Alexandrian and Gregorian Paschal
Calendars (in the sense of time reckoning systems, and not in terms of typographic products) were my hobby at a junior school age. Therefore, when I read the article Calculation of Easter , I immediately began to itch my hands to write the “revised and amended wording” of this article. And khdavid (its author) seemed to support me. In general, if you are reading this article, it means that the arising impulse of motivation is still enough to overcome my usual bad infinity of perfectionist reflection.
Without going into history, I will give the basic principle of calculating the Christian Easter. Easter is celebrated on the first Sunday after the first full moon in spring, that is, the first full moon that was not earlier than the day of the vernal equinox. Both Orthodox and Catholics use not the astronomical, but the calculated spring full moon. The only difference is in this very calculation.
Let's start with Easter, which is used by the majority of Orthodox churches (despite the fact that most of them switched from the Julian calendar to the new Julian). A complete description of all the calculations associated with it in traditional sources looked quite impressive and used half a dozen mysterious and intricate terms, such as “epacta” or “vruceleto”. But in fact, it is quite simple. It is based on the schedule of new moon and full moon, which in turn is based on the well-known 19-year metonic cycle (19 years ≈ 235 months ≈ 6940 days).
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On the day of the vernal equinox in Alexandrian Paschal, it was adopted on March 21 according to the Julian calendar. The first spring full moon can be calculated by a surprisingly simple scheme. Starting with the day cycle set for the first year (specifically, April 5), each next year we either take 11 days off the previous date or, in order not to leave before the equinox, we add 19 days to it. This simple scheme almost works. For 19 years, we subtract 11 times 12 times and add 7 times 19. 19 * 7-11 * 12 = 1 day we must additionally subtract when moving to the first year of the next 19-year cycle. This violation of the scheme at the boundary of the cycles was called the “jump of the moon” (saltus lunae).
To better understand how this works, we must mention the cycle of Kallippe, which in disguised form actually underlies this paschalia. This does not contradict the statement of the proton cycle, because the 76-year Kallippe cycle is just 4 metonic cycles: three for 6940 days and one of 6939 days. One could call the Kallippe cycle an adaptation of the metonic cycle to the Julian calendar, if it had not been invented about 3 centuries earlier than this calendar.
When calculating the Alexandrian Paschalion, all leap days and years are counted as 365 days, and the intervals between the Easter full moons (in fact, the years of the lunar-solar calendar embedded in Easter) are counted as 354 and 384 days. Due to this simplification, on the one hand, 15 days are lost for the Kallippe cycle (3.75 days per meton cycle), but on the other, missed February 29 gives an extra 19 days during this time (4.75 days per meton cycle). One extra day on metonov cycle and leads to the "jump of the moon."
The Julian calendar is not particularly accurate. The accuracy of the Kallippe cycle in relation to the phases of the moon is not much better. In the end, the deviations of the calculated equinoxes and full moons from astronomical concerns disturbed the Catholic Church and in 1582 Pope Gregory XIII adopted a new calendar and a new Paschal, later named after him.
Gregorian Paschalia is the "patched" version of Alexandria. In addition to correcting errors that have already accumulated since the time of the Nicene Council in 325 (which adopted the basic principles for calculating Easter), two “patches” were added to correct the inaccuracy of the Kallippe cycle in relation to the sun and moon due to periodic corrections.
The “solar patch” takes 3 days every 400 years due to the fact that the years that are multiples of 100, but not multiples of 400, are simple in the Gregorian calendar.
The Lunar Patch takes 8 days every 2500 years from the virtual lunar-solar calendar embedded in Passover, shifting the dates of the calculated phases of the moon back by 1 day in years, when divided by 2500, giving residues of 200, 500, 800, 1100, 1400, 1800, 2100 and 2400.
On top of these two was added a third “patch”, which I will write about below.
The rest of the Gregorian Easter is no different from the Alexandrian one. The spring equinox is considered March 21 of the Gregorian calendar, and the same hack is used with simplification and the "jump of the moon".
1. Before I found out what exactly the “moon patch” introduces corrections, I imagined it to be related to the Gregorian calendar, that is, applied over the “solar” one. It is important to understand that this is not the case. Both “patches” - “lunar” and “sunny” - are applied specifically to the Alexandrian Paschalia independently of each other. Their interaction leads to interesting consequences. The exclusion of one day by the "solar patch" means the shift of the calculated phases of the moon a day ahead . If this coincided with the shift introduced by the "moon patch", they compensate each other and no amendment for the paschalium occurs. Finally, if the "moon patch" shifts the leap year of the Gregorian calendar, the phases of the moon are shifted a day ago . That is, the movements of the calculated phases caused by the “patches” are nonmonotonic! On average, dates move forward, but sometimes, for example, in the year 2400, they will move backwards.
2. When correcting the accumulated error in the phases of the moon, an error was left in 1 day. Apparently, this was done intentionally to prevent the coincidence of the Catholic Easter with the Judean. However, because of the irregularities present in both the Gregorian Paschal and the Jewish calendar, such coincidences sometimes do occur. With every such coincidence, some sectarians are surely expecting the end of the world :) By the way, the fact that the Catholic Easter happens before the Jewish one did not seem to bother anyone especially. In any case, in the long run, this could not be prevented - simply because the average Jewish year’s duration in the Jewish calendar is longer than in the Gregorian one. Actually, the average speed of the phases of the moon due to the amendments in the first approximation is the average speed at which the Jewish calendar lags behind the Gregorian.
3. After March 21 is taken as the day of the vernal equinox, 30 possible days remain for the first spring full moon - from March 21 to April 19. But in Alexandrian Paschalia from these 30 days only 19 can be the days of the calculated full moon. In particular, it cannot be April 19, the latest day for it is April 18. Accordingly, Easter may be at the earliest on March 22 (if the calculated full moon is March 21 on Saturday) and at the latest - on April 25 (if the calculated full moon is April 18 on Sunday). But in the project of Paschalia, developed by Luigi Lilio (aka Aloisy Lily), which later became the Gregorian Easter, due to the amendments, the calculated full moon could already occur on any of the 30 possible days, including April 19. If it falls on Sunday April 19th, Easter will be April 26th. Paschal with such a "violation of backward compatibility" did not dare to release, and before the "release" it was "processed with a file." The third “patch”, which I announced earlier, shifted the calculated full moon from April 19 to 18. If among the current (between amendments) 19 possible days for the full moon were April 18 and 19, it moved from 17 to 18 so that the dates were not repeated within the 19-year cycle. Fortunately, out of these 19 days there can not be three in a row, so there’s no need to worry about April 17th.
Without going into history, I will give the basic principle of calculating the Christian Easter. Easter is celebrated on the first Sunday after the first full moon in spring, that is, the first full moon that was not earlier than the day of the vernal equinox. Both Orthodox and Catholics use not the astronomical, but the calculated spring full moon. The only difference is in this very calculation.
Alexandria easter
Let's start with Easter, which is used by the majority of Orthodox churches (despite the fact that most of them switched from the Julian calendar to the new Julian). A complete description of all the calculations associated with it in traditional sources looked quite impressive and used half a dozen mysterious and intricate terms, such as “epacta” or “vruceleto”. But in fact, it is quite simple. It is based on the schedule of new moon and full moon, which in turn is based on the well-known 19-year metonic cycle (19 years ≈ 235 months ≈ 6940 days).
')
On the day of the vernal equinox in Alexandrian Paschal, it was adopted on March 21 according to the Julian calendar. The first spring full moon can be calculated by a surprisingly simple scheme. Starting with the day cycle set for the first year (specifically, April 5), each next year we either take 11 days off the previous date or, in order not to leave before the equinox, we add 19 days to it. This simple scheme almost works. For 19 years, we subtract 11 times 12 times and add 7 times 19. 19 * 7-11 * 12 = 1 day we must additionally subtract when moving to the first year of the next 19-year cycle. This violation of the scheme at the boundary of the cycles was called the “jump of the moon” (saltus lunae).
To better understand how this works, we must mention the cycle of Kallippe, which in disguised form actually underlies this paschalia. This does not contradict the statement of the proton cycle, because the 76-year Kallippe cycle is just 4 metonic cycles: three for 6940 days and one of 6939 days. One could call the Kallippe cycle an adaptation of the metonic cycle to the Julian calendar, if it had not been invented about 3 centuries earlier than this calendar.
When calculating the Alexandrian Paschalion, all leap days and years are counted as 365 days, and the intervals between the Easter full moons (in fact, the years of the lunar-solar calendar embedded in Easter) are counted as 354 and 384 days. Due to this simplification, on the one hand, 15 days are lost for the Kallippe cycle (3.75 days per meton cycle), but on the other, missed February 29 gives an extra 19 days during this time (4.75 days per meton cycle). One extra day on metonov cycle and leads to the "jump of the moon."
Gregorian Easter
The Julian calendar is not particularly accurate. The accuracy of the Kallippe cycle in relation to the phases of the moon is not much better. In the end, the deviations of the calculated equinoxes and full moons from astronomical concerns disturbed the Catholic Church and in 1582 Pope Gregory XIII adopted a new calendar and a new Paschal, later named after him.
Gregorian Paschalia is the "patched" version of Alexandria. In addition to correcting errors that have already accumulated since the time of the Nicene Council in 325 (which adopted the basic principles for calculating Easter), two “patches” were added to correct the inaccuracy of the Kallippe cycle in relation to the sun and moon due to periodic corrections.
The “solar patch” takes 3 days every 400 years due to the fact that the years that are multiples of 100, but not multiples of 400, are simple in the Gregorian calendar.
The Lunar Patch takes 8 days every 2500 years from the virtual lunar-solar calendar embedded in Passover, shifting the dates of the calculated phases of the moon back by 1 day in years, when divided by 2500, giving residues of 200, 500, 800, 1100, 1400, 1800, 2100 and 2400.
On top of these two was added a third “patch”, which I will write about below.
The rest of the Gregorian Easter is no different from the Alexandrian one. The spring equinox is considered March 21 of the Gregorian calendar, and the same hack is used with simplification and the "jump of the moon".
Bagofits and crutches
1. Before I found out what exactly the “moon patch” introduces corrections, I imagined it to be related to the Gregorian calendar, that is, applied over the “solar” one. It is important to understand that this is not the case. Both “patches” - “lunar” and “sunny” - are applied specifically to the Alexandrian Paschalia independently of each other. Their interaction leads to interesting consequences. The exclusion of one day by the "solar patch" means the shift of the calculated phases of the moon a day ahead . If this coincided with the shift introduced by the "moon patch", they compensate each other and no amendment for the paschalium occurs. Finally, if the "moon patch" shifts the leap year of the Gregorian calendar, the phases of the moon are shifted a day ago . That is, the movements of the calculated phases caused by the “patches” are nonmonotonic! On average, dates move forward, but sometimes, for example, in the year 2400, they will move backwards.
2. When correcting the accumulated error in the phases of the moon, an error was left in 1 day. Apparently, this was done intentionally to prevent the coincidence of the Catholic Easter with the Judean. However, because of the irregularities present in both the Gregorian Paschal and the Jewish calendar, such coincidences sometimes do occur. With every such coincidence, some sectarians are surely expecting the end of the world :) By the way, the fact that the Catholic Easter happens before the Jewish one did not seem to bother anyone especially. In any case, in the long run, this could not be prevented - simply because the average Jewish year’s duration in the Jewish calendar is longer than in the Gregorian one. Actually, the average speed of the phases of the moon due to the amendments in the first approximation is the average speed at which the Jewish calendar lags behind the Gregorian.
3. After March 21 is taken as the day of the vernal equinox, 30 possible days remain for the first spring full moon - from March 21 to April 19. But in Alexandrian Paschalia from these 30 days only 19 can be the days of the calculated full moon. In particular, it cannot be April 19, the latest day for it is April 18. Accordingly, Easter may be at the earliest on March 22 (if the calculated full moon is March 21 on Saturday) and at the latest - on April 25 (if the calculated full moon is April 18 on Sunday). But in the project of Paschalia, developed by Luigi Lilio (aka Aloisy Lily), which later became the Gregorian Easter, due to the amendments, the calculated full moon could already occur on any of the 30 possible days, including April 19. If it falls on Sunday April 19th, Easter will be April 26th. Paschal with such a "violation of backward compatibility" did not dare to release, and before the "release" it was "processed with a file." The third “patch”, which I announced earlier, shifted the calculated full moon from April 19 to 18. If among the current (between amendments) 19 possible days for the full moon were April 18 and 19, it moved from 17 to 18 so that the dates were not repeated within the 19-year cycle. Fortunately, out of these 19 days there can not be three in a row, so there’s no need to worry about April 17th.
Source: https://habr.com/ru/post/368917/
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