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Kalender


Til 'Lav din egen kalender'  Lav din egen kalender
 

10 udregninger - og så er påskesøndag fundet.


En grundig gennemgang af hvordan og hvorfor påske falder som den gør
       
 
         
Ten Divisions Lead to Easter
 
 
Back rooms and their problems
 
 
The variation of Easter from year to year is regarded as something of a puzzle--or paradox--by many: we propose to discuss some of the considerations which lead to the apparently arbitrary variations; and we shall give a purely numerical rule--subject to no exceptions of any kind--which calculates the date of Easter from a knowledge only of the year number, with no reference to any other rules or tables. This rule is an arithmetical equivalent of the various types of tables which have been employed by religious authorities and others: it includes a mathematical formulation of the procedure by which these tables have been constructed. (Simpler rules are sometimes found, and may be useful if limitations to their validity are known and acceptable. Such rules may apply to some particular century only, or to a longer--but still limited--period: they may include exceptional procedures for certain rare cases; or errors may arise in these cases, because these are omitted.)
 
 
The Easter rules may be regarded as an entirely creditable piece of applied science: here--as sometimes elsewhere--we have to remember that the technologists responsible for the solution were not themselves responsible for the specification of the problem; for they had to meet requirements laid down by others to whom the scientific aspects were by no means the only questions at issue.
 
 
The basic problem derives from the wishes of the Christian churches to commemorate the Resurrection annually on a suitable date: but difficulties arise because of certain astronomical considerations which are implicitly involved, though not immediately apparent; and further complications have arisen from various causes which are attributable to various human imperfections.
 
 
Wheels within wheels within wheels
 
 
Astronomical considerations become important in various ways, as soon as increasing civilization requires days to be arranged in an ordered succession, related to periodicities of natural phenomena. Agriculture requires attention to annual events like sowing and harvesting, or major variations of water supply: hunting and travelling are activities in which the presence or absence of light from the moon may be important--and this is true also of warfare, even in modern times.
 
 
In arranging days in a calendar, the natural units to employ are therefore months as defined by successive new (or full) moons, and years as defined by the seasons--or, more precisely, by equinoxes or solstices: mean values of these periods define the synodic months and tropical years of the astronomer. As time progresses, it becomes increasingly necessary to take note of the facts that the year does not contain an exact number of lunar months, and that neither of the latter periods contains an exact number of days.
 
 
In the very long term, astronomers have to take note of slight changes in the tropical year and synodic month: and values applicable to the year 1900 are 365 242l99 days and 29 5305882 days, giving l2 3682670 months in the year. In the course of a century, the three values stated will all reduce by a few units of the last figure given; and these modern estimates can be in error only by still smaller amounts.
 
 
Some calendars purchase simplicity at the price of giving up one of the objectives: in Muslim countries, where seasonal variations are less noticeable, the months are kept in step with the moon, and a l2-month year causes the seasons to run through a cycle of the months about three times in every century; the Julian and Gregorian civil calendars do precisely the opposite, keeping the year correct by following the Roman procedure of having twelve calendar months each of which--on the average--includes about one day more than a lunar month. The Jewish practice is that of the Babylonians, which was adopted also in ancient Greece: both months and years are here kept astronomically significant by having seven 13-month years and twelve 12-month years in every nineteen years; for the agreement of 19 years with 235 months is much better than might perhaps be expected in a comparison which involves such relatively small numbers.
 
 
The Jewish Passover is celebrated for a week following the first full moon of spring: this therefore involves a connexion both with the seasons and with the phases of the moon; and the Jewish calendar conveniently provides for both needs. The Christian Easter involves all this--in a calendar less directly suitable for it--and more besides; for it introduces additional considerations dependent on the cycle of the seven days of the week.
 
 
Numerical orthodoxies and heresies
 
 
The special features of the ecclesiastical calendar arise because all four gospels agree in stating that the Crucifixion occurred on the day of preparation--the Friday before the Saturday of the Jewish Sabbath in the week of the Passover; likewise in stating that the Resurrection took place on the first day of the week--that is, on the Sunday: hence we derive Good Friday and Faster Sunday, and also the choice of Sunday as the holy day of the Christian religion. (The phrase ‘on the third day’ involves the older practice of including both limits when specifying an interval.) These references established a relation between the Christian holy days and the Jewish Passover; and Christian teaching wished to emphasize a connexion between the sacrifice of Christ and the Passover sacrifice.
1.
År 325 - Nicaea.
 

På kirkemødet blev det vedtaget at:

Påskesøndag er første søndag efter fuldmåne efter forårsjævndøgn.
 
Controversy arose in the early Church between those who wished to celebrate Easter on the day of the Passover, regardless of the day of the week, and those who preferred the Sunday following. (The alternative of a fixed Easter was not then in question.) The first day of the Jewish month was intended to coincide with the earliest visibility of the crescent of the new moon--one or two days after the true instant of astronomical new moon--and this caused the full moon to be taken as the fourteenth day of the month: fourteen here became an unlucky number, for those who wished a common day of celebration were finally stigmatized as Quartodeciman heretics; and the Council of Nicaea in A.D.325 laid down that Easter was to be the Sunday which followed the full moon which occurred on, or next after, the day of the spring equinox.
 
 
The participants thus achieved some of the unanimity hoped for by their patron, the Emperor Constantine: but their pronouncements still left scope for argument as to when the full moon and equinox occurred--or were to be held to have occurred. National differences of practice arose, to be gradually resolved, as, for instance, at the Synod of Whitby in A.D. 664, when agreement was secured between the Latin and Celtic Churches; and in A.D. 725 the Venerable Bede wrote what became a standard work on the subject.
 
Påskedag afhænger så bl.a. af lændegraden :-(
Rules of calculation were rightly regarded as preferable to astronomical observations, since observations might in critical cases be unable to decide whether the equinox or the full moon occurred the earlier, or whether the full moon occurred just before midnight, or just after: a month of difference in Easter could arise in the first case, and a further week in the second. Even worse difficulties arise from the dependence of time upon longitude; for an event like the instant of occurrence of an equinox or of a full moon could be indisputably before midnight in Rome, but definitely after midnight in Constantinople or Alexandria.
 
 
Where the limits are set
 
Definition: Forårsjævndøgn er den 21. marts !
The rules adopted were that the date of the equinox was to be taken as 21 March (this was approximately correct for the period at which the Council took place, though leap-year variations could then take the true equinox into 22 March and back); the Julian civil calendar, with leap year every fourth year, was to be used for the year. The full moons were assumed to recur after an exact period of nineteen Julian years of 365 1/4 days: this was made equivalent to 235 lunar months by arranging positions for a corresponding number of new moons spaced some with 30-day gaps and others with 29-day gaps between their calendar dates--with an occasional 31-day gap owing to the extra day of a leap year, which received no special consideration; and the cycle repeated exactly, after seventy-six years.
 
 
The Easter full moon could be either on or after the assumed date of the equinox, but Easter Sunday had to be definitely later than the day of the full moon--being the following Sunday if the full moon was itself a Sunday. This meant that if the full-moon date was Saturday, 21 March, Easter would be Sunday, 22 March, and it could never be earlier: when the full-moon date was Saturday, 20 March, the next full moon was 29 days later, on Sunday 18 April; this then made Easter fall on Sunday, 25 April, its latest possible date. These considerations became matters of orthodox doctrine, and were promulgated--with other matters perhaps of more evident religious importance--by various authorities.
 
532 års periode - og så er det forfra igen.
The 19-year period was called the Lunar Cycle, and the year’s place in this cycle was the Golden Number, which ran from 1 to 19 in each cycle: associations of dates with days of the week recurred in a 28-year cycle compounded from the joint effect of leap-year changes and the cycle of seven days in the week. In a rather forced parallelism this 28-year cycle was called the Solar Cycle: concern with the year involves the sun in both cycles; and the week has no solar connexion or direct astronomical basis. Easter dates then recurred in a 532-year cycle--the least common multiple of the Lunar and Solar cycles.
 
 
The errors of these approximations then began to accumulate: both equinox and full moon could be observed to be occurring ever earlier than their nominal dates. Discussion continued for about 1,000 years; looking some further millennia ahead, Dante could foresee January no longer a winter month; and by the sixteenth century the error was ten days for the sun and four days for the moon.
 
 
Reformations and their problems
2.
år 1582 -
 
Gregoriansk kalender med skudår
In 1582, Pope Gregory XIII introduced a reformation of the calendar, to correct both of the errors: ten days were omitted between Thursday, 4 October, and Friday, 15 October, at a time of year when an exceptional rearrangement of the ecclesiastical calendar was judged to produce least inconvenience; and the dates of the full moons were moved seven days on, in the new calendar--the equivalent of three days back in the old. A bias of one day was intentionally introduced to keep Easter still more clear of the Passover date, for better avoidance of religious riots.
 
 
Three century years out of four were made ordinary years instead of leap years, to correct the accumulating error of the equinox; and eight one-day shifts of the full-moon dates were to occur in each 2,500 years, in century years three apart at first, with a four-century gap at the end of the cycle: the first lunar adjustment was to take place at the assumed termination of such a cycle, in the year 1800.
 
 
The lunar information in the ecclesiastical calendar received a new definition in terms of the epact: this is a whole number between 1 and 30 inclusive which gives the assumed age of the moon, in days, immediately prior to the first day of January of the year concerned. (An epact of 30 is sometimes represented by an asterisk, intended to stand for 0 or 30 indifferently.) In the Julian calendar, the epact had a fixed relation to the Golden Number, and only nineteen of the thirty possible epacts could occur: in the Gregorian calendar, this relation is in effect reviewed at the start of each new century, and the Golden Numbers continue their l9-year cycle undisturbed. The two Gregorian century-year corrections can affect the epact by one day each, in different directions, at different intervals.
 
 
Three times in each four-century period, a leap-year day which was present in the Julian calendar is omitted--in the century years where the year number will not divide exactly by 400--to correct the length of the year: but this omission has to be compensated by a reduction made to the epact, from then onwards; for the epact remains elsewhere conditioned to Julian-calendar assumptions. Apart from this, a less frequent change in the other direction is required, to make a correction for the actual length of the month: because 235 lunar months very slightly fall short of the nineteen Julian years with which--in the cycle--they would otherwise be equated. At present we are in the first of three successive centuries of stable epacts: for neither change arises in the year 2000; and both occur--only to cancel each other--in the year 2100.
 
 
We should note here that in calendars we concern ourselves with a fictitious mean sun and mean moon which have uniform motions, but no accumulating discrepancies with their real astronomical counter-parts. This is similar to the distinction whereby time is determined with a uniform rate suitable for a clock, rather than with the irregularities which affect the time given by a sundial: these irregularities arise because the Earth’s axis is not perpendicular to the plane of its orbit, and because the orbit is an ellipse, and not a circle. Analogous considerations can cause the mean full moons to differ by a day or more from the true full moons of the heavens--or of almanacs which correctly take note of all calculable variations. Moreover, for ecclesiastical purposes the equinox and the full moons are assigned only to dates, and not to any particular times--to avoid disputes dependent on differences of local time, or arguable cases in the vicinity of date changes at midnight: and we noted earlier that the ecclesiastical full moon was purposely placed later than the mean full moon--for a wholly praiseworthy reason.
 
 
Discrepancies between ecclesiastical and astronomical data need not therefore imply any errors in either: though paradoxes can indeed arise if this consideration is not kept in mind. Owing to varying lack of exact compensation between whole numbers of leap-year days and the excesses which they are intended to correct, the almanacs can show the astronomical equinox to occur as early as 19 March: an extreme case of this will arise in 2096. The opposite limit is an equinox late in the day on 21 March, and an extreme case of this occurred in 1903: the lack of balance is greatest over the intervening 193 years, in the course of which forty-nine leap-year days will have been inserted, although the true excess--over this period--is approximately 46 3/4 days only; and similar extremes occur at four-century intervals from these years. Arguments involving astronomical moons or equinoxes--or both--could occasionally give scope for believing that Easter should be either four or five weeks different from the chosen date.
 
 
Other cases can suggest that the date of Easter is wrong by exactly a week: in these cases, a difference of date in different parts of the world will usually allow local astronomical defence of the accepted date of Easter--but not always. In 1845 and 1923 the true full moon fell on Easter Sunday everywhere except for localities of large easterly longitude, which then had a full moon on Easter Monday: in 1744 there was the opposite case of a full moon which most of the world would say was on a Saturday eight days before Easter Sunday, while a few places of extreme westerly longitude would place it one day earlier still, on a Friday. There may even be an eclipse, to prove that the moon is full on Easter Sunday, as in Europe in 1903. These cases can arise when the various sources of variation combine to make the real moon differ sufficiently from the ecclesiastical moon.
 
 
In fact there were few relevant considerations--if any--which escaped the notice of Gregory’s principal adviser, the German Jesuit astronomer Clavius (after whom a crater on the Moon has been named). In his advocacy of proposals made first by his deceased Italian colleague Lilius, he emphasized the administrative advantages of arithmetical tables, and he stated quite explicitly that two distinct moons were in question: one astronomical, or real; and the other ecclesiastical, or notional. He compared ecclesiastical data with astronomical data ostensibly for the meridian of Rome (these are apparently based on astronomical data for the meridian of Venice; but this is more a matter of illustrative convenience than religious principle, in view of the day or so of lunar displacement which was inserted to avoid coincidences with the Passover): and he directed that those in the newly discovered Southern hemisphere should base their Easter on spring as in the Northern hemisphere--where Christ had lived--rather than on their own spring, with a six-month difference.
 
 
The equinox was to be assumed to occur on 21 March in all cases, and care was taken to ensure that only a 29-day gap and not a 30-day gap could take place between a full moon of 20 March and the next full moon. Easter then was confined to the same limits as in the Julian calendar: and the full moons were also arranged to have different dates in each year of any 19-year cycle. Well-known properties of the old calendar were thus maintained; and the good repute of earlier tables and authors was not irresponsibly disturbed.
 
 
This involved suitably placing some otherwise necessary exceptions at various points in a regular interweaving of 30-day and 29-day lunar months; and Clavius is himself responsible for a further improvement--a method which secures all these results with the minimum necessity of making exceptions to simple rules for Easter.
 
 
In the published explanation of the changes, it was explicitly stated that the introduction of a fixed date for Easter would not have been a contradiction of any essential consideration of fundamental doctrine; but that it had been held preferable to secure continuity with the past--so far as possible--while arranging to avoid uncertainty and lack of unanimity for the future. The need for an occasional contradiction of astronomical truth was recognized and accepted as the price to be paid for simplicity and uniformity: and--subject to these considerations--the need to minimize the discrepancies received careful attention.
 
 
The Catholic states of Europe all adopted the new calendar within a few years; but renewed confusion arose from conflict with another Reformation. Some Protestant authorities disagreed with the change, to be ridiculed for preferring to be ‘wrong with the sun rather than right with the Pope‘--by opponents who had chosen to be wrong with the moon rather than be right with the Jews: and Protestant preferences for astronomical observations drew from Kepler the comment that ‘Easter is a feast, and not a star’. In Western Europe, the advantages of uniformity finally prevailed; and most of the Protestant states adopted the new calendar in 1700 or 1701, leaving Britain and Sweden to do the like about fifty years later.
 
 
Britain changed over in 1752 between Wednesday, 2 September, and Thursday, 14 September, amid some clamour of 'Give us back our eleven days!’ (omission of an extra day was required because the year 1700--but not 1600--had been of different length in the two calendars). Provision was made for determining Easter by methods which differed in form from those of the Roman church, but which nevertheless produced an identical result. By the same Act of 1751, the calendar year was made to begin in England with 1 January, from 1752 onwards: it had done this in Scotland since the beginning of 1600. Before 1752 the English year began on 25 March: the Act provided that financial and legal matters were not to be anticipated or postponed, but were to continue for their due terms as if there had been no change; and as a result there is now a British financial year whose first day is 5 April.
 
 
By the early twentieth century the spread of communications had brought ever-increasing adoption of the Gregorian calendar for civil purposes, even in Oriental countries. World-wide agreement on civil-calendar dates was virtually achieved by the accession of Greece in 1924 and Turkey in 1927: except that the Greek proposals were in fact for an ‘improved Julian’ calendar which would gain slightly in accuracy by having two century leap-years out of nine, rather than the Gregorian choice of one out of four; and failing concessions by one side or other (or more drastic reform) this would produce rival civil calendars in Europe--once again--from the year 2800 onwards. Some other Greek proposals envisaged the Orthodox churches determining Easter in terms of astronomical data for the meridian of Jerusalem: these--if adhered to--would give opportunities for religious disagreements, occasionally, even in the present century.
 
 
The tables of the law
 
Før Gauss
The churches have usually employed tabular methods for determining Easter: these mostly make explicit or implicit use of a calendar in which successive days from 1 January onwards are labelled with the letters A, B, C, D, E, F, G, continuing in cyclic succession; and the cycle pays no regard to a date of 29 February, which should be considered to have the same letter as the date 1 March. (For many centuries, use continued to be made of the Roman calendar, in which the days of the months were numbered backwards from the next Kalends, Nones, or Ides. The extra day of February was then arranged differently, as the term ‘bissextile’ commemorates: here there was a repeat of the twenty-fourth day of February which (by inclusive reckoning) was referred to as the sixth day before the Kalends--or first day--of March.)
 
 
Relatively to any common year, one of the seven letters can then be specified as the Dominical Letter of the year, to identify which dates are the Sundays. In a leap year, two letters have to be provided: adjacent pairs of letters may occur in inverted alphabetical order, or we may have AG; the first-written of the two letters applies to January and February only, and the other letter is used after the extra day, in leap years. Since 365 days include one day more than an exact number of weeks, the elapse of a common year serves to displace the Dominical Letter one step backwards in the alphabet (or from A to G): so a leap year will result in a two-step displacement, overall--and in fact the original reason for the name is this leap of the Dominical Letter.
 
 
In the Julian calendar it then was necessary only to insert series of Golden Numbers in various places, to show which dates were new-moon dates in the corresponding years of every 19-year cycle; alternatively, these Golden Numbers could all be placed in positions thirteen days later--moving them to the fourteenth day of the month, instead of the first--to indicate the corresponding full-moon dates: in either event some days would thus receive no Golden Number. The Golden Number for any year could then be used to find the full moon which was on, or next after, the date of 21 March: the first subsequent appearance of the Dominical Letter of the year then identified the date of Easter Sunday for that year.
 
 
In the Julian calendar, the Dominical Letters required a cycle of twenty-eight years for their exact recurrence, since four years which include a leap year do not contain an exact number of weeks: but the new leap-year cycle of four Gregorian centuries does in fact contain a number of days which is an exact multiple of 7; and so a four-century period here serves as a cycle for the Dominical Letter as well. (One consequence is that week-days no longer are distributed to dates with absolute impartiality--in the long run--as they were before: and the thirteenth of a month is now a little more likely to be a Friday than anything else!)
 
 
The Gregorian calendar uses epacts in place of Golden Numbers, to determine the lunar months: these follow the Julian practice, exceptwhen the special corrections are made in century years. Apart from these special corrections, the epact is made to increase by 11 from one year to the next--with multiples of 30 neglected--unless the Golden Number changes from 19 to 1, when the epact is made to increase by 12: the increases represent the number of days by which a year exceeds twelve lunar months; and they secure a return to the same full-moon dates after nineteen years.
 
 
In the Julian calendar, only nineteen values of the epact were possible: in the Gregorian calendar, the century-year corrections can operate to make unit changes--in the same direction--in all the nineteen epacts of a previous cycle. This causes thirty different sets of nineteen epacts to be required as time progresses; and every number from 1 to 30 can then be an epact at some time or other. Whereas before there were some dates which received no Golden Number, there here have to be some dates which receive two epacts: because thirty possibilities have to be distributed over a range of thirty days, in some cases, and over a range of twenty-nine days, in others, for the following reasons.
 
 
The epact is the assumed age of the moon at the start of the year, and so an epact of 30 is the indication of a new moon on 1 January: epacts which diminish by units can then be associated with later dates, in similar fashion, because each unit decrease in the age of the moon at the start of the year will cause a day’s postponement to the date of the first new moon; but this system cannot continue throughout the year, or all months would have thirty days.
 
Nu bliver det spændende !
So a particular set of three consecutive epacts is chosen, and on alternate occasions--at later sets of dates in the year--these are made to correspond sometimes to three different days, and sometimes only to two. The epacts so selected are taken to be 26, 25, 24, for a reason we shall later discuss: and 25 is used in two forms, sometimes written and 25 and 25'. When we attach epacts to dates which are to be their corresponding new-moon dates, we then arrange that alternately at later dates in the year we have 25 and 25' coinciding on a day uniquely of their own, or else being split--with 25' coinciding with 26 for one day, and 25 coinciding with 24 for the following day: in all other cases, the epacts decrease by units, or change from 1 to 30, from one day to the next; and no epact is given to the extra day of February in leap years. To preserve the feature that no two years of any 19-year cycle can have identical new-moon dates, it is then necessary to arrange that epacts 24 and 25 cannot be in use together in any such cycle--and similarly for epacts 25' and 26.
 
 
This is secured when a rule is made that the epact 25 is to be used only when years which have Golden Numbers from 1 to 11 are in question, whereas 25' is to be used as the epact for years with Golden Numbers from 12 to 19. No three consecutive epacts can be in simultaneous use, in any 19-year cycle; and two consecutive epacts can occur only eleven years apart--with the smaller occurring in one of the first eight years of the cycle, and the larger occurring in one of the last eight years--when both are present in the same cycle: so 25 is always available for use with 26, if both are needed, and similarly for 25' with 24.
 
 
No pairs of epacts which can be of equivalent effect are then required in any of the thirty sets of nineteen epacts which can become current epacts together (for single centuries, or for up to three centuries at a time): assigning the three middle years of the cycle to the earlier sub- division then ensures that every cycle includes either an epact 24--with or without 25’--or alternatively an epact 25; and 25' is required only when 24 is present in the same cycle, which happens only in eight of the thirty possible sets.
 
 
This system will be found to place an epact of 23 to indicate a new moon on 8 March, and therefore a full moon on 21 March; and a full cycle of epacts continues thereafter, until the epacts 26 and 25' indicate new moons for 4 April and full moons for 17 April, and finally the epacts 25 and 24 indicate new moons for 5 April and full moons for 18 April.
 
 
Notwithstanding the thirty possibilities which replace the previous nineteen, the Easter full moon is then restricted to a range of only twenty-nine possible dates, within the same limits as before; and no need arises to allow Easter to occur on any date which was known to be impossible--and authoritatively stated to be so--in relation to the earlier calendar. (Similar consideration for earlier authorities caused the spring equinox to be restored to a date of 21 March as at the Council of Nicaea--rather than to the later date which it had at the time of the Crucifixion.) The greatest irregularity in the months occurs when--occasionally--four consecutive months include a total of 120 days, or when three months each of twenty-nine days occur in succession, at a change from one year to the next.
 
 
Continuing relevance of earlier pronouncements could equally well be provided by treating any other epact similarly to 25; and the original proposal (by Lilius) was to do this for the epact 30. By choosing 25 for the ambiguous epact, Clavius secured that strict numerical sequence continues for as long as possible through the range of dates for the Easter full moon: for the break in serial order then applies to 25' but not to 25--and otherwise only to 24. Simpler rules (analogous to familiar rules for the Julian calendar) are then able to cover nearly all cases; and there is a minimum need to define and make use of exceptional procedures--the replacement of one crudely calculated epact by another, or a shift of a week in the derived date of Easter--to give them universal validity.
 
 
Gaussian error and its correction
3.
År 1800 - Gauss
justeringer år 1807
 
Gauss algotitme - også kaldet påske-formlen - blev lavet af den tyske matematiker Gauss (1777-1855)
'Published by von Zach in the Monatliche Correspondenz, II (August, 1800), 121'.
Forenklet var formlen:
Påskesøndag er x-1 dage efter 1.marts i år y
hvor x = (22+ (19(y mod 19)+m)mod 30) + ((2(y mod 4)+4(y mod 7)+6 (19(y mod 19)+m)mod 30) +n)mod 7) ).
mod betyder modulus - altså den positive rest efter en division.
m og n afhænger om det er en Julianske eller Gregorianske kalender (før 1583 eller efter).
Men der var mange undtagelser og Gauss kom selv med korrektionerne.
'Published in 1807 (September 12) in the Braunschweigisches Magazin'.
 
Gauss lavede algoritmen, fordi hans mors ikke kendte hans fødselsdag. Hun kunne kun dagen i forhold til påske.
 
Kilde: Carl Friedrich Gauss: Titan of science: A study of his life and work - Exposition Press, New York, 1955
 
 
Rules for Easter have been considered by various mathe­maticians, including Gauss--perhaps the greatest of all--who published a rule that ignores the change from the 300-year interval to the 400-year interval, in the lunar correction: this will give 13 April, instead of the correct 20 April, for Easter in A.D. 4200; similar errors occur inter­mittently, with slowly increasing frequency, in later years.
4.
År 1876
 
Anonym N.Y. korrespondent udtænkte den rigtige formel uden undtagelser
A wholly correct procedure of a purely arithmetical type was first given in Nature in 1876 by an anonymous New York correspondent: this includes an arithmetical device which takes care of cases which more usually had caused numerical rules to be complicated by special exceptions. The rule uses all the letter-symbols from a to o inclusive, apart from j: it becomes tempting to speculate whether its author was a printer, or was more familiar with an Italian (or Latin) alphabet. Based on this procedure, we have developed similar methods of our own, which are now published for the first time.
 
 
There are three points of difference between our present procedures and the one given in Nature. We have replaced one step of the other rule by two, so that we can calculate the actual date in the month, instead of a date which has to be increased by a unit (because zero is a possible result); but we have been able to replace two steps of the other rule by one, by taking account of a manuscript note which Gauss made to correct his error, in his own copy of his published paper. Apart from this, our procedures now give a closer parallel to the use of tables, and allow the easy determination of the correct epact and correct date of the Easter full moon, in all cases without exception.
 
 
We give one set of calculation rules in Table 10a: the only initial information needed is the year number for a Gregorian year; and all else depends only on the quotients and remainders which are obtained by ten successive division operations, in which there are specified divisors and dividends which are formed--with no exceptional provisions--from previous results. Our earlier discussion will allow us to explain what is concerned in the different stages which are involved.
 
 
Commentary on a decalogue
5.
År 1965 - T.H. O’Beirne algoritme.
 
Skotten T.H. O’Beirne og hans team omsatte formlen fra 1876 til god logik jvf tabel 10b.
 

Algoritmen har jeg omsat til C# programmering, og ser ud som:
 
int Y = valgte årstal
int G = (Y % 19) + 1;
int C = (Y / 100) + 1;
int X = (3 * C / 4) - 12;
int Z = ((8 * C + 5) / 25) - 5;
int D = (5 * Y / 4) - X - 10;
int E = (11 * G + 20 + Z - X) % 30;
if ((E == 25 && G > 11) || E == 24) { E++; }
int N = 44 - E;
if (N < 21) { N = N + 30; }
N = N + 7 - ((D + N) % 7);
int maned = 3; // marts default
if (N > 31) { maned++; N = (N - 31); }
Påskedag = new DateTime(Y, maned, N)
 
PS: T.H. O’Beirne algoritme er beskrevet af Donald E. Knuth.
Kilde: Knuth, Donald E. The art of computer programming, Fundamental algorithms - Addison-Wesley, 1968
(Flere angiver fejlagtigt, at det er  Donald E. Knuth algoritme - men det er det ikke. Han har taget den fra skotten T.H. O’Beirne og hans team !)
Desværre er der en fejl i formlen. År deleligt med 3600 er IKKE skudår (selvom det er deleligt med 400). Det skyldes, at det jordiske kalenderår afviger ca. 1 døgn fra det astronomiske solår i løbet af 3571 år.
Den tid den sorg.

 
 
In step (1) of the procedure of Table 10a we identify the position of the year in a 19-year cycle, to use this later in (5) to introduce the principal constituent of the change of full-moon dates from year to year: the value of a is one unit less than the Golden Number; for Golden Numbers run from 1 to 19 in cases where our remainders will run from 0 to 18. In (2) we prepare to take note of the special corrections which the Gregorian calendar introduces in century years: the value of b increases by one unit at century years only, and the value of c is used later in (7), to take note of the effect which the ordinary cycle of leap years has on the succession of week-days for the same date in successive years.
 
 
From (3) we derive d, which increases only in century leap-years, and e, which gives the number of century years which have not been leap years, subsequently to the immediately previous century leap-year. From (4) we obtain g, which increases only when there is an increase of the epact because of the correction of the month; and b--d has a value which increases only when a reduction of the epact is required because of the omissions which gave the Gregorian correction of the year: at (5), we can then compute a number h which is an equivalent of the epact.
 
 
The epact itself is in fact either 53--h or 23--h, whichever lies between 1 and 30 inclusive; and we thus ensure that the critical epacts 24 and 25 are represented by the largest values of h: in (6) we can then arrange to have my= 1 only when a critical case has arisen.
Step
Dividend
Di­vi­sor
Quo­ti­ent
Re­main­der
(1)
x
19
-
a
(2)
x
100
b
c
(3)
b
4
d
e
(4)
8b+13
25
g
-
(5)
19a+b-d-g+15
30
-
h
(6)
a+11h
319
μ
-
(7)
c
4
i
k
(8)
2e+2i-k-h+μ+32
7
-
λ
(9)
h-μ+λ+90
25
n
-
(10)
h-μ+λ+n+19
32
-
p
Table 10 a
 
 
 
 
 
Step
Dividend
Di­vi­sor
Quo­ti­ent
Re­main­der
(1)
x
100
b
c
(2)
5b+c
19
-
a
(3)
3(b+25)
4
σ
ε
(4)
8(b+11)
25
λ
-
(5)
19a+σ-λ
30
-
h
(6)
a+11h
319
μ
-
(7)
60(5-ε)+c
4
j
k
(8)
2j-k-h+μ
7
-
λ
(9)
h-μ+λ+110
30
n
q
(10)
q+5-n
(32)
(o)
p
Table 10 b

 
 
 
Explanations are easier with the upper rules, but computation may be simpler with the lower rules. In the xth year A.D. of the Gregorian calendar, Easter Sunday is the pth day of the nth month, with both rules. The date of the Easter full moon is obtained if--1 is substituted for λ, in steps (9) and (10). The Golden Number is a+1, and the epact is either 23--h or 53--h, whichever lies between 1 and 30 inclusive. To find the Dominical Letter, divide 2e+2i-k or 2j-k+4 by 7 --with multiples of 7 added, if need be, since we wish to have the smallest non-negative remainder. The letter is A if this remainder is zero, and it moves on in the alphabet as the remainder increases: for leap years, prefix this by the next following letter (or G by A).
 
 
We always have μ = 0 except in two circumstances: when h = 29, we have μ = 1 regardless of the value of a, and the epact then is 24; we again have μ = 1 when we have h = 28 and a > 10 simultaneously, and then the epact is 25'. By suitable use of the value of μ, we then can always pass from the epact to the correct date of the ecclesiastical Easter full moon (which we can in fact obtain if we replace + λ by -1, in the expressions which give the dividends for the steps (9) and (10) of our procedure).
 
 
We next have to find the day of the week for the Easter full-moon date. With (7) we arrange to take note of ordinary leap-years; and we can then derive an equivalent of the Dominical Letter: for the Dominical Letter is A when 2e+2i-k leaves a zero remainder when divided by 7, and advances by one step in the alphabet for each unit increase in the remainder, in other cases. (This applies to the letter which we use in March and thereafter, in the case of leap-years: this is the one with which we are concerned in the determination of Easter.)
 
 
Ordinarily, e will not change, and there will be a single backward step in a common year, due to a unit advance in k; or a fivefold step forward--equivalent to a double backward step--because i increases by 1 and k reduces by 3, for a leap year. At century years, i will reduce by 24, and k by 3; and e will increase by 1 for a common year, but reduce by 3 for a leap year: the equivalent of a single or a double backward step is then correctly provided, as required, in the respective cases.
 
 
In (8), a full-moon date correctly derived from the epact--in terms of h and μ--is implicitly compared with a Dominical Letter derived from 2e+2i-k: we thus obtain λ, with λ = 0 if the full-moon date is Saturday, but λ = 6 if it is a Sunday; in all cases, λ is one unit less than the number of days which must elapse before there is a Sunday which is strictly subsequent to the date of the ecclesiastical full moon. We then have everything necessary to enable steps (9) and (10) to produce n = 3 for March or n = 4 for April, with p giving the actual date of Easter Sunday in the month concerned.
 
 
In our explanation, we have concentrated on the cyclic features which are required if accuracy, once introduced, is to be subsequently maintained. The initial accuracy is introduced by the constants 15 and 32 in steps (5) and (8); these in effect introduce the age of the moon and the day of the week for some particular date: the constant 13 in (4) takes note that a lunar correction was made at the start of a 2,500-year lunar-correction cycle, in the year 1800. Constants have been chosen in such a way that the rules will not require division of negative numbers.
 
 
In Table 10b we give an alternative procedure, in which various sophistications have been introduced to simplify the calculation of various dividends. Some readers may care to check that the two sets of rules are arithmetically equivalent.
 
 
Hard cases and bad laws
 
 
The rules we have given here are based on slight modifications of a set of equally correct rules which we have published elsewhere: in the earlier form they gave the correct date of Easter; but they did not then allow a direct calculation of the epact, or of the date of the Easter full moon--for all cases without exception--as they now do.
 
 
We shall find that we have μ = 1 with the new rules more often than we have the corresponding case of m = 1 in the Nature rule (and in our earlier rule); the reason is that we now find every Easter full moon correctly. In the other rules we implicitly use an incorrect epact to find this date with an error of one day, in the cases when we would have μ = 1 here: the other rules pay no regard to this except when the date of Easter would be affected because a Sunday was taken in place of a Saturday; and corrections dependent on a term 7m then become effective. (We have chosen the symbols in such a fashion that identical symbols are used in identical senses, in our procedures and in the earlier procedure as given in Nature.)
 
 
Since only minor alterations are involved--with no increase in the computational effort--we have thought it desirable to make the changes. They provide a better correspondence between the arithmetical data and equivalent tabular data: and they will make clear that the validity of the earlier limiting dates was preserved by a means somewhat more subtle--and rational--than simply having the date of Easter earlier by a week when the limit would otherwise have been exceeded.
 
 
If Easter is to have its earliest possible date--22 March--this requires the year to have an epact of 23, to give a full moon on 21 March, and a Dominical Letter D (or letters ED, if a leap-year), to make the full- moon date a Saturday: this occurs for 1818 and 2285, but not for any intervening year. If Easter is to have its latest possible date--25 April--the epact must be either 24, or 25 which is not treated as 25', either of which will give a full moon on 18 April (and in the second event the Golden Number must be between 1 and 11 inclusive): in both cases, there must also be a Dominical Letter C (or letters DC), to make the full-moon date a Sunday. This happens for 1886 (with epact 25) and for 1943 (with epact 24), but not again until 2038.
 
 
The latest date comes up about once per century. It may occur twice, as in the thirty-first century, with 3002 and 3097; and it may not occur at all, as in the forty-fifth century. The earliest date occurs more rarely: during several consecutive centuries it may occur once per century (rarely twice, as in 3401 and 3496), and then for several consecutive centuries it may not occur at all; there are no examples between 3716 and 4308. The latest date is more frequent than the earliest, because--unlike the earliest--it can arise with either of two epacts.
 
 
The cases which require a value m = 1 in the Nature rule are those where the epact is either 24, or a 25 which is treated as 25' (and in the second event the Golden Number must then lie between 12 and 19 inclusive), provided that the Dominical Letter is given by D (or ED) in the first case, and by C (or DC) in the second. The first alternative occurred for 1609 and will occur for 1981, but not again until 2076, to give an Easter date of 19 April: the second alternative occurred 1954 and will not occur again until 2049, giving an Easter date of 18 April.
 
 
If we have an epact of 25 and a Dominical Letter of C (or DC), the vital question therefore is whether the epact 25 is to be treated as 25', or not: for if not, we will have an Easter date of 25 April, as in 1886, and if so, we will have an Easter date of 18 April, as in 1954. (The Golden Number is 6 for 1886, but 17 for 1954: and the full-moon dates are 18 April 1886, but 17 April 1954, because the full-moon date is obtained differently from the same numerical epact, in the two cases.)
 
 
These critical cases--and the case of the year 4200--can form useful test examples for various rules for Easter. (They will show that numerical rules in earlier editions of the Encyclopaedia Britannica are in error for 1886 and critical cases of similar type.)
 
 
Future prospects for Easter
 
 
With the Gregorian rules, the completion of a full cycle of Easter dates would require 5,700,000 years; these would include 70,499,183 months and 2,081,822,250 days. The cycle has this duration because a period of 10,000 years is required before there is an exact repeat of the century-year epact shifts; and in any such period there are seventy-five one-day shifts in one direction, and thirty-two one-day shifts in another--giving an overall lunar displacement of forty-three days. This has to be repeated 30 x 19 times before the cycle closes with a repeat of the original epact and original Golden Number, when 43 x 19 months will have been omitted, as compared with an exact allowance of 235 months for every nineteen years.
 
 
Any astronomical relevance would, however, disappear long before the completion of the cycle: and the new rules will themselves be in error by several days, after the passage of a few tens of thousands of years. This arises from two causes. Residual imperfections of the revised rules would still accumulate even if the day, month, and year continued to have unaltered duration. Apart from this, the periods concerned, and their ratios, are not strictly constant: tidal friction makes each successive day longer, in the average, by a few millimicroseconds--equivalent to the time required for light to travel several feet; and various gravitational perturbations combine to affect the other periods also, over very long periods of time.
 
 
At the limit of present-day accurate time-keeping, there is already a need to have two quite distinct measures of time: one--essentially for civil purposes and other matters conditioned to the Earth’s rotation--has to remain in mean agreement with the progressive increase of the day; but another--more suited to measurement of unvarying repetitive physical processes such as atomic vibrations, and to consideration of the dynamics of planetary motions--is now based on an appropriate subdivision of a specially selected year. Failing this, we should have to consider that precise clocks have slightly increasing rates--and planets an unaccountable acceleration--because we chose to ignore the simpler fact that the rate of the Earth’s rotation is gradually diminishing.
 
 
In concluding, we should remark that the Easter Act, 1928, makes provision that an Order in Council can fix Easter to be the first Sunday after the second Saturday in April, after regard duly paid to opinions of churches or other Christian bodies--most of whom have already pronounced that no point of religious principle need make them oppose a uniformly agreed change.
 
 
If uniformity both of date and of practice could come soon, there would be attendant advantages to holiday-makers and to statisticians, in particular: and for our part we would prefer to see our rules become mathematical or historical curiosities, rather than to have them continue as something applicable to contemporary use.
 
 
 
 
Kilde: Puzzles and paradoxes by T. H. O'Beirne
London: Oxford University Press, 1965
 
 


 
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