{"id":115,"date":"2006-04-14T15:06:17","date_gmt":"2006-04-14T15:06:17","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=115"},"modified":"2007-10-31T14:38:14","modified_gmt":"2007-10-31T14:38:14","slug":"calendars","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=115","title":{"rendered":"Calendars"},"content":{"rendered":"

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<\/a> (Java applet<\/i>) and details of these systems are given in a fascinating book Calendrical Calculations<\/a>.<\/p>\n

Algorithms to calculate Easter dates have been given by mathematicians through the ages, including Gauss<\/a> (see for example Mind Over Mathematics: How Gauss Determined The Date of His Birth<\/a>) 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<\/a> 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).<\/p>\n

O’Beirne’s algorithm (based on an 1876 article in Nature<\/a>) 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<\/a><\/p>\n

O’Beirne’s algorithm<\/u>
\nThe following process gives the date of Easter Sunday as the pth<\/sup> day of the nth<\/sup> month in year x. It also gives the Golden Number<\/a> a+1 and the epact<\/a> (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<\/a>.<\/p>\n

\\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}
\n(Table produced by LaTable<\/a>)<\/small><\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-115","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/115","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=115"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/115\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}