Subject: Re: [HM] Calendrical questions
From: John Conway (conway@math.Princeton.edu)
Date: Tue Feb 01 2000 - 11:24:26 EST
On Mon, 31 Jan 2000, Ivan Van Laningham wrote:
> Raymond Ayoub wrote:
> .... In these
> days of high speed computers, there's not much reason to choose one over
> the other. All correct algorithms can, with few modifications, provide
> accurate proleptic Easter dates.
However, if one doesn't want to use a computer, (and it's not
necessary to), then the choice of algorithm is important. I regularly
work out such thinks mentally, using the fact that the Paschal Full Moon
is given by the formula
(April 19 = March 50) - (11G + C)_mod30 (**)
except that when (**) gives April 19 one should take April 18, and
when (**) gives April 18 and G >= 12 one should take April 17. Easter
day is the first Sunday strictly later than the date of the Paschal Full
Moon.
In the formula, the Golden number for Y A.D. is G = Y_mod19 + 1 and
C = +3 for all years in the Julian calendar
-4 for years 15xx, 16xx \
-5 for years 17xx, 18xx ) in the Gregorian calendar
-6 for years 19xx, 20xx, 21xx /
and in general C = -H + [H/4] + [8(H+11)/25] for years Hxx.
So for example for Y = 2000 we find G = 6, 11G-6 = 60 which
is 0 mod 30, so (**) gives April 19, which must be corrected to April 18,
and since this is a Tuesday in 2000, Easter day is April 23.
Gauss's error was to suppose the proemptosis correction for the moon
was to take place at 300 year intervals, rather than 8 times in every 2500
years in accordance with the last term in the formula for C above. Errors
in some other algorithms arise from incorrect treatment of the exceptions to
(**).
I'm puzzled by the fact that people give unnecessarily complicated rules
for Easter. O'Beirne, for instance, seems pleased by the fact that his
formula involves 10 divisions! Of course, it helps that it is now
trivial to compute weekdays mentally by the Doomsday rule.
John Conway
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