## Thursday, March 24, 2016

### Computing the date of Easter

 Raphael - Resurrection of Christ
The first Ecumenical Council was held in Nicaea, Asia Minor, in 325, convened by Emperor Constantine. Ecumenical councils could not be held before that date, for Christians were persecuted and had to meet in secret. This first Council had very important consequences: Aryanism, which denied the divinity of Christ, was condemned; the version of the creed still recited in mass today was defined; and the Council established the algorithm to calculate the date of Easter, which is still in use. This algorithm is so complicated that Donald Knuth included a program to perform it in his famous and classic computer encyclopedia, The Art of Computer Programming.
Easter was fixed on the first Sunday after the first full moon after the spring equinox, which that year fell in March 21st. To calculate the exact date one must take into account that the solar cycle (the year) contains 365.2421988 days, while the lunar cycle (the phases of the moon) lasts 29.530588 days, two numbers with many figures that do not have a simple relationship with one another, so the calculation is quite complex.

The astronomers of the ancient civilizations were deeply disturbed because the cycle of the sun is not an exact multiple of the cycle of the moon. The quotient of the two numbers is 12.368267, which means that a year lasts for 12 lunar months plus over a third. In Babylon they divided the year into 12 months and chose 12 as the basis of their numbering system, but every three years or so they had to introduce an extra month, so that the sun would not be left behind. The years with 13 months were considered abnormal, weird, bad luck. Number 13 still drags that disreputable character.
By the sixth century B.C., Babylonian astronomers discovered that every 19 years the sun and the moon are back in almost the same relative positions. Indeed, 19 solar years have 19´365.2421988 = 6939.60 days, while 235 lunar months have 235´29.530588 = 6939.69 days. In other words, if 235 lunar months are distributed between 19 years, the resulting cycle introduces an error of 0.09 days, one day per 220 years. Around 432 B.C., the Greek astronomer Meton rediscovered the cycle of 19 years (we call it the Metonic cycle), and even improved it a bit, reducing the error to about one day every 300 years.
As the date of Easter depends on the full moon, the algorithm to calculate that date depends on 19, the number of years in the Metonic cycle. I will not insert the full algorithm here, just a table that helps to calculate the date with very little effort.

 0 1 2 3 4 5 6 7 8 9 Up to 1582 5/4 25/3 13/4 2/4 22/3 10/4 30/3 18/4 7/4 27/3 1583-1699 12/4 1/4 21/3 9/4 29/3 17/4 6/4 26/3 14/4 3/4 1700-1899 13/4 2/4 22/3 10/4 30/3 18/4 7/4 27/3 15/4 4/4 1900-2199 14/4 3/4 23/3 11/4 31/3 18/4 8/4 28/3 16/4 5/4

 10 11 12 13 14 15 16 17 18 Up to 1582 15/4 4/4 24/3 12/4 1/4 21/3 9/4 29/3 17/4 1583-1699 23/3 11/4 31/3 18/4 8/4 28/3 16/4 5/4 25/3 1700-1899 24/3 12/4 1/4 21/3 9/4 29/3 17/4 6/4 26/3 1900-2199 25/3 13/4 2/4 22/3 10/4 30/3 17/4 7/4 27/3

To compute the date of Easter in a given year using the table, just find the remainder of the year in question when divided by 19, which is always a number between 0 and 18. Then look at the corresponding column in the table, in the row corresponding to the year. The table gives the date of the first full moon after the spring equinox, in the day/month format.
There is just one thing missing: the date thus obtained can fall on any day of the week. Easter will be the next Sunday. This last rule applies even if the date on the table falls on a Sunday, to prevent the Christian Easter to fall on the same day as the Jewish Passover.

Manuel Alfonseca