POUVILLON RESTORATION PROJECT - November 2011

A discussion of the Easter calculator's functions, the computus.

Front elevation view of the Easter calculator set within the movement.

These photos of the front and side elevations of the Easter calculator show how diminuitive the mechanism is.

The calculator has 7 indications. Starting from the top dial and working clockwise we have the following:

I will explain what each dial means and if it is used in the calculation of the date of Easter; not all dials are used for this purpose. It is an ecclesiastical calculator involved with the Catholic and Protestant Church’s celebration of Easter. While Christmas is what many consider to be the most celebrated religious holiday, Easter is the most important religious event for the church. Easter is what validates all of Christ’s teachings.

These religious functions are not restricted to just the calculator, although the calculator is the most obvious and unusual mechanism in connection with this. Another portion of the clock is used in this calculation as well as other various movable and fixed feasts as they relate to the church’s main celebration of Easter. The rolling moon on the right side of the clock has an unusual two month duration dial with one sector having 29 divisions and the other 30. This was difficult to understand until viewed within the ecclesiastical context. About a quarter of the clock Pouvillon designed is with the ecclesiastical functions in mind. This movement is very much a religious device with many other fancy complications added in. It appears that the calculator was added quite late in the life of the clock before the passing of Pouvillon in 1969 at the age of 91. We have a date stamped on the Easter disc of 1946 and two newspaper articles from 1953 and 1955 indicating that he was still perfecting the Easter calculator. It is also possible that he added the movable feast indicators on the tellurian dial during this time. In other words, most of the religious functions could have been added to the clock when Pouvillon was in his late 60’s. The clock could have taken a more religious tone in Pouvillon’s later life. I have left all of the links originally provided in the Wikipedia references so one can further expand on those items highlighted in blue.

In principle, the date of Easter is defined as the Sunday following the full moon following vernal equinox (the so-called Paschal Full Moon). In the current year it’s pretty easy to find the vernal equinox, it’s commonly known as the first day of Spring, March 21st, when the length of the day as well as the night are equal. Then look up the current table of the moon’s phases, find when the first full moon occurs after the first day of Spring and then pick the following Sunday. BUT, here the simplicity ends. The phases of the moon change over the calendar and there are irregularities in the calendar which require corrections from time to time; the leap year every four years, and the 400 and 1000 year corrections. So to be able to predict the day Easter will fall in the future get’s quite complex.

The church turned to a series of complicated steps and tables to take these variations into account using the Dominical Letter, Golden Number, Solar Cycle and Epact all of which are meant to take into account the variations of the calendar and celestial events to reliably predict Easter. This they called the computus or system of reckoning. The computus is thus the procedure of determining the first Sunday after the first ecclesiastical full moon that falls on or after March 21st.

The dial descriptions below are taken from references in Wikipedia as well as the book Jens Olsen’s Clock, Otto Mortensen. Please be aware that I do not personally have full command of the following calculations and if you find inconsistencies, let me know.

Dominical Letter :

Dominical letters are letters A, B, C, D, E, F and G assigned to days in a cycle of seven with the letter A always set against January 1st as an aid for finding the day of the week of a given calendar date and in calculating Easter.

A common year is assigned a single dominical letter, indicating which letter is Sunday (hence the name, from Latin dominica for Sunday). Thus, 2011 is B, indicating that B days are Sunday. Leap years are given two letters, the first indicating the dominical letter for January 1 - February 28 (or February 24, see below), the second indicating the dominical letter for the rest of the year.

In leap years, the leap day may or may not have a dominical letter. In the original 1582 Catholic version, it did, but in the 1752 Anglican version it did not. The Catholic version caused February to have 29 days by doubling the sixth day before 1 March, inclusive, because 24 February in a common year is marked "duplex", thus both halves of the doubled day had a dominical letter of F. The Anglican version added a day to February that did not exist in common years, 29 February, thus it did not have a dominical letter of its own.

In either case, all other dates have the same dominical letter every year, but the days of the weeks of the dominical letters change within a leap year before and after the intercalary day, 24 February or 29 February. The dominical letter of a year determines the days of week in its calendar:

Dominical letters were a device adopted from the Romans by chronologers to aid them in finding the day of the week corresponding to any given date, and indirectly to facilitate the adjustment of the "Proprium de Tempore" to the "Proprium Sanctorum" when constructing the ecclesiastical calendar for any year. The Christian Church, due

to its complicated system of movable and immovable feasts, has long been concerned with the regulation and measurement of time. To secure uniformity in the observance of feasts and fasts, it began, even in the patristic age, to supply a system of reckoning (computus) by which the relation of the solar and lunar years might be accommodated and the celebration of Easter determined. It adopted the astronomical methods that were available at the time, and these methods and their methodology have become traditional and are perpetuated in a measure to this day, even the reform of the calendar, in the prolegomena to the Breviary and Missal.

Calculation

The dominical letter of a year can be calculated based on any method for calculating the day of the week, with letters in reverse order compared to numbers indicating the day of the week.

For example:

Epact:

The epact (Latin epactae, from Greek: epaktai hèmerai = added days) was originally defined as the age of the moon in days on January 1, and occurs primarily in connection with tabular methods for determining the date of Easter. The Epact arises from the inequality of the solar and lunar year. It varies (usually by 11 days) from year to year, because of the difference between the solar year of 365 days and the lunar year of 354 days.

 Epacts as they relate to the Lunar calendar

Epacts are used to find the date in the lunar calendar from the date in the common solar calendar. As the epact calculation gives 30 different epacts, while the time between two consecutive mean new moons is only a little over 29 ½  days, it has been established the cyclic lunar months should change at a rate of 29 and 30 days, respectively. This explains why the lunar dial has the unusual arrangement of the moon cycling over a two month period with the ring dial bisected by divisions 1 through 29 on one sector and 1 through 30 on the other.

Solar and lunar years

A (solar) calendar year has 365 days (366 days in leap years). A lunar year has 12 lunar months which alternate between 30 and 29 days (in leap years, one of the lunar months has a day added).

If a solar and lunar year start on the same day, then after one year, the start of the solar year is 11 days after the start of the lunar year; after two years, it is 22 days after. These excess days are epacts, and are added to the day of the solar year to determine the day of the lunar year.

Whenever the epact reaches or exceeds 30, an extra (embolismic or intercalary) month is inserted into the lunar calendar, and the epact is reduced by 30.

Leap days extend both the solar and lunar year, so they do not affect epact calculations for any other dates.

The 19-year cycle

The tropical year is about 365¼ days, while the synodic month is also slightly longer than 29½ days on average. This gets corrected in the following way. Nineteen tropical years are as long as 235 synodic months (Metonic cycle). A cycle can last 6939 or 6940 full days, depending on whether there are 4 or 5 leap days in this 19-year period.

After 19 years the lunations should fall the same way in the solar years, so the epact should repeat after 19 years. However, 19 × 11 = 209 , and this is not an integer multiple of the full cycle of 30 epact numbers (209 modulo 30 = 29, not 0). So after 19 years the epact must be corrected by +1 in order for the cycle to repeat over 19 years. This is the saltus lunae (leap of the moon). The sequence number of the year in the 19-year cycle is called the Golden Number. The extra 209 days fill 7 embolismic months, for a total of 19×12 + 7 = 235 lunations. This correction is discussed first in our October installment, photos 49-002 through 49-006 and the correction is implemented in the November installment photos 56-001 through 56-016.

Golden Number:

The number of the year in the lunar cycle is the golden number and was found by Meton, the Athenian who in the year 430 discovered that 19 tropic years coincided very nearly with 235 synodic periods of the moon, in other words the moon is in the same position in the sky with respect to the surrounding stars every 19 years (the Metonic cycle), and the ancient Greeks thought this so extraordinary that they are said to have cut these numbers on buildings and highlighted them in gold. This term was first used in the computistic poem Massa Compoti by Alexander de Villa Dei in 1200. A later scribe added the Golden Number to tables originally composed by Abbo of Fleury in 988.

The lunar cycle is a period of 19 years, after which the phases of the moon again fall on approximately the same dates in the solar year. The number of the year in the period is found by adding 1 to the number of the year and dividing by 19; the remainder is the number of the year. If there is no remainder, the number of the year-or the golden number is 19. (In mathematics this can be expressed as (year number modulo 19) +1.)

For example, 2011 divided by 19 gives 105, remainder 16. Adding the number before to the number that you are using gives a golden number of 17. In computing, the modulo operation finds the remainder of division of one number by another.

Solar Cycle:

The number of the year in the solar cycle is found according to this rule:

The solar cycle is the period of 4 times 7 = 28 years, after which period of days of the week in the Julian calendar again fall on the same dates in the solar year. The number of the year in the period is found by adding 9 to the golden number of the year and dividing by 28; the remainder is then the number of the year. If there is no remainder, the number is 28.

Cycle of Indiction:

The cycle of indiction is a period of 15 years but has no connection with astronomical periods. Nothing is known for certain about its origin, perhaps it was used by the Romans, possibly as an interest or fiscal term, but it is continuous from that time throughout the reckoning. The number of the year in the indiction is found by adding 3 to the number of the year and dividing by 15. The remainder is the number of the year. If there is no remainder, the number of the year is 15.

The three cycles, the solar cycle, golden number and indiction together form a period of 28 x 19 x 15 = 7,980 years. This period has often been used in ancient times for the dating of documents; in addition to the rather uncertain date, the number of the year in the three cycles were given. In this way the year can be determined with perfect certainty. If a year is denoted by, say, solar cycle 12, indiction 3 and golden number 9, it will be an easy matter to fix the year at 255 after our reckoning. The three cycle numbers will only fit in for that year or for years 7,980 years before or after it. The period 7,980 years is called the Julian period. Year 1933, which has solar cycle 10, indiction 1 and golden number 15, is the 6,645^th year in the Julian period. This dial is not needed for the Easter calculation

Day of the week that January 1^st falls on

This dial is self explanatory and not used for the Easter calculation.

Theory behind the calculations

To each day in a calendar year, the Easter cycle implicitly assigns a lunar age, which is a whole number from 1 to 30. The moon's age starts at 1 and increases to 29 or 30, then starts over again at 1. Each period of 29 (or 30) days of the moon's age makes up a lunar month. With occasional exceptions, 30-day lunar months alternate with 29-day months. So a lunar year of 12 lunar months is reckoned to have 354 days. The solar year is 11 days longer than the lunar year. Supposing a solar and lunar year start on the same day, with a crescent new moon indicating the beginning of a new lunar month on 1 January, then the lunar year will finish first, and 11 days of the new lunar year will have already passed by the time the new solar year starts. After two years, the difference will have accumulated to 22: the start of lunar months fall 11 days earlier in the solar calendar each year. These days in excess of the solar year over the lunar year are called epacts (Greek: epakta hèmerai). It is necessary to add them to the day of the solar year to obtain the correct day in the lunar year. Whenever the epact reaches or exceeds 30, an extra (so-called embolismic or intercalary) month of 30 days has to be inserted into the lunar calendar; then 30 has to be subtracted from the epact.

Note that leap days are not counted in the schematic lunar calendar: The cycle assigns to the first day of March after the leap-day the same age of the moon that the day would have had if there had been no leap-day. The nineteen-year cycle (Metonic cycle) assumes that 19 tropical years are as long as 235 synodic months. So after 19 years the lunations should fall the same way in the solar years, and the epacts should repeat. However, 19 × 11 = 209 ≡ 29 (modulo 30), not 0 (mod 30); that is, 209 divided by 30 leaves a remainder of 29 instead of being an even multiple of 30. So after 19 years, the epact must be corrected by +1 day in order for the cycle to repeat. This is the so-called saltus lunae or moon's leap. The extra 209 days fill seven embolismic months, for a total of 19 × 12 + 7 = 235 lunations. The sequence number of the year in the 19-year cycle is called the "Golden Number", and is given by the formula:

                   GN = Y modulo 19 + 1

That is, the remainder of the year number Y in the Christian era when divided by 19, plus one.

Using the method just described, a period of 19 calendar years is also divided into 19 lunar years of 12 or 13 lunar months each. In each calendar year (beginning on 1 January) one of the lunar months must be the first one within the calendar year to have its 14th day (its formal full moon) on or after 21 March. This lunar month is the Paschal or Easter-month, and Easter is the Sunday after its 14th day (or, saying the same thing, the Sunday within its third week.) The Paschal lunar month always begins on a date in the 29-day period from 8 March to 5 April inclusive. Its 14th day, therefore, always falls on a date between 21 March to 18 April inclusive, and the following Sunday then necessarily falls on a date in the range 22 March to 25 April inclusive. In the solar calendar Easter is called a moveable feast since its date varies within a 35-day range. But in the lunar calendar, Easter is always the third Sunday in the Paschal lunar month, and is no more "moveable" than any holiday that is fixed to a particular day of the week and week within a month.

Tabular methods - background

The claim by the Roman Catholic Church in the 1582 papal bull Inter gravissimas, which promulgated the Gregorian calendar, that it restored "the celebration of Easter according to the rules fixed by ... the great ecumenical council of Nicæa" was based on a false claim by Dionysius Exiguus (525) that "we determine the date of Easter Day ... in accordance with the proposal agreed upon by the 318 Fathers of the Church at the Council in Nicaea." The First Council of Nicaea (325) only stated that Easter was to be celebrated by all Christians on the same Sunday—it did not fix any rules to determine which Sunday. The medieval computus was based on the Alexandrian computus, which was developed by the Church of Alexandria during the first decade of the 4th century using the Alexandrian calendar. The Eastern Roman Empire accepted it shortly after 380 after converting the computus to the Julian calendar. Rome accepted it sometime between the sixth and 9th centuries. The British Isles accepted it during the 7th century except for a few monasteries. Francia (all of Western Europe except Scandinavia (pagan), the British Isles, the Iberian peninsula, and southern Italy) accepted it during the last quarter of the 8th century. The last Celtic monastery to accept it, Iona, did so in 716, whereas the last English monastery to accept it did so in 931. Before these dates other methods were used which resulted in dates for Easter Sunday that sometimes differed by up to five weeks.

This method for the computation of the date of Easter was introduced with the Gregorian calendar reform in 1582.

 The general method of working was given by Clavius in the Six Canons (1582), and a full explanation followed in his "Explicatio" (1603).

Easter Sunday is the Sunday following the Paschal Full Moon date. The Paschal Full Moon date is the Ecclesiastical Full Moon date following 20 March. The Gregorian method derives Paschal Full Moon dates by determining the epact for each year. The epact can have a value from * (=0 or 30) to 29 days. The first day of a lunar month is considered the day of the crescent new moon. The 14th day is considered the day of the full moon.

Historically the Paschal Full Moon date for a year was found from its sequence number in the Metonic cycle, called the golden number, which cycle repeats the lunar phase on a certain date every 19 years. This method was abandoned in the Gregorian reform because the tabular dates go out of sync with reality after about two centuries, but from the epact method a simplified table can be constructed that has a validity of one to three centuries.

The epacts for the (2009) Metonic cycle are:          

 (M=March, A=April)

This table can be extended for previous and following 19-year periods, and is valid from 1900 to 2199.

The epacts are used to find the dates of the New Moon in the following way: Write down a table of all 365 days of the year (the leap day is ignored). Then label all dates with a Roman number counting downwards, from "*" (= 0 or 30), "xxix" (29), down to "i" (1), starting from 1 January, and repeat this to the end of the year. However, in every second such period count only 29 days and label the date with xxv (25) also with xxiv (24). Treat the 13th period (last eleven days) as long, though, and assign the labels "xxv" and "xxiv" to sequential dates (26 and 27 December respectively). Finally, in addition, add the label "25" to the dates that have "xxv" in the 30-day periods; but in 29-day periods (which have "xxiv" together with "xxv") add the label "25" to the date with "xxvi". The distribution of the lengths of the months and the length of the epact cycles is such that each civil calendar month starts and ends with the same epact label, except for February and for the epact labels xxv and 25 in July and August. This table is called the calendarium. The ecclesiastical new moons for any year are those dates at which the epact for the year is entered. If the epact for the year is for instance 27, then there is an ecclesiastical new moon on every date in that year that has the epact label xxvii (27).

Also label all the dates in the table with letters "A" to "G", starting from 1 January, and repeat to the end of the year. If, for instance, the first Sunday of the year is on 5 January, which has letter E, then every date with the letter "E" will be a Sunday that year. Then "E" is called the Dominical Letter for that year (from Latin: dies domini, day of the Lord). The Dominical Letter cycles backward one position every year. However, in leap years after 24 February the Sundays will fall on the previous letter of the cycle, so leap years have two Dominical Letters: the first for before, the second for after the leap day.

In practice, for the purpose of calculating Easter, this need not be done for all 365 days of the year. For the epacts, you will find that March comes out exactly the same as January, so one need not calculate January or February. To also avoid the need to calculate the Dominical Letters for January and February, start with D for 1 March. You need the epacts only from 8 March to 5 April. This gives rise to the following table:

Now that we have gone through the very tortured path of using the tabular method to determine Easter, the second table below is a mathematical algorithm that will give us the result we are looking for where the only information needed is the year we are looking to determine Easter’s date. Note the mathematical expressions of mod and floor. Mod or modulo has been defined. The floor function is also called the greatest integer function, and its value at x is called the integral part or integer part of x. An example below will clarify:

We are only concerned with the floor function and positive expressions. And now for the algorithm to determine Easter in the Gregorian calendar.

Gregorian algorithm

"A New York correspondent" submitted the algorithm below for determining the Gregorian Easter to the journal Nature in 1876. It has been reprinted many times, in 1877 by Samuel Butcher in The Ecclesiastical Calendar, in 1922 by H. Spencer Jones in General Astronomy, in 1977 by the Journal of the British Astronomical Association,  in 1977 by The Old Farmer's Almanac, in 1988 by Peter Duffett-Smith in Practical Astronomy with your Calculator, and in 1991 by Jean Meeus in Astronomical Algorithms.

Just for fun, I present the graphic above. It shows the percentage of time easter will fall on a certain date in March or April throughout the complete 5,700,000 year cycle which is calculated as follows: The solar and lunar corrections repeat after 4 × 25 = 100 centuries. In that period, the epact has changed by a total of −1 × (3/4) × 100 + 1 × (8/25) × 100 = −43 ≡ 17 mod 30. This is prime to the 30 possible epacts, so it takes 100 × 30 = 3000 centuries before the epacts repeat; and 3000 × 19 = 57,000 centuries before the epacts repeat at the same Golden Number. This period has (5,700,000/19) × 235 + (−43/30) × (57,000/100) = 70,499,183 lunations. So the Gregorian Easter dates repeat in exactly the same order only after 5,700,000 years = 70,499,183 lunations = 2,081,882,250 days. However, the calendar will already have to have been adjusted after some millennia because of changes in the length of the vernal equinox year, the synodic month, and the day due to the slowing of the Earth's rotation.