[In the following excerpt, Richards summarizes the types, characteristics, and sources of various calendars.]

EMPIRICAL CALENDARS

Men have ordered their affairs by the phases of the moon and the seasons for as long as records exist. Even before calendars had been invented they could have told their wives that they would be back three days after the next full moon or remarked that their son was born three winters ago. Such perhaps were the beginnings of calendars. Later, men counted the days between moons and solstices, and were better able to anticipate a day in the future or to remember the past. But they still needed to observe the moon or the solstices to determine when a new month or a year began. As astronomy developed, it became possible to foresee the new moons or the solstices with increasing precision, and calendars as we know them today began to take shape.

For as long as estimates of the length of a lunation or of a year were insufficiently precise, or the calendar inadequate to its task, the months would tend to get out of synchronization with the moon, or the seasons with the year. The remedy was to insert (or even remove) extra days in a month or a year to bring them back into register. This was intercalation (or extracalation).

We call calendars which required actual observations of the sun or the moon, empirical calendars, and note that they had several defects. First, they were dependent on the weather; storms or heavy cloud might well obscure a new moon or the rising of the sun at the summer solstice. A protracted spell of bad weather could throw the calendar into disarray. No doubt such occurrences were less likely in the deserts of Egypt or Mesopotamia than further north, but there were other problems.

As human communities increased in size and empires spread, a second defect became apparent. The time and day on which the new moons and solstices were observed would depend on where the observer stood. A man might first glimpse a new moon on one day; another man, a few miles west, would see it some minutes later, maybe after the start of the next day; after the Diaspora this discrepancy was particularly troublesome to the Jews. The whole question of exact timings became problematic. One solution was for the calendar priests, who were in charge of watching the heavens at some definitive point of observation (such as the Temple in Jerusalem), to communicate their sightings throughout the land by bonfires or messengers. But as empires grew larger, the news of the start of a new month might take days to arrive.

Yet another problem arose from the need to adjust the calendar by intercalating extra months or days to keep it in synchrony with the heavens. The occasion at which such intercalations were made had to be left to the discretion of the officials or calendar priests, who were entrusted with running the calendar. These might be ignorant, incompetent, or venal people; there was money to be made or other advantages to be gained by accepting bribes to delay or advance an intercalation for political or financial reasons. By the time of Julius Caesar, the Roman calendar was seriously affected by such practices.

In time, however, precise astronomical knowledge and ingenious calendrical systems made it possible to design a calendar that could be calculated well in advance, and which kept in reasonable synchrony with the sun or the moon. Any intercalations required were made according to a rule and were not at the discretion of the priests.

It was at this point, to grossly oversimplify a complicated swathe of history, that the calendars of the East and of the West diverged. The calendar makers of India and China designed calendars that kept in step with the sun and the moon; they depended on calculations of the exact time of a new moon or a solstice. The lengths of the months or the years were subservient to these events. Such calendars have been called astronomical calendars. In contrast the Western nations—Egypt, Rome, Christendom, and Islam—opted for arithmetic calendars in which the lengths of the months and years were fixed according to simple but rigid rules, which were designed to keep the calendar more or less in step with the heavens, but did not require either astronomical observations or calculations.

Both the astronomical and arithmetic calendars depend on precise knowledge of the length of a lunation or a year. Within the limitations of this knowledge the astronomical calendars are self adjusting and are always in agreement with the heavens, whereas the calculated calendars only keep in step on the average.

The basic element of almost all calendars is the day, and all longer calendrical periods, such as weeks, months, and years, contain a whole number of days—even if the astronomical lunation and year do not. The problem besetting calendar makers, whether they opt for an astronomical or an arithmetic calendar, is how to reconcile the day with the lunation and the year. Different peoples managed this in different ways, but before we turn to the different strategies that were adopted, we must first examine the day, month, and year more closely. For each of these periods of time, it is necessary to agree on the moment when each begins and to know its precise length as measured in days and fractions of a day.

THE DAY

The day is the period defined to include a night and a day. It is mostly unproblematic to count the days as they pass, and only people who live within the Arctic Circle where the sun does not set in summer or rise in winter have difficulties. Nevertheless, when we equate a day in one calendar to a day in another, we must pay close attention to when the day starts and when it ends.

In the history of most nations, including the Greeks, the day has started either at sunrise or at sunset. For the Romans, however, it began at midnight. After the collapse of the Roman Empire, many people reverted to starting their day at sunrise; whilst today we have gone back to the Roman custom. Astronomers once preferred their day to start at noon, so that the whole of their viewing time (at night) fell in the same day, but since 1925 they have started their days at midnight. Only their Julian days … start at noon.

In the absence of clocks there was very little alternative to starting the day at daybreak or sunset, despite the disadvantages. Whether the day began at noon or midnight, sunrise or sunset, the precise moment had to be determined from the sun. The instants of time when you observe these events depends on your location—they vary with longitude and also with latitude. Thus, two people living not far apart could disagree as to on which day an event took place, if it occurred close to the start of a day.

The length of the day measured by clocks is not constant and only a mean can be precisely defined. In this century it has been found that even the average length of the day is slowly increasing—but very, very slowly.

THE ASTRONOMICAL MONTH AND YEAR

The lunation—the period defined by the cycle of phases of the moon—is the basis of the month, and the moment when each cycle starts is a matter of convention. The start of a month was once determined by observations of the phase of the moon: some people began their months with the first sighting of a new moon; others with the disappearance of the old. Others preferred a gibbous moon, and yet others used an interpolated date of conjunction with the sun, when the moon is not actually visible.

In order to anticipate the day of the start of the month, it is necessary to know how many days there are in a lunation. The ancients found that this interval was sometimes 29 days and sometimes 30. Eventually, they accepted the fact that the exact number was not integral; we know today that it is not even constant, varying by several hours from lunation to lunation. Only the period of a mean lunation—the mean synodic period of the moon—can be stated precisely, and that too is increasing, albeit very slowly.

The astronomical year is the period in which the sun moves round the full circle of the zodiac, or, which is the same thing, the earth revolves around the sun in its orbit. Like the lunation, the moment when it starts is a matter of convention. Some have taken either the vernal or autumnal equinox as this moment; others, the summer or winter solstice. Astronomers opt for the vernal equinox. The history of the calendar is largely the history of the invention of better methods for detecting the instant of an equinox or solstice and of measuring the length of the year.

The ancients found the year to contain 365 or 366 days. Today we know that, like the lunation, it varies by several minutes from year to year; only the mean tropical year can be precisely specified, and that is slowly decreasing. …

LUNAR CALENDARS

Most early people used a lunar calendar and no doubt they began by counting the days each time a new lunation was judged to have started. If the new moon (or whatever portent indicated a new month) could not be seen because of bad weather, there was uncertainty. In time this became intolerable and formal systems in which a fixed number of days were assigned to each month were developed. Maybe extra days were intercalated or extracalated if the count got out of phase with the astronomical moon.

… [The] obvious approximation [is to assign] 29 and 30 days to alternate months, and all lunar and lunisolar calendars employ variations on this theme. Taking 29 and 30 in strict alternation, the average number of days in a month is 29.5, with a discrepancy of 0.030 589 days per month; this amounts to a full day in about 32 months or a bit under three years. Such a calendar would get badly out of synchronization with the moon within a human lifetime.

One solution to this problem was to intercalate an extra day from time to time. There are two difficulties with this however. First, there is the question of when exactly to intercalate—regularly according to some rule, or after examination of the heavens to see if one were needed? Then, there is the complication that the individual lunations are not all of exactly the same length.

Here we note the approximation used in the only surviving lunar calendar of note, the Islamic calendar. This intercalates 11 extra days in each cycle of 30 years of 12 months each, to give an average month of (29[frac12] x 360 + 11) / 360 = 29.530 556 days; the synchronization is much better, and the small remaining discrepancy amounts to a day in about 2500 years. The Islamic calendar has so far been running for a little over half of this time. This rule, invented by mediaeval Arabian astronomers is simple and accurate. The period of 30 years is less than a human lifetime, so there is little difficulty in remembering how long the current cycle has been running. A more accurate rule, particularly for cycles of less than 200 years, would be to insert 29 days every 79 years. This would reduce the discrepancy some sixfold, but it is more complicated and spans longer than an average human lifetime.

It might be thought that all lunar calendars whose months began when some phase of the moon is observed, would remain in perfect synchronization with the great ‘clock in the sky’, in all parts of world, at all times; but this is not so. The instant when, for example, a new moon is first observed depends on the latitude and longitude of the observer, to say nothing of the state of the weather.

LUNISOLAR CALENDARS

A purely lunar calendar is not acceptable to farming communities who run their lives according to the seasons, and the search started for ways of reconciling the lunation with the tropical year—that is, for a workable lunisolar calendar.

Keeping a calendar in step with both the moon and the seasons is difficult. One solution was to introduce or intercalate a thirteenth month into the year from time to time. The problem then was to decide when to do this. Some left it as an arbitrary decision of the local priest or astronomer; other people developed rules based upon simple natural or astronomical phenomena. One interesting example of the latter is the practice of the fishermen of Botel-Tobago Island (near Taiwan) who operated a lunisolar calendar. When the calendar fell too far behind the season, they failed to observe the seasonal rising of the flying fish in the month expected; then was the time to postpone their fishing season by inserting an intercalary month. There were many variations on this theme practised across the world.

The Babylonians and later the Greek astronomers, put their minds to the problem and came up with purely arithmetical rules. Over the years the accuracy of their astronomical knowledge and the adequacy of the rules improved. The aim of these rules was to provide a practical scheme for intercalating the extra month. The average length of the month should be such as to keep the months in step with the lunations, and the frequency of intercalation, such as to keep the average year in step with the seasons. It took time for a scheme which adequately satisfied both of these criteria to evolve.

The initial steps were taken in Babylon, from where development passed to the Greek world. Our knowledge of Greek calendars is all too fragmentary and often we must rely on classical authors, some writing much later. The following account must be viewed as being schematic and not necessarily historically accurate. An early Greek calendar attributed to Solon of Athens (638-558 bc) specified a year of 12 months containing alternately 30 days (full months) and 29 days (deficient months) to give a total of 354 days. The calendar was reconciled with the year by introducing an intercalated month of 30 days every other year. Thus two lunar years would contain 738 (= 2 x 254 + 30) days, whereas two tropical years contain about 730[frac12] days. The average length of the month was 738/25 = 29.52 days, so that the calendar would get a day out of step with the moon after about eight years.

The discrepancy of seven and a half days was ultimately unacceptable and a little later Cleostratus of Tenedos (c. 520 bc) suggested that the intercalated month be dropped once every eight years. Thus an eight-year cycle, the octaeteris, which contained 2922 days and 99 months, was developed. Miraculously, eight solar years of 365[frac14] days also contained 2922 days.

The matter seemed to be settled, but, alas, the actual length of these 99 lunations was 2923.528 days. This meant that Cleostratus' calendar was a day and a half out of step with the moon at the end of eight years; after 160 years the discrepancy would be 30 days. It was suggested therefore that one of the intercalated months be dropped every 160 years. In practice the Athenians reverted to the older and unsatisfactory method of dropping the month when they felt like it, with the inevitable result of corruption and chaos in the calendar. Nevertheless, the octaeteris came to be considered a fundamental time period; for instance, the Olympic Games were held (and still are) every four years (half an octaeteris). There the matter stood till Meton and Euctemon of Athens introduced a new method in 432 bc. This method was almost certainly invented by the Babylonians and in use there by 499 bc. When contact between the East and the West was opened up by Alexander the Great, it became known in India and, later still, in China. It is, however, difficult to be certain who first discovered it or even if it was discovered independently more than once. As we shall see, the Indian and Chinese versions differ in subtle ways from the Babylonian method.

Meton introduced a cycle of 19 years—the Metonic cycle. Each of these 19 years were to contain 12 months apiece, but seven extra months were to be intercalated in the 3rd, 5th, 8th 11th, 13th, 16th, and 19th years of the cycle, to make a total of 235 months. Of these months, 125 were to be full (30 days) and the remaining 110 deficient (29 days), to give a total of 6940 days. The distribution of full and deficient days was calculated as follows. First suppose that all 235 months were full (30 days each), to give a total of 7050 days; from this, 110 days must be dropped or 110 months made deficient. This was done by dropping every 64th day and demoting the month in which this day occurred. The average length of a month was thus 29.5319 days, whereas the astronomical average was about 29.5306. The discrepancy was only about 24 minutes in a year. Likewise, the average length of the year was 365.2632 days—about 30 minutes too long.

Meton was eventually honoured for his accomplishment by an Olympic crown and his name, and that of his co-author, Euctemon, are said to have been inscribed in letters of gold in the Temple of Minerva in Athens, together with the numbers in the calendar which gave the positions of the years in the 19-year cycle. In medieval calendars these numbers were still being written in gold, and to this day they are known as ‘golden numbers’. This medieval usage dates back at least to the Massa compoti of d'Alexandre de Villediew, which was written in 1170. One thirteenth-century scholar wrote: ‘This number excels all other lunar ratios as gold excels all other metals’. Curiously the Athenians themselves only used the Metonic calendar for a short period, but the Seleucid Empire later adapted it from Babylonian practices.

The small discrepancies in Meton's calendar were tolerated, by those who used it, for about 100 years. To correct them, Calippus of Cyzicus (c. 370-300 bc) proposed replacing Meton's cycle of 19 years by one of 76 years—the Calippic cycle—and to omit just one day (in the fourth 19-year period). The average month was now 29.5308 days and the discrepancy reduced to about 22 seconds a year. The average year became exactly 365[frac14] days long (about 11 minutes too long still, but the Greeks of the time believed that the true length of the year was 365[frac14]days). The first Calippic cycle was reckoned to have started in 330 bc.

Later still, in 143 BC, Hipparchus of Nicea (c. 180-125 bc) recommended dropping one further day every four Calippic cycles, or 304 years. This gave an average year too long by about 6 minutes and an average lunation of only half a second less than the true synodic period. This was the best that the ancients were able to do for people who wished to continue with a lunar calendar. A variation on the method is still used in the Jewish calendar and in the Christian calculation of the date of Easter. …

At this point we may ask whether a better rate of intercalation of a thirteenth month is available. It is readily shown that the rate of intercalation (R), expressed as the average number of months intercalated in a year (this is some proper fraction), should depend only on the ratio of the lengths of a mean tropical year (Y) to that of a mean lunation (L). In fact, we can show that ideally R = Y/L − 12. Estimates of Y/L − 12 vary from 0.368270 for accurate modern values of Y and L, to 0.368530 for the old estimate of Y = 365[frac14] days. The theory of continued fraction can be used to provide the following successively better approximations (whatever reasonable value of R (Y/L − 12) we select): 1/3, 3/8, 4/11, 7/19, 123/334. The first approximation gives us an early but crude intercalation rule; 3/8 gives us the octaeteris rule; as far as I know the 4/11 has never been entertained; 7/19 is Meton's rule; and 123/334 involves intercalating 123 months in every 334 years (but it is not really practical to use a cycle longer than the human lifetime). More detailed analysis shows that the 7/19 rule cannot be improved upon for cycles of less than 100 years.

The Indians and the Chinese also used what is in effect a 19-year cycle, but based upon more subtle astronomical considerations. …

The Indians also used a radically different approach to the non-integral number of days in the lunation. They effectively abandoned the day and used instead the tithi. This was, on average, 1/30th part of the lunar synodic period, or 0.984353 days. The tithis were numbered consecutively; the first one of each lunar month began at the moment of conjunction. Each day was allocated the tithi which prevailed at sunrise. Not every tithi corresponded to a day; since the tithi varied in length (as does the solar day), sometimes one was skipped and sometimes two days had the same tithi.

SOLAR CALENDARS

Those who, like the Romans, were prepared to abandon the moon and organize their calendar to keep in synchrony only with the sun were presented with a simpler problem. This was first solved by the ancient Egyptians, with their 365-day wandering year. Later, the Romans used a year of 365 days with an extra day intercalated every four years. This gave a four-year cycle of 1461 days, with an average length of the year of 365[frac14]days—a little too long. The date on which the equinoxes and solstices occurred would advance one day after about 128 years. This was the calendar that Julius Caesar introduced on the advice of Sosigenes. It became known as the Julian calendar and was used throughout Christendom till the sixteenth century.

The discrepancy between the Julian and astronomical years eventually became unacceptable, and in the sixteenth century further refinements were introduced by Pope Gregory XIII. The intercalated day was to be dropped from the years of the Christian era which were divisible by 100, but not by 400. The resulting 400-year cycle had 146097 days and the year contained an average of 365.2425 days. Estimates available to Gregory of the length of the astronomical year were within a few seconds of this; modern-day estimates show that it is about 26 seconds too long—but this discrepancy would not amount to a day till about the 51st century. …

Several other proposals have been made to reduce this remaining small discrepancy. The French astronomer, Delambre, suggested dropping a leap day every 3600 years—an excellent suggestion, provided someone was able to remember when exactly to do it. The astronomer Sir John Herschel (1792-1871), son of Sir William, the discoverer of the seventh planet, made a suggestion which is easier to grasp and remember, although it had been anticipated by Gilbert Romme in 1795 in his modification of the French revolutionary calendar. The proposal was to drop the leap day in years divisible by 4000. This would lead to an average length of a year of 365.24225 days, which, so he thought, was a mere five seconds too long and would lead to a discrepancy of a day only in the remote future.

We may ask whether it is possible, using all the paraphernalia of modern mathematics and computers, to design a more accurate scheme of intercalation. The short answer is no. The problem is to discover the number of leap years required in a given period to best approximate the length of the year. Suppose that we decide to have A leap years in a period of B years. A and B must be chosen so that the ratio A/B approximates to the fractional part of a year expressed in days; this I shall take to be 0.242 189. The Gregorian calendar has A = 97 and B = 400. One may set a computer to take all values from 1 to a 1000 for B, and all appropriate values of A, and calculate the ratio A/B for all such pairs of values. The best solution that can be found in this way gives A = 225 and B = 929; the difference between the assumed length of the year and the ratio A/B then amounts to a tiny 0.6 seconds a year.

However, there is a difficulty in deciding exactly when to intercalate the 225 days. Both the Julian and Gregorian calendars use very simple rules for deciding this issue. But first it is worth restating the problem. Suppose we intercalate a day regularly once every N years and then, as well, drop or insert a day K times in 100N years. For the Gregorian scheme N = 4 and K = −3, meaning that three leap days are dropped every 400 years. It then turns out that, within these constraints and examining all values of N up to 10, the method proposed by Lilio and Clavius is by far the best. The only other scheme that equals or improves on its accuracy requires a leap day every five years and the insertion of 21 more days within the space of 500 years; one way of doing this is to add another leap year every 25 years, and an extra one still every 500 years. This has several disadvantages, not the least being the possibility of having more than one leap day in some years. There is little compensating advantage, for the discrepancy is only reduced from 26 to 16 seconds a year.

In 1923, the Soviet Union adopted a somewhat more accurate, but slightly more complicated scheme. In this there are leap years every four years except in certain centurial years: the leap year is dropped in every centurial year unless it leaves a remainder of two or six when divided by nine. Thus seven out of nine of the centurial years are common. The mean calendar year is now a little under three seconds too long.

However, we now know, since the work of Simon Newcomb at the beginning of this century, that the length of the year is slowly decreasing by about half a second each century. This makes nonsense of any attempt to design a calendar that will last forever. Because of this effect the Gregorian calendar will be a day slow in about the year 4000; Herschel's adjustment would only extend this by about 1000 years. The year will presumably continue to get shorter and days will have to be dropped from the calendar year on a permanent basis from time to time—not that this need concern you or me. The effect will not be noticeable for many centuries, and at this point we will let the matter rest.

A CLASSIFICATION OF CALENDARS

EMPIRICAL CALENDARS

(a) The start of the months or years is determined by direct observation.

(b) The number of days in the month and year is fixed save that extra days are intercalated when this is deemed appropriate.

CALCULATED CALENDARS

(a) Lunar calendars: no attempt is made to keep the start of the year in synchrony with the sun, but the months keep in reasonable step with the moon by intercalating days according to a rule; for example, the Islamic calendar.

(b) Solar calendars: the moon is ignored but the year keeps in step with the sun.

(i) Astronomical calendars: the start of the year is determined from the calculated time of an equinox or solstice; for example, the original forms of the French Revolutionary and Bahá'i calendars.

(ii) Arithmetic solar calendars: the length of the year is adjusted by intercalating days according to precise rules to keep it in synchrony with the sun; for example, the Julian, Gregorian, Coptic, Ethiopian calendars.

(c) Lunisolar calendars: the months are geared to the moon and extra months are intercalated to keep the year, on average, in synchrony with the sun.

(i) Astronomical lunisolar calendars: these attempt to keep in synchrony with both the moon and the sun, but the start of the month or year is determined by astronomical calculation; for example, the Indian and Chinese calendars.

(ii) Arithmetic lunisolar calendars: these attempt to keep in synchrony with both the moon and the sun by rule-based intercalations; for example, the Jewish calendar and the Christian ecclesiastical calendar.

(d) Calendars with a wandering year: both the moon and the sun are abandoned. Each year contains a fixed number of days—365; for example, the Egyptian civil calendar; the Mayan year.

INTERCALATIONS

We have seen how the operation of a lunisolar calendar requires the intercalation of an extra month from time to time. The year which contains such a month has been called ‘embolismic’ and those that do not, ‘common’; months with more days than usual have been called ‘full’ or ‘abundant’ and those with fewer, ‘deficient’. Likewise, the Roman year that contained an extra day, ‘annus bissextile’, because in such years both 24 and 25 February were called VI Kal. Mart.; the first of these was called ante diem bissextum Kal. Mart or bis VI Kal. Mart.

In the Christian calendar, 29 February is the intercalated day in leap years. Leap years were so called because, as was written in the 1604 edition of the Anglican prayer book, ‘On every fourth year, the Sunday Letter leapeth’. The years which are not leap years are generally called ‘common’ years. In the French Revolutionary calendar, as in several other calendars, the extra day came at the end of the year. The French called theirs the ‘fête du Revolution’, and other calendars also had special names for it.

There is a subtle difference between Anglican and Roman Catholic practices concerning leap day. In the old Roman Julian calendar, the extra day was, as we have seen, inserted between the VII Kal. Mart. and VI Kal. Mart.—that is, between 24 and 25 February. That practice was taken over by the Roman Catholic Church and continued by the English until 1662, when the extra day was moved to a place between 28 February and 1 March and called 29 February. All this makes little discernible difference, except that in the Roman Catholic practice the Sunday, or dominical letter, is changed after 24 February rather than after 29 February. This can never affect the date of Easter but it does lead to celebrations of the feast of St Matthias taking place on different dates in leap years in the two Churches. In leap years, Roman Catholics celebrate this feast on 25 February, whereas the Anglicans continue to celebrate it on 24 February. Likewise, Roman Catholics celebrate the feast of St Gabriel of the Seven Dolours one day later, on 27 February, in leap years. It is as if the leap day and the day following were counted as one.

CONTRIVED CYCLES

So far we have discussed natural cycles of days based upon astronomical events and man's attempts to reconcile the lengths of the year, month, and day. Further cycles were, however, introduced by various nations from time to time.

The most ubiquitous cycle is the seven-day week which has been running in one form or another for 3000 years or more; more than 150 000 weeks have gone by. Other nations had short periods resembling the week but containing five, six, eight, or ten days. The Romans had an eight-day cycle, and the Mayan people of Central America had both a 13-day and a 20-day cycle running together.

Another cycle of some historical importance was the 15-year cycle of indiction. The pomp and splendour of Rome, as well as its army, were supported by taxes. Every 15 years, the wealth of the landowners, who paid these taxes, was assessed by an indiction. The first is said to have been held in ad 312, after Constantine had moved the capital of the Roman Empire to Constantinople, but one may extrapolate the 15-year cycles back in time; in this way the year ad 1 is generally regarded to be the fourth year of the then current cycle of indiction. The year of indiction is often taken to have begun on 1 September, but other dates have been used.

Most other ancient cultures—the Indians, the Chinese, the Mayans in particular—entertained cycles of years; some encompassed millions of years. The Chinese employed two cycles of 12 and 10 years.

INTERLOCKING CYCLES

An interesting effect, which has been called a resonance, occurs when two or more cycles with different periods operate simultaneously. Any particular day (or year if they are cycles of years) has several positions as counted from the start of each of the cycles. The day is thus characterized by a set of numbers, one for each cycle. In time the same set of numbers will repeat, thus forming a longer cycle. The period of this longer period is quite simply the lowest common multiple (LCM) of the periods of its components. For instance, the Mayans used three cycles of 13, 20, and 365 days. The LCM of these is 18 980 (13 x 20 x 365/5) days, which is about 52 years. They combined the 13- and 20-day cycles into a larger 260-day cycle, usually called a tzolkin. A similar device was employed by the Chinese in numbering their years (and days). They used a cycle of 12 years in which every year of the cycle was given the name of an animal, and a cycle of 10 years named by a celestial sign. Thus a year was characterized by the name of an animal and a celestial sign. The sequence of these pairs of names repeated every 60 years.

In the Christian Julian calendar, the first day of the year can fall on any of the seven days of the week, but there is a leap year every four years. This means that there is a complete cycle of 28 years before the days of the week fall on the same dates over an entire year. This cycle of 28 years is known as the solar cycle or cycle of the sun. It was invented about the time of the Council of Nicaea in ad 325, but the first cycle is regarded as having started with the year 9 bc, so that ad 1 was year 10 of the first solar cycle. Since the first day of a year may fall on any one of the seven days of the week, and be either a leap or a common year, there are 14 possible calendars. If you have not filled up your diary and are more than usually thrifty, keep it for a few years and used it again.

Following the Gregorian reform, the solar cycle is now interrupted at the end of three centuries in four, so that although there are still only 14 possible calendars, the sequence only repeats every 400 years.

The date of Easter in the Christian calendar is fixed using a variant of Meton's scheme. According to this, the full moon falls on the same day of the year every 19 Julian years (sometimes termed the lunar cycle). The lunar and solar cycles combine in the Julian calendar to give the paschal or dionysian cycle of 532 (19 x 28) years. The position of any year within the lunar cycle is known as its ‘golden number’.

The Renaissance polymath Joseph Justus Scaliger (1540-1609) introduced a ‘monster’ cycle by combining the solar and lunar cycles, and the cycle of indiction, to give a cycle of 7980 years. This he named the Julian cycle since, to quote his own words (in translation), ‘… it is laid out for Julian years only …’. (There is a widespread belief that he also named it in honour of his brilliant but eccentric father, Julius Caesar Scaliger (1484-1558).) Any particular year within this Julian cycle is uniquely defined by three numbers which give it its position in the solar, lunar, and indiction cycles. Thus, the position of ad 1 in the Julian cycle is 4714; and the first year of the Julian cycle is 4713 years before this, or bc 4713.

The Julian cycle is still used by astronomers, with the small difference that the year ad 1 is preceded by the year 0, and that by the year −1, and so forth. In their scale the Julian cycle began with year −4712. In fact, they also define a Julian date—each day, which begins at noon, Greenwich Mean Time, is given a unique number; day zero is 1 January 4713 bc in the Julian calendar.

Nearly as long as the Julian cycle is the cycle implicit in the long count of the Mayas; this was 13 baktuns or about 5125 years long (1 872 000 days exactly). At the end of each great cycle the world comes to an end and is recreated. Even longer periods, up to the hablatun of about 1[frac14] billion years, were considered. The Chinese had similar beliefs and had numerous cycles. … [After] a period of 31 920 years, which they called a chi, the world would come to an end only to be created anew. There are other instances of a belief in cycles of destruction and renewal.

The chi cycle did not exhaust the liking of the Chinese for large cycles. They entertained a cycle of 4617 years (which was constructed on principles similar to those of Scaliger's Julian period) and another period, 138 240 years, on whose completion all the planets were supposed to be simultaneously in conjunction. Finally, the resonance of these two gave a period of 23 539 040 years.

The Hindus had a series of long cycles based on early astronomical and cosmological speculation. The longest of these, the mahayuga, was of 4 320 000 years, but there is disagreement about the lengths of the shorter krta-, treta-, dvapara-, and kali-yugas. The kali-yuga is said to mark a grand conjunction of all the planets. It was argued (unconvincingly) that such an event occurred in 3102 BC, initiating an era which would last for 432 000 years. The numerological reasoning behind these numbers is unclear but could be based on 10 800—the number of muhurta … in 360 days. …

REGNAL YEARS AND ERAS

The position of a year within a cycle is one way of specifying a year, but a variety of other methods have been used. You or I might use the years of salient events in our lives as landmarks, and describe other events with reference to them: ‘the year I was born’, ‘the year I first went to America’, ‘the year my father died’, and so forth. The method is as old as records exist; the Babylonians compiled lists of notable events, one for each year.

Events of particular importance, such as the accession of kings, acted as beacons for the naming of years. Thus an agreement might be noted as being signed in the third year of King Nebuchanezzar II. This method of specifying regnal years, sometimes called eponymous dating, has persisted into the twentieth century in England, where Acts of Parliament are dated in this way, for example, as 3 Elizabeth II (meaning the third regnal year in the reign of Elizabeth II).

Yet another method is to name rather than number the year with reference to some regularly occurring event. The Greeks of Athens dated events by mentioning the current archon—a new one was elected every year. Similarly, the Romans mentioned the current consuls—again new ones were appointed each year. So Alexandria was founded in the year of the consulship of C. Valerius, L. F. Potitus, M. Claudius, and C. F. Marcellus. The Greeks sometimes dated events by the Olympiad; Olympic games were held every four years (as they are today following their revival in the last century). Thus Alexandria was founded in the second year after the 112th Olympiad (written as 112.2) in 331 bc. The first Olympiad is said to have been held in 776 bc.

Yet another method was to specify the year with reference to some cycle. Thus Hipparchus, when comparing the dates of the summer solstice determined by himself and by Aristarchus, several years earlier, notes that his observation was made in year 43 of the third Calippic cycle, whereas that of Aristarchus was in year 50 of the first cycle. … [The] Calippic cycle was 76 years long; the two observations were therefore 145 years apart. Since the first cycle started in 330 bc, Aristarchus made his measurement in 280 bc (when he was aged about 30), and Hipparchus made his in 135 bc (when he was about 55). The Chinese developed a system for naming the years that involved a cycle of 60 years (about the expected lifetime of a man), composed of interlocking 12- and 10-year cycles. In time however, the 60-year cycle of names returns to the start, leaving an ambiguity.

Perhaps the most important method, however, was to date all events from the year (or presumed year) of some very notable event, such as the Creation, the birth of Christ, or the flight of the Prophet from Mecca. These events mark the first year of eras. This method, in many cases, developed naturally from the practice of regnal dating. Instead of starting the count anew with each successive king, the counting continued through reign after reign, for as long as the dynasty lasted. The later Romans dated events from the foundation of Rome, though there was some disagreement as to when this took place; a popular date was 753 bc. Other eras were used by various nations at particular times. …

Of particular importance is the Christian era, which now pervades most of the world—despite occasional attempts to abandon it and begin a new era; one such attempt was made in the early days of the French Revolution, which gave us the Republican era (er). The Christian era was first proposed by Dionysius Exiguus in ad 532, and the first year, ad 1, is supposed to be the year of the birth of Christ—though it is almost certainly incorrect. … Years after the birth of Christ are termed ‘Anno Domini’ (ad), ‘in the year of the Lord’; years before are counted backwards and termed ‘before Christ’ (bc). The year before ad 1 is 1 bc, and there is no year zero—to the mild annoyance of astronomers and other mathematically minded people. The terms ‘bc’ and ‘ad’ are often not used by people of different religious persuasions; for instance, the Jews refer to ‘Anno Domini’ as the ‘Common era’ (ce), and ‘before Christ’ as ‘before the Common era’ (bce). Some writers refer to dates in the past as ‘before present’ (bp). Many people would write, in a consistent manner, years before and after Christ as 4 bc and 1998 ad, for instance, though it makes better grammatical sense to write ad 1998 because ad stands for ‘Anno Domini’ or ‘In the year of the Lord’.

The start of an era may occur many years before the first date on which it is used; for instance, the Christian era began on ad 1, but was not invented till ad 532, and was not generally in use till later still. Calendar dates specifying a day before the invention of the calendar are sometimes call ‘proleptic dates’.

… [Some] eras begin with the fanciful date of the Creation; in fact there have been over 200 different calculations of this momentous event. Many of these are based upon the ages of the patriarchs recorded in the Bible. Such calculations were abandoned in the nineteenth century when Lord Rayleigh demonstrated that the earth was every much older than any of these calculations had suggested, and its age was to be measured in millions rather than thousands of years. Many now believe that it all started with a ‘big bang’ some 10 or 20 billion years ago, and that the solar system coalesced out of the debris several billion years later. Curiously, the biblical estimates of the date of the Creation more or less correspond to that of the start of the earliest civilizations.

The start of other eras might signal the foundation of a new religion, whilst others could relate to a new dynasty or an important battle. A remarkable cluster of eras start in the eighth century bc. …

The starting date of an era is sometimes called an epoch. Epochs should be used with care when converting dates from one era to another; in some cases the years counted vary in length, and in others they start in different seasons.