# On the Units of Time, Part III: The Year

Time Machine September 25, 2002 admin

# On the Units of Time

## Part III: The Year

by Edward Hahn

February 19, 2001

#### Uh, Oh…He’s Going to Ask That Question Again…

So, what, exactly, is a year? The standard answer is that it’s the time it takes for the seasonal cycle to repeat. Since the seasons are determined largely by the axial tilt of the earth, and the earth’s axis points in the same direction throughout the entire year (otherwise we wouldn’t have a north star), the cycle takes about the same time as a single revolution of the earth around the sun.

Figure 1: The seasons are caused by the tilt of the earth’s axis. A year is one complete cycle of the seasons.

However, when it comes to designing a calendar which is supposed to work for hundreds or thousands of years, additional rigor is required. The problem is that the direction which the earth’s axis points at isn’t entirely static. Like a wobbling top, the axis of the earth describes a circle in the sky, over the course of 25,800 years:

Figure 2: The earth’s axis rotates, like a wobbly top, making a full wobble in about 25,800 years. That means that the direction that the earth’s axis points at changes over time.

Figure 3: The approximate circle in the sky that the earth’s axis inscribes over time. The red dots represent the approximate spot in the sky where the earth’s axis is pointing on that date . Thus, at the time when the pyramids were built (~3000 BC, or a full cycle before 22,800 AD), the earth’s axis pointed toward Thuban, in the constellation of Draco – this was the “pole star” for the pharaohs.

The significance of this precession of the equinoxes is that, since the beginning of the year is set relative to the vernal equinox (more about this later), and the equinox is slowly changing relative to the stars because the earth is wobbling, then the calendar must be designed to measure something slightly different than a single orbit of the earth around the sun, relative to the stars.

The Vernal Equinox Year (365.242374 days), therefore, is slightly different from the Sidereal Year (365.256363 days). This may not seem like much of a difference: it amounts to about 20.4 minutes per year, but enough to make a day error in about 70 years – clearly enough to have to account for it in a calendar system. It also means that a simple observation of the stars couldn’t be used by ancient astronomers as a long-term reference to align the calendar, and instead requires more subtle observational methods. Amazingly, the precession of the equinoxes was discovered by Greek astronomer Hipparchus in 130 BC!

You’ll note that I used the Vernal Equinox Year. This is because the Gregorian calendar, which is de facto the worldwide secular calendar, is also the Roman Catholic calendar, and was designed to keep the date of Easter bouncing around within a fixed period of time in the calendar. The date of Easter is determined by the vernal equinox and the full moon (more about this later), so it was correct for the designers of the calendar to use the vernal equinox year as the time to optimize the calendar across.

(A side note: Some people out there might be wondering why I didn’t use the Tropical Year of 365.24219 days length? In Duncan Steel’s book Marking Time, he points out that both the US Naval Observatory and the Royal Greenwich Observatory make a mistake when saying that, “the tropical year is defined as the mean interval between vernal equinoxes.” This is not astronomically correct – the tropical year is the average time between consecutive occurrences of any date on the calendar. Since the earth on average orbits the sun in the same amount of time regardless of when you start your measurement, shouldn’t they be the same? Well…they aren’t – the vernal equinox year is an average over some number of orbits measured on a single calendar day, while the tropical year averages once more across all the days of the year. The difference is about 16 seconds a year – not much, but even so the people who designed the Gregorian calendar knew enough to aim for the vernal equinox year since they were trying to stabilize the date of Easter, and not the entire year.)

### The Julian Calendar – a Recap

The details of the Julian calendar were discussed in Part II. The main motivation for the Julian calendar was to straighten out the previous calendars of the Roman republic. These had drifted back and forth relative to the seasons, often due to political whim; by the time Julius Caesar finished fighting both Rome’s enemies and his own rivals for power in 47 BC, it was fast by about 90 days.

As a result, in addition to changing the lengths of months to our modern values in his rules for a new calendar, Caesar also declared that 46 BC would be 445 days long, to get the calendar back in step with the seasons. But it is not precisely known exactly what event (celestial or secular) was he trying to tie to a particular date in the Julian calendar.

Nevertheless, he and his advisor, Sosigenes, did a pretty good job of eliminating the confusion in the Roman year. Aside from some initial missteps about how the leap year would be observed, the Julian calendar far outlasted the Roman Empire. Each year was 365 days long, except for every fourth year, which was given an additional day in February – essentially what we (usually) still do today – and in fact is still being used today as the ecclesiastical calendar for the Orthodox Christian church.

Although the best astronomers of his day had measured a year length of slightly less than the Julian average of 365.25 days, Caesar probably felt that his simple rule was much better than the confusion that had gone before – and perhaps figured that some future ruler would make the necessary adjustments. In some sense it is ironic that his calendrical legacy lasted unchanged for over 1600 years.

### The Gregorian Reform

#### Religious Considerations

After the conversion of Constantine to Christianity in the early 4th century AD, there was suddenly a new, religious target for the calendar – to observe Christian rites at the correct time. In his decree of 321 AD, Constantine instituted the Christian seven-day week with Sunday as the holy day throughout the Roman empire. This appears to have been chosen both from religious considerations (i.e. the fact that the resurrection seems to have occurred on a Sunday) as well as practical considerations (i.e. the popular cult of Mithraism (sun worship) in the Army already held Sunday as a holy day.) By 401 AD, the official Roman state religion was changed to Christianity.

Figure 4: Constantine I, Emperor of Rome

The adoption of Sunday as the weekly holy day illustrates the fact that some early Christians desired to make their religion as distinct from Judaism as possible. However, it was obvious that the most important Christian rite, Easter, was dependent on the Jewish calculation of Passover, since Jesus appears to have been crucified on a Passover that fell on a Friday, and was resurrected on the “third” day (Sunday).

(Recall that the Passover festival is a commemoration of the Exodus of the Jews from Egypt, under the leadership of Moses. According to custom, this occurs on the first full moon after the vernal equinox, and can fall on any day of the week. Passover occurs on the same date in the Jewish calendar (Nisan 14) because their calendar is lunar.)

The fact that Passover was not fixed relative to the Julian calendar caused consternation among some, but not all, of the various Christian sects, who saw three alternatives: (1) observing the crucifixion on Passover and the resurrection on the third day after (regardless of the day of the week), (2) observing the crucifixion on Passover, but observing the resurrection on Sunday (regardless of the number of days between), and (3) the solution which was final chosen, observing the crucifixion on Friday, the resurrection on Sunday, and ignoring the Passover full moon.

In 325 AD, the Council of Nicaea was convened with representatives from all branches of the Christian Church. Incorporating an earlier church resolution that all Christians should observe Easter on the same day, the council then agreed that “all brethren in the East, who formerly celebrated [Easter] at the same time as the Jews [celebrated Passover], will in the future conform to the [practice of the Church of Alexandria.]” Unfortunately, while this resolution was adopted to unify the church, there were still squabbles among the different Christian sects on when Easter was to occur, which helped lead to a final split between Eastern (Orthodox) and Western (Catholic) churches around 1054 AD which still exists today.

#### Astronomical Considerations

It should be noted that there was agreement on the basic formula for Easter: it should occur on the first Sunday after the first full moon after the vernal equinox. The problem was that the various churches had adopted different calculations for both the date of the vernal equinox as well as when the full moon was predicted to occur.

From 275 AD to 343 AD, the equinox was defined by different sects to occur at different times, within the range of March 18-25. As an aftermath of the Council of Nicaea, by 343 AD, the date of the ecclesiastical vernal equinox was fixed at March 21. However, the Julian calendar yielded a vernal equinox year that was slightly long on average, which meant that the actual date the vernal equinox occurred was moving farther and farther away from March 21. By the time of the Gregorian reform in 1582, it was off by about nine days.

Thus, one requirement for a reformed calendar would be, if not to keep the vernal equinox on the same date, at least hovering around the same date without drifting.

The second problem, that of calculating the full moon, was a much more difficult problem. Given the fact that Christianity was becoming quite widespread geographically by the fourth century, visual observation of the traditional new moon was not practical if Christians everywhere were to celebrate Easter on the same day (see Part II for more information on the vagaries of the moon’s motion.) In addition, the Christians were hesitant to follow the exact method of the Jews for predicting the date of the full moon, even though it was also their desire to ensure that Easter and Passover did not coincide. Even putting aside the issue of what method was used to calculate full moons, the Venerable Bede in the 8th century first pointed out in writing that the predicted and observed full moons were getting out of sync systematically.

Therefore, the second requirement for a reformed calendar was to design a more accurate method of predicting the dates of future full moons.

#### The Shaky Road to Reform

The observation by Bede was followed by several others over the next several hundred years, including a direct plea by Roger Bacon to Pope Clement IV in the thirteenth century that calendar reform was needed. A different Pope Clement (VI) in 1344 asked for advice on ways to reform the calendar, but was unable to implement them due to other matters, including the Black Plague of 1348/49. This was followed by a period of many years where there were repeated political upheavals within the church, with rival factions in the church crowning different Popes, and ultimately the sundering of the Catholic Church by the Protestant Reformation.

Pope Sixtus IV was prepared to take up the question of calendar reform in 1475, but the most qualified man, Johannes Müller (a.k.a. Regiomontanus) died before he could provide a solution to the pontiff. The issue was taken up again in the early 1500s by Popes Julius II, Leo X, Paul III, Pius IV, and Pius V, with the later Pius making a stop-gap reform of the method used to predict full moons, based on recommendations made by the 3rd Council of Trent.

#### Ugo Buoncompagni Steps In

The Councils of Trent had many other things to discuss other than calendar reform; these included dealing with the upstart Protestant Reformation and the demand by English King Henry VIII for an annulment of his marriage to Catherine of Aragon. (Henry VIII eventually broke with Rome over this issue and created the Anglican church.) One of the players who distinguished himself as a vigorous man of action against the Protestants in these debates was an Italian bishop, Ugo Buoncompagni.

Figure 5: Pope Gregory XIII (born Ugo Buoncompagni). Pope Gregory used the advice of Aloysius Lilius and Christopher Clavius in the reform of the calendar.

This same Ugo Buoncompagni was elected Pope Gregory XIII after the death of Pius V. In addition to continuing a vigorous front against the Protestants, he became convinced that calendar reform had been languishing for too long, and was determined to leave his mark on the calendar.

His main advisors were Aloysius Lilius, who died early in the reform process but whose plan was represented by his brother Antonio, and Christopher Clavius, who expanded Lilius’ plans into an 800 page volume. In addition to modifying the average length of the year, thus stabilizing the date of the vernal equinox, the key feature of the Lilius/Clavius plan was a scheme to settle the date of Easter in a manner that was irrefutably superior. This plan was approved by Gregory XIII, and was decreed via a “Papal Bull” to be implemented in October of 1582.

The changing of the length of the year to keep the vernal equinox from drifting too far was simple enough. The papal bull eliminated the scheduled leap year in the century years (e.g. 1700 or 1800), except when the century years are divisible by 400 (e.g. 2000 or 2400). Thus, 1900 and 2100 are not a leap years, but 2000 is a leap year. This gives an average year length of 365.2425 days, pretty close to the vernal equinox year of 365.242374 (but more about this later.) It should be noted that with this scheme, even though the ecclesiastical vernal equinox is fixed to 21 March, the astronomical equinox actually falls in the range from 19 to 21 March on the calendar, but stays in this range reasonably well.

The second reform was to put the vernal equinox back to the 21st of March – this required 10 days to be deleted from the calendar. This was required to keep a feasts tied to Easter from colliding with other feasts at the beginning of January. The papal bull instructed these days to be eliminated thus: Thursday 3 October 1582 was followed by Friday 15 October 1582 – thus causing a jump in the month and day cycle, but keeping the continuity of the days of the week.

The deletion of the ten days was adopted throughout the Catholic world gradually, with only Italy, Portugal, Spain, and Poland following it to the letter. Luxembourg, France, Belgium, and Catholic Holland followed before the end of 1583, and with the remaining Catholic world (parts of Germany, Switzerland, modern Czech Republic/Slovakia, and Hungary) performing the switch by 1587. Protestant countries started changing in 1700, with England and her colonies switching in September of 1752 (by which time eleven days needed to be deleted). Despite legends to the contrary, there were no riots or other widespread civil disturbances of note during the initial switch in 1582, nor in the switch for the English speaking world in 1752.

The third reform was in the calculation of the lunar cycle, which is used to calculate the date of Easter. Here, the method used is quite complex, involving Tables of Epacts and Golden Numbers to precisely determine which day Easter falls on, in perpetuity. Again, one of the objectives of this reform was to ensure that Passover and Easter did not fall on the same day. Despite their best efforts, Lilius and Clavius were unable to do so perfectly, and Easter and Passover coincide from time to time (most recently in 1981, with the next occurrence in 2123). If you are really interested in the details of the Easter reform, I recommend that you read Duncan Steel’s book which describes it more fully than would be appropriate here.

#### Some Comments on the Gregorian Calendar

First, it should be noted that the Orthodox Church still uses the Julian calendar as its basis to calculate Easter and other feasts. Other countries, such as the former Soviet Union, did not change over until the early part of the 20th century. Thus, despite the agreement back in 325 AD that all Christians should observe Easter on the same day, this wish has yet to be realized.

From time to time there is a suggestion that there should be a millennial rule added to the Gregorian calendar (there emphatically is not one in the papal bull of 1582), in order to make the calendar (current average year length of 365.2425 days) correspond more correctly to the tropical year (365.242190 days). One example of this is a proposal is to drop the leap year in 4000 AD, first suggested in 1849 by John Herschel, son of astronomer William Herschel and an astronomer in his own right.

Figure 6: Sir John Herschel. Herschel proposed a “millenial” rule to the calendar; as an astronomer, he should have known better…

(This early photograph of him is by the famous pioneer British photographer Julia Margaret Cameron, and is owned by TZer Michael Friedberg, who states, “It is a carbon print in my collection, that I bought from a London auction house. It had been in the Herschel family until the time of auction.”)

There are two things wrong with this reasoning: first as already stated, the Gregorian calendar was designed by Christians to keep the date of Easter fixed – thus, the objective of the calendar is to minimize the drift of the vernal equinox, not to minimize the overall drift of all of the seasons. The mean year measured between vernal equinoxes is 365.242374 days; since this value is much closer to the unmodified Gregorian year length, any anticipated “adjustment” that would be necessary is put off for several thousand years beyond 4000 AD.

More importantly, the length of the day and year are changing, such that over millennial timescales any corrections calculated on the current value for the tropical year (or vernal equinox year for that matter) are meaningless.

That being said, the Gregorian reform was not the best solution to keeping the calendar tied to the vernal equinox – and there is evidence that both Lilius and Clavius knew it. First, it should be recognized that even though the ecclesiastical vernal equinox is fixed to 21 March, the astronomical equinox actually falls in a 53 hour window from 19 to 21 March on the calendar, and stays in this range reasonably well. The main reason for this window is because the Gregorian calendar kept the Julian calendar’s four-year leap year cycle, and merely tacked on an additional correction at three out of every four century marks. This results in a cycle of 400 years, with 97 leap years over that period. Thus, while the error is small, it has 400 years in which to oscillate before being reset.

In contrast, a 33-year cycle which includes 8 leap years and 25 common years yields an average year length of 365.242424… days – both closer to the target of 365.242374, and with a short enough cycle to keep the vernal equinox on the same calendar day! Furthermore, the 33-year cycle can be tied to the traditional 33-years that Jesus lived as a man between his birth and his execution – surely something the laity would see as “divinely” inspired. (There is also an additional simplification to the calculation of Easter, having to do with simplification of the epacts, but again, see Duncan Steel’s book for more information.)

Despite the fact that 8-of-33 cycle would have been more accurate, and the fact that Clavius and Lilius almost surely knew about it, the 97-of-400 cycle was chosen. This was likely due to the fact that it was a small modification to the existing Julian calendar, and would thus be easier for the local priesthood to understand and calculate. (By the way, Duncan Steel also proposes a hypothesis, based on plausible evidence, that the English knew about the superiority of the 33 year cycle, and may have had calendrical oneupmanship at least partially in mind in their colonization attempts at Jamestown and at the “Lost Colony” of Roanoke Island.)

### The Calendar of Islam

I’ve spent quite a lot of time on the Gregorian calendar, mainly because it has essentially become the single worldwide secular calendar. However, a description of the religious calendars of the other major western religions is worthwhile, not least because they are all structured fundamentally differently from each other.

As discussed above, the Gregorian calendar is strictly a solar/seasonal calendar, designed to keep the calendar tied to the vernal equinox. However, this is not the only way to organize a perfectly usable calendar. Instead of sticking to the 365-and-change seasonal cycle, it is certainly possible to base your calendar entirely on the 29.5+ day lunar cycle, which is what the calendar of Islam does. However, unlike most Christian churches, in Islam there is no priestly hierarchy with which to propagate changes from a central authority. Furthermore, since Allah (via Mohammed) did not say anything about the details of the calendar in the Koran, there is room for debate within Muslim society without worry of going against the Koran. Thus, the practices outlined in the chapter vary somewhat among the various Muslim communities across the globe.

For example, Muslims in some communities follow a “popular” calendar, based entirely on empirical observations of the new moon. However, there is a calculated Muslim calendar used for official business, based on one that was set down by Caliph Umar I in 642, ten years after the death of Mohammed. It calls for 12 months, each of which corresponds to a single cycle of lunar phases. Since each lunation lasts 29.53059 days on average, the Muslim year lasts on average 354.367 days – or about eleven days shorter than the Gregorian calendar.

Rather than worry about this, the year is left to drift its way through the seasons, making a complete round of the seasons in 33.6 Muslim years. This leads to interesting consequences. For example, Allah calls for all Muslims to fast (including not drinking water) during daylight hours in the month of Ramadan. This is clearly much less of a burden when Ramadan falls in winter (with shorter daylight and cooler temperatures in the middle east) rather than summer (with longer hours of daylight and hotter temperatures.)

In the calculated Muslim calendar, each year contains 12 months of 30 or 29 days, which alternate. An extra day, analogous to a leap year day, is periodically added to the end of the last month. These are inserted 11 times over the course of 30 years, and take care of the fraction of extra time over 29.5 days which occur in each lunation. The margin of error is very small, with the calendar expected to stay within one day of the actual moon in 2500 Muslim years.

 Muslim Month Name Duration Muharram 30 days Safar 29 days Rabi’ I 30 days Rabi’ II 29 days Jumada I 30 days Jumada II 29 days Rajab 30 days Sha’ban 29 days Ramadan 30 days Shawwal 29 days Dhu’l-qu’da 30 days Dhu’l-hejji 29 or 30 days

### The Jewish Calendar

Notwithstanding the calendar of the Roman Republic (see Part II), a calendar which can keep in synchrony with the moon and the seasons is possible, and in fact is what the Jewish faith uses as the basis for their calendar. These “lunisolar” calendars are much like the Muslim calendar, except that an extra month is added in every so often to keep the entire year more-or-less lined up with the seasons.

From evidence in the Torah from the time of Solomon (ca. 1000 BC), the Jewish calendar was basically a lunar calendar and was empirical: months began after the physical sighting of a new moon in Jerusalem. An extra month was added ad hoc when it looked like the traditional sheaf of barley offered to God the day after Passover would not be ripe in time.

After this, it appears that the Jewish calendar was then influenced by the Babylonian captivity, which began in 597 BC: the modern names of the Jewish months are clearly related to the Babylonian month names. The Jewish calendar remained empirical until the Diaspora (i.e. the destruction of the Temple of Jerusalem by Roman general Titus in 70 AD), when the dispersal of the Jewish community led to much confusion, as groups of Jews could not count on the timely arrival of information from Jerusalem, and needed to figure out the calendar locally to celebrate feasts.

By the 4th century AD, a more structured calendar was set down (according to tradition) by Rabbi Hillel II, a patriarch within the Jewish faith. His calendar used the Metonic Cycle, discovered by Meton of Athens in 432 BC (the ancient Greeks also used a lunisolar calendar). This calls for a cycle of 19 years; 12 normal years of twelve months, and 7 years with an extra month (called embolismic years).

In addition, the months were no longer empirical, but instead used a theoretical moon with a fixed period between lunations.

There are two “starts” to the year in the Jewish calendar. For religious purposes, the Jewish year begins with Rosh Hashanah in the month of Tishri. For other purposes, the year begins in the spring with Nisan (recall that Passover falls on Nisan 14). The months normally alternate between 30 and 29 days, except in an embolismic year, where the sixth month (Adar) is given an extra day, and the extra month (Adar II) is given 29 days.

Finally, there are some rules to prevent, for example, Yom Kippur falling immediately before or after a Sabbath. These are for practical reasons; since food preparation, etc. are prohibited both on Yom Kippur and on the Sabbath – food prepared two days in advance might be spoiled by the second day. Thus, the calendar allows for delaying the beginning of a year by up to two days to prevent this from happening for a couple of Jewish holidays. These show up as varying lengths for the months of Cheshvan and Kislev for the previous year.

These rules combine to give a year of 365.24682 days on average, which is about 6 minutes long per year (i.e. causes one extra day every 216 years).

 Jewish Month Names Duration Nisan (civil new year) 30 days Iyar 29 days Sivan 30 days Tammuz 29 days Av 30 days Elul 29 days Tishri (religious new year) 30 days Cheshvan 29 or 30 days Kislev 30 or 29 days Tevet 29 days Shevat 30 days Adar 29 or 30 days Adar II (extra month) 29 days

Primary references for writing this article has been provided largely by two excellent works which have been written for the millennium: Mapping Time: the Calendar and Its History by E.G. Richards (Oxford University Press, 1998, ISBN 0-19-286205-7), and Marking Time: the Epic Quest to Invent the Perfect Calendar by Duncan Steel (John Wiley and Sons, 2000, ISBN 0-471-29827-1). These two works are written primarily for the layman, although they both contain a good dose of mathematics and astronomy.

Of the two, Steel’s book appears to be more definitive, as he has written this book in response to incorrect statements about the calendar made by such august authorities as the US Naval Observatory and the Royal Greenwich Observatory – in addition to those in more “popular” accounts of the calendar.

Richard’s book may appeal to both the sociologists and the mathematicians out there, as he investigates several ancient and modern calendars, and includes a section on algorithms to convert between these calendars.

Of particular utility in understanding how the stars and planets move is a freeware program, called Home Planet, which has a host of astronomical functions available. This program can be found for free at www.fourmilab.ch.

Image credits – all images are copyright © 2001 by Edward Hahn, except as follows:

• Bust of Constantine from the Palazzo dei Conservatori, Rome
• Image of Pope Gregory XIII in the public domain
• Image of Sir John Herschel courtesy of Michael Friedberg