Welcome to Computus Paschalis & Other Calculations

My name is Karl-Heinz Lewin. On this website I would like to deal with the Computus Paschalis, the ancient and medieval Easter calculation, with the modern Easter calculation, with calendars, calendar calculations, calendar conversions from one calendar system to another, with planetary astronomy, as far as it is relevant for the Easter or calendar calculation, with programming languages and their respective quirks, with natural languages and their interconnections, with the German language and its flattenings, with scripts and alphabets and their decipherments.

31.07.2024

Today I am publishing my analyses of the Easter calculations of Carl Friedrich Gauss, Donald Knuth and Claus Tøndering on the Internet, in the two variants for the Julian and Gregorian calendars.
Carl Friedrich Gauss is certainly known to anyone who has ever seriously studied the Easter calculation.
Donald Knuth belonged to the first generation of IT scientists who, among other things, was involved in efficient programming, programming languages and in this field also with Edgar (actually Edsger) Dijkstra's verdict against a "goto" statement [(1968): Go To Statement Considered Harmful; Communications of the ACM, Volume 11, Issue 3, March 1998; pp. 147-148; ACM is the ‘Association of Computer Manufacturers’] and countered it with an article [(1974): Structured Programming with go to Statements; Computing Surveys, 6 (4), 261-301], in which he showed that a good software developer could create good software even with bad tools – which of course flattered me as a good software developer! This same Donald Knuth had also published the first formulations of an Easter calculation algorithm in the then common programming languages Algol 60 and Cobol [(1962): The Calculation of Easter...; Communications of the ACM, Volume 5, Issue 4, April 1962; pp. 209-210; https://doi.org/10.1145/366920.366980], which I have now transcribed into JavaScript.
Claus Tøndering is the author of the algorithm that the German Wikipedia published under the keyword “Osterdatum” in the period from 22 April 2003 to 29 February 2009 and which I had taken up in my first article on the Easter calendar stone in Ravenna and unfortunately cited incorrectly at the time (see below under 28.02.2024). At the time, his algorithm seemed to me to be the shortest and most elegant (I will have to verify this again when I get the chance).
After that, the German-language Wikipedia had moved the Easter calculation from the article “Osterdatum” to the articles Computus (Osterrechnung), Gaußsche Osterformel and Spencers Osterformel referenced there. The latter seems unnecessarily complicated to me and only offers a solution for the Gregorian calendar, but not for the Julian calendar. I have also included the Gaussian Easter formula in my presentation, but Tøndering's ingenious algorithm has been removed from the German-language Wikipedia without a greeting. As a further point of criticism, I note that the article on the “Computus (Osterrechnung)” does not provide a description of the antique computus, but without any reference to it merely portrays a variant of the medieval computus, which was designed for people who cannot or do not like to calculate, but prefer to add data from different tables (which we programmers preferred to call “vectors” or “arrays”) – or rather just count them further – the only mathematical operation they could be bothered with. In fact, these methods were used at the time, and they worked, but transferring them to a modern programming language would simply require too much writing. How much simpler were the algorithmic specifications of Beda, Dionysius and the Easter Calendar Stone of Ravenna!
You can find the descriptions of the algorithms for the Easter calculation by Carl Friedrich Gauss, Donald Knuth and Claus Tøndering at [Computus Paschalis: Antique and modern Easter calculation], and calculator programs with selectable start years and selectable number of years under [Interactive antique and modern Easter calculators]. Under the last link you will also find two new comparison programs, one that compares the ancient algorithms of Ravenna, Dionysius and Beda with the modern algorithms of Gauss, Knuth and Tøndering for the Julian calendar, and one that compares the algorithms of Gauss, Knuth and Tøndering for the Gregorian calendar.
[*** Translated from German with www.DeepL.com/Translator (free version) ***]

21.05.2024

In The Easter Calendar Stone of Ravenna, my current analysis, and in the JavaScript file osterrechner/cyclopaschalisravenna.js, the starting value for the Easter moon had to be corrected, because the starting value 36 is of course only valid if for the starting year 'annus' the expression ( annus - 1 ) modulo 19 yields 0. For other start years, the algorithm returned nonsensical values.
To ensure that the calculation works for every starting year, the starting value for the Easter moon is calculated using a formula that cannot be derived directly from the values on the Easter calendar stone, but requires a mathematical abstraction: The rule for calculating the Easter moon in the next year by subtracting 11 or conditional addition of 19 given in the first paragraph of this chapter corresponds to a subtraction of ( 11 times the number of the previous year modulo 19 ) modulo 30, the two constants involved can be found by trial and error. This results in a start value for the Easter moon as
Easter moon start = 50 - ( 11 * ( py mod 19 ) + 14 ) mod 30, where py = the year number of the previous year.
With this change, The Ravenna Easter calendar stone as a table calculator now provides correct Easter Moon and Easter Sunday dates for any starting year from 1 to 4996.

28.02.2024

The algorithm for calculating the Easter date in the Julian calendar published in 2005 in my first article on the Easter calendar stone in Ravenna in Zeitensprünge 2005.2 and since 17 June 2023 on this website was faulty, a column was missing and thus an intermediate step in the calculation. (But nobody tried the algorithm, so nobody noticed that it was wrong.) The error has now been corrected, see under [Computus Paschalis: Antique Easter calculation].

19.02.2024

A presentation of the algorithm of the Easter calculation of Beda Venerabilis can now be found at [Computus Paschalis: Antique Easter calculation], and an Easter calculator that implements this algorithm, as well as a comparison programme that shows the results of the Easter calculation algorithms of the Easter calendar stone of Ravenna and the Easter tables of Dionysius Exiguus and Beda Venerabilis side by side for comparison, can be found at [Interactive Antique Easter calculators].

12.02.2024

A presentation of the algorithm of the Easter calculation of Dionysius Exiguus can now be found at [Computus Paschalis: Antique Easter calculation], and an Easter calculator that implements this algorithm, as well as a comparison programme that shows the results of the Easter calculation algorithms of the Easter calendar stone of Ravenna and Dionysius Exiguus side by side for comparison, can be found at [Interactive Antique Easter calculators].

30.11.2023

Three years ago, I came across writings by Kiev professor Iurii Mosenkis on the platform academia.edu, in which he deals with the decipherment of the Cretan linear script A and comes to the conclusion that the language is a Greek dialect.

In fact, there is no lack of attempts to decipher the few surviving written documents in Linear A on the assumption of one or other language or language family - however “many consider themselves qualified, but only a few are predestined” [Tabary in the comic series “Iznogoud”]:

• Gretchen Leonhardt sees similarities between Linear A characters and characters in Japanese hiragana and kanji.
• Kjell Aartun (* 6. July 1925 – † 28. May 2023) considered the Minoan language written in Linear A to be a Semitic language.
• Peter Revesz, computer science professor at the University of Nebraska in Lincoln and specialist in data mining, summarises the Minoan texts he “deciphered” in Linear A with Hattic and Hungarian to form a Western-Ugric language family.
• Peter van Soesbergen reads Linear A tablets as Hurrian texts.

I will leave it at the examples given. Presumably each of these authors has recognised one or other of the word equations correctly – there are probably Hattic, Hurrian and Semitic (Akkadian) words in Minoan texts (I consider Hungarian and Japanese words to be far-fetched), just as there are Hebrew (Jubel), Arabic (Magazin) and Turkish (Kiosk) words in the German language. However, the suffixes cited by Mosenkis and the prefix a as well as the pictorial representations (puppy = skylax for Scylla, knee for Knossos, spider = Arachna for Ariadne / Arihagna, etc.) have convinced me that Mosenkis interpreted the texts correctly.

The results of the works of

point into the same direction.

I have summarised the work of Mosenkis and his findings in the article “Linear A is deciphered – the language is Greek” (to be found on the navigation bar under [Scripts and Languages]).

03.11.2023 / 26.06.2023

I have completely revised my article about the Easter calendar stone of Ravenna and limited it to the topic of this marble tablet. Finally, I explain the algorithm found in this tablet and present two implementations (as an Excel spreadsheet and in JavaScript) with the possibility to visually compare the recalculation of the Easter tablet by the JavaScript implementation with the summarised reading of the original, the Easter calendar stone. Today, also the English version of this revision is ready. (To be found on the topic sheet under [Computus Paschalis: Antique Easter calculation].)

21.06.2023

Today I put my first Easter calculator on my website (see under [Interactive Antique Easter calculators]). It implements the algorithm of the Ravenna Easter calendar stone with selectable start year and selectable time span (number of years), once in modern representation and the other one in antique representation. Note that the start year 1 almost certainly corresponds to the year 532 C.E. With the default setting (start year: 1, number of years: 95), the antique representation should correspond exactly to the values of the Ravenna Easter calendar stone with a few exceptions. The antique representation allows a tabulation up to the year MDCCCCLXXXXↅIII (1999) at the most. Since this corresponds to the year 2530 C.E., I have refrained from implementing the representation of even larger numerical values in Roman numerals according to the notation of the Easter calendar stone.

Translated with www.DeepL.com/Translator (free version)

17.06.2023

As a first article, I present here an English translation of an updated German version of my first publication dealing with the marble Easter tablet in the Archbishop’s Museum at Ravenna and its possible implications for the doubts expressed elsewhere <karl-der-große.de> about the validity of our believed chronology. (To be found on the topic sheet under [Computus Paschalis: Antique Easter calculation].)

Kind regards
Karl-Heinz Lewin