Computus: Algorithms to compute Easter

With Easter arriving this weekend, I thought it might be interesting to mention something on the algorithms used to compute Easter. As we know, Easter is a not a fixed date each year, it is determined by the rule

Easter is the first Sunday after the first full moon occurring on or after the vernal equinox

It is not immediately obvious how one might compute the date of Easter at some point in future and yet we need to know such dates in finance when valuing a derivative contract which pays out an amount at some point in the future, the payment is on a good business day, and so one needs to know future public holidays. In some western countries, there are several public holidays associated with Easter; for example, Good Friday, Easter Monday, Ascension Thursday, Whit Monday, and Corpus Christi. For practical purposes a lookup table for a few hundred years is quite sufficient (this is exactly how the Quantlib library manages the calculation), and here I am assuming one even wants to spend time on determining calendars, there are specific vendors who specialise in supplying business holiday calendars.

However, recently I was curious to understand the rules a little better for a different reason and very quickly uncovered, to my surprise, that some serious mathematicians have been involved in algorithms to compute Easter, most notably Carl Friedrich Gauss (R. Bien, 2004).

Johann Heinrich Lambert

Lambert, known for introducing hyperbolic functions, also produced a rule for the Eastern Easter in 1776.
J. H. Lambert, “Einige Anmerkungen über die Kirchenrechnung,” Astronomisches Jahrbuch oder Ephemeriden für das Jahr 1778, Berlin 1776, 210–226

Carl Friedrich Gauss

Gauss, known for the famous normal distribution, produced his Osterformel in 1800. However it was not perfect and was corrected in later years by Gauss and one of his students.
C. F. Gauss, “Berechnung des Osterfestes,” Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, Aug. 1800 (= Werke VI, 73–79) (Note: Werke = Gauss’collected work, Göttingen 1863)

Augustus De Morgan

De Morgan, known for his work in mathematical logic, cleared up a mis-understanding of the term ‘full moon’ in the rule, it is not an Astronomical moon but a so-called Paschal Moon (it was this aspect of the calculation that sparked my curiosity).
A. de Morgan, “A Budget of Paradoxes,” Volume 1, Chicago and London 1915, 356-357. The first edition appeared in 1872

Donald E. Knuth

Knuth, known for his work in computer science and algorithms, implemented an algorithm to compute Easter.
D. Knuth, “The Calculation of Easter…,” Communications of the Association for Computing Machinery, Volume 5, Number 4, April, 1962, 209–210.

Much more detail concerning the algorithms, and the history of the algorithms, to calculate Easter is found in
Bien, Reinhold. “Gauß and Beyond: The Making of Easter Algorithms.” Archive for history of exact sciences 58, no. 5 (2004): 439-452.

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One thought on “Computus: Algorithms to compute Easter”

1. Matt says:

I looked at Easter calculations in the past but lost interest a little when I discovered they were essentially driven by a fixed table of offsets produced by the Vatican rather than an exact formula. As a young man Bede got interested in this problem and wrote a report of the famous Northumbrian Synod of Whitby (664) where proponents of the old and new methods of calculating Easter were invited to put their case. The upshot was that the traditionalist Ionian monks were sent home with a flea in their ear for not following Roman practice and that pretty much settled the matter as far as Britain was concerned.

Of course Easter simply commemorates Passover and one of the earliest practical methods for determining its date was to ask the local Jewish community! For obvious reasons the Church fathers preferred to calculate their own date and actually made a fetish of not being the same date as Passover. Indeed the main objection to the Eastern Orthodox Church shifting to the Gregorian calendar was that Easter would sometimes fall on the same day as Passover! To add to the confusion some Orthodox churches decided to switch to the Gregorian calendar in 1923 and thus be out of synch (13 days) with the traditional Orthodox churches except that, as a “compromise” they celebrate Easter on the same day (although not date).