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Calculating the Date of Easter


Date: 04/04/2001 at 15:33:16
From: james
Subject: Calculated dates

I was browsing your FAQ section and reading why we have leap year 
every four years. But Easter is a calculated date. How is Easter 
calculated?


Date: 04/04/2001 at 17:06:49
From: Doctor Rob
Subject: Re: Calculated dates

Thanks for writing to Ask Dr. Math, James.
 
According to a decision made by the Council of Nicea in 325 A.D., 
Easter is the first Sunday after the first full moon on or after
March 21st. This means that both the tropical solar year of 365 days, 
5 hours, 48 minutes, and 46 seconds, and the length of a lunation, 29
days, 12 hours, 44 minutes, and 2.8 seconds, must be taken into
account.

It was noticed by the ancient Babylonians and other cultures that 19
tropical solar years is very nearly equal to 235 lunations, the
difference being about 2 hours, 4 minutes. That and other very
complicated considerations led Carl Friederich Gauss to give the
following rule for the computation of the date of Easter.

Let N be the number of the year. Let C = [[N/100]], where [[x]] means 
the greatest integer less than or equal to x (that is, x rounded down 
to the nearest integer). Set

   L = 4 + C - [[C/4]],
   M = 11 + L - [[(8*C+13)/25]].

Divide N by 4, 7, and 19, and call the resulting remainders a, b, and
c. 

Divide 19*c + M by 30, calling the remainder d.  
Divide L + 2*a + 4*b + 6*d by 7, calling the remainder e.  

Then the date of Easter is either March 22+d+e, or April d+e-9, with 
the following two exceptions:

  (1) If e = 6 and d = 29, Easter is on April 19; and
  (2) If e = 6, d = 28, and M = 2, 5, 10, 13, 16, 21, 24, or 29, then
      Easter is on April 18.

The explanation and proof of this rule is given by J. V. Uspensky and
M. A. Heaslet in _Elementary Number Theory_ (New York: McGraw-Hill,
1939), pages 212-221.

Example: What day is Easter in the current year, 2001? 
N = 2001, C = 20.

   L = 4 + 20 - 5 = 19,
   M = 11 + 19 - [[173/25]] = 24.

(L and M are the same for all years from 1900 to 2099.)

   2001 =  4*500 + 1,
   2001 =  7*285 + 6,
   2001 = 19*105 + 6,

so a = 1, b = 6, and c = 6. Then

   19*c + M = 138,
            = 4*30 + 18,

so d = 18, and

   L + 2*a + 4*b + 6*d = 19 + 2 + 24 + 108,
                       = 153,
                       = 21*7 + 6,

so e = 6. Neither exception occurs, and d + e - 9 = 15, so Easter must 
fall on April 15. (That's correct, by the way.)

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
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