Calendars – keeping track of time
Key text
This topic is sponsored by Australian university mathematical sciences departments and the Australian Government's National Innovation Awareness Strategy.
At midnight on New Year's Eve 2008 an extra second was added to clocks around the world to synchronise them to Earth's rotation. But keeping time wasn't always so precise.
A calendar can be defined as a systematic way of organising days into weeks, months, years, and millennia. By such ordering we know when the young Egyptian pharoah Tutankhamun died (1323 BCE), when Napoleon met his Waterloo (18 June 1815), and when the next school holidays will be. The calendar is a very useful device, providing us with essential information for both the study of history and the ordering of our daily lives. Few of us ever think about the science that underlies the calendar – but it has a history of its own stretching back thousands of years.
Lunar calendar
Humans have probably always recognised certain cycles in the passage of time. Perhaps the most obvious is that of the moon. At the start of its cycle (‘new moon’) the moon lies directly between the sun and the Earth and its illuminated face cannot be seen from the Earth. As the moon moves in its orbit around the Earth, a crescent of its illuminated face becomes visible. The crescent grows over a period of nights until the entire face can be seen: this is called ‘full moon’. The face then wanes until once more it can’t be seen from Earth. This cycle takes an average 29.530589 days. Most of the early calendars were based on this moon cycle, also known as a 'lunation'.
There were all sorts of problems with such calendars, due partly to the fact that the average lunation is not a whole number. If ‘29’ were the number used to mark the lunar month, the calendar would very quickly get out of synchrony with the actual phases of the moon. The first month would be out of synchrony by about half a day and the next month by a full day.
This problem could be solved by alternating the length of the month between 29 and 30 days, giving an average month of 29.5 days. Even this would get out of step pretty quickly, since the actual length of a lunation is a bit more than 29.5. Thus, such a calendar must be ‘adjusted’ from time to time. This is usually done by a series of additions or subtractions of days known as intercalations or extracalations.
Muslims have been using a lunar calendar for more than a thousand years. It keeps in step with the moon by the intercalation of 11 extra days over a period of 30 years, each year consisting of 12 lunar months. The average length of a month over the 30-year period therefore becomes:
(29.5 × 360 + 11)/360 = 29.530556 days
where 29.5 is the average number of days in the calendar month, ie (29+30)/2; 360 is the number of months in the 30-year cycle; and 11 is the number of intercalated days.
This calendar gets ‘out of step’ at the rate of about one day every 2500 years.
The solar year
Calendars based on the solar cycle must deal with similar issues. The solar cycle, or solar year, is the length of time it takes the Earth to complete one circuit of the sun. More than 2500 years ago, people were already combining astronomy with mathematics to measure the solar year. The first thing they needed was a starting and finishing point. Early astronomers used solstices (when the sun was the furthest from the equator) and equinoxes (when the sun crossed the plane of the Earth's equator) as starting and finishing points:
- One of the most common ways of measuring the length of a year in ancient times involved the use of a gnomon (a structure that casts a shadow). The direction of the shadow was used to tell the time. The shadow cast by a vertical gnomon is shortest at noon on the day of the summer solstice. Thus, a count of the days between two summer solstices would give an estimate of the length of the year. This estimate could be refined by interpolation between readings on successive days around the summer solstice, and by the construction of ever-larger gnomons, which provided increasingly accurate estimates of the exact time of the solstice.
- Year lengths were also determined by counting the days between two equinoxes. In about 135 BCE, the Greek astronomer, Hipparchus improved the accuracy of such estimates by counting the days between his own estimate of the moment of the vernal (March) equinox with that of another astronomer some 145 years earlier. By averaging, he arrived at an estimate of 365.24667 days – an error of only about 6 minutes and 16 seconds. Not bad, considering the distinct lack of accurate clocks in those days!
As the measurement of lunar and solar cycles became more accurate, calendars became increasingly sophisticated. Many different cultures derived their own calendars. Some were lunar, some were solar, and some were ‘lunisolar’, which attempted to keep in step with both the moon and the solar year. This was not an easy task, since there are about 12.368 lunations in a solar year. A lunar calendar consisting of 354 days (12 lunations) would keep in step with the moon – with some days intercalated from time to time – but would very soon get out of step with the year and, therefore, the seasons. All calendars were – and still are – plagued by the lack of synchrony between the moon’s cycle and the length of the year, and by the fact that neither the length of the solar year nor the length of the lunar month is a whole number.
The Roman calendar
The precursor of the calendar in common use today was the Roman calendar. According to legend, it was first used at the time of the founding of Rome, around 750 BCE. At first, the Roman calendar contained 10 months starting in March. Two further months – January and February – were added over time as the calendar was progressively reformed.
A complex series of intercalations was required to keep this calendar in step with the moon, the year and the seasons. However, some of the intercalations were at the discretion of certain officials, who, it seems, didn’t always do their job adequately.
By the time of Julius Caesar (100-44 BCE), it had all become quite muddled. Caesar requested an astronomer called Sosigenes to advise him on reforming the calendar. Sosigenes recommended abandoning the lunar calendar and adopting one that focussed solely on the solar year. Caesar decreed that henceforth each year would consist of 365 days, with an extra day added to every fourth year (this later became known as a ‘leap’ year) in the month of February. To accommodate the change, a once-off adjustment was needed: the year 46 BCE was decreed to be 445 days long – giving some indication of how confused the Roman calendar had become. The month of July was renamed in honour of the reformer, and the new calendar has been known as the Julian calendar ever since.
The Gregorian calendar
But Caesar’s reform didn’t quite end the confusion. His calendar assumed that each year was 365¼ days long – so that the addition of one extra day every four years would be adequate compensation. However, even then it was known that the actual length of a year was slightly shorter than this – the modern estimate is 365.24219 days. The difference between this and 365.25 is not much – 0.00781 days, or about 11¼ minutes. But over time it adds up: in a thousand years the discrepancy is 0.00781 × 1000 = 7.8 days.
By the Middle Ages, the Julian calendar was well entrenched in Europe. The system of counting the years since the birth of Christ had been introduced by Dionysius Exiguus (Box 1: Zeroing in on nothing), and leap years were deemed to be all years divisible by 4 (the year 1212 for example, was a leap year). But the cumulative error was beginning to be noticed. The vernal equinox, held traditionally to occur on 21 March, was actually taking place earlier and earlier and other dates of religious significance were becoming similarly confused.
Calendar reform was talked about in the Catholic Church for more than 300 years. But it wasn’t until 1582 that Pope Gregory took the advice of mathematicians and astronomers and decreed that the problem would be addressed by omitting 3 leap years every 400 years. This was done by declaring that new centuries would not be leap years unless divisible by 400 (thus, 1900 was not a leap year but 2000 was). This became known as the Gregorian calendar, and is the one we use today. In most European countries they adjusted for the accumulated errors of the Julian calendar by omitting 10 days from the year 1582. In fact, people living in what is now Belgium missed out on Christmas because of these cancelled days.
Not all the countries of Europe adopted the Gregorian reform immediately. Gregory was Catholic, and Protestant countries largely ignored his decree. Nevertheless, the problem of the extra days had become so acute in England by the 1700s that an adjustment was eventually decreed by Parliament. Eleven days were omitted from the month of September in 1752 and Pope Gregory's system for dealing with century-years was adopted.
Problem solved?
Most countries have now adopted the Gregorian calendar for the purposes of international trade, although some simultaneously maintain their traditional calendars. But the adjustments made to the calendar by Gregory are still not perfect.
The discrepancy between the calendar and the real length of the year was only about 0.00028 days (around 24 seconds) in 1582, but even this will be noticed eventually. Compounding the problem is the fact that the years are actually getting shorter. Since 1582, the year has decreased from 365.24222 days to 365.24219 days – a real decline of about 2.5 seconds.
Why are the years getting shorter?
The length of the year is less than the time the Earth takes to circle the sun due to a slow wobble in the Earth's motion, called precession. A gradual increase in this wobble is shortening the year. The wobble is increasing because the tides that are raised by the sun and moon act like brakes on a wheel and gradually slow the Earth's daily spin. As the Earth slows, the wobble increases – reducing the length of the year.
We no longer measure a year by the Earth's orbit around the sun. Using atomic clocks, measurement of time has become more precise. Too precise perhaps – we now have to add on 'leap seconds' every few years to keep atomic clocks synchronised with Earth's rotation.
Posted November 1999.