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

by Karl Palmen :: Rate this Message:

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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 Bromberg, Toronto, Canada

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

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