Why was Pope Gregory's adjustment 10 days not 8 days?

[Moongazer]

I wrote an article recently on a rare calendric event of the Jewish Calendar. The article, "Myths and Maths of the Blessing of Sun" discusses some aspects of the Julian calendar and the Gregorian reform in the following terms:

Because the mean year-length of the Julian calendar is longer than a tropical year, each season begins on progressively earlier dates in the calendar. Their start dates regress in the calendar at the rate of 7.8 days per 1000 years.

This was a problem for the Church, because it caused Easter to drift ever closer toward summer. ... Easter is a northern spring festival and must occur shortly after the equinox. In 532, the council of Nicaea had irreversibly linked Easter not to the equinox itself, but to its presumed date, March 20. However by 1582 it was occurring on March 10. To correct this, Pope Gregory 13th reformed the calendar. As a one-off adjustment he dropped 10 days from that year, and ...

However, on checking the maths, I find that it doesn't quite add up. The above was based on an estimate of the mean tropical year as 365.24219 days.

Accordingly, the actual regression of the seasons in the Julian calendar is 7.81 days per 1000 years. The calculation is: 365.25 - 365.24219 = 0.00781 days/year. From 532 (the year of the council of Nicaea) to 1582 (the year of Pope Gregory's reform) is 1,050 years, and 1050 years x 0.00781 days/year = 8.2005 days. So why did Gregory drop 10 days rather than 8 days?

[Karl Palmen]

Dear Moongazer and Calendar People

The year of the council of Nicea was 325 rather than 532, so making the
reckoned drift equal to 9 or 10 days.

If the Gregorian calendar were extrapolated back, it'd match the Julian
calendar in the 200s century and be one day ahead in the 300s century
when the council of Nicea took place. So an adjustment of 9 days would
be needed for a perfect match at the Council of Nicea.

However the equinox normally occurs on the 20 March Gregorian rather
than 21 March and 11 day adjustment would have been needed to put it on
21 March normally.

10 days seems to be a compromise between the 9 and 11 days argued here.

Why the difference between the 9 and 11 days? This is because an
inaccurate equinox date of 21 March Julian Calendar (from Ptolemy) was
used at the Council of Nicea.

Karl

10(08(06 till noon

-----Original Message-----
From: East Carolina University Calendar discussion List
[mailto:CALNDR-L@...] On Behalf Of Moongazer
Sent: 01 May 2009 06:22
To: CALNDR-L@...
Subject: Why was Pope Gregory's adjustment 10 days not 8 days?

I wrote an article recently on a rare calendric event of the Jewish
Calendar.
The article, " http://geocities.com/calendar.luchot Myths and Maths of
the
Blessing of Sun " discusses some aspects of the Julian calendar and the
Gregorian reform in the following terms:

Because the mean year-length of the Julian calendar is longer than a
tropical year, each season begins on progressively earlier dates in the
calendar. Their start dates regress in the calendar at the rate of 7.8
days
per 1000 years.

This was a problem for the Church, because it caused Easter to drift
ever
closer toward summer. ... Easter is a northern spring festival and must
occur shortly after the equinox. In 532, the council of Nicaea had
irreversibly linked Easter not to the equinox itself, but to its
presumed
date, March 20. However by 1582 it was occurring on March 10. To correct
this, Pope Gregory 13th reformed the calendar. As a one-off adjustment
he
dropped 10 days from that year, and ...

However, on checking the maths, I find that it doesn't quite add up. The
above was based on an estimate of the mean
http://en.wikipedia.org/wiki/Tropical_year tropical year as 365.24219
days .

Accordingly, the actual regression of the seasons in the Julian calendar
is
7.81 days per 1000 years. The calculation is: 365.25 - 365.24219 =
0.00781
days/year. From 532 (the year of the council of Nicaea) to 1582 (the
year of
Pope Gregory's reform) is 1,050 years, and 1050 years x 0.00781
days/year =
8.2005 days. So why did Gregory drop 10 days rather than 8 days?

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[Mark J. Reed]

On Fri, May 1, 2009 at 4:21 AM, Palmen, KEV (Karl)
<karl.palmen@...> wrote:
> Why the difference between the 9 and 11 days? This is because an
> inaccurate equinox date of 21 March Julian Calendar (from Ptolemy) was
> used at the Council of Nicea.

This reminds me of a related issue...

I've read that Caesar/Sisogenes intended the equinoxes and solstices
to fall on the 25th of their respective months, but that seems to be
an inference from Christmas (and Lady Day, but that's not a separate
data point since it's derived from Christmas...).  It also implies
that someone was bad at math, since the calendar as implemented had
the spring equinox falling usually on March 23rd for the first century
of its inception...

So what was the goal of the calendar's design?  Given the Year of
Confusion, there was obviously a concerted effort to bring the Julian
calendar into alignment with the seasons, but what were the reference
points?

--
Mark J. Reed <markjreed@...>

[Irv Bromberg]

On 2009 May 1, at 01:22 , Moongazer wrote:

Because the mean year-length of the Julian calendar is longer than a
tropical year, each season begins on progressively earlier dates in the
calendar. Their start dates regress in the calendar at the rate of 7.8 days
per 1000 years.

Irv replies:  The "tropical year" is irrelevant, and too short, and in the wrong time units (atomic time).

The appropriate year to compare to is the mean northward equinoctial year in terms of mean solar time, which is presently about 365d 5h 49m 0s, but in 325 AD it was slightly shorter, about 365d 5h 48m 54s.  See <http://www.sym454.org/seasons/>.

Moongazer continued:  However, on checking the maths, I find that it doesn't quite add up. The

above was based on an estimate of the mean
http://en.wikipedia.org/wiki/Tropical_year tropical year as 365.24219 days .

Irv replies:  As above, that is too short = 365d 5h 48m 45s or 365+31/128 days, and neither the duration of the so-called "tropical year" nor the mean northward equinoctial year has been constant over the elapsed interval, so you can't accurately calculate the drift using constants as you have done.  Likewise the mean tropical year is not the appropriate year length to use in evaluation of Hebrew calendar drift, see <http://www.sym454.org/hebrew/drift.htm>.

Moongazer continued:  Accordingly, the actual regression of the seasons in the Julian calendar is

7.81 days per 1000 years. The calculation is: 365.25 - 365.24219 = 0.00781
days/year. From 532 (the year of the council of Nicaea) to 1582 (the year of
Pope Gregory's reform) is 1,050 years, and 1050 years x 0.00781 days/year =
8.2005 days. So why did Gregory drop 10 days rather than 8 days?

Irv replies:  You don't need to know the timing of any equinox or length of solar year.

The difference arises simply from the difference in the calendar mean years.

1258 years elapsed from 325 to 1583 AD.

Julian elapsed days = 1258 * (365+1/4)

Gregorian elapsed days = 1258 * (365+97/400)

The difference is 9+87/200 days = exactly 9d 10h 26m 24s.

Clearly 9 days was an insufficient correction, so they rounded it up to 10 days.

Also note that (mean northward equinox in 325 plus 1258*(365+97/400) minus mean northward equinox in 1583) is accurate to within only about 5h 23m 4s = <1/4 day error.

Compare with the more accurate fraction 365+71/293 (for that elapsed interval), the difference would be 9+387/586 days = 9d 15h 50m 59+113/293s or simply 9d 15h 51m.

Clearly the 10-day Gregorian reform adjustment was calendrically and astronomically appropriate.

On 2009 May 1, at 04:21 , Palmen, KEV (Karl) wrote:

an inaccurate equinox date of 21 March Julian Calendar (from Ptolemy) was used at the Council of Nicea.

Irv replies:  On the contrary, I have shown that March 21st was the correct date, assuming it was reckoned at Alexandria with the calendar day starting at sunset, as was the practice at the time, see <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to the first day that is in the spring season, which was without doubt March 21st in 325 AD in Alexandria.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear Irv and Calendar People

 

Irv replies:  On the contrary, I have shown that March 21st was the correct date, assuming it was reckoned at Alexandria with the calendar day starting at sunset, as was the practice at the time, see <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to the first day that is in the spring season, which was without doubt March 21st in 325 AD in Alexandria.

 

If this were so for the Julian Calendar, it would have been March 22 for the Proleptic Gregorian calendar. Four 400-year cycles later in 1925 the equinox was March 21^st 03:12 according to http://stellafane.org/misc/equinox.html . ,so giving a date of March 21^st    (12^th hour) in Alexandria. So a drift of over half a day would be necessary to make it 22^nd March in the proleptic Gregorian 1600 years earlier (I’ve chosen a multiple of 400 years to eliminate calendar jitter). This is sufficient argument to show that the date was wrong. Also I recall another E-mail that said that the date was got from Ptolemy who reckoned that the tropical year was 1/300 day short of 365 ¼ days.

 

Karl

 

10(08(07

 

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 01 May 2009 16:12
To: CALNDR-L@...
Subject: Re: Why was Pope Gregory's adjustment 10 days not 8 days?

 

On 2009 May 1, at 01:22 , Moongazer wrote:

Because the mean year-length of the Julian calendar is longer than a
tropical year, each season begins on progressively earlier dates in the
calendar. Their start dates regress in the calendar at the rate of 7.8 days
per 1000 years.

 

Irv replies:  The "tropical year" is irrelevant, and too short, and in the wrong time units (atomic time).

 

The appropriate year to compare to is the mean northward equinoctial year in terms of mean solar time, which is presently about 365d 5h 49m 0s, but in 325 AD it was slightly shorter, about 365d 5h 48m 54s.  See <http://www.sym454.org/seasons/>.

 

 

Moongazer continued:  However, on checking the maths, I find that it doesn't quite add up. The

above was based on an estimate of the mean
http://en.wikipedia.org/wiki/Tropical_year tropical year as 365.24219 days .

 

Irv replies:  As above, that is too short = 365d 5h 48m 45s or 365+31/128 days, and neither the duration of the so-called "tropical year" nor the mean northward equinoctial year has been constant over the elapsed interval, so you can't accurately calculate the drift using constants as you have done.  Likewise the mean tropical year is not the appropriate year length to use in evaluation of Hebrew calendar drift, see <http://www.sym454.org/hebrew/drift.htm>.

 

 

Moongazer continued:  Accordingly, the actual regression of the seasons in the Julian calendar is

7.81 days per 1000 years. The calculation is: 365.25 - 365.24219 = 0.00781
days/year. From 532 (the year of the council of Nicaea) to 1582 (the year of
Pope Gregory's reform) is 1,050 years, and 1050 years x 0.00781 days/year =
8.2005 days. So why did Gregory drop 10 days rather than 8 days?

 

Irv replies:  You don't need to know the timing of any equinox or length of solar year.

The difference arises simply from the difference in the calendar mean years.

1258 years elapsed from 325 to 1583 AD.

 

Julian elapsed days = 1258 * (365+1/4)

Gregorian elapsed days = 1258 * (365+97/400)

The difference is 9+87/200 days = exactly 9d 10h 26m 24s.

Clearly 9 days was an insufficient correction, so they rounded it up to 10 days.

 

Also note that (mean northward equinox in 325 plus 1258*(365+97/400) minus mean northward equinox in 1583) is accurate to within only about 5h 23m 4s = <1/4 day error.

 

Compare with the more accurate fraction 365+71/293 (for that elapsed interval), the difference would be 9+387/586 days = 9d 15h 50m 59+113/293s or simply 9d 15h 51m.

 

Clearly the 10-day Gregorian reform adjustment was calendrically and astronomically appropriate.

 

 

On 2009 May 1, at 04:21 , Palmen, KEV (Karl) wrote:

an inaccurate equinox date of 21 March Julian Calendar (from Ptolemy) was used at the Council of Nicea.

 

Irv replies:  On the contrary, I have shown that March 21st was the correct date, assuming it was reckoned at Alexandria with the calendar day starting at sunset, as was the practice at the time, see <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to the first day that is in the spring season, which was without doubt March 21st in 325 AD in Alexandria.

 

-- Irv Bromberg, Toronto, Canada

 

<http://www.sym454.org/>

 

---

Scanned by iCritical.

[Brillig]

Dear Karl and Calendar People,

I did a google search for ptolemy year length, and the first hit was
the contents of a book called "Wrong for the right reasons
 By Jed Z. Buchwald, Allan Franklin". According to the text, Ptolemy
used equinox observations of his own and of Hipparchus over a 285 year
span and found that the times of the events were 19/20 day short. Thus
for 1 day, we get 20/19 * 285 years = 300 years.

Victor

On Fri, May 1, 2009 at 10:41 AM, Palmen, KEV (Karl)
<karl.palmen@...> wrote:

> date was wrong. Also I recall another E-mail that said that the date was got
> from Ptolemy who reckoned that the tropical year was 1/300 day short of 365
> ¼ days.

[Karl Palmen]

Some parts of this message have been removed. Learn more about Nabble's security policy.

Dear Irv and Calendar People

 

Reading Irv’s article, I see what was meant my equinox day is actually the day after the equinox (days beginning sunset in Alexandria). My argument applies to the actual day of the equinox not the day after.

 

Nevertheless,  the equinox time (quoted by Irv) was about noon 21 March 325  Proleptic Gregorian in Alexandrian time (about 14:00 UT). 1600 years later it’s 03:00, so giving a drift of 11 hours, which I think is excessive for 1600 years. However, even an equinox time as early  as 03:00 UT would make no difference to Irv’s argument. The day after the equinox (which was called the equinox day) would still be 22 March proleptic Gregorian and so 21 March Julian.

 

Is there any independent confirmation of Irv’s argument?

Did they reckon equinox as Irv asserts could the direction of sunset be measured sufficiently accurately?

 

Karl

 

10(08(07

 

From: Palmen, KEV (Karl)
Sent: 01 May 2009 16:41
To: 'East Carolina University Calendar discussion List'
Subject: RE: Why was Pope Gregory's adjustment 10 days not 8 days?

 

Dear Irv and Calendar People

 

Irv replies:  On the contrary, I have shown that March 21st was the correct date, assuming it was reckoned at Alexandria with the calendar day starting at sunset, as was the practice at the time, see <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to the first day that is in the spring season, which was without doubt March 21st in 325 AD in Alexandria.

 

If this were so for the Julian Calendar, it would have been March 22 for the Proleptic Gregorian calendar. Four 400-year cycles later in 1925 the equinox was March 21^st 03:12 according to http://stellafane.org/misc/equinox.html . ,so giving a date of March 21^st    (12^th hour) in Alexandria. So a drift of over half a day would be necessary to make it 22^nd March in the proleptic Gregorian 1600 years earlier (I’ve chosen a multiple of 400 years to eliminate calendar jitter). This is sufficient argument to show that the date was wrong. Also I recall another E-mail that said that the date was got from Ptolemy who reckoned that the tropical year was 1/300 day short of 365 ¼ days.

 

Karl

 

10(08(07

 

 

From: East Carolina University Calendar discussion List [mailto:CALNDR-L@...] On Behalf Of Irv Bromberg
Sent: 01 May 2009 16:12
To: CALNDR-L@...
Subject: Re: Why was Pope Gregory's adjustment 10 days not 8 days?

 

On 2009 May 1, at 01:22 , Moongazer wrote:

Because the mean year-length of the Julian calendar is longer than a
tropical year, each season begins on progressively earlier dates in the
calendar. Their start dates regress in the calendar at the rate of 7.8 days
per 1000 years.

 

Irv replies:  The "tropical year" is irrelevant, and too short, and in the wrong time units (atomic time).

 

The appropriate year to compare to is the mean northward equinoctial year in terms of mean solar time, which is presently about 365d 5h 49m 0s, but in 325 AD it was slightly shorter, about 365d 5h 48m 54s.  See <http://www.sym454.org/seasons/>.

 

 

Moongazer continued:  However, on checking the maths, I find that it doesn't quite add up. The

above was based on an estimate of the mean
http://en.wikipedia.org/wiki/Tropical_year tropical year as 365.24219 days .

 

Irv replies:  As above, that is too short = 365d 5h 48m 45s or 365+31/128 days, and neither the duration of the so-called "tropical year" nor the mean northward equinoctial year has been constant over the elapsed interval, so you can't accurately calculate the drift using constants as you have done.  Likewise the mean tropical year is not the appropriate year length to use in evaluation of Hebrew calendar drift, see <http://www.sym454.org/hebrew/drift.htm>.

 

 

Moongazer continued:  Accordingly, the actual regression of the seasons in the Julian calendar is

7.81 days per 1000 years. The calculation is: 365.25 - 365.24219 = 0.00781
days/year. From 532 (the year of the council of Nicaea) to 1582 (the year of
Pope Gregory's reform) is 1,050 years, and 1050 years x 0.00781 days/year =
8.2005 days. So why did Gregory drop 10 days rather than 8 days?

 

Irv replies:  You don't need to know the timing of any equinox or length of solar year.

The difference arises simply from the difference in the calendar mean years.

1258 years elapsed from 325 to 1583 AD.

 

Julian elapsed days = 1258 * (365+1/4)

Gregorian elapsed days = 1258 * (365+97/400)

The difference is 9+87/200 days = exactly 9d 10h 26m 24s.

Clearly 9 days was an insufficient correction, so they rounded it up to 10 days.

 

Also note that (mean northward equinox in 325 plus 1258*(365+97/400) minus mean northward equinox in 1583) is accurate to within only about 5h 23m 4s = <1/4 day error.

 

Compare with the more accurate fraction 365+71/293 (for that elapsed interval), the difference would be 9+387/586 days = 9d 15h 50m 59+113/293s or simply 9d 15h 51m.

 

Clearly the 10-day Gregorian reform adjustment was calendrically and astronomically appropriate.

 

 

On 2009 May 1, at 04:21 , Palmen, KEV (Karl) wrote:

an inaccurate equinox date of 21 March Julian Calendar (from Ptolemy) was used at the Council of Nicea.

 

Irv replies:  On the contrary, I have shown that March 21st was the correct date, assuming it was reckoned at Alexandria with the calendar day starting at sunset, as was the practice at the time, see <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to the first day that is in the spring season, which was without doubt March 21st in 325 AD in Alexandria.

 

-- Irv Bromberg, Toronto, Canada

 

<http://www.sym454.org/>

 

---

Scanned by iCritical.

[Irv Bromberg]

On 2009 May 1, at 11:41 , Palmen, KEV (Karl) wrote:

Irv wrote:  On the contrary, I have shown that March 21st was the correct date, assuming it was reckoned at Alexandria with the calendar day starting at sunset, as was the practice at the time, see <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to the first day that is in the spring season, which was without doubt March 21st in 325 AD in Alexandria.

 

If this were so for the Julian Calendar, it would have been March 22 for the Proleptic Gregorian calendar. Four 400-year cycles later in 1925 the equinox was March 21st 03:12 according to http://stellafane.org/misc/equinox.html . ,so giving a date of March 21st    (12th hour) in Alexandria. So a drift of over half a day would be necessary to make it 22nd March in the proleptic Gregorian 1600 years earlier (I’ve chosen a multiple of 400 years to eliminate calendar jitter). This is sufficient argument to show that the date was wrong. Also I recall another E-mail that said that the date was got from Ptolemy who reckoned that the tropical year was 1/300 day short of 365 ¼ days.

Irv replies:  I've never heard of there ever having been any intention of making the equinox land on March 21st in 325 AD on the proleptic Gregorian calendar.  This seems implausible to me.

Notwithstanding Karl's argument, the actual northward equinox reckoned for the meridian of Alexandria was just before noon on March 21st on the proleptic Gregorian calendar, so no problem there.

Julian March 21st would have started at the sunset about 6+1/4 hours later, making Julian March 21st the ecclesiastical first day of spring.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/mar21/>

<http://www.sym454.org/seasons/>

<http://www.sym454.org/leap/>

[Brillig]

Dear Calendar People,

Just out of curiosity, using the information at
http://en.wikipedia.org/wiki/Hipparchus I decided to calculate the
mean year length between Hipparchus' observation in 146 BC at dawn and
this year's equinox as tabulated in wikipedia. I get a mean year
length of 365.242415 days (assuming dawn at Rhodes is 7.86 UT,
calculated from 28 degrees east). If I use instead the one hour before
noon time also mentioned in the article, I get instead 365.242318
days.

Victor

On Fri, May 1, 2009 at 11:26 AM, Irv Bromberg <irv.bromberg@...> wrote:

> On 2009 May 1, at 11:41 , Palmen, KEV (Karl) wrote:
>
> Irv wrote:  On the contrary, I have shown that March 21st was the correct
> date, assuming it was reckoned at Alexandria with the calendar day starting
> at sunset, as was the practice at the time, see
> <http://www.sym454.org/mar21/>.  Also, the ecclesiastical equinox refers to
> the first day that is in the spring season, which was without doubt March
> 21st in 325 AD in Alexandria.
>
> If this were so for the Julian Calendar, it would have been March 22 for the
> Proleptic Gregorian calendar. Four 400-year cycles later in 1925 the equinox
> was March 21st 03:12 according to http://stellafane.org/misc/equinox.html .
> ,so giving a date of March 21st    (12th hour) in Alexandria. So a drift of
> over half a day would be necessary to make it 22nd March in the proleptic
> Gregorian 1600 years earlier (I’ve chosen a multiple of 400 years to
> eliminate calendar jitter). This is sufficient argument to show that the
> date was wrong. Also I recall another E-mail that said that the date was got
> from Ptolemy who reckoned that the tropical year was 1/300 day short of 365
> ¼ days.
>
> Irv replies:  I've never heard of there ever having been any intention of
> making the equinox land on March 21st in 325 AD on the proleptic Gregorian
> calendar.  This seems implausible to me.
> Notwithstanding Karl's argument, the actual northward equinox reckoned for
> the meridian of Alexandria was just before noon on March 21st on the
> proleptic Gregorian calendar, so no problem there.
> Julian March 21st would have started at the sunset about 6+1/4 hours later,
> making Julian March 21st the ecclesiastical first day of spring.
>
> -- Irv Bromberg, Toronto, Canada
> <http://www.sym454.org/mar21/>
> <http://www.sym454.org/seasons/>
> <http://www.sym454.org/leap/>

[RDoug]

Mark J. Reed wrote:

So what was the goal of the calendar's design?  Given the Year of
Confusion, there was obviously a concerted effort to bring the Julian
calendar into alignment with the seasons, but what were the reference
points?

The Julian Calendar restored the Spring Equinox to somewhere around the traditional date in late March.  It is likely that the actual set-point was the New Moon (conjunction) occurring just before Roman daybreak on Julian Day 1704988.  We know that day as January 2nd of 45 BC (also called -44).  But if that year (the first year on the Julian Calendar) were not considered as a Leap Year (after all, why put in a correction already in the very first year?), then that day would be January 1st, the First Day of the Julian Calendar.  Fitting in a way that the Solar Calendar should have a Lunar set-point as its origin.

The considerations for the Gregorian Calendar were that the Equinox (as observed at Rome, for a day beginning at Midnight) should never occur LATER than March 21.  In fact, due to the extreme jitter of that Calendar, there were Equinoxes occurring on the 21st, the 20th, and the 19th already in the first century from its introduction.  But never on the 22nd, for several millennia at least.

-- Robert H. Douglass

[Mark J. Reed]

On Fri, May 1, 2009 at 1:34 PM, RDoug <rdouglass001@...> wrote:
> The Julian Calendar restored the Spring Equinox to somewhere around the
> traditional date in late March.

That's a bit vague. :) You're saying the "spring equinox in late
March" was already a tradition on the old Roman calendar (even if
political manipulations of intercalation had prevented it from being
reality for a long time)?

> It is likely that the actual set-point was the New Moon (conjunction) occurring
> just before Roman daybreak on Julian Day 1704988.

Hm.  I thought archaeological discoveries had pretty much established
the observed new year sequence (under the erroneous interpretation of
the rules) in the early years of the Julian reform and thereby placed
the first observed January 1 of the Julian era (45 BCE) on the
*previous* proleptic Julian date: BCE 46 December 31 =JD 1704986.

> The considerations for the Gregorian Calendar were that the Equinox (as
> observed at Rome, for a day beginning at Midnight) should never occur LATER
> than March 21.

Ah, an upper bound.  That makes sense.

--
Mark J. Reed <markjreed@...>

[RDoug]

"Mark J. Reed" wrote:
>You're saying the "spring equinox in late
> March" was already a tradition on the old Roman calendar?

Yes, I think March 25th was the traditional date.  No hard evidence to support this, however.

> Hm.  I thought archaeological discoveries had pretty much established...
> the *previous* proleptic Julian date: BCE 46 December 31 =JD 1704986.

I don't think the evidence is persuasive.  There should be plenty of room to shift things by a day or so in either direction.  It is interesting that there really was a New Moon within a day or so of the correct time.  I have seen it suggested that this occurrence would have helped the adoption of the Julian Calendar by making it look "more natural" to folks in the Senate and other positions of power.  Again, an idea of interest but no hard evidence in support.

-- Robert H. Douglass

[Irv Bromberg]

On 2009 May 1, at 13:19 , Victor Engel wrote:

Just out of curiosity, using the information at
http://en.wikipedia.org/wiki/Hipparchus I decided to calculate the
mean year length between Hipparchus' observation in 146 BC at dawn and
this year's equinox as tabulated in wikipedia. I get a mean year length of 365.242415 days

Irv says:  .242415 = 5h 49m 4.7s, which is almost 5 seconds too long today, and >14 seconds too long in 146 BC.

(assuming dawn at Rhodes is 7.86 UT,
calculated from 28 degrees east). If I use instead the one hour before
noon time also mentioned in the article, I get instead 365.242318 days.

Irv says:  .242318 = 5h 48m 56.3s, similar to my favorite 293-year cycle.

Those were observations of the same equinox, differing by only 5 hours yet taken at nearly the same longitude.

Even though >2000 years elapsed and therefore tended to average away that 5h difference, it still makes a highly significant difference.

Calculating the mean northward equinoctial year length based on only two points is not very reliable.

We have no idea how uncertain is Hipparchus' observation.  They had no accurate clocks at the time.  And we have no extant original record of his observation, only Ptolemy's record of it.

The mean year length changed from about 365d 5h 48m 50.6s in 146 BC to about 365d 5h 49m today.

The actual equinox varies ±15 minutes from the mean, mainly due to Moon causing Earth to careen around the Earth-Moon center of gravity, but also with non-negligible contributions due to Jupiter and Venus.

If Hipparchus' observed equinox was exceptionally early and today's is exceptionally late, then the calculated year length will be too long, even though >2000 years elapsed.

If Hipparchus' was exceptionally late and today's is exceptionally early, then the calculated year length will be too short.

This year the northward equinox was almost 2m 10s earlier than the mean equinox.

Assuming that the year 146 BC was reckoned without a year zero then in that year the actual equinox was about 4m 59s after the mean equinox.

The JD of the actual equinox was 1668179.04619997, the mean equinox was 1668179.04273973, is that the correct equinox?

Thus the distances between the ancient equinox and this year's has been stretched by about 7m 9s compared to the mean difference, if I got the correct equinox.

On the other hand, with 2154 elapsed years, that stretch makes only about 2/10 second difference.

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/>

[Brillig]

Dear Irv and Calendar People,

On Fri, May 1, 2009 at 6:42 PM, Irv Bromberg <irv.bromberg@...> wrote:

> On 2009 May 1, at 13:19 , Victor Engel wrote:
>
> Just out of curiosity, using the information at
> http://en.wikipedia.org/wiki/Hipparchus I decided to calculate the
> mean year length between Hipparchus' observation in 146 BC at dawn and
> this year's equinox as tabulated in wikipedia. I get a mean year length of
> 365.242415 days
>
> Irv says:  .242415 = 5h 49m 4.7s, which is almost 5 seconds too long today,
> and >14 seconds too long in 146 BC.

Based upon what? Incidentally, based on my reading that Ptolemy used a
285 year span of time (between his an Hipparchus' observations), I
decided to see how much variation there might be in mean year lengths
over 285 years. I took a block of just over 1000 years and compared
all possible contiguous 285 year means, and they ranged in length by
up to 8 seconds from each other. Was this due to changes in delta-T? I
didn't check, but I don't think so. I think it's simply due to the
variability in year length due to planetary dynamics.

> (assuming dawn at Rhodes is 7.86 UT,
> calculated from 28 degrees east). If I use instead the one hour before
> noon time also mentioned in the article, I get instead 365.242318 days.
>
> Irv says:  .242318 = 5h 48m 56.3s, similar to my favorite 293-year cycle.

My favorite, too.

> Those were observations of the same equinox, differing by only 5 hours yet
> taken at nearly the same longitude.

In fact, I assumed the same longitude in my calculations.

> Even though >2000 years elapsed and therefore tended to average away that 5h
> difference, it still makes a highly significant difference.

Right. That was the main point of my post.

> Calculating the mean northward equinoctial year length based on only two
> points is not very reliable.

True. Those were the only two ancient observations immediately at hand
to me. If you know of more, please share, especially if you consider
them reliable and accurate.

> We have no idea how uncertain is Hipparchus' observation.  They had no
> accurate clocks at the time.  And we have no extant original record of his
> observation, only Ptolemy's record of it.
> The mean year length changed from about 365d 5h 48m 50.6s in 146 BC to about
> 365d 5h 49m today.
> The actual equinox varies ±15 minutes from the mean, mainly due to Moon
> causing Earth to careen around the Earth-Moon center of gravity, but also
> with non-negligible contributions due to Jupiter and Venus.

Yeah. If I remember correctly, 1999 - 2000 was almost exactly 365.25
days, close to the high end of the extremes.

> If Hipparchus' observed equinox was exceptionally early and today's is
> exceptionally late, then the calculated year length will be too long, even
> though >2000 years elapsed.

If Hipparchus and Ptolemy both made observations at the same lunar
phase, then the chaotic nature of the data would be greatly reduced. I
think Ptolemy at least, paid attention to the lunar phase, but I don't
know the details of that.

> If Hipparchus' was exceptionally late and today's is exceptionally early,
> then the calculated year length will be too short.
> This year the northward equinox was almost 2m 10s earlier than the mean
> equinox.
> Assuming that the year 146 BC was reckoned without a year zero then in that

That's what I was assuming, too.

> year the actual equinox was about 4m 59s after the mean equinox.
> The JD of the actual equinox was 1668179.04619997, the mean equinox
> was 1668179.04273973, is that the correct equinox?
> Thus the distances between the ancient equinox and this year's has been
> stretched by about 7m 9s compared to the mean difference, if I got the
> correct equinox.
> On the other hand, with 2154 elapsed years, that stretch makes only about
> 2/10 second difference.

Yep.

> -- Irv Bromberg, Toronto, Canada
> <http://www.sym454.org/>

[Moongazer]

Wow! I find it amazing and exciting how a question based on nothing more than a dyslexic transcription error has engendered so much discussion about other matters surrounding the subject of my original question. Thanks, Karl for pointing out my error. I can now correct it in my article.

Your answer also explains so much more. (Like why the maths hadn't been a problem for me earlier. In my previous writings on this subject, from which my recent article was drawn, I had written the date of Nicaea correctly.)

It also clarifies why March 21 is mentioned in so many sources in connection with Nicaea. Until now, I had thought that the authors were simply a day out in their assumptions as to when the equinox occurred in 325, but your little bit of history about the council using the Ptolemyean date, makes it clear that the authors were not mistaken, they were simply reporting the date that had been selected by the council.

However, your characterization of the 10 days as a compromise doesn't seem right to me. Side-stepping the ensuing debate here as to whether March 20 or 21 was the correct date of the equinox in 325, I think that either way, Gregory would have had to adjust the calendar by 10 days to restore things to the way they were at the time of the first council of Nicaea. For as far as Gregory was concerned, his hands were tied. Constantine's council of 325 had subsequently come to be regarded as the first ecumenical council of the Christian church and its decisions were held to be unalterable. If not for that, Gregory would have had a much simpler adjustment open to him. He could simply have severed the nexus with a specific calendar date altogether and linked Easter to the actual equinox instead. For the council was in error in linking Easter to any Julian date on the assumption that that was the permanent date of the equinox. But Gregory could not correct that error, and if so, neither could he correct their choice of date, even if it were found to be wrong.

So, in 1582, to restore things to the way they were 1257 years earlier in 325, we have: 365.25 - 365.2419 = 0.0081 days per year, and 1257 * 0.0081 = 10.1817, which, rounded, is 10 days. This is the way we would work it out now using our present estimation of the tropical year length. In Gregory's time what happened historically is that in 1576, Luigi Lilio Ghiraldi, better known as Aloysius Lilius, a physician of Naples, while working on a scheme for a new calendar, found that the equinox of that year occurred on March 11, 10 days earlier than March 21. Aloysius Lilius died before he could inform the authorities of his work, but his brother Antonio submitted it to Pope Gregory, who appointed commissioners to check Aloysius's work and to frame the new calendar's rules, and they completed their work some time before early 1581. (This information comes from Rev S. B. Burnaby's Elements of the Jewish and Mohammadan Calendars ... and Explanatory Notes on the Julian and Gregorian Calendars, London, 1901.)

It seems clear that to this day the Church is stuck with "March 21 as their assumed ecclesiastical equinox".

In the first lunation whose 14th day occurs on or after that date, that 14th day is called the Paschal full moon and Easter is the Sunday following that "full moon".
That lunation is not an astronomical lunation but an artificially computed calendric-lunation, whose method of calculation was not specified at Nicaea but was entrusted (at first) to the Bishop of Alexandria. Although there have been changes to both the process and the computus itself, the Church's official definition of the date of the March equinox for the purpose of fixing the date of Easter remains March 21.

[Moongazer]

Sorry, slight correction to my last post: The calculation should have read:

So, in 1582, to restore things to the way they were 1257 years earlier in 325, we have: 365.25 - 365.2419 = 0.0081 days per year, and 1257 * 0.0081 = 10.1817, which, rounded, is 10 days.

(I have edited the post accordingly.)

Same result though.

[Mark J. Reed]

> So, in 1582, to restore things to the way they were 1257 years earlier in
> 325, we have: 365.25 - 365.2419

Why do you keep using that value?  As it says on the WIkipedia page
you got it from, the length of the "tropical year" depends entirely on
when in the year you start measuring from.   The 365.2419 value is a
mean value averaged across the entire year - a sort of average of
averages.  It's notably smaller than the northward equinoctial year,
which is what matters when we're talking about the date of the
northward (March) equinox - the "vernal equinox" row of the table on
the Wikipedia page.  The northward equinoctial year is only about 11
minutes smaller than the Julian mean - it rounds to 365.2424.   Sure,
1257 years at 0.0076 days/year is 9.6 days, which still rounds to
10... but if you're going to quote the figure to four decimals, I
think you should use the one that matters to the calculation at hand.

In any case.  A 9-day correction would have put the equinox mostly on
March 19th and occasionally on the 20th, but never on the desired date
of the 21st.  An 11-day correction would have put the equinox mostly
on target, but occasionally it would fall on the 22nd, which Robert
pointed out was unacceptable. So the 10-day correction had the desired
result: it fixed the latest date of the equinox on the
Nicaean/Ptolemaiac date of March 21st.

--
Mark J. Reed <markjreed@...>

[Moongazer]

Mark J. Reed wrote:

> So, in 1582, to restore things to the way they were 1257 years earlier in
> 325, we have: 365.25 - 365.2419

Why do you keep using that value?
...
An 11-day correction would have put the equinox mostly on target, but occasionally it would fall on the 22nd, which Robert pointed out was unacceptable. So the 10-day correction had the desired result: So the 10-day correction had the desired result: it fixed the latest date of the equinox on the Nicaean/Ptolemaiac date of March 21st.
---------
> The considerations for the Gregorian Calendar were that the Equinox (as
> observed at Rome, for a day beginning at Midnight) should never occur LATER
> than March 21.

Ah, an upper bound.  That makes sense.

Just did a browser refresh and found some posts that did not show up before. Finally the penny drops. (So THAT was the compromise that Karl was hinting at.) Thanks for the explanation.

[Tom Peters-6]

Op 2-mei-2009, om 14:24 heeft Moongazer het volgende geschreven:

> Mark J. Reed wrote:
>>
>>> So, in 1582, to restore things to the way they were 1257 years  
>>> earlier in
>>> 325, we have: 365.25 - 365.2419
>>
>> Why do you keep using that value?
>> ...
>> An 11-day correction would have put the equinox mostly on target, but
>> occasionally it would fall on the 22nd, which Robert pointed out was
>> unacceptable. So the 10-day correction had the desired result: So the
>> 10-day correction had the desired result: it fixed the latest date  
>> of the
>> equinox on the Nicaean/Ptolemaiac date of March 21st.
>> ---------
>>> The considerations for the Gregorian Calendar were that the  
>>> Equinox (as
>>> observed at Rome, for a day beginning at Midnight) should never  
>>> occur
>>> LATER
>>> than March 21.
>>
>> Ah, an upper bound.  That makes sense.
>>
>>
>
> Just did a browser refresh and found some posts that did not show  
> up before.
> Finally the penny drops. (So THAT was the compromise that Karl was  
> hinting
> at.) Thanks for the explanation.

I fall late into the discussion, and most appears to have been said.  
I think the main points are:

- The Gregorian reformers intended to restore the situation as they  
thought it was at the Nicaean council.

- At that council, no actual decisions were made on how to compute  
the date of Easter; indeed, no people with adequate astronomical  
knowledge appear to have been present.  The fight on the computus  
(see the Wikipedia article http://en.wikipedia.org/wiki/Computus)  
raged in the following centuries, and were essentially based on  
Ptolemy's values, which were based on his own observations before 150  
AD, Hipparchos's after 150 BC, and much earlier observations.  In any  
case, the 325 AD date of the Nicaean council is hardly relevant as a  
zero point, except as far as Lilius and Clavius attempted to reckon  
back to that year.

- The proper solar year length is that of the mean period between  
Northward aequinoxes, in mean solar days; not some mean mean tropical  
year measured in SI days.  So 365.2424.. days , not 365.24219...  
days.  The Gregorian year length of 365.2425 is not so bad, but its 3  
nested cycles (4, 100, 400) cause unnecessary jitter of over 2 days.
   Incidentally, the value may have been chosen as a truncation of  
365;14:33:... which were the sexagesimal digits that the three main  
astronomical tables of the time had in common: Clavius c.s. wouldn't  
choose a particular cosmological model, and went for the largest  
common denominator.  It also could conveniently be implemented as a  
modification of the traditional 4-year leap cycle at easily to  
remember century years (unlike the more accurate Khayyam year of 365
+8/33 days, requiring a postponement of the leap day by a year in odd  
years); and the 400-year cycle contains an integer number of weeks,  
keeping the so-called solar cycle within bounds.

- The main reason for the Gregorian reform was to correct the  
computus (computation of the date of Easter), and its associated  
lunar calendar, which was conspicuously wrong.  The seasonal error  
appears to have been less relevant, and so was the 21 March date -  
implicit in the computus, rather than as a canonical date of its own  
for the aequinox.  Specifically, Easter should fall no earlier than  
22 March and no later than 25 April, as it had been in the original  
computus.  A 10-day re-adjustment of the solar calendar was necessary  
to make that happen in concord with the new epact tables.  So if you  
reckon that they should have used 11 days to make the aequinox fall  
on 21 March more of the time (- for which meridian?  Alexandria?  
Jerusalem?  Rome? - that was never specified), then that is  
irrelevant because that was not their main goal.

   As an aside, I like to address the story of the "Blue Moon" here  
(http://en.wikipedia.org/wiki/Blue_moon).  In the 16th century people  
would have to fast during Lent because of necessity, and they were  
really looking forward to springtime, heralded by the Spring Full  
Moon, after which Easter Sunday would follow and they could eat again  
(bird eggs, for instance).  So they were very pissed off when they  
could clearly see a Full Moon in the sky and they could see that  
birds were nesting, and the clergy would tell them that they had to  
fast for another month because their obsolete computus told them so.


For almost everything you never wanted to know about the Gregorian  
reform, see:
        http://articles.adsabs.harvard.edu//full/book/grc../ 
1983//0000001,001.html
which I think deals too much with the correction of the solar  
calendar, and too little with that of the lunar calendar and the  
computus.

--
Tom Peters

[Deckers, Michael-2]

   A small amendment concerning the Nicene council:
   Tom Peters wrote on 2009-05-04:  

>  - At that council, no actual decisions were made on how to compute  
>  the date of Easter; indeed, no people with adequate astronomical  
>  knowledge appear to have been present.

   Dionysisus Exiguus tells us that Athanasius of Alexandria, as
   well as his successors Theophilus and Cyrillus have been present
   at the council. All three have been involved with Easter tables,
   using the Metonic cycle. I am afraid that is probably as much
   astronomy as one could expect at that time from a cleric.

   Michael Deckers