mynameisben
Half man, half-wit
- Joined
- Apr 18, 2003
- Posts
- 50,215
The Easter season is nigh upon us, so I thought I'd post a few lesser known trivia facts associated with Easter for those who so choose to find them read-worthy. No attempt is made to promote one religious belief over another.
Defining the Date of Easter
Long ago, the Christian church defined the date on which Easter would be celebrated thusly:
“Easter shall be celebrated throughout the world on the first Sunday after the first full moon after the vernal equinox.”
An enduring myth maintains the above definition is still the correct method for determining the date of Easter, but it has several flaws which deserve to be pointed out.
First, to avoid an obvious ambiguity in the above definition, the entire world has always accepted the assumption that “vernal equinox” refers to the equinox of the northern hemisphere. The vernal equinox in the northern hemisphere occurs at the same time as the autumnal equinox in the southern hemisphere and vice-versa, but this inconvenience of terminology has no effect on when Easter occurs on either half of the globe. Easter is not celebrated after the full Moon that follows September 21 in Australia and never has been.
Still, the ancient definition is confusing. Even way back when, people folk started scratching their heads and saying, “Easter... uhh, what date is that?” Church officials were called upon to announce the date Easter would fall on for the coming year so that celebrations could be planned in advance. Soon, a single year's advance notice wasn't enough, and church officials were tasked with calculating the date of Easter for several years in advance. And as it turned out, determining the date Easter would fall on was hard! Determining the date was difficult for a variety of reasons, one of which arises from the fact that the vernal equinox date differs from year to year. In Rome, the equinox can usher in Spring as early as March 19, and as late as March 21. If you take time zones into account, then the Spring equinox in New Zealand (autumn there) can come as late as March 22. This four-day span for the equinox throws a real monkey wrench into the ancient definition of the Easter date.
Also unclear in the ancient definition of the Easter date is what is meant by the word “after.” Since the astronomical equinox happens at a precise instant in time on the day it arrives, and because the Moon technically goes full at a precise time as well, it is possible for the Moon to go full on the same day the vernal Equinox arrives yet still occur after it. And if both these celestial events occur on a Sunday, then Easter could be celebrated on that same day. Or, it could be celebrated a week later or even two weeks later, depending on one's interpretation of “after.”
Yet another difficulty with the ancient definition has to do with the phases of the Moon. As it so happens, the exact date the Moon goes full depends on which patch of the Earth one is standing on when looking up at the night sky, and this date can vary by as much as two calendar days. Because of these astronomical difficulties, it wasn't long before people in differing parts of the world were rightfully using the Church definition of the Easter date and celebrating Easter in different weeks. Worse, Easter was being celebrated in some places in some years on the exact same Sunday as the Jewish Passover. Christian church elders wanted Easter to be celebrated everywhere on the exact same date, and they did not like the idea of confusing the celebrations of Easter and Passover one little bit.
To shore up the problems with the ancient definition of the Easter date, the Catholic church held a Council in Nicaea in 325 AD. The council defined “vernal equinox” to mean the set date of March 21 for the purposes of calculating the Easter date, regardless of the the actual calendar date the true astronomical equinox arrives on. The council also decided upon an unambiguous method for determining the date Easter would fall on, which was immediately put into affect by papal bull. The method chosen was based on a system of tables worked out centuries earlier by Greek astronomers, tables which were already in use by the Jews for determining the date of their Passover celebration. The new definition of the Easter date, while substantively the same as the old, included a proviso to ensure it would never again be celebrated on the same date as Jewish Passover. The revised definition is as follows:
“Easter shall be celebrated throughout the world on the first Sunday following the Paschal Full Moon date, the Paschal Full Moon date occurring on or after the vernal equinox [March 21].”
And this is the correct definition of the Easter date in use by the Christian world today.
The Paschal Full Moon date is the full moon date that defines the Jewish Passover (the word “paschal” meaning, 'of or relating to Passover'). While the Paschal Full Moon date frequently coincides with an actual astronomical full Moon date (in Rome), often times it does not. The Paschal full Moon basically falls on whatever date the Church says it does, according to their system of tables. (By “Church,” I mean either Christian or Jewish, since they agree upon the Paschal Full Moon date, using the same tabular methods). Since the tabular system is directly tied to mathematics, equivalent methods for determining the date Easter falls on may come from equations and computer programs which model the mathematics on which the ancient tables were constructed. Church officials and astronomers need not be consulted.
Basis for the Range of Dates for Easter
The Moon goes through its various phases and may become “full” on any day of the month, including March 21. When the Moon is full on March 21, and when it agrees with the “Paschal Full Moon” date derived from the ancient Greek tables, then it has done so at the earliest possible calendar date consistent with the “on or after the vernal equinox [March 21].” Should a March 21 Paschal Full Moon occur on a Saturday, then Easter would be the very next day—Sunday, March 22, the earliest possible date Easter can be. This confluence of events is quite rare, making March 22 the most unlikely date for Easter to fall on. The last time Easter fell on March 22 was in 1918, and before that there was a March 22 Easter in 1761. Easter will not come as early as March 22 again until the year 2285.
Since the Moon takes 29 1/2 days to make one full orbit around the Earth, there are 30 distinct phases the Moon can be in on any given day. The last half-day of the Moon's orbit is rounded up to make 30 Moon phases. To lower round-off errors that result from turning fractions into integers, it is more precise to round the number of Moon phases Up while rounding the number of days over which they occur Down. Within the confines of this simple but fairly accurate model, the Moon can go full on any of 29 days that begin with the Mar 21st “equinox” to be declared the Paschal Full Moon for a given year. Those 29 days span the dates of March 21 through April 18, and the first Sunday thereafter (Easter) will come 1 to 7 days later, making the full range of possible Easter dates March 22 through April 25.
April 25 is the latest date Easter can fall on, and it is also rare, though not as rare as the earliest possible Easter date of March 22. Easter fell on April 25th in 1734, 1886, 1943, and will come again next in 2038.
Why a March 22 Easter date is rarer than an April 25 Easter date
Intuition might suggest the earliest possible Easter date (March 22) is just as rare an occurrence as the latest possible Easter date (April 25). But in this case, intuition would be wrong. The present day definition of the Easter date, in affect since 326 AD, makes an April 25 Easter approximately 50% more frequent an occurrence than a March 22 Easter. The answer to this intuition defying puzzle lies within the ancient Greek tables that define the date of the Paschal Full Moon.
When the ancient Greek astronomers constructed their tables to predict the dates of the full Moon, they used a “30 phases in 29 days” model, mentioned previously. The phase of the Moon on March 21 in an arbitrary year (known as an “epact”) can be any of 30 different values. No matter the phase of the Moon on March 21, it is known with certainty the Moon will be full on some day 0 to 28 days later. To make tables out of such knowledge, Greek mathematicians were required to construct an equation that takes each of 30 input epact values (Moon phases on March 21) and maps it to one of 29 possible output values (dates of the next full Moon). 30 epact values mapped to 29 possible Full Moon dates is not a one-to-one relationship, which means that one epact value would have to be mapped to an output Full Moon date already mapped to from some other epact value. The output value to be mapped to twice was chosen to be the latest date in the Lunar cycle, because careful observations over many years showed that round-off error approximations in the model were affected most in the final day of the Lunar cycle (because the final day in the Lunar cycle is the invisible New Moon). The equations that support the ancient Greek tables therefore slightly bias the Paschal Full Moon date towards the latest of possible days, which therefore makes the date of Easter slightly more likely to fall at the end of its possible range than at its beginning.
Why April 19 is the most common date for Easter
It's a quirky trivia fact that Easter falls most often on April 19. To the few who are even aware of this trivia fact, it is a profound mystery why April 19, is the “preferred” date, a date which is nowhere near the middle of the possible range of Easter dates nor all that close to either end of the spectrum. Here's why:
As stated in the previous section, the “30 moon phases in 29 days” model for the ancient Greek tables requires one of the Paschal Full Moon dates to be mapped to by two different epact values, and this date occurs at the end of the Lunar cycle. The end of the Lunar cycle is on the last possible date for the Paschal Full Moon, which is April 18.
Therefore, whenever the Paschal Full Moon falls on April 18 AND that day is also a Saturday, then Easter is on April 19 for one of two possible moon phases that occurred on March 21. Easter can also fall on April 19, in any year, because the Paschal Full Moon date happened to fall on any of the the other six dates, April 12 through April 17. This gives April 19 eight possible chances to be Easter in some years.
Looking at April 20 as a possible Easter date, there are sill two March 21 moon phase dates that can lead to an April 18 Paschal Full Moon date. In years when April 18 falls on a Friday, that gives two possible chances for Easter to be on April 20. But now there are only 5 other Paschal Full Moon dates that can lead to an April 20 Easter, those being April 13 through April 17. (April 12 is too early because even if the next Sunday is 7 days later that only takes you to April 19). This gives April 20 at most 7 possible chances to be Easter. And with each advancing date for Easter beyond April 19, one more possible date for the Paschal Full Moon drops off the calendar, lessening the chances for Easter to fall on that date.
For all dates in which Easter falls on April 18 or earlier, those dates never get the “bonus chance” of being Easter that comes from following the extra chance provided by the April 18th Paschal Full Moon date. Therefore, that makes April 19th the most commonly occurring Easter date.
Will the sequence of dates on on which Easter falls ever repeat itself in a predictable pattern?
Yes. Because the Easter date can be predicted by deterministic equations, then there must be a periodic repetition of the sequence of dates on which Easter falls. What may be surprising to many is that the length of this sequence spans 5.7 million years! The figure of 5.7 million years looks like a rounded figure, but it is not. It is the product of four different influencing factors:
30 – the number of different Moon phases that occur on March 21 of any given year, which directly influences the Paschal Full Moon date.
19 – the number of years in the Metonic Cycle. Observations show that the dates within a year that the phases of the moon fall on repeat themselves every 19 years. This cycle assumes that no adjustments to the Solar calendar or to the Lunar calendar are made to accommodate “Leap Days.”
2500 – the number of years it takes for the Lunar Calendar to repeat its sequence of dates, after taking into account that it must be tweaked to include 8 different Leap Days distributed through each sequence of 2500 years.
400 – the number of years it takes the Solar Calendar to repeat its sequence of dates, after taking into account that it must be tweaked to include a Leap Day every 4 years, except for the 3 out of 4 years that are evenly divisible by 100 but not by 400.
The 400-year cycle and the 2500-year cycle have a common factor of 100 years which cancels out. Therefore the cycle of years in which Easter dates repeat themselves is 30 x 19 x 2500 x 400 / 100 = 5.7 million years, exactly.
Easter falls on April 8th this year. Go forth and amaze friends and family with this fascinating collection of numerical Easter facts. Happy Easter everybody!
Defining the Date of Easter
Long ago, the Christian church defined the date on which Easter would be celebrated thusly:
“Easter shall be celebrated throughout the world on the first Sunday after the first full moon after the vernal equinox.”
An enduring myth maintains the above definition is still the correct method for determining the date of Easter, but it has several flaws which deserve to be pointed out.
First, to avoid an obvious ambiguity in the above definition, the entire world has always accepted the assumption that “vernal equinox” refers to the equinox of the northern hemisphere. The vernal equinox in the northern hemisphere occurs at the same time as the autumnal equinox in the southern hemisphere and vice-versa, but this inconvenience of terminology has no effect on when Easter occurs on either half of the globe. Easter is not celebrated after the full Moon that follows September 21 in Australia and never has been.
Still, the ancient definition is confusing. Even way back when, people folk started scratching their heads and saying, “Easter... uhh, what date is that?” Church officials were called upon to announce the date Easter would fall on for the coming year so that celebrations could be planned in advance. Soon, a single year's advance notice wasn't enough, and church officials were tasked with calculating the date of Easter for several years in advance. And as it turned out, determining the date Easter would fall on was hard! Determining the date was difficult for a variety of reasons, one of which arises from the fact that the vernal equinox date differs from year to year. In Rome, the equinox can usher in Spring as early as March 19, and as late as March 21. If you take time zones into account, then the Spring equinox in New Zealand (autumn there) can come as late as March 22. This four-day span for the equinox throws a real monkey wrench into the ancient definition of the Easter date.
Also unclear in the ancient definition of the Easter date is what is meant by the word “after.” Since the astronomical equinox happens at a precise instant in time on the day it arrives, and because the Moon technically goes full at a precise time as well, it is possible for the Moon to go full on the same day the vernal Equinox arrives yet still occur after it. And if both these celestial events occur on a Sunday, then Easter could be celebrated on that same day. Or, it could be celebrated a week later or even two weeks later, depending on one's interpretation of “after.”
Yet another difficulty with the ancient definition has to do with the phases of the Moon. As it so happens, the exact date the Moon goes full depends on which patch of the Earth one is standing on when looking up at the night sky, and this date can vary by as much as two calendar days. Because of these astronomical difficulties, it wasn't long before people in differing parts of the world were rightfully using the Church definition of the Easter date and celebrating Easter in different weeks. Worse, Easter was being celebrated in some places in some years on the exact same Sunday as the Jewish Passover. Christian church elders wanted Easter to be celebrated everywhere on the exact same date, and they did not like the idea of confusing the celebrations of Easter and Passover one little bit.
To shore up the problems with the ancient definition of the Easter date, the Catholic church held a Council in Nicaea in 325 AD. The council defined “vernal equinox” to mean the set date of March 21 for the purposes of calculating the Easter date, regardless of the the actual calendar date the true astronomical equinox arrives on. The council also decided upon an unambiguous method for determining the date Easter would fall on, which was immediately put into affect by papal bull. The method chosen was based on a system of tables worked out centuries earlier by Greek astronomers, tables which were already in use by the Jews for determining the date of their Passover celebration. The new definition of the Easter date, while substantively the same as the old, included a proviso to ensure it would never again be celebrated on the same date as Jewish Passover. The revised definition is as follows:
“Easter shall be celebrated throughout the world on the first Sunday following the Paschal Full Moon date, the Paschal Full Moon date occurring on or after the vernal equinox [March 21].”
And this is the correct definition of the Easter date in use by the Christian world today.
The Paschal Full Moon date is the full moon date that defines the Jewish Passover (the word “paschal” meaning, 'of or relating to Passover'). While the Paschal Full Moon date frequently coincides with an actual astronomical full Moon date (in Rome), often times it does not. The Paschal full Moon basically falls on whatever date the Church says it does, according to their system of tables. (By “Church,” I mean either Christian or Jewish, since they agree upon the Paschal Full Moon date, using the same tabular methods). Since the tabular system is directly tied to mathematics, equivalent methods for determining the date Easter falls on may come from equations and computer programs which model the mathematics on which the ancient tables were constructed. Church officials and astronomers need not be consulted.
Basis for the Range of Dates for Easter
The Moon goes through its various phases and may become “full” on any day of the month, including March 21. When the Moon is full on March 21, and when it agrees with the “Paschal Full Moon” date derived from the ancient Greek tables, then it has done so at the earliest possible calendar date consistent with the “on or after the vernal equinox [March 21].” Should a March 21 Paschal Full Moon occur on a Saturday, then Easter would be the very next day—Sunday, March 22, the earliest possible date Easter can be. This confluence of events is quite rare, making March 22 the most unlikely date for Easter to fall on. The last time Easter fell on March 22 was in 1918, and before that there was a March 22 Easter in 1761. Easter will not come as early as March 22 again until the year 2285.
Since the Moon takes 29 1/2 days to make one full orbit around the Earth, there are 30 distinct phases the Moon can be in on any given day. The last half-day of the Moon's orbit is rounded up to make 30 Moon phases. To lower round-off errors that result from turning fractions into integers, it is more precise to round the number of Moon phases Up while rounding the number of days over which they occur Down. Within the confines of this simple but fairly accurate model, the Moon can go full on any of 29 days that begin with the Mar 21st “equinox” to be declared the Paschal Full Moon for a given year. Those 29 days span the dates of March 21 through April 18, and the first Sunday thereafter (Easter) will come 1 to 7 days later, making the full range of possible Easter dates March 22 through April 25.
April 25 is the latest date Easter can fall on, and it is also rare, though not as rare as the earliest possible Easter date of March 22. Easter fell on April 25th in 1734, 1886, 1943, and will come again next in 2038.
Why a March 22 Easter date is rarer than an April 25 Easter date
Intuition might suggest the earliest possible Easter date (March 22) is just as rare an occurrence as the latest possible Easter date (April 25). But in this case, intuition would be wrong. The present day definition of the Easter date, in affect since 326 AD, makes an April 25 Easter approximately 50% more frequent an occurrence than a March 22 Easter. The answer to this intuition defying puzzle lies within the ancient Greek tables that define the date of the Paschal Full Moon.
When the ancient Greek astronomers constructed their tables to predict the dates of the full Moon, they used a “30 phases in 29 days” model, mentioned previously. The phase of the Moon on March 21 in an arbitrary year (known as an “epact”) can be any of 30 different values. No matter the phase of the Moon on March 21, it is known with certainty the Moon will be full on some day 0 to 28 days later. To make tables out of such knowledge, Greek mathematicians were required to construct an equation that takes each of 30 input epact values (Moon phases on March 21) and maps it to one of 29 possible output values (dates of the next full Moon). 30 epact values mapped to 29 possible Full Moon dates is not a one-to-one relationship, which means that one epact value would have to be mapped to an output Full Moon date already mapped to from some other epact value. The output value to be mapped to twice was chosen to be the latest date in the Lunar cycle, because careful observations over many years showed that round-off error approximations in the model were affected most in the final day of the Lunar cycle (because the final day in the Lunar cycle is the invisible New Moon). The equations that support the ancient Greek tables therefore slightly bias the Paschal Full Moon date towards the latest of possible days, which therefore makes the date of Easter slightly more likely to fall at the end of its possible range than at its beginning.
Why April 19 is the most common date for Easter
It's a quirky trivia fact that Easter falls most often on April 19. To the few who are even aware of this trivia fact, it is a profound mystery why April 19, is the “preferred” date, a date which is nowhere near the middle of the possible range of Easter dates nor all that close to either end of the spectrum. Here's why:
As stated in the previous section, the “30 moon phases in 29 days” model for the ancient Greek tables requires one of the Paschal Full Moon dates to be mapped to by two different epact values, and this date occurs at the end of the Lunar cycle. The end of the Lunar cycle is on the last possible date for the Paschal Full Moon, which is April 18.
Therefore, whenever the Paschal Full Moon falls on April 18 AND that day is also a Saturday, then Easter is on April 19 for one of two possible moon phases that occurred on March 21. Easter can also fall on April 19, in any year, because the Paschal Full Moon date happened to fall on any of the the other six dates, April 12 through April 17. This gives April 19 eight possible chances to be Easter in some years.
Looking at April 20 as a possible Easter date, there are sill two March 21 moon phase dates that can lead to an April 18 Paschal Full Moon date. In years when April 18 falls on a Friday, that gives two possible chances for Easter to be on April 20. But now there are only 5 other Paschal Full Moon dates that can lead to an April 20 Easter, those being April 13 through April 17. (April 12 is too early because even if the next Sunday is 7 days later that only takes you to April 19). This gives April 20 at most 7 possible chances to be Easter. And with each advancing date for Easter beyond April 19, one more possible date for the Paschal Full Moon drops off the calendar, lessening the chances for Easter to fall on that date.
For all dates in which Easter falls on April 18 or earlier, those dates never get the “bonus chance” of being Easter that comes from following the extra chance provided by the April 18th Paschal Full Moon date. Therefore, that makes April 19th the most commonly occurring Easter date.
Will the sequence of dates on on which Easter falls ever repeat itself in a predictable pattern?
Yes. Because the Easter date can be predicted by deterministic equations, then there must be a periodic repetition of the sequence of dates on which Easter falls. What may be surprising to many is that the length of this sequence spans 5.7 million years! The figure of 5.7 million years looks like a rounded figure, but it is not. It is the product of four different influencing factors:
30 – the number of different Moon phases that occur on March 21 of any given year, which directly influences the Paschal Full Moon date.
19 – the number of years in the Metonic Cycle. Observations show that the dates within a year that the phases of the moon fall on repeat themselves every 19 years. This cycle assumes that no adjustments to the Solar calendar or to the Lunar calendar are made to accommodate “Leap Days.”
2500 – the number of years it takes for the Lunar Calendar to repeat its sequence of dates, after taking into account that it must be tweaked to include 8 different Leap Days distributed through each sequence of 2500 years.
400 – the number of years it takes the Solar Calendar to repeat its sequence of dates, after taking into account that it must be tweaked to include a Leap Day every 4 years, except for the 3 out of 4 years that are evenly divisible by 100 but not by 400.
The 400-year cycle and the 2500-year cycle have a common factor of 100 years which cancels out. Therefore the cycle of years in which Easter dates repeat themselves is 30 x 19 x 2500 x 400 / 100 = 5.7 million years, exactly.
Easter falls on April 8th this year. Go forth and amaze friends and family with this fascinating collection of numerical Easter facts. Happy Easter everybody!
so I have attached the wiki article about the controversy of dating Easter.
) Jewish and the Last Supper was actually a Passover Seder. (Jesus and the disciples must have eaten wine, matzoh, bitter herbs, and lamb, if they could afford it. Gefilte fish and chicken soup were probably optional.) 
