BYZANTINE ART (continued 9)

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, and x the corresponding year of the Christian era, we shall have

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

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

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 his De Anni Ratione; and by Roger Bacon, in a treatise De 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 great éclat on 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 the Gregorian Calendar or New Style The 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 called Epacts, 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 (11 n / 30) in which n is 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 the Extended 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 day of 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 the Gregorian 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 letters b, 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.

Table III. 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 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.

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. 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. In this case,

x = 1839,x 4_w =1839 4_w = 459, c = 18, c - 16 = 2, andc - 16 4_w = 0.

Hence

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

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:

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 x/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 {(x - ((x+1) / 25)_w) / 3}_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 312½, 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

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, P = 24 - E

l = 27 - E, or27 - E 7_r,

When E > 23, P = 54 - E

l = 57 - E, or57 - 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

For the golden number we have N = ((1840 + 1) / 19)_r; therefore N = 17 . . . (2).

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

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

l =57 - E 7_r = 57 - 26 7_r = 31 7_r = 3;

consequently, since p = P + (L - l),

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 is 9 weeks before Easter.

First Sunday in Lent 6 weeks

Ash Wednesday 46 days

Rogation Sunday is 5 weeks 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.

Hebrew Calendar.—In the construction of the Jewish calendar numerous details require attention. The calendar is dated from the Creation, which is considered to have taken place 3760 years and 3 months before the commencement of the Christian era. The year is luni-solar, and, according as it is ordinary or embolismic, consists of twelve or thirteen lunar months, each of which has 29 or 30 days. Thus the duration of the ordinary year is 354 days, and that of the embolismic is 384 days. In either case, it is sometimes made a day more, and sometimes a day less, in order that certain festivals may fall on proper days of the week for their due observance. The distribution of the embolismic years, in each cycle of 19 years, is determined according to the following rule:—

The number of the Hebrew year (Y) which has its commencement in a Gregorian year (x) is obtained by the addition of 3761 years; that is, Y = x + 3761. Divide the Hebrew year by 19; then the quotient is the number of the last completed cycle, and the remainder is the year of the current cycle. If the remainder be 3, 6, 8, 11, 14, 17 or 19 (0), the year is embolismic; if any other number, it is ordinary. Or, otherwise, if we find the remainder

R=7Y+1 19_r

the year is embolismic when R < 7.

The calendar is constructed on the assumptions that the mean lunation is 29 days 12 hours 44 min. 3⅓ sec., and that the year commences on, or immediately after, the new moon following the autumnal equinox. The mean solar year is also assumed to be 365 days 5 hours 55 min. 25-25/57 sec., so that a cycle of nineteen of such years, containing 6939 days 16 hours 33 min. 3⅓ sec., is the exact measure of 235 of the assumed lunations. The year 5606 was the first of a cycle, and the mean new moon, appertaining to the 1st of Tisri for that year, was 1845, October 1, 15 hours 42 min. 43⅓ sec., as computed by Lindo, and adopting the civil mode of reckoning from the previous midnight. The times of all future new moons may consequently be deduced by successively adding 29 days 12 hours 44 min. 3⅓ sec. to this date.

To compute the times of the new moons which determine the commencement of successive years, it must be observed that in passing from an ordinary year the new moon of the following year is deduced by subtracting the interval that twelve lunations fall short of the corresponding Gregorian year of 365 or 366 days; and that, in passing from an embolismic year, it is to be found by adding the excess of thirteen lunations over the Gregorian year. Thus to deduce the new moon of Tisri, for the year immediately following any given year (Y), when Y is

ordinary, subtract10 11days 15 hours 11 min. 20 sec.,

embolismic, add18 17days 21 hours 32 min. 43½ sec.

the second-mentioned number of days being used, in each case, whenever the following or new Gregorian year is bissextile.

Hence, knowing which of the years are embolismic, from their ordinal position in the cycle, according to the rule before stated, the times of the commencement of successive years may be thus carried on indefinitely without any difficulty. But some slight adjustments will occasionally be needed for the reasons before assigned, viz. to avoid certain festivals falling on incompatible days of the week. Whenever the computed conjunction falls on a Sunday, Wednesday or Friday, the new year is in such case to be fixed on the day after. It will also be requisite to attend to the following conditions:—

If the computed new moon be after 18 hours, the following day is to be taken, and if that happen to be Sunday, Wednesday or Friday, it must be further postponed one day. If, for an ordinary year, the new moon falls on a Tuesday, as late as 9 hours 11 min. 20 sec., it is not to be observed thereon; and as it may not be held on a Wednesday, it is in such case to be postponed to Thursday. If, for a year immediately following an embolismic year, the computed new moon is on Monday, as late as 15 hours 30 min. 52 sec., the new year is to be fixed on Tuesday.

After the dates of commencement of the successive Hebrew years are finally adjusted, conformably with the foregoing directions, an estimation of the consecutive intervals, by taking the differences, will show the duration and character of the years that respectively intervene. According to the number of days thus found to be comprised in the different years, the days of the several months are distributed as in Table VI.

The signs + and - are respectively annexed to Hesvan and Kislev to indicate that the former of these months may sometimes require to have one day more, and the latter sometimes one day less, than the number of days shown in the table—the result, in every case, being at once determined by the total number of days that the year may happen to contain. An ordinary year may comprise 353, 354 or 355 days; and an embolismic year 383, 384 or 385 days. In these cases respectively the year is said to be imperfect, common or perfect. The intercalary month, Veadar, is introduced in embolismic years in order that Passover, the 15th day of Nisan, may be kept at its proper season, which is the full moon of the vernal equinox, or that which takes place after the sun has entered the sign Aries. It always precedes the following new year by 163 days, or 23 weeks and 2 days; and Pentecost always precedes the new year by 113 days, or 16 weeks and 1 day.

Table VI.—Hebrew Months.

Hebrew Month. Ordinary Year. Embolismic Year.

Tisri 30 30

Hesvan 29+ 29+

Kislev 30- 30-

Tebet 29 29

Sebat 30 30

Adar 29 30

(Veadar) (...) (29)

Nisan 30 30

Yiar 29 29

Sivan 30 30

Tamuz 29 29

Ab 30 30

Elul 29 29

Total 354 384

The Gregorian epact being the age of the moon of Tebet at the beginning of the Gregorian year, it represents the day of Tebet which corresponds to January 1; and thus the approximate date of Tisri 1, the commencement of the Hebrew year, may be otherwise deduced by subtracting the epact from

Sept. 24 Oct. 24after anordinary embolismicHebrew year.

The result so obtained would in general be more accurate than the Jewish calculation, from which it may differ a day, as fractions of a day do not enter alike in these computations. Such difference may also in part be accounted for by the fact that the assumed duration of the solar year is 6 min. 39-25/57 sec. in excess of the true astronomical value, which will cause the dates of commencement of future Jewish years, so calculated, to advance forward from the equinox a day in error in 216 years. The lunations are estimated with much greater precision.

The following table is extracted from Woolhouse's Measures, Weights and Moneys of all Nations:—

Table VII.—Hebrew Years.

296 Cycle. Jewish Year Number of Days Commencement (1st of Tisri). 5606 354 Thur. 2 Oct. 1845 07 355 Mon. 21 Sept. 1846 08 383 Sat. 11 Sept. 1847 09 354 Thur. 28 Sept. 1848 10 355 Mon. 17 Sept. 1849 11 385 Sat. 7 Sept. 1850 12 353 Sat. 27 Sept. 1851 13 384 Tues. 14 Sept. 1852 14 355 Mon. 3 Oct. 1853 15 355 Sat. 23 Sept. 1854 16 383 Thur. 13 Sept. 1855 17 354 Tues. 30 Sept. 1856 18 355 Sat. 19 Sept. 1857 19 385 Thur. 9 Sept. 1858 20 354 Thur. 29 Sept. 1859 21 353 Mon. 17 Sept. 1860 22 385 Thur. 5 Sept. 1861 23 354 Thur. 25 Sept. 1862 24 383 Mon. 14 Sept. 1863 297 Cycle. 5625 355 Sat. 1 Oct. 1864 26 354 Thur. 21 Sept. 1865 27 385 Mon. 10 Sept. 1866 28 353 Mon. 30 Sept. 1867 29 354 Thur. 17 Sept. 1868 30 385 Mon. 6 Sept. 1869 31 355 Mon. 26 Sept. 1870 32 383 Sat. 16 Sept. 1871 33 354 Thur. 3 Oct. 1872 34 355 Mon. 22 Sept. 1873 35 383 Sat. 12 Sept. 1874 36 355 Thur. 30 Sept. 1875 37 354 Tues. 19 Sept. 1876 38 385 Sat. 8 Sept. 1877 39 355 Sat. 28 Sept. 1878 40 354 Thur. 18 Sept. 1879 41 383 Mon. 6 Sept. 1880 42 355 Sat. 24 Sept. 1881 43 383 Thur. 14 Sept. 1882 298 Cycle. 5644 354 Tues. 2 Oct. 1883 45 355 Sat. 20 Sept. 1884 46 385 Thur. 10 Sept. 1885 47 354 Thur. 30 Sept. 1886 48 353 Mon. 19 Sept. 1887 49 385 Thur. 6 Sept. 1888 50 354 Thur. 26 Sept. 1889 51 383 Mon. 15 Sept. 1890 52 355 Sat. 3 Oct. 1891 53 354 Thur. 22 Sept. 1892 54 385 Mon. 11 Sept. 1893 55 353 Mon. 1 Oct. 1894 56 355 Thur. 19 Sept. 1895 57 384 Tues. 8 Sept. 1896 58 355 Mon. 27 Sept. 1897 59 353 Sat. 17 Sept. 1898 60 384 Tues. 5 Sept. 1899 61 355 Mon. 24 Sept. 1900 62 383 Sat 14 Sept. 1901 299 Cycle. 5663 355 Thur. 2 Oct. 1902 64 354 Tues. 22 Sept. 1903 65 385 Sat. 10 Sept. 1904 66 355 Sat. 30 Sept. 1905 67 354 Thur. 20 Sept. 1906 68 383 Mon. 9 Sept. 1907 69 355 Sat. 26 Sept. 1908 70 383 Thur. 16 Sept. 1909 71 354 Tues. 4 Oct. 1910 72 355 Sat. 23 Sept. 1911 73 385 Thur. 12 Sept. 1912 74 354 Thur. 2 Oct. 1913 75 353 Mon. 21 Sept. 1914 76 385 Thur. 9 Sept. 1915 77 354 Thur. 28 Sept. 1916 78 355 Mon. 17 Sept. 1917 79 383 Sat. 7 Sept. 1918 80 354 Thur. 25 Sept. 1919 81 385 Mon. 13 Sept. 1920 300 Cycle. Jewish Year Number of Days Commencement (1st of Tisri). 5682 355 Mon. 3 Oct. 1921 83 353 Sat. 23 Sept. 1922 84 384 Tues. 11 Sept. 1923 85 355 Mon. 29 Sept. 1924 86 355 Sat. 19 Sept. 1925 87 383 Thur. 9 Sept. 1926 88 354 Tues. 27 Sept. 1927 89 385 Sat. 15 Sept. 1928 90 353 Sat. 5 Oct. 1929 91 354 Tues. 23 Sept. 1930 92 385 Sat. 12 Sept. 1931 93 355 Sat. 1 Oct. 1932 94 354 Thur. 21 Sept. 1933 95 383 Mon. 10 Sept. 1934 96 355 Sat. 28 Sept. 1935 97 354 Thur. 17 Sept. 1936 98 385 Mon. 6 Sept. 1937 99 353 Mon. 26 Sept. 1938 5700 385 Thur. 14 Sept. 1939 301 Cycle. 5701 354 Thur. 3 Oct. 1940 02 355 Mon. 22 Sept. 1941 03 383 Sat. 12 Sept. 1942 04 354 Thur. 30 Sept. 1943 05 355 Mon. 18 Sept. 1944 06 383 Sat. 8 Sept. 1945 07 354 Thur. 26 Sept. 1946 08 385 Mon. 15 Sept. 1947 09 355 Mon. 4 Oct. 1948 10 353 Sat. 24 Sept. 1949 11 384 Tues. 12 Sept. 1950 12 355 Mon. 1 Oct. 1951 13 355 Sat. 20 Sept. 1952 14 383 Thur. 10 Sept. 1953 15 354 Tues. 28 Sept. 1954 16 355 Sat. 17 Sept. 1955 17 385 Thur. 6 Sept. 1956 18 354 Thur. 26 Sept. 1957 19 383 Mon. 15 Sept. 1958 302 Cycle. 5720 355 Sat. 3 Oct. 1959 21 354 Thur. 22 Sept. 1960 22 383 Mon. 11 Sept. 1961 23 355 Sat. 29 Sept. 1962 24 354 Thur. 19 Sept. 1963 25 385 Mon. 7 Sept. 1964 26 353 Mon. 27 Sept. 1965 27 385 Thur. 15 Sept. 1966 28 354 Thur. 5 Oct. 1967 29 355 Mon. 23 Sept. 1968 30 383 Sat. 13 Sept. 1969 31 354 Thur. 1 Oct. 1970 32 355 Mon. 20 Sept. 1971 33 383 Sat. 9 Sept. 1972 34 355 Thur. 27 Sept. 1973 35 354 Tues. 17 Sept. 1974 36 385 Sat. 6 Sept. 1975 37 353 Sat. 25 Sept. 1976 38 384 Tues. 13 Sept. 1977 303 Cycle. 5739 355 Mon. 2 Oct. 1978 40 355 Sat. 22 Sept. 1979 41 383 Thur. 11 Sept. 1980 42 354 Tues. 29 Sept. 1981 43 355 Sat. 18 Sept. 1982 44 385 Thur. 8 Sept. 1983 45 354 Thur. 27 Sept. 1984 46 383 Mon. 16 Sept. 1985 47 355 Sat. 4 Oct. 1986 48 354 Thur. 24 Sept. 1987 49 383 Mon. 12 Sept. 1988 50 355 Sat. 30 Sept. 1989 51 354 Thur. 20 Sept. 1990 52 385 Mon. 9 Sept. 1991 53 353 Mon. 28 Sept. 1992 54 355 Thur. 16 Sept. 1993 55 384 Tues. 6 Sept. 1994 56 355 Mon. 25 Sept. 1995 57 383 Sat. 14 Sept. 1996 304 Cycle. Jewish Year Number of Days Commencement (1st of Tisri). 5758 354 Thur. 2 Oct. 1997 59 355 Mon. 21 Sept. 1998 60 385 Sat. 11 Sept. 1999 61 353 Sat. 30 Sept. 2000 62 354 Tues. 18 Sept. 2001 63 385 Sat. 7 Sept. 2002 64 355 Sat. 27 Sept. 2003 65 383 Thur. 16 Sept. 2004 66 354 Tues. 4 Oct. 2005 67 355 Sat. 23 Sept. 2006 68 383 Thur. 13 Sept. 2007 69 354 Tues. 30 Sept. 2008 70 355 Sat. 19 Sept. 2009 71 385 Thur. 8 Sept. 2010 72 354 Thur. 29 Sept. 2011 73 353 Mon. 17 Sept. 2012 74 385 Thur. 5 Sept. 2013 75 354 Thur. 25 Sept. 2014 76 385 Mon. 14 Sept. 2015 305 Cycle. 5777 353 Mon. 3 Oct. 2016 78 354 Thur. 21 Sept. 2017 79 385 Mon. 10 Sept. 2018 80 355 Mon. 30 Sept. 2019 81 353 Sat. 19 Sept. 2020 82 384 Tues. 7 Sept. 2021 83 355 Mon. 26 Sept. 2022 84 383 Sat. 16 Sept. 2023 85 355 Thur. 3 Oct. 2024 86 354 Tues. 23 Sept. 2025 87 385 Sat. 12 Sept. 2026 88 355 Sat. 2 Oct. 2027 89 354 Thur. 21 Sept. 2028 90 383 Mon. 10 Sept. 2029 91 355 Sat. 28 Sept. 2030 92 354 Thur. 18 Sept. 2031 93 383 Mon. 6 Sept. 2032 94 355 Sat. 24 Sept. 2033 95 385 Thur. 14 Sept. 2034 306 Cycle. 5796 354 Thur. 4 Oct. 2035 97 353 Mon. 22 Sept. 2036 98 385 Thur. 10 Sept. 2037 99 354 Thur. 30 Sept. 2038 5800 355 Mon. 19 Sept. 2039 01 383 Sat. 8 Sept. 2040 02 354 Thur. 26 Sept. 2041 03 385 Mon. 15 Sept. 2042 04 353 Mon. 5 Oct. 2043 05 355 Thur. 22 Sept. 2044 06 384 Tues. 12 Sept. 2045 07 355 Mon. 1 Oct. 2046 08 353 Sat. 21 Sept. 2047 09 384 Tues. 8 Sept. 2048 10 355 Mon. 27 Sept. 2049 11 355 Sat. 17 Sept. 2050 12 383 Thur. 7 Sept. 2051 13 354 Tues. 24 Sept. 2052 14 385 Sat. 13 Sept. 2053 307 Cycle. 5815 355 Sat. 3 Oct. 2054 16 354 Thur. 23 Sept. 2055 17 383 Mon. 11 Sept. 2056 18 355 Sat. 29 Sept. 2057 19 354 Thur. 19 Sept. 2058 20 383 Mon. 8 Sept. 2059 21 355 Sat. 25 Sept. 2060 22 385 Thur. 15 Sept. 2061 23 354 Thur. 5 Oct. 2062 24 353 Mon. 24 Sept. 2063 25 385 Thur. 11 Sept. 2064 26 354 Thur. 1 Oct. 2065 27 355 Mon. 20 Sept. 2066 28 383 Sat. 10 Sept. 2067 29 354 Thur. 27 Sept. 2068 30 355 Mon. 16 Sept. 2069 31 383 Sat. 6 Sept. 2070 32 355 Thur. 24 Sept. 2071 33 384 Tues. 13 Sept. 2072

Mahommedan Calendar.—The Mahommedan era, or era of the Hegira, used in Turkey, Persia, Arabia, &c., is dated from the first day of the month preceding the flight of Mahomet from Mecca to Medina, i.e. Thursday the 15th of July A.D. 622, and it commenced on the day following. The years of the Hegira are purely lunar, and always consist of twelve lunar months, commencing with the approximate new moon, without any intercalation to keep them to the same season with respect to the sun, so that they retrograde through all the seasons in about 32½ years. They are also partitioned into cycles of 30 years, 19 of which are common years of 354 days each, and the other 11 are intercalary years having an additional day appended to the last month. The mean length of the year is therefore 354-11/30 days, or 354 days 8 hours 48 min., which divided by 12 gives 29-191/360 days, or 29 days 12 hours 44 min., as the time of a mean lunation, and this differs from the astronomical mean lunation by only 2.8 seconds. This small error will only amount to a day in about 2400 years.

To find if a year is intercalary or common, divide it by 30; the quotient will be the number of completed cycles and the remainder will be the year of the current cycle; if this last be one of the numbers 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29, the year is intercalary and consists of 355 days; if it be any other number, the year is ordinary.

Or if Y denote the number of the Mahommedan year, and

R =11 Y + 14 30_r,

the year is intercalary when R < 11.

Also the number of intercalary years from the year 1 up to the year Y inclusive = ((11 Y + 14) / 30)_w; and the same up to the year Y - 1 = (11 Y + 3 / 30)_w.

To find the day of the week on which any year of the Hegira begins, we observe that the year 1 began on a Friday, and that after every common year of 354 days, or 50 weeks and 4 days, the day of the week must necessarily become postponed 4 days, besides the additional day of each intercalary year.

Hence if w = 1 indicate Sun. 2 Mon. 3 Tue. 4 Wed. 5 Thur. 6 Frid. 7 Sat.

the day of the week on which the year Y commences will be

w = 2 + 4Y 7_r +11 Y + 3 30_w (rejecting sevens).

But, 3011 Y + 3 30_w +11 Y + 3 30_r = 11 Y + 3

gives 12011 Y + 3 30_w = 12 + 44 Y - 411 Y + 3 30_r,

or11 Y + 3 30_w = 5 + 2 Y + 311 Y + 3 30_r (rejecting sevens).

So that

w = 6Y 7_r + 311 Y + 3 30_r (rejecting sevens),

the values of which obviously circulate in a period of 7 times 30 or 210 years.

Let C denote the number of completed cycles, and y the year of the cycle; then Y = 30 C + y, and

w = 5C 7_r + 6y 7_r + 311 y +3 30_r (rejecting sevens).

From this formula the following table has been constructed:—

Table VIII.

Year of the Current Cycle (y) Number of the Period of Seven Cycles = (C/7)_r

0 1 2 3 4 5 6

0 8 Mon. Sat. Thur. Tues. Sun. Frid. Wed.

1 9 17 25 Frid. Wed. Mon. Sat. Thur. Tues. Sun.

*2 *10 *18 *26 Tues. Sun. Frid. Wed. Mon. Sat. Thur.

3 11 19 27 Sun. Frid. Wed. Mon. Sat. Thur. Tues.

4 12 20 28 Thur. Tues. Sun. Frid. Wed. Mon. Sat.

*5 *13 *21 *29 Mon. Sat. Thur. Tues. Sun. Frid. Wed.

6 14 22 30 Sat. Thur. Tues. Sun. Frid. Wed. Mon.

*7 15 23 Wed. Mon. Sat. Thur. Tues. Sun. Frid.

*16 *24 Sun. Frid. Wed. Mon. Sat. Thur. Tues.

To find from this table the day of the week on which any year of the Hegira commences, the rule to be observed will be as follows:—

Rule.—Divide the year of the Hegira by 30; the quotient is the number of cycles, and the remainder is the year of the current cycle. Next divide the number of cycles by 7, and the second remainder will be the Number of the Period, which being found at the top of the table, and the year of the cycle on the left hand, the required day of the week is immediately shown.

The intercalary years of the cycle are distinguished by an asterisk.

For the computation of the Christian date, the ratio of a mean year of the Hegira to a solar year is

Year of Hegira Mean solar year=354-11/30 365.2422= 0.970224.

The year 1 began 16 July 622, Old Style, or 19 July 622, according to the New or Gregorian Style. Now the day of the year answering to the 19th of July is 200, which, in parts of the solar year, is 0.5476, and the number of years elapsed = Y - 1. Therefore, as the intercalary days are distributed with considerable regularity in both calendars, the date of commencement of the year Y expressed in Gregorian years is

This formula gives the following rule for calculating the date of the commencement of any year of the Hegira, according to the Gregorian or New Style.

Rule.—Multiply 970224 by the year of the Hegira, cut off six decimals from the product, and add 621.5774. The sum will be the year of the Christian era, and the day of the year will be found by multiplying the decimal figures by 365.

The result may sometimes differ a day from the truth, as the intercalary days do not occur simultaneously; but as the day of the week can always be accurately obtained from the foregoing table, the result can be readily adjusted.

Example.—Required the date on which the year 1362 of the Hegira begins.

970224

1362

————

1940448

5821344

2910672

970224

—————

1321.445088

621.5774

—————

1943.0225

365

——

1125

1350

675

———

8.2125

Thus the date is the 8th day, or the 8th of January, of the year 1943.

To find, as a test, the accurate day of the week, the proposed year of the Hegira, divided by 30, gives 45 cycles, and remainder 12, the year of the current cycle.

Also 45, divided by 7, leaves a remainder 3 for the number of the period.

Therefore, referring to 3 at the top of the table, and 12 on the left, the required day is Friday.

The tables, page 571, show that 8th January 1943 is a Friday, therefore the date is exact.

For any other date of the Mahommedan year it is only requisite to know the names of the consecutive months, and the number of days in each; these are—

Muharram 30

Saphar 29

Rabia I. 30

Rabia II. 29

Jomada I. 30

Jomada II 29

Rajab 30

Shaaban 29

Ramadān 30

Shawall (Shawwāl) 29

Dulkaada (Dhu'l Qa'da) 30

Dulheggia (Dhu'l Hijja) 29

- and in intercalary years 30

The ninth month, Ramadān, is the month of Abstinence observed by the Moslems.

The Moslem calendar may evidently be carried on indefinitely by successive addition, observing only to allow for the additional day that occurs in the bissextile and intercalary years; but for any remote date the computation according to the preceding rules will be most efficient, and such computation may be usefully employed as a check on the accuracy of any considerable extension of the calendar by induction alone.

The following table, taken from Woolhouse's Measures, Weights and Moneys of all Nations, shows the dates of commencement of Mahommedan years from 1845 up to 2047, or from the 43rd to the 49th cycle inclusive, which form the whole of the seventh period of seven cycles. Throughout the next period of seven cycles, and all other like periods, the days of the week will recur in exactly the same order. All the tables of this kind previously published, which extend beyond the year 1900 of the Christian era, are erroneous, not excepting the celebrated French work, L'Art de vérifier les dates, so justly regarded as the greatest authority in chronological matters. The errors have probably arisen from a continued excess of 10 in the discrimination of the intercalary years.

Table IX.—Mahommedan Years.

43rd Cycle. Year of Hegira. Commencement (1st of Muharram). 1261 Frid. 10 Jan. 1845 1262* Tues. 30 Dec. 1845 1263 Sun. 20 Dec. 1846 1264 Thur. 9 Dec. 1847 1265* Mon. 27 Nov. 1848 1266 Sat. 17 Nov. 1849 1267* Wed. 6 Nov. 1850 1268 Mon. 27 Oct. 1851 1269 Frid. 15 Oct. 1852 1270* Tues. 4 Oct. 1853 1271 Sun. 24 Sept. 1854 1272 Thur. 13 Sept. 1855 1273* Mon. 1 Sept. 1856 1274 Sat. 22 Aug. 1857 1275 Wed. 11 Aug. 1858 1276* Sun. 31 July 1859 1277* Frid. 20 July 1860 1278* Tues. 9 July 1861 1279 Sun. 29 June 1862 1280 Thur. 18 June 1863 1281* Mon. 6 June 1864 1282 Sat. 27 May 1865 1283 Wed. 16 May 1866 1284* Sun. 5 May 1867 1285 Frid. 24 April 1868 1286* Tues. 13 April 1869 1287 Sun. 3 April 1870 1288 Thur. 23 Mar. 1871 1289* Mon. 11 Mar. 1872 1290 Sat. 1 Mar. 1873 44th Cycle. 1291 Wed. 18 Feb. 1874 1292* Sun. 7 Feb. 1875 1293 Frid. 28 Jan. 1876 1294 Tues. 16 Jan. 1877 1295* Sat. 5 Jan. 1878 1296 Thur. 26 Dec. 1878 1297* Mon. 15 Dec. 1879 1298 Sat. 4 Dec. 1880 1299 Wed. 23 Nov. 1881 1300* Sun. 12 Nov. 1882 1301 Frid. 2 Nov. 1883 1302 Tues. 21 Oct. 1884 1303* Sat. 10 Oct. 1885 1304 Thur. 30 Sept. 1886 1305 Mon. 19 Sept. 1887 1306* Frid. 7 Sept. 1888 1307 Wed. 28 Aug. 1889 1308* Sun. 17 Aug. 1890 1309 Frid. 7 Aug. 1891 1310 Tues. 26 July 1892 1311* Sat. 15 July 1893 1312 Thur. 5 July 1894 1313 Mon. 24 June 1895 1314* Frid. 12 June 1896 1315 Wed. 2 June 1897 1316* Sun. 22 May 1898 1317 Frid. 12 May 1899 1318 Tues. 1 May 1900 1319* Sat. 20 April 1901 1320 Thur. 10 April 1902 45th Cycle. 1321 Mon. 30 Mar. 1903 1322* Frid. 18 Mar. 1904 1323 Wed. 8 Mar. 1905 1324 Sun. 25 Feb. 1906 1325 Thur. 14 Feb. 1907 1326 Tues. 4 Feb. 1908 1327* Sat. 23 Jan. 1909 1328 Thur. 13 Jan. 1910 1329 Mon. 2 Jan. 1911 1330* Frid. 22 Dec. 1911 45th Cycle.—continued. Year of Hegira. Commencement (1st of Muharram). 1331 Wed. 11 Dec. 1912 1332 Sun. 30 Nov. 1913 1333* Thur. 19 Nov. 1914 1334 Tues. 9 Nov. 1915 1335 Sat. 28 Oct. 1916 1336* Wed. 17 Oct. 1917 1337 Mon. 7 Oct. 1918 1338* Frid. 26 Sept. 1919 1339 Wed. 15 Sept. 1920 1340 Sun. 4 Sept. 1921 1341* Thur. 24 Aug. 1922 1342 Tues. 14 Aug. 1923 1343 Sat. 2 Aug. 1924 1344* Wed. 22 July 1925 1345 Mon. 12 July 1926 1346* Frid. 1 July 1927 1347 Wed. 20 June 1928 1348 Sun. 9 June 1929 1349* Thur. 29 May 1930 1350 Tues. 19 May 1931 46th Cycle. 1351 Sat. 7 May 1932 1352* Wed. 26 April 1933 1353 Mon. 16 April 1934 1354 Frid. 5 April 1935 1355* Tues. 24 Mar. 1936 1356 Sun. 14 Mar. 1937 1357* Thur. 3 Mar. 1938 1358 Tues. 21 Feb. 1939 1359 Sat. 10 Feb. 1940 1360* Wed. 29 Jan. 1941 1361 Mon. 19 Jan. 1942 1362 Frid. 8 Jan. 1943 1363* Tues. 28 Dec. 1943 1364 Sun. 17 Dec. 1944 1365 Thur. 6 Dec. 1945 1366* Mon. 25 Nov. 1946 1367 Sat. 15 Nov. 1947 1368* Wed. 3 Nov. 1948 1369 Mon. 24 Oct. 1949 1370 Frid. 13 Oct. 1950 1371* Tues. 2 Oct. 1951 1372 Sun. 21 Sept. 1952 1373 Thur. 10 Sept. 1953 1374* Mon. 30 Aug. 1954 1375 Sat. 20 Aug. 1955 1376* Wed. 8 Aug. 1956 1377 Mon. 29 July 1957 1378 Frid. 18 July 1958 1379* Tues. 7 July 1959 1380 Sun. 26 June 1960 47th Cycle. 1381 Thur. 15 June 1961 1382* Mon. 4 June 1962 1383 Sat. 25 May 1963 1384 Wed. 13 May 1964 1385* Sun. 2 May 1965 1386 Frid. 22 April 1966 1387* Tues. 11 April 1967 1388 Sun. 31 Mar. 1968 1389 Thur. 20 Mar. 1969 1390* Mon. 9 Mar. 1970 1391 Sat. 27 Feb. 1971 1392 Wed. 16 Feb. 1972 1393* Sun. 4 Feb. 1973 1394 Frid. 25 Jan. 1974 1395 Tues. 14 Jan. 1975 1396* Sat. 3 Jan. 1976 1397 Thur. 23 Dec. 1976 1398* Mon. 12 Dec. 1977 1399 Sat. 2 Dec. 1978 1400 Wed. 21 Nov. 1979 47th Cycle.—continued. Year of Hegira. Commencement (1st of Muharram). 1401* Sun. 9 Nov. 1980 1402 Frid. 30 Oct. 1981 1403 Tues. 19 Oct. 1982 1404* Sat. 8 Oct. 1983 1405 Thur. 27 Sept. 1984 1406* Mon. 16 Sept. 1985 1407 Sat. 6 Sept. 1986 1408 Wed. 26 Aug. 1987 1409* Sun. 14 Aug. 1988 1410 Frid. 4 Aug. 1989 48th Cycle. 1411 Tues. 24 July 1990 1412* Sat. 13 July 1991 1413 Thur. 2 July 1992 1414 Mon. 21 June 1993 1415* Frid. 10 June 1994 1416 Wed. 31 May 1995 1417* Sun. 19 May 1996 1418 Frid. 9 May 1997 1419 Tues. 28 April 1998 1420* Sat. 17 April 1999 1421 Thur. 6 April 2000 1422 Mon. 26 Mar. 2001 1423 Frid. 15 Mar. 2002 1424 Wed. 5 Mar. 2003 1425 Sun. 22 Feb. 2004 1426* Thur. 10 Feb. 2005 1427 Tues. 31 Jan. 2006 1428* Sat. 20 Jan. 2007 1429 Thur. 10 Jan. 2008 1430 Mon. 29 Dec. 2008 1431* Frid. 18 Dec. 2009 1432 Wed. 8 Dec. 2010 1433 Sun. 27 Nov. 2011 1434* Thur. 15 Nov. 2012 1435 Tues. 5 Nov. 2013 1436* Sat. 25 Oct. 2014 1437 Thur. 15 Oct. 2015 1438 Mon. 3 Oct. 2016 1439* Frid. 22 Sept. 2017 1440 Wed. 12 Sept. 2018 49th Cycle. 1441 Sun. 1 Sept. 2019 1442* Thur. 20 Aug. 2020 1443 Tues. 10 Aug. 2021 1444 Sat. 30 July 2022 1445* Wed. 19 July 2023 1446 Mon. 8 July 2024 1447* Frid. 27 June 2025 1448 Wed. 17 June 2026 1449 Sun. 6 June 2027 1450* Thur. 25 May 2028 1451 Tues. 15 May 2029 1452 Sat. 4 May 2030 1453* Wed. 23 April 2031 1454 Mon. 12 April 2032 1455 Frid. 1 April 2033 1456* Tues. 21 Mar. 2034 1457 Sun. 11 Mar. 2035 1458* Thur. 28 Feb. 2036 1459 Tues. 17 Feb. 2037 1460 Sat. 6 Feb. 2038 1461* Wed. 26 Jan. 2039 1462 Mon. 16 Jan. 2040 1463 Frid. 4 Jan. 2041 1464* Tues. 24 Dec. 2041 1465 Sun. 14 Dec. 2042 1466* Thur. 3 Dec. 2043 1467 Tues. 22 Nov. 2044 1468 Sat. 11 Nov. 2045 1469* Wed. 31 Oct. 2046 1470 Mon. 21 Oct. 2047

Table X.—Principal Days of the Hebrew Calendar.

Tisri 1, New Year, Feast of Trumpets.

" 3, ^[1] Fast of Guedaliah.

" 10, Fast of Expiation.

" 15, Feast of Tabernacles.

" 21, Last Day of the Festival.

" 22, Feast of the 8th Day.

" 23, Rejoicing of the Law.

Kislev 25, Dedication of the Temple.

Tebet 10, Fast, Siege of Jerusalem.

Adar 13, ^[2] Fast of Esther, In embolismic years, Veadar.

" 14, Purim,

Nisan 15, Passover.

Sivan 6, Pentecost.

Tamuz 17, ^[1] Fast, Taking of Jerusalem.

Ab 9, ^[1] Fast, Destruction of the Temple.

Table XI.—Principal Days of the Mahommedan Calendar.

Muharram 1, New Year.

" 10, Ashura.

Rabia I. 11, Birth of Mahomet.

Jornada I. 20, Taking of Constantinople.

Rajab 15, Day of Victory.

" 20, Exaltation of Mahomet.

Shaaban 15, Borak's Night.

Shawall 1,2,3, Kutshuk Bairam.

Dulheggia 10, Qurban Bairam.

Table XII.—Epochs, Eras, and Periods.

Name. Christian Date of Commencement.

Grecian Mundane era 1 Sep. 5598 B.C.

Civil era of Constantinople 1 Sep. 5508 "

Alexandrian era 29 Aug. 5502 "

Ecclesiastical era of Antioch 1 Sep. 5492 "

Julian Period 1 Jan. 4713 "

Mundane era Oct. 4008 "

Jewish Mundane era Oct. 3761 "

Era of Abraham 1 Oct. 2015 "

Era of the Olympiads 1 July 776 "

Roman era 24 April 753 "

Era of Nabonassar 26 Feb. 747 "

Metonic Cycle 15 July 432 "

Grecian or Syro-Macedonian era 1 Sep. 312 "

Tyrian era 19 Oct. 125 "

Sidonian era Oct. 110 "

Caesarean era of Antioch 1 Sep. 48 "

Julian year 1 Jan. 45 "

Spanish era 1 Jan. 38 "

Actian era 1 Jan. 30 "

Augustan era 14 Feb. 27 "

Vulgar Christian era 1 Jan. 1 A.D.

Destruction of Jerusalem 1 Sep. 69 "

Era of Maccabees 24 Nov. 166 "

Era of Diocletian 17 Sep. 284 "

Era of Ascension 12 Nov. 295 "

Era of the Armenians 7 July 552 "

Mahommedan era of the Hegira 16 July 622 "

Persian era of Yezdegird 16 June 632 "

For the Revolutionary Calendar see French Revolution ad fin.

The principal works on the calendar are the following:—Clavius, Romani Calendarii a Gregorio XIII. P.M. restituti Explicatio (Rome, 1603); L'Art de vérifier les dates; Lalande, Astronomie tome ii.; Traité de la sphère et du calendrier, par M. Revard (Paris, 1816); Delambre, Traité de l'astronomie théorique et pratique, tome iii.; Histoire de l'astronomie moderne; Methodus technica brevis, perfacilis, ac perpetua construendi Calendarium Ecclesiasticum, Stylo tam novo quam vetere, pro cunctis Christianis Europae populis, &c., auctore Paulo Tittel (Gottingen, 1816); Formole analitiche pel calcolo delta Pasgua, e correzione di quello di Gauss, con critiche osservazioni sù quanta ha scritto del calendario il Delambri, di Lodovico Ciccolini (Rome, 1817); E.H. Lindo, Jewish Calendar for Sixty-four Years (1838); W.S.B. Woolhouse, Measures, Weights, and Moneys of all Nations (1869).

(T. G.; W. S. B. W.)

CALENDER, (1) (Fr. calendre, from the Med. Lat. calendra, a corruption of the Latinized form of the Gr. κύλινδρος, a cylinder), a machine consisting of two or more rollers or cylinders in close contact with each other, and often heated, through which are passed cotton, calico and other fabrics, for the purpose of having a finished smooth surface given to them; the process flattens the fibres, removes inequalities, and also gives a glaze to the surface. It is similarly employed in paper manufacture (q.v.). (2) (From the Arabic qalandar), an order of dervishes, who separated from the Baktashite order in the 14th century; they were vowed to perpetual travelling. Other forms of the name by which they are known are Kalenderis, Kalenderites, and Qalandarites (see Dervish).

CALENUS, QUINTUS FUFIUS, Roman general. As tribune of the people in 61 B.C., he wa$ chiefly instrumental in securing the acquittal of the notorious Publius Clodius when charged with having profaned the mysteries of Bona Dea (Cicero, Ad. Att. i. 16). In 59 Calenus was praetor, and brought forward a law that the senators, knights, and tribuni aerarii, who composed the judices, should vote separately, so that it might be known how they gave their votes (Dio Cassius xxxviii. 8). He fought in Gaul (51) and Spain (49) under Caesar, who, after he had crossed over to Greece (48), sent Calenus from Epirus to bring over the rest of the troops from Italy. On the passage to Italy, most of the ships were captured by Bibulus and Calenus himself escaped with difficulty. In 47 he was raised to the consulship through the influence of Caesar. After the death of the dictator, he joined Antony, whose legions he afterwards commanded in the north of Italy. He died in 41, while stationed with his army at the foot of the Alps, just as he was on the point of marching against Octavianus.

Caesar, B.G. viii. 39; B.C. i. 87, iii. 26; Cic. Philippicae, viii. 4.

CALEPINO, AMBROGIO (1435-1511), Italian lexicographer, born at Bergamo in 1435, was descended of an old family of Calepio, whence he took his name. Becoming an Augustinian monk, he devoted his whole life to the composition of a polyglott dictionary, first printed at Reggio in 1502. This gigantic work was afterwards augmented by Passerat and others. The most complete edition, published at Basel in 1590, comprises no fewer than eleven languages. The best edition is that published at Padua in seven languages in 1772. Calepino died blind in 1511.

CALES (mod. Calvi), an ancient city of Campania, belonging Originally to the Aurunci, on the Via Latina, 8 m. N.N.W. of Casilinum. It was taken by the Romans in 335 B.C., and, a colony with Latin rights of 2500 citizens having been established there, it was for a long time the centre of the Roman dominion in Campania, and the seat of the quaestor for southern Italy even down to the days of Tacitus.^[1] It was an important base in the war against Hannibal, and at last refused further contributions for the war. Before 184 more settlers were sent there. After the Social War it became a municipium. The fertility of its territory and its manufacture of black glazed pottery, which was even exported to Etruria, made it prosperous. At the end of the 3rd century it appears as a colony, and in the 5th century it became an episcopal see, which (jointly with Teano since 1818) it still is, though it is now a mere village. The cathedral, of the 12th century, has a carved portal and three apses decorated with small arches and pilasters, and contains a fine pulpit and episcopal throne in marble mosaic. Near it are two grottos which have been used for Christian worship and contain frescoes of the 10th and 11th centuries (E. Bertaux, L'Art dans l'Italie méridionale (Paris, 1904), i. 244, &c.). Inscriptions name six gates of the town: and there are considerable remains of antiquity, especially of an amphitheatre and theatre, of a supposed temple, and other edifices. A number of tombs belonging to the Roman necropolis were discovered in 1883.