of chronology, is formed by taking the continued product of the three cycles of the sun, of the moon, and of the indiction, and is consequently 28 × 19 × 15 = 7980 years. In the course of this long period no two years can be expressed by the same numbers in all the three cycles. Hence, when the number of any proposed year in each of the cycles is known, its number in the Julian period can be determined by the resolution of a very simple problem of the indeterminate analysis. It is unnecessary, however, in the present case to exhibit the general solution of the problem, because when the number in the period corresponding to any one year in the era has been ascertained, it is easy to establish the correspondence for all other years, without having again recourse to the direct solution of the problem. We shall therefore find the number of the Julian period corresponding to the first of our era.
We have already seen that the year 1 of the era had 10 for its number in the solar cycle, 2 in the lunar cycle, and 4 in the cycle of indiction; the question is therefore to find a number such, thatwhen it is divided by the three numbers 28, 19, and 15 respectively the three remainders shall be 10, 2, and 4.
Letx,y, andzbe the three quotients of the divisions; the number sought will then be expressed by 28x+ 10, by 19y+ 2, or by 15z+ 4. Hence the two equations
28x+ 10 = 19y+ 2 = 15z+ 4.
28x+ 10 = 19y+ 2 = 15z+ 4.
28x+ 10 = 19y+ 2 = 15z+ 4.
x= 18m′ + 16 +m′ = 19m′ + 16 . . . (1).
x= 18m′ + 16 +m′ = 19m′ + 16 . . . (1).
x= 18m′ + 16 +m′ = 19m′ + 16 . . . (1).
Again, since 28x+ 10 = 15z+ 4, we have
x= 14n′ + 3 +n′ = 15n′ + 3 . . . (2).
x= 14n′ + 3 +n′ = 15n′ + 3 . . . (2).
x= 14n′ + 3 +n′ = 15n′ + 3 . . . (2).
Equating the above two values ofx, we have
whencem′ = 16p′ - 52 -p′ = 15p′ - 52.
whencem′ = 16p′ - 52 -p′ = 15p′ - 52.
whencem′ = 16p′ - 52 -p′ = 15p′ - 52.
Now in this equationp′ may be any number whatever, provided 15p′ exceed 52. The smallest value ofp′ (which is the one here wanted) is therefore 4; for 15 × 4 = 60. Assuming thereforep′ = 4, we havem′ = 60 - 52 = 8; and consequently, sincex= 19m′ + 16,x= 19 × 8 + 16 = 168. The number required is consequently 28 × 168 + 10 = 4714.
Having found the number 4714 for the first of the era, the correspondence of the years of the era and of the period is as follows:—
Era,
1,
2,
3, ...
x,
Period,
4714,
4715,
4716, ...
4713 +x;
from which it is evident, that if we take P to represent the year of the Julian period, andxthe corresponding year of the Christian era, we shall have
P = 4713 +x, andx= P - 4713.
P = 4713 +x, andx= P - 4713.
P = 4713 +x, andx= P - 4713.
With regard to the numeration of the years previous to the commencement of the era, the practice is not uniform. Chronologists, in general, reckon the year preceding the first of the era -1, the next preceding -2, and so on. In this case
Era,
-1,
-2,
-3, ...
-x,
Period,
4713,
4712,
4711, ...
4714 -x;
whence
P = 4714 -x, andx= 4714 - P.
P = 4714 -x, andx= 4714 - P.
P = 4714 -x, andx= 4714 - P.
But astronomers, in order to preserve the uniformity of computation, make the series of years proceed without interruption, and reckon the year preceding the first of the era 0. Thus
Era,
0,
-1,
-2, ...
-x,
Period,
4713,
4712,
4711, ...
4713 -x;
therefore, in this case
P = 4713 -x, andx= 4713 - P.
P = 4713 -x, andx= 4713 - P.
P = 4713 -x, andx= 4713 - P.
Reformation of the Calendar.—The ancient church calendar was founded on two suppositions, both erroneous, namely, that the year contains 365¼ days, and that 235 lunations are exactly equal to nineteen solar years. It could not therefore long continue to preserve its correspondence with the seasons, or to indicate the days of the new moons with the same accuracy. About the year 730 the venerable Bede had already perceived the anticipation of the equinoxes, and remarked that these phenomena then took place about three days earlier than at the time of the council of Nicaea. Five centuries after the time of Bede, the divergence of the true equinox from the 21st of March, which now amounted to seven or eight days, was pointed out by Johannes de Sacro Bosco (John Holywood,fl.1230) in hisDe Anni Ratione; and by Roger Bacon, in a treatiseDe Reformatione Calendarii, which, though never published, was transmitted to the pope. These works were probably little regarded at the time; but as the errors of the calendar went on increasing, and the true length of the year, in consequence of the progress of astronomy, became better known, the project of a reformation was again revived in the 15th century; and in 1474 Pope Sixtus IV. invited Regiomontanus, the most celebrated astronomer of the age, to Rome, to superintend the reconstruction of the calendar. The premature death of Regiomontanus caused the design to be suspended for the time; but in the following century numerous memoirs appeared on the subject, among the authors of which were Stoffler, Albert Pighius, Johann Schöner, Lucas Gauricus, and other mathematicians of celebrity. At length Pope Gregory XIII. perceiving that the measure was likely to confer a greatéclaton his pontificate, undertook the long-desired reformation; and having found the governments of the principal Catholic states ready to adopt his views, he issued a brief in the month of March 1582, in which he abolished the use of the ancient calendar, and substituted that which has since been received in almost all Christian countries under the name of theGregorian CalendarorNew StyleThe author of the system adopted by Gregory was Aloysius Lilius, or Luigi Lilio Ghiraldi, a learned astronomer and physician of Naples, who died, however, before its introduction; but the individual who most contributed to give the ecclesiastical calendar its present form, and who was charged with all the calculations necessary for its verification, was Clavius, by whom it was completely developed and explained in a great folio treatise of 800 pages, published in 1603, the title of which is given at the end of this article.
It has already been mentioned that the error of the Julian year was corrected in the Gregorian calendar by the suppression of three intercalations in 400 years. In order to restore the beginning of the year to the same place in the seasons that it had occupied at the time of the council of Nicaea, Gregory directed the day following the feast of St Francis, that is to say the 5th of October, to be reckoned the 15th of that month. By this regulation the vernal equinox which then happened on the 11th of March was restored to the 21st. From 1582 to 1700 the difference between the old and new style continued to be ten days; but 1700 being a leap year in the Julian calendar, and a common year in the Gregorian, the difference of the styles during the 18th century was eleven days. The year 1800 was also common in the new calendar, and, consequently, the difference in the 19th century was twelve days. From 1900 to 2100 inclusive it is thirteen days.
The restoration of the equinox to its former place in the year and the correction of the intercalary period, were attended with no difficulty; but Lilius had also to adapt the lunar year to the new rule of intercalation. The lunar cycle contained 6939 days 18 hours, whereas the exact time of 235 lunations, as we have already seen, is 235 × 29.530588 = 6939 days 16 hours 31 minutes. The difference, which is 1 hour 29 minutes, amounts to a day in 308 years, so that at the end of this time the new moons occur one day earlier than they are indicated by the golden numbers. During the 1257 years that elapsed between the council of Nicaea and the Reformation, the error had accumulated to four days, so that the new moons which were marked in the calendar as happening, for example, on the 5th of the month, actually fell on the 1st. It would have been easy to correct this error by placing the golden numbers four lines higher in the new calendar; and the suppression of the ten days had already rendered it necessary to place them ten lines lower, and to carry those which belonged, for example, to the 5th and 6th of the month, to the 15th and 16th. But, supposing this correction to have been made, it would have again become necessary, at the end of 308 years, to advance them one line higher, in consequence of the accumulation of the error of the cycle to a whole day. On the other hand, as the golden numbers were only adapted to the Julian calendar, every omission of the centenary intercalation would require them to be placed one line lower, opposite the 6th, for example, instead of the 5th of the month; so that, generally speaking, the places of the golden numbers would have to be changed every century. On this account Lilius thought fit to reject the golden numbers from the calendar, and supply their place by another set of numbers calledEpacts, the use of which we shall now proceed to explain.
Epacts.—Epact is a word of Greek origin, employed in the calendar to signify the moon's age at the beginning of the year.The common solar year containing 365 days, and the lunar year only 354 days, the difference is eleven; whence, if a new moon fall on the 1st of January in any year, the moon will be eleven days old on the first day of the following year, and twenty-two days on the first of the third year. The numbers eleven and twenty-two are therefore the epacts of those years respectively. Another addition of eleven gives thirty-three for the epact of the fourth year; but in consequence of the insertion of the intercalary month in each third year of the lunar cycle, this epact is reduced to three. In like manner the epacts of all the following years of the cycle are obtained by successively adding eleven to the epact of the former year, and rejecting thirty as often as the sum exceeds that number. They are therefore connected with the golden numbers by the formula (11n/ 30) in whichnis any whole number; and for a whole lunar cycle (supposing the first epact to be 11), they are as follows:—11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29. But the order is interrupted at the end of the cycle; for the epact of the following year, found in the same manner, would be 29 + 11 = 40 or 10, whereas it ought again to be 11 to correspond with the moon's age and the golden number 1. The reason of this is, that the intercalary month, inserted at the end of the cycle, contains only twenty-nine days instead of thirty; whence, after 11 has been added to the epact of the year corresponding to the golden number 19, we must reject twenty-nine instead of thirty, in order to have the epact of the succeeding year; or, which comes to the same thing, we must add twelve to the epact of the last year of the cycle, and then reject thirty as before.
This method of forming the epacts might have been continued indefinitely if the Julian intercalation had been followed without correction, and the cycle been perfectly exact; but as neither of these suppositions is true, two equations or corrections must be applied, one depending on the error of the Julian year, which is called the solar equation; the other on the error of the lunar cycle, which is called the lunar equation. The solar equation occurs three times in 400 years, namely, in every secular year which is not a leap year; for in this case the omission of the intercalary day causes the new moons to arrive one day later in all the following months, so that the moon's age at the end of the month is one day less than it would have been if the intercalation had been made, and the epacts must accordingly be all diminished by unity. Thus the epacts 11, 22, 3, 14, &c., become 10, 21, 2, 13, &c. On the other hand, when the time by which the new moons anticipate the lunar cycle amounts to a whole day, which, as we have seen, it does in 308 years, the new moons will arrive one day earlier, and the epacts must consequently be increased by unity. Thus the epacts 11, 22, 3, 14, &c., in consequence of the lunar equation, become 12, 23, 4, 15, &c. In order to preserve the uniformity of the calendar, the epacts are changed only at the commencement of a century; the correction of the error of the lunar cycle is therefore made at the end of 300 years. In the Gregorian calendar this error is assumed to amount to one day in 312½ years or eight days in 2500 years, an assumption which requires the line of epacts to be changed seven times successively at the end of each period of 300 years, and once at the end of 400 years; and, from the manner in which the epacts were disposed at the Reformation, it was found most correct to suppose one of the periods of 2500 years to terminate with the year 1800.
The years in which the solar equation occurs, counting from the Reformation, are 1700, 1800, 1900, 2100, 2200, 2300, 2500, &c. Those in which the lunar equation occurs are 1800, 2100, 2400, 2700, 3000, 3300, 3600, 3900, after which, 4300, 4600 and so on. When the solar equation occurs, the epacts are diminished by unity; when the lunar equation occurs, the epacts are augmented by unity; and when both equations occur together, as in 1800, 2100, 2700, &c., they compensate each other, and the epacts are not changed.
In consequence of the solar and lunar equations, it is evident that the epact or moon's age at the beginning of the year, must, in the course of centuries, have all different values from one to thirty inclusive, corresponding to the days in a full lunar month. Hence, for the construction of a perpetual calendar, there must be thirty different sets or lines of epacts. These are exhibited in the subjoined table (Table III.) called theExtended Table of Epacts, which is constructed in the following manner. The series of golden numbers is written in a line at the top of the table, and under each golden number is a column of thirty epacts, arranged in the order of the natural numbers, beginning at the bottom and proceeding to the top of the column. The first column, under the golden number 1, contains the epacts, 1, 2, 3, 4, &c., to 30 or 0. The second column, corresponding to the following year in the lunar cycle, must have all its epacts augmented by 11; the lowest number, therefore, in the column is 12, then 13, 14, 15 and so on. The third column corresponding to the golden number 3, has for its first epact 12 + 11 = 23; and in the same manner all the nineteen columns of the table are formed. Each of the thirty lines of epacts is designated by a letter of the alphabet, which serves as its index or argument. The order of the letters, like that of the numbers, is from the bottom of the column upwards.
In the tables of the church calendar the epacts are usually printed in Roman numerals, excepting the last, which is designated by an asterisk (*), used as an indefinite symbol to denote 30 or 0, and 25, which in the last eight columns is expressed in Arabic characters, for a reason that will immediately be explained. In the table here given, this distinction is made by means of an accent placed over the last figure.
At the Reformation the epacts were given by the line D. The year 1600 was a leap year; the intercalation accordingly took place as usual, and there was no interruption in the order of the epacts; the line D was employed till 1700. In that year the omission of the intercalary day rendered it necessary to diminish the epacts by unity, or to pass to the line C. In 1800 the solar equation again occurred, in consequence of which it was necessary to descend one line to have the epacts diminished by unity; but in this year the lunar equation also occurred, the anticipation of the new moons having amounted to a day; the new moons accordingly happened a day earlier, which rendered it necessary to take the epacts in the next higher line. There was, consequently, no alteration; the two equations destroyed each other. The line of epacts belonging to the present century is therefore C. In 1900 the solar equation occurs, after which the line is B. The year 2000 is a leap year, and there is no alteration. In 2100 the equations again occur together and destroy each other, so that the line B will serve three centuries, from 1900 to 2200. From that year to 2300 the line will be A. In this manner the line of epacts belonging to any given century is easily found, and the method of proceeding is obvious. When the solar equation occurs alone, the line of epacts is changed to the next lower in the table; when the lunar equation occurs alone, the line is changed to the next higher; when both equations occur together, no change takes place. In order that it may be perceived at once to what centuries the different lines of epacts respectively belong, they have been placed in a column on the left hand side of the table on next page.
The use of the epacts is to show the days of the new moons, and consequently the moon's age on any day of the year. For this purpose they are placed in the calendar (Table IV.) along with the days of the month and dominical letters, in a retrograde order, so that the asterisk stands beside the 1st of January, 29 beside the 2nd, 28 beside the 3rd and so on to 1, which corresponds to the 30th. After this comes the asterisk, which corresponds to the 31st of January, then 29, which belongs to the 1st of February, and so on to the end of the year. The reason of this distribution is evident. If the last lunation of any year ends, for example, on the 2nd of December, the new moon falls on the 3rd; and the moon's age on the 31st, or at the end of the year, is twenty-nine days. The epact of the following year is therefore twenty-nine. Now that lunation having commenced on the 3rd of December, and consisting of thirty days, will end on the 1st of January. The 2nd of January is therefore the dayof the new moon, which is indicated by the epact twenty-nine. In like manner, if the new moon fell on the 4th of December, the epact of the following year would be twenty-eight, which, to indicate the day of next new moon, must correspond to the 3rd of January.
When the epact of the year is known, the days on which the new moons occur throughout the whole year are shown by Table IV., which is called theGregorian Calendar of Epacts. For example, the golden number of the year 1832 is ((1832 + 1) / 19)r= 9, and the epact, as found in Table III., is twenty-eight. This epact occurs at the 3rd of January, the 2nd of February, the 3rd of March, the 2nd of April, the 1st of May, &c., and these days are consequently the days of the ecclesiastical new moons in 1832. The astronomical new moons generally take place one or two days, sometimes even three days, earlier than those of the calendar.
There are some artifices employed in the construction of this table, to which it is necessary to pay attention. The thirty epacts correspond to the thirty days of a full lunar month; but the lunar months consist of twenty-nine and thirty days alternately, therefore in six months of the year the thirty epacts must correspond only to twenty-nine days. For this reason the epacts twenty-five and twenty-four are placed together, so as to belong only to one day in the months of February, April, June, August, September and November, and in the same months another 25′, distinguished by an accent, or by being printed in a different character, is placed beside 26, and belongs to the same day. The reason for doubling the 25 was to prevent the new moons from being indicated in the calendar as happening twice on the same day in the course of the lunar cycle, a thing which actually cannot take place. For example, if we observe the line B in Table III., we shall see that it contains both the epacts twenty-four and twenty-five, so that if these correspond to the same day of the month, two new moons would be indicated as happening on that day within nineteen years. Now the three epacts 24, 25, 26, can never occur in the same line; therefore in those lines in which 24 and 25 occur, the 25 is accented, and placed in the calendar beside 26. When 25 and 26 occur in the same line of epacts, the 25 is not accented, and in the calendar stands beside 24. The lines of epacts in which 24 and 25 both occur, are those which are marked by one of the eight lettersb,e,k,n,r, B, E, N, in all of which 25′ stands in a column corresponding to a golden number higher than 11. There are also eight lines in which 25 and 26 occur, namely,c,f,l,p,s, C, F, P. In the other 14 lines, 25 either does not occur at all, or it occurs in a line in which neither 24 nor 26 is found. From this it appears that if the golden number of the year exceeds 11, the epact 25, in six months of the year, must correspond to the same day in the calendar as 26; but if the golden number does not exceed 11, that epact must correspond to the same day as 24. Hence the reason for distinguishing 25 and 25′. In using the calendar, if the epact of the year is 25, and the golden number not above 11, take 25; but if the golden number exceeds 11, take 25′.
Another peculiarity requires explanation. The epact 19′ (also distinguished by an accent or different character) is placed in the same line with 20 at the 31st of December. It is, however, only used in those years in which the epact 19 concurs with the golden number 19. When the golden number is 19, that is to say, in the last year of the lunar cycle, the supplementary month contains only 29 days. Hence, if in that year the epact should be 19, a new moon would fall on the 2nd of December, and the lunation would terminate on the 30th, so that the next new moon would arrive on the 31st. The epact of the year, therefore, or 19, must stand beside that day, whereas, according to the regular order, the epact corresponding to the 31st of December is 20; and this is the reason for the distinction.
TableIII.Extended Table of Epacts.
Years.
Index.
Golden Numbers.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
1700 1800 8700
C
*
11
22
3
14
25
6
17
28
9
20
1
12
23
4
15
26
7
18
1900 2000 2100
B
29
10
21
2
13
24
5
16
27
8
19
*
11
22
3
14
25′
6
17
2200 2400
A
28
9
20
1
12
23
4
15
26
7
18
29
10
21
2
13
24
5
16
2300 2500
u
27
8
19
*
11
22
3
14
25
6
17
28
9
20
1
12
23
4
15
2600 2700 2800
t
26
7
18
29
10
21
2
13
24
5
16
27
8
19
*
11
22
3
14
2900 3000
s
25
6
17
28
9
20
1
12
23
4
15
26
7
18
29
10
21
2
13
3100 3200 3300
r
24
5
16
27
8
19
*
11
22
3
14
25′
6
17
28
9
20
1
12
3400 3600
q
23
4
15
26
7
18
29
10
21
2
13
24
5
16
27
8
19
*
11
3500 3700
p
22
3
14
25
6
17
28
9
20
1
12
23
4
15
26
7
18
29
10
3800 3900 4000
n
21
2
13
24
5
16
27
8
19
*
11
22
3
14
25′
6
17
28
9
4100
m
20
1
12
23
4
15
26
7
18
29
10
21
2
13
24
5
16
27
8
4200 4300 4400
l
19
*
11
22
3
14
25
6
17
28
9
20
1
12
23
4
15
26
7
4500 4600
k
18
29
10
21
2
13
24
5
16
27
8
19
*
11
22
3
14
25′
6
4700 4800 4900
i
17
28
9
20
1
12
23
4
15
26
7
18
29
10
21
2
13
24
5
5000 5200
h
16
27
8
19
*
11
22
3
14
25
6
17
28
9
20
1
12
23
4
5100 5300
g
15
26
7
18
29
10
21
2
13
24
5
16
27
8
19
*
11
22
3
5400 5500 5600
f
14
25
6
17
28
9
20
1
12
23
4
15
26
7
18
29
10
21
2
5700 5800
e
13
24
5
16
27
8
19
*
11
22
3
14
25′
6
17
28
9
20
1
5900 6000 6100
d
12
23
4
15
26
7
18
29
10
21
2
13
24
5
16
27
8
19
*
6200 6400
c
11
22
3
14
25
6
17
28
9
20
1
12
23
4
15
26
7
18
29
6300 6500
b
10
21
2
13
24
5
16
27
8
19
*
11
22
3
14
25′
6
17
28
6600 6800
a
9
20
1
12
23
4
15
26
7
18
29
10
21
2
13
24
5
16
27
6700 6900
P
8
19
*
11
22
3
14
25
6
17
28
9
20
1
12
23
4
15
26
7000 7100 7200
N
7
18
29
10
21
2
13
24
5
16
27
8
19
*
11
22
3
14
25′
7300 7400
M
6
17
28
9
20
1
12
23
4
15
26
7
18
29
10
21
2
13
24
7500 7600 7700
H
5
16
27
8
19
*
11
22
3
14
25
6
17
28
9
20
1
12
23
7800 8000
G
4
15
26
7
18
29
10
21
2
13
24
5
16
27
8
19
*
11
22
7900 8100
F
3
14
25
6
17
28
9
20
1
12
23
4
15
26
7
18
29
10
21
8200 8300 8400
E
2
13
24
5
16
27
8
19
*
11
22
3
14
25′
6
17
28
9
20
1500 1600 8500
D
1
12
23
4
15
26
7
18
29
10
21
2
13
24
5
16
27
8
19
As an example of the use of the preceding tables, suppose it were required to determine the moon's age on the 10th of April 1832. In 1832 the golden number is ((1832 + 1) / 19)r= 9 and the line of epacts belonging to the century is C. In Table III, under 9, and in the line C, we find the epact 28. In the calendar, Table IV., look for April, and the epact 28 is found opposite the second day. The 2nd of April is therefore the first day of the moon,and the 10th is consequently the ninth day of the moon. 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 ((1916 + 1) / 19)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 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 mustfollowthe 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;4ththe 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.
TableIV.—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