# On the Units of Time, Part II: The Months and Weeks

Time Machine September 25, 2002 admin

On the Units of Time

Part II: The Months and Weeks

by Edward Hahn

February 13, 2001

### The Lunar Connection

The month is, of course, related, both horologically and etymologically to the moon – each modern month (with the exception of February) is slightly longer than the average length of a “lunation”, or the full cycle of moon phases from one new moon to the next new moon. Lasting a good deal longer than a single day but a good deal shorter than a year, and being marked by something easily observable, the month is a pretty handy unit of time to have in the calendar.

The term “new moon” in modern usage denotes that point in the moon’s cycle when the moon is completely dark – that is when the moon is exactly between the earth and the sun. However, the term was originally described to denote a true observed “new” moon – the day when the first sliver of moon is visible, just after sunset, which occurs a couple of days later than the modern new moon.

Astronomers now know enough about the lunar orbit to be able to calculate exactly where in its orbit the moon is – which leads to the modern usage of the term.

Since a 29.5 day period won’t divide equally into an approximate 365 day year something has to give when defining a calendar; how various cultures have decided to deal with this is the focus of this and the next part of this article.

### Ignore That Shiny White Thing in the Sky

As with the length of the day in Part I, the earth’s motion around the sun makes the month a bit longer than a single orbit of the moon around the earth:

Figure 1: The Lunar Orbit. When moving from the new moon at position A, the motion of the earth around the sun causes a single orbit of the moon (position B) to fall a bit short of the angle required for another new moon. (Distances and sizes are exaggerated.)

Because we’re working with a longer duration, the difference between a Sidereal Month (a lunar orbit of 360 degrees around the earth, 27.32158 days) and a Synodic Month (a single lunation or full cycle of lunar phases, 29.53059 days on average) is much larger than for a sidereal vs. solar day.

The phrase “on average” is very important – because the period of time between any two new moons (modern usage) can vary between 29.2 and 29.8 days, a variation of up to 14 hours! How can this be?

When defining an equation that describes the orbital motion of the moon over time, the number of elements to the equation is related to the different influences in the orbit – for example, the shape of the orbit, the effect of the pull of gravity from the sun and planets, etc. Because the moon is quite small relative to the earth and sun, and also is located very close to the earth, the moon’s orbit is going to be influenced by quite a few things. In fact, the complete equation which describes the motion of the moon has over 1,500 elements that influence the orbit(!)

On top of that, because the traditional synodic period of the moon is based entirely on the observation of the lunar phases, the exact start or end of a synodic month is going to vary with the observer’s location and altitude on the earth. When one considers that the lunar phases often have religious significance, the possibility that different groups of worshipers at different locations could celebrate the new moon on different days obviously could cause a lot of friction.

Some societies have settled the matter empirically (i.e. kept it completely “up to the sky”), and declared new months according to observation of a new moon, sometimes at a designated location to prevent squabbles. None of the major existing religious or civil calendars are run this way, although the calendar of Islam comes closest. It declares the start and end of particularly significant months (e.g. Ramadan) by priestly observation in Mecca of the slender crescent of the new moon; consequently, the symbol of Islam is the crescent. Note that according to tradition, if the sky is cloudy the night where an observed new moon would otherwise happen, the new moon is not declared, and must be put off until the next day the moon is visible. In some ways, these are the most “scientific” of calendars, because they are tied solely to observation rather than prediction of events.

Other societies have decided to deal with the problem by creating a fictional “mean moon” – similar to mean solar time. Many religious calendars which depend on the moon are based on this method, such as the Jewish and Chinese calendars, as well as the Christian calendar when it comes to determining the date of Easter only. It should be noted that some of these calendars make a bigger effort to keep the mean moon and the real moon in sync, while others let it go (e.g. the Christian religious calendar, which sometimes deliberately moves Easter out of sync with the moon.)

Finally, quite a few societies have abandoned the real moon altogether when defining a month. This includes the familiar Gregorian and Julian calendars, as well as the Bahá’i and the now-defunct French Revolutionary calendar.

As watch collectors, what we should take away from this is the fact that our moonphase watches are never going to be quite right with respect to the real moon. The standard method of using a 59 day wheel (i.e. two 29.5 day cycles) and advancing it once per day is probably good enough for practical purposes, in the sense that trying to improve the accuracy of the display is a futile task.

### The Moon’s Orbit and Eclipses

Because of the amazing accomplishments of Ludwig Oechslin at Ulysse Nardin, I’ll discuss one other aspect of the moon’s orbit before we move on – specifically, predicting lunar and solar eclipses:

Figure 3: The Astrolabium Galileo Galilei by Ulysse Nardin. In addition to the regular hour and minute baton hands, note the sun hand, moon hand, and dragon hand.

This watch shows the essence of the Trilogy’s ability to predict solar and lunar eclipses. The sun hand describes the position of the sun relative to the ecliptic, or line circling the sky where the sun, moon, and planets appear to lie. Similarly, the moon hand shows the position of the moon relative to the ecliptic. The dragon hand represents the location of the “first point of Draco”, whose significance I’ll describe in a minute.

First, imagine if the earth and moon shared the same orbital plane (i.e. the flat surface which contains the circle of the orbit.) If this were the case, then every time the moon was new, it would cast a shadow on the earth (i.e. a solar eclipse), and every time the moon was full, it would fall into the earth’s shadow (i.e. a lunar eclipse):

Figure 4: Hypothetical arrangement where the earth and moon share the same orbital plane. In this case, every New Moon would cause a solar eclipse, and Full Moon would be a lunar eclipse.

Obviously, this doesn’t happen. The reason is that the moon’s orbit is tilted relative to the earth’s orbit, by about five degrees:

Figure 5: The real arrangement. The moon’s orbit is tilted by about 5 degrees to the earth’s orbit, which means that the moon’s and earth’s shadow usually miss each other. (Drawing is not to scale.)

The moon’s orbit, therefore, intersects the earth’s orbit in two places, which lie along a line:

Figure 6: Perspective drawing of the earth and moon orbit planes and their line of intersection. This line of intersection is what the Dragon Hand in the UN Astrolabium represents.

This line doesn’t stay in the same place, but spins slowly around the earth like a compass needle. It takes about 18.61 years relative to the stars for this line to make one rotation. The line is known as the First Point of Draco – Draco being Latin for “dragon” – and is what the dragon hand on the Ulysse Nardin watches represent. (More correctly, the half of the line which points to where the moon’s crosses from underneath to above the earth’s orbit plane is the First Point of Draco, but the full dragon hand represents both halves of the line depicted above.)

The significance of this line is that when the moon, earth, and sun are aligned with each other, and this dragon hand is also pointing toward or away from the sun, the moon is lying in the same plane as the earth, and therefore an eclipse will occur. It will be a solar eclipse if the moon is between the sun and earth, and will be a lunar eclipse when the earth is between the moon and the sun. These eclipses will repeat in a cycle of 18.03 years, known as a Saros.

In other words, the Astrolabium is a moving model of Figure 6; the center of the watch represents the earth, and when the sun, moon, and dragon hands are aligned (either point in the same or opposite directions), an eclipse will occur. The Tellurium Johannes Kepler is also equipped with a dragon hand, so it can also be used in a similar fashion.

Figure 7: The Ulysse Nardin Tellurium Johannes Kepler

### Back to the Months (The Romans Get Involved)

Since the synodic period of the moon is about 29.5 days, and it makes no sense to have months end on half days, the most convenient way to have months with whole numbers of days is to alternate months between 29 and 30 days. So, how did we end up with months of 30 and 31 days, with a February of 28 or 29 days? The answer is to be found (somewhat hazily) in Roman history.

Figure 8: Romulus and Remus, the mythical founders of Rome. The legend is that these twin brothers were offspring of the god Mars and a Vestal Virgin; the brother of the virgin ordered the twins drowned, but they floated down the Tiber and were suckled by a she-wolf.

The first Roman calendar was attributed to Romulus, one of the mythical founders of Rome. For some reason, the unknown designer put forth a 304 day calendar, with 10 months of 30 and 31 days. Note that the other 61 days in the year still seem to have existed, but were not counted in the calendar. This year began in the spring (just before the vernal equinox), and gives us the names of the months from March to December:

1. March (Martius, from the god of war Mars)
2. April (Aprilis, possibly from the goddess Aphrodite (using the Greek name), but uncertain)
3. May (Maius, from Maia the goddess of the Spring)
4. June (Iunius, from Juno, the wife of Jupiter)
5. Quintilis (fifth, later changed by the Roman senate to Iulius/July after Julius Caesar’s assassination)
6. Sextilis (sixth, later changed by the Roman senate to Augustus/August during Augustus’ reign)
7. September (seventh)
8. October (eighth)
9. November (ninth)
10. December (tenth)

Clearly, a 304 day calendar would be pretty confusing, so the next king Numa Pompilius (ruled ~715 to 673 BC) added the months of February (Februarius, possibly from the Sabine word Februare, “to purify”) and January (Ianuarius, from the Ianus the two-faced god of doors) to the end of the year. Note that the order appears to have been reversed from our modern usage, but this fact is not certain. (It should also be noted that some attribute the original 304 day calendar to Numa, which would then make this his second attempt at a calendar.)

At this point, the Romans got superstitious – by this time it was thought that even numbers were unlucky; thus Numa removed one day from each of the 30-day months, making them 29. Finally, February was given 28 days (if you’re going to be unlucky, might as well be short). This yields a 355 day year, which appears to be lunar in origin (a more correct number, 354, was not used because it was even – thus unlucky.) Some have Numa adding an additional month (Mercedonius) of 22 or 23 days every other year, yielding an average year length of 365 days – but this “broke” the tie to the lunar cycles.

If nothing else, it’s pretty clear that the calendar was a pretty confusing thing for the first several hundred years of the Roman republic, as the Romans tried to do the impossible by keeping the calendar referenced to both the moon and the sun, without a more sophisticated scheme to keep them aligned.

During the next several hundred years, the calendar was kept by the college of pontiffs (literally the “bridge builders”) in Rome. Minor small reforms were tried, and extra months were added, supposedly to keep the calendar aligned with the seasons. However, these were often political devices to help cronies by arbitrarily adding a month to the length of years (thus, for example, adding days to existing contracts). The result is that the calendar was off by as many as 119 days (this can be determined by noting the date on which things such as eclipses were recorded, and comparing them to when they ought to have occurred if the Julian calendar had been in force.) At some time during this period, it appears that the beginning of the year was moved to January 1, on or before 153 BC by documentary evidence.

Figure 9: Julius Caesar (100-44BC)

By the time Julius Caesar got around to worrying about domestic affairs (47 BC), he decided that something radical needed to be done to the calendar. He enlisted the help of a Greek Egyptian, Sosigenes, whom he had met during his excursions in Egypt. Sosigenes dug up some proposals by the Egyptian pharaoh Ptolemy III Euergetes, which called for a year of 365.25 days to keep the calendar in line with the seasons. These became the basis for the Julian Calendar (three 365-day years, followed by a leap year of 366 days) – which abandoned the lunar cycle for good. Finally, to top it all off, to get the seasons back in line with the calendar Julius Caesar decreed an annus ultimus confusionis (“final year of confusion”), which made 46 BC a whopping 445 days long!

Caesar also changed the length of the days of the months to their modern values (the myth that it was Augustus that made August 31 days long, so as to be equal to July, appears to be a 13th century English invention.) It should also be noted that the leap year day was declared to be the 24th of February, not the 29th – the 24th was counted twice (see the section on the Roman dating system for additional information about this.)

Unfortunately for Caesar, Sosigenes’ instructions for the Julian calendar were misinterpreted (he was assassinated in 44 BC, prior to the first scheduled leap year.) It turns out that the Romans, but not the Greeks, counted off to a different convention than we do (the Roman “fourth” is our “third”), and leap years were observed every three years until Augustus was able to straighten things out between 9 BC and 8 AD. Since then, the leap years and months have been stable.

Augustus’ successor Tiberius, by the way, in a fit of rational thinking declined to have a month named after him – he thought that continuing the tradition would make things get really out of hand after the twelfth emperor. Later emperors tried to appropriate months later on, but Julius and Augustus are the only Roman rulers whose names have stuck to the present day.

Figure 10: The Ulysse Nardin GMT Perpetual Calendar: Despite the invention of the perpetual calendar watch in the era of the Gregorian calendar, most perpetual calendars actually implement the year according to the rules of the Julian calendar.

The Julian Calendar, by the way, is what most perpetual calendars implement – the most well-known difference between it and the Gregorian calendar is that the Gregorian calendar omits a leap year on century years, except when divisible by 400. (Other differences will be discussed in Part III.)

### The Seven Day Week

The seven day week is the only unit of time that we use which is not tied to a celestial event, and its origin can be traced at least as far back to the Jewish calendar. Strangely enough, the week is also the unit of time with the longest unbroken string – 7-day weeks have been observed without interruption for upwards of 3000 years. Finally, the English names of the days can be attributed to pagan Norse gods and their associated planets.

In the book of Genesis, God created the world in 6 days and rested on the seventh: “And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made.” (Genesis 2:3)

The practice of keeping every seventh day as a holy day was further codified as Commandment #4: “Remember the Sabbath day to keep it holy. Six days shalt thou labour and do all thy work. But the seventh day is the Sabbath of the Lord thy God: in it thou shalt not do any work, thou, nor they son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates: For in six days the Lord made Heaven and Earth, the sea, and all that in them is, and rested the seventh day: whereby the Lord blessed the Sabbath day and hallowed it.” (Exodus 20:8-11)

The Jewish seven-day week was in turn adopted by the Christians. The early Christians merely moved the holy day to Sunday, because it is the day of the resurrection of Christ (along with a desire to distance them from Jewish tradition). Finally, the Roman emperor Constantine began his conversion to Christianity in 312 AD, and in 321 AD decreed that Sunday should be an official day of rest throughout the empire. Thus ends our documented chain of evidence for the origin of the seven-day week. Now for some speculation…

This week appears to have been used from around the time of the sacking of Jerusalem by the Babylonian Emperor Nebuchadnezzar II in 597 BC; there is some evidence that the seven day week is actually Babylonian in origin, with the story of the Jewish creation having been influenced by captivity in Babylon. However, the evidence is contradictory – the books of Genesis and Exodus appear to have been written down before the Babylonian exile.

By the way, the modern Moslem observance of Friday as a holy day may echo the similar status of Friday in ancient Babylon.

In any case, no one knows how far from the middle east this seven-day week spread, but there is evidence of a seven-day calendar being used in parallel with other calendars in Pompeii, which pre-dates the conversion of the Roman empire to Christianity.

In addition to possible Babylonian influence, there was also the coincidence that there were seven “planets” of the ancient world (in descending order of observed orbital period):

1. Saturn
2. Jupiter
3. Mars
4. Sun
5. Venus
6. Mercury
7. Moon

When rearranged by astrological calculations (i.e. by assigning a planetary sign to each hour of the day, and then noting under which sign each subsequent day begins), one gets the familiar pattern of: Saturn, Sun, Moon, Mars, Mercury, Jupiter, Venus. Germanic language influence has modified this to: Saturn, Sun, Moon, Tiw (Norse god of war, as is the Roman Mars), Odin/Woden (Norse god responsible for escorting warriors to Valhalla, role similar to Mercury), Thor (Norse god of thunder, as is the Roman Jupiter), and Freya (Norse goddess of love).

Finally, the equivalence of Saturn’s day and the Jewish Sabbath was made sometime before the 1st century AD – by this time, Jews called the planet Saturn “Shabtai”, which is related to Sabbath. By the way, the Hebrew names of the days of the week are largely numerical in origin, with Sunday assigned (naturally) as the first day of the week (although the archangels Gabriel and Michael also appear.)

Figure 11: The Ulysse Nardin Planetarium Nicolaus Copernicus. This watch shows the relative motion of the seven “planets” of the ancient world: sun, moon, Mercury, Venus, Mars, Jupiter, Saturn

### Beware The Ides of March

The seven day week is pretty much universally used today because of the worldwide usage of the Gregorian calendar as the secular calendar. Interestingly, this is not the only grouping of days used in the ancient world.

The Romans used a strange method of counting down the days in a month, which had nothing to do with regular weeks. The first day of, say, July was known as the kalends Julius, from the Latin calare – “to call”. This is where the English word “calendar” comes from. The day before the kalends of July (June 30) is known as pridie kalends Julius, literally “the day before the kalends of July”.

Figure 12: The Roman Method of Dates within a month. The Romans counted down days to “landmarks” within each month, rather than numbering them consecutively. (The numbers in each day are for comparison with modern dating.)

The day before that day (June 29) is known as ante diem III kalends Julius – “third day before the kalends of July”. (Recall from above that the Romans used a different counting convention – thus, two days before the kalends of July is the “third” day.) This continues until up to 19 days prior to the kalends, depending on the month: June 14 is called ante diem XVIII kalends Julius – “18th day before the kalends of July”.

At this point, the Romans put a second landmark in the month: Idus, or the Ides. For March, May, July, and October, the Ides fall on the 15th (Ides of March = March 15th). For all other months, they fall on the 13th. Continuing our example, June 13 is Idus Junius. The 12th of June, therefore, is … you guessed it, pridie idus Junius – “day before the Ides of June”. June 11 is ante diem III idus Junius, etc.

For March, May, July, and October, the next landmark falls on the 7th of the month: the Nonae; the other months put the Nonae on the 5th. Continue counting backwards/forwards until the Kalends is reached again.

How on earth did this system get invented? It appears that it had to do with the month during the early Roman republic. These were supposed to be lunar months, so the Kalends of the month was marked by the observed new moon (i.e. when the moon was first visible as a sliver after sunset – a couple of days after the modern “new moon”), and the Ides were on the full moon. The first quarter occurred about eight days before the full moon (Romans would have counted it the ninth, hence Nonae.) The last quarter moon rose late in the evening, and thus would only have been visible when most people were asleep. Thus, the Roman names for days was in fact a means to count down to the next lunar phase.

However, due to the fact that the Roman calendar couldn’t keep both the lunar phases and seasonal cycles reconciled, the names had lost their meanings by 450 BC. Despite this, the usage of these terms not only survived to the founding of the Roman empire, but outlived the Roman empire altogether! Jules Verne has a 16th century alchemist named Arne Saknussemm use the Kalends of July in Journey to the Center of the Earth. Similarly, common people attending the first run of Shakespeare’s play Julius Caesar would have had no difficulty understanding when the “Ides of March” were.

Finally, recall from above that the leap year day set down by Caesar occurred on 24 February, with the same date being applied to two consecutive days. In Latin parlance, this was known as the bissextile day, and has its origins because the first 24 February (leap day) is officially ante diem sextum (VI) kalendas Martius – “the sixth day before the kalends of March”. The second 24 February was known as ante diem bis sextum kalendas Martius – or “the second sixth day before the kalends of March” – hence, the term bissextile. Bissextile has since come to be used as Latin for any reference to the leap day, even though its reference to the Gregorian practice of a 29th of February makes no literal sense at all. (It should be noted that the Roman Catholic religious calendar still uses bissextile in the correct manner – making the feasts which might occur on 25-28 February one day later (i.e.
26-29 February) during a leap year.)

### Other Weeks

In addition to the unwieldy system described above, there was also a Roman market week of eight days (Nundinae – taking into account the Roman counting convention.) This 8 day cycle defined when towns could hold markets, and appears to have been used by the Etruscans – predecessors to the Romans on the Italian peninsula.

Ten day weeks were adopted by the ancient Egyptians, the Chinese (in conjunction with a parallel 12 day week), and the French Revolutionaries.

The Bahá’i calendar uses a 19 day cycle – one each for the Bab and his 18 disciples in this religion of recent origin (1844 AD).

The ancient Mayans used three parallel weeks: one of 13 days, one of 9 days, and one of 20 days.

Possibly the most complex system of weeks was the Waku system, developed by the Javanese people in the tenth century, with parallel cycles of 2, 3, 4, 5, 6, 7, 8, 9, and 10 days!

The Soviet Union flirted with several different weeks. In 1917, in addition to adopting the Gregorian calendar, it was proposed that a ten day week replace the seven day week, but this was not adopted. Later, in 1929, Stalin decreed a five week day – the theory that the people’s productivity would be increased (1 day off in 5 = 80% vs. two days off in 7 = 71.4%). In addition to reducing the number of days off, different people were assigned different days off, to ensure factories remained open continuously. Unfortunately (?), social pressures prevented this system from taking off, as people who were assigned different days off never had an opportunity to socialize. In addition, there were never days where the all the workers were able to hold meetings, in good Soviet style.

By late 1931, the staggered days off were abolished, but the Soviets instituted a six day week – five work days followed by a common day off. Finally, in 1940, the seven-day week was reinstated, but with only a single day off.

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:

• Images of Ulysse Nardin Astrolabium, Tellurium, Planetarium, and GMT Perpetual Calendar by Mike Disher
• Image of Romulus and Remus from the Capitoline Museum, Rome
• Image of Julius Caesar from the British Museum, from the book The Art of the Romans by HP Walters (1911)