The oldest possibly astronomically inspired monuments on record consist of a series of aligned megalithic structures and stone circles found adjacent to Middle and Late Neolithic (7000–5000 BCE) settlements at Nabta in the Western Desert of Egypt just north of Sudan. One of these, a 4-meter (12-foot) circle of small upright and recumbent stone slabs, also contains four sets of narrower pillar-like stones arranged in pairs that form slots at opposite ends of a diameter that appears to have been aligned along the summer solstice sunrise—sunset line of about 6000 BCE. In addition, two similar opposing pairs of uprights marked the north—south axis of the monument (Malville, Wendorf, et al. 1998).

To the ancient Egyptians, astronomy became integrated into their concept of self-identity. Their observations of moving star patterns in the night sky more than seven thousand years ago resulted in a variety of myths that influenced their religious beliefs. Their principal deities were specific heavenly bodies, whose actions governed many aspects of daily Egyptian life. People, institutions, and the state were considered reflections of the greater cosmos from which they evolved and toward which they aspired. That relationship was a consequence of the ability of those early sky watchers to predict the time and place of their gods' appearances. They developed an elementary form of positional astronomy that led to an annual time unit of 365 days; the division of night into segments that predicted the time of sunrise; and the formation of a relatively complicated lunar calendar, used to determine recurrent times of feasts and offerings to their gods. As the cohesiveness of the state strengthened, economic progress forced the simplification of this religious calendar into a secular one that the ordinary Egyptian could use for business transactions. It comprised twelve 30-day months with a special 5-day unit added to bring the total to 365 days. Known as the civil calendar, it is the earliest predecessor of our modern Western counterpart. These early evolutionary trends and their effect on later generations are summarized here.

Predynastic Astronomy.

The preeminent Egyptian god was the sun god Re. The early Egyptians therefore followed the annual motion of the sun disk along the horizon at sunrise and sunset, the northern and southernmost turning points of which are called the summer and winter solstices, respectively. Much of Egyptian astronomy and religion can ultimately be traced back to that simple movement of the Sun along the horizon.

Two principal collections of Predynastic legends had a profound effect on Egyptian culture. One was the renowned myth of the sky goddess Nut giving birth to Re, which catalyzed both timekeeping and calendar development and endowed the concept of divine royalty. The other, a compilation of stories related to the safe passage of Re through the underworld, referred to as the Book of Gates, became the first markers of the night intervals that were used to predict the time of sunrise for daily morning offerings to Re. The specific effects of each of these legends are discussed below.

The mythology of Nut and the birth of Re.

The goddess Nut was depicted as a naked female stretching across the sky between the horizons. The Sun is shown passing into her mouth, coursing through her star speckled body, and emerging again from her birth canal. The faint outer arm of the Milky Way fortuitously resembles a female and was clearly perceived by early Egyptians as the goddess Nut. An expansion of the star patches in the vicinity of the constellation Gemini forms her head. A division in the star patterns at the cross shape of the constellation Cygnus represents her legs; this particular location of Cygnus, reminiscent of primitive female figurines marked with cross-shaped genitalia, provides the female identification in the heavenly image. Cygnus' principal bright star, Deneb, coincides with the birth canal's exit.

The apparent path of the Sun in the sky is directed through Gemini, in an area of the Milky Way that corresponds with the head's mouth. After twilight on the vernal (spring) equinox, the head of Nut may be seen dropping below the horizon, with her face upward and her mouth open at or very close to the position where the Sun had set earlier. This configuration must have impressed early Egyptians as a faintly glowing head on the horizon that just consumed the Sun.

The Egyptian perception of the Sun emerging from Nut at birth occurred at winter solstice sunrise, 272 days later. The lower half of the goddess then reached a point where a great circle from the north celestial pole drawn through Deneb would intersect the horizon at exactly the spot where the Sun would appear. This great circle marked the shortest path that the infant Re would follow after birth, from Deneb to the point of its appearance on the horizon at sunrise. That picture reflected the ancient Egyptian method of birthing, where the woman assumed a squatting position with her feet supported on bricks, to deposit the infant directly onto the ground. This particular winter solstice geometry does not repeat on any other day, since the sunrise horizon point would then be farther north. The correlation is supported by the number of days between the vernal equinox and the winter solstice, which is also the period of human gestation.

Two related aspects of Egyptian culture have been clarified by the astronomical associations discussed above. The first is the religious aspect, that Re was considered to be a self-creating god; Re does enter Nut at sunset on the vernal equinox, at which time the goddess presumably conceives. Oral conception was implied, but that is not an unusual belief in a primitive society. Nut then gives birth to Re nine months later, on the morning of the winter solstice. The second aspect is the assistance of the goddess Nut in performing the task of self-creation, which appears to have later governed the behavior of Egyptian kings, who maintained that their origin was divine—a result of the sun god intervening in the form of the reigning pharaoh to impregnate the queen to produce the next heir to the throne. Therefore, the king and queen might justify the line of succession throughout dynastic times as a reflection of the ancient tradition incorporated in the Nut cosmogony (Wells 1992).

The hours of night and the Book of Gates.

The stories of the nocturnal passage of Re in his solar bark through the twelve portals of the underworld were also related to time-keeping (Wells, 1993). The Book of Gates was concerned with his safety on this voyage. Each gate, corresponding to one of the 12 hours of night, was in the charge of a demonic gatekeeper and several attendants. Re had to recite correctly the names of the gate, the gatekeeper, and his assistants in a spell before he could pass on to the next one. In later periods, both the deceased pharaoh and even his subjects were expected to utter the recitations correctly in order to reach the immortal gods.

Although the earliest form of the Book of Gates is unknown, it probably originated as a device for telling the time at night, by using stars to predict the time of sunrise. A particularly bright star would have been noticed to rise a given interval ahead of the Sun in sufficient time to permit the daily offering preparations before sunrise. A single star would be an inadequate herald of sunrise, since its interval would continuously increase—because stars rise 4 minutes later each night. As an example, for a star clock to work for the whole year, with 12 assumed equal hours of 60 minutes each, a series of twenty-four bright stars would be required. Each star, spaced across the sky in equal hourly intervals, would serve 15 days in turn as the marker of the hour before sunrise. The 4-minute delay would then shift it to spend 15 days marking the previous hour, and so on, in turn, until that star returned to its duty as the herald of sunrise again, a year later.

Bright stars may be difficult to distinguish from one another at rising. As part of a recognizable configuration of three or four other, fainter ones, however, which rose a few minutes before the principal bright star, the star pattern might forewarn the bright star's rising. The ancient Egyptians would have identified the faint stars as the attendants; the bright star as the gatekeeper; and the place of their appearance, the gate itself. The recitation requiring the names of all the gates and the demigods was just a mnemonic device to ensure that each star group would be recognized and remembered in the correct sequence. Only the group just rising would be at the horizon gate, marking the current hour. Others would be distributed overhead, toward the western horizon, while the remainder would still be in the underworld. As each group dominated the horizon gate, the gate would be associated with the stories that characterized the order of the sequence. Later, when the Egyptians wrote down those narratives, they depicted each group as having its own separate gate. Since twelve star groups came out of the underworld during the night, then the underworld would be illustrated as having twelve gates. Variants of the Book of Gates also have eight, ten, or other numbers of gates, which reflect the different number of patterns used in the different star clocks. Such differences indicate that the first Egyptian “hours” were of unequal length, since they depended on the sizes of the star configurations.

Astronomical Origin of Egyptian Calendars.

Although specific documented predynastic evidence is lacking, the astronomical origin of the two myths discussed above along with the surviving structure of the later lunar calendar both serve as important guidelines in the reconstruction of the way the precursor calendar system might have been originated (Wells 1994).

Calendars are usually based on Earth's orbital period. One sacrosanct cycle for the Egyptians was the time required for Re to return to his birthplace on the southeastern horizon each year, at the winter solstice. The accurate determination of this period as 365 days, by a simple counting procedure—probably using wooden stakes, cords, and shadows—can be dated to sometime around 4500 BCE. At that time, the mouth area of Nut in Gemini was aligned more directly to the west; due west was interpreted by the ancient Egyptians as the entrance to the underworld, a concept that probably originated because of this alignment of Nut's mouth.

A desire to know when to celebrate the annual Birth of Re festival, which the Egyptians called mswt Rʿ, would be a strong incentive for the creation of a calendar. The calendar was probably originated in Lower Egypt, where the chief cult center was later established at Heliopolis, by counting the number of lunar months (i.e., the occurrences of a specific lunar phase) between each winter solstice. Normally, there would be twelve 29- or 30-day months, the first one beginning after the festival had occurred. Because such a calendar averages only 354 days, whenever the first day of a new year occurred within 11 days after a winter solstice feast, an intercalary thirteenth month would have to be placed at its beginning, so that the next celebration of the festival would occur within the last month, which was named for the feast. Such intercalations were necessary about once every 2 or 3 years. In this manner, although the festival might occur on different days during the last month, the embarrassment of having to celebrate the festival in a differently named month would be avoided.

Because of the agrarian dependence of Upper Egypt on the Nile River floods, a different lunar calendar appears to have originated in the South. Instead of the Birth of Re festival, its regulating factor was the heliacal rising of the star Sirius—the first appearance of this star at dawn, just prior to sunrise, after an absence from the sky of about 70 days. Sirius rose heliacally in 4500 BCE, near the time the Nile itself began to rise. The Egyptians called this rising prt spdt (“the going forth of Sothis”), and it became a marker associated with the variable onset of the flood because its period of 365 days was easier to count. The cult center was at the temple of Satet (Sothis) on the island of Elephantine, in the Nile, near the present-day town of Aswan (Wells 1985); this was the fabled source of the Nile, according to ancient Egyptian tradition. The lunar calendar of Upper Egypt, therefore, appears to have used the festival celebrating prt spdt as the principal feast in its last month, governing the intercalation scheme.

The dynastic lunar calendar, from which recorded dates survive, was apparently the product of the political consolidation of Upper and Lower Egypt. The relationships between the northern “lunisolar” and the southern “lunistellar” calendars made their combining relatively easy. The key factors behind the religious festivals that regulated both calendars were theologically related—the star Sirius was equated with the deity Isis, the daughter of the sun god Re. At the beginning of the first dynasty, on their festival days, they rose within 2 degrees of each other on the horizon, although these occurred 6 months apart. During the merging of Upper and Lower Egypt, the use of the southern calendar spread northward, to usurp many of the functions of the northern calendar. Although the newly combined lunar calendar was regulated by Sirius, the name of its last month, normally derived from its principal festival, instead retained the old northern name, mswt Rʿ, in deference to the “Birth of Re.”

This dynastic Egyptian lunar calendar was impractical for use by the general public. It contained either 12 or 13 months, and the beginnings of each had to be determined by observation. A special class of priests, called the “Overseers of the Hours,” were needed to ensure its regulation. Yet as the ability to write spread throughout Egypt during the early dynasties, the need for dated records, for business transactions, forced the adoption of a simpler calendrical method. It rounded the numbers evenly to 12 months of 30 days each, none beginning observationally. This new civil calendar was divided into three 4-month seasons—corresponding to the periods of growth, harvest, and inundation—just like the lunar calendar, which was not abandoned but was used to regulate the observance of religious feasts. Since the civil calendar's total was only 360 days, a special interval of 5 days, known as the epagomenal days, preceded the onset of each year; the ancient Egyptians called them “the 5 days upon the year.”

The Old Kingdom.

By the onset of the Old Kingdom, observations of the Sun, Moon, and stars had led to a sophisticated time-keeping system intimately bound with a series of religious orders. Both contributed to the massive building programs of the Old Kingdom pharaohs, which promoted a growing economy. Astronomical/religious zeal reached an apex with the pyramids and sun temples of the fourth dynasty and the fifth.

The pyramids were stone pathways to the gods, conduits by which the soul of the deceased pharaoh could reach the northern circumpolar stars, identified as the immortal gods. The Egyptians called these stars ikhemu-sek (“the ones not knowing destruction”), an indication that they understood their circumpolar nature—or rather, perhaps, recognized that they did not pass into the underworld, as did the stars that rose and set. The sanctity of the circumpolar stars was apparently incorporated into the Giza pyramids, all of which had northern entrances with corridors sloping downward at an angle such that the circumpolar stars were visible from the interior (Edwards 1993). Consequently, it is easy to explain the peculiar northeastern–southwestern diagonal orientation of these three pyramids with respect to one another; they were simply offset away from the river, each by one full width, so that their northern faces did not block another's view toward the circumpolar stars.

Of the nine fifth dynasty kings, six built huge sun temples, expressly to honor Re, and they incorporated the title “Son of Re” into their official names, to acknowledge their divine origin according to the cosmogony of Nut (Edwards 1994). Their temples were important not only for religious reasons but also because the administration of their vast temple estates had a significant influence on the state economy. Two of the temples have been excavated, revealing a special architectural design, which facilitated the measurement of night hours for the prediction of sunrise.

The lower (so-called valley) temples in the priests' towns, located below the main sun temples, were each axially oriented toward a separate series of stars on the northeastern horizon; in that way, their roofs could have been used as observation platforms for marking the hours at night. Analysis of the stars (Wells 1993) has shown that they measured unequal segments of the night. In addition, Userkaf's temple, the first one of the six built, was associated with a series containing Deneb as its brightest star, appropriately the one from which Re himself was born. The Overseers of the Hours would have sighted the stars with an instrument called the bay (a palm rib with a notch cut into one end). The bay, with another instrument called the merkhet (essentially a plumb bob for determining the local vertical), would also have been used to ascertain the orientation of the building and the axial line on the roof. Similar devices were probably used to determine the orientation of pyramids.

These six temples from the fifth dynasty eventually operated simultaneously and, judging from the different orientations of the associated two known valley temples, each probably used different series of stars to predict the time of sunrise (Wells 1993). One king's priests probably did not want to usurp the ritual duties of another king's; this circumstance suggests that at one point there were six local times (i.e., nocturnal “hours” of differing lengths not having synchronous onsets). Yet since the communities were separated, and their clocks had the common purpose of predicting sunrise, these differences would not have mattered.

The Middle Kingdom.

Primary knowledge of Middle Kingdom astronomy has been provided by a series of ninth and tenth dynasty wooden coffins, on whose interior lids were painted tables of rising stars. Those tables suggest that attempts had been made to regularize the measurement of hours with portable written records.

The surviving coffin lid tables record thirty-six rising stars, each successively marking an “hour” for an interval of 10 days; those hours were, however, only of a duration of forty minutes (24/36 × 60). Referred to as the “Decans,” the thirty-six stars were located in a belt south of the ecliptic, all having the same invisible interval of seventy days prior to their heliacal rising, the property of the principal star in the group, Sirius (Neugebauer and Parker 1960). This new method of denoting the night hours was based on combining only those stars that behaved like Sirius with 10-day weeks of the civil calendar. In that way, one star's heliacal rising was replaced by the next after 10 days; alternatively, they were spaced in forty-minute intervals across the sky. (In other words, 18 decanal hours × 40/60 = 12 of our standard hours.) Although 18 Decans marked the period from sunset to sunrise, three Decans were assigned to each interval of twilight, leaving 12 Decans to mark the hours of total darkness. The 12-unit division of the night may, therefore, have originated in this combination of Decan stars with the civil calendar decades.

Although having a specific Egyptian name, the only one of the Decans unambiguously identified today is Sirius (Sepdet). The constellation of Orion (Sah) is also identifiable, although the association of names with its individual stars remains undetermined. Some of the coffin-lid vignettes depict the seven stars of the Big Dipper (Ursa Major), in approximately the known configuration. The Egyptians also depicted this star pattern as the hind leg of an ox, called it Meskhetiu, and identified it with the goddess Hathor.

Scholars usually refer to the coffin-lid tables as diagonal star clocks, because a given star appears one line higher in adjacent decade columns, marking an earlier hour. Consulting one of these decanal tables while viewing the night sky at a given moment, the time would be known by noting the tabular position of a specific star. The tables included an extra entry to account for the five epagomenal days of the civil calendar. Nevertheless, they failed to utilize an extra day for leap year. Every 4 years, the tables were therefore one day in error; after 120 years, they were off by a whole month. The Egyptians apparently attempted to solve this problem by shifting the star names by the appropriate amounts, which reset the clock with the civil calendar. Yet that procedure appears to have been abandoned by the time of the New Kingdom, when new observational procedures were instituted.

The New Kingdom.

Astronomy during the New Kingdom was characterized by more complicated meridional transits, which required calibration by the newly introduced water clock. A variety of tomb paintings has provided this information. Two new elements appeared in these tomb scenes. One depicted men, facing the viewer and seated in front of grid patterns that contained stars, with the star names to the side; the other is an arrangement of zodiac-like representations of star groups, in the form of animals and deities.

Those stellar groupings have as yet defied attempts at decipherment, except for the already known examples of Sepdet, Sah, and Meskhetiu. Although certain vague relationships are evident, correct identifications cannot be made without independent, contemporary texts that would have similar descriptive elements.

The star grids behind seated priests in the tombs of Ramesses VI, VII, and IX, however, represent the last major stage in telling time with stars. Instead of using the rising stars, the new procedure involved any stars that were transiting both the meridian and the several adjacent pre- and postmeridional lines. The technique was derived from earlier tables of transiting decanal stars that were found in the cenotaph of Sety I and in the tomb of Ramesses IV. The Ramesses IV example consisted of twelve Decans in thirty-six tables that transited only the local meridian, thereby measuring equal “hours” during the night. The final stage exhibited in the later Ramessid tombs, however, was not limited to decanal stars. It consisted of thirteen stars (one marking the beginning of night), in twenty-four tables, and used transits over three lines that were parallel with the meridian on either side, in addition to the meridian; that resulted in unequal “hours,” depending on the altitude of the star—hardly an observational improvement.

Actual observation of the stars required two priests, with one seated facing north, who performed the function of the bay, and the other seated facing the one watching the stars behind him. The viewer of the tomb scene figure was assumed to be the pharaoh's spirit, which would be able to move safely from hour to hour, as with the Book of Gates. The inscriptions listed particular stars for the beginning and the twelve hours of the night, and they indicated that the stars might be seen “over the left shoulder,” “over the right shoulder,” “over the left ear,” “over the right ear,” and so on. The vertical lines in the grid denoted those same positions, both before and after the meridian transit. The horizontal grid lines denoted the hours of night.

Astronomy

Astronomy. Astronomical scenes based on a star clock. The twelve discs represent the months. This is a copy by C. K. Wilkinson of the ceiling painting in the tomb of Senenmut, eighteenth dynasty, reign of Hatshepsut. (The Metropolitan Museum of Art.)

The Ramessid tables suggest that a water clock was probably required to regulate which of the transiting stars should be used. An inscription dated to about 1520 BCE, written by Amenemhet, a nobleman under Amenhotpe I, is the earliest description of the use of a water clock. Those devices were shaped like vases, with a scale on the inside to denote hours. The base had a plug with an exit hole no larger than one made by a needle (Cotterell, Dickson, and Kamminga 1986). They were filled with water that escaped through this tiny hole. Hours were measured against the scale as the water level dropped.

In daylight, time was measured by shadow clocks, a device probably dating to the earliest periods of telling time. The extant examples all resemble the merkhet surveying instrument, which probably served dual purposes. When facing the Sun, it could be quickly leveled by hand with the aid of the plumb bob, so that the short L-shaped upright arm, or a small crossbeam attached to it, would accurately cast its shadow to the “hour” mark on the long arm. A text from the cenotaph of Sety I indicates that it could measure 4 “hours” before and after noon but that there was an “hour” after sunrise and another before sunset during which it could not be used. Shadows would be too long at that time, because of the low Sun angle, making the instrument unwieldy. There was then an hour of twilight before sunrise and again after sunset. Those numbers divided the day period, including twilight, into 12 “hours.” Hence, the day division into twelve segments is probably at least as old as the night division into twelve, which has been traced to the diagonal star-clock tables of the Middle Kingdom.

The Ptolemaic Period.

The character of Egyptian astronomy changed significantly when the Ptolemies became the rulers of Egypt (304–30 BCE). Both Greek and Babylonian influences were soon visible in the temples, monuments, and papyri that are known. The Greco-Babylonian zodiac was incorporated into Egyptian stylistic art, to appear on temples and monuments, and Greek and Demotic papyri included astrological horoscopes and Babylonian-like omens.

See also ASTROLOGY; and CALENDARS.

Bibliography

  • Clagett, M. Ancient Egyptian Science, vol. 2: Calendars, Clocks, and Astronomy. American Philosophical Society, Philadelphia, 1995. A treatise by a historian of science, with volume I consisting of two books that deal with science and engineering. Volume II provides source material and discusses interpretations of some of the problems related to calendar studies.
  • Cotterell, B., F. P. Dickson, and J. Kamminga. “Ancient Egyptian Water-Clocks: A Reappraisal.” Journal of Archaeological Science 14(1986), 31–50. The only technical engineering study of the workings of Egyptian outflow clocks.
  • Edwards, I. E. S. The Pyramids of Egypt. Rev. ed. Harmondsworth, 1993. Still the standard work on Egyptian pyramids.
  • Edwards, I. E. S. “Do the Pyramid Texts Suggest an Explanation for the Abandonment of the Subterranean Chamber of the Great Pyramid?” In Hommages à Jean Leclant, edited by C. Berger, G. Clerc, and N. Grimal. Bibliothèque d'Étude 106.1 (1994), 159–167. Discusses, among other things, the title “Son of Re” from Khufu onward.
  • Malville, J. McKim, Fred Wendorf, et al. “Megaliths and Neolithic Astronomy in Southern Egypt.” Nature 392 (1998), 488–491. Discussion of Nabta settlements and megaliths, with several good photos of a stone circle.
  • Mengoli, P. “Some Considerations of Egyptian Star-Clocks.” Archivder Geschichte der Naturwissenschaften, 22/23/24 (1988), 1127–1150. A good summary (in English) of the manner in which Egyptian star clocks worked.
  • Neugebauer, O. The Exact Sciences in Antiquity. New York, Dover edition, 1969. The revised, published version of his six 1949 Cornell lectures; easier reading than the 1960 volumes.
  • Neugebauer, O. A History of Ancient Mathematical Astronomy, 3 vols. New York, Heidelberg, Berlin, 1975. Not for the casual reader (background in mathematics and astronomy, as well as historical background in several ancient cultures are recommended). The short section on Egypt is of interest from a purely mathematical, as opposed to a cultural, viewpoint; the general discussions on chronological concepts, calendars, and the year provide useful interrelationships.
  • Neugebauer, O., and R. A. Parker. Egyptian Astronomical Texts. vol. 1: The Early Decans. Providence (1960); vol. 2: The Ramesside Star Clocks. Providence (1964); vol. 3: Decans, Planets, Constellations and Zodiacs. Providence (1969). Specialist books, requiring a knowledge of Egyptian grammar and a minimal background in positional astronomy for full appreciation.
  • Parker, R. A. The Calendars of Ancient Egypt. Oriental Institute of Chicago, Studies in Ancient Oriental Civilization, No. 26. Chicago, 1950. Knowledge of Egyptian grammar and some background in positional astronomy and calendrics are needed for comprehension; some sections are dated, but many basic concepts are still valid.
  • Quirke, S. Ancient Egyptian Religion. British Museum Press, London, 1992. A valuable reader on the intricacies of Egyptian religion.
  • Wells, R. A. “Sothis and the Satet Temple on Elephantine: A Direct Connection.” Studien zur Altägyptischen Kultur 12(1985), 255–302. A study of the orientation of the eighteenth dynasty temple, with references to earlier temples on the site that date to Predynastic times; the first evidence is discussed of a temple named for a stellar deity and actually aligned toward the rising of that star (and toward the winter solstice sunrise).
  • Wells, R. A. “The 5th Dynasty Sun Temples at Abu Ghurab as Old Kingdom Star Clocks: Examples of Applied Ancient Egyptian Astronomy.” Beiheft zu Studien zur Altägyptischen Kultur, Akten des Vierten Internationalen Ägyptologen Kongresses München 1985 4(1990), 95–104. A discussion of the use of the sun temples as star clocks.
  • Wells, R. A. “The Mythology of Nut and the Birth of Ra.” Studien zur Altägyptischen Kultur 19(1992), 305–321. The correlation study that demonstrated an astronomical connection to the Nut cosmogony.
  • Wells, R. A. “Origin of the Hour and the Gates of the Duat.” Studien zur Altägyptischen Kultur 20(1993):305–326. A study of the series of axial-rising stars in front of the fifth dynasty sun temples and the manner in which the Book of Gates might have arisen.
  • Wells, R. A. “Re and the Calendars.” In Revolutions in Time: Studies in Egyptian Calendrics, edited by A. J. Spalinger. San Antonio, 1994. A detailed discussion of the origin and evolution of the Predynastic and dynastic lunar and civil calendars; the other chapters are concerned with complex discussions of the fixing of feasts in the lunar calendar or with the intricacies of the operation of ancient Near Eastern lunar calendars.

Ronald A. Wells