The purpose of a calendar is to reckon time in advance, to show how many days have to elapse until a certain event takes place – the harvest, a religious festival, or whatever. The earliest calendars, naturally, were crude, and they

But in warmer countries, where the seasons are less pronounced, the moon became the basic unit for time reckoning; an old Jewish book actually makes the statement that “the Moon was created for the counting of the days.” All the oldest calendars for which we have reliable information were lunar calendars, based on the time interval from one new moon to the next – a so-called lunation. But even in a warm climate there are annual events that pay no attention to the phases of the moon. In some areas it was a rainy season; in Egypt it was the annual flooding of the Nile. It was, therefore, necessary to regulate daily life and religious festivals by lunations, but to take care of the annual event in some other manner.

The calendar of the Assyrians was based on the phases of the moon. The month began with the first appearance of the lunar crescent, and since this can best be observed in the evening, the day began with sunset. They knew that a lunation was 29 1/2 days long, so their lunar year had a duration of 354 days, falling 11 days short of the solar year. The correct figures are lunation: 29 d, 12 h, 44 min, 2.8 sec (29.530585 d); solar year: 365 d, 5 h, 48 min, 46 sec (365.242216 d); 12 lunations: 354 d, 8 h, 48 min, 34 sec (354.3671 d).

After three years, such a lunar calendar would be off by 33 days, or more than one lunation. We know that the Assyrians added an extra month from time to time, but we do not know whether they had developed a special rule for doing so or whether the priests proclaimed the necessity for an extra month from observation. If they made every third year a year of 13 lunations, their three-year period would cover 1,091 1/2 days (using their value of 29 1/2 days for one lunation), or just about four days too short. In one century this mistake would add up to 133 days by their reckoning (in reality closer to 134 days), requiring four extra lunations per century.

We now know that an eight-year period, consisting of five years with 12 months and three years with 13 months, would lead to a difference of only 20 days per century, but we do not know whether such a calendar was actually used.

The best approximation that was possible in antiquity was a 19-year period, with seven of these 19 years having 13 months. This means that the period contained 235 months. This, still using the old value for a lunation, made a total of 6,932 1/2 days, while 19 solar years added up to 6,939.7 days – a difference of just one week per period and about five weeks per century. Even the 19-year period required constant adjustment, but it was the period that became the basis of the religious calendar of the Jews. The Arabs used the same calendar at first, but Muhammad forbade shifting from 12 months to 13 months, so that the Islamic religious calendar, even today, has a lunar year of 354 days. As a result the Islamic religious festivals run through all the seasons of the year three times per century.

The Egyptians had a traditional calendar with 12 months of 30 days each. At one time they added five extra days at the end of every year. These turned into a five-day festival because it was thought to be unlucky to work during that time.

When Rome emerged as a world power, the difficulties of making a calendar were well known, but the Romans complicated their lives because of their superstition that even numbers were unlucky. Hence their months were 29 or 31 days long, with the exception of February, which had 28 days. However, four months of 31 days, seven months of 29 days, and one month of 28 days added up to only 355 days. Therefore, the Romans invented an extra month – called Mercedonius – of 22 or 23 days. It was added every second year.

Even with Mercedonius, the Roman calendar was so far off that Julius Caesar, advised by the astronomer Sosigenes, ordered a sweeping reform in 45 B.C. One year, made 445 days long by imperial decree, brought the calendar back in step with the seasons. Then the solar year (with the value of 365 days and 6 hours) was made the basis of the calendar. The months were 30 or 31 days in length, and to take care of the 6 hours, every fourth year was made a 366-day year. Moreover, Caesar decreed the year began with the first of January, not with the vernal equinox in late March.

This was the Julian calendar, named after Julius Caesar. It is still the calendar of the Eastern Orthodox churches. However, the year is 11 1/2 minutes shorter than the figure written into Caesar's calendar by Sosigenes, and after a number of centuries, even 11 1/2 minutes add up.

While Caesar could decree that the vernal equinox should not be used as the first day of the new year, the vernal equinox is still a fact of nature that could not be disregarded. One of the first (as far as we know) to become alarmed about this was Roger Bacon. He sent a memorandum to Pope Clement IV, who apparently was not impressed. But Pope Sixtus IV (who reigned from 1471 to 1484) decided that another reform was needed and called the German astronomer Regiomontanus to Rome to advise him. Regiomontanus arrived in 1475, but one year later he died in an epidemic, one of the recurrent outbreaks of the plague. The pope himself survived, but his reform plans died with Regiomontanus.

Less than a hundred years later, in 1545, the Council of Trent authorized the then pope, Paul III, to reform the calendar once more. Most of the mathematical and astronomical work was done by Father Christopher Clavius, S.J. The immediate correction, advised by Father Clavius and ordered by Pope Gregory XIII was that Thursday, October 4, 1582 was to be the last day of the Julian calendar. The next day was Friday, with the date of October 15. For long-range accuracy, a formula suggested by the Vatican librarian Aloysius Giglio (latinized into Lilius) was adopted: every fourth year is a leap year unless it is a century year like 1700 or 1800. Century years can be leap years only when they are divisible by 400 (e.g., 1600). This rule eliminates three leap years in four centuries, making the calendar sufficiently correct for all ordinary purposes.

Unfortunately, all the Protestant princes in 1582 chose to ignore the papal bull; they continued with the Julian calendar. It was not until 1698 that the German professor Erhard Weigel persuaded the Protestant rulers of Germany and of the Netherlands to change to the new calendar. In England the shift took place in 1752, and in Russia it needed the revolution to introduce the Gregorian calendar in 1918.

The average year of the Gregorian calendar, in spite of the leap year rule, is about 26 seconds longer than the earth's orbital period. But this discrepancy will need 3,323 years to build up to a single day.

Modern proposals for calendar reform do not aim at a “better” calendar, but at one that is more convenient to use, especially for commercial purposes. A 365-day year cannot be divided into equal halves or quarters; the number of days per month is haphazard; the months begin or end in the middle of a week; a holiday fixed by date (e.g., the Fourth of July) will wander through a week; a holiday fixed in another manner (e.g., Easter) can fall on 35 possible dates. The Gregorian calendar, admittedly, keeps the calendar dates in reasonable unison with astronomical events, but it still is full of minor annoyances. Moreover, you need a calendar every year to look up dates; an ideal calendar should be one that you can memorize for one year and that is valid for all other years, too.

In 1834 an Italian priest, Marco Mastrofini, suggested taking one day out of every year. It would be made a holiday and not be given the name of a weekday. That would make every year begin with January 1 as a Sunday. The leap-year day would be treated the same way, so that in leap years there would be two unnamed holidays at the end of the year. About a decade later the philosopher Auguste Comte also suggested a 364-day calendar with an extra day, which he called Year Day. Since then there have been other unsuccessful attempts at calendar reform.