>Actually, it depends on whether or not the March equinox is the most
>important marking-point of the year.  I do not consider it so, and neither
>should most people unless the Easter setting date is paramount as it was
>with Gregory.

As I explained in my messages of 03Oct and 04Oct [both titled "Re: Misnamed
months, (Amos asked)"] and in my message of  10Oct [titled "Thirty-three year
calendars"], there is a NATURAL ASTRONOMICAL reason why the vernal equinox
year is preferable to the other three natural regulating points. See herein,
below, for a numerical demonstration, in Chris Carrier's terms, of the natural
superiority of the spring eqinox as a calendar regulator.

Chris Carrier then attempted to estimate the four year-lengths by starting
with the relevant instants in ephemeris time for 200 A.D., as follows:

>AD  200 March equinox: 21d 11h 06m 13s                 WRONG!
>        June solstice: 23d  8h 01m 41s                 WRONG!
>        Sept. equinox: 23d 22h 21m 28s                 WRONG!
>        Dec. solstice: 21d 17h 03m 51s                 WRONG!
>

Unfortunately Chris has used the times (as given in Meeus 1983) for
201 A.D.! The times for 200 A.D. are actually as follows (in Meeus 1983):

AD  200 March equinox: 21d  5h 18m 41s
        June solstice: 23d  2h 26m 06s
        Sept. equinox: 23d 16h 43m 06s
        Dec. solstice: 21d 11h 20m 09s

Chris continued with the ephemeris times expected for 3000 A.D. and the
discrepancies from whole numbers of Gregorian Years based on his WRONG
200 A.D. times:
 
>AD 3000 March equinox: 20d 17h 33m 33s,  62980s earlier after 2800 years
>        June solstice: 20d 16h 58m 49s, 226972s earlier after 2800 years
>        Sept. equinox: 22d 14h 54m 14s, 113174s earlier after 2800 years
>        Dec. solstice: 22d  5h 23m 17s,  44366s LATER   after 2800 years
>

The above, 3000 A.D. table, thus needs to be ammended as follows:

AD 3000 March equinox: 20d 17h 33m 33s,  42308s earlier after 2800 years
        June solstice: 20d 16h 58m 49s, 206837s earlier after 2800 years
        Sept. equinox: 22d 14h 54m 14s,  92932s earlier after 2800 years
        Dec. solstice: 22d  5h 23m 12s,  64983s LATER   after 2800 years

Then continuing with Chris' procedure:

>So, using 365/05:49:12 as a Gregorian year, and dividing these variances
>from it by 2800, we get a year length of:

ACTUALLY

365d 5h 48m 56.89s for the March equinox.   [or 365.242325 ephemeris days]
365d 5h 47m 58.13s for the June solstice.   [or 365.241645 ephemeris days]
365d 5h 48m 38.81s for the September equinox.  [365.242116 ephemeris days]
365d 5h 49m 35.21s for the December solstice.  [365.242769 ephemeris days]

365d 5h 48m 47.26s for the average of four. [or 365.242214 epehmeris days]

NOT as Chris gives:

>365d 5h 48m 49.54s measured against the March equinox.              WRONG!
>365d 5h 47m 50.94s measured against the June solstice.              WRONG!
>365d 5h 48m 31.58s measured against the September equinox.          WRONG!
>365d 5h 49m 27.85s measured against the December solstice.          WRONG!
>
>365d 5h 48m 39.97s measured against the average of all four of them.WRONG!

So in fact, the Exigian and the Soviet year are less accurate than Dee's year,
against the vernal equinox, in Chris's 2800 year run, EVEN IN EPHEMERIS DAYS!

>Gregorian Year, 365+(97/400) days, 365/05:49:12.
>Khayyam's (and Dee's) Year, 365+(8/33) days, 365/05:49:05.454545....
>Soviet Year, 365.25-(7/900) days, 365/05:48:48.
>Exigius's (and my[Carrier]) Year, 365+(31/128) days, 365/05:48:45.

While Chris Carrier's conclusion still, APPARENTLY, holds:

>So the Exigian or Soviet year is more accurate, even against the vernal
>equinox, than the Gregorian year over a 2800 year run, which we are a
>little past the middle of.

this is only because he has centered his 2800 years on 1600 A.D. instead
of the present century, and because he has ignored the Delta-T correction
necessary to translate ephemeris days into REAL calendrical Universal days.

If we consult Meeus 1995, where he redoes his tables in the more accurate
Dynamical Time of current astronomers, and gives (on page 7) appropriate
Delta-T corrections for finding Universal time from the new Dynamical times,
we obtain the following results, using Chris' method, for the same 2800 year
period, ca. 1600AD, in real calendrical Universal days.

365d 5h 48m 59.2s for the March equinox.   [or 365.242352 Universal days]
365d 5h 48m  0.5s for the June solstice.   [or 365.241673 Universal days]
365d 5h 48m 41.2s for the September equinox. [ 365.242143 Universal days]
365d 5h 49m 37.5s for the December solstice. [ 365.242795 Universal days]

which demonstrates that even circa 1600 A.D. the V.E. year was 365.2424 days
(Dee's year to the nearest ten-thousandth of a day) and that the Gregorian
mean year was more accurate than the Exigian mean calendar year, on its own,
Nicene, terms!

Furthermore if we compare these four values ca. 1600 AD with those for the
2000 year period, (1000 AD to 3000 AD), ca. 2000 AD, using the same method,
we get:

365d 5h 48m 59.6s for the March equinox.   [or 365.242357 Universal days]
365d 5h 47m 56.1s for the June solstice.   [or 365.241621 Universal days]
365d 5h 48m 29.1s for the September equinox. [ 365.242003 Universal days]
365d 5h 49m 30.5s for the December solstice. [ 365.242714 Universal days]

which demonstrates that the Vernal Equinox year length is relatively constant
compared to the other three values, and is thus to be preferred for regulating
solar calendars that use a fixed mean calendar year-length!

The slow upward trend of the Vernal Equinox year-length (when averaged over
a couple of millenia) will continue to make the Dee Calendar-Year ever more
accurate in the centuries ahead, thus making it (the Stonehenge-Khayyam-Dee
year-length) the preferable mean calendar year of all candidates proposed to
date.

An average taken over a period of a few decades (rather than centuries or
millenia) reveals a small perturbation of the year-length with a period of
about 240 years and an amplitude of about 0.00002 days. This and the
non-linearity of the function being averaged conspire to make the
average Vernal Equinox year-length be 365.24237 Universal days, ca. 1923 AD.
(when the Greek Orthodox church adopted the "Soviet" mean calendar
year-length) thus vindicating my assertion that the Gregorian year-length
is more accurate than the Soviet (Greek Orthodox) value on their own Nicene
terms! The average length of the Vernal Equinox year is currently 365.24238
days (to the mearest one-hundred-thousandth of a day) and will probably only
ever drop below the mean betwen the Gregorian and Orthodox value for a brief
few decades after 2060 AD. The Dee value will, of course, continue to be
superior to both these for the foreseeable future.
 
--
Dee's Yrs, Simon Cassidy, 1053 47th.St. Emeryville Ca.94608. ph.510-547-0684.