Celestial bodies provide the basic standards for determining the periods of a calendar. Their movement as they rise and set is now known to be a reflection of the Earth’s rotation, which, although not precisely uniform, can conveniently be averaged out to provide a suitable calendar day. The day can be measured either by the stars or by the Sun. If the stars are used, then the interval is called the sidereal day and is defined by the period between two passages of a star (more precisely of the vernal equinox, a reference point on the celestial sphere) across the meridian: it is 23 hours 56 minutes 4.10 seconds of mean solar time. The interval between two passages of the Sun across the meridian is a solar day. In practice, since the rate of the Sun’s motion varies with the seasons, use is made of a fictitious Sun that always moves across the sky at an even rate. This period of constant length, far more convenient for civil purposes, is the mean solar day, which has a duration in sidereal time of 24 hours 3 minutes 56.55 seconds. It is longer than the sidereal day because the motion of the Earth in its orbit during the period between two transits of the Sun means that the Earth must complete more than a whole revolution to bring the Sun back to the meridian. The mean solar day is the period used in calendar computation.

The month is determined by the Moon’s passage around the Earth, and, as in the case of the day, there are several ways in which it can be defined. In essence, these are of two kinds: first, the period taken by the Moon to complete an orbit of the Earth and, second, the time taken by the Moon to complete a cycle of phases. Among primitive societies, the month was determined from the phases; this interval, the synodic month, is now known to be 29.53059 days. The synodic month grew to be the basis of the calendar month.

The year is the period taken by the Earth to complete an orbit around the Sun and, again, there are a number of ways in which this can be measured. But for calculating a calendar that is to remain in step with the seasons, it is most convenient to use the tropical year, since this refers directly to the Sun’s apparent annual motion. The tropical year is defined as the interval between successive passages of the Sun through the vernal equinox (i.e., when it crosses the celestial equator late in March) and amounts to 365.242199 mean solar days.

The tropical year and the synodic month are incommensurable, 12 synodic months amounting to 354.36706 days, almost 11 days shorter than the tropical year. Moreover, neither is composed of a complete number of days, so that to compile any calendar that keeps in step with the Moon’s phases or with the seasons it is necessary to insert days at appropriate intervals; such additions are known as intercalations.

In primitive lunar calendars, intercalation was often achieved by taking alternately months of 29 and 30 days. When, in order to keep dates in step with the seasons, a solar calendar was adopted, some greater difference between the months and the Moon’s phases was bound to occur. And the solar calendar presented an even more fundamental problem—that of finding the precise length of the tropical year. Observations of cyclic changes in plant or animal life were far too inaccurate, and astronomical observations became necessary. Since the stars are not visible when the Sun is in the sky, some indirect way had to be found to determine its precise location among them. In tropical and subtropical countries it was possible to use the method of heliacal risings. Here the first task was to determine the constellations around the whole sky through which the Sun appears to move in the course of a year. Then, by observing the stars rising in the east just after sunset it was possible to know which were precisely opposite in the sky, where the Sun lay at that time. Such heliacal risings could, therefore, be used to determine the seasons and the tropical year. In temperate countries, the angle at which stars rise up from the horizon is not steep enough for this method to be adopted, so that there wood or stone structures were built to mark out points along the horizon to allow analogous observations to be made.

The most famous of these is Stonehenge in Wiltshire, England, where the original structure appears to have been built about 2000 bce and additions made at intervals several centuries later. It is composed of a series of holes, stones, and archways arranged mostly in circles, the outermost ring of holes having 56 marked positions, the inner ones 30 and 29, respectively. In addition, there is a large stone—the heel stone—set to the northeast, as well as some smaller stone markers. Observations were made by lining up holes or stones with the heel stone or one of the other markers and watching for the appearance of the Sun or Moon against that point on the horizon that lay in the same straight line. The extreme north and south positions on the horizon of the Sun—the summer and winter solstices—were particularly noted, while the inner circles, with their 29 and 30 marked positions, allowed “hollow” and “full” (29- or 30-day) lunar months to be counted off. More than 600 contemporaneous structures of an analogous but simpler kind have been discovered in Britain, in Brittany, and elsewhere in Europe and the Americas. It appears, then, that astronomical observation for calendrical purposes was a widespread practice in some temperate countries three to four millennia ago.

Today a solar calendar is kept in step with the seasons by a fixed rule of intercalation. But although the Egyptians, who used the heliacal rising of Sirius to determine the annual inundation of the Nile, knew that the tropical year was about 365.25 days in length, they still used a 365-day year without intercalation. This meant that the calendar date of Sirius’s rising became increasingly out of step with the original dates as the years progressed. In consequence, while the agricultural seasons were regulated by the heliacal rising of Sirius, the civil calendar ran its own separate course. It was not until well into Roman times that an intercalary day once every four years was instituted to retain coincidence.

Complex cycles

The fact that neither months nor years occupied a whole number of days was recognized quite early in all the great civilizations. Some observers also realized that the difference between calendar dates and the celestial phenomena due to occur on them would first increase and then diminish until the two were once more in coincidence. The succession of differences and coincidences would be cyclic, recurring time and again as the years passed. An early recognition of this phenomenon was the Egyptian Sothic cycle, based on the star Sirius (called Sothis by the ancient Egyptians). The error with respect to the 365-day year and the heliacal risings of Sirius amounted to one day every four tropical years, or one whole Egyptian calendar year every 1,460 tropical years (4 × 365), which was equivalent to 1,461 Egyptian calendar years. After this period the heliacal rising and setting of Sothis would again coincide with the calendar dates (see below The Egyptian calendar).

The main use of cycles was to try to find some commensurable basis for lunar and solar calendars, and the best known of all the early attempts was the octaëteris, usually attributed to Cleostratus of Tenedos (c. 500 bce) and Eudoxus of Cnidus (390–c. 340 bce). The cycle covered eight years, as its name implies, and so the octaëteris amounted to 8 × 365, or 2,920 days. This was very close to the total of 99 lunations (99 × 29.5 = 2,920.5 days), so this cycle gave a worthwhile link between lunar and solar calendars. When in the 4th century bce the accepted length of the year became 365.25 days, the total number of solar calendar days involved became 2,922, and it was then realized that the octaëteris was not as satisfactory a cycle as supposed.

Another early and important cycle was the saros, essentially an eclipse cycle. There has been some confusion over its precise nature because the name is derived from the Babylonian word shār or shāru, which could mean either “universe” or the number 3,600 (i.e., 60 × 60). In the latter sense it was used by Berosus (c. 290 bce) and a few later authors to refer to a period of 3,600 years. What is now known as the saros and appears as such in astronomical textbooks (still usually credited to the Babylonians) is a period of 18 years 11 1/3 days (or with one day more or less, depending on how many leap years are involved), after which a series of eclipses is repeated.

In Central America an independent system of cycles was established (see below The Americas). The most significant of all the early attempts to provide some commensurability between a religious lunar calendar and the tropical year was the Metonic cycle. This was first devised about 432 bce by the astronomer Meton of Athens. Meton worked with another Athenian astronomer, Euctemon, and made a series of observations of the solstices, when the Sun’s noonday shadow cast by a vertical pillar, or gnomon, reaches its annual maximum or minimum, to determine the length of the tropical year. Taking a synodic month to be 29.5 days, they then computed the difference between 12 of these lunations and their tropical year, which amounted to 11 days. It could be removed by intercalating a month of 33 days every third year. But Meton and Euctemon wanted a long-term rule that would be as accurate as they could make it, and they therefore settled on a 19-year cycle. This cycle consisted of 12 years of 12 lunar months each and seven years each of 13 lunar months, a total of 235 lunar months. If this total of 235 lunations is taken to contain 110 hollow months of 29 days and 125 full months of 30 days, the total comes to (110 × 29) + (125 × 30), or 6,940 days. The difference between this lunar calendar and a solar calendar of 365 days amounted to only five days in 19 years and, in addition, gave an average length for the tropical year of 365.25 days, a much-improved value that was, however, allowed to make no difference to daily reckoning in the civil calendar. But the greatest advantage of this cycle was that it laid down a lunar calendar that possessed a definite rule for inserting intercalary months and kept in step with a cycle of the tropical years. It also gave a more accurate average value for the tropical year and was so successful that it formed the basis of the calendar adopted in the Seleucid empire (Mesopotamia) and was used in the Jewish calendar and the calendar of the Christian church; it also influenced Indian astronomical teaching.

The Metonic cycle was improved by both Callippus and Hipparchus. Callippus of Cyzicus (c. 370–300 bce) was perhaps the foremost astronomer of his day. He formed what has been called the Callippic period, essentially a cycle of four Metonic periods. It was more accurate than the original Metonic cycle and made use of the fact that 365.25 days is a more precise value for the tropical year than 365 days. The Callippic period consisted of 4 × 235, or 940 lunar months, but its distribution of hollow and full months was different from Meton’s. Instead of having totals of 440 hollow and 500 full months, Callippus adopted 441 hollow and 499 full, thus reducing the length of four Metonic cycles by one day. The total days involved therefore became (441 × 29) + (499 × 30), or 27,759, and 27,759 ÷ (19 × 4) gives 365.25 days exactly. Thus the Callippic cycle fitted 940 lunar months precisely to 76 tropical years of 365.25 days.

Hipparchus, who flourished in Rhodes about 150 bce and was probably the greatest observational astronomer of antiquity, discovered from his own observations and those of others made over the previous 150 years that the equinoxes, where the ecliptic (the Sun’s apparent path) crosses the celestial equator (the celestial equivalent of the terrestrial Equator), were not fixed in space but moved slowly in a westerly direction. The movement is small, amounting to no more than 2° in 150 years, and it is known now as the precession of the equinoxes. Calendrically, it was an important discovery because the tropical year is measured with reference to the equinoxes, and precession reduced the value accepted by Callippus. Hipparchus calculated the tropical year to have a length of 365.242 days, which was very close to the present calculation of 365.242199 days; he also computed the precise length of a lunation, using a “great year” of four Callippic cycles. He arrived at the value of 29.53058 days for a lunation, which, again, is comparable with the present-day figure, 29.53059 days.

The calendar dating of historical events and the determination of how many days have elapsed since some astronomical or other occurrence are difficult for a number of reasons. Leap years have to be inserted, but, not always regularly, months have changed their lengths and new ones have been added from time to time and years have commenced on varying dates and their lengths have been computed in various ways. Since historical dating must take all these factors into account, it occurred to the 16th-century French classicist and literary scholar Joseph Justus Scaliger (1540–1609) that a consecutive numbering system could be of inestimable help. This he thought should be arranged as a cyclic period of great length, and he worked out the system that is known as the Julian period. He published his proposals in Paris in 1583 under the title Opus de emendatione temporum.

The Julian period is a cycle of 7,980 years. It is based on the Metonic cycle of 19 years, a “solar cycle” of 28 years, and the Indiction cycle of 15 years. The so-called solar cycle was a period after which the days of the seven-day week repeated on the same dates. Since one year contains 52 weeks of seven days, plus one day, the days of the week would repeat every seven years were no leap year to intervene. A Julian calendar leap year cycle is four years, therefore the days of the week repeat on the same dates every 4 × 7 = 28 years. The cycle of the Indiction was a fiscal, not astronomical, period. It first appears in tax receipts for Egypt in 303 ce and probably took its origin in a periodic 15-year taxation census that followed Diocletian’s reconquest of Egypt in 297 ce. By multiplying the Metonic, solar, and Indiction cycles together, Scaliger obtained his cycle of 7,980 years (19 × 28 × 15 = 7,980), a period of sufficient length to cover most previous and future historical dates required at any one time.

Scaliger, tracing each of the three cycles back in time, found that all coincided in the year 4713 bce, on the Julian calendar reckoning. On the information available to him, he believed this to be a date considerably before any historical events. He therefore set the beginning of the first Julian period at January 1, 4713 bce. The years of the Julian period are not now used, but the day number is still used in astronomy and in preparing calendar tables, for it is the only record where days are free from combination into weeks and months.