Introduction

One of the most famous and beautiful medieval calendars begins the Trés Riches Houres of Duke de Berry.  Each month of the calendar is represented by a full-page illumination depicting a detailed scene of medieval life during that month, along with astrological data along the concentric circles at the top of the page.  The facing page is a text calendar.

 The top circular section of the illuminated calendar pages is not filled in for the months of January, April, May, and August.  This indicates that the illumination and calligraphy were done by different artisans.  In order to fill in the top section of the missing months, I examined these calendar pages in more detail to determine the purpose of each section.

First, the facing text page of each month is complete.  So, that can be used as a reference to determine the information needed for the illuminated circular sections.  Here are the first five days of the text page for January:

Each text page starts with an illuminated KL which stands for the Latin kalendas which was the name of the first day of each month.  Dates were not specified using the same system we use in modern times.  Instead, the complex ancient Roman system was used, based on lunar cycles.  Days are counted down backwards from the nones (the fifth or seventh day of the month) of the month, then from the ides (either the thirteenth or fifteenth day), and again from the first day of the following month (the kalends).  These three special days were also feast days in the month (as in the familiar "Ides of March").

You can see in the example above that the nonas is on the fifth day of January, and the days preceding are numbered two, three, four going backwards from the nonas.  This day number is shown in red in the third column.  The longer text in the middle of the page indicates the names of saints for each day.  To the right of the saints is the length of the day between sunrise and sunset in hours (blue) and minutes (red).  To the far right is the "New Golden Number" which I will describe in detail later.  It is related to the normal "Golden Number" shown in gold in the first column.  Finally, the second column (in black) is the dominical letter indicating the day of the week.  This perpetual calendar starts with the letter A for January first.  For a given year, you can compute the dominical letter for that year which indicates the letter on which Sunday falls for that year. 

Here is a closeup view of the semi-circular region above the illuminated calendar page for the month of February:

At the center is a monochrome blue illustration of a chariot holding the sun being drawn by horses.  This same picture appears on all twelve calendar pages.  The innermost ring indicates the day of the month in arabic numbers starting at one.  In the case of February, it goes from one to twenty-eight.  Each number alternates in gold and red.  Notice that they are not showing the roman day as described above on the text page, but rather the more modern absolute day number.

The next two rings are used to show the dates on which new moons occur and are related to the Golden Number and will be discussed at length below.  The next ring has the text "primacones Lune mensis novembus dies xxx".  primacones Lune basically translates to "beginning phase of the moon" or "new moon"; mensis is Latin for month which is followed by the name of the month in Latin;  dies is Latin for days which is followed by the number of days in the month in Roman numerals.  

The next ring of blue illumination depicts the falling and rising sign of the zodiac for the month.  In this case, the Aquarius is falling (ending) and Pisces  is rising (beginning).  Gold stars illuminate the blue background.  Outside of this is the text "finis graduum aquari initium pisaum gradus xix".  This roughly translates as "the final steps of Aquarius.  The initial steps of Pisces" with the xix indicating the day of the rising sign that falls at the end of the month.  In this case, the end of the month is the nineteenth day of Pisces.  On the very outside ring is the days within the solar calendar of the zodiac numbered sequentially from one to thirty with arabic numbers.  

Golden Numbers

Calendars were often included as part of a Book of Hours because of the religious significance placed on many dates.  In particular, determining what day of the year Easter would fall upon was of major importance to the Church.  Based upon the bible, the Church defined Easter as: 

"the Sunday after the first full moon of Spring.  The full moon may be on the first day of Spring (March 21st), but Easter will never be, and if the full moon is on a Sunday, Easter is on the following Sunday."  

In 325 AD, the Council of Niceae hammered out a set of compromises to deal with the potential problems determining the exact moment of the full moon, and problems with variance depending upon your longitude in the world.  Their compromise states that the Easter full moon happens on a date, not a particular time of day, and it is on the same date everywhere in the world. It also states that the date of the Easter full moon is determined by the motions of a "mathematical moon" which only approximate the real moon.  For clarity, the calculated date of the Easter full moon is called the Paschal full moon.

So, in order to calculate when Easter would occur, one needed to know when the Paschal full moon after the first day of spring (March 21st) would occur.  In developing the perpetual calendar of the Trés Riches Heures, the dates of all such computed full moons are shown.  But how were these full moon dates computed, and how does one decode their number and letter representations on the calendar?

In Medieval times, they used the Julian Calendar, which was divided into the twelve months that we use today.  They employed a leap year every four years by adding a day to February.  Astronomers as early as Babylonian times noticed that 235 lunar months (29.53 days in a lunar month) was almost exactly 19 years of 365.25 days (235 * 29.53 = 6939.55, 19 * 365.25 = 6939.75).  In other words, if a full moon occurred on a specific date,  it would occur on that same date nineteen years later.  Since the lunar cycle repeated every 19 years, they assigned a "Golden Number" to each year which indicated it's position in this 19 year cycle.  To compute the Golden Number for a particular year, simply divide the year by 19, determine the remainder and add one.  For example, the Golden Number for 1492 is 11.  Thus, Easter occurs on only 19 specific dates, and this date is given by the Golden Number for that year.

Since the 19 year cycle is true for any full moon, not just the Easter full moon, you can construct a perpetual calendar that will tell you when each full moon will occur.  The date of the full moons will be the same for any two years with the same Golden Number.  The same is true for any particular phase of the moon.  While Easter is defined based upon a full moon, it is easier to observe more accurately the new moon by looking for the first sliver of the moon.  These new moons were observed and tabulated as the Metonic cycle producing the following table:

To read this table, compute the Golden Number for a particular year, then read down the column to determine what day(s) of the month will have a new moon.  For example, if the Golden Number turns out to be 2, then there will be a new moon on January 1st and 31st, then on March 1st and 31st, then on April 29th, and so on.  The full moon is simply thirteen days after the new moon (the 14th day of the lunar cycle).

For the purposes of creating a perpetual calendar, one looks at the row for a particular month.  For example, in January there will be a new moon on January 23rd when the Golden Number is one, or a new moon on January 12th if the Golden Number is two, and so on.  

Go back up and look at the first five days of the text calendar page for the month of January.  Look at the gold numbers in the first column.  You'll see an iii next to January 1st, a xi (a bit hard to read in this smaller version, but clear in the original) next to January 3rd, and a xix next to January 5th.  Now look at the new moon table above: a new moon occurs on January 1st when the Golden Number is 3, on January 3rd when the Golden Number is 11, and on January 5th when the Golden Number is 19, and so on.  So this first column of gold numbers indicates the Golden Number for which a new moon occurs on that specific date.  If one looks at each text page for each month, you'll find that it exactly matches the table of new moons shown above.

However, this calendar is not exact.  235 lunar months isn't quite exactly 19 years.  And this table was derived in the year 325 AD.  Over time, the discrepancy between the computed  new moon and the observed new moon became greater and greater.   In fact, the error amounts to one day every 310 years.  So, by the year 1416 or so when the Trés Riches Hours was created, this discrepancy was about 3 days between the computed new moon and observed new moon.  That was quite an obvious difference and was causing much friction between scholars and the Church.

Eventually, in 1582, the Church would commission Christoph Clavius to come up with an improved system for calculating Easter, which resulted in the Gregorian Calendar that we use today (many non-papal countries delayed the adoption of this new calendar until the 18th century).  However, even before the official change of standard from the Church, many scholars of the late middle ages were tackling this problem and trying to devise more accurate systems or  ways to adjust the existing system.  The Duke of Berry was obviously aware of this issue.  In fact, an inventory of his library lists several maps and scientific works, including an astrological treatise on the seven planets. Such works provided tables of the observed new moon cycle, so in his calendar he has his artisans list the "New Golden Number", which is shown in gold on the right-most column of the text calendar.  

For example, January 1st is labeled with xix, January 3rd with viii and so on.  Looking carefully at this "New Golden Number" you will see that it follows the same pattern as the original Golden Number, but is simply shifted by approximately 3 days.  But this isn't consistent.  Sometimes the shift is by 2 days and sometimes by 4 days.  For example, on a year with Golden Number of 19, the original computation would have a new moon on January 5th, but the new list shows a new moon on January 1st, which is 4 days earlier.  Clearly these new Golden Numbers were taken from some astronomical work of the time, but the exact source is not known.  Several scholars in Oxford had published works within the previous hundred years discussing the problem, tabulating lunar cycles, and proposing ways to correct the existing calendar system.

This brings us, finally, to the decoding of the two inner rings of the illuminated calendar pages.  The determination of Easter was still based on the original Golden Numbers at this time.  But the Duke of Berry decided to show the actual dates of the new moon on his perpetual calendar's illuminated pages based upon the New Golder Number, rather than the new moons computed by the Church.  Each crescent moon symbol represents a date upon which there can be a new moon.  The New Golden Number for those years that have a new moon on that date are shown inside the moon symbols by a letter rather than a number.  So, the letter 'a' represents a year with a New Golden Number of one, 'b' for two, and so on.  Two things need to be kept in mind when performing this conversion between numbers and letters:  first, there was no letter 'j' at this time in period.  So, while the letter 'i' was used for 9, the letter 'k' was used for 10.  So, they have the letter 't' representing 19, even though it's the 20th letter in the modern alphabet.  The second consideration is that they used the long form of the 's', so the symbol for 18 is sometimes confused with an 'l' or 'f'. 

The Scroll

For the scroll done based upon these calendar pages (see separate documentation), I needed to show the full calendar arches for the months of May and January.  Both of these months were left blank in the original document.  Instead of learning how to decode these calendar pages, I could have simply taken the New Golden Numbers from the facing text pages and used that information to fill in the arches.  However, once I learned the purpose of the New Golden Number and understood that the Duke had purposely displayed the actual new moon cycle, rather than the Church-computed new moon cycle, I realized I had another alternative.  Rather than filling in the pages on the scroll with the new-moon cycle from the fifteenth century, I further updated the New Golden Number to the present day.  So, the new moon cycle shown on the scroll are computed for the year 2004 AD, resulting in an additional correction of nearly two days as compared to the New Golden Number used in the original manuscript.

Conclusion

Jean, Duke of Berry, was obviously an extraordinary noble in his time.  During the tumultuous times of the fifteenth century, nearly one hundred years before the reformation of the Church, and one hundred and sixty years before the development of the Gregorian calendar, the Duke was reading scholarly works from Oxford and publishing in his own Book of Hours the fact that the Church was wrong about the date of Easter.  Perhaps only a Noble could get away with such heresy.  The fact that the right-most column of the text pages matches the encoded letters within the illuminated arch pages clearly shows that this column was not added at a later date but was done at the time the original manuscript was created.  This blend of art and science makes the Trés Riches Heures an even more extraordinary manuscript.

Bibliography

Cazelles, Raymond and Rathofer, Johannes, Illuminations of Heaven and Earth (The Glories of the Tres Riches Heures du Duc de Berry), 1988, Harry N. Abrams, Inc., New York

A very high-quality reproduction and study of many pages from the Trés Riches Heures including a basic description of the calendar pages.  This source has very clear and detailed enlargements of the illuminated pages of the calendar.

http://www.ortelius.de/kalender/east_en.php, The Calculation of Easter

Excellent background on the history of computing Easter.  The Golden Number table comes from this reference and has been verified with other calculations.  Many sources on the Internet do not compute this table correctly for the old Julian calendar.

http://www.davros.org/misc/easter.html, Whence Easter

Another good source for the computation of Easter from the old calendar.

http://www.iclasses.org/assets/math/scripts/science/new_and_full_moon_calculator.html

An online calculator for determining the new moon cycle for any modern year.