Keeping Track of Time

From almost the dawn of civilisation, man noticed that many things in nature came in cycles. The most basic cycle was the alternation of day and night with the rising and setting of the sun.

As the nights passed, the moon waxed from a crescent to a circle and then waned again to a crescent. This lunar cycle gave rise fo a natural notion of a month.

Over a period of months, the seasons changed, following a fixed pattern. Near the tropics -- for instance, over most of India -- the hot summer gave way to rain, which in turn was followed by a cool winter. Further away from the equator, there were four distinct seasons -- spring, summer, autumn, winter.

The seasons were intimately related to man's survival. As he began to farm the land, he needed to know when to sow and when to reap. In other words, he needed a way of keeping track of what time of year it was.

Measuring a year

In the beginning, it was sufficient to just keep track of the seasons. However, the seasons vary slightly in length from year to year. The beginning of summer in one region may not always coincide with the beginning of summer in a distant country.

As trading spread and goods were exported from one area to another, it became important to be able to agree on a common measure of time. One needed to be able to define a civil year which could be used to regulate trade and commerce.

Meanwhile, many religious and social customs also became tied to the cycle of the seasons. Most religious festivals were fixed with respect to the phases of the moon, since these were relatively easy to observe. The civil calendar which standardised the length of the year also had to take this into account.

The problem with designing an accurate civil calendar is that the three natural units of time -- the day, the month and the year -- are based on different movements -- the earth's rotation about its axis, the moon's revolution around the earth and the earth's revolution around the sun.

There is no convenient way to relate these three concepts of time. A solar year is about 365.25 days long, while a lunar month is about 29.53 days long. Twelve lunar months add up to 354 days and thus fall a little short of a year. On the other hand, thirteen lunar months are 18 days too many.

Hours, minutes, seconds....

A day is a natural unit of time. But how did we come to divide a day into hours, minutes, seconds? The main reason for dividing a day into smaller units was commercial. Many useful tasks take less than a full day to complete and man needed a way to measure the value of such work. Initially, the daylight hours were divided into a fixed number of parts -- say, a quarter of a day each. However, since the length of day varies during the year, it was necessary to fix a standard unit of measure. This gave rise to the hour, minute and second.

Why do we have 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute? The most likely reason is that 12 and 60 are "nice" numbers to compute fractions of. 12 is evenly divisible by 2,3 and 4. So, a 12 hour working day could easily be broken up into halves, thirds, and quarters, 60 goes one better than 12 in that it is divisible by 5 as well.

The week

Unlike the day and the month, the week is a completely artificial unit of time. The week, like the civil year, probably first came into being when man began trading regularly. A week marked the interval between market days.

The week varied in length in different parts of the world. Some Africal tribes used a 4 day week, the Central American week was 5 days long and the ancient Romans used 8 day intervals. Most societies eventually settled on a 7 day week.

Why 7? Perhaps because many ancient civilisations associated mystical properties with the number 7. Or because 7 days is approximately one fourth of a month (or approximately half the time between a new moon and full moon). The origin of the 7 day week seems to be lost in the mists of time.

The calendar which we follow for most of our daily activities developed mostly in Europe. This so called Western or Gregorian calendar (more on the Gregorian later) is purely solar -- it ignores moon completely and only relates days and years. Later in this article, we will explain how this Western calendar came to take on its present form. But first we need to understand the concept of a year -- this is not as simple as it appears!

When does the day begin

We are used to the fact that the day officially begins at midnight. This ensures that during our normal waking (and hence working) hours, the date is fixed.

However, not everyone uses midnight as the beginning of day. Astronomers, for instance, begin their day at noon, since they work primarily at night and would like the entire night's observations to be recorded as being on the same date.

In the Indian calendar, the day begins with sunrise. This convention holds for most traditional calendars. However the Jewish calendar is different -- the Jewish day begins at sunset. So the weekly Jewish holiday, the Sabbath , runs from Friday evening to Saturday evening.

How long is a year

The seasons arise because the axis of the earth about which the earth rotates is at an angle of about 23.5 degrees from the vertical. As the earth goes around the sun, the earth alternately tilts towards and away from the sun.

At the summer solstice June 21, the tilt is maximum towards the sun while at the winter solstice December 22, it is maximum away from the sun. Between the soltices fall the equinoxes, when the axis is not tilted with respect to the sun. The vernal equinox is on March 21 and the autumnal equinox on September 23.

To keep track of the cycle of the seasons accurately, we need to measure the time between corresponding equinoxes (or solstices). This is known as the tropical year.

Interestingly, this is different from a sidereal year -- the time the earth takes to complete a revolution around the sun. To understand this, consider a spinning top which is not entirely vertical. The axis of the top wobbles around the vertical -- the centre point at the crown of the top describes a circle around the vertical line above the point where it is balanced. This circular movement is called precession.

In the same way, the earth's axis which is tilted, is not pointed at a constant spot in the sky. Instead, it describes a large circle in the sky. It takes 26,000 years for the axis to complete one full circle.

One sidereal year after an equinox, the earth's axis will have moved 1/26000 of the way around the circle due to precession. So it will not be an equinox -- the the axis will be titled slightly towards the sun. The actual equinox will have occurred earlier.

So the tropical year is shorter than a sidereal year -- the tropical year is 365.2422 days long while a sidereal year is 365.2564 days long. As a result, a calendar based on the sidereal year (like the Indian solar calendars which we will read about later) will gradually begin to err with respect to the seasons -- the "real" equinoxes and solstices will shift away from the original dates specified by the calendar.

We see this in the two prominent Sankrantis (see below) -- Makara Sankranti (January 14) and Mesha Sankranti (April 14). These two dates are supposed to denote the winter solstice and vernal equinox respectively. However, since the time these dates were fixed, the equinoxes have shifted back by about 24 days due to precession.

Gradually the point corresponding to the vernal equinox shifts further and further back along the trajectory of the earth. Eventually, after a full cycle of 26,000 years, the equinox will return to the same point as when it started and the dates predicted by the Indian calendar will once again coincide with the observed equinoxes and solstices.

The Indian Calendar(s)

For the moment we shall concentrate on Indian calendars. There are, of course, many Indian calendars. The most common one is luni-solar, taking both the sun and moon into account. It tries to fit together the cycle of lunar months and the solar year in a single framework.

As we saw earlier, 12 lunar months add up to less than a full year whereas 13 lunar months are more than a year. To solve this problem, the Indian calendar defines a normal year to have 12 lunar months. Every few years, an extra lunar month is intercalated to keep in step with the solar year.

Adding 7 extra lunar months over a period of 19 years gives a remarkably close approximation of 19 solar years.

But how exactly does the Indian luni-solar calendar work? How does one decide when to add the extra lunar months? This is the subject of the next section.

The Solar calendar

At the heart of the Indian calendar is the solar calendar based on the Indian equivalent of the zodiac. A year is measured by keeping track of the relative motion of the sun with respect to the stars.

As the earth revolves around the sun, the sun as seen from the earth appears to move with respect to the stars in the background. The apparent path of the sun lies on a fixed line (orbit) through the sky called the ecliptic.

Names of solar months in

Most of India Tamil Nadu Kerala Starts on
Vaisakha Chittrai Mesha April 14
Jyeshth Vaikasi Vrishabha May 15
Ashadha Ani Mithuna June 15
Shravana Adi Karkataka July 17
Bhadrapada Avani Simha August 17
Ashvina Purattasi Kanya September 17
Kartika Aipasi Tula October 18
Margashira Kartigai Vrishchia November 17
Pushya Margazhi Dhanus December 16
Magha Thai Makara January 14
Phalguna Masi Kumbha February 13
Chaitra Panguni Meena March 15
The name of each month in Kerala corresponds to the Zodiac constellation that that sun is in during that month. Thus, during the month of Mesha, the sun is in the constellation Mesha.

By measuring the progress of the sun along the ecliptic, one can keep track of the passage of the year. If we fix the point on the ecliptic where the sun is now, the sun will next return there exactly a year later.

Of course we cannot directly observe which stars the sun is passing in front of, since we cannot see the stars when the sun is shining. However, we can indirectly infer the position of the sun by observing which stars on the ecliptic are visible near the horizon just before sunrise and just after sunset.

For convenience, the ecliptic (which is a circle) is divided into twelve equal segments. Each segment is associated with a constellation. These twelve special constellations are the signs of the zodiac.

A solar month is the time taken for the sun to pass through one of the twelve segments. The time when the sun crosses from one sign to the next is called a sankranti and marks the beginning of the solar month.

Two well known sankrantis are Makara Sankranti or Pongal around January 14 and Mesh Sankranti on April 14. Mesha Sankranti marks the beginning of the new year in Assam, Bengal, Kerala, Orissa and Tamil Nadu -- these states follow a purely solar calendar for fixing the length of the year.

The lunar months

The lunar months are defined with respect to the solar months -- in fact, they have the same names as the solar months. In Andhra Pradesh, Karnataka, Maharashtra and Gujarat, the lunar month begins and ends with the new moon (amavasya). In most of North India, the month runs from full moon to full moon (poornima).

The first lunar month of the year in Chaitra. In Andhra Pradesh, Karnataka, Maharashtra and Gujarat, Chaitra begins with the last amavasya before Mesha Sankranti (April 14).

The next lunar month is Vaisakha beginning with the first amavasya during the solar month Vaisakha. Similarly each amavasya falling between two sankrantis marks the beginning of the lunar month. The lunar month inherits the same name as the solar month during which amavasya falls.

Typically the correspondence between a solar year and a lunar year is as follows. The upper row denotes the solar months, with the vertical lines denoting sankrantis. The lower row denotes lunar months, with vertical lines denoting amavasyas.

A solar month is normally 30 to 31 days in length whereas the lunar month is only 29.5 days long. Thus, as the year goes by, each lunar month starts a little earlier within the corresponding solar month.

Eventually, an entire lunar month will lie within a solar month -- in other words, there will be two amavasyas between a pair of sankrantis. In such a case we get an extra intercalated month, called an adhika masaa.

For instance, consider a year like the following when there are two amavasyas within the solar month of Bhadrapada. The first amavasya begins an extra month called Adhika Bhadrapada while the second one begins the "real" month Nija Bhadrapada.

A year with an adika maasa occurs around 7 times in 19 years. The adhika maasa could come at almost any time during the year, depending on which solar month happens to have a double amavasya.

Occasionally, a very peculiar situation occurs -- a lunar month spans two sankrantis. This, for example, is what happened in 1991-92. There is no amavasya during the solar month Magha. As a result, the lunar month Magha was "lost" and became a kshaya maasa.

How can this happen? Isn't a lunar month always shorter than a solar month?

It so happens that a solar month is normally 30 to 31 days long. However, since the earth moves at varying speeds around the sun, the sun's apparent motion through the ecliptic is not uniform. If the earth is moving exceptionally fast, the sun may pass through a sign of the zodiac in less than a lunar month.

Note that in 1991-92, there were two adhika maasas -- Ashvina and Phalguni. This is always the case -- a year with a kshaya maasa will have two adhikamaasa.

Though it seems fairly complicated, the luni-solar system does manage to cope with the tedious problem of reconciling the solar and lunar calendars rather well. However because of the complication involving the earth's rotation called precession, the Indian solar calendar does not keep track of the seasons accurately. This is explained elsewhere in this article.

Many different societies have developed their own calendars -- for instance, the Jews and the Chinese. Ancient civilisations which came up with reasonably sophisticated calendars include the Babylonians, Egyptians, Assyrians, Mayans. All of these were luni-solar, although each had a different way of reconciling the lunar month with the solar year.

Surprisingly, there is still a widely used calendar which is purely lunar -- the Islamic calendar. The Islamic calendar consists of twelve lunar months, with no correction for the extra days in a solar year. As a result, the Islamic months move forward by about 11 days every solar year. So, for instance, the month of Ramzan (or Ramadan) keeps shifting.It used to occur in mid summer during the 1980s but moved to February--March in the mid nineties and is now in November in the year 2005.

The Western Calendar

Finally we come to the calendar most familiar to us -- the Western or Gregorian calendar. Unlike the Indian calendar(s) which measure the sidereal year, the Western calendar directly measures the tropical year. So the goal of the Western calendar is to keep the dates of the solstices and equinoxes fixed -- the vernal equinox always falls on March 21 and the calendar is designed to ensure that it never shifts from this date. (In this way of keeping track of the solstices and equinoxes, the sun is directly overhead at noon, on the summer solstice June 21, at the Tropic of Cancer and directly overhead on the winter solstice December 22 at the Tropic of Capricorn. Similarly the sun is directly overhead at the Equator during the two equinoxes (vernal and autumnal) on March 21 and September 23).

The Western calendar had its origin in the ancient calendars of the Babylonians, Egyptians and Greeks. These civilisations based their calendars on the relative movement of the sun against the stars.

However, by the year 50 BCE, the observed equinox had moved away from the equinox predicted by the calendar by around 3 months.

In 46 BCE, Julius Caesar sought the advice of the Egyptian astronomer Sosigenes to rectify this problem. Based on Sosigenes's recommendation, 67 days were added to the year 46 BCE to reset the actual dates of the equinoxes to those specified in the calendar.

Julius Caesar went further. Realising that a permanent solution was needed to prevent this problem from recurring, he decided to reform the calendar. By then, astronomers had calculated the tropical year to be 365.25 days long. So Julius Caesar began the practice of having normal years of 365 days with an extra day added every fourth year to take care of the extra quarter day which had accumulated each year. In the new Julian calendar, the year began with January 1 and the vernal equinox fell on March 21.

However, it turns out as we have mentioned earlier, that a tropical year is not 365.25 days long but 365.2422 days long. This tiny extra fraction of .0078 days a year adds up to approximately three days every four centuries.

This discrepancy was so small that it went unnoticed for several hundred years. By the 13th century, some scientists and philosophers had begun to suggest that a reform was needed in the calendar. However, the actual reform had to wait another 300 years, till March 1582, when Pope Gregory XIII announced a new calendar.

Pope Gregory's solution was to declare that a century year was not a leap year unless it was divisible by 400. Thus 1600 and 2000 are leap years but 1700, 1800, 1900 are not, though they are divisible by 4 like normal leap years. In this way 3 days were dropped every 400 years and the error in the Julian calendar was corrected.

When the new calendar was put in use, to correct the error already accumulated in the thirteen centuries since the Council of Nicaea, a deletion of ten days was made in the solar calendar. The last day of the Julian calendar was 4 October 1582 and this was followed by the first day of the Gregorian calendar 15 October 1582.

Today most of the world follows the Gregorian calendar as its civil calendar, though many countries including India follow their own calendars for other purposes including religious festivals. The only change made in the last 400 years has been to fine tune the calendar to make the years 4000, 8000,... non leap years. With this correction, the Gregorian calendar is in synchrony with the tropical year to an accuracy of one day in 20,000 years.

An interesting aside: Many non Catholic countries like England, Scotland and thereby the rest of the British Empire (including part of what is now the United States) refused to adopt what was considered a Catholic invention, and did not adopt it until 1752, by which time it was necessary to correct by eleven days (2 September 1752 being followed by 14 September 1752). This is reflected these days in the fact that if you use the calendar command on your computer (UNIX/LINUX system) for September 1752 ("cal 9 1752") you will find that the dates 3 to 13 September are missing.

One final complication: On timescales of thousands of years, the Gregorian calendar falls behind the seasons drastically because the slowing down of the Earth's rotation makes each day slightly longer over time while the year maintains a more uniform duration. The equinox will occur earlier than now by a number of days approximately equal to [years into future/5000]^2. This is a problem that the Gregorian calendar shares with any rule-based calendar.

This article was first written by Madhavan Mukund and me and appeared in the Jan-Feb 1995 issue of the magazine Jantar Mantar. Some of the details in this article were taken from Man and the Universe by M. S. Sriram. If you have any constructive suggestions, comments or corrections, I would like to hear from you. The contact details are below. An excellent overview of various ancient and modern calendars is given in the book Mapping Time - the calendar and its history by E. G. Richards (Oxford University Press).

Rahul Basu
The Institute of Mathematical Sciences
Chennai 600113