The Ravenna Easter Calendar Stone

Karl-Heinz Lewin

Fig.1: Easter Calendar stone (“Calendario Liturgico”) in the Museo Arcivescovile at Ravenna
[Photo found on the Internet on 19.11.2017, source can no longer be found in 2023]

Reading of the Easter calendar stone

The reading was taken sector by sector and line by line from the copper engraving in Dissertatio de latinorum paschali cyclo [Noris, cited after Voigt 2003], for lack of a complete illustration of the original (Fig. 1), a comparison with the section of the calendar stone illustrated in [Voigt 2003, 113], which shows at least three sectors completely and two further sectors approximately halfway, did not show any deviations. I was able to verify the data personally by eye in Ravenna in 2006.

The reading begins with the Sector marked with a small cross as Sector 1, which is also referred to in the original by the large cross in the center.

Striking is the use of the numeral sign ↅ (Unicode character U+2185, late Roman six, probably originating from the Greek letter and numeral sign digamma, stigma or episemon) instead of “VI”, “Xↅ” instead of “XVI”, “XↅI” instead of “XVII” etc. (however only predominantly, not consistently, as the “XVIII” in Sector 1, line 5 shows). Pridie, the day before, is written once as PR (Sector 6, line 3), otherwise as PD (cf. Sector 7, right next to it).

Anticipating the interpretation, it should be noted that the entry in Sector 12, Cycle 3 (cf. Fig. 3), highlighted above with a yellow background, is incorrect. According to the Julian calendar it must read either “ↅII ID APR LU Xↅ” (6.4. Moon 16) or “ↅI ID APR LU XↅI” (7.4. Moon 17). In Sector 10, Cycle 5 (highlighted with cyan above) the “I” of “LU XXI” (3.4. Moon 21) is only faintly visible on the stone – apparently a correction (cf. Fig. 4) – and is probably therefore missing in the engraving. Since these are not systematic errors and the Easter dates will prove to be consistent, I assume that in the first case it is a “printing error” of the stonemason in indicating the lunar day (for the meaning see below), in the second case a “reading error” of the engraver, just as the spelling “CII” instead of the meant “ↅII” in line 7 of Sector 16, whereby the hook of the “ↅ” on the stone (Fig. 5) is only recognizable as a very short tick.

In summary, we get the following table, the rows of which correspond to the sectors of the Easter calendar stone:

For each of the five cycles the Easter date and the lunar day are listed. The entry with two values for the lunar days (XↅI/Xↅ in year XII of cycle III) first shows the incorrect value taken over from the Easter calendar stone and after the slash the correct one (cf. Fig. 3 and the assessment on this above).

Interpretation of the calendar stone

The following information can be found in each of the 19 sectors:

Heading (sector 9 only): The meaning of “DIVISIO CYCLI II” at this point is unclear to me.

Line 1: Lunar year of the 19-year lunar cycle. The cycle begins with the lunar year 17; this corresponds to a Byzantine counting [Voigt, personal communication], but the number has no meaning for the further interpretation.

Line 2: Year number of the first 19-year cycle in the 95-year cycle and date of the Easter full moon (Luna 14). The year number is at the same time the “golden number” (= year number mod 19 + 1). The “Year 1” (AN I) represents therefore the Julian year in the used era modulo 532 + 1, so it can mean the year 532 of the “Era of Christ”, as Noris writes (see fig. 1), or the year 0 (!), that would mean the year 1 before the beginning of the era (1 B.C.), or also the years 1064, 1596 ...

The Easter full moon is the calculated spring full moon.⁹ Its date repeats every 19 years, so it is valid for the whole sector.

Line 3: Easter date and lunar day of Easter (in the first 19-year cycle). The lunar day is the consecutive day number in the lunar month.

Lines 4 and 5: Easter date and lunar day of Easter in the 2^nd 19-year cycle.

Lines 6 to 8: Easter date and lunar day of Easter in the 3^rd 19-year cycle.

Lines 9 to 11: Easter date and lunar day of Easter in the 4^th 19-year cycle.

Lines 12 to 14: Easter date and lunar day of Easter in the 5^th 19-year cycle.

Line 15: Indication whether the lunar year in question is a leap year (“EB” for “annus embolismalis”, a year with 13 lunar months.) or a normal year (“CM” for “anno communis”, a year with 12 lunar months). (This has nothing to do with the Julian leap years – solar years).

Conversion to modern dates

To convert the dates, I use the following small table for the relevant time period so that I don’t need to know all the intricacies of the calendar calculation:

This gives us the following table, whose rows correspond to the sectors of the calendar stone:

The indication of the lunar leap year was taken from the last line of the original as “(L)” into the second column.

For each of the five cycles the Easter date and the lunar day are listed. The entry with two values for the lunar days (17/16 in year 12 of the 3^rd cycle) shows as first value the incorrect value taken over from the calendar stone and after the slash the correct one (cf. fig. 3 and the assessment on this above).

The algorithm of the Easter calendar stone

The Easter cycle of the Easter Calendar Stone begins with AN I LU XIV NO AP (annus I luna XIV nonae aprilis) [details see Lewin 2005, ²2023]. The date corresponds to 5 April. In the following years (in the following sectors), the date of the Easter Moon is 11 days earlier or, if this would be before 21 March, 19 days later than the date of the previous year. This results in 19 Easter Moon dates in the total circle, which remain constant over all five cycles.

This is followed in each sector by the dates for Easter Sunday and the lunar day (the consecutive day number in the lunar month) for a total of five 19-year cycles. From the lunar day of Easter Sunday, the weekday of each date in the corresponding year can be deduced. The author of the Easter table, however, had to proceed in reverse: he first had to determine the weekday of a fixed date, deduce from this the weekday at the date of the Easter moon, and calculate or count the Easter date and the lunar day of the Easter date from the number of days until the next following Sunday.

The information on the lunar year in line 1 and the information on the lunar leap year in line 15 of each sector are not required for the calculation.

The year in line 2 is helpful for deducing the days of the week of other dates from knowing the day of the week of a date on the chart. If, starting from the date of the first Easter Sunday on the table, I determine the day of the week of a particular date, such as 21 March, in each year, I recognise that the years 1, 5, 9, 13, 17 in the 1st cycle, 2, 6, 10, 14, 18 in the 2nd cycle, 3, 7, 11, 15, 19 in the 3rd cycle, 4, 8, 12, 16 in the 4th cycle and again 1, 5, 9, 13, 17 in the 5th cycle are leap years in the Julian calendar. With this knowledge, I can use the algorithm outlined above to calculate all the other values on the Easter table starting from two basic values, year number 1 and the date of the first Easter moon (whereby the incorrect value in sector 12, line 7, 3rd cycle is corrected) and, with a slight modification of the algorithm, extend the calculation to all 532 years of a complete Easter cycle.

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.

Implementation in a spreadsheet program

(Here Microsoft Excel 2007, German licence):

Note: The counting of the weekdays W={"Sun", "Mon", "Tue", "Wed", "Thu", "Fri", "Sat"} from 0 to 6 is chosen so that W[5×J÷4 mod 7] in the Julian calendar results in the weekday of 21 March. This also results in the simplest calculation for reaching the Sunday following the Easter moon (see field G2).

Implementation in JavaScript

function DaynumberToDayAndMonth( daynumber ) {
	// assert( daynumber > 0 && daynumber < 62 );
	if ( daynumber <= 31 ) {
		this.dd = daynumber;
		this.mm = 3;
	} else {
		this.dd = daynumber - 31;
		this.mm = 4;
	}
	return this;
}

function RavennaEasterTableLine( annus, lunaXIV ) {
	// assert( annus > 0 && annus <= 9999 );
	// assert( lunaXIV >= 21 && lunaXIV <= 50 );
	var date;
	this.an = annus;
	this.weekday21 = Math.floor( 5 * ( annus - 1 ) / 4 ) % 7 ; // 5×J÷4 mod 7
	this.om = lunaXIV;
	date = DaynumberToDayAndMonth( lunaXIV );
	this.omdd = date.dd;
	this.ommm = date.mm;
	this.weekdayOM = ( weekday21 + lunaXIV - 21 ) % 7 ; // =REST(E2+(B2-21);7)
	this.lunaOS = 21 - this.weekdayOM;
	this.os = lunaXIV + 7 - this.weekdayOM;
	date = DaynumberToDayAndMonth( this.os );
	this.osdd = date.dd;
	this.osmm = date.mm;
	return this;
}

const OsterMondStart = 36;
var osterMond = OsterMondStart;

function nextOsterMond( jj ) {
	var newOM = osterMond - 11;  // =WENN(B2-11>=21;B2-11;B2+19)
	osterMond = ( jj % 19 ) ? ( newOM >= 21 ? newOM : newOM + 30 )
                    : OsterMondStart;
            return osterMond;
}

function RavennaEasterTable( annus, times ) {
	// assert( annus > 0 && annus < 9999 );
	// assert( times >= 4 && times <= 532 );
	osterMond = 50 - ( 11 * (( annus - 1 ) % 19 ) + 14 ) % 30 ; // =50-REST(11*REST(A2-1;19)+14;30)
	for ( let j = annus; j < annus + times; j++ ) {
		var line = RavennaEasterTableLine( j, osterMond );
		osterMond = nextOsterMond( j );
		generateRavennaEasterTableOutput( line );
	}
}

Formal verification of the algorithm

If you click on the button below, the summarised table of the reading of the Easter calendar stone shown above and a new calculation of the data generated according to the algorithm just developed with the JavaScript functions shown appear one after the other for comparison, whereby in this newly calculated table the first column with the Byzantine lunar year and the last column with the display of the lunar leap year are omitted as unnecessary for the calculation.
An Easter calculator with this algorithm, where you can choose the start year, the number of years and the output format, can be found at The Easter calendar stone of Ravenna as a table calculator.

Literature

The author is a mathematician and worked as a software developer.

Karl-Heinz Lewin, Haar: Karl-Heinz.Lewin@t-online.de

Copyright © Karl-Heinz Lewin, 2023

Translated from a German version by DeepL with authoritative assistance by the author.