Talk:Epact

Age of the Moon
User:Karl Palmen writes:

The section on the age of the moon is not aparently NPOV. So I have moved it here.

It asserts that the age of the moon equals the day of the lunation. Also the concluding paragraph contradicts the Catholic encyclopeadia article referenced and the end of the article.


 * I observe that an epact of "*" (0 or 30 or nulla) for a year, implies a 1st day of the lunar month on the 1st day of the year. If the epact is an age, this would mean that the age is 0 on day 1.  Hardly consistent.  All this involves the usual confusion between ordinal counting and cardinal (or real) numbers - people used to be "in their first year", nowadays your age is 1 year *after completing* your 1st year.  So what is the age of the Moon on the 1st day of the month, traditionally?  For that matter, what is the age of J.C. today?  This is related to the milennium controversy (was 1 Jan 2000 or 1 Jan 2001 the start of "the" new milennium?).
 * And incidentally, the Catholic Encyclopedia can have it wrong too... -- Tom Peters 13:07, 15 Aug 2003 (UTC)

age of the Moon
The epact is often defined as the age of the Moon at the begin of the (solar) year. Now when a new lunar month begins on 1 January, the epact is 0. However we count the days of the lunar calendar as ordinal numbers, i.e. in this case the 1st day of January is also the 1st day of the lunar year and the 1st day of the lunar month. We can accordingly assign an age of 1 day to the Moon, if we follow the ancient convention that the lunar month starts when the New Moon is first visible. This is actually some time after the astronomical conjunction of Sun and Moon (also called Dark Moon), when the "astronomical age of the Moon" is 0. Indeed the lunar calendar used for Calculating the date of Easter in the Gregorian calendar has its months starting systematically a day after the astronomical New Moon. In this reckoning the astronomical opposition of Sun and Moon indeed usually falls on the 14th day of the lunar month, which is consistent with the tradition of having the Paschal Full moon on the 14th day of the spring month.

The epact can then be defined as the age of the Moon on the day before the begin of the (solar) year. This is not the same as the age of the Moon on the last day of the previous year, because the epact may be corrected by 1 at the start of the new year.

Karl August 15 10h UT

Oh well, how about this text then:

"The epact is often defined as the age of the Moon at the begin of the (solar) year. Now when the epact is 0, 1 January is also the 1st day (i.e. the day of first visibility of the lunar crescent) of the 1st lunar month.  However age used to be expressed as an ordinal number, starting with 1 (a newborn baby was said to be "in its 1st year").  A supposed age of 0 of the Moon is translated to the 1st ordinal number, which is an apparent discrepancy.

Also there has been a historical change in the definition of "age". A baby that was said to be "in its first year", is now called "0 years old". Events used to be timed with a hierarchy of periods labeled with ordinals numbers (e.g starting point is 1st day of 1st month of 1st year). We now interpret time as a continuous variable starting from 0 at some specific moment, and count fully completed days and years for age. But calendar dates are labels that are still expressed the old way, as ordinal numbers. This is the underlying reason for the milennium controversy (start at 1 January 2000 or 2001?).

Also it has become custom to define New Moon at the moment of astronomical conjunction, which is at the same moment everywhere on Earth, and take that as the zero point for counting the (continuous) age of the Moon. In contrast the crescent Moon becomes first visible one or two days later, and the moment of first visibility depends on the place on Earth.

The epacts of the ecclesiastic calendar are usually still consistent with the age of the Moon in the modern sense, if interpreted in the following way: "
 * 1) define New Moon in the modern astronomical sense;
 * 2) count age continuous from that moment;
 * 3) define the epact as the age in the modern sense (i.e. count completed days) at midnight of 1 January.

-- 194.109.250.130 21:00, 15 Aug 2003 (UTC)

Tom has given here two alternative interpretations of the concept of age of moon, which are both different from the one normally used in the description of the Calculation of Easter.
 * 1) Use Ordinal Numbers for Age. This is never done as far as I know in the West, but is done in the East. For example in China a new born baby is age 1 and becomes age 2 at the nexct new year.
 * 2) Define the age of the moon as the number of days AFTER the day as the (reckoned) astronomical new moon.

Karl 18 Aug 2003 09h UT


 * Karl misunderstands. Age has always been counted in ordinal numbers of some unit of time in the West too (until the 19th cy. I think).  Painted portraits for instance typically have a note "in aetas sui xxxxi", i.e. "in his 41st year"; nowadays we call that person 40 years old.  Pertinent to this discussion: Dionysius always refers to "the 14th moon".  Presumably the lunar month then started with the "1st moon", which was on the 1st of January when the epact is 0 ("nulla" with D.E.).  There is however never an "0th moon".  That is the inconsistency I point out when interpreting "epact = age of the Moon on 1 Jan." using that old (ordinal) definition of age.


 * But nowadays we do count age from 0, and as a continuous variable rather than an ordinal number (even when counting only completed days or years). This causes 3 other problems:


 * you have to define where this 0-point lies;
 * you have to define when you evaluate the age, in case you wish to assign a single age to a whole day;
 * in any case the traditional tables of epacts may match age starting from 0 at the time of first observation of the lunar crescent, but then the "1st moon" has age 0.x, and the "14th moon" has age 13.x, which is as confusing as the original inconsistency as I noted above.


 * So the statement "epact = age at 1 Jan." is valid only if we interpret the concepts in the way I specified in the proposed paragraph.


 * -- 167.202.196.71 12:50, 18 Aug 2003 (UTC)

So it seems that the concept of age of moon is far from clear-cut. Therefore, in discussions about such things as epacts or Easter Calculations, it is better to use something else, such as day of moon (defined to begin with new moon) or number of days after new moon. -- Karl 19 Aug, 13h UT.

Epacts in Book of Common Prayer
In the Preface in the Book of Common Prayer of the Church of England, in a section decreed by the Calendar Act, there is a page listing the Moveable Feasts for about 40-50 years. Its third column contains Epacts. The Act and Book do not seem to have any definition of the Epact. Just what formal definition does the Church use?

Note that such a table is in Statutes at Large 1765, but not in the database which represents the Act as amended to date.

I suggest that the Article should contain a short section explicitly answering this point.

82.163.24.100 (talk) 10:01, 3 May 2008 (UTC)

Lilian (Gregorian) epacts
The current article text under "Lilian (Gregorian) epacts" I suggest that the relevant sources here are those that refer to the actual work that culminated in the Gregorian reform of 1582, especially the papers given in the 1983 commemorative conference. These sources (primarily the papers by O Pedersen, J D North, G Moyer and A Ziggelaar) appear to show that the lunar equation in the Lilian system of epacts was a systematic empirical adjustment, intended to reflect the fact that the Moon's mean rate of motion had turned out to be just a little faster than the expectation embodied in the traditional form of 19-year cycle. In the second half of the sixteenth century, full moons were occurring much sooner than the dates indicated on the traditional tabular basis (which by then had been in use for over a thousand years). According to the text of "Inter gravissimas" the paschal full moons needed to be "put back in place" from a deviation of something more than 4 days ("et XIV paschalem suo in loco, a quo quatuor et eo amplius dies hoc tempore distat, reponendam".) Lilius' periodical adjustment was intended as a correction to bring the tabular epact value closer to the real age of the Moon, and prevent any repeat of the long-term divergence which had been seen with the traditional tables.
 * (a) seems to get it just right where it comes to the actual rules of the solar equation and lunar equation; but
 * (b) it also contains a quantity of material that seems completely unsupported, and at best dubious, where it comes to the 'Metonic relation', 'pure' or otherwise, and to the unit called the 'tithi'.

A temporary 'patch' had been applied in the 1568 edition of the Breviary, namely, to move the Golden Numbers up four places and to warn that they would need to be reviewed again and moved every 300 years or so. But the text of "Inter gravissimas" complains that previously-proposed solutions were not long-term ("neque perennes erant"), and it praises Lilio's effort for providing in principle a perpetual arrangement to restore "all the things in the calendar that had got into disarray" so that there would be no future "mutation" ("omnia quæ in calendario collapsa sunt, constanti ratione et sæculis omnibus duratura, sic restitui posse ostendit ut calendarium ipsum nulli umquam mutationi in posterum expositum esse videatur.").

In all of this there seems to be no reference to Metonic relations or tithis. It seems appropriate to take out the material about these factors, unless good sources are given for their relevance to the 16th-century work. Terry0051 (talk) 01:01, 4 December 2009 (UTC)


 * Terry, I notice your edit a bit late. I think you misunderstand.  Your text does not explain why the reformers bothered to have both a solar equation and a lunar equation, instead of just 43 epact adjustments evenly spread over 10000 years.  The only explanation I can think of is that they tried to undo the effect of the skipped Gregorian leap days, and then adjust the ratio of the synodic month to the (Julian) calendar year.  But as explained, if this was their intention, then it is mistaken.  It is a mathematical fact, as explained in my text in 2 different ways, that Lilian epact corrections are 1/30th of a synodic month, and not full days on average.  Unfortunately, both in the Canons and in his explicatio, Clavius explains the use of the epacts but is vague on the reasons and background of the construction.  Likewise the commemorative conferences spent much attention to the tropical year and the solar calendar and the history of the computus, but lacks a thorough analysis of the lunar calendar.

Tom Peters (talk) 10:22, 20 February 2010 (UTC)

definition of epacts
I see the quality of the article has eroded since 2007. First, someone "cleaned" the prose of my translation of the original verbose Latin quote for style, and then removed the original text altogether. Then it was claimed that this text from the Gregorian calendar Canon was the first to define epacts as the excess of the solar over the lunar year: but it was only an explanation, not a formal original definition, of a concept that had been widely known among computists for over a millennium before the reform. Then Terry picked up the definition of "age at 1 January" from the paper by Perdersen in the commemorative conference: he defines it as age at 1 January, and shows that epacts were introduced in the computus with the latercula of 354: Terry here incorrectly assumes that the "1 January" is then the older, original definition. However from the very name it is obvious that the "excess" definition is the original one of the concept of epacts, and Pedersen only uses the equivalent "1 January" definition for computist convenience. Moreover, as you could have seen on this discussion page in the thread from 2003, the "1 January" definition is problematic, and therefore is best avoided. Also see the first sentence in the lemma on epacts in the Catholic encyclopedia, which succinctly supports my view: http://www.newadvent.org/cathen/05480b.htm. So I am inclined to revert this article back to its 2007 version! Tom Peters (talk) 19:28, 20 February 2010 (UTC)


 * Sorry, I only just noticed this. This looks like a matter on which not only sources are important but also careful reporting of their content and area of application.  The 'added days' definition does not look at all compatible with the usages of 'epact' that have been current for the last few hundred years e.g. in the Prayer Book and other easter-calendar uses; these current usages appear clearly to refer to 'age of the moon on day x', generally January 1.  Maybe, in the current lead para text, 'original' could be disputed, but the cited source shows that the 'age on day x' usage is clearly older by a few centuries than the Dionysian easter calculus. It clearly goes back to a time when the records can scarcely be expected to be more than fragmentary, so it seems unlikely that an earlier specimen of the etymological usage could be found.


 * The older state of the article (I don't know if this includes 2007) contained material about time units in relation to epact (especially Gregorian epact) that seemed to be unsourced, unreliable, and anachronistic if not also irrelevant. A proposal just to wind the clock back on the article text to 2007 therefore looks to be highly undesirable from the point of view of article reliability and quality.  But I'd agree that the 'added days' etymology seems to have some claim to early mention in the lead paragraph as long as it is differentiated from the recent and current 'age on day x' usage which also needs to be explained there.  A considered amendment along those lines might strengthen the quality of the article usefully. Terry0051 (talk) 00:29, 28 March 2010 (UTC)


 * Late response (it's getting tedious to repair mistakes again and again in Wikipedia).
 * 1) The Catholic encyclopedia says: "The surplus days of the solar over the lunar year; hence, more freely, the number of days in the age of the moon on 1 January of any given year." So The "1 January" definition is a supposedly equivalent definition for computistic convenience.  Mind that in the Middle Ages many "styles" for the start of the year were used, 1 January has become a standard recently.  Epacts were used before the Dyonisian computus of 525 (which made them unnecessary for the computus): Pedersen as quoted states that they were introduced by some Augustalis (possibly) in the early 3rd century, and he mentions 1 January, but since the table itself apparently has not been preserved, it is not completely clear if Pedersen explains the concept using current practice, or that we actually know that Augustalis used 1 January already.  Anyway the current description does not explain the etymology.  So the initial text of this page should be changed to something more like the original one.
 * I have to add that the"age of the Moon at 1 January" is problematic or simply wrong. "Age of the Moon" is a problematic concept when used with a calendar counting ordinal days, on a round world with a dateline.  Moreover consider the following: in Judeo-Christian computus the 14th day of the lunar month is the day of the Full Moon.  The 1st day of the month has always been that of the first visible crescent, since the Babylonians unto present-day muslims: so not that of the conjunction, which has the astronomical age 0.0, and precedes the Full Moon by almost 15 days.  Now an epact of '*' = day 30 or equivalently day 0.  A year with epact * has the first day of the lunar month, the ecclesiastic New Moon, on 1 January: not the 30th or 0th day.  So the epact is the age of the Moon on 31 December of the preceding year, not the age on 1 January.
 * Like it or not, but the Gregorian reformers did not define the epact in a way that you can conveniently quote in Wikipedia. They presented tables and how to use them, and definitions must be deduced from them.  The "age at 1 January" is bogus, whatever the Catholic Encyclopdia and Pedersen say.  If you use that definition you get the wrong Easter dates.
 * 2) The second statement of the introduction (that the epacts differ by 11 days from year to year BECAUSE a solar year is longer than a lunar year) is a false interpretation of the quote from the canon of the Gregorian calendar. It says that epacts are CALLED this way because the solar year exceeds the lunar year by so many days, which also is why I think it should go into the initial definition.  This is also a matter of perspective: epacts deal with reckoning solar years and lunar YEARS, where the lunar year is central and the solar year is deviant.  The way it is put now, the solar year is central, and there is some funny number mysteriously related to the Moon that changes by 11 days every year for a reason that does not become clearly apparent.
 * 3) The fact remains that Lillian / Gregorian epacts, in contrast to the original "Dyonisian / Julian" expacts" are NOT measured in days, even though the reformers apparently were not aware of that (and apparently many modern calendrists still do not either). This has already been pointed out by Viete around 1600, and why this is was explained and self-evident in the original Wiki text, but the current addition dropped all that and contines the confusion by implying that Lillian epacts are days.  This section must be repaired.
 * Tom Peters (talk) 08:46, 15 August 2010 (UTC)

Reopening old discussion
If I may reopen an old debate, I just came across this passage in Bede, On the Reckoning of Time, cap. 50 (p. 131, Wallis Translation):
 * "For example, if as I write today the Moon is five days old, on this same day one year from now the Moon will be 16 days old; after three years, 8 days; nor does it revert to what it is now until the cycle of 19 years is finished. But the epacts noted in the 19-year cycle specifically stand for the age of the moon on the 11th kalends of April [22 March], the beginning of the Paschal feast.  The [epacts] always observe this rule in connection with [the Moon's] course: whenever they are less than the number 16, they announce the Paschal lunation, but whenever they are more, they direct us to look for Easter in the next lunation."

It seems quite clear that for Bede, and for those who followed him, the epact was related to the age of the Moon. I'll have to do a bit of thinking about Pedersen's referring the epact to 1 January, but I wouldn't dismiss him out of hand. He was a leading student of medieval astronomy. --SteveMcCluskey (talk) 23:00, 23 January 2017 (UTC)
 * As it happens, I don't have to do any calculations, Bede has already worked out the problem: In Chapter 20 where he discusses computing the age of the Moon on any given first day of the month (p. 64 Wallis translation) he writes:
 * "In the first year of the 19-year cycle, in which the epact is zero, the Moon on the kalends of January is 9 days old, on the first of February, 10, on the first of March 9,…"
 * On the 22nd of March (21 days after the first) the age of the moon will be 9 + 21 = 30 (indicating and epact of zero). It's quite clear that the epact cannot refer both to the age of the Moon on 22 March (as Bede specifies) and on 1 January (the kalends of January) as Pedersen claims. Pedersen clearly slipped up here.
 * It seems appropriate to replace the discussion of epact as the age of the Moon on 1 January, citing Pedersen, with a discussion of the epact as the age of the moon on 22 March, citing Bede. --SteveMcCluskey (talk) 00:13, 24 January 2017 (UTC)
 * Further investigation has shown that a number of modern sources agree with Pedersen's definition of epact as the age of the Moon on 1 January. Most authoritative is the Explanatory Supplement to the Astronomical Almanac, which provides full detail of modern calculations as used in the Astronomical Almanacs of the United States and Great Britain.  Unfortunately, they don't state when the modern usage was introduced, but it seems it was part of the Gregorian reform of  the calendar.  Further checking is needed to verify that point, so for the time I have used the phrase "modern usage".  --SteveMcCluskey (talk) 02:58, 24 January 2017 (UTC)
 * I've always taken it that the epacts counting as the age of the moon on 22 March are for the Julian Calendar even today, and the epacts counting as the age of the moon on 1 January are for the Gregorian calendar. These two epact definitions would produce different moon phases, so one could not change from one to another in the same calendar without also changing the epact tables. So I think it is safe to assume "modern usage" applies to the Gregorian Calendar and never the Julian Calendar. Karl (talk) 12:51, 24 January 2017 (UTC)
 * I've looked at the second reference and find it refers to epacts only in the section about the Gregorian calendar and never in the section about the Julian calendar and so one can take it to apply to the Gregorian Calendar. Karl (talk) 13:12, 24 January 2017 (UTC)

'Computistical' - What does this mean?
The word 'computistical is used a number of times in this article. Does anyone out there know what this word means?Duncan.france (talk) 00:58, 19 October 2018 (UTC) > The article already has a cross-reference to 'computus'. The explanation is there. [Terry0051, April 30, 2020]

Original definition of epact, and modern use of daily, monthly and yearly epacts
The original definition of the epact (ἐπακται ἡμεραι, "added days") seems to have been the difference between the lunar date (either observational, or in any pre-calculated lunisolar or purely lunar calendar) and the solar date (in any arithmetical solar calendar; nowadays it would also be possible to implement epacts with any minor observational solar calendar, such as e.g. the Persian calendar). With this definition, it is very understandable why this "quantity" is different from the actual lunar date, and also why the modern Gregorian epact equals the lunar date on the day before the beginning of the Gregorian solar year. Originally in Alexandria, when pope Demetrios I wrote about epacts and used them for paschal calculations, his epacts were most certainly defined as the difference between a "schematic" lunar calendar (very similar or identical to the one used in modern Ethiopia, as described by Neugebauer 1979) and the Alexandrian solar calendar as instituted by Augustus. When these epacts (and their associated pre-calculated lunar calendar) were implemented by Dionysius Exiguus into the roman Julian solar calendar, the Alexandrian "sedes epactarum" (on 1 Thoth = 29/30 August) was not redefined to 1 January (or any other Julian date, as e.g. 1 March), probably because Dionysius Exiguus himself did not understand the basis for the calculations he made, and it was left to later computists, as Beda Venerabilis, to find out empirically that the value of the Alexandrian epact is equal to "the age of the moon on 22 March" in the Julian calendar. But the modern Gregorian epacts have their "sedes epactarum" on 1 January in the Gregorian solar calendar; therefore, they are equal during most years to the "schematic" lunar date of 31 December in the year before. However, these "schematic" lunar dates can be compared to real, observational lunar dates, from e.g. (the true, observational version of) the Islamic calendar (preferably from the Middle East - e.g. Cairo, Medina or Mecca) or the observational Qaraite calendar (from Israel). It can then be observed that, in most cases, there is a slight difference between the observational lunar dates and the Gregorian ones. True values of the original meaning of epact (daily, monthly, or yearly) can be calculated from the observational lunar dates and the Gregorian solar dates, as given by this simple algorithm:

• Calculate the difference between the true, observational lunar date L and the Gregorian solar date S, i.e. E1 = L-S

• If E1 should happen to be a negative number, add the number of days in the current Gregorian month to the difference, i.e. E2 = L-S+28/29/30/31

• If E2 should happen to be equal to 30, subtract 30 from it, giving the epact the value of "0" (or, in traditional computistic style, "*"), i.e. E3 = L-S+28/29/30/31-30

• The last calculated value of "E" (in most cases E1 or E2, seldom E3) is the "true" epact of the date considered, and also of all days in the whole lunar month in which it is situated

Today, the Gregorian solar date is "17" (17 June 2021 C.E.) and the observational lunar date is "6" (in e.g. "Chodesh revii" or "Dhu al-Qadah"). Thus, E1 = 6-17 = -11, which is negative. Therefore we will add the number of days in June, "30", to get E2 = 6-17+30 = -11+30 = 19, which is therefore the true epact for today. It is also the monthly value of the epact for the current lunar month, which started with the observation of the Crescent New Moon on the evening of 11 June 2021 C.E. However, the true yearly epact of 2021 C.E. is equal to the true daily epact of 1 January 2021 C.E. which was E1 = 16-1 = 15, but the "schematic" Gregorian epact of 2021 C.E. is "16", a difference by one day.

In the same way other "true epacts" can be defined e.g. by comparision of (the true observational version of) the Islamic calendar and the Persian official solar calendar, in which all years begin at Nowruz, situated as close as possible to the "true astronomical vernal equinox" as calculated for some meridian in Iran (possibly the capital, Teheran). In the same way, in Egypt, other "true epacts" could be calculated from the difference between the (observational) Islamic lunar dates and the Coptic (Alexandrian) solar dates, but these, if calculated for "1 Tut", would differ much more from the traditional Alexandrian calculated epact values; as is well known, the original Alexandrian lunar calendar, still used by the christian Copts of Egypt, and also used in both Ethiopia and Eritrea, has accumulated very grave errors during the centuries of its use, because of too high values both for the average tropical solar year (365.25 days, compared to the "true" average value at about 365.2422 days), and the average synodic lunar month (about 29.53085 days, compared to the "true" average value of about 29.53059 days). /Erik Ljungstrand (Sweden)

Leap day or leap month?
At present, the section Solar and lunar years begins: "A solar calendar year has 365 days (366 days in leap years). A lunar calendar year has 12 lunar months which alternate between 30 and 29 days for a total of 354 days (in leap years, one of the lunar months has a day added; since a lunar year lasts a little over $354 1/3$ days, a leap year arises every third year rather than every fourth.)"

Is the reference to a single leap day every three years correct? It certainly would not bring the lunar year back into synchronisation with the solar year. Later in the section, there is a much more convincing statement: "After two years the difference is 22 days, and after 3 years, 33. Whenever the epact reaches or exceeds 30, an extra (embolismic or intercalary) month is inserted into the lunar calendar, and the epact is reduced by 30." and at the Intercalation article, the section on lunisolar calendars says "The solar year does not have a whole number of lunar months (it is about 365/29.5 = 12.37 lunations), so a lunisolar calendar must have a variable number of months in a year. Regular years have 12 months, but embolismic years insert a 13th "intercalary" or "leap" or "embolismic" month every second or third year (see blue moon). Whether to insert an intercalary month in a given year may be determined using regular cycles such as the 19-year Metonic cycle (Hebrew calendar and in the determination of Easter) or using calculations of lunar phases (Hindu lunisolar and Chinese calendars). The Buddhist calendar adds both an intercalary day and month on a usually regular cycle."

In a nutshell, I really can't make sense of the statement that a single leap day is added every three years. What am I missing? --John Maynard Friedman (talk) 16:41, 4 October 2021 (UTC)

Does anyone here speak medieval Latin?
In this article and at Gregorian calendar, the decision to change the base to 1 January is either uncited or inadequately cited. The info needed is almost certainly somewhere in the Canons that accompanied Inter gravissimas, specifically this one: So all we need is a medievalist who can find out exactly where.

If it helps to show that the work is likely to be fruitful, the preceding Canon has the gem "The year of the ten-year cycle, which is the golden number 6, ends at the same time in the year of the Lord 1582 in the month of December. And in the month of January begins another year of the Lord, that is, 1583. And in the same month of January also, another year of the golden number is ushered in, namely 7." (which, after I corrected the mistranscription of $⟨ſ⟩$ as $⟨f⟩$ and a few other obvious typos, Google translate made an astonishingly good job of converting into credible English). But to transcribe and google-translate the whole chapter would take more time that I am willing to give. 𝕁𝕄𝔽 (talk) 10:20, 23 April 2023 (UTC)
 * On further consideration, I decided to remove discussion of the epact from the Gregorian Calendar article as being too detailed and arguably off-topic. So I have no further interest in this translation but decided to leave the citation here in case anyone has other reasons to investigate further. --𝕁𝕄𝔽 (talk) 10:54, 23 April 2023 (UTC)