(6) The Calendar

Lesson Plan for (6) The Calendar

So familiar has the calendar become that people tend to forget that it, too, had to be invented. Early farmers needed to know when to plow and sow ahead of rainy seasons, and to time other seasonal activities. Early priests in Babylonia, Egypt, China and other countries, even among the Maya in America, examined therefore the motions of the Sun, Moons and planets across the sky, and came up with a variety of calendars, some still in use.

The Day

The basic unit is obviously the day: 24 hours, 1440 minutes, 86400 seconds, each second slightly longer than the average heartbeat. The day is defined by the motion of the Sun across the sky, and a convenient benchmark is noon, the time when the Sun is at its highest (i. e. most distant from the horizon) and is also exactly south or north of the observer.

"One day" can therefore be conveniently defined as the time from one noon to the next. A sundial can track the Sun's motion across the sky by the shadow of a rod or fin ("gnomon") pointing to the celestial pole (click here for construction of a folded-paper sundial), allowing the day to be divided into hours and smaller units. Noon is the time when the shadow points exactly south (or north) and is at its shortest.

What then is the period of the Earth's rotation around its axis? A day, you say? Not quite.

Suppose we observe the position of a star in the sky--for instance Sirius, the brightest of the lot. One full rotation of the Earth is the time it takes for the star to return to its original position (of course, we are the ones that move, not the star). That is almost how the day is defined, but with one big difference: for the day, the point of reference is not a star fixed in the firmament, but the Sun, whose position in the sky slowly changes. During the year the Sun traces a full circle around the sky, so that if we keep a separate count of "Sirius days" and "Sun days", at the end of the year the numbers will differ by 1. We will get 366. 2422 "star days" but only 365. 2422 Sun days.

It is the "star day" (sidereal day) which gives the rotation period of the Earth, and it is about 4 minutes shy of 24 hours. A clockwork designed to make a telescope follow the stars makes one full rotation per sidereal day.

The clocks we know and use, though, are based on the solar day--more precisely, on the average solar day, because the time from noon to noon can vary as the Earth moves in its orbit around the Sun. By Kepler's laws (discussed in a later section) that orbit is slightly elliptical. The distance from the Sun therefore varies slightly, and by Kepler's second law, the motion speeds up when nearer to the Sun and slows down when further away. Such variations can make "sun-dial time" fast or slow, by up to about 15 minutes.

Very precise atomic clocks nowadays tell us that the day is gradually getting longer. The culprits are the tides, twin waves raised in the Earth's ocean by (mainly) the Moon's gravitational pull. As the waves travel around the Earth, they break against shorelines and shallow seas, and thus give up their energy: theory suggests that this energy comes out of the (kinetic) energy of the Earth's rotational motion.

The Year

The year is the time needed by the Earth for one full orbit around the Sun. At the end of that time, the Earth is back to the same point in its orbit, and the Sun is therefore back to the same apparent position in the sky.

It takes the Earth 365. 2422 days to complete its circuit (average solar days), and any calendar whose year differs from this number will gradually wander through the seasons. The ancient Roman calendar had 355 days but added a month every 2 or 4 years: it wasn't good enough, and by the time Julius Caesar became ruler of Rome, it had slipped by three months.

In 46 BC Caesar introduced a new calendar, named after him the Julian calendar. It is similar to the one used today: the same 12 months, and an added day at the end of February every 4th year ("leap year"), on years whose number is divisible by 4. Two years afterwards the 5th month of the Roman year was renamed July, in honor of Julius. The name of his successor, Augustus Caesar, was later attached to the month follwing July.

The Julian calendar thus assumes a year of 365. 25 days, leaving unaccounted a difference of 0. 0078 days or about 1/128 of a day. Thus the calendar still slips, but at a very slow rate, about one day in 128 years. By 1582 that slippage was approaching two weeks and Pope Gregory the 13th therefore decreed a modified calendar, named after him the Gregorian calendar. Henceforth years ending in two zeros, such as 1700, 1800, 1900--would not be leap years, except when the number of centuries was divisible by 4, such as 2000. This took away 3 "leap days" every 400 years, i. e. one day per 133 1/3 years--close enough to the required correction of one day per 128 years.

But it was not enough to modify the calendar: a one-time jump of dates was also needed, to get rid of the accumulated difference. In Italy this was done soon after the pope's edict, and "Tibaldo and the Hole in the Calendar" by Abner Shimony spins the story of a boy whose birthday was on a day skipped by that jump. Another birthday affected was that of George Washington, born 11 February 1732: when the British empire shifted its calendar, in 1751, the 11th of February "old style" became the 22nd of February "new style," and nowadays that is when Washington's birthday is usually celebrated.

In Russia the change came only after the revolution, which is why the Soviet government used to celebrate the anniversary of the "October Revolution" on November 7th. The Russian orthodox church continues to use the Julian calendar and celebrates Christmas and Easter about 2 weeks later than most of the Christian world.

The Moon

The Moon's orbital period, measured by the stars ("sidereal period") is 27. 321662 days. However, the monthly cycle of the Moon--thin crescent to half-moon, to full and back to crescent--takes 29. 530589 days, because it depends on the position of the Sun in the sky, and that position changes appreciably in the course of each orbit. The different shapes of the Moon represent different angles of illumination, and the appearance of the Moon in the night sky gives a fair idea of where the Sun would be (e. g. the Moon observed in the east before sunrise appears illuminated from below). The duration of the Moon's cycle ("synodic period") gave rise to the division of time known as month.

Many ancient calendars were based on the month. The most successful of these is the "Metonic" calender, named after the Greek Meton, who noted that adding 7 months in the course of 19 years kept the calendar almost exactly in step with the seasons. That would make the length of the average year (12 + 7/19) months, and with a calculator you can easily find its value as

(12 + 7/19) x 29.530589 = 365.2467 days

pretty close to the full value 365. 2422. The Metonic calendar is thus more accurate than the Julian one, though less so than the Gregorian. It is still used by Jews, on whose calendar each month begins at or near the new moon, when the Moon's position in the sky is nearest to the Sun's. The traditional Chinese calendar also uses of a formula like Meton's, which was probably invented by the ancient Babylonians.

Muslims use an uncorrected lunar calendar, and as a result their holidays slip through the seasons at a rate of about 11 days per year. The reason is not ignorance of astronomy--early Muslim culture included distinguished astronomers--but a deliberate effort to follow a different schedule from that of any other faith.

This creates a problem with the month of Ramadan, during which faithful Muslims are expected not do eat or drink from sunrise to sunset. When Ramadan falls in mid-winter, this imposes no great hardship, since days are short and cool. Fifteen years later, however, Ramadan falls in mid-summer, when days are long and the heat makes people quite thirsty. That is when Arab cities wait impatiently for the boom of the cannon which traditionally announces every evening the end of the fast.

The Maya Calendar

The Maya Indians in Central America, living on the Yucatan peninsula in Mexico, Belize and Guatemala (where Maya languages are still spoken), created an extensive civilization which peaked around the years 1200-1450. They developed an early system of symbolic writing ("glyphs") and simple mathematics, using a system like ours (including the zero!) based not on the number 10 but on 20. They did not, however, use fractions.

Their astronomy was well developed, and they noted the "zenial days" when the Sun was directly overhead ("at zenith") and a vertical stick cast no shadow. Their year had 365 days, but in the absence of leap years it slowly shifted with respect to the solstices. That year was divided into 18 named "months" of 20 days each (numbered from 0 to 19), plus the "short month" of Wayeb, whose days were considered unlucky.

Yucatan does not experience summer and winter the way middle latitudes do (e.g. Europe or most of the US), and therefore the Maya calendar was not strongly tied to the seasons the way ours is. The planet Venus received major attention, and its cycles were accurately measured by Maya astronomers. In addition the Maya also observed a "ritual year" of 260 days, consisting of 20 named "long weeks" of 13 numbered days each.

Next Stop: #7 Precession of the Equinoxes

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Author and curator: David P. Stern
Last updated 10 July 1999