any system for dividing time over extended periods, such as days, months, or years, and arranging such divisions in a definite order. A calendar is convenient for regulating civil life and religious observances and for historical and scientific purposes. The word is derived from the Latin calendarium, meaning interest register, or account book, itself a derivation from calendae (or kalendae), the first day of the Roman month, the day on which future market days, feasts, and other occasions were proclaimed.

The development of a calendar is vital for the study of chronology, since this is concerned with reckoning time by regular divisions, or periods, and using these to date events. It is essential, too, for any civilization that needs to measure periods for agricultural, business, domestic, or other reasons. The first practical calendar to evolve from these requirements was the Egyptian (Egypt, ancient), and it was this that the Romans developed into the Julian calendar that served western Europe for more than 1,500 years. The Gregorian calendar was a further improvement and has been almost universally adopted because it satisfactorily draws into one system the dating of religious festivals based on the phases of the Moon and seasonal activities determined by the movement of the Sun. Such a calendar system is complex, since the periods of the Moon's phases and the Sun's (Sun) motion are incompatible; but by adopting regular cycles of days and comparatively simple rules for their application, the calendar provides a year with an error of less than half a minute.

The basic unit of computation in a calendar is the day, and although days are now measured from midnight to midnight this has not always been so. Astronomers (astronomy), for instance, from about the 2nd century AD until 1925 counted days from noon to noon. In earlier civilizations and among primitive peoples, where there was less communication between different settlements or groups, different methods of reckoning the day presented no difficulties. Most primitive tribes used a dawn-to-dawn reckoning, calling a succession of days so many dawns, or suns. Later, the Babylonians, Jews, and Greeks counted a day from sunset to sunset, whereas the day was said to begin at dawn for the Hindus and Egyptians and at midnight for the Romans. The Teutons counted nights, and from them the grouping of 14 days called a fortnight is derived.

There was also great variety in the ways in which the day was subdivided. In Babylonia, for example, the astronomical day was divided differently than the civil day, which, as in other ancient cultures, was composed of “watches.” The length of the watches was not constant but varied with the season, the day watches being the longer in summer and the night watches in the winter. Such seasonal (season) variations in divisions of the day, now called seasonal or temporal hours (hour), became customary in antiquity because they corresponded to the length of the Sun's time above the horizon, at maximum in summer and at minimum in winter. Only with the advent of mechanical clocks in western Europe at the end of the 13th century did seasonal (unequal) hours become inconvenient.

Most early Western civilizations used 24 seasonal hours in the day—12 hours of daylight and 12 of darkness. This was the practice of the Greeks, the Sumerians and Babylonians, the Egyptians, and the Romans, and of Western Christendom so far as civil reckoning was concerned. The church adopted its own canonical hours (divine office) for reckoning daily worship: there were seven of these—matins, prime, terce, sext, none, vespers, and compline—but in secular affairs the system of 24 hours held sway. This number, 2 × 12, or 24, was derived in Babylonia from the Sumerian sexagesimal method of reckoning, based on gradations of 60 (5 × 12 = 60) rather than on multiples of 10. In Babylonia, for most purposes, both daylight and night were divided into three equal watches, and each watch was subdivided into half- and quarter-watches. Babylonian astronomers, perhaps in preference to the variable civil system, divided every day into 12 equal units, called bēru, each of which was subdivided into 30 gesh. The oldest known astronomical texts are from the Old Babylonian period, but this dual system may be attributable to earlier Sumerian society.

Once the day is divided into parts, the next task is to gather numbers of days into groups. Among primitive peoples, it was common to count moons (months) rather than days, but later a period shorter than the month was thought more convenient, and an interval between market days was adopted. In West Africa some tribes used a four-day interval; in central Asia five days was customary; the Assyrians adopted five days and the Egyptians, 10 days, whereas the Babylonians attached significance to the days of the lunation that were multiples of seven. In ancient Rome, markets were held at eight-day intervals; because of the Roman method of inclusive numeration, the market day was denoted nundinae (“ninth-day”) and the eight-day week, an inter nundium.

The seven-day week may owe its origin partly to the four (approximately) seven-day phases of the Moon and partly to the Babylonian belief in the sacredness of the number seven, which was probably related to the seven planets. Moreover, by the 1st century BC the Jewish seven-day week seems to have been adopted throughout the Roman world, and this influenced Christendom. The names in English of the days of the week are derived from Latin or Anglo-Saxon names of gods.

The month is based on the lunation, that period in which the Moon completes a cycle of its phases. The period lasts approximately 29 ¹/₂ days, and it is easy to recognize and short enough for the days to be counted without using large numbers. In addition, it is very close to the average menstrual period of women and also to the duration of cyclic behaviour in some marine creatures. Thus, the month possessed great significance and was often the governing period for religious observances, of which the dating of Easter is a notable example. Most early calendars were, essentially, collections of months, the Babylonians using 29- and 30-day periods alternately, the Egyptians fixing the duration of all months at 30 days, with the Greeks copying them, and the Romans in the Julian calendar having a rather more complex system using one 28-day period with the others of either 30 or 31 days.

The month is not suitable for determining the seasons, for these are a solar, not a lunar, phenomenon. Seasons vary in different parts of the world—in tropical countries there are just the rainy and dry periods, but elsewhere there are successions of wider changes. In Egypt the annual flooding of the Nile (Nile River) was followed by seeding and then harvest, and three seasons were recognized; but in Greece and other more northern countries there was a succession of four seasons of slightly different lengths. However many there seemed to be, it was everywhere recognized that seasons were related to the Sun and that they could be determined from solar observations. These might consist of noting the varying length of the midday shadow cast by a stick thrust vertically into the ground or follow the far more sophisticated procedure of deducing from nocturnal observations the Sun's position against the background of the stars. In either case the result was a year of 365 days, a period incompatible with the 29 ¹/₂-day lunation. To find some simple relationship between the two periods was the problem that faced all calendar makers from Babylonian times onward.

A number of nonastronomical natural signs have also been used in determining the seasons. In the Mediterranean area, such indications change rapidly, and Hesiod (c. 800 BC) mentions a wide variety: the cry of migrating cranes, which indicated a time for plowing and sowing; the time when snails climb up plants, after which digging in vineyards should cease; and so on. An unwitting approximation to the tropical year may also be obtained by intercalation, using a simple lunar calendar and observations of animal behaviour. Such an unusual situation has grown up among the Yami fishermen of Botel-Tobago Island, near Taiwan. They use a calendar based on phases of the Moon, and some time about March (the precise date depends on the degree of error of their lunar calendar compared with the tropical year) they go out in boats with lighted flares. If flying fish appear, the fishing season is allowed to commence, but if the lunar calendar is too far out of step with the seasons, the flying fish will not rise. Fishing is then postponed for another lunation, which they insert in the lunar calendar, thus having a year of 13 instead of the usual 12 lunations.

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 (star) or by the Sun. If the stars are used, then the interval is called the sidereal (sidereal period) 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 (see below). 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 (intercalation).

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 BC 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 (solstice)—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' 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.

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 the section below The Egyptian calendar (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 BC) and Eudoxus of Cnidus (390–c. 340 BC). 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 BC, 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 BC) 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 Ancient and religious calendar systems: The Americas (calendar)). 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 BC 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 BC) 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.

The calendar dating of historical events (historiography) and the determination of how many days have elapsed since some astronomical or other occurrence are difficult for a number of reasons. Leap years (leap year) 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 (Scaliger, Joseph Justus) (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 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 (see below) 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 an astronomical, period. It first appears in tax receipts for Egypt in AD 303, and probably took its origin in a periodic 15-year taxation census that followed Diocletian's reconquest of Egypt in AD 297. 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 BC, 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 BC. 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.

The lunisolar calendar, in which months are lunar but years are solar—that is, are brought into line with the course of the Sun—was used in the early civilizations of the whole Middle East, except Egypt, and in Greece. The formula was probably invented in Mesopotamia (Mesopotamia, history of) in the 3rd millennium BC. Study of cuneiform tablets found in this region facilitates tracing the development of time reckoning back to the 27th century BC, near the invention of writing. The evidence shows that the calendar is a contrivance for dividing the flow of time into units that suit society's current needs. Though calendar makers put to use time signs offered by nature—the Moon's phases, for example—they rearranged reality to make it fit society's constructions.

In Mesopotamia the solar year was divided into two seasons, the “summer,” which included the barley harvest in the second half of May or in the beginning of June, and the “winter,” which roughly corresponded to today's fall–winter. Three seasons (Assyria) and four seasons (Anatolia) were counted in northerly countries, but in Mesopotamia the bipartition of the year seemed natural. As late as c. 1800 BC the prognoses for the welfare of the city of Mari, on the middle Euphrates, were taken for six months.

The months began at the first visibility of the New Moon, and in the 8th century BC court astronomers still reported this important observation to the Assyrian kings. The names of the months differed from city to city, and within the same Sumerian (Sumer) city of Babylonia a month could have several names, derived from festivals, from tasks (e.g., sheepshearing) usually performed in the given month, and so on, according to local needs. On the other hand, as early as the 27th century BC, the Sumerians had used artificial time units in referring to the tenure of some high official—e.g., on N-day of the turn of office of PN, governor. The Sumerian administration also needed a time unit comprising the whole agricultural (agriculture, origins of) cycle; for example, from the delivery of new barley and the settling of pertinent accounts to the next crop. This financial year began about two months after barley cutting. For other purposes, a year began before or with the harvest. This fluctuating and discontinuous year was not precise enough for the meticulous accounting of Sumerian scribes, who by 2400 BC already used the schematic year of 30 × 12 = 360 days.

At about the same time, the idea of a royal year took precise shape, beginning probably at the time of barley harvest, when the king celebrated the new (agricultural) year by offering first fruits to gods in expectation of their blessings for the year. When, in the course of this year, some royal exploit (conquest, temple building, and so on) demonstrated that the fates had been fixed favourably by the celestial powers, the year was named accordingly; for example, as the year in which “the temple of Ningirsu was built.” Until the naming, a year was described as that “following the year named (after such and such event).” The use of the date formulas was supplanted in Babylonia by the counting of regnal years in the 17th century BC.

The use of lunar reckoning began to prevail in the 21st century BC. The lunar year probably owed its success to economic progress. A barley loan could be measured out to the lender at the next year's threshing floor. The wider use of silver as the standard of value demanded more flexible payment terms. A man hiring a servant in the lunar month of Kislimu for a year knew that the engagement would end at the return of the same month, without counting days or periods of office between two dates. At the city of Mari in about 1800 BC, the allocations were already reckoned on the basis of 29- and 30-day lunar months. In the 18th century BC, the Babylonian Empire standardized the year by adopting the lunar calendar of the Sumerian sacred city of Nippur. The power and the cultural prestige of Babylon assured the success of the lunar year, which began on Nisanu 1, in the spring. When, in the 17th century BC, the dating by regnal years became usual, the period between the accession day and the next Nisanu 1 was described as “the beginning of the kingship of PN,” and the regnal years were counted from this Nisanu 1.

It was necessary for the lunar year of about 354 days to be brought into line with the solar (agricultural) year of approximately 365 days. This was accomplished by the use of an intercalated month. Thus, in the 21st century BC, a special name for the intercalated month iti dirig appears in the sources. The intercalation was operated haphazardly, according to real or imagined needs, and each Sumerian city inserted months at will; e.g., 11 months in 18 years or two months in the same year. Later, the empires centralized the intercalation, and as late as 541 BC it was proclaimed by royal fiat. Improvements in astronomical knowledge eventually made possible the regularization of intercalation; and, under the Persian kings (c. 380 BC), Babylonian calendar calculators succeeded in computing an almost perfect equivalence in a lunisolar cycle of 19 years and 235 months with intercalations in the years 3, 6, 8, 11, 14, 17, and 19 of the cycle. The new year's day (Nisanu 1) now oscillated around the spring equinox within a period of 27 days.

The Babylonian month names were Nisanu, Ayaru, Simanu, Duʾuzu, Abu, Ululu, Tashritu, Arakhsamna, Kislimu, Tebetu, Shabatu, Adaru. The month Adaru II was intercalated six times within the 19-year cycle but never in the year that was 17th of the cycle, when Ululu II was inserted. Thus, the Babylonian calendar until the end preserved a vestige of the original bipartition of the natural year into two seasons, just as the Babylonian months to the end remained truly lunar and began when the New Moon was first visible in the evening. The day began at sunset. Sundials and water clocks served to count hours.

The influence of the Babylonian calendar was seen in many continued customs and usages of its neighbour and vassal states long after the Babylonian Empire had been succeeded by others. In particular, the Jewish calendar (Jewish religious year) in use at relatively late dates employed similar systems of intercalation of months, month names, and other details (see below The Jewish calendar (calendar)). The Jewish adoption of Babylonian calendar customs dates from the period of the Babylonian Exile in the 6th century BC.

Of the calendars of other peoples of the ancient Near East, very little is known. Thus, though the names of all or of some months are known, their order is not. The months were probably everywhere lunar, but evidence for intercalation is often lacking; for instance, in Assyria. For accounting, the Assyrians also used a kind of week, of five days, as it seems, identified by the name of an eponymous official. Thus, a loan could be made and interest calculated for a number of weeks in advance and independently of the vagaries of the civil year. In the city of Ashur, the years bore the name of the official elected for the year; his eponym was known as the limmu. As late as about 1070 BC, his installation date was not fixed in the calendar. From about 1100 BC, however, Babylonian month names began to supplant Assyrian names, and, when Assyria became a world power, it used the Babylonian lunisolar calendar.

The calendar of the Hittite Empire is known even less well. As in Babylonia, the first Hittite month was that of first fruits, and, on its beginning, the gods determined the fates.

At about the time of the conquest of Babylonia in 539 BC, Persian kings made the Babylonian cyclic calendar standard throughout the Persian Empire, from the Indus to the Nile. Aramaic documents from Persian Egypt, for instance, bear Babylonian dates besides the Egyptian. Similarly, the royal years were reckoned in Babylonian style, from Nisanu 1. It is probable, however, that at the court itself the counting of regnal years began with the accession day. The Seleucids and, afterward, the Parthian rulers of Iran maintained the Babylonian calendar. The fiscal administration in northern Iran, from the 1st century BC, at least, used Zoroastrian month and day names in documents in Pahlavi (the Iranian language of Sāsānian Persia). The origin and history of the Zoroastrian calendar year of 12 months of 30 days, plus five days (that is, 365 days), remain unknown. It became official under the Sāsānian dynasty, from about AD 226 until the Arab conquest in 621. The Arabs introduced the Muslim lunar year, but the Persians continued to use the Sāsānian solar year, which in 1079 was made equal to the Julian year by the introduction of the leap year.

The ancient Egyptians (Egypt, ancient) originally employed a calendar based upon the Moon, and, like many peoples throughout the world, they regulated their lunar calendar by means of the guidance of a sidereal calendar. They used the seasonal appearance of the star Sirius (Sothis); this corresponded closely to the true solar year, being only 12 minutes shorter. Certain difficulties arose, however, because of the inherent incompatibility of lunar and solar years. To solve this problem the Egyptians invented a schematized civil year of 365 days divided into three seasons, each of which consisted of four months of 30 days each. To complete the year, five intercalary days were added at its end, so that the 12 months were equal to 360 days plus five extra days. This civil calendar was derived from the lunar calendar (using months) and the agricultural, or Nile, fluctuations (using seasons); it was, however, no longer directly connected to either and thus was not controlled by them. The civil calendar served government and administration, while the lunar calendar continued to regulate religious affairs and everyday life.

In time, the discrepancy between the civil calendar and the older lunar structure became obvious. Because the lunar calendar was controlled by the rising of Sirius, its months would correspond to the same season each year, while the civil calendar would move through the seasons because the civil year was about one-fourth day shorter than the solar year. Hence, every four years it would fall behind the solar year by one day, and after 1,460 years it would again agree with the lunisolar calendar. Such a period of time is called a Sothic cycle.

Because of the discrepancy between these two calendars, the Egyptians established a second lunar calendar based upon the civil year and not, as the older one had been, upon the sighting of Sirius. It was schematic and artificial, and its purpose was to determine religious celebrations and duties. In order to keep it in general agreement with the civil year, a month was intercalated every time the first day of the lunar year came before the first day of the civil year; later, a 25-year cycle of intercalation was introduced. The original lunar calendar, however, was not abandoned but was retained primarily for agriculture because of its agreement with the seasons. Thus, the ancient Egyptians operated with three calendars, each for a different purpose.

The only unit of time that was larger than a year was the reign of a king. The usual custom of dating by reign was: “year 1, 2, 3 . . . , etc., of King So-and-So,” and with each new king the counting reverted back to year One. King lists recorded consecutive rulers and the total years of their respective reigns.

The civil year was divided into three seasons, commonly translated: Inundation, when the Nile overflowed the agricultural (agriculture, origins of) land; Going Forth, the time of planting when the Nile returned to its bed; and Deficiency, the time of low water and harvest.

The months of the civil calendar were numbered according to their respective seasons and were not listed by any particular name—e.g., third month of Inundation—but for religious purposes the months had names. How early these names were employed in the later lunar calendar is obscure.

The days in the civil calendar were also indicated by number and listed according to their respective months. Thus a full civil date would be: “Regnal year 1, fourth month of Inundation, day 5, under the majesty of King So-and-So.” In the lunar calendar, however, each day had a specific name, and from some of these names it can be seen that the four quarters or chief phases of the Moon were recognized, although the Egyptians did not use these quarters to divide the month into smaller segments, such as weeks. Unlike most people who used a lunar calendar, the Egyptians began their day with sunrise instead of sunset because they began their month, and consequently their day, by the disappearance of the old Moon just before dawn.

As was customary in early civilizations, the hours were unequal, daylight being divided into 12 parts, and the night likewise; the duration of these parts varied with the seasons. Both water clocks and sundials were constructed with notations to indicate the hours for the different months and seasons of the year. The standard hour of constant length was never employed in ancient Egypt.

The earliest sources (clay tablets of the 13th century BC, the writings of Homer and Hesiod) imply the use of lunar months; Hesiod also uses reckoning determined by the observation of constellations and star groups; e.g., the harvest coincides with the visible rising of the star group known as the Pleiades before dawn. This simultaneous use of civil and natural calendars is characteristic of Greek (Greek calendar) as well as Egyptian time reckoning. In the classical age and later, the months, named after festivals of the city, began in principle with the New Moon. The lunar year of 12 months and about 354 days was to be matched with the solar year by inserting an extra month every other year. The Macedonians (Macedonia) used this system as late as the 3rd century BC, although 25 lunar months amount to about 737 days, while two solar years count about 730 days. In fact, as the evidence from the second half of the 5th century BC shows, at this early time the calendar was already no longer tied in with the phases of the Moon. The cities, rather, intercalated months and added or omitted days at will to adjust the calendar to the course of the Sun and stars and also for the sake of convenience, as, for instance, to postpone or advance a festival without changing its traditional calendar date. The calendric New Moon could disagree by many days with the true New Moon, and in the 2nd century BC Athenian (Athens) documents listed side by side both the calendar date and that according to the Moon. Thus, the lunar months that were in principle parallel might diverge widely in different cities. Astronomers such as Meton, who in 432 BC calculated a 19-year lunisolar cycle, were not heeded by the politicians, who clung to their calendar-making power.

The civil year (etos) was similarly dissociated from the natural year (eniautos). It was the tenure term of an official or priest, roughly corresponding to the lunar year, or to six months; it gave his name to his time period. In Athens, for instance, the year began on Hecatombaion 1, roughly midsummer, when the new archon entered his office, and the year was designated by his name; e.g., “when Callimedes was archon”—that is, 360–359 BC. There was no New Year's festival.

As the archon's year was of indefinite and unpredictable length, the Athenian administration for accounting, for the dates of popular assemblies, etc., used turns of office of the sections (prytanies) of the Council (Boule), which each had fixed length within the year. The common citizen used, along with the civil months, the seasonal time reckoning based on the direct observation of the Moon's phases and on the appearance and setting of fixed stars. A device (called a parapēgma) with movable pegs indicated the approximate correspondence between, for example, the rising of the star Arcturus and the civil date.

After Alexander's conquest of the Persian Empire, the Macedonian calendar came to be widely used by the Greeks in the East, though in Egypt it was supplanted by the Egyptian year at the end of the 3rd century BC. The Seleucids, from the beginning, adapted the Macedonian year to the Babylonian 19-year cycle (see above Babylonian calendars (calendar)). Yet, Greek cities clung to their arbitrary system of time reckoning even after the introduction of the Julian calendar throughout the Roman Empire. As late as c. AD 200, they used the antiquated octaëteris (see above Complex cycles (calendar)).

The Athenian months were called Hecatombaion (in midsummer), Metageitnion, Boedromion, Pyanopsion, Maimacterion, Poseideion, Gamelion, Anthesterion, Elaphebolion, Mounychion, Thargelion, and Scirophorion. The position of the intercalary month varied. Each month, in principle, consisted of 30 days, but in roughly six months the next to last day, the 29th, was omitted. The days were numbered within each of the three decades of the month. Thus, for example, Hecatombaion 16th was called “6th after the 10th of Hecatombaion.” The Macedonian months were Dios (in fall), Apellaios, Audynaios, Peritios, Dystros, Xanthicos, Artemisios, Daisios, Panemos, Loos, Gorpiaios, and Hyperberetaios. In the Seleucid calendar, Dios was identified with the Babylonian Tashritu, Apellaios with Arakhsamna, and so on.

Similar to the Babylonian civil pattern, the daylight time and the night were divided into four “watches” and 12 (unequal) hours each. Thus, the length of an hour oscillated between approximately 45 and 75 present-day minutes, according to the season. Water clocks, gnomons, and, after c. 300 BC, sundials roughly indicated time. The season division was originally bipartite as in Babylonia—summer and winter—but four seasons were already attested by about 650 BC.

This originated as a local calendar in the city of Rome (Roman republican calendar), supposedly drawn up by Romulus (Romulus and Remus) some seven or eight centuries before the Christian Era. The year began in March and consisted of 10 months, six of 30 days and four of 31 days, making a total of 304 days: it ended in December, to be followed by what seems to have been an uncounted winter gap. Numa Pompilius, according to tradition the second king of Rome (715?–673? BC), is supposed to have added two extra months, January and February, to fill the gap and to have increased the total number of days by 50, making 354. To obtain sufficient days for his new months, he is then said to have deducted one day from the 30-day months, thus having 56 days to divide between January and February. But since the Romans had, or had developed, a superstitious dread of even numbers, January was given an extra day; February was still left with an even number of days, but as that month was given over to the infernal gods, this was considered appropriate. The system allowed the year of 12 months to have 355 days, an uneven number.

The so-called Roman republican calendar was supposedly introduced by the Etruscan Tarquinius (Tarquin) Priscus (616–579 BC), according to tradition the fifth king of Rome. He wanted the year to begin in January since it contained the festival of the god of gates (later the god of all beginnings), but expulsion of the Etruscan dynasty in 510 BC led to this particular reform's being dropped. The Roman republican calendar still contained only 355 days, with February having 28 days; March, May, July, and October 31 days each; January, April, June, August, September, November, and December 29 days. It was basically a lunar calendar and short by 10 ¹/₄ days of a 365 ¹/₄-day tropical year. In order to prevent it from becoming too far out of step with the seasons, an intercalary month, Intercalans, or Mercedonius (from merces, meaning wages, since workers were paid at this time of year), was inserted between February 23 and 24. It consisted of 27 or 28 days, added once every two years, and in historical times at least, the remaining five days of February were omitted. The intercalation was therefore equivalent to an additional 22 or 23 days, so that in a four-year period the total days in the calendar amounted to (4 × 355) + 22 + 23, or 1,465: this gave an average of 366.25 days per year.

Intercalation was the duty of the Pontifices (pontifex), a board that assisted the chief magistrate in his sacrificial functions. The reasons for their decisions were kept secret, but, because of some negligence and a measure of ignorance and corruption, the intercalations were irregular, and seasonal chaos resulted. In spite of this and the fact that it was over a day too long compared with the tropical year, much of the modified Roman republican calendar was carried over into the Gregorian calendar now in general use.

Present knowledge of the Jewish calendar in use before the period of the Babylonian Exile is both limited and uncertain. The Bible refers to calendar matters only incidentally, and the dating of components of Mosaic Law remains doubtful. The earliest datable source for the Hebrew calendar is the Gezer Calendar, written probably in the age of Solomon, in the late 10th century BC. The inscription indicates the length of main agricultural tasks within the cycle of 12 lunations. The calendar term here is yereaḥ, which in Hebrew denotes both “moon” and “month.” The second Hebrew term for month, ḥodesh, properly means the “newness” of the lunar crescent. Thus, the Hebrew months were lunar. They are not named in pre-exilic sources except in the biblical report of the building of Solomon's Temple in I Kings, where the names of three months, two of them also attested in the Phoenician calendar, are given; the months are usually numbered rather than named. The “beginning of the months” was the month of the Passover (see Judaism: The cycle of the religious year (Judaism)). In some passages, the Passover month is that of ḥodesh ha-aviv, the lunation that coincides with the barley being in the ear. Thus, the Hebrew calendar is tied in with the course of the Sun, which determines ripening of the grain. It is not known how the lunar year of 354 days was adjusted to the solar year of 365 days. The Bible never mentions intercalation. The year shana, properly “change” (of seasons), was the agricultural and, thus, liturgical year. There is no reference to the New Year's day in the Bible.

After the conquest of Jerusalem (587 BC), the Babylonians introduced their cyclic calendar (see above Babylonian calendars (calendar)) and the reckoning of their regnal years from Nisanu 1, about the spring equinox. The Jews now had a finite calendar year with a New Year's day, and they adopted the Babylonian month names, which they continue to use. From 587 BC until AD 70, the Jewish civil year was Babylonian, except for the period of Alexander the Great and the Ptolemies (332–200 BC), when the Macedonian calendar was used. The situation after the destruction of the Temple in Jerusalem in AD 70 remains unclear. It is not known whether the Romans introduced their Julian calendar or the calendar that the Jews of Palestine used after AD 70 for their business transactions. There is no calendar reference in the New Testament; the contemporary Aramaic documents from Judaea are rare and prove only that the Jews dated events according to the years of the Roman emperors. The abundant data in the Talmudic (Talmud and Midrash) sources concern only the religious calendar.

In the religious calendar, the commencement of the month was determined by the observation of the crescent New Moon, and the date of the Passover was tied in with the ripening of barley. The actual witnessing of the New Moon and observing of the stand of crops in Judaea were required for the functioning of the religious calendar. The Jews of the Diaspora, or Dispersion, who generally used the civil calendar of their respective countries, were informed by messengers from Palestine about the coming festivals. This practice is already attested for 143 BC. After the destruction of the Temple in AD 70, rabbinic leaders took over from the priests the fixing of the religious calendar. Visual observation of the New Moon was supplemented and toward AD 200, in fact, supplanted by secret astronomical calculation. But the people of the Diaspora were often reluctant to wait for the arbitrary decision of the calendar makers in the Holy Land. Thus, in Syrian Antioch in AD 328–342, the Passover was always celebrated in (Julian) March, the month of the spring equinox, without regard to the Palestinian rules and rulings. To preserve the unity of Israel, the patriarch Hillel II, in 358/359, published the “secret” of calendar making, which essentially consisted of the use of the Babylonian 19-year cycle with some modifications required by the Jewish ritual.

The application of these principles occasioned controversies as late as the 10th century AD. In the 8th century, the Karaites (Karaism), following Muslim practice, returned to the actual observation of the crescent New Moon and of the stand of barley in Judaea. But some centuries later they also had to use a precalculated calendar. The Samaritans, likewise, used a computed calendar.

Because of the importance of the Sabbath as a time divider, the seven-day week served as a time unit in Jewish worship and life. As long as the length of a year and of every month remained unpredictable, it was convenient to count weeks. The origin of the biblical septenary, or seven-day, week remains unknown; its days were counted from the Sabbath (Saturday for the Jews and Sunday for Christians). A visionary, probably writing in the Persian or early Hellenistic age under the name of the prediluvian Enoch, suggested the religious calendar of 364 days, or 52 weeks, based on the week, in which all festivals always fall on the same weekday. His idea was later taken up by the Qumrān community.

The Jewish calendar in use today is lunisolar, the years being solar and the months lunar, but it also allows for a week of seven days. Because the year exceeds 12 lunar months by about 11 days, a 13th month of 30 days is intercalated in the third, sixth, eighth, 11th, 14th, 17th, and 19th years of a 19-year cycle. For practical purposes—e.g., for reckoning the commencement of the Sabbath—the day begins at sunset; but the calendar day of 24 hours always begins at 6 PM. The hour is divided into 1,080 parts (ḥalaqim; this division is originally Babylonian), each part (ḥeleq) equalling 3 ¹/₃ seconds. The ḥeleq is further divided into 76 regaʿim.

The synodic month is the average interval between two mean conjunctions of the Sun and Moon, when these bodies are as near as possible in the sky, which is reckoned at 29 days 12 hours 44 minutes 3 ¹/₃ seconds; a conjunction is called a molad. This is also a Babylonian value. In the calendar month, however, only complete days are reckoned, the “full” month containing 30 days and the “defective” month 29 days. The months Nisan, Sivan (Siwan), Av, Tishri, Shevaṭ, and, in a leap year, First Adar are always full; Iyyar, Tammuz, Elul, Ṭevet, and Adar (known as Second Adar, or Adar Sheni, in a leap year) are always defective, while Ḥeshvan (Ḥeshwan) and Kislev (Kislew) vary. The calendar, thus, is schematic and independent of the true New Moon. The number of days in a year varies. The number of days in a synodic month multiplied by 12 in a common year and by 13 in a leap year would yield fractional figures. Hence, again reckoning complete days only, the common year has 353, 354, or 355 days and the leap year 383, 384, or 385 days. A year in which both Ḥeshvan and Kislev are full, called complete (shelema), has 355 or (if a leap year) 385 days; a normal (sedura) year, in which Ḥeshvan is defective and Kislev full, has 354 or 384 days; while a defective (ḥasera) year, in which both these months are defective, has 353 or 383 days. The character of a year (qeviʾa, literally “fixing”) is described by three letters of the Hebrew alphabet, the first and third giving, respectively, the days of the weeks on which the New Year occurs and Passover begins, while the second is the initial of the Hebrew word for defective, normal, or complete. There are 14 types of qeviʿot, seven in common and seven in leap years. The New Year begins on Tishri 1, which may be the day of the molad of Tishri but is often delayed by one or two days for various reasons. Thus, in order to prevent the Day of Atonement (Yom Kippur) (Tishri 10) from falling on a Friday or a Sunday and the seventh day of Tabernacles (Sukkoth) (Tishri 21) from falling on a Saturday, the New Year must avoid commencing on Sundays, Wednesdays, or Fridays. Again, if the molad of Tishri occurs at noon or later, the New Year is delayed by one or, if this would cause it to fall as above, two days. These delays (deḥiyyot) necessitate, by reason of the above-mentioned limits on the number of days in the year, two other delays.

The mean beginning of the four seasons is called tequfa (literally “orbit,” or “course”); the tequfa of Nisan denotes the mean Sun at the vernal equinox, that of Tammuz at the summer solstice, that of Tishri at the autumnal equinox, and that of Ṭevet at the winter solstice. As 52 weeks are the equivalent to 364 days, and the length of the solar year is nearly 365 ¹/₄ days, the tequfot move forward in the week by about 1 ¹/₄ days each year. Accordingly, reckoning the length of the year at the approximate value of 365 ¹/₄ days, they are held to revert after 28 years (28 × 1 ¹/₄ = 35 days) to the same hour on the same day of the week (Tuesday, 6 PM) as at the beginning. This cycle is called the great, or solar, cycle (maḥzor gadol or ḥamma). The present Jewish calendar is mainly based on the more accurate value 365 days, 5 hours, 55 minutes, 25 ²⁵/₅₇ seconds—in excess of the true tropical year by about 6 minutes 40 seconds. Thus, it is advanced by one day in about 228 years with regard to the equinox.

To a far greater extent than the solar cycle of 28 years, the Jewish calendar employs, as mentioned above, a small, or lunar, cycle (maḥzor qaṭan) of 19 years, adjusting the lunar months to the solar years by intercalations. Passover, on Nisan 14, is not to begin before the spring tequfa, and so the intercalary month is added after Adar. The maḥzor qaṭan is akin to the Metonic cycle described above.

The Jewish Era in use today is that dated from the supposed year of the Creation (creation myth) (designated anno mundi or AM) with its epoch, or beginning, in 3761 BC. For example, the Jewish year 5745 AM, the 7th in the 303rd lunar cycle and the 5th in the 206th solar cycle, is a regular year of 12 months, or 354 days. The qeviʿa is, using the three respective letters of the Hebrew alphabet as two numerals and an initial in the manner indicated above, HKZ, which indicates that Rosh Hashana (New Year) begins on the fifth (H = 5) and Passover on the seventh (Z = 7) day of the week and that the year is regular (K = ke-sidra); i.e., Ḥeshvan is defective—29 days, and Kislev full—30 days. The Jewish year 5745 AM corresponds with the Christian Era period that began September 27, 1984, and ended September 15, 1985. Neglecting the thousands, current Jewish years AM are converted into years of the current Christian Era by adding 239 or 240—239 from the Jewish New Year (about September) to December 31 and 240 from January 1 to the eve of the Jewish New Year. The adjustment differs slightly for the conversion of dates of now-antiquated versions of the Jewish Era of the Creation and the Christian Era, or both. Tables for the exact conversion of such dates are available.

The months of the Jewish year and the notable days are as follows:

The Muslim Era is computed from the starting point of the year of the emigration ( Hijrah 【Hegira】); that is, from the year in which Muhammad, the prophet of Islam, emigrated from Mecca to Medina, AD 622. The second caliph, ʿUmar I, who reigned 634–644, set the first day of the month Muḥarram as the beginning of the year; that is, July 16, 622, which had already been fixed by the Qurʾān as the first day of the year.

The years of the Muslim calendar are lunar and always consist of 12 lunar months alternately 30 and 29 days long, beginning with the approximate New Moon. The year has 354 days, but the last month (Dhū al-Ḥijjah) sometimes has an intercalated day, bringing it up to 30 days and making a total of 355 days for that year. The months do not keep to the same seasons in relation to the Sun, because there are no intercalations of months. The months regress through all the seasons every 32 ¹/₂ years.

The era of the Hijrah is the official era in Saudi Arabia, Yemen, and the principalities of the Persian Gulf. Egypt, Syria, Jordan, and Morocco use both the Muslim and the Christian eras. In all Muslim countries, people use the Muslim Era in private, even though the Christian Era may be in official use.

Some Muslim countries have made a compromise on this matter. Turkey, as early as 1088 AH (AD 1677), took over the solar (Julian) year with its month names but kept the Muslim Era. March 1 was taken as the beginning of the year (commonly called marti year, after the Turkish word mart, for March). Late in the 19th century, the Gregorian calendar was adopted. In the 20th century, President Mustafa Kemal Atatürk (Atatürk, Kemal) ordered a complete change to the Christian Era. Iran, under Reza Shah Pahlavi (reigned 1925–41), also adopted the solar year but with Persian names for the months and keeping the Muslim Era. March 21 is the beginning of the Iranian year. Thus, the Iranian year 1359 began on March 21, 1980. This era is still in use officially. (See also Islam: Sacred places and days (Islām).)

While the Republic of India has adopted the Gregorian calendar for its secular life, its Hindu religious life continues to be governed by the traditional Hindu calendar. This calendar, based primarily on the lunar revolutions, is adapted to solar reckoning.

The oldest system, in many respects the basis of the classical one, is known from texts of about 1000 BC. It divides an approximate solar year of 360 days into 12 lunar months of 27 (according to the early Vedic text Taittirīya Saṃhitā 4.4.10.1–3) or 28 (according to the Atharvaveda, the fourth of the Vedas, 19.7.1.) days. The resulting discrepancy was resolved by the intercalation of a leap month every 60 months. Time was reckoned by the position marked off in constellations on the ecliptic in which the Moon rises daily in the course of one lunation (the period from New Moon to New Moon) and the Sun rises monthly in the course of one year. These constellations (nakṣatra) each measure an arc of 13° 20′ of the ecliptic circle. The positions of the Moon were directly observable, and those of the Sun inferred from the Moon's position at Full Moon, when the Sun is on the opposite side of the Moon. The position of the Sun at midnight was calculated from the nakṣatra that culminated on the meridian at that time, the Sun then being in opposition to that nakṣatra. The year was divided into three thirds of four months, each of which would be introduced by a special religious rite, the cāturmāsya (four-month rite). Each of these periods was further divided into two parts (seasons or ṛtu): spring (vasanta), from mid-March until mid-May; summer (grīṣma), from mid-May until mid-July; the rains (varṣa), from mid-July until mid-September; autumn (śarad ), from mid-September until mid-November; winter (hemanta), from mid-November until mid-January; and the dews (śiśira), from mid-January until mid-March. The spring months in early times were Madhu and Mādhava, the summer months Śukra and Śuci, the rainy months Nabhas and Nabhasya, the autumn months Īṣa and Ūrja, the winter months Sahas and Sahasya, and the dewy months Tapas and Tapasya. The month, counted from Full Moon to Full Moon, was divided into two halves (pakṣa, “wing”) of waning (kṛṣṇa) and waxing (śukla) Moon, and a special ritual (darśapūrṇamāsa, “new and full moon rites”) was prescribed on the days of New Moon (amāvasya) and Full Moon (pūrṇimās). The month had theoretically 30 days (tithi), and the day (divasa) 30 hours (muhūrta).

This picture is essentially confirmed by the first treatise on time reckoning, the Jyotiṣa-vedāṅga (“Vedic auxiliary 【text】 concerning the luminaries”) of about 100 BC, which adds a larger unit of five years (yuga) to the divisions. A further old distinction is that of two year moieties, the uttarāyaṇa (“northern course”), when the Sun has passed the spring equinox and rises every morning farther north, and the dakṣiṇāyana (“southern course”), when it has passed the autumnal equinox and rises progressively farther south.

In its classic form (Sūrya-siddhānta, 4th century AD) the calendar continues from the one above with some refinements. With the influence of Hellenism, Greek and Mesopotamian astronomy and astrology were introduced. Though astronomy and time reckoning previously were dictated by the requirements of rituals, the time of which had to be fixed correctly, and not for purposes of divination, the new astrology came into vogue for casting horoscopes and making predictions. Zodiacal time measurement was now used side by side with the older nakṣatra one. The nakṣatra section of the ecliptic (13° 20′) was divided into four parts of 3° 20′ each; thus, two full nakṣatras and a quarter of one make up one zodiac period, or sign (30°). The year began with the entry of the Sun (saṃkrānti) in the sign of Aries. The names of the signs (rāśi) were taken over and mostly translated into Sanskrit: meṣa (“ram,” Aries), vṛṣabha (“bull,” Taurus), mithuna (“pair,” Gemini), karkaṭa (“crab,” Cancer), siṃha (“lion,” Leo), kanyā (“maiden,” Virgo), tulā (“scale,” Libra), vṛścika (“scorpion,” Scorpius), dhanus (“bow,” Sagittarius), makara (“crocodile,” Capricornus), kumbha (“water jar,” Aquarius), mīna (“fish,” Pisces).

The precession of the vernal equinox (equinoxes, precession of the) from the Sun's entry into Aries to some point in Pisces, with similar consequences for the summer solstice, autumnal equinox, and winter solstice, has led to two different methods of calculating the saṃkrānti (entry) of the Sun into a sign. The precession (ayana) is not accounted for in the nirayana system (without ayana), which thus dates the actual saṃkrānti correctly but identifies it wrongly with the equinox or solstice, and the sāyana system (with ayana), which thus dates the equinox and solstice correctly but identifies it wrongly with the saṃkrānti.

While the solar system has extreme importance for astrology, which, it is claimed, governs a person's life as an individual or part of a social system, the sacred time continues to be reckoned by the lunar nakṣatra system. The lunar day (tithi), a 30th part of the lunar month, remains the basic unit. Thus, as the lunar month is only about 29 ¹/₂ solar days, the tithi does not coincide with the natural day (ahorātra). The convention is that that tithi is in force for the natural day that happened to occur at the dawn of that day. Therefore, a tithi beginning after dawn one day and expiring before dawn the next day is eliminated, not being counted in that month, and there is a break in the day sequence.

The names of the nakṣatras, to which correspond the tithis in the monthly lunar cycle and segments of months in the annual solar cycle, are derived from the constellations on the horizon at that time and have remained the same. The names of the months have changed: Caitra (March-April), Vaiśākha (April-May), Jyaiṣṭha (May-June), Āṣāḍha (June-July), Śrāvaṇa (July-August), Bhādrapada (August-September), Āśvina (September-October), Kārttika (October-November), Mārgaśīrṣa (November-December), Pauṣa (December-January), Māgha (January-February), and Phālguna (February-March).

In this calendar the date of an event takes the following form: month, fortnight (either waning or waxing Moon), name (usually the number) of the tithi in that fortnight, and the year of that era which the writer follows. Identification, particularly of the tithi, is often quite complicated, since it requires knowledge of the time of sunrise on that day and which 30th of the lunar month was in force then.

Eventually, India also adopted the seven-day week (saptāha) from the West and named the days after the corresponding planets: Sunday after the Sun, ravivāra; Monday after the Moon, somavāra; Tuesday after Mars, maṅgalavāra; Wednesday after Mercury, budhavāra; Thursday after Jupiter, bṛhaspativāra; Friday after Venus, śukravāra; and Saturday after Saturn, śanivāra.

A further refinement of the calendar was the introduction into dating of the place of a year according to its position in relation to the orbital revolution of the planet Jupiter, called bṛhaspati in Sanskrit. Jupiter has a sidereal period (its movement with respect to the “fixed” stars) of 11 years, 314 days, and 839 minutes, so in nearly 12 years it is back into conjunction with those stars from which it began its orbit. Its synodic period brings it into conjunction with the Sun every 398 days and 88 minutes, a little more than a year. Thus, Jupiter in a period of almost 12 years passes about the same series of nakṣatras that the Sun passes in one year and, in a year, about the same nakṣatras as the Sun in a month. A year then can be dated as the month of a 12-year cycle of Jupiter, and the date is given as, for example, grand month of Caitra. This is extended to a unit of five cycles, or the 60-year cycle of Jupiter (bṛhaspaticakra), and a “century” of 60 years is formed. This system is known from the 6th century AD onward.

At the other end of the scale, more precision is brought to the day. Every tithi is divided into two halves, called karaṇas. The natural day is divided into units ranging from a vipala (0.4 second) to a ghaṭik (24 minutes) and an “hour” (muhūrta) of 48 minutes; the full natural day has 30 such hours. The day starts at dawn; the first six ghaṭikās are early morning, the second set of six midmorning, the third midday, the fourth afternoon, the fifth evening. Night lasts through three units (yāma) of time: six ghaṭikās after sundown, or early night; two of midnight; and four of dawn.

There are a few secular state holidays (e.g., Independence Day) and some solar holidays, such as the entry of the Sun into the sign of Aries (meṣa-saṃkrānti), marking the beginning of the new astrological year; the Sun's entry into the sign of Capricornus (makara-saṃkrānti), which marks the winter solstice but has coalesced with a hoary harvest festival, which in southern India is very widely celebrated as the Poṅgal festival; and the mahāviṣuva day, which is New Year's Eve. But all other important festivals are based on the lunar calendar. As a result of the high specialization of deities and events celebrated in different regions, there are hundreds of such festivals, most of which are observed in smaller areas, though some have followings throughout India. A highly selective list of the major ones, national and regional, follows. (See also Hinduism: Sacred times and places (Hinduism).)

Not before the first century BC is there any evidence that the years of events were recorded in well-defined eras, whether by cycles, as the Olympic Games in Greece and the tenures of consuls in Rome, or the Roman year dating from the foundation of the city. Perhaps under outside influence, the recording of eras was begun at various times, but these were without universal appeal, and few have remained influential. Among those are (1) the Vikrama era, begun 58 BC; (2) the Śaka era, begun AD 78 (these two are the most commonly used); (3) the Gupta era, begun AD 320; (4) the Harṣa era, begun AD 606. All these were dated from some significant historical event. Of more mythological interest is the Kali era (Kali being the latest and most decadent period in the system of the four Yugas), which is thought to have started either at dawn on February 18, 3102 BC, or at midnight between February 17 and 18 in that year.

Evidence from the Shang oracle bone inscriptions shows that at least by the 14th century BC the Shang Chinese (China) had established the solar year at 365 ¹/₄ days and lunation at 29 ¹/₂ days. In the calendar that the Shang used, the seasons of the year and the phases of the Moon were all supposedly accounted for. One of the two methods that they used to make this calendar was to add an extra month of 29 or 30 days, which they termed the 13th month, to the end of a regular 12-month year. There is also evidence that suggests that the Chinese developed the Metonic cycle (see above Complex cycles (calendar))—i.e., 19 years with a total of 235 months—a century ahead of Meton's first calculation (no later than the Spring and Autumn period, 770–476 BC). During this cycle of 19 years there were seven intercalations of months. The other method, which was abandoned soon after the Shang started to adopt it, was to insert an extra month between any two months of a regular year. Possibly, a lack of astronomical and arithmetical knowledge allowed them to do this.

By the 3rd century BC, the first method of intercalation was gradually falling into disfavour, while the establishment of the meteorological cycle, the erh-shih-ssu chieh-ch'i (Pinyin ershisi jieqi), during this period officially revised the second method. This meteorological cycle contained 24 points, each beginning one of the periods named consecutively the Spring Begins, the Rain Water, the Excited Insects, the Vernal Equinox, the Clear and Bright, the Grain Rains, the Summer Begins, the Grain Fills, the Grain in Ear, the Summer Solstice, the Slight Heat, the Great Heat, the Autumn Begins, the Limit of Heat, the White Dew, the Autumn Equinox, the Cold Dew, the Hoar Frost Descends, the Winter Begins, the Little Snow, the Heavy Snow, the Winter Solstice, the Little Cold, and the Severe Cold. The establishment of this cycle required a fair amount of astronomical understanding of the Earth as a celestial body, and without elaborate equipment it is impossible to collect the necessary information. Modern scholars acknowledge the superiority of pre-Sung Chinese astronomy (at least until about the 13th century AD) over that of other, contemporary nations.

The 24 points within the meteorological cycle coincide with points 15° apart on the ecliptic (the plane of the Earth's yearly journey around the Sun or, if it is thought that the Sun turns around the Earth, the apparent journey of the Sun against the stars). It takes about 15.2 days for the Sun to travel from one of these points to another (because the ecliptic is a complete circle of 360°), and the Sun needs 365 ¹/₄ days to finish its journey in this cycle. Supposedly, each of the 12 months of the year contains two points, but, because a lunar month has only 29 ¹/₂ days and the two points share about 30.4 days, there is always the chance that a lunar month will fail to contain both points, though the distance between any two given points is only 15°. If such an occasion occurs, the intercalation of an extra month takes place. For instance, one may find a year with two “Julys” or with two “Augusts” in the Chinese calendar. In fact, the exact length of the month in the Chinese calendar is either 30 days or 29 days—a phenomenon which reflects its lunar origin. Also, the meteorological cycle means essentially a solar year. The Chinese thus consider their calendar as yin-yang li, or a “lunar–solar calendar.”

Although the yin-yang li has been continuously employed by the Chinese, foreign calendars were introduced to the Chinese, the Hindu calendar, for instance, during the T'ang (Tang) dynasty (Tang dynasty) (618–907), and were once used concurrently with the native calendar. This situation also held true for the Muslim calendar, which was introduced during the Yüan dynasty (Yuan dynasty) (1206–1368). The Gregorian calendar was taken to China by Jesuit missionaries in 1582, the very year that it was first used by Europeans. Not until 1912, after the general public adopted the Gregorian calendar, did the yin-yang li lose its primary importance.

One of the most distinguished characteristics of the Chinese calendar is its time-honoured day-count system. By combining the 10 celestial stems, kan (gan), and the 12 terrestrial branches, chih (zhi), into 60 units, the Shang Chinese counted the days with kan–chih (gan–zhi) combinations cyclically. For more than 3,000 years, no one has ever tried to discard the kan–chih day-count system. Out of this method there developed the idea of hsün (xun), 10 days, which some scholars would render into English as “week.” The kan–chih combinations probably were adopted for year count by Han (Han dynasty) emperors during the 2nd century AD.

The yin-yang li may have been preceded by a pure lunar calendar because there is one occurrence of the “14th month” and one occurrence of the “15th month” in the Shang oracle bone inscriptions. Unless there was a drastic change in the computation, it is quite inconceivable that an extra 90 days should have been added to a regular year. Julius Caesar had made 45 BC into a year of 445 days for the sake of the adoption of the Julian calendar in the next year. Presumably, the Shang king could have done the same for similar reasons. From the above discussion on the intercalation of months, it is clear that within the yin-yang li the details of the lunar calendar are more important than those of the solar calendar. In a solar calendar the 24 meteorological points would recur on the same days every year. Moreover, if a solar calendar were adopted first, then the problem of intercalation would be more related to the intercalation of days rather than intercalation of months.

Many traditional Chinese scholars tried to synchronize the discrepancy between the lunation and the solar year. Some even developed their own ways of computation embodying accounts of eclipses and of other astronomical phenomena. These writings constitute the bulk of the traditional almanacs (almanac). In the estimation of modern scholars, at least 102 kinds of almanacs were known, and some were used regularly. The validity or the popularity of each of these almanacs depends heavily on the author's proficiency in handling planetary cycles. In the past these authors competed with one another for the position of calendar master in the Imperial court, even though mistakes in their almanacs could bring them punishment, including death.

The basic structure of the Mayan calendar is common to all calendars of Mesoamerica (Mesoamerican civilization) (i.e., the civilized part of ancient Middle America). It consists of a ritual cycle of 260 named days and a year of 365 days. These cycles, running concurrently, form a longer cycle of 18,980 days, or 52 years of 365 days, called a “Calendar Round,” at the end of which a designated day recurs in the same position in the year.

The native Mayan name for the 260-day cycle is unknown. Some authorities call it the Tzolkin (Count of Days); others refer to it as the Divinatory Calendar, the Ritual Calendar, or simply the day cycle. It is formed by the combination of numerals 1 through 13, meshing day by day with an ordered series of 20 names. The names of the days differ in the languages of Mesoamerica, but there is enough correspondence of meaning to permit the correlation of the known series, and there is reason to think that all day cycles were synchronous. The days were believed to have a fateful character, and the Tzolkin was used principally in divination. Certain passages in the Dresden Codex, one of the three Mayan manuscripts that survived the conquest, show various Tzolkins divided into four parts of 65 days each, or into five parts of 52 days. The parts are in turn subdivided into a series of irregular intervals, and each interval is accompanied by a group of hieroglyphs and by an illustration, usually depicting a deity performing some simple act. The hieroglyphs apparently give a prognostication, but just how the Maya determined the omens is not known.

The 365-day year was divided into 18 named months of 20 days each, with an additional five days of evil omen, named Uayeb. In late times, the Maya named the years after their first days. Since both the year and the number (20) of names of days are divisible by five, only four names combined with 13 numbers could begin the year. These were called Year Bearers and were assigned in order to the four quarters of the world with their four associated colours. Unlike day cycles, years were not synchronous in all regions. They began at different times and in different seasons, and even among Maya-speaking peoples there was imperfect concordance of the months. Some differences may be due to postconquest attempts to keep the native year in step with the Christian calendar; others no doubt have an earlier origin.

The manner of recording historical dates is peculiar to the ancient Mayan calendar. The Maya did not use the names of years for this purpose. To identify a date of the Calendar Round, they designated the day by its numeral and name, and added the name of the current month, indicating the number of its days that had elapsed by prefixing one of the numerals from 0 through 19. A date written in this way will occur once in every Calendar Round, at intervals of 52 years.

This was not good enough to link events over longer periods of time. Mayan interest in history, genealogy, and astrology required accurate records of events far in the past. To connect dates to one another, the Maya expressed distances between them by a count of days and their multiples. They used what was essentially a vigesimal place-value system of numeration, which is one based on a count of 20, but modified it by substituting 18 for 20 as the multiplier of units of the second order, so that each unit in the third place had the value of 360 days instead of 400. In monumental inscriptions, the digits are usually accompanied by the names of the periods their units represent, although in the manuscripts the period names are omitted and placement alone indicates the value of the units. The period names in ascending order are: kin (day); uinal (20 days); tun (18 uinals or 360 days); katun (20 tuns or 7,200 days); baktun or cycle—native name unknown—(20 katuns or 144,000 days); and so on up to higher periods. By introducing an odd multiplier to form the tun, the Maya made multiplication and division difficult, and there are in the Dresden Codex long tables of multiples of numbers that could be more simply manipulated by addition and subtraction.

To correlate all historical records and to anchor dates firmly in time, the Maya established the “Long Count,” a continuous count of time from a base date, 4 Ahau 8 Cumku, which completed a round of 13 baktuns far in the past. There were several ways in which one could indicate the position of a Calendar Round dated in the Long Count. The most direct and unambiguous was to use an Initial-Series (IS) notation. The series begins with an outsized composition of signs called the Initial-Series-introducing glyph, which is followed by a count of periods written in descending order. On the earliest known monument, Stela 29 from Tikal in Guatemala, the Initial Series reads: 8 baktuns, 12 katuns, 14 tuns, 8 uinals, 15 kins, which is written: IS. 8.12.14.8.15. It shows that the Calendar Round date that follows falls 1,243,615 days (just under 3,405 years) after the 4 Ahau 8 Cumku on which the Long Count is based. Stela 29 is broken, and its Calendar Round date is missing, but from the information above, it can be calculated to have been 13 Men 3 Zip (the 195th day of the Tzolkin, the 44th of the year).

Normally, only the opening date of an inscription is written as an Initial Series. From this date, distance numbers, called Secondary Series (SS), lead back or forward to other dates in the record, which frequently ends with a Period-Ending (PE) date. This is a statement that a given date completes a whole number of tuns or katuns in the next higher period of the Long Count. Such a notation identifies the date unambiguously within the historic period. The latest Period Ending recorded on a given monument is also known as its Dedicatory Date (DD), for it was a common custom to set up monuments on the completion of katuns of the Long Count and sometimes also at the end of every five or 10 tuns. The Maya also celebrated katun and five-tun “anniversaries” of important dates and recorded them in much the same way as the period endings.

Period-Ending dates gradually took the place of Initial Series, and, in northern Yucatán (Yucatán Peninsula), where Mayan sites of the latest period are located, a new method of notation dispensed with distance numbers altogether by noting after a Calendar Round date the number of the current tun in a Long Count katun named by its last day. Long Count katuns end with the name Ahau (Lord), combined with one of 13 numerals; and their names form a Katun Round of 13 katuns. This round is portrayed in Spanish colonial manuscripts as a ring of faces depicting the Lords. There are also recorded prophesies for tuns and katuns, which make many allusions to history, for the Maya seem to have conceived time, and even history itself, as a series of cyclical, recurring events.

The discontinuance of Initial-Series notations some centuries before the conquest of Mexico by Spain makes all attempted correlations of the Mayan count with the Christian calendar somewhat uncertain, for such correlations are all based on the assumption that the Katun Round of early colonial times was continuous with the ancient Long Count. The correlation most in favour now equates the 4 Ahau 8 Cumku that begins the Mayan count with the Julian day 584,283 (see above Complex cycles (calendar)). According to this correlation, the katun 13 Ahau that is said to have ended shortly before the foundation of Mérida, Yucatán, ended on November 14, 1539, by the Gregorian calendar, and it was the Long Count katun 11.16.0.0.0. 13 Ahau 8 Xul. Some tests of archaeological material by the radiocarbon method corroborate this correlation; but results are not sufficiently uniform to resolve all doubts, and some archaeologists would prefer to place the foundation of Mérida in the neighbourhood of 12.9.0.0.0. in the Mayan count. Correlations based on astronomical data so far have been in conflict with historical evidence, and none has gained a significant degree of acceptance.

The basic elements of the Mayan calendar have little to do with astronomy. A lunar count was, however, included in a Supplementary Series appended to Initial-Series dates. The series is composed of hieroglyphs labelled Glyphs G, F, E or D, C, B, and A, and a varying number of others. Glyph G changes its form daily, making a round of nine days, possibly corresponding to the nine gods of the night hours (hour) or Mexican Lords of the Night. Glyph F is closely associated with Glyph G and does not vary. Glyphs E and D have numerical coefficients that give the age of the current Moon within an error of two or three days; Glyph C places it in a lunar half year; and Glyph A shows whether it is made up of 29 or 30 days. The meaning of Glyph B is unknown. There are discrepancies in the lunar records from different sites, but during a period of about 80 years, called the Period of Uniformity, a standard system of grouping six alternating 29- and 30-day moons was used everywhere.

Occasionally included with the Supplementary Series is a date marking the conclusion of an 819-day cycle shortly before the date of Initial Series. The number of days in this cycle is obtained by multiplying together 13, 9, and 7, all very significant numbers in Mayan mythology.

It has been suggested that certain other dates, called determinants, indicate with a remarkable degree of accuracy how far the 365-day year had diverged from the solar year since the beginning of the Long Count, but this hypothesis is questioned by some scholars. The identification of certain architectural assemblages as observatories of solstices and equinoxes is equally difficult to substantiate. So far,it has not been demonstrated how the Maya reckoned the seasons of their agricultural cycle or whether they observed the tropical or the sidereal year.

In colonial times, the star group known as the Pleiades was used to mark divisions of the night, and the constellation Gemini was also observed. A computation table in the Dresden Codex records intervals of possible eclipses (eclipse) of the Sun and Moon. Another correlates five revolutions of the planet Venus around the Sun with eight 365-day years and projects the count for 104 years, when it returns to the beginning Tzolkin date. Three sets of month positions associated with the cycle suggest its periodic correction. Other computations have not been adequately explained, among them some very long numbers that transcend the Long Count. Such numbers appear also on monuments and indicate a grandiose conception of the complexity and the almost infinite extent of time. (See also pre-Columbian civilizations: The Maya calendar and writing system (pre-Columbian civilizations).)

The Aztec year of 365 days was also similar to the year of the Maya, though probably not synchronous with it. It had 18 named months of 20 days each and an additional five days, called nemontemi, which were considered to be very unlucky. Though some colonial historians mention the use of intercalary days, in Aztec annals there is no indication of a correction in the length of the year. The years were named after days that fall at intervals of 365 days, and most scholars believe that these days held a fixed position in the year, though there appears to be some disagreement as to whether this position was the first day, the last day of the first month, or the last day of the last month, as was suggested by a distinguished Mexican scholar. Since 20 and 365 are both divisible by five, only four day names—Acatl (Reed), Tecpatl (Flint), Calli (House), and Tochtli (Rabbit)—figure in the names of the 52 years that form a cycle with the tonalpohualli. The cycle begins with a year 2 Reed and ends with a year 1 Rabbit, which was regarded as a dangerous year of bad omen. At the end of such a cycle, all household utensils and idols were discarded and replaced by new ones, temples were renovated, and human sacrifice was offered to the Sun at midnight on a mountaintop as people awaited a new dawn.

The year served to fix the time of festivals, which took place at the end of each month. The new year was celebrated by the making of a new fire, and a more elaborate ceremony was held every four years, when the cycle had run through the four day names. Every eight years was celebrated the coincidence of the year with the 584-day period of the planet Venus, and two 52-year cycles formed “One Old Age,” when the day cycle, the year, and the period of Venus all came together. All these periods were noted also by the Maya.

Where the Aztec differed most significantly from the Maya was in their more primitive number system and in their less precise way of recording dates. Normally, they noted only the day on which an event occurred and the name of the current year. This is ambiguous, since the same day, as designated in the way mentioned above, can occur twice in a year. Moreover, years of the same name recur at 52-year intervals, and Spanish colonial annals often disagree as to the length of time between two events. Other discrepancies in the records are only partially explained by the fact that different towns started their year with different months. The most widely accepted correlation of the calendar of Tenochtitlán with the Christian Julian calendar is based on the entrance of Cortés into that city on November 8, 1519, and on the surrender of Cuauhtémoc on August 13, 1521. According to this correlation, the first date was a day 8 Wind, the ninth day of the month Quecholli, in a year 1 Reed, the 13th year of a cycle.

So little is known about the calendar used by the Inca that one can hardly make a statement about it for which a contrary opinion cannot be found. Some workers in the field even assert that there was no formal calendar but only a simple count of lunations. Since no written language was used by the Inca, it is impossible to check contradictory statements made by early colonial chroniclers. It is widely believed that the quipus of the Inca contain calendrical notations, but no satisfactory demonstration of this is possible.

Most historians agree that the Inca had a calendar based on the observation of both the Sun and the Moon, and their relationship to the stars. Names of 12 lunar months are recorded, as well as their association with festivities of the agricultural cycle; but there is no suggestion of the widespread use of a numerical system for counting time, although a quinary decimal system, with names of numbers at least up to 10,000, was used for other purposes. The organization of work on the basis of six weeks (week) of nine days suggests the further possibility of a count by triads that could result in a formal month of 30 days.

A count of this sort was described by Alexander von Humboldt for a Chibcha tribe living outside of the Inca Empire, in the mountainous region of Colombia. The description is based on an earlier manuscript by a village priest, and one authority has dismissed it as “wholly imaginary,” but this is not necessarily the case. The smallest unit of this calendar was a numerical count of three days, which, interacting with a similar count of 10 days, formed a standard 30-day “month.” Every third year was made up of 13 moons, the others having 12. This formed a cycle of 37 moons, and 20 of these cycles made up a period of 60 years, which was subdivided into four parts and could be multiplied by 100. A period of 20 months is also mentioned. Although the account of the Chibcha system cannot be accepted at face value, if there is any truth in it at all it is suggestive of devices that may have been used also by the Inca.

In one account, it is said that the Inca Viracocha established a year of 12 months, each beginning with the New Moon, and that his successor, Pachacuti (Pachacuti Inca Yupanqui), finding confusion in regard to the year, built the sun towers in order to keep a check on the calendar. Since Pachacuti reigned less than a century before the conquest, it may be that the contradictions and the meagreness of information on the Inca calendar are due to the fact that the system was still in the process of being revised when the Spaniards first arrived.

Despite the uncertainties, further research has made it clear that at least at Cuzco, the capital city of the Inca, there was an official calendar of the sidereal–lunar type, based on the sidereal month of 27 ¹/₃ days. It consisted of 328 nights (12 × 27 ¹/₃) and began on June 8/9, coinciding with the heliacal rising (the rising just after sunset) of the Pleiades; it ended on the first Full Moon after the June solstice (the winter solstice for the Southern Hemisphere). This sidereal–lunar calendar fell short of the solar year by 37 days, which consequently were intercalated. This intercalation, and thus the place of the sidereal–lunar within the solar year, was fixed by following the cycle of the Sun as it “strengthened” to summer (December) solstice and “weakened” afterward, and by noting a similar cycle in the visibility of the Pleiades.

No North American Indian tribe had a true calendar—a single integrated system of denoting days and longer periods of time. Usually, intervals of time were counted independently of one another. The day was a basic unit recognized by all tribes, but there is no record of aboriginal names for days. A common device for keeping track of days was a bundle of sticks of known number, from which one was extracted for every day that passed, until the bundle was exhausted. Longer periods of time were usually counted by moons, which began with the New Moon, or conjunction of the Sun and Moon. Years were divided into four seasons, occasionally five, and when counted were usually designated by one of the seasons; e.g., a North American Indian might say that a certain event had happened 10 winters ago. Among sedentary agricultural tribes, the cycle of the seasons was of great ritual importance, but the time of the beginning of the year varied. Some observed it about the time of the vernal equinox, others in the fall. The Hopi tribe of northern Arizona held a new-fire ceremony in November. The Creek ceremony, known as the “Busk,” was held late in July or in August, but it is said that each Creek town or settlement set its own date for the celebration.

As years were determined by seasons and not by a fixed number of days, the correlation of moons and years was also approximate and not a function of a daily count. Most tribes reckoned 12 moons to a year. Some northern tribes, notably those of New England, and the Cree tribes, counted 13. The Indians of the northwest coast divided their years into two parts, counting six moons to each part, and the Kiowa split one of their 12 moons between two unequal seasons, beginning their year with a Full Moon.

The naming of moons is perhaps the first step in transforming them into months. The Zuni Indians of New Mexico named the first six moons of the year, referring to the remainder by colour designations associated with the four cardinal (horizontal) directions, and the zenith and the nadir. Only a few Indian tribes attempted a more precise correlation of moons and years. The Creeks (Creek) are said to have added a moon between each pair of years, and the Haida from time to time inserted a “between moon” in the division of their year into two parts. It is said that an unspecified tribe of the Sioux or the Ojibwa (Chippewa) made a practice of adding a “lost moon” when 30 moons had waned.

A tally of years following an important event was sometimes kept on a notched stick. The best known record commemorates the spectacular meteoric shower (the Leonids) of 1833. Some northern tribes recorded series of events by pictographs, and one such record, said to have been originally painted on a buffalo robe and known as the “Lone-dog Winter Count,” covers a period of 71 years beginning with 1800.

Early explorers had little opportunity to learn about the calendrical devices of the Indians, which were probably held sacred and secret. Contact with Europeans and their Christian calendar doubtless altered many aboriginal practices. Thus, present knowledge of the systems used in the past may not reflect their true complexity.

The calendar now in general worldwide use had its origin in the desire for a solar calendar that kept in step with the seasons and possessed fixed rules of intercalation. Because it developed in Western Christendom, it had also to provide a method for dating movable religious feasts, the timing of which had been based on a lunar reckoning. To reconcile the lunar and solar schemes, features of the Roman republican calendar and the Egyptian calendar were combined.

The Roman republican calendar was basically a lunar reckoning and became increasingly out of phase with the seasons (season) as time passed. By about 50 BC the vernal equinox that should have fallen late in March fell on the Ides of May, some eight weeks later, and it was plain that this error would continue to increase. Moreover, the behaviour of the Pontifices (see above The early Roman calendar (calendar)) made it necessary to seek a fixed rule of intercalation in order to put an end to arbitrariness in inserting months.

In addition to the problem of intercalation, it was clear that the average Roman republican year of 366.25 days would always show a continually increasing disparity with the seasons, amounting to one month every 30 years, or three months a century. But the great difficulty facing any reformer was that there seemed to be no way of effecting a change that would still allow the months to remain in step with the phases of the Moon and the year with the seasons. It was necessary to make a fundamental break with traditional reckoning to devise an efficient seasonal calendar.

In the mid-1st century BC Julius Caesar (Caesar, Julius) invited Sosigenes, an Alexandrian (Sosigenes Of Alexandria) astronomer, to advise him about the reform of the calendar, and Sosigenes decided that the only practical step was to abandon the lunar calendar altogether. Months must be arranged on a seasonal basis, and a tropical (solar) year used, as in the Egyptian calendar, but with its length taken as 365 ¹/₄ days.

To remove the immense discrepancy between calendar date and equinox, it was decided that the year known in modern times as 46 BC should have two intercalations. The first was the customary intercalation of the Roman republican calendar due that year, the insertion of 23 days following February 23. The second intercalation, to bring the calendar in step with the equinoxes, was achieved by inserting two additional months between the end of November and the beginning of December. This insertion amounted to an addition of 67 days, making a year of no less than 445 days and causing the beginning of March, 45 BC in the Roman republican calendar, to fall on what is still called January 1 of the Julian calendar.

Previous errors having been corrected, the next step was to prevent their recurrence. Here Sosigenes' suggestion about a tropical year was adopted and any pretense to a lunar calendar was rejected. The figure of 365.25 days was accepted for the tropical year, and, to achieve this by a simple civil reckoning, Caesar directed that a calendar year of 365 days be adopted and that an extra day be intercalated every fourth year. Since February ordinarily had 28 days, February 24 was the sixth day (using inclusive numbering) before the Kalendae, or beginning of March, and was known as the sexto-kalendae; the intercalary day, when it appeared, was in effect a “doubling” of the sexto-kalendae and was called the bis-sexto-kalendae. This practice led to the term bissextile being used to refer to such a leap year. The name leap year is a later connotation, probably derived from the Old Norse hlaupa (“to leap”) and used because, in a bissextile year, any fixed festival after February leaps forward, falling on the second weekday from that on which it fell the previous year, not on the next weekday as it would do in an ordinary year.

Apparently, the Pontifices (pontifex) misinterpreted the edict and inserted the intercalation too frequently. The error arose because of the Roman practice of inclusive numbering, so that an intercalation once every fourth year meant to them intercalating every three years, because a bissextile year was counted as the first year of the subsequent four-year period. This error continued undetected for 36 years, during which period 12 days instead of nine were added. The emperor Augustus then made a correction by omitting intercalary days between 8 BC and AD 8. As a consequence, it was not until several decades after its inception that the Julian calendar came into proper operation, a fact that is important in chronology but is all too frequently forgotten.

It seems that the months of the Julian calendar were taken over from the Roman republican calendar but were slightly modified to provide a more even pattern of numbering. The republican calendar months of March, May, and Quintilis (July), which had each possessed 31 days, were retained unaltered. Although there is some doubt about the specific details, changes may have occurred in the following way. Except for October, all the months that had previously had only 29 days had either one or two days added. January, September, and November received two days, bringing their totals to 31, while April, June, Sextilis (August), and December received one day each, bringing their totals to 30. October was reduced by one day to a total of 30 days and February increased to 29 days, or 30 in a bissextile year. With the exception of February, the scheme resulted in months having 30 or 31 days alternately throughout the year. And in order to help farmers, Caesar issued an almanac showing on which dates of his new calendar various seasonal astronomical phenomena would occur.

These arrangements for the months can only have remained in force for a short time, because in 8 BC changes were made by Augustus. In 44 BC, the second year of the Julian calendar, the Senate proposed that the name of the month Quintilis be changed to Julius (July), in honour of Julius Caesar, and in 8 BC the name of Sextilis was similarly changed to Augustus (August). Perhaps because Augustus felt that his month must have at least as many days as Julius Caesar's, February was reduced to 28 days and August increased to 31. But because this made three 31-day months (July, August, and September) appear in succession, Augustus is supposed to have reduced September to 30 days, added a day to October to make it 31 days, reduced November by one day to 30 days, and increased December from 30 to 31 days, giving the months the lengths they have today.

Several scholars, however, believe that Caesar originally left February with 28 days (in order to avoid affecting certain religious rites observed in honour of the gods of the netherworld) and added two days to Sextilis for a total of 31; January, March, May, Quintilis, October, and December also had 31 days, with 30 days for April, June, September, and November. The subsequent change of Sextilis to Augustus therefore involved no addition of days to the latter.

The Julian calendar retained the Roman republican calendar method of numbering the days of the month. Compared with the present system, the Roman numbering seems to run backward, for the first day of the month was known as the Kalendae, but subsequent days were not enumerated as so many after the Kalendae but as so many before the following Nonae (“nones”), the day called nonae being the ninth day before the Ides (from iduare, meaning “to divide”), which occurred in the middle of the month and were supposed to coincide with the Full Moon. Days after the Nonae and before the Ides were numbered as so many before the Ides, and those after the Ides as so many before the Kalendae of the next month.

It should be noted that there were no weeks in the original Julian calendar. The days were designated either dies fasti or dies nefasti, the former being business days and days on which the courts were open; this had been the practice in the Roman republican calendar. Julius Caesar designated his additional days all as dies fasti, and they were added at the end of the month so that there was no interference with the dates traditionally fixed for dies comitiales (days on which public assemblies might be convened) and dies festi and dies feriae (days for religious festivals and holy days). Originally, then, the Julian calendar had a permanent set of dates for administrative matters. The official introduction of the seven-day week by Emperor Constantine I in the 4th century AD disrupted this arrangement.

It appears, from the date of insertion of the intercalary month in the Roman republican calendar and the habit of designating years by the names of the consuls, that the calendar year had originally commenced in March, which was the date when the new consul took office. In 222 BC the date of assuming duties was fixed as March 15, but in 153 BC it was transferred to the Kalendae of January, and there it remained. January therefore became the first month of the year, and in the western region of the Roman Empire (Roman republican calendar), this practice was carried over into the Julian calendar. In the eastern provinces, however, years were often reckoned from the accession of the reigning emperor, the second beginning on the first New Year's day after the accession; and the date on which this occurred varied from one province to another.

The Julian calendar year of 365.25 days was too long, since the correct value for the tropical year is 365.242199 days. This error of 11 minutes 14 seconds per year amounted to almost one and a half days in two centuries, and seven days in 1,000 years. Once again the calendar became increasingly out of phase with the seasons. From time to time, the problem was placed before church councils, but no action was taken because the astronomers who were consulted doubted whether enough precise information was available for a really accurate value of the tropical year to be obtained.

By 1545, however, the vernal equinox, which was used in determining Easter, had moved 10 days from its proper date; and in December, when the Council of Trent (Trent, Council of) met for the first of its sessions, it authorized Pope Paul III to take action to correct the error. Correction required a solution, however, that neither Paul III nor his successors were able to obtain in satisfactory form until nearly 1572, the year of election of Pope Gregory XIII. Gregory found various proposals awaiting him and agreed to issue a bull that the Jesuit astronomer Christopher Clavius (1537–1612) began to draw up, using suggestions made by the astronomer and physician Luigi Lilio (also known as Aloysius Lilius; died 1576).

The papal bull Inter gravissimas (“In the gravest concern”) was issued on February 24, 1582. First, in order to bring the vernal equinox back to March 21, the day following the Feast of St. Francis (that is, October 5) was to become October 15, thus omitting 10 days. Second, to bring the year closer to the true tropical year, a value of 365.2422 days was accepted. This value differed by 0.0078 days per year from the Julian calendar reckoning, amounting to 0.78 days per century, or 3.12 days every 400 years. It was therefore promulgated that three out of every four centennial years should be common years, that is, not leap years; and this practice led to the rule that no centennial years should be leap years unless exactly divisible by 400. Thus, 1700, 1800, and 1900 were not leap years, as they would have been in the Julian calendar, but the year 2000 was. The reform, which established what became known as the Gregorian calendar and laid down rules for calculating the date of Easter, was well received by such astronomers as Johannes Kepler (Kepler, Johannes) and Tycho Brahe (Brahe, Tycho) and by the Catholic princes of Europe. Many Protestants, however, saw it as the work of the Antichrist and refused to adopt it. Eventually all of Europe, as late as 1918 in the case of Russia, adopted the Gregorian calendar.

Easter was the most important feast of the Christian Church, and its place in the calendar determined the position of the rest of the church's movable feasts (see church year (Christianity)). Because its timing depended on both the Moon's phases and the vernal equinox, ecclesiastical authorities had to seek some way of reconciling lunar and solar calendars. Some simple form of computation, usable by nonastronomers in remote places, was desirable. There was no easy or obvious solution, and to make things more difficult there was no unanimous agreement on the way in which Easter should be calculated, even in a lunar calendar.

Easter, being the festival of the resurrection, had to depend on the dating of the Crucifixion, which occurred three days earlier and just before the Jewish Passover. The Passover was celebrated on the 14th day of Nisan, the first month in the Jewish religious year—that is, the lunar month the 14th day of which falls on or next after the vernal equinox. The Christian churches in the eastern Mediterranean area celebrated Easter on the 14th of Nisan on whatever day of the week it might fall, but the rest of Christendom adopted a more elaborate reckoning to ensure that it was celebrated on a Sunday in the Passover week.

To determine precisely how the Resurrection and Easter Day should be dated, reference was made to the Gospels; (Gospel) but, even as early as the 2nd century AD, difficulties had arisen, because the synoptic Gospels (Matthew, Mark, and Luke) appeared to give a different date from the Gospel According to John for the Crucifixion. This difference led to controversy that was later exacerbated by another difficulty caused by the Jewish reckoning of a day from sunset to sunset. The question arose of how the evening of the 14th day should be calculated, and some—the Quintodecimans—claimed that it meant one particular evening, but others—the Quartodecimans—claimed that it meant the evening before, since sunset heralded a new day. Both sides had their protagonists, the Eastern churches supporting the Quartodecimans, the Western churches the Quintodecimans. The question was finally decided by the Western church in favour of the Quintodecimans, though there is debate whether this was at the Council of Nicaea in 325 or later. The Eastern churches (Eastern Orthodoxy) decided to retain the Quartodeciman position, and the church in Britain, which had few links with European churches at this time, retained the Quartodeciman position until Roman missionaries arrived in the 6th century, when a change was made. The dating of Easter in the Gregorian calendar was based on the decision of the Western church, which decreed that Easter should be celebrated on the Sunday immediately following the (Paschal) Full Moon that fell on or after the vernal equinox, which they took as March 21. The church also ordered that if this Full Moon fell on a Sunday, the festival should be held seven days later.

With these provisions in mind, the problem could be broken down into two parts: first, devising a simple but effective way of calculating the days of the week for any date in the year and, second, determining the date of the Full Moons in any year. The first part was solved by the use of a letter code derived from a similar Roman system adopted for determining market days. For ecclesiastical use, the code gave what was known as the Sunday, or dominical, letter.

The seven letters A through G are each assigned to a day, consecutively from January 1 so that January 1 appears as A, January 2 as B, to January 7 which appears as G, the cycle then continuing with January 8 as A, January 9 as B, and so on. Then in any year the first Sunday is bound to be assigned to one of the letters A–G in the first cycle, and all Sundays in the year possess that dominical letter. For example, if the first Sunday falls on January 3, C will be the dominical letter for the whole year. No dominical letter is placed against the intercalary day (leap year), February 29, but, since it is still counted as a weekday and given a name, the series of letters moves back one day every leap year after intercalation. Thus, a leap year beginning with the dominical letter C will change to a year with the dominical letter B on March 1; and in lists of dominical letters, all leap years are given a double letter notation, in the example just quoted, CB. It is not difficult to see what dominical letter or letters apply to any particular year, and it is also a comparatively simple matter to draw up a table of dominical letters for use in determining Easter Sunday. The possible dates on which Easter Sunday can fall are written down—they run from March 22 through April 25—and against them the dominical letters for a cycle of seven years. Once the dominical letter for a year is known, the possible Sundays for celebrating Easter can be read directly from the table. This system does not, of course, completely determine Easter; to do so, additional information is required.

This must provide dates for Full Moons throughout the year, and for this a lunar cycle like the Metonic cycle was originally used. Tables were prepared, again using the range of dates on which Easter Sunday could appear, and against each date a number from one through 19 was placed. This number indicated which of the 19 years of the lunar cycle would give a Full Moon on that day. From medieval times these were known as golden numbers (golden number), possibly from a name used by the Greeks for the numbers on the Metonic cycle or because gold is the colour used for them in manuscript calendars.

The system of golden numbers was introduced in 530, but the numbers were arranged as they should have been if adopted at the Council of Nicaea two centuries earlier; and the cycle was taken to begin in a year when the New Moon fell on January 1. Working backward, chronologers found that this date had occurred in the year preceding AD 1, and therefore the golden number for any year is found by adding one to the year and dividing that sum by 19. The golden number is the remainder or, if there is no remainder, 19.

To compute the date of Easter, the medieval chronologer computed the golden number for the year and then consulted his table to see by which date this number lay. Having found this date, that of the first Full Moon after March 20, he consulted his table of dominical letters and saw the next date against which the dominical letter for that year appeared; this was the Sunday to be designated Easter. The method, modified for dropping centennial leap years as practiced in the Gregorian calendar, is still given in the English prayer book, although it was officially discarded when the Gregorian calendar was introduced.

The system of golden numbers was eventually rejected because the astronomical Full Moon could differ by as much as two days from the date they indicated. It was Lilius who had proposed a more accurate system based on one that had already been in use unofficially while the Julian calendar was still in force. Called the epact—the word is derived from the Greek epagein, meaning “to intercalate”—this was again a system of numbers concerned with the Moon's phases, but now indicating the age of the Moon on the first day of the year, from which the age of the Moon on any day of the year may be found, at least approximately, by counting, using alternately months of 29 and 30 days.

The epact as previously used was not, however, completely accurate because, like the golden number, it had been based on the Metonic cycle. This 19-year cycle was in error, the discrepancy amounting to eight days every 2,500 years. A one-day change on certain centennial years was then instituted by making the computed age of the Moon one day later seven times, at 300-year intervals, and an eighth time after a subsequent 400 years. This operation was known as the lunar correction, but it was not the only correction required; there was another.

Because the Gregorian calendar used a more accurate value for the tropical year than the Julian calendar and achieved this by omitting most centennial leap years, Clavius decided that, when the cycle of epacts reached an ordinary centennial year, the number of the epact should be reduced by one; this reduction became known as the solar correction.

One advantage of the epact number was that it showed the age of the Moon on January 1 and so permitted a simple calculation of the dates of New Moon and Full Moon for the ensuing year. Another was that it lent itself to the construction of cycles of 30 epact numbers, each diminishing by one from the previous cycle, so that, when it became necessary at certain centennial years to shift from one cycle to another, there would still be a cycle ready that retained a correct relationship between dates and New Moons.

For determining Easter, a table was prepared of the golden numbers, one through 19, and below them the cycles of epacts for about 7,000 years; after this time, all the epact cycles are repeated. A second table was then drawn up, giving the dates of Easter Full Moons for different epact numbers. Once the epact for the year was known, the date of the Easter Full Moon could be immediately obtained, while consultation of a table of dominical letters showed which was the next Sunday. Thus, the Gregorian system of epacts, while more accurate than the old golden numbers, still forced the chronologer to consult complex astronomical tables.

The derivation of the term style for a type of calendar seems to have originated some time soon after the 6th century as a result of developments in calendar computation in the previous 200 years. In AD 463, Victorius (or Victorinus) of Aquitaine, who had been appointed by Pope Hilarius to undertake calendar revision, devised the Great Paschal (i.e., Passover) period, sometimes later referred to as the Victorian period. It was a combination of the solar (Sun) cycle of 28 years and the Metonic 19-year cycle, bringing the Full Moon back to the same day of the month, and amounted to 28 × 19, or 532 years. In the 6th century, this period was used by Dionysius Exiguus (Denis the Little) in computing the date of Easter, because it gave the day of the week for any day in any year, and so it also became known as the Dionysian period. Dionysius took the year now called AD 532 as the first year of a new Great Paschal period and the year now designated 1 BC as the beginning of the previous cycle. In the 6th century it was the general belief that this was the year of Christ's birth, and because of this Dionysius introduced the concept of numbering years consecutively through the Christian Era. The method was adopted by some scholars but seems only to have become widely used after its popularization by the Venerable Bede of Jarrow (Bede the Venerable, Saint) (673?–735), whose reputation for scholarship was very high in Western Christendom (Christianity) in the 8th century. This system of BC/AD numbering threw into relief the different practices, or styles, of reckoning the beginning of the year then in use. When the Gregorian calendar firmly established January 1 as the beginning of its year, it was widely referred to as the New Style calendar, with the Julian the Old Style calendar. In Britain, under the Julian calendar, the year had first begun on December 25 and then, from the 14th century onward, on March 25.

Because of the division of the Eastern and Western Christian churches and of Protestants (Protestantism) and Roman Catholics, the obvious advantages of the Gregorian calendar were not accepted everywhere, and in some places adoption was extremely slow. In France, Italy, Luxembourg, Portugal, and Spain, the New Style calendar was adopted in 1582, and it was in use by most of the German Roman Catholic states as well as by Belgium and part of the Netherlands by 1584. Switzerland's change was gradual, on the other hand, beginning in 1583 and being completed only in 1812. Hungary adopted the New Style in 1587, and then there was a pause of more than a century before the first Protestant countries made the transition from the Old Style calendar. In 1699–1700, Denmark and the Dutch and German Protestant states embraced the New Style, although the Germans declined to adopt the rules laid down for determining Easter. The Germans preferred to rely instead on astronomical tables and specified the use of the Tabulae Rudolphinae (1627; “Rudolphine Tables”), based on the 16th-century observations of Tycho Brahe (Brahe, Tycho). They acceded to the Gregorian calendar rules for Easter only in 1776. Britain adopted the New Style in 1752 and Sweden in 1753, although the Swedes, because they had in 1740 followed the German Protestants in using their astronomical (astronomy) methods for determining Easter, declined to adopt the Gregorian calendar rules until 1844. Japan adopted the New Style in 1873; Egypt adopted it in 1875; and between 1912 and 1917 it was accepted by Albania, Bulgaria, China, Estonia, Latvia, Lithuania, Romania, and Turkey. The now-defunct Soviet Union adopted the New Style in 1918, and Greece in 1923.

In Britain and the British dominions, the change was made when the difference between the New and Old Style calendars amounted to 11 days: the lag was covered by naming the day after September 2, 1752, as September 14, 1752. There was widespread misunderstanding among the public, however, even though legislation authorizing the change had been framed to avoid injustice and financial hardship. The Alaskan territory retained the Old Style calendar until 1867, when it was transferred from Russia to the United States.

In late 18th-century France, with the approach of the French Revolution, demands began to be made for a radical change in the civil calendar that would divorce it completely from any ecclesiastical connections. The first attacks on the Gregorian calendar and proposals for reform came in 1785 and 1788, the changes being primarily designed to divest the calendar of all its Christian associations. After the storming of the Bastille in July 1789, demands became more vociferous, and a new calendar, to start from “the first year of liberty,” was widely spoken about. In 1793 the National Convention appointed Charles-Gilbert Romme, president of the committee of public instruction, to take charge of the reform. Technical matters were entrusted to the mathematicians Joseph-Louis Lagrange (Lagrange, Joseph-Louis, comte de l'Empire) and Gaspard Monge (Monge, Gaspard, comte de Péluse) and the renaming of the months to the Paris deputy to the convention, Philippe Fabre d'Églantine (Fabre d'Églantine, Philippe). The results of their deliberations were submitted to the convention in September of the same year and were immediately accepted, it being promulgated that the new calendar should become law on October 5.

The French republican calendar, as the reformed system came to be known, was taken to have begun on September 22, 1792, the day of the proclamation of the Republic and, in that year, the date also of the autumnal equinox. The total number of days in the year was fixed at 365, the same as in the Julian and Gregorian calendars, and this was divided into 12 months of 30 days each, the remaining five days at year's end being devoted to festivals and vacations. These were to fall between September 17 and 22 and were specified, in order, to be festivals in honour of virtue, genius, labour, opinion, and rewards. In a leap year an extra festival was to be added—the festival of the Revolution. Leap years were retained at the same frequency as in the Gregorian calendar, but it was enacted that the first leap year should be year 3, not year 4 as it would have been if the Gregorian calendar had been followed precisely in this respect. Each four-year period was to be known as a Franciade.

The seven-day week was abandoned, and each 30-day month was divided into three periods of 10 days called décades, the last day of a décade being a rest day. It was also agreed that each day should be divided into decimal parts, but this was not popular in practice and was allowed to fall into disuse.

The months themselves were renamed so that all previous associations should be lost, and Fabre d'Églantine chose descriptive names as follows (the descriptive nature and corresponding Gregorian calendar dates for years 1, 2, 3, 5, 6, and 7 are given in parentheses):

The French republican calendar was short-lived, for while it was satisfactory enough internally, it clearly made for difficulties in communication abroad because its months continually changed their relationship to dates in the Gregorian calendar. In September 1805, under the Napoleonic regime, the calendar was virtually abandoned, and on January 1, 1806, it was replaced by the Gregorian calendar.

When Soviet Russia undertook its calendar reform in February 1918, it merely moved from the Julian calendar to the Gregorian. This move resulted in a loss of 13 days, so that February 1, 1918, became February 14.

The current calendar is not without defects, and reforms are still being proposed. Astronomically, it really calls for no improvement, but the seven-day week and the different lengths of months are unsatisfactory to some. Clearly, if the calendar could have all festivals and all rest days fixed on the same dates every year, as in the original Julian calendar, this arrangement would be more convenient, and two general schemes have been put forward—the International Fixed Calendar and the World Calendar.

The International Fixed Calendar is essentially a perpetual Gregorian calendar, in which the year is divided into 13 months, each of 28 days, with an additional day at the end. Present month names are retained, but a new month named Sol is intercalated between June and July. The additional day follows December 28 and bears no designation of month date or weekday name, while the same would be true of the day intercalated in a leap year after June 28. In this calendar, every month begins on a Sunday and ends on a Saturday.

It is claimed that the proposed International Fixed Calendar does not conveniently divide into quarters for business reckoning; and the World Calendar is designed to remedy this deficiency, being divided into four quarters of 91 days each, with an additional day at the end of the year. In each quarter, the first month is of 31 days and the second and third of 30 days each. The extra day comes after December 30 and bears no month or weekday designation, nor does the intercalated leap year day that follows June 30. In the World Calendar January 1, April 1, July 1, and October 1 are all Sundays. Critics point out that each month extends over part of five weeks, and each month within a given quarter begins on a different day. Nevertheless, both these proposed reforms seem to be improvements over the present system that contains so many variables.