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By the Gregorian rule of intercalation the coincidence of the solar and civil year is restored very nearly every 400 years.Appendix I.
A SIMPLE METHOD FOR FINDING THE DAY OF THE WEEK OF EVENTS WHICH OCCUR QUADRENNIALLY.
The inaugural of the Presidents. The day of the week on which they have occurred, and on which they will occur for the next one hundred years:
Any one understanding what has been said in a preceding chapter concerning the dominical letter, can very easily make out such a table without going through the process of making calculations for everyyear. As every succeeding year, or any day of the year, commences one day later in the week than the year preceding, and two days later in leap-year, which makes five days every four years, and as the Presidential term is four years, so every inaugural occurs five days later in the week than it did in the preceding term.
Now, as counting forward five days is equivalent to counting back two, it will be much more convenient to count back two days every term. There is one exception, however, to this rule; the year which completes the century is reckoned as a common year (that is, three centuries out of four), consequently we count forward only four days or back three.
Commencing, then, with the second inaugural of Washington, which occurred on Monday, March 4,1793, and counting back two days to Saturday in 1797, three days to Wednesday in 1801 and two days to 1805, and so on two days every term till 1901, when, for reasons already given, we count back three days again for one term only, after which it will be two days for the next two hundred years; hence anyone can make his calculations as he writes, and as fast as he can write. See table on 61st page.
SOME PECULIARITIES CONCERNING EVENTS WHICH FALL ON THE TWENTY-NINTH DAY OF FEBRUARY.
The civil year and the day must be regarded as commencing at the same instant. We cannot well reckon a fraction of a day, giving to February 28 days and 6 hours, making the following month to commence six hours later every year; if so, then March, for example, in
again, and so on.
Instead of doing so, we wait until the fraction accumulates to a whole day; then give to February 29 days, and the year 366. Therefore, events which fall on the 29th of February cannot be celebrated annually, but only quadrennially; and at the close of those centuries in which the intercalations are suppressed only octennially. For example: From the year 1696 to 1704, 1796 to 1804, and 1896 to 1904, there is no 29th day of February; consequently no day of the month in the civil year on which an event falling on the 29th of February could be celebrated. Therefore, a person born on the 29th of February, 1896, could celebrate no birthday till 1904, a period of eight years.
In every common year February has 28 days, each day of the week being contained in the number of days in the month four times; but in leap-year, when February has 29 days, the day which begins and ends themonth is contained five times. Let us suppose that in a certain year, when February has 29 days, the month comes in on Friday; it also must necessarily end on Friday.
After four years it will commence on Wednesday, and end on Wednesday, and so on, going back two days in the week every four years, until after 28 years we come back to Friday again. This, as has already been explained, is the dominical or solar cycle. For example: February in
So that after 28 years we come back to Friday again; and so on every 28 years, until change of style in 1582, when the Gregorian rule of intercalation being adopted by suppressing the intercalations in three centurial years out of four interrupts this order at the close of these three centuries. For example—1700, 1800 and 1900, after which the cycle of 28 years will be continued until 2100, and so on. The cycle being interrupted by the Gregorian rule of intercalation, causes all events which occur between 28 and 12 years of the close of the centuries to fall on the same day of the week again in 40 years; and those events that fall within 12 years of the close of these centuries, to fall on the same day of the week again in 12 years; afterwhich the cycle of 28 years will be continued during the century. See following table:
It will be seen from this table that in 1804 February had five Wednesdays; and then again in 1832, 1860 and 1888; then suppressing the intercalation in the year 1900 suppresses the 29th of February; so opposite 1900 in the table is blank, and the 29th of February does not occur again till 1904, and the five Wednesdays do not occur again till 1928—that is, 40 years from 1888, when it last occurred.
Again taking the five Mondays which occurred first in this century, in 1808, and then again in 1836, 1864 and in 1892, you will see, for reasons already given, that it will occur again in 12 years, that is, in 1904; and so on with all the days of the week, when it will be seen what is peculiar concerning the 29th of February.
But attention is particularly called to the five Thursdays, which occur first in this century 1816, and then again in 1844 and 1872, the last date being within 28 years of the close of the century. Suppressing the intercalation suppresses the 29th of February; consequently the five Thursdays do not occur again till 1912, that is 40 years from the preceding date, after which the cycle will be continued for two hundred years.
Hence it may be seen that the dominical or solar cycle of 28 years is so interrupted at the close of these centuries by the suppression of the leap-year, that certain events do not occur again on the same day of the week under 40 years, while others are repeated again on the same day of the week in 12 years; also the number of years in the cycle, that is 28 + 12 = 40.
And again the change of style in 1582, causes all events which occur between 28 and 8 years of thatchange, to fall again on the same day of the week in 36 years, and all that occur within 8 years of that change to be repeated again on the same day of the week in 8 years, after which the cycle of 28 years is continued for 100 years; also that the number of years in the cycle, that is, 28 + 8 = 36.
RULES FOR FINDING THE DAY OF THE WEEK OF EVENTS THAT TRANSPIRED PRIOR TO THE CHRISTIAN ERA.
First, it should be understood that the year 4 is the first leap year in our era, reckoning from the year 1 B. C., which must necessarily be leap year; so that the odd numbers 1, 5, 9, 13, etc., are leap years. Hence every year that is divisible by four and one remainder, is leap year; if no remainder, it is the first year after leap-year; if 3, the second; if 2, the third, thus:
and so on, every year being divided by four and 1 remainder is leap-year of 366 days. It should be borne in mind that the same calendar was in use without any correction from the days of Julius Cæsar 46 B. C. to Pope Gregory XIII in 1582; consequently the method of finding the dominical letter is, in some respects, similarto the one already given on the 44th page. But in some respects the one is the reverse of the other, for we reckon backward and forward from a fixed point—the era; that is the numbers increase each way from the era. Also the dominical letters occur in the natural order of the letters in reckoning backward, but exactly the reverse in reckoning forward. See table on the 73d page, where the dominical letter is placed opposite each year from 45 B. C. to 45 A. D. Now we use the same number 3, because C, the third letter is dominical letter for the year 1 B. C., the point from which we reckon. But instead of taking the remainder, after dividing by 7,from3 or 10, to find the number of the letter, as inPart Second, Chapter IV, (q. v.) we add the remainderto3; hence we have the following:
RULE.
Divide the number of the given year by 4, neglecting the remainders, and add the quotient to the given number, divide this amount by 7, and add the remainder to 3, and that amount will give the number of the letter, calling A, 1; B, 2; C, 3, etc.; except the first year after leap-year, (which is the year exactly divisible by 4), the number of the letter is one less than is indicated by the rule.
This rule gives the dominical letter for January and February only, in leap-year, while the letter that precedes it, is the letter for the rest of the year. If the amount be greater than seven, we should reckon from A to A or B again.
It has already been stated inPart First, Chapter III, (q. v.), that a change was made by Augustus Cæsar about 8 B. C., in the number of days in the month; and,as this change effects the day of the week on which certain events fall, it becomes necessary that they should be presented as they were arranged by Julius Cæsar, and as corrected by Augustus. Julius Cæsar gave to February 29 days in common years, and in leap-year 30. This arrangement was the very best that could possibly be made, but, as has already been shown, it was interrupted to gratify the vanity of Augustus.
The left hand column in the table on the 72d page represents the number of days in each month from the days of Julius Cæsar to Augustus, a period of 37 years. The right hand column represents the number of days in the months as they now are, and have been since the change was made by Augustus, 8 B. C. Consequently the rule for finding the day of the week on which events have fallen for the 37 years prior to the last mentioned date, is not perfectly exact, and needs a little explanation here.
The rule itself is given, and fully explained inPart Second, Chapter V, (q. v.) but cannot be applied to the 37 years without some correction. In all the months marked with a star, events fall one day later in the week than that which is indicated by the rule. This should be borne in mind, and make the event one day later in the week than that which is found by the rule. For example, Julius Cæsar was assassinated on the 15th of March, 44 B. C. By giving to February 28 days the first day of March would fall on Wednesday, and, of course, the 15th would be Wednesday. But Cæsar gave to February 29 days, so that the first day of March fell on Thursday, and the 15th was Thursday.
Hence, every event from March to September will fall one day later in the week than the rule indicates. But the rule is applicable to September, for it will make no difference whether there are 29 days in February or 31 in August, there are the same number of days from February to September. But the 31 days in September will cause all events to fall one day later in the week during the month of October, but theycoincideagain during the month of November. The order is interrupted again in December by giving 31 days to November. See following table:
PART THIRD.
CYCLES—JULIAN PERIOD—EASTER.
HEBREW CALENDAR.
THE SOLAR CYCLE.
Cycle, (LatinCyclus, ring or circle). The revolution of a certain period of time which finishes and re-commences perpetually. Cycles were invented for the purpose of chronology, and for marking the intervals in which two or more periods of unequal length are each completed a certain number of times, so that both begin exactly in the same circumstance as at first. Cycles used in chronology are three: The solar cycle, the lunar cycle, and the cycle of indiction.
The solar cycle is a period of time after which the same days of the year recur on the same days of the week. If every year contained 365 days, then every year would commence one day later in the week than the year preceding, and the cycle would be completed in seven years. For if the first day of January, in any given year, fall on Sunday, then the following year on Monday, the third on Tuesday, and so on to Sunday again in seven years.
But this order is interrupted in the Julian calendar every four years by giving to February 29 days, andconsequentlythe year 366. Now the number of years in the intercalary period being four and the days of the week being seven, their product is 4 × 7 = 28; twenty-eight years then is a period after which the first day of the year and the first day of every month recur again in the same order on the same day of the week. This period is called the solar cycle or the cycle of the sun, the origin of which is unknown; but is supposed to have been invented about the time of the Council of Nice, in the year of our Lord 325; but the first year of the cycle is placed by chronologists nine years before the commencement of the Christian era.
Hence the year of the cycle corresponding to any given year in the Julian calendar is found by the following rule: Add nine to the date and divide the sum by twenty-eight; the quotient is the number of cycles elapsed, and the remainder is the year of the cycle. Should there be no remainder, the proposed year is the twenty-eighth, or last of the cycle. Thus, for the year 1892, we have (1892 + 9) ÷ 28 = 67, remainder 25. Therefore, 67 is the number of cycles, and 25 the number in the cycle.
THE LUNAR CYCLE.
The Lunar cycle, or the cycle of the moon, is a period of nineteen years, after which the new and full moons fall on the same days of the year as they did nineteen years before. This cycle was invented by Meton, a celebrated astronomer of Athens, and may be regarded as the masterpiece of ancient astronomy. In nineteen solar years there are 235 lunations, a number which, on being divided by nineteen, gives twelve lunations, with seven of a remainder, to be distributed among the years of the period.
The period of Meton, therefore, consisted of twelve years containing twelve months each, and seven years containing thirteen mouths each, and these last formed the third, fifth, eighth, eleventh, thirteenth, sixteenth, and nineteenth years of the cycle. As it had now been discovered that the exact length of the lunation is a little more than twenty-nine and a half days, it became necessary to abandon the alternate succession of full and deficient months; and, in order to preserve a more accurate correspondence between the civil month and the lunation, Meton divided the cycle into 125 full months of 30 days, and 110 deficient months of 29 days each. The number of days in the period was, therefore, 6940; for (125 × 30) + (110 × 29) = 6940.
In order to distribute the deficient months through the period in the most equable manner, the whole period may be regarded as consisting of 235 full monthsof thirty days, or 7050 days, from which 110 days are to be deducted; for (235 × 30) = 7050; 7050 - 110 = 6940, as before. This gives one day to be suppressed in sixty-four, so that if we suppose the months to contain each thirty days, and then omit every sixty-fourth day in reckoning from the beginning of the period, those months in which the omission takes place will, of course, be the deficient months.
The number of days in the period being known, it is easy to ascertain its accuracy both in respect of the solar and lunar motions. The exact length of nineteen solar years is (365d, 5h, 48m, 49.62s.) × 19 = 6939d, 14h, 27m, 42.78s.; hence, the period, which is exactly 6940 days, exceeds nineteen annual revolutions of the earth by a little more than nine and a half hours. On the other hand, the exact time of the synodic revolution of the moon is 29d, 12h, 44m, 2.87s.; 235 lunations, therefore, contain 235 × (29d, 12h, 44m, 2.87s.) = (6939d, 16h, 31m, 14.45s.), so that the period exceeds 235 lunations by nearly seven and a half hours.
At the end of four cycles, or seventy-six years, the accumulation of the seven and a half hours of difference between the cycle and 235 lunations amounts to thirty hours, or one whole day and six hours. Calippus, therefore, in order to make a correction of the Metonic cycle, proposed to quadruple the period of Meton, and deduct one day at the end of that time by changing one of the full months into a deficient month. The period of Calippus, therefore, consisted of three Metonic cycles of 6940 days each, and a period of 6939 days; and its error in respect to the moon, consequently,amounted to only six hours, or to one day in 304 years. This period exceeds seventy-six true solar years by 14h, 9m, 8.88s., but coincides exactly with seventy-six Julian years; and in the time of Calippus the length of the solar year was almost universally supposed to be exactly 365¼ days.
THE LUNAR CYCLE AND GOLDEN NUMBER.
In connecting the lunar month with the solar year, the framers of the ecclesiastical calendar adopted the period of Meton, or lunar cycle, which they supposed to be exact. A different arrangement has, however, been followed with respect to the distribution of the months. The lunations are supposed to consist of twenty-nine and thirty days alternately, or the lunar year of 354 days; and in order to make up nineteen solar years, six embolismic or intercalary months, of thirty days each, are introduced in the course of the cycle, and one of twenty-nine days is added at the end. This gives (19 × 354) + (6 × 30) + 29 = 6935 days, to be distributed among 235 lunar months.
But every leap-year one day must be added to the lunar month in which the 29th of February is included. Now if leap-year happened on the first, second or third year of the period, there will be five leap-years in the period, but only four when the first leap-year falls on the fourth. In the former case the number of days in the period becomes 6940, and in the latter 6939. The mean length of the cycle is, therefore, 6939¾ days,agreeing exactly with nineteen Julian years. By means of the lunar cycle the new moons of the calendar were indicated before the reformation in 1582. 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 nineteen years will serve for any year whatever when we know its number in the cycle.
The number of the year in the cycle is called the Golden Number; either because it was so termed by the Greeks, who, on account of its utility, ordered it to be inscribed in letters of gold in their temples, or more probably because it was usual to distinguish it by 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 Nice. The cycle is supposed to commence with the year in which the new moon falls on the first day of January, which took place the year preceding the commencement of our era.
Hence to find the Golden Number for any year, we have the following rule: Add one to the date, divide the sum by nineteen; the quotient is the number of cycles elapsed, and the remainder is the Golden Number. Should there be no remainder, the proposed year is, of course, the last or nineteenth of the cycle. Thus, for the year 1892, we have (1892 + 1) ÷ 19 = 99, remainder 12; therefore, 99 is the number of cycles, and 12 the number in the cycle, or the Golden number.
It ought to be remarked that the new moons determined in this manner, may differ from the astronomicalnew 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 degrees and thereby cause the new moon to arrive on the second day before or after its mean time.
The cycle of the sun brings back the days of the month to the same day of the week; the cycle of the moon 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 letter, 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 the Dionysian or Great Paschal Period, from its having been employed by Dionysius Exiguus in determining Easter Sunday.
CYCLE OF INDICTION, AND THE JULIAN PERIOD.
The cycle of Indiction or Roman Indiction, is a period of fifteen years; not astronomical like the two former, but entirely arbitrary. Its origin and the purpose for which it was established are alike uncertain; but it is conjectured that it was introduced by Constantine the Great, about the year 312 of the common era, and had reference to certain judicial acts that took place under the Greek emperors at stated intervals of fifteen years. In chronological reckoning, it is considered as having commenced on the first day of January of the year 313.
By extending it backward, it will be found that the cycle commenced three years before the beginning of our era. In order, therefore, to find the number of any year in the cycle of indiction, we have this rule: Add three to the date, divide the number by fifteen; and the remainder is the year of the indiction. Should there be no remainder, the proposed year is the fifteenth or last of the cycle. Thus, for the year 1892, we have (1892 + 3) ÷ 15 = 126, remainder 5. Therefore, 5 is the number in the cycle.
The Julian period, proposed by the celebrated Joseph Scaliger, as an universal measure of chronology, is a period of 7980 years, and is formed by the continual multiplication of the three numbers, 28, 19 and 15; that is, of the cycle of the sun, the cycle of the moon, and the cycle of indiction. Thus, (28 × 19 × 15) = 7980. In the course of this long period no two years can be expressed by the same numbers in all the three cycles.
The first year of the Christian era had 10 for its number in the cycle of the sun, 2 in the cycle of the moon, and 4 in the cycle of the indiction. Now, it is found by actual calculation, that the only number less than 7980 which, on being divided successively by 28, 19, and 15, leaves the respective remainders 10, 2 and 4, is 4714. Hence, the first year of the Christian era corresponded with the year 4714 of the Julian period, which period coincides with the 710th before the common mundane era 4004 B. C.; for 4004 + 710 = 4714. Hence, also, the year of our era corresponding to any other year of the period, orvice versa, is found by the following rule:
When the given year is anterior to the commencementof the era, subtract the number of the year of the Julian period from 4714, and the remainder is the year before Christ; or, subtract the year before Christ from 4714, and the remainder is the corresponding year in the Julian period. Thus, Rome was founded in the year 3960 of the Julian period. What was the year before Christ? We have then 4714 - 3960 = 754. Julius Cæsar was assassinated 44 years before Christ, what was the corresponding year of the Julian period? We have then, 4714 - 44 = 4670.
When the given year is after Christ, subtract 4713 from the year of the period, and the remainder is the year of the era; or add 4713 to the year of the era, and the sum is the corresponding year in the Julian period. Thus, the Council of Nice was convened in the year 5038 of the Julian period, what was the year of our era? We have then, 5038 - 4713 = 325. What year of the Julian period corresponds with the present year, 1892? We have then, 4713 + 1892 = 6605.
EASTER.
Easter (Germ.Ostern, Old SaxonOster, fromOsten, signifying rising). The English name is probably derived from Ostera or Eostre, the Teutonic goddess of spring, whose festival occurred about the same time of the year as the celebration of Easter. The Hebrew-Greek word Pascha has passed into the name given to this feast by most Christian nations. This festival is held in commemoration of our Lord’s resurrection.
The Jews celebrated their passover, in conformity with the directions given them by Moses, on the 14th day of the month Nisan, being the lunar month of which the 14th day either falls on or next follows the day of the vernal equinox. In the year of our Lord’s crucifixion this fell on a Friday; the resurrection, therefore, took place on the first day of the week, which from thence is denominated the Lord’s Day.
The primitive Christians, in celebrating this anniversary, fell into two different systems. The Western churches observed the nearest Sunday to the full moon of Nisan, taking no account of the day on which the passover would be celebrated. The Asiatics, on the other hand, following the Jewish calendar, adopted the 14th of Nisan upon which to commemorate the crucifixion, and observed the festival of Easter on the third day following, upon whatever day of the week that might fall, hence they obtained the name of Quartodecimans, (from quarto, four, and decem, ten,) thefourteenth day men. The former appealed to the authority of St. Peter and St. Paul, the latter to that of St. John.
The dispute which took place upon this point in the second and third centuries of our era is remarkable, as connected with perhaps the first event which can be brought to bear upon the question of the primacy of the Roman bishop; and it is the more interesting as both parties are accustomed to claim it as a testimony in favor of their own views. Victor, bishop of Rome, wrote an imperious letter to the Asiatic bishops, requiring their conformity to the Western rule; which was answered by Polycrates, bishop of Ephesus, in the name of the rest, expressing their resolution to maintain the custom handed down to them by their ancestors. The Roman bishop thereupon broke off communion with them; but he was rebuked by Irenaeus, of Lyons, and it was agreed by his mediation that each party should retain its customs. Such continued to be the practice till the time of Constantine, when the Council of Nice determined the matter by the following Canons:
a—Easter must be celebrated on a Sunday.
b—This Sunday must follow the 14th day of the paschal moon, so that if the 14th day of the paschal moon fall on a Sunday, then Easter must be celebrated on the Sunday following.
c—The paschal moon is that moon of which the 14th day either falls on or next follows the day of the vernal equinox.
d—The 21st of March is to be accounted the day of the vernal equinox. (Appendix L.)
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 if the paschal full moon fall on Saturday, the 21st of March, then Easter will happen on the 22d, its earliest possible date. For if the full moon arrive on the 20th, it would not be the paschal full moon, which cannot happen before the 21st, consequently the following moon is the paschal full moon, which happens 30 days after the 20th of March, which is the 19th of April. Now, if in this case the 19th of April is Sunday, then Easter must be celebrated the following Sunday, or the 26th of April. Hence, Easter Sunday cannot happen earlier than the 22d of March, or later than the 26th of April.
The observance of these rules renders it necessary to reconcile three periods which have no common measure, namely, the week, the lunar month, and the solar year; and as this can be done only approximately, and within certain limits, the determination of Easter is an affair of considerable nicety and complication. It has already been shown that the lunar cycle contained 6939 days and 18 hours; also, that the exact time of 235 lunations is 6939d, 16h, 31m, 14.45s. The difference, which is 1h, 28m, 45.55s., 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 Nice and the reformation, the error had accumulated to four days, so thatthe 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, but the suppression of 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 become necessary, at the end of 308 years, to place 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 places by another set of numbers called Epacts, the use of which we shall now proceed to explain.
Epact, (Greekepaktos, added or introduced). The excess of the solar year beyond the lunar, employed in the calendar to signify the moon’s age at the beginning of the year. The common solar year consisted of 365 days and the lunar year only 354 days, the difference is eleven; whence, if a new moon fall on the first day of January in any year, the moon will be eleven days old on the first day of the following year, andtwenty-two days old 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; for 33 - 30 = 3. 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 or equals that number.
In order to show how the epacts are connected with the Golden Numbers, let a cypher represent the new moon on the first day of January in any year, then the epacts and Golden Numbers for a whole lunar cycle would be represented thus:
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 18 + 11 = 29, whereas it ought to be a cipher to correspond with the moon’s age and the Golden Number 1. The reason for 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. Thus,18 + 12 = 30; 30 - 30 = 0; the cipher corresponding with the Golden Number 1.
This method of forming the epacts might have been continued indefinitely if the Julian intercalation had been followed without correction and the cycle had 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 errors 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, etc., become 10, 21, 2, 13, etc.
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, etc., in consequence of the lunar equation, becomes 12, 23, 4, 15, etc. In order to preserve the uniformity of the calendar, the epacts are changed only at the commencement of the 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 a day in312½ 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, etc. 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, etc., they compensate each other, and the epacts are not changed.