Epact

The epact (Latin: epactae, from Ancient Greek: ἐπακται ἡμεραιepaktai hēmerai 'added days') was originally the term used by medieval computists for the age of a phase of the Moon, in days since the new moon, as determined on 22 March of a given year.^[1] In the newer Gregorian calendar, however, the epact is reckoned as the age of the ecclesiastical moon on 1 January.^[2] Its principal use is in determining the date of Easter by computistical methods. It varies (usually by 11 days) from year to year, because of the difference between the solar year of 365–366 days and the lunar year of 354–355 days.^[3]

Lunisolar calendar

Epacts can also be used to relate dates in a lunisolar calendar to dates in the common solar calendar.

Solar and lunar years

A solar calendar year has 365 days (366 days in leap years). A lunisolar calendar year has 12 lunar months which typically alternate between 30 and 29 days for a total of 354 days: to resynchronise with the solar year, a thirteenth leap month is added. This is because a lunar year lasts a little over 354+1⁄3 days and consequently a leap year arises every second or third year rather than every fourth.)

Si un año solar y un año lunar comienzan el mismo día, entonces, después de un año, el inicio del año solar se produce 11 días después del inicio del año lunar. Estos días "excedentes" se denominan epactas y deben sumarse al año lunar para completar el año solar; o, desde la perspectiva complementaria, se suman al día del año solar para determinar el día del año lunar.

Después de dos años, la diferencia es de 22 días, y después de 3 años, de 33 días. Siempre que el epacto alcance o supere los 30 días, se añade un mes lunar adicional (embolismo o intercalar ) al calendario lunisolar, y el epacto se reduce en 30 días.

El cálculo de las epactas no tiene en cuenta los años bisiestos del calendario solar. Este principio exige añadir un día extra a cualquier año lunar que contenga el 29 de febrero, de modo que el siguiente año solar comience con el número de días calculado después del siguiente año lunar.

Un calendario lunar puro tiene doce meses lunares de forma invariable, por lo que se desfasa con respecto al año solar: el concepto de epactas no es relevante.

ciclo de 19 años

Un año tropical (año solar ) es ligeramente más corto que 365 + 1/4 días , mientras que el mes sinódico ( mes lunar) es, en promedio, ligeramente más largo que 29 + 1/2 días , lo que significa que ambos son números no enteros. Esto se corrige de la siguiente manera: diecinueve años tropicales se consideran equivalentes a 235 meses sinódicos ( ciclo metónico ). Un ciclo puede durar 6939 o 6940 días completos, dependiendo de si hay 4 o 5 días bisiestos en este período de 19 años.

Después de 19 años, las lunaciones deberían caer de la misma manera en los años solares, por lo que el epacto debería repetirse después de 19 años. Sin embargo, 19 × 11 = 209, y esto no es un múltiplo entero del ciclo completo de 30 números epactos ( 209 módulo 30 = 29, no 0). Por lo tanto, después de 19 años, el epacto debe corregirse en +1 para que el ciclo se repita durante 19 años. Esto es el saltus lunae ("salto de la luna"). El número de secuencia del año en el ciclo de 19 años se llama número áureo . Los 209 días adicionales llenan 7 meses embolísmicos , para un total de 19 × 12 + 7 = 235 lunaciones.

epactas lilianas (gregorianas)

When the Gregorian calendar reform was instituted in 1582, the lunar cycle previously used with the Julian calendar to complete the calculation of Easter dates was adjusted also, in accordance with a (modification of the) scheme devised by Aloysius Lilius.^[4] There were two adjustments to the old lunar cycle:

a "solar equation", decrementing the epact by 1, whenever the Gregorian calendar drops a leap day (3 times in 400 calendar years), and
a "lunar equation", incrementing the epact by 1, 8 times in 2500 calendar years (7 times after an interval of 300 years, and the 8th time after an interval of 400 years).

The revised "solar equation" was intended to adjust for the Gregorian change in the solar calendar, if they were applied at 1 January of the Julian calendar instead of the Gregorian calendar as the reformers implemented it; moreover the corrections to the solar calendar are leap days, whereas there are 30 epact values for a mean lunar month of 29+1⁄2 days and a bit: Therefore changing the epact by 1 day does not exactly compensate for a dropped leap day. The "lunar equation" only approximately adjusts for what had (by 1582) been seen after many centuries of recording, that the Moon moves a little faster than the expectation of the rate used for it in the old lunar cycle. By 1582 it was noted (for example, in the text of the bull Inter gravissimas itself) that the new and full moons were at that point occurring "four days and something more" sooner than the old lunar cycle indicated.

The explicit formula for the numerical value of the Gregorian epact can be derived from the solar and lunar equations.^[5] For a given year $y,$ define, respectively, the century number $C_{y}$ and the golden number $G_{y}$ ,

${\begin{aligned}C_{y}&=\lfloor y/100\rfloor +1,\\G_{y}&=y\mod 19+1.\end{aligned}}$

The epact $E_{y}$ is then given by

$E_{y}=11G_{y}-\left\lfloor {\frac {3C_{y}}{4}}\right\rfloor +\left\lfloor {\frac {8C_{y}+5}{25}}\right\rfloor +27\mod 30.$

For 2026 the century number $C_{y}$ is 21.

For 2026 the golden number $G_{y}$ is 13.

The Gregorian epact of 2026 is (143 - 15 + 6 + 27) mod 30 = 161 mod 30 = 11.

History

El descubrimiento del epacto para calcular la fecha de la Pascua ("el computus ") se ha atribuido al patriarca Demetrio I de Alejandría , quien ocupó el cargo entre 189 y 232 d. C. En el año 214, utilizó el epacto para producir un calendario pascual, que no se ha conservado, que utilizaba un ciclo lunisolar de ocho años . ^{[ 6 ]} Una aplicación posterior del epacto a un calendario pascual, que utiliza un ciclo de dieciséis años, se encuentra en la Tabla Pascual de Hipólito, una lista de 112 años de fechas de Pascua que comienza en el año 222 y que está inscrita en el lateral de una estatua encontrada en Roma. ^{[ 6 ]} Augustalis, cuyas fechas habían sido disputadas desde el siglo III hasta el siglo V, ^{[ 7 ]}^{(pp. 224-228)} calculó un laterculus ("pequeña tablilla") de fechas de Pascua. Tal como se ha reconstruido, utiliza epactos (en este caso, la edad de la luna el 1 de enero) y un ciclo lunisolar de 84 años. ciclo para calcular las fechas de Pascua usando como fecha base el año 213 d. C. Si aceptamos las fechas anteriores de Augustalis, su laterculus se extiende desde el 213 hasta el 312 d. C., y Augustalis fue quien originó el uso de epactas para calcular la fecha de Pascua. ^{[ 8 ]}^{(págs. 40-45)}

As early as the fourth century, an Easter computus using the epact and the nineteen-year Metonic cycle was used in Alexandria. Subsequent computational tables were influenced by the structure of the Alexandrian calendar. The epact was taken as the age of the Moon on 26 Phamenoth (22 March in the Julian calendar) but that value of the epact also corresponded to the age of the Moon on the last epagomenal day of the preceding year. Thus the epact can be seen as having been established at the beginning of the current year.^[7]^(pp75–80) Subsequent Easter tables, such as those of Bishop Theophilus of Alexandria, which covered 100 years beginning in 380 AD, and of his successor Bishop Cyril, which covered 95 years beginning in 437 AD discussed the computation of the epact in their introductory texts. Under the influence of Dionysius Exiguus and later, of Bede, the Alexandrian Easter Tables were adopted throughout Europe where they established the convention that the epact was the age of the Moon on 22 March.^[8]^{(p 52)} This Dionysian epact fell into disuse after the introduction of a perpetual calendar based on the golden number, which made the calculation of epacts unnecessary for ordinary computistical calculations.^[9]

Two factors led to the creation of three new forms of the epact in the fifteenth and sixteenth centuries. The first was the increasing error of computistical techniques, which led to the introduction of a new Julian epact around 1478 AD, to be used for practical computations of the phase of the Moon for medical or astrological purposes. With the Gregorian reform of the calendar in 1582, two additional epacts came into use. The first was the Lilian epact, developed by Aloisius Lilius as an element of the ecclesiastical computations using the Gregorian calendar. The Lilian epact included corrections for the motions of the Sun and the Moon that broke the fixed relationship between the epact and the golden number. The second new epact was a simple adjustment of the practical Julian epact to account for the ten-day change produced by the Gregorian Calendar.^[9]

References

↑Bede the Venerable (1999) [725], "Lunar Epacts", The Reckoning of Time, Translated Texts for Historians, vol. 29, translated by Wallis, Faith, Liverpool: Liverpool University Press, p. 131, ISBN 0-85323-693-3, 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.
↑Richards, E. G. (2012), "Calendars"(PDF), in Urban, S. E.; Seidelman, P. K. (eds.), Explanatory Supplement to the Astronomical Almanac, Mill Valley, CA: University Science Books, pp. 599–601, ISBN 978-1-891389-85-6, archived from the original(PDF) on 2019-04-30, retrieved 2017-01-24, The Epact of a year … is the age in days (0 to 29) of the ecclesiastical moon on the first day of the year (January 1).
↑Latin text and French translation of the Second Canon of the Gregorian calendar
↑Moyer, G. (1983). "Aloysius Lilius and the Compendium novae rationis restituendi kalendarium"(PDF). In Coyne, George V.; Hoskin, Michael A.; Pedersen, O. (eds.). Gregorian Reform of the Calendar. Vatican Conference to commemorate [the Gregorian Reform's] 400th Anniversary, 1582–1982(PDF). Vatican City, IT: Pontifical Academy of Sciences, Vatican Observatory. p. 171. Archived from the original(PDF) on 16 November 2019. Retrieved 20 January 2024.
↑Fukś, Henryk (2022), "Mathematics in the liturgical books of the Catholic Church: phases of the ecclesiastical moon", Antiquitates Mathematicae, 16: 259--282, arXiv:2601.00474, doi:10.14708/am.v16i1.7138
1 2Mosshammer, Alden A. (2008). "The 8 year cycle and the invention of the epacts". The Easter Computus and the Origins of the Christian Era. Oxford Early Christian Studies. Oxford, UK: Oxford University Press. pp. 109–125. ISBN 978-0-19-954312-0.
1 2 Mosshammer, Alden A. (2008). El Cáutus de Pascua y los orígenes de la era cristiana . Estudios cristianos primitivos de Oxford. Oxford, Reino Unido: Oxford University Press. págs. 39–40 , 109–125 , 224–228 . ISBN 978-0-19-954312-0.
1 2 Pedersen, O. (c. 1982). "El calendario eclesiástico y la vida de la Iglesia" (PDF) . En Coyne, GV ; Hoskin, MA; Pedersen, O. (eds.). Reforma gregoriana del calendario . Conferencia vaticana para conmemorar el 400 aniversario [ de la Reforma gregoriana ] , 1582–1982 (PDF) . Ciudad del Vaticano, IT: Academia Pontificia de las Ciencias , Observatorio Vaticano (publicado en 1983). pp. 39–59 . Archivado del original (PDF) el 25 de mayo de 2015. Recuperado el 13 de mayo de 2023 .
1 2 Dekker, Elly (1993). "Tablas de impacto en instrumentos: su definición y uso". Annals of Science . 50 (4): 303– 324. doi : 10.1080/00033799300200251 .

Enlaces externos

Epactos de la Enciclopedia Católica

[1] Bede the Venerable (1999) [725], "Lunar Epacts", The Reckoning of Time, Translated Texts for Historians, vol. 29, translated by Wallis, Faith, Liverpool: Liverpool University Press, p. 131, ISBN 0-85323-693-3, 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.

[2] Richards, E. G. (2012), "Calendars"(PDF), in Urban, S. E.; Seidelman, P. K. (eds.), Explanatory Supplement to the Astronomical Almanac, Mill Valley, CA: University Science Books, pp. 599–601, ISBN 978-1-891389-85-6, archived from the original(PDF) on 2019-04-30, retrieved 2017-01-24, The Epact of a year … is the age in days (0 to 29) of the ecclesiastical moon on the first day of the year (January 1).

[3] Latin text and French translation of the Second Canon of the Gregorian calendar

[4] Moyer, G. (1983). "Aloysius Lilius and the Compendium novae rationis restituendi kalendarium"(PDF). In Coyne, George V.; Hoskin, Michael A.; Pedersen, O. (eds.). Gregorian Reform of the Calendar. Vatican Conference to commemorate [the Gregorian Reform's] 400th Anniversary, 1582–1982(PDF). Vatican City, IT: Pontifical Academy of Sciences, Vatican Observatory. p. 171. Archived from the original(PDF) on 16 November 2019. Retrieved 20 January 2024.

[5] Fukś, Henryk (2022), "Mathematics in the liturgical books of the Catholic Church: phases of the ecclesiastical moon", Antiquitates Mathematicae, 16: 259--282, arXiv:2601.00474, doi:10.14708/am.v16i1.7138

[Mosshammer-ch8year-6] 1 2Mosshammer, Alden A. (2008). "The 8 year cycle and the invention of the epacts". The Easter Computus and the Origins of the Christian Era. Oxford Early Christian Studies. Oxford, UK: Oxford University Press. pp. 109–125. ISBN 978-0-19-954312-0.

[Mosshammer-2008-7] 1 2 Mosshammer, Alden A. (2008). El Cáutus de Pascua y los orígenes de la era cristiana . Estudios cristianos primitivos de Oxford. Oxford, Reino Unido: Oxford University Press. págs. 39–40 , 109–125 , 224–228 . ISBN 978-0-19-954312-0.

[Pedersen-1983-8] 1 2 Pedersen, O. (c. 1982). "El calendario eclesiástico y la vida de la Iglesia" (PDF) . En Coyne, GV ; Hoskin, MA; Pedersen, O. (eds.). Reforma gregoriana del calendario . Conferencia vaticana para conmemorar el 400 aniversario [ de la Reforma gregoriana ] , 1582–1982 (PDF) . Ciudad del Vaticano, IT: Academia Pontificia de las Ciencias , Observatorio Vaticano (publicado en 1983). pp. 39–59 . Archivado del original (PDF) el 25 de mayo de 2015. Recuperado el 13 de mayo de 2023 .

[Dekker-9] 1 2 Dekker, Elly (1993). "Tablas de impacto en instrumentos: su definición y uso". Annals of Science . 50 (4): 303– 324. doi : 10.1080/00033799300200251 .

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