Calendars – keeping track of timeAt 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.
Key text
Key textA 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:
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:
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.
You may be surprised to learn that zero is one of the most important concepts in mathematics and has a long and controversial history. What makes a zero so special?
Zero as a place-holder
One of zero's many useful functions is as a ‘place-holder’. For example, in the number 600, the zero immediately to the right of 6 informs us that the ‘tens’ column is empty; the zero to the right of that tells us that the ‘units’ column is also empty. The only column that has any values in it is the ‘hundreds’ column. But if we did not somehow indicate that the two right-hand columns were empty we would write ‘600’ as ‘6’, which is not the number that we mean. Because zero means ‘nothing’, it is a concept that took a long time to be accepted. Historians argue about who first came to terms with it, although they seem to agree that the first recorded use of it was as a place-holder in the ancient civilisation of Babylon around 300 BCE. But it took much longer before it became a meaningful mathematical concept. Part of the reason was an apparent problem of logic: how could ‘nothing’ have any meaning?
Zero as a symbol
Zero really came into its own via India, where it first gained recognition perhaps 1500 years ago. Various words with meanings similar to ‘nothing’ became incorporated into the poems of Hindu mathematicians (yes, they wrote their mathematical problems in verse). For example, sunya meant void, kha meant sky and akasa meant space. Eventually, as the verse evolved to something closer to the way we write our maths today, a dot or open circle came to symbolise zero in mathematical equations.
Solving quadratic equations
The importance of zero in modern mathematics can be illustrated by a simple – although not so modern – example. In the late 1500s, a Scottish baron by the name of John Napier found a way of solving quadratic equations of the form: x2 + 2x = 24 (Equation 1)The aim, of course, is to work out the value of x. Napier realised that he could rewrite such an equation as follows: x2 + 2x – 24 = 0 (Equation 2)The left-hand side of the equation, in turn, could be rewritten to give: (x – 4)(x + 6) = 0 (Equation 3)For the product of two values to equal zero, as in Equation 3, one of them must be zero. So either (x – 4) equals zero (thus, x = 4), or (x + 6) equals zero (thus, x = –6). This way of solving quadratic equations is now commonplace in schools – but its discovery helped give mathematics the power to solve real-world problems such as those encountered in engineering. Related sites
First, divide the date into four parts: (1) the number of centuries, (2) the number of years left over, (3) the month, and (4) the day of the month.
Leap-year adjustment The total arrived at must be corrected by deducting 1 (first adding 7 if the total is zero) if the date is January or February in a leap-year (remembering that every year divisible by 4 is a leap-year except in century-years, which are only leap-years if divisible by 400). The corrected number gives the day of the week, if the days of the week are numbered such that 1 = Monday and 7 = Sunday. Thus, the value becomes 7 minus 1 = 6. You might like to try this procedure. Test it first on a few recent dates to make sure you’ve got it right, then calculate the day of the week on which important events have occurred throughout history (eg, the first lunar landing on 20 July 1969, or the date of your birth). You can check your calculations by entering the date you have chosen at the following site: How to calculate mentally the day of the week for any date (Guy Rimmer)
Australasian Science November/December 1999, pages 26-28 2000 and all that (by David Bromage) The history of our calendar.
New Scientist 4 February 2009, pages 39-41 A clock more accurate than time itself (by Matthew Chalmers) Reviews developments in precision time keeping using atomic clocks.
20 December 2008, page 10 Calls to scrap the 'leap second' grow (by Devin Powell) Explains the controversy surrounding the addition of the leap second to clocks every few years.
23 December 2006, pages 40-41 When a calendar finally settles down (by Chris Turney) Reviews the historical development of the calendar.
18 May 2005, page 25 The most accurate clock of all time (by Paul Marks) If time waits for no man, then neither does human ingenuity in measuring its passing.
22 November 2003, page 30 Time warp (by Stephen Battersby) Describes the technical problems caused by leap seconds.
13 July 2001 World's most accurate clock created (by Eugenie Samuel) Reports on a more accurate optical atomic clock used for satellite navigation.
19 December 1998, pages 74-75 The magic clock (by Sean Roberts) Describes a large clockwork device that will keep time for the next 10,000 years.
Scientific American April 2008, pages 52-59 Rulers of light (by Steven Cundiff, Jun Ye and John Hall) Explores a new kind of laser light and its applications, including a more precise atomic clock.
5 March 2007 Ask an expert Answers the question, ‘Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?’
14 November 2005 Wait a second (by Wendy Grossman) Describes the adjustment to time to synchronise the Earth's rotation with calendar.
Calendars (University of Sydney Library)
Provides information on the history of calendars, an opportunity to make your own calendar, and links to perpetual, ecclesiastical, historical, and literary calendars. You can also search for world public holidays and festivals.
The measurement of time (National Physics Laboratory, UK)
This site gives the definition of time and time scales (eg, calendars, the Earth's rotation, atomic time, the World time standard). It also answers the questions 'Why do we need accurate time?' and 'What are leap seconds and why do we need them?'
Keeping time (In Depth, 18 December 2008, Australian Broadcasting Corporation)
Looks at historical and proposed ways of measuring time (by Carmelo Amalfi).
How time works (How Stuff Works, USA)
Information about measuring time, time zones and daylight-saving time.
Time keeping (National Maritime Museum, Greenwich, UK)
Covers a number of topics relating to time. Of particular interest are 'Leap years' and 'The calendar'.
Calendars (Astronomy Department, New Mexico State University, USA)
By L.E. Doggett and reprinted from the Explanatory Supplement to the Astronomical Almanac, this article provides a detailed look at the Gregorian, Hebrew, Islamic, Indian, Chinese and Julian calendars.
The numbering of the Gregorian calendar was instituted by Dionysius Exiguus in 532. He investigated the date of the birth of Jesus Christ and set that as the start of the year 1. Thus, the year 1999 is referred to as 1999 CE (Common Era) or 1999 AD (anno Domini – ‘in the year of our Lord’). Years before the birth of Christ are designated as BCE – ‘before the Common Era’, or BC – ‘before Christ’. Dates can also be designated as the number of years before the present (BP). equinox. The times of the year when the sun crosses the celestial equator (the projection of the Earth's equator onto the sky) making the length of day and night nearly equal at all latitudes. There are two equinoxes each year, one in March, known in the northern hemisphere as the 'vernal' equinox, and one in September, known in the northern hemisphere as the 'autumnal' equinox. The dates of the equinoxes do not occur precisely when the lengths of the day and night are equal, but are out of step by a few days. This discrepancy is because of the finite size of the sun and the bending of sunlight by the atmosphere. More information can be found at FAQ–Equinoxes (US Naval Observatory). gnomon. A column that can indicate the time of day by the shadow that it casts on a marked surface. On a sundial, the pin or vertical triangular plate that casts the shadow is called a gnomon. More information can be found at Design of the Richard D. Swensen sundial (University of Wisconsin-River Falls, USA) interpolation. To estimate a value for a function between values that are already known or determined. This is in contrast to extrapolation, which is estimating the value of a function lying outside the range of known values. More information can be found at Linear interpolation – how it works (The Greenhouse, National Science Foundation's Graphics and Visualization Center, USA). solstice. The time of year when the sun is furthest from the celestial equator (the projection of the Earth's equator onto the sky). The summer solstice occurs in mid-summer and the winter solstice in mid-winter. For more information see The equinoxes and solstices (National Maritime Museum, Greenwich, UK).
Posted November 1999. The Australian Foundation for Science is also a supporter of Nova.
This topic is sponsored by Australian university mathematical sciences departments and the Australian Government's National Innovation Awareness Strategy.
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