Encyclopædia Britannica, Ninth Edition/Calendar
A CALENDAR is a method of distributing time into certain periods adapted to the purposes of civil life, as hours, days, weeks, months, years, &c.
Of all the periods marked out by the motions of the celestial bodies, the most conspicuous, and the most intimately connected with the affairs of mankind, are the solar day, which is distinguished by the diurnal revolution of the earth and the alternation of light and darkness, and the solar year, which completes the circle of the seasons. But in the early ages of the world, when mankind were chiefly engaged in rural occupations, the phases of the moon must have been objects of great attention and interest,—hence the month, and the practice adopted by many nations of reckoning time by the motions of the moon, as well as the still more general practice of combining lunar with solar periods. The solar day, the solar year, and the lunar month, or lunation, may therefore be called the natural divisions of time. All others, as the hour, the week, and the civil month, though of the most ancient and general use, are only arbitrary and conventional.
Day.—The true solar day is the interval of time which elapses between two consecutive returns of the same terrestrial meridian to the sun. By reason of the inclined position of the ecliptic, and the unequal progressive motion of the earth in its orbit, it is not always of the same absolute length. But as it would be hardly possible, in the artificial measurement of time, to have regard to this small inequality which is besides constantly varying, the mean solar day is employed for all civil purposes. This is the time in which the earth would make one revolution on its axis, as compared with the sun, if the earth moved at an equable rate in the plane of the equator. The mean solar day is therefore a result of computation, and is not marked precisely by any astronomical phenomenon; but its difference from the true solar or apparent day is so small as to escape ordinary observation.
The subdivision of the day into twentyfour parts, or hours, has prevailed since the remotest ages, though different nations have not agreed either with respect to the epoch of its commencement or the manner of distributing the hours. Europeans in general, like the ancient Egyptians, place the commencement of the civil day at midnight, and reckon twelve morning hours from midnight to midday, and twelve evening hours from midday to midnight. Astronomers, after the example of Ptolemy, regard the day as commencing with the sun's culmination, or noon, and find it most convenient for the purposes of computation to reckon through the whole twentyfour hours. Hipparchus reckoned the twentyfour hours from midnight to midnight. Some nations, as the ancient Chaldeans and the modern Greeks, have chosen sunrise for the commencement of the day; others, again, as the Italians and Bohemians, suppose it to commence at sunset. In all these cases the beginning of the day varies with the seasons at all places not under the equator. In the early ages of Rome, and even down to the middle of the 5th century after the foundation of the city, no other divisions of the day were known than sunrise, sunset, and midday, which was marked by the arrival of the sun between the Rostra and a place called Græcostasis, where ambassadors from Greece and other countries used to stand. The Greeks divided the natural day and night into twelve equal parts each, and the hours thus formed were denominated temporary hours, from their varying in length according to the seasons of the year. The hours of the day and night were of course only equal at the time of the equinoxes. The whole period of day and night they called νυχθήμερον.
The English names of the days are derived from the Saxon. The ancient Saxons had borrowed the week from some Eastern nation, and substituted the names of their own divinities for those of the gods of Greece. In legislative and justiciary acts the Latin names are still retained.
Latin.  English.  Saxon. 
Dies Solis.  Sunday.  Sun's day. 
Dies Lunæ.  Monday.  Moon's day. 
Dies Martis.  Tuesday.  Tiw's day. 
Dies Mercurii.  Wednesday.  Woden's day. 
Dies Jovis.  Thursday.  Thor's day. 
Dies Veneris.  Friday.  Friga's day. 
Dies Saturni.  Saturday.  Seterne's day. 
Month.—Long before the exact length of the year was determined, it must have been perceived that the synodic revolution of the moon is accomplished in about 2912 days. Twelve lunations, therefore, form a period of 354 days, which differs only by about 1114 days from the solar year. From this circumstance has arisen the practice, perhaps universal, of dividing the year into twelve months. But in the course of a few years the accumulated difference between the solar year and twelve lunar months would become considerable, and have the effect of transporting the commencement of the year to a different season. The difficulties that arose in attempting to avoid this inconvenience induced some nations to abandon the moon altogether, and regulate their year by the course of the sun. The month, however, being a convenient period of time, has retained its place in the calendars of all nations; but, instead of denoting a synodic revolution of the moon, it is usually employed to denote an arbitrary number of days approaching to the twelfth part of a solar year.
Among the ancient Egyptians the month consisted of thirty days invariably; and in order to complete the year, five days were added at the end, called supplementary days. They made use of no intercalation, and by losing a fourth of a day every year, the commencement of the year went back one day in every period of four years, and consequently made a revolution of the seasons in 1461 years. Hence 1461 Egyptian years are equal to 1460 Julian years of 36514 days each. This year is called vague, by reason of its commencing sometimes at one season of the year, and sometimes at another.
The Greeks divided the month into three decades, or periods of ten days,—a practice which was imitated by the French in their unsuccessful attempt to introduce a new calendar at the period of the Revolution. This division offers two advantages: the first is, that the period is an exact measure of the month of thirty days; and the second is, that the number of the day of the decade is connected with and suggests the number of the day of the month. For example, the 5th of the decade must necessarily be the 5th, the 15th, or the 25th of the month; so that when the day of the decade is known, that of the month can scarcely be mistaken. In reckoning by weeks, it is necessary to keep in mind the day of the week on which each month begins.
The Romans employed a division of the month and a method of reckoning the days which appear not a little extraordinary, and must, in practice, have been exceedingly inconvenient. As frequent allusion is made by classical writers to this embarrassing method of computation, which is carefully retained in the ecclesiastical calendar, we here give a table showing the correspondence of the Roman months with those of modern Europe.
Days of the Month. 
March. May. July. October. 
January. August. December. 
April. June. September. November. 
February. 
1  Calendæ.  Calendæ.  Calendæ.  Calendæ. 
2  6  4  4  4 
3  5  3  3  3 
4  4  Prid. Nonas.  Prid. Nonas.  Prid. Nonas. 
5  3  Nonæ  Nonæ  Nonæ 
6  Prid. Nonas.  8  8  8 
7  Nonæ  7  7  7 
8  8  6  6  6 
9  7  5  5  5 
10  6  4  4  4 
11  5  3  3  3 
12  4  Prid. Idus.  Prid. Idus.  Prid. Idus. 
13  3  Idus  Idus  Idus 
14  Prid. Idus.  19  18  16 
15  Idus  18  17  15 
16  17  17  16  14 
17  16  16  15  13 
18  15  15  14  12 
19  14  14  13  11 
20  13  13  12  10 
21  12  12  11  9 
22  11  11  10  8 
23  10  10  9  7 
24  9  9  8  6 
25  8  8  7  5 
26  7  7  6  4 
27  6  6  5  3 
28  5  5  4  Prid. Calen. Mart. 
29  4  4  3  
30  3  3  Prid. Calen.  
31  Prid. Calen.  Prid. Calen. 
Year.—The year is either astronomical or civil. The solar astronomical year is the period of time in which the earth performs a revolution in its orbit about the sun, or passes from any point of the ecliptic to the same point again; and consists of 365 days 5 hours 48 min. and 46 sec. of mean solar time. The civil year is that which is employed in chronology, and varies among different nations, both in respect of the season at which it commences and of its subdivisions. When regard is had to the sun's motion alone, the regulation of the year, and the distribution of the days into months, may be effected without much trouble; but the difficulty is greatly increased when it is sought to reconcile solar and lunar periods, or to make the subdivisions of the year depend on the moon, and at the same time to preserve the correspondence between the whole year and the seasons.
Of the Solar Year.—In the arrangement of the civil year, two objects are sought to be accomplished,—first, the equable distribution of the days among twelve months; and secondly, the preservation of the beginning of the year at the same distance from the solstices or equinoxes. Now, as the year consists of 365 days and a fraction, and 365 is a number not divisible by 12, it is impossible that the months can all be of the same length, and at the same time include all the days of the year. By reason also of the fractional excess of the length of the year above 365 days, it likewise happens that the years cannot all contain the same number of days if the epoch of their commencement remains fixed; for the day and the civil year must necessarily be considered as beginning at the same instant; and therefore the extra hours cannot be included in the year till they have accumulated to a whole day. As soon as this has taken place, an additional day must be given to the year.
The civil calendar of all European countries has been borrowed from that of the Romans. Romulus is said to have divided the year into ten months only, including in all 304 days, and it is not very well known how the remaining days were disposed of. The ancient Roman year commenced with March, as is indicated by the names September, October, November, December, which the last four months still retain. July and August, likewise, were anciently denominated Quintilis and Sextilis, their present appellations having been bestowed in compliment to Julius Cæsar and Augustus. In the reign of Numa two months were added to the year, January at the beginning, and February at the end; and this arrangement continued till the year 452 B.C., when the Decemvirs changed the order of the months, and placed February after January. The months now consisted of twentynine and thirty days alternately, to correspond with the synodic revolution of the moon, so that the year contained 354 days; but a day was added to make the number odd, which was considered more fortunate, and the year therefore consisted of 355 days. This differed from the solar year by ten whole days and a fraction; but, to restore the coincidence, Numa ordered an additional or intercalary month to be inserted every second year between the 23rd and 24th of February, consisting of twentytwo and twentythree days alternately, so that four years contained 1465 days, and the mean length of the year was consequently 36614 days. The additional month was called Mercedinus, or Mercedonius, from merces, wages, probably because the wages of workmen and domestics were usually paid at this season of the year. According to the above arrangement, the year was too long by one day, which rendered another correction necessary. As the error amounted to twentyfour days in as many years, it was ordered that every third period of eight years, instead of containing four intercalary months, amounting in all to ninety days, should contain only three of those months, consisting of twentytwo days each. The mean length of the year was thus reduced to 36514 days; but it is not certain at what time the octennial periods, borrowed from the Greeks, were introduced into the Roman calendar, or whether they were at any time strictly followed. It does not even appear that the length of the intercalary month was regulated by any certain principle, for a discretionary power was left with the pontiffs, to whom the care of the calendar was committed, to intercalate more or fewer days according as the year was found to differ more or less from the celestial motions. This power was quickly abused to serve political objects, and the calendar consequently thrown into confusion. By giving a greater or less number of days to the intercalary month, the pontiffs were enabled to prolong the term of a magistracy, or hasten the annual elections; and so little care had been taken to regulate the year, that, at the time of Julius Cæsar, the civil equinox differed from the astronomical by three months, so that the winter months were carried back into autumn, and the autumnal into summer.
In order to put an end to the disorders arising from the negligence or ignorance of the pontiffs, Cæsar abolished the use of the lunary year and the intercalary month, and regulated the civil year entirely by the sun. With the advice and assistance of Sosigenes, he fixed the mean length of the year at 36514 days, and decreed that every fourth year should have 366 days, the other years having each 365. In order to restore the vernal equinox to the 25th of March, the place it occupied in the time of Numa, he ordered two extraordinary months to be inserted between November and December in the current year, the first to consist of thirtythree, and the second of thirtyfour days. The intercalary month of twentythree days fell into the year of course, so that the ancient year of 355 days received an augmentation of ninety days; and the year on that occasion contained in all 445 days. This was called the last year of confusion. The first Julian year commenced with the 1st of January of the 46th before the birth of Christ, and the 708th from the foundation of the city.
In the distribution of the days through the several months, Cæsar adopted a simpler and more commodious arrangement than that which has since prevailed. He had ordered that the first, third, fifth, seventh, ninth, and eleventh months, that is January, March, May, July, September, and November, should have each thirtyone days, and the other months thirty, excepting February, which in common years should have only twentynine, but every fourth year thirty days. This order was interrupted to gratify the vanity of Augustus, by giving the month bearing his name as many days as July, which was named after the first Cæsar. A day was accordingly taken from February and given to August; and in order that three months of thirtyone days might not come together, September and November were reduced to thirty days, and thirtyone given to October and December. For so frivolous a reason was the regulation of Cæsar abandoned, and a capricious arrangement introduced, which it requires some attention to remember.
The regulations of Cæsar were not at first sufficiently understood; and the pontiffs, by intercalating every third year instead of every fourth, at the end of thirtysix years had intercalated twelve times, instead of nine. This mistake having been discovered, Augustus ordered that all the years from the thirtyseventh of the era to the forty eighth inclusive should be common years, by which means the intercalations were reduced to the proper number of twelve in fortyeight years. No account is taken of this blunder in chronology; and it is tacitly supposed that the calendar has been correctly followed from its commencement.
Although the Julian method of intercalation is perhaps the most convenient that could be adopted, yet, as it supposes the year too long by 11 minutes 14 seconds, it could not without correction very long answer the purpose for which it was devised, namely, that of preserving always the same interval of time between the commencement of the year and the equinox. Sosigenes could scarcely fail to know that this year was too long; for it had been shown long before, by the observations of Hipparchus, that the excess of 36514 days above a true solar year would amount to a day in 300 years. The real error is indeed more than double of this, and amounts to a day in 128 years; but in the time of Cæsar the length of the year was an astronomical element not very well determined. In the course of a few centuries, however, the equinox sensibly retrograded towards the beginning of the year. When the Julian calendar was introduced, the equinox fell on the 25th of March. At the time of the Council of Nice, which was held in 325, it fell on the 21st; and when the reformation of the calendar was made in 1582, it had retrograded to the 11th. In order to restore the equinox to its former place, Pope Gregory XIII. directed ten days to be suppressed in the calendar; and as the error of the Julian intercalation was now found to amount to three days in 400 years, he ordered the intercalations to be omitted on all the centenary years excepting those which are multiples of 400. According to the Gregorian rule of intercalation, therefore, every year of which the number is divisible by four without a remainder, is a leap year, excepting the centurial years, which are only leap years when divisible by four after omitting the two ciphers. Thus 1600 was a leap year, but 1700, 1800, and 1900 are common years; 2000 will be a leap year, and so on.
As the Gregorian method of intercalation has been adopted in all Christian countries, Russia excepted, it becomes interesting to examine with what degree of accuracy it reconciles the civil with the solar year. According to the best determinations of modern astronomy (Le Verrier's Solar Tables, Paris, 1858, p. 102), the mean geocentric motion of the sun in longitude, from the mean equinox during a Julian year of 365·25 days, the same being brought up to the present date, is 360° + 27″ ·685. Thus the mean length of the solar year is found to be 360°360° + 27″ ·685 × 365·25 = 365·2422 days, or 365 days 5 hours 48 min. 46 sec. Now the Gregorian rule gives 97 intercalations in 400 years; 400 years therefore contain 365 × 400 + 97, that is, 146,097 days; and consequently one year contains 365·2425 days, or 365 days 5 hours 49 min. 12 sec. This exceeds the true solar year by 26 seconds, which amount to a day in 3323 years. It is perhaps unnecessary to make any formal provision against an error which can only happen after so long a period of time; but as 3323 differs little from 4000, it has been proposed to correct the Gregorian rule by making the year 4000 and all its multiples common years. With this correction the rule of intercalation is as follows:—
Every year the number of which is divisible by 4 is a leap year, excepting the last year of each century, which is a leap year only when the number of the century is divisible by 4; but 4000, and its multiples, 8000, 12,000, 16,000, &c. are common years. Thus the uniformity of the intercalation, by continuing to depend on the number four, is preserved, and by adopting the last correction the commencement of the year would not vary more than a day from its present place in two hundred centuries.
In order to discover whether the coincidence of the civil and solar year could not be restored in shorter periods by a different method of intercalation, we may proceed as follows:—The fraction 0·2422, which expresses the excess of the solar year above a whole number of days, being converted into a continued fraction, becomes
1  
4  +  1  
7  +  1  
1  +  1  
3  +  1  
4  +  1  
1  +,  &c. 
which gives the series of approximating fractions,
14,  729,  833,  31128,  132545,  163673,  &c. 
The third, 833, gives eight intercalations in thirtythree years, or seven successive intercalations at the end of four years respectively, and the eighth at the end of five years. This supposes the year to contain 365 days 5 hours 49 min. 5·45 sec.
The fraction 833 offers a convenient and very accurate method of intercalation. It implies a year differing in excess from the true year only by 19·45 seconds, while the Gregorian year is too long by 26 seconds. It produces a much nearer coincidence between the civil and solar years than the Gregorian method; and, by reason of its shortness of period, confines the evagations of the mean equinox from the true within much narrower limits. It has been stated by Scaliger, Weidler, Montucla, and others, that the modern Persians actually follow this method, and intercalate eight days in thirtythree years. The statement has, however, been contested on good authority; and it seems proved (see Delambre, Astronomie Moderne, tom. i. p. 81) that the Persian intercalation combines the two periods 729 and 833. If they follow the combination 7 + 3 x 829 + 3 x 33 = 31128, their determination of the length of the tropical year has been extremely exact. The discovery of the period of thirtythree years is ascribed to Omar Cheyam, one of the eight astronomers appointed by GelalEddin Malech Shah, sultan of Khorassan, to reform or construct a calendar, about the year 1079 of our era.
If the commencement of the year, instead of being retained at the same place in the seasons by a uniform method of intercalation, were made to depend on astronomical phenomena, the intercalations would succeed each other in an irregular manner, sometimes after four years and sometimes after five; and it would occasionally, though rarely indeed, happen, that it would be impossible to determine the day on which the year ought to begin. In the calendar, for example, which was attempted to be introduced in France in 1793, the beginning of the year was fixed at the midnight preceding the day in which the true autumnal equinox falls. But supposing the instant of the sun's entering into the sign Libra to be very near midnight, the small errors of the solar tables might render it doubtful to which day the equinox really belonged; and it would be in vain to have recourse to observation to obviate the difficulty. It is therefore infinitely more commodious to determine the commencement
of the year by a fixed rule of intercalation; and of the various methods which might be employed, no one, perhaps is on the whole more easy of application, or better adapted for the purpose of computation, than the Gregorian now in use. But a system of 31 intercalations in 128 years would be by far the most perfect as regards mathematical accuracy. Its adoption upon our present Gregorian calendar would only require the suppression of the usual bissextile once in every 128 years, and there would be no necessity for any further correction, as the error is so insignificant that it would not amount to a day in 100,000 years.
Of the Lunar Year and Lunisolar Periods.—The lunar year, consisting of twelve lunar months, contains only 354 days; its commencement consequently anticipates that of the solar year by eleven days, and passes through the whole circle of the seasons in about thirtyfour lunar years. It is therefore so obviously illadapted to the computation of time, that, excepting the modern Jews and Mahometans, almost all nations who have regulated their months by the moon have employed some method of intercalation by means of which the beginning of the year is retained at nearly the same fixed place in the seasons.
In the early ages of Greece the year was regulated entirely by the moon. Solon divided the year into twelve months, consisting alternately of twentynine and thirty days, the former of which were called deficient months, and the latter full months. The lunar year, therefore, contained 354 days, falling short of the exact time of twelve lunations by about 8⋅8 hours. The first expedient adopted to reconcile the lunar and solar years seems to have been the addition of a month of thirty days to every second year. Two lunar years would thus contain 25 months, or 738 days, while two solar years, of 36514 days each, contain 73012 days. The difference of 712 days was still too great to escape observation; it was accordingly proposed by Cleostratus of Tenedos, who flourished shortly after the time of Thales, to omit the biennary intercalation every eighth year. In fact, the 712 days by which two lunar years exceeded two solar years, amounted to thirty days, or a full month, in eight years. By inserting, therefore, three additional months instead of four in every period of eight years, the coincidence between the solar and lunar year would have been exactly restored if the latter had contained only 354 days, inasmuch as the period contains 354 × 8 + 3 × 30 = 2922 days, corresponding with eight solar years of 36514 days each. But the true time of 99 lunations is 2923⋅528 days, which exceeds the above period by 1⋅528 days, or thirtysix hours and a few minutes. At the end of two periods, or sixteen years, the excess is three days, and at the end of 160 years, thirty days. It was therefore proposed to employ a period of 160 years, in which one of the intercalary months should be omitted; but as this period was too long to be of any practical use, it was never generally adopted. The common practice was to make occasional corrections as they became necessary, in order to preserve the relation between the octennial period and the state of the heavens; but these corrections being left to the care of incompetent persons, the calendar soon fell into great disorder, and no certain rule was followed till a new division of the year was proposed by Meton and Euctemon, which was immediately adopted in all the states and dependencies of Greece.
The mean motion of the moon in longitude, from the mean equinox, during a Julian year of 365⋅25 days (according to Hansen's Tables de la Lune, London, 1857, pages 15, 16) is, at the present date, 13 × 360° + 477644″⋅409; that of the sun being 360° + 27″⋅685. Thus the corresponding relative mean geocentric motion of the moon from the sun is 12 × 360° + 477616″⋅724; and the duration of the mean synodic revolution of the moon, or lunar month, is therefore 360°12 × 360° + 477616″⋅724 × 365⋅25 = 29⋅530588 days, or 29 days, 12 hours, 44 min. 2⋅8 sec.
The Metonic Cycle, which may be regarded as the chefd'œuvre of ancient astronomy, is a period of nineteen solar years, after which the new moons again happen on the same days of the year. In nineteen solar years there are 235 lunations, a number which, on being divided by nineteen, gives twelve lunations for each year, 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 months 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 twentynine 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 thirty days, and 110 deficient months of twentynine days each. The number of days in the period was therefore 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 months of thirty days, or of 7050 days, from which 110 days are to be deducted. This gives one day to be suppressed in sixtyfour; so that if we suppose the months to contain each thirty days, and then omit every sixtyfourth 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 19 × 365⋅2422 = 6939⋅6018 days, or 6939 days 14 hours 26⋅592 minutes; hence the period, which is exactly 6940 days, exceeds nineteen revolutions of the sun by nine and a half hours nearly. On the other hand, the exact time of a synodic revolution of the moon is 29⋅530588 days; 235 lunations, therefore, contain 235 × 29⋅530588 = 6939⋅68818 days, or 6939 days 16 hours 31 minutes, so that the period exceeds 235 lunations by only seven and a half hours.
After the Metonic cycle had been in use about a century, a correction was proposed by Calippus. At the end of four cycles, or seventysix 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, 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 of the moon, consequently, amounted only to six hours, or to one day in 304 years. This period exceeds seventysix true solar years by fourteen hours and a quarter nearly, but coincides exactly with seventysix Julian years; and in the time of Calippus the length of the solar year was almost universally supposed to be exactly 36514 days. The Calippic period is frequently referred to as a date by Ptolemy.
Dominical Letter.—The first problem which the construction of the calendar presents is to connect the week with the year, or to find the day of the week corresponding to a given day of any year of the era. As the number of days in the week and the number in the year are prime to one another, two successive years cannot begin with the same day; for if a common year begins, for example, with Sunday, the following year will begin with Monday, and if a leap year begins with Sunday, the year following will begin with Tuesday. For the sake of greater generality, the days of the week are denoted by the first seven letters of the alphabet, A, B, C, D, E, F, G, which are placed in the calendar beside the days of the year, so that A stands opposite the first day of January, B opposite the second, and so on to G, which stands opposite the seventh; after which A returns to the eighth, and so on through the 365 days of the year. Now, if one of the days of the week, Sunday for example, is represented by E, Monday will be represented by F, Tuesday by G, Wednesday by A, and so on; and every Sunday through the year will have the same character E, every Monday F, and so with regard to the rest. The letter which denotes Sunday is called the Dominical Letter, or the Sunday Letter; and when the dominical letter of the year is known, the letters which respectively correspond to the other days of the week become known at the same time.
Solar Cycle.—In the Julian calendar the dominical letters are readily found by means of a short cycle, in which they recur in the same order without interruption. The number of years in the intercalary period being four, and the days of the week being seven, their product is 4 × 7 = 28; twentyeight years is therefore a period which includes all the possible combinations of the days of the week with the commencement of the year. This period is called the Solar Cycle, or the Cycle of the Sun, and restores the first day of the year to the same day of the week. At the end of the cycle the dominical letters return again in the same order on the same days of the month; hence a table of dominical letters, constructed for twentyeight years, will serve to show the dominical letter of any given year from the commencement of the era to the reformation. The cycle, though probably not invented before the time of the Council of Nice, is regarded as having commenced nine years before the era, so that the year one was the tenth of the solar cycle. To find the year of the cycle, we have therefore the following rule:—Add nine to the date, divide the sum by twentyeight; 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 twentyeighth or last of the cycle. This rule is conveniently expressed by the formula , in which x denotes the date, and the symbol r denotes that the remainder, which arises from the division of x + 9 by 28, is the number required. Thus, for 1840, we have 1840 + 928 = 66128; therefore = 1, and the year 1840 is the first of the solar cycle. In order to make use of the solar cycle in finding the dominical letter, it is necessary to know that the first year of the Christian era began with Saturday. The dominical letter of that year, which was the tenth of the cycle, was consequently B. The following year, or the 11th of the cycle, the letter was A; then G. The fourth year was bissextile, and the dominical letters were F, E; the following year D, and so on. In this manner it is easy to find the dominical letter belonging to each of the twentyeight years of the cycle. But at the end of a century the order is interrupted in the Gregorian calendar by the secular suppression of the leap year; hence the cycle can only be employed during a century. In the reformed calendar the intercalary period is four hundred years, which number being multiplied by seven, gives two thousand eight hundred years as the interval in which the coincidence is restored between the days of the year and the days of the week. This long period, however, may be reduced to four hundred years; for since the dominical letter goes back five places every four years, its variation in four hundred years, in the Julian calendar, was five hundred places, which is equivalent to only three places (for five hundred divided by seven leaves three); but the Gregorian calendar suppresses exactly three intercalations in four hundred years, so that after four hundred years the dominical letters must again return in the same order.
Hence the following table of dominical letters for four hundred years will serve to show the dominical letter of any year in the Gregorian calendar for ever. It contains four columns of letters, each column serving for a century. In order to find the column from which the letter in any given case is to be taken, strike off the two last figures of the date, divide the preceding figures by four, and the remainder will indicate the column. The symbol X, employed in the formula at the top of the column, denotes the number of centuries, that is, the figures remaining after the last two have been struck off. For example, required the dominical letter of the year 1839? In this case X = 18, therefore = 2; and in the second column of letters, opposite 39, in the table we find F, which is the letter of the proposed year.
It deserves to be remarked, that as the dominical letter of the first year of the era was B, the first column of the following table will give the dominical letter of every year from the commencement of the era to the reformation. For this purpose divide the date by 28, and the letter opposite the remainder, in the first column of figures, is the dominical letter of the year. For example, supposing the date to be 1148. On dividing by 28, the remainder is 0, or 28; and opposite 28, in the first column of letters, we find D, C, the dominical letters of the year 1148.
Table I.—Dominical Letters.
Years of the Century.  
0  C  E  G  B, A  
1  29  57  85  B  D  F  G 
2  30  58  86  A  C  E  F 
3  31  59  87  G  B  D  E 
4  32  60  88  F, E  A, G  C, B  D, C 
5  33  61  89  D  F  A  B 
6  34  62  90  C  E  G  A 
7  35  63  91  B  D  F  G 
8  36  64  92  A, G  C, B  E, D  F, E 
9  37  65  93  F  A  C  D 
10  38  66  94  E  G  B  C 
11  39  67  95  D  F  A  B 
12  40  68  96  C, B  E, D  G, F  A, G 
13  41  69  97  A  C  E  F 
14  42  70  98  G  B  D  E 
15  43  71  99  F  A  C  D 
16  44  72  E, D  G, F  B, A  C, B  
17  45  73  C  E  G  A  
18  46  74  B  D  F  G  
19  47  75  A  C  E  F  
20  48  76  G, F  B, A  D, C  E, D  
21  49  77  E  G  B  C  
22  50  78  D  F  A  B  
23  51  79  C  E  G  A  
24  52  80  B, A  D, C  F, E  G, F  
25  53  81  G  B  D  E  
26  54  82  F  A  C  D  
27  55  83  E  G  B  C  
28  56  84  D, C  F, E  A, G  B, A 
Table II.—The Day of the Week.
Month.  Dominical Letter.  
Jan.Oct.  A  B  C  D  E  F  G  
Feb.Mar.Nov.  D  E  F  G  A  B  C  
AprilJuly  G  A  B  C  D  E  F  
May  B  C  D  E  F  G  A  
June  E  F  G  A  B  C  D  
August  C  D  E  F  G  A  B  
Sept.Dec.  F  G  A  B  C  D  E  
1  8  15  22  29  Sun.  Sat.  Frid.  Thur.  Wed.  Tues.  Mon. 
2  9  16  23  30  Mon.  Sun.  Sat.  Frid.  Thur.  Wed.  Tues. 
3  10  17  24  31  Tues.  Mon.  Sun.  Sat.  Frid.  Thur.  Wed. 
4  11  18  25  Wed.  Tues.  Mon.  Sun.  Sat.  Frid.  Thur.  
5  12  19  26  Thur.  Wed.  Tues.  Mon.  Sun.  Sat.  Frid.  
6  13  20  27  Frid.  Thur.  Wed.  Tues.  Mon.  Sun.  Sat.  
7  14  21  28  Sat.  Frid.  Thur.  Wed.  Tues.  Mon.  Sun.  
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 twentynine 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 twentynine days is added at the end. This gives 19 x 354 + 6 x 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 happens 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 693934 days, agreeing exactly with nineteen Julian years.
By means of the lunar cycle the new moons of the calendar were indicated before the reformation. As the cycle restores these phenomena to the same days of the civil month, they will fall on the same days in any two years which occupy the same place in the cycle; consequently a table of the moon's phases for 19 years will serve for any year whatever when we know its number in the cycle. This number is called the Golden Number, either because it was so termed by the Greeks, or because it was usual to mark it with red letters in the calendar. The Golden Numbers were introduced into the calendar about the year 530, but disposed as they would have been if they had been inserted at the time of the Council of Nice. The cycle is supposed to commence with the year in which the new moon falls on the 1st of January, which took place the year preceding the commencement of our era. Hence, to find the Golden Number N, for any year x, we have , which gives the following rule: Add 1 to the date, divide the sum by 19; the quotient is the number of cycles elapsed, and the remainder is the Golden Number. When the remainder is 0, the proposed year is of course the last or 19th of the cycle. It ought to be remarked that the new moons, determined in this manner, may differ from the astronomical new moons sometimes as much as two days. The reason is, that the sum of the solar and lunar inequalities, which are compensated in the whole period, may amount in certain cases to 10°, and thereby cause the new moon to arrive on the second day before or after its mean time.
Dionysian Period.—The cycle of the sun brings back the days of the month to the same day of the week; the lunar cycle restores the new moons to the same day of the month; therefore 28 x 19 = 532 years, includes all the variations in respect of the new moons and the dominical letters, and is consequently a period after which the new moons again occur on the same day of the month and the same day of the week. This is called the Dionysian or Great Paschal Period, from its having been employed by Dionysius Exiguus, familiarly styled “Denys the Little,” in determining Easter Sunday. It was, however, first proposed by Victorius of Aquitain, who had been appointed by Pope Hilary to revise and correct the church calendar. Hence it is also called the Victorian Period. It continued in use till the Gregorian reformation.
Cycle of Indiction.—Besides the solar and lunar cycles, there is a third of 15 years, called the cycle of indiction, frequently employed in the computations of chronologists. This period is not astronomical, like the two former, but has reference to certain judicial acts which took place at stated epochs under the Greek emperors. Its commencement is referred to the 1st of January of the year 313 of the common era. By extending it backwards, it will be found that the first of the era was the fourth of the cycle of indiction. The number of any year in this cycle will therefore be given by the formula , that is to say, add 3 to the date, divide the sum by 15, and the remainder is the year of the indiction. When the remainder is 0, the proposed year is the fifteenth of the cycle.
We have already seen that the year 1 of the era had 10 for its number in the solar cycle, 2 in the lunar cycle, and 4 in the cycle of indiction; the question is therefore to find a number such, that when it is divided by the three numbers 28, 19, and 15 respectively, the three quotients shall be 10, 2, and 4.
28 x + 10 = 19 y + 2 = 15 z + 4.
To resolve the equation 28 x + 10 = 19 y + 2, or y = x + 9 x + 819, let m = 9 x + 819, we have then x = 2 m + m − 89.
Let m − 89 = m′; then m = 9 m′ + 8; hence
x = 18 m′ + 16 + m′ = 19 m′ + 16. 
(1.) 
Again, since 28 x + 10 = 15 z + 4, we have
15 z = 28 x + 6, or z = 2 x − x − 615.
Let x − 615 = n; then 2 x = 15 n + 6, and x = 7 n + 3 + n2.
Let n2 = n′; then n′ = 2 n′; consequently
x = 14 n′ + 3 + n′ = 15 n′ + 3. 
(2.) 
Equating the above two values of x, we have
15 n′ + 3 = 19 m′ + 16; whence n′ = m′ + 4 m′ + 1315.
Let 4 m′ + 1315 = p; we have then
4 m′ = 15 p − 13, and m′ = 4 p − p + 134.
Let p + 134 = p′; then p = 4 p′ − 13;
whence m′ = 16 p′ − 52 − p′ = 15 p′ − 52.
Now in this equation p′ may be any number whatever, provided 15 p′ exceed 52. The smallest value of p′ (which is the one here wanted) is therefore 4; for 15 × 4 = 60. Assuming therefore p′=4, we have m′=60−52=8; and consequently, since x=19 m′+16, x=19×8+16=168. The number required is consequently 28×168+10=4714.
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
P = 4713 + x, and x = P − 4713.
Era, 
−1, 
−2, 
−3,... 
−x, 

Period, 
4713, 
4712, 
4711,... 
4714 
− x; 
whence
P = 4714 − x, and x = 4714 − P.
But astronomers, in order to preserve the uniformity of computation, make the series of years proceed without interruption and reckon the year preceding the first of the era 0. Thus
Era, 
0 
−1 
−2,... 
−x, 

Period, 
4713, 
4712, 
4711,... 
4713 
− x; 
therefore, in this case
P = 4713 − x, and x = 4713 − P.
Reformation of the Calendar.—The ancient church calendar was founded on two suppositions, both erroneous, namely, that the year contains 36514 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 Nice. 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 John of Sacrobosco, in a work published under the title 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 Sextus 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 Stöffler, Albert Pighius, John Schöner, Lucas Gauricus, and other mathematicians of celebrity. At length Pope Gregory XIII. perceiving that the measure was likely to confer a great eclat on his pontificate, undertook the longdesired 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 commencement of the year to the same place in the seasons that it had occupied at the time of the Council of Nice, 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 present century is twelve days. From 1900 to 2100 inclusive it will be thirteen days.
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 twentytwo days on the first of the third year. The numbers eleven and twentytwo are therefore the epacts of those years respectively. Another addition of eleven gives thirtythree 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 , 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 twentynine 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 twentynine 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 31212 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.
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 following table.
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 
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 twentynine days. The epact of the following year is therefore twentynine. 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 twentynine. In like manner, if the new moon fell on the 4th of December, the epact of the following year would be twentyeight, 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 = 9, and the epact, as found in Table III., is twentyeight. 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.
Table IV.—Gregorian Calendar.
Days  January  February  March  April  May  June  July  August  September  October  November  December  
E  L  E  L  E  L  E  L  E  L  E  L  E  L  E  L  E  L  E  L  E  L  E  L  
1  *  A  29  D  *  D  29  G  28  B  27  E  26  G  25 24  C  23  F  22  A  21  D  20  F 
2  29  B  28  E  29  E  28  A  27  C  25′26  F  25′25  A  23  D  22  G  21  B  20  E  19  G 
3  28  C  27  F  28  F  27  B  26  D  25 24  G  24  B  22  E  21  A  20  C  19  F  18  A 
4  27  D  25′26  G  27  G  25′26  C  25′25  E  23  A  23  C  21  F  20  B  19  D  18  G  17  B 
5  26  E  25 24  A  26  A  25 24  D  24  F  22  B  22  D  20  G  19  C  18  E  17  A  16  C 
6  25′25  F  23  B  25′25  B  23  E  23  G  21  C  21  E  19  A  18  D  17  F  16  B  15  D 
7  24  G  22  C  24  C  22  F  22  A  20  D  20  F  18  B  17  E  16  G  15  C  14  E 
8  23  A  21  D  23  D  21  G  21  B  19  E  19  G  17  C  16  F  15  A  14  D  13  F 
9  22  B  20  E  22  E  20  A  20  C  18  F  18  A  16  D  15  G  14  B  13  E  12  G 
10  21  C  19  F  21  F  19  B  19  D  17  G  17  B  15  E  14  A  13  C  12  F  11  A 
11  20  D  18  G  20  G  18  C  18  E  16  A  16  C  14  F  13  B  12  D  11  G  10  B 
12  19  E  17  A  19  A  17  D  17  F  15  B  15  D  13  G  12  C  11  E  10  A  9  C 
13  18  F  16  B  18  B  16  E  16  G  14  C  14  E  12  A  11  D  10  F  9  B  8  D 
14  17  G  15  C  17  C  15  F  15  A  13  D  13  F  11  B  10  E  9  G  8  C  7  E 
15  16  A  14  D  16  D  14  G  14  B  12  E  12  G  10  C  9  F  8  A  7  D  6  F 
16  15  B  13  E  15  E  13  A  13  C  11  F  11  A  9  D  8  G  7  B  6  E  5  G 
17  14  C  12  F  14  F  12  B  12  D  10  G  10  B  8  E  7  A  6  C  5  F  4  A 
18  13  D  11  G  13  G  11  C  11  E  9  A  9  C  7  F  6  B  5  D  4  G  3  B 
19  12  E  10  A  12  A  10  D  10  F  8  B  8  D  6  G  5  C  4  E  3  A  2  C 
20  11  F  9  B  11  B  9  E  9  G  7  C  7  E  5  A  4  D  3  F  2  B  1  D 
21  10  G  8  C  10  C  8  F  8  A  6  D  6  F  4  B  3  E  2  G  1  C  *  E 
22  9  A  7  D  9  D  7  G  7  B  5  E  5  G  3  C  2  F  1  A  *  D  29  F 
23  8  B  6  E  8  E  6  A  6  C  4  F  4  A  2  D  1  G  *  B  29  E  28  G 
24  7  C  6  F  7  F  5  B  5  D  3  G  3  B  1  E  *  A  29  C  28  F  27  A 
25  6  D  4  G  6  G  4  C  4  E  2  A  2  G  *  F  29  B  28  D  27  G  26  B 
26  5  E  3  A  5  A  3  D  3  F  1  B  1  D  29  G  28  C  27  E  25′26  A  25′25  C 
27  4  F  2  B  4  B  2  E  2  G  *  C  *  E  28  A  27  D  26  F  25 24  B  24  D 
28  3  G  1  C  3  C  1  F  1  A  29  D  29  F  27  B  25′26  E  25′25  G  23  C  23  E 
29  2  A  2  D  *  G  *  B  28  E  28  G  26  C  25 24  F  24  A  22  D  22  F  
30  1  B  1  E  29  A  29  C  27  F  27  A  25′25  D  23  G  23  B  21  E  21  G  
31  *  C  *  F  28  D  25′26  B  24  E  22  C  19′20  A 
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 twentynine and thirty days alternately, therefore in six months of the year the thirty epacts must correspond only to twentynine days. For this reason the epacts twentyfive and twentyfour 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 twentyfour and twentyfive, 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, I, 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 V.—Perpetual Table, showing Easter.
Epact.  Dominical Letter.  
For Leap Years use the second Letter.  
A  B  C  D  E  F  G  
*  Apr.  16  Apr.  17  Apr.  18  Apr.  19  Apr.  20  Apr.  14  Apr.  15 
1  ,,  16  ,,  17  ,,  18  ,,  19  ,,  20  ,,  14  ,,  15 
2  ,,  16  ,,  17  ,,  18  ,,  12  ,,  13  ,,  14  ,,  15 
3  ,,  16  ,,  17  ,,  11  ,,  12  ,,  13  ,,  14  ,,  15 
4  ,,  16  ,,  10  ,,  11  ,,  12  ,,  13  ,,  14  ,,  15 
5  ,,  9  ,,  10  ,,  11  ,,  12  ,,  13  ,,  14  ,,  15 
6  ,,  9  ,,  10  ,,  11  ,,  12  ,,  13  ,,  14  ,,  8 
7  ,,  9  ,,  10  ,,  11  ,,  12  ,,  13  ,,  7  ,,  8 
8  ,,  9  ,,  10  ,,  11  ,,  12  ,,  6  ,,  7  ,,  8 
9  ,,  9  ,,  10  ,,  11  ,,  5  ,,  6  ,,  7  ,,  8 
10  ,,  9  ,,  10  ,,  4  ,,  5  ,,  6  ,,  7  ,,  8 
11  ,,  9  ,,  3  ,,  4  ,,  5  ,,  6  ,,  7  ,,  8 
12  ,,  2  ,,  3  ,,  4  ,,  5  ,,  6  ,,  7  ,,  8 
13  ,,  2  ,,  3  ,,  4  ,,  5  ,,  6  ,,  7  ,,  1 
14  ,,  2  ,,  3  ,,  4  ,,  5  ,,  6  Mar.  31  ,,  1 
15  ,,  2  ,,  3  ,,  4  ,,  5  Mar.  30  ,,  31  ,,  1 
16  ,,  2  ,,  3  ,,  4  Mar.  29  ,,  30  ,,  31  ,,  1 
17  ,,  2  ,,  3  Mar.  28  ,,  29  ,,  30  ,,  31  ,,  1 
18  ,,  2  Mar.  27  ,,  28  ,,  29  ,,  30  ,,  31  ,,  1 
19  Mar.  26  ,,  27  ,,  28  ,,  29  ,,  30  ,,  31  ,,  1 
20  ,,  26  ,,  27  ,,  28  ,,  29  ,,  30  ,,  31  Mar.  25 
21  ,,  26  ,,  27  ,,  28  ,,  29  ,,  30  ,,  24  ,,  25 
22  ,,  26  ,,  27  ,,  28  ,,  29  ,,  23  ,,  24  ,,  25 
23  ,,  26  ,,  27  ,,  28  ,,  22  ,,  23  ,,  24  ,,  25 
24  Apr.  23  Apr.  24  Apr.  25  Apr.  19  Apr.  20  Apr.  21  Apr.  22 
25  ,,  23  ,,  24  ,,  25  ,,  19  ,,  20  ,,  21  ,,  22 
26  ,,  23  ,,  24  ,,  18  ,,  19  ,,  20  ,,  21  ,,  22 
27  ,,  23  ,,  17  ,,  18  ,,  19  ,,  20  ,,  21  ,,  22 
28  ,,  16  ,,  17  ,,  18  ,,  19  ,,  20  ,,  21  ,,  22 
29  ,,  16  ,,  17  ,,  18  ,,  19  ,,  20  ,,  21  ,,  15 
We will now show in what manner this whole apparatus of methods and tables may be dispensed with, and the Gregorian calendar reduced to a few simple formulæ of easy computation.
And, first, to find the dominical letter. Let L denote the number of the dominical letter of any given year of the era. Then, since every year which is not a leap year ends with the same day as that with which it began, the dominical letter of the
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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, tom. ii.; Traité de la Sphère et du Calendrier, par M. Revard, Paris, 1816; Delambre, Traité de l'Astronomie Théoretique et Practique, tom. iii.; Histoire de l'Astronomie Moderne; Methodus technica brevis, perfacilis, ac perpetua construendi Calendarium Ecclesiasticum, Stylo tam novo quam vetere, pro cunctis Christianis Europæ populis, &c. auctore Paulo Tittel, Göttingen, 1816; Formole analitiche pel calcolo della Pasqua, e correzione di quello di Gauss, con critiche osservazioni sù quanto ha scritto del Calendario il Delambri di Lodovico Ciccolini, Rome, 1817; E. H. Lindo, Jewish Calendar for Sixtyfour Years, 1838; W. S. B. Woolhouse, Measures, Weights, and Moneys of all Nations, 1869.