Calendars

Friday 14 April 2006 at 3:06 pm | In Articles | 2 Comments

It is this time of year that has attracted a lot of attention devoted to finding the dates of religious festivals. A very comprehensive calendar calculator for 25 different calendar systems can be found at Calendrica (Java applet) and details of these systems are given in a fascinating book Calendrical Calculations.

Algorithms to calculate Easter dates have been given by mathematicians through the ages, including Gauss (see for example Mind Over Mathematics: How Gauss Determined The Date of His Birth) but it does produce a few errors. In 1961 the Scottish mathematician T.H. O’Beirne published an algorithm in his Puzzles and Paradoxes column in the New Scientist subsequently reprinted in his book of the same name published by the Oxford University Press (sadly out of print but I throughly recommend looking for a second-hand copy).

O’Beirne’s algorithm (based on an 1876 article in Nature) has the merit of always giving the correct date as well as being easy to use. It is a simple exercise to write a program to do the work for you. Simple Example

O’Beirne’s algorithm
The following process gives the date of Easter Sunday as the pth day of the nth month in year x. It also gives the Golden Number a+1 and the epact (23-h or 53-h whichever is between 1 and 30 inclusive). All you have to do is start with the year x and perform 10 division operations noting the quotients and remainders.

\begin{tabular}{|c|l|r|c|c|}
\cline{1-5}
\multicolumn{1}{|l|}{Step} & Divide & By & Quotient & Remainder \\ 
\cline{1-5}
1 & $x$ & 100 & $b$ & $c$ \\ \cline{1-5}
2 & $5b+c$ & 19 & - & $a$ \\ \cline{1-5}
3 & $3(b+25)$ & 4 & $\delta$ & $\epsilon$ \\ \cline{1-5}
4 & $8(b+11)$ & 25 & $\gamma$ & - \\ \cline{1-5}
5 & $19a+\delta-\gamma$ & 30 & - & $h$ \\ \cline{1-5}
6 & $a+11h$ & 319 & $\mu$ & - \\ \cline{1-5}
7 & $60(5-\epsilon)+c$ & 4 & $j$ & $k$ \\ \cline{1-5}
8 & $2j-k-h+\mu$ & 7 & - & $\lambda$ \\ \cline{1-5}
9 & $h-\mu+\lambda+110$ & 30 & $n$ & $q$ \\ \cline{1-5}
10 & $q+5-n$ & 32 & 0 & $p$ \\ \cline{1-5}
\end{tabular}
(Table produced by LaTable)

2 Comments »

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  1. Your homepage is wonderful!

    But I’m curious about your printing mathematical formula in the website.

    Can you tell me how to do this?

    Thank you for your help!

    Maybe you can just e-mail to me,give me some hints:the software for this,some tutorial.

    Comment by zhangzujin — Tuesday 23 May 2006 4:22 pm #

  2. I use LatexRender to show the mathematics

    Comment by Steve — Tuesday 23 May 2006 4:29 pm #

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