Ecclesiastical new moon

An ecclesiastical new moon is the first day of a lunar month (an ecclesiastical moon) in an ecclesiastical lunar calendar. Such months have a whole number of days, 29 or 30, whereas true synodic months can vary from about 29.27 to 29.83 days in length. Medieval authors equated the ecclesiastical new moon with a new crescent moon, but it is not a phase of the true moon. If the ecclesiastical lunar calendar is accurate, the ecclesiastical new moon can be any day from the day of the astronomical new moon or dark moon to two days later (see table). The ecclesiastical calendar valid for the Julian and Gregorian calendar are described in detail by Grotefend,[1] Ginzel[2] and in the Explanatory Supplement to The Astronomical Ephemeris.[3]

The ecclesiastical new moon which falls on or next after March 8 is of special importance, since it is the paschal new moon that begins the paschal lunar month (see table). The fourteenth day of the same lunar month is the first of the calendar year to occur on or next after March 21. This fourteenth day was called the paschal full moon by medieval computists. Easter is the following Sunday.

Calendar pages in medieval liturgical books indicated the ecclesiastical new moons by writing the Golden Number to the left of the day of the month on which the ecclesiastical new moon would occur in the year of that Golden Number. In some places the age of the moon was announced daily in the office of Prime at the reading of the martyrology.[4]

When in the 13th century Roger Bacon complained about the discrepancy between the ecclesiastical moon and the observed lunar phases, he specifically mentioned the discrepancy involving the ecclesiastical new moon

> Quilibet computista novit quod fallit primatio per tres dies vel quatuor his temporibus; et quilibet rusticus potest in coelo hunc errorem contemplari. (Any computist knows that the prime [of the moon] is off by three or four days in our time; and any rustic can see this error in the sky.)[5]

These complaints were finally addressed by the construction of the Gregorian calendar.

A check can be made on the difference between the astronomical new moon and the ecclesiastical new moon. The following table gives a comparison for 2010. All times are Greenwich Mean Time. Coordinated Universal Time is the same with a tolerance of 0.9 seconds either way.

New moon in 2010[6] | Ecclesiastical new moon, 2010[7]

|

January 15 07h | January 17
February 14 03h | February 15
March 15 21h | March 17
April 14 12h | April 15
May 14 01h | May 15
June 12 11h | June 13
July 11 20h | July 13
August 10 03h | August 11
September 8 10h | September 10
October 7 19h | October 9
November 6 05h | November 8
December 5 18h | December 7

The long term accuracy of the Gregorian ecclesiastical lunar calendar is remarkable. It will be in error by one day in about 73 500 years while the error with respect to the tropical year will be one day in about 3320 years.[8]

The following table gives the ecclesiastical new moon used for determining Easter (in the Gregorian system) for a range of years.

Year  | Gregorian paschal new moon  | Days in paschal lunar month

|
|

2014  | April 1  | 29
2015  | March 21  | 29
2016  | March 10  | 29
2017  | March 29  | 29
2018  | March 18  | 29
2019  | April 5  | 30
2020  | March 26  | 29
2021  | March 15  | 29
2022  | April 3  | 29
2023  | March 23  | 29
2024  | March 12  | 29
2025  | March 31  | 29
2026  | March 20  | 29
2027  | March 9  | 29
2028  | March 28  | 29
2029  | March 17  | 29
2030  | April 4  | 30
2031  | March 25  | 29
2032  | March 14  | 29

### Determining the age of the eccesiastical moon

In the liturgical books of the Catholic Church, the age of the moon (defined as the time elapsed since the last new moon) is determined by using tabular methods. For explicit calculations of the age of the ecclesiastical moon, a convenient formula has been derived by H. Fukś[9].

For a given day d of the month m and the year y, define the century number Cy, the golden number Gy, the epact Ey and the day number Nm,d, respectively as follows,

Cy=⌊y/100⌋+1,Gy=ymod19+1,Ey=11Gy−⌊3Cy4⌋+⌊8Cy+525⌋+27mod30,Nm,d=d−1+30(m−1)+⌊7m−212⌋−2⌊m+912⌋.

Then the age of the ecclesiastical moon Ay,m,d on that day is given by

Ay,m,d={g(Ey+Nm,d)if Ey<25 or (Ey=25 and Gy≥12),g(Ey+Nm,d+29)+𝟏Nm,d+Ey<30otherwise,

where

g(x)=(x+⌊x59⌋)mod30+1.

and

𝟏P={1if P is true,0otherwise.

This formula reflects the age of the moon as given in tables of Martyrologium Romanum . On some years with golden number 1 these tables give incorrect result in the fist cycle of January. Liturgical books stipulate that in such cases the appropriate correction should be applied, and this is called `"pronounced age of the moon", defined as

Ay,m,dpron={Ay,m,d−1if Gy=1,Ey>0,m=1 and d+Ey≤30,Ay,m,dotherwise.

Even with the above correction, the formula is applicable till year 16399. Afterwards, another correction needs to be applied because further anomalies [10] may appear. Fully corrected formula for the age of ecclesiastical moon, eliminating all known anomalies, has been proposed by H. Fukś[9] as well:

Jy=(Ey−Ey−1)mod30−11,Ay,m,dcorr={Ay,m,d−Jy+30⋅𝟏Ay,m,d−Jy≤0if m=1 and d+Ey≤30,Ay,m,dotherwise.

Python implementation of all of these formulas is available. [11]

## References

1. ↑ Grotefend, Hermann (1891). Zeitrechnung des deutschen Mittelalters und der Neuzeit, Bd. 1. Hannover, Germany: Hahn'sche Buchhandlung.
2. ↑ Ginzel, Freidrich Karl (1914). Handbuch der mathematischen und technischen Chronologie. Leipzig, Germany: Hinrichs.
3. ↑ Explanatory Supplement to The Astronomical Ephemeris. 1961.
4. ↑ At medieval Exeter Cathedral, it was the next day's date and age of the moon that were announced. Et omnibus in locis suis sedentibus sit ibi quidam puer...paratus ad legendum leccionem de Martilogio, absque Iube domine, sed pronunciondo primo loco numerum Nonarum, Iduum, Kalendarum, et etatem lune qualis erit in crastino... (And when all are sitting in their places let a boy be there ready to read the Martyrology beginning with Iube domine, but first saying the number of Nones, Ides, Kalends, and what the age of the moon will be on the morrow...) J.N. Dalton, ed., Ordinale Exon. vol. 1, Henry Bradshaw Society, London, 1909, p. 37.
5. ↑ Roger Bacon, Opus Tertium LXX, in J. S. Brewer, ed., Fr. Rogeri Bacon Opera quaedam hactenus Inedita. Vol. 1. H.M. Stationery Office, 1859 (Kraus Reprint 1965), p. 282.
6. ↑ From the U.S. Naval Observatory's Principal phase calculator
7. ↑ From the Explanatory Supplement to the Ephemeris and the American Ephemeris and Nautical Almanac, H. M. Stationery Office, London, 1966, fourth revised printing, 1977, page 426.
8. ↑ Lichtenberg, Heiner (1994). "Die Struktur des Gregorianischen Kalenders anhand der Schwankungen des Osterdatums entschlüsselt". Sterne und Weltraum 33 (3): 194–201. Bibcode: 1994S&W....33..194L.
9. ↑ 9.0 9.1 Fukś, Henryk (2022), "Mathematics in the liturgical books of the Catholic Church: phases of the ecclesiastical moon", Antiquitates Mathematicae 16: 259--282, doi:10.14708/am.v16i1.7138, https://wydawnictwa.ptm.org.pl/index.php/antiquitates-mathematicae/article/view/7138/6598
10. ↑ Roegel, D. (2004), The missing new moon of A.D. 16399 and other anomalies of the Gregorian calendar, Rapport de recherche A04-R-436, INRIA, https://hal.inria.fr/inria-00099868
11. ↑ Fukś, Henryk, Program constructing lunar tables for ecclesiastical moon, Brock University, https://brocku.scholaris.ca/items/570bbab6-37d0-4a25-9945-5f1c07d861d5, retrieved 2026-01-02

## External links

* U.S. Naval Observatory Data Services
* Gregorian ecclesiastical new moons
* Oxford, Bodleian Library MS. Lat. liturg. e. 6. (14th century)

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