Pergamon

Vist~ in Astronomy, Vol. 36, pp. 363-373, 1993 01994 Elsevier Science Ltd

Printed in Great Britain. All rights re~rved. 0083.6656/93 $24.00

0098-2997(93)E0005-9

PERCEIVED NIGHT LENGTH RATIOS IN ANCIENT EGYPT

John Fermor Department of Language and Media, Glasgow Caledonian University,

Cowcaddens Road, Glasgow G4 0BA, U.K.

A b s t r a c t

The first record we have of a seasonal night length ratio for Egypt is from the mid 16th century BC. The origin of this estimate is traced to observations made three centuries previously, and the later reinterpretation and instrumental use of this ratio is traced down to 100AD. Extended comment is made on the astronomical dating involved in this description of events, and an attempt is made to reconstruct the alleged confirmation (or calibration) of the new timepiece that plays a central part in the story. It is believed that this is the earliest example of this fundamental scientific practice on record.

1 I n t r o d u c t i o n

The development of astronomical science by the ancient Egyptians was, on present evi- dence, of a limited kind.Not until the Ptolemaic period under the influence of Hellenistic science do we have anything approaching a theoretical treatise, neither is there any evi- dence for tile development of a technical vocabulary in the subject, nor of any systematic observations of the moon or the planets throughout the three millennia. Some part of this lack of interest may stem from their early downgrading of calendars based on astronomy. In earliest times Sirius was used to regulate a hmar calendar, but fi'om sometime in the Old Kingdom this was replaced for civil purposes with a fixed calendar of 365 days divided into twelve 30 day months and one of 5 days which thereafter cycled with total disregard to sky or season, and even in time came to take over the role of regulator of the lunar year. This change must greatly have reduced the motivation for close observation and theory concerning the motions of the sun and moon. It is true that a plausible case can be made for the importance of total solar eclipses as an enactment of the Osiris myth (1), but we have no record of a prediction, and it may be that a lack of close observation of these bodies under other circumstances coupled with the narrowness of the path of totality put this beyond their competence.

363

364 J. Fermor

Some stellar observations were certainly made. The Egyptians distinguished constel- lations such as the Thigh (our plough or dipper) and Orion, and would align buildings to the pole star (of the day). However the zodiac, and by implication an interest in the ecliptic, was a late importation from other cultures. One star, Spdt (Sirius), was of lasting interest to them as its first rising in the dawn was recognised as the herald of the Nile flood. This star provided the archetype for a selection of hour stars also initially based on dawn risings. The selection was made on the first day of each 10 day week of the civil calendar, the star chosen becoming the marker of the end of the final hour of the night for a 10 day period or decan. It was then replaced by the next selection but took over as penultimate hour marker, etc; until it had signalled each night hour in turn. These decan stars must often have been quite faint ones and the restriction of nightlength to full darkness that this necessitated limited the number of decanal hours to 12. From this beginning stems a parallel 12 fold division of the day, and its present transmutation into our 24 fixed hour system. Yet this great success in its future history should not hide a drawback to the original system which was wedded to a civil calendar adrift against the siderial year so that the decan star selection needed periodic revision. Moreover because Sirius with its 70 day absence from the night sky was the standard for all decan stars these seem likely to have been roughly the same distance south of the ecliptic rather than along a constant declination. Such factors may explain why nowhere in Egyptian astronomy is there any hint of angular measure.

This preamble suggests that the measure of seasonal changes in night length was cer- tainly consonant with the main thrust of Egyptian astronomy. Nevertheless they were not best placed theoretically to arrive at the answer. Other cultures tackled the complemen- tary problem of daylength changes, involving a careful study of the sun's path.Thus we find Ptolemy importing Greek and Babylonian theory into his estimation of the longest day for Alexandria as 14 hours (2). In dynastic Egypt by contrast the matter was approached from the difficult end, with results that leave us with a pretty problem to decipher.

2 The ratio and its meaning

An inscription concerning Amenemhet, chief astronomer to Ahmose, Amenhotep I and Thutmose I, claims for him the invention of a water clock of the outflow type around the year 1550BC, of a pattern found from the reign of Amenhotep III and from the Ptolemaic and Roman periods. Fragmentary though it is, this inscription describes some sort of testing or calibration of the instrument, but ill one respect this is seen as confirming prior knowledge, for Amenemhet derived his ratio for the longest to the shortest night and a rough dating for the latter, from his reading of past attthorities (3). The ratio he gave, of 14 to 12, seems likely to have originated in tile Middle Kingdom, and was still in use two millennia later. Yet it owes nothing to veracity, for if night is considered to run from sunset to sunrise the correct ratio is about 16 to 12 for Upper Egypt and 17 to 12 in the delta. If stellar visibility provides the measure of night, full astronomical twilight gives ratios of 18 and 20 to 12 for the two regions of Egypt. Explanations for this anomaly

Night Length Ratios in Ancient Egypt 365

are hard to find. Neugebauer and Parker gloss over the problem by remarking that the 7 to 6 ratio is a rather poor approximation for the 7 to 5 actuality (4), but this minimizes the divergence by assuming the longer definition of night. R.W. Sloley (5) tenders an explanation based on the changing viscosity of water whereby slow winter outflows require the scale ratio in water clocks to be less than the time ratio they represent. This would require Amenemhet's received ratio to have also been derived from water clocks.

Accordingly this was the first line of enquiry pursued despite Amenemhet's claim for his clock that "Never was one made like it since the beginning of time" .The near complete water clock discovered at Karnak and carrying a dedication to Amenhotep III,appears to be a slavish copy of his original design since the shortest scale is for II smw, a month that contained the summer solstice in Amenemhet's time, but did not do so two centuries later. This clock shows equally divided scales and an inclined side wall, so that the volumes of water between lower (later) hour markers decreased in rough compensation for the declining discharge rate of the vessel as the head of water was reduced. Each monthly scale is divided into twelve hours but a linear increase in total scale length occurred month by month between extremes of 19 . and 14 Fingerbreadths (6). Mills describes this instrument as a sophisticated device, indicative of many predecessors (7). Certainly the features described would justify a claim to invention, without implying that the measurement of time by the emptying of a standard vessel was itself unknown prior to Amenemhet. The cylindrical water clocks of the Babylonians may well have been in use before this time, and one might hypothesise a cultural diffusion of the idea via the Hyksos invasion of Egypt from the east in the previous century. However, even if Amenemhet's clock was an improvement on a Babylonian style device he caai hardly have received the seasonal ratio from them since they appear to have erred in this respect in the opposite direction imputing a 2 to l ratio to their city which was only altered to a more nearly correct 3 to 2 ratio at a later date (8).

His actual claim is that the ratio came from an Egyptian source, and since it is described as 14 to 12 not 7 to 6 we are drawn to texts preserved in the cenotaph of Seti I at Abydos which can be shown by internal evidence to stem from originals written about 1870 BC (9). These texts describe a system in which both day and night were divided into 12 hours, but whereas the hours of the day were perfectly elastic with the seasons, so that no measure of seasonal change in duration is possible, those of the night bear a more complex relationship to the total period of darkness. These nocturnal hours were delimited by the culminations of selected stars rather than their risings but in all other respects we have the decanal system as previously described. A late commentary on the texts, Papyrus Carlsberg I, remarks that each star is born (i.e rises in the dawn), spends 8 decades in the eastern sky before work, 12 decades working (during which its culminations are hour markers), and 9 decades in the west Mter work, before an absence from the night sky of 7 decades. It is also stated that 29 of the decan stars (of the total 36) are visible on any one night, which is implicit in the above. This must necessarily describe an average condition since the seasonal change in nightlength would restrict or increase the number of hour stars visible. However, the 12 decades of work must always apply, and 120 visible culminations must be the minimum for any decan star.

366 J. Fermor

This proposition allows a test to be made. Making the simplifying assumption of a constant minimum solar depression necessary for the decan stars to be seen (their arcus visionis), we can find an approximate value of this which allows 12 decades of 'work' throughout the year while precluding 13 decades of work for at least part of the year. This is shown diagramatically in Figure 1 and proves to be compatible with astronomical twilight (18 degrees of solar depression). Of considerable interest is the fact that these same bounds allow 14 decades of work (but not 1.5) for part of the year.In 1870 BC the period of the year when only 12 decades of work can be done before stellar extinction in the dusk, was centred on late August - early September. In the civil calendar of that date this corresponds to II smw, thus agreeing with the heading of the shortest scale on the Karnak clock, and the more general statement "the night of smw is 12 hours" in Amenemhet's inscription.

Outwith civil smw, sometimes 13. sometimes 14 decades of visible culminations were possible for some decan stars, as distinct from the 12 decades of actual employment asso- ciated with timely offerings to the gods. The Egyptians appear to have used a principle of parsimony in these matters - what could not be done at all times, was not done at any time. This is also implicit in Papyrus Carlsberg I, and explains the curious asymmetry between the 8 decades a decan star spent in the eastern sky, and the 9 allegedly spent in the west. The one decan star identified is Sirius, which in 1870 BC had a declination of 18S. Such a declination coupled with observation from 30 degrees North (Memphis), gives a semi-diurnal arc of 79 degrees. If Sirius rose at say 4 am on a certain day, it would culminate at 4 am 80 days later (10). If all the decan stars followed this pattern then on average their births would precede the beginning of their work by 80 days. However by the same token we might expect them to be visible in the western sky, but not at their culmination, for 80 days. The 90 days claimed seems to have included 10 days of visible culminations which were not working days, which is indeed the average excess of such culminations over those required for the 12 hours of the night. A strong case has been presented that the 14 to 12 hour ratio, discovered in ancient literature by Amen- emhet, refers to decanal hours. The implication by the early writers, or the inference of Amenemhet, that this ratio was of night length in standardized terms, is negated by the realization that decanal hours are unequal.This inequality arises from the seasonal change of dawn, but since it does not also incorporate the change of the time of dusk, it follows that night length will not always be of 12 decanal hours.

Amenemhet adopted the ratio, and placed the shortest scale of his water clock in the second month of civil stow. This apparent conservatism masks considerable conceptual change. The drift in the civil calendar against the seasons had changed the significance of II smw, from being the period when only twelve decan stars could be seen culminating through the night, to being the month containing the summer solstice. Moreover, though the midsummer and midwinter scales were made 12 Fingers and 14 Fingers in total length respectively, each monthly scale was divided into 12 hours only, so that seasonally unequal hours had become the plain intention.

The bounds of the night also underwent revision. Whereas in the Middle Kingdom the official count of any decan star's cuhninations was kept at 120 days , and the number

Night Length Ratias in Ancient Egypt 367

i,o- u_

o 20 LLI m

Z m

20-

m 40- _a z

6 0 - u_

80 -

Z m Z

100-

/ / approx 15*

150

A.V.= 18 8

~ 1 3 0 ~ 1 5 0 ~ Days after first transit at s a m e A . V .

1201 Solstice . ~ , 12 hours only

J I F I H I A I M I J I J I A I S I o I N I D

Figure 1: The visibility of decan star transits at Memphis, 1870 BC

368 J. Fermor

of culminations actually visible varied from 120 to 140 days, in star tables of early New Kingdom origin we find summer stars such as the "the giants hip' visibly culminating over a 140 day period and winter stars such as 'the lion's head' extending this period to 150 days. Something close to astronomical twilight was succeeded by a bounding convention midway between nautical and astronomical twilight in modern terms. Thereafter the Egyptians' view of nocturnal time shifted to making the basic 12 hours equinoctal rather than summer solsticia]. A document of Ramesside age outlines an 18 to 6 hour seasonal contraction alternatively affecting both da b" and night hours (11), while a similar schema from the reign of Necho has the ratio reduced to 14 to l0 (12), a more or less correct value for lower Egypt if the bounds of night and day are sunrise and sunset. A water clock dedicated to Ptolemy II employs a 14 to 10 scale ratio (13), while an Egyptian manuscript from early Roman times tabulates day length ratios for the known world including a ratio for Ethiopia (presumably Egypt) of 53 to 37 i.e just over 17 to 12 (14).

Yet the 14 to 12 ratio survived all these rival schemata to appear as the solsticial scale lengths in the receiving vessel of an inflow clock discovered at Edfu in upper Egypt and dated by its style to around 100 AD. Perhaps even more remarkably the shortest scale is that given for II smw, echoing the Narnak outflow clock, the summer solstice at the time of Count Amenemhet, and the period of most restricted decan star culminations visible in the reign of Sesostris Ill (15).

It would be mistaken to see preservation of the scale ratio merely as the persistence of error in an obsolete instrument. The Edfll clock was far from obsolete in another respect as it incorporated an uneven spacing of the monthly scales producing a more sophisticated notion of the change of night length with season than the linear interpolations between extremes previously in use. The anomalous scale ratio rnust surely have accorded with the behaviour of the clock, just as Amenenlhet's original outflow clock of 1600 years before is recorded as confirming the ratio upon testing. Inflow and outflow clocks were designed differently, the receiving vessel at Edfu being a cylinder, while the discharging vessel of the Karnak clock, and of all subsequent outflow clocks, was a fl'ustrum of a cone. This very contre~st suggests that these clocks were designed in their combination of known and missing parts (16) to deliver a preconceived scale ratio while simultaneously coinciding with the bounds of the night as these changed with the season. Initially scale ratios in water clocks were thought to reflect time ratios, but by the Roman period a certain understanding of the sensitivity of water flow rates to temperature changes prevailed. Thus Atheneus (17) in 300 AD could write "water used in hour glasses does not make the hours in winter the same as in summer, but longer, for the flow is slower on account of the increased density of the water".

An instrument such as the Edfu clock was built to instructions, which could not be varied in one part (the scale) without complementary changes that were difficult to determine in other parts (inflow spout characteristics, siting instructions) if the fit between scale reading and dawn was to be maintained. Thus we may conclude that the 14 to 12 ratio was born in valid, if misinterpreted, observations of the stars, that it dictated the initial designs of both outflow and inflow clock iu the belief that the ratio was true, and that it survived a greater understanding both of the real ratio and of water clock function

Night Length Ratios in Ancient Egypt 369

because in its instrumental aspect it represented a hard won congruence with nature.

3 The Egyptian civil calendar and Astronomical dat- ing

The Egyptians' employment of a civil calendar of exactly 365 days from before the Middle Kingdom until the reforms of Augustus and Julius has been a boon to chronologists in the modern era since if an event is tied in some way to the seasonal year and an instance of it is dated in the civil calendar terms then the year of that instance can be known. The commonest subject of this is the star Sirius (18}.

Two examples concern us here. The inscription in the cenotaph of Seti I places the first morning observation of Sirius on the 16th day of the 8th month, as it was at the time of origin of the revered formula concerning the decan stars. An identical date has been obtained from the seventh year of the reign of Sesostris III. Usually the year calculable from this is subject to minor uncertainty since the seasons only move a day in civil calendar terms every four years,and this margin can be widened by failures of observation to which such data are prone. In this case however R.A.Parker has refined the calculation to the year 1872 BC by using civil dates of hmar events fi'om the same period (19).

The astronomical dating of the Ramesside star clocks is rather less satisfactory. Three published attempts to date the time of origin of these tables give answers of 1450 (Re- nouf),1502 (Schackenberg), and 1470 (Neugebauer & Parker) (20).None of these attempts is above reproach.The first two employ the datum that the train of Sirius' culminates at hour 6 - clock midnight - on the first clay of the fourth month. However Renouf asked the Astronomer Royal to calculate "..in what year did Sirius culminate at midnight at Thebes within the first 15 dab, s of the month...", whereas Schackenberg used the plain meaning of the table. Both authors clearly considered that the train of Sirius' was very close in right ascension to Sirius itself.The data available to them did indeed seem to use the two stars interchaageably, but the later discovery of a table for the 16th day of the sixth month in which these stars were employed in consecutive hours closed this line of enquiry. This extra table contained the observation of Sirius culminating at the 'beginning of the uight'.Neugebauer & Parker wished to use this for dating purposes but eschewed direct retrocalculation as the time of this bcginning was obscure. Instead they employed the formula in Papyrus Carlsberg I, interpreting this observation of Sirius as if it was its last day of work as a decan star, which is followed by 90 days in the western sky and 70 days of invisibility to arrive at the first morning observation on the 26th day of the l l t h month. The going forth of Sirius was an important festival to the Egyptians and we have an independently dated record of it on July 20th Julian 139AD which was also the first day of the first month in the civil calendar (21).Around 1500BC Sirius rose on July 16th Julian, and the equating of the Julian and Civil dates places the record at 1470BC. This calculation assumes that the observation was made at Memphis (the other authors supposed Thebes) else the date would be 1450BC.It should also be noted that Papyrus Carlsberg I describes a year of 360 days leaving open the alternative that we

370 J. Fennor

could subtract the 120 days of work and the alleged 80 days from first culmination to first rising adding 20 years to the age of the record. Yet other refinements have latterly been added to the concept of the 'Sothic cycle'(22). All this is dwarfed by the error in equating the beginning of night with the end of work, for the Ramesside tables record 150 days of visible culminations not the 120 of the decan star scheme. In 1550BC the 16th day of the sixth month fell on 1st February (Greg) and suggests a considerable time difference between the two concepts.

The major faults of previous dating attempts err in the same direction. If the star following Sirius culminated at midnight on a certain civil date then Sirius will have done so at an earlier date. Similarly Sirius culminated at the end of work in decanal terms some days before it culminated at the beginning of the night. As an earlier civil date indicates an earlier year of origin for the tables this approaches that of the invention of the waterclock - perhaps Amenemhet himself was the author of them.

4 Testing the waterclock

In the foregoing account Amenemhet claims to have invented a clock with a 14 to 12 scale ratio which was successfully calibrated against the moon. What might this have entailed? The problem has two aspects, the first being the calibration itself. The even scale division indicates it was not calibrated directly, and likewise the realization that the wall angle stems from a simple doubling of the basal width to give the rim width rules out this variable as the active one. This leaves the outlet characteristics of spout bore and length. Experiment has shown that the former is of the order of lmm and highly sensitive to fractional changes thus rendering it beyond Amenemhet's technical ability to control. This suggests pipe length as the means of fine tuning (23).One wonders from what materials these spouts were made. Curiously enough we know of a very practicable material. A glass factory has been discovered at Tel el Amarna dating from this period wherein fine glass tubes were made and drawn in the manner of our own capillary tubing. The main use seems to have been for chopping up into glass beads but at least the potential existed for manufacturing small diameter spouts in any quantity (24).

The second aspect concerns the natural reference. The moon is mentioned and so is the division of the scales into halves. This could indicate the use of the culmination of the moon on the 15th or 16th night of each hmar month (the first night being that following the moon's absence) as a check on the end of the sixth hour of the night. The culmination of the full moon is not a perfect indicator of midnight but that is its central tendency and if its employment near the solstices gave half night scale lengths in a 6 to 7 ratio this would provide Amenemhet with a valuable confirmation of ancient writ. This speculative reconstruction from fragmentary allusions is fortunately not the only evidence for waterclock timekeeping around this time. for a second test can use the observations made sometime in the next few decades that were later copied by Ramesside scribes (the so called star clocks).

The modern consensus is that the stellar appearances recorded textually and diagra-

Night Length Ratios in Ancient Egypt 371

Table 1 - The fit of clock hour 12 to a solar depression of 15 degrees in 1500BC

Star Date (hr 0) night begins Date (hr 12) night ends Hr 12 hr min hr rain hr min

The pedestal 3 Nov 18 26 16 June 3 46 3 38 Star of Orion 17 Jun 18 47 20 Aug 4 28 4 38 Stars of the water 3 Mar 19 07 4 Oct 4 54 4 58 Head of the lion 18 Mar 19 13 19 Oct 5 02 5 04 The Hippo's buttocks 1 July 19 58 1 Feb 5 43 5 49 The giant's hip 4 Oct 18 51 17 Mar 3 56 4 03 The giant's knee 19 Oct 18 38 1 Jun 3 48 3 50

matically in these tables are of culminations or near culminations. On the first and 16th of each civil month 13 observations are'recorded,the first record at the "beginning of the night", the others at the end of each hour. Unlike the decan star system the variation in the stars named and their failure to be always at culmination indicates independent timing which must surely be by waterclock. However a waterclock cannot provide its own starting time, nor were the stars in the tables so employed else each night would begin with a culmination not a near miss as is often the case. The likely solution is that startup was by a cue available at all seasons such as the first visibility of a circumpolar star or constellation. Such a cue amounts to a standard solar depression and the night the clock purports to measure should be bounded by it.There are seven cases in the tables where particular stars are recorded as being in the same position in the sky at the end of hour 12 and some months later at the beginning of the night. This allows us to estimate the value of the solar depression trigger and how near hour 12 comes to coinciding with the same bounds in the dawn. Taking the received rounded date of 1500BC and a rounded solar depression of 15 degrees we arrive at Table 1 which seems to show a remarkable agreement between the clock hour and dawn especially in view of the inevitable imprecision of these markers in practice. It. should be said that though this supports the value for the solar depression chosen it does not pin down the date so exactly a.s a trial with 1550BC as the assumed date also gives a close fit between hour 12 and dawn.

A sceptic might think that the observers ignored the hour 12 marker and recorded the last star as dawn approached. However we should recall that this was no routine set of observations but one thought worthy of preservation for centuries making such blatant fudgeing less likely. It is more likely that the approach of dawn would affect a judgment on the exact level of a meniscus and vice versa but such lesser fudgeing in no way detracts from the conclusion that the waterclock coukl deliver a close approximation to real night lengths. That the clock used also had a 14 to 12 scale ratio is very likely as this is found in another such clock a century later. In sum there is good evidence that Amenemhet and his followers did not ignore discrepancies between their clocks and nature but produced and used versions that were in close agreement with it.

372 J. Fermor

5 N o t e s a n d R e f e r e n c e s

1. Sellers,J.B.(1992) The death of gods in Ancient Egypt. Penguin Books. While Sellers' evidence concerning total solar eclipses is persuasive, her further claim

concerning the early calculation of the precessional period seems more speculative. If it should eventually turn out to be true then a major revision of current views on early Egyptian astronomy must result and much of the argument of this introduction falls. 2. Chabas, J. & Tihon, A.(1993) Verification of parallax in Ptolemy's Handy Tables JHA 24, 125 3. Sloley, R.W.(1924) Ancient Clepsydrae. Ancient Egypt. J (2 ) , 43-50

"...while reading in all the books of the divine words .... 14...if the night of smw was 12 hours .... fi'om month to month and decrease month by month ....... I made an instrument reckoned from the zero of the year .... never was one made like it since the beginning of time. I made it in honour of the deceased king. Amenhotep I, divided in half....It was correct at the beginning of smw,in winter at the .... of the moon in its times...every hour to its time." 4. Neugebauer, O.& Parker. R.A.(1960) Egyptian Astronomical tezts LThe early de- cans.p119, Brown IIniversity Press. 5. Sloley, R.W. op.cit.47 6. Borchardt, L.(1920) Altagyptische Zeitmessung. In G.Z.U, Ed. Von Basserman-Jordan Berlin.

Although Borchardt and successive commenlators give the scale ratio of this clock as 14 to 12, his Table B shows that only one scale (the shortest) has an hour 12 marker to avoid disfiguring the signs of life and stability that girdle the inside base. The longest scales have been foreshortened at hour 11 for the same reason. It is only by comparing earlier hours that the reasonableness of Borchardt's inference can be shown. Amenemhet's genius contrasts with the unfortunate priorities of his successors. 7. Mills, A.A.(1982) Newton's Water Clocks and the fluid mechanics of Clepsydrae. Notes (J Records of the Royal Society of Los~don 37, 35-61 8. Van der Waerden, B.L.(1974) Science Awakening II. pp87-88, Oxford University Press. 9. Neugebauer, O.& Parker, R.A.(1960) op cit. p54 10. Because nights lengthen between the birth of Sirius and its first visible culmination this actually occurs only seven decades later. The average condition of a set of stars near this declina.tion will however conform to the formula of Papyrus Carlsberg I. 11. Carny, J (1943) The origin of the month Tybi. ASAE 43, 179-180 This extreme ratio has defied explanation. 12. Clare, J.J.(1949) h'emi 10.3-25 13. Borchardt. L. op.cit, BI3 14. Neugebauer, O.(19,12) On some astronomical papyri and related problems in ancient geography. Trans. AM. Phil. Soc 32, 251-263 15. This feature was presumably a superficial conservation to be combined with pragmatic use. A conversion table could readily correct for the drift in the civil calendar. Other aspects are discussed by Pogo.A.(1936) Egyptian water clocks Isis 25, 403-425

Night length Ratios in Ancient Egypt 373

16. No spout has ever been recovered fi'om an Egyptian outflow or inflow clock. 17. Atheneus (300) II, 16 18. Parker, R.A.(1952) Sothic dates and calendar "Adjustment'.Revue d'Egyptologie 9, 101-108. 19. Parker, R.A.(1950) The calendars of a l~cient Egypt .The University of Chicago Press. 20. Renouf, P.Le Page.(1874) Calendar of astronomical observations found in the royal tombs of the XXth dynast3". TSBA 2, 400-421

Schackenberg, H. Schack.(1902) D ie Sternnetzabscissen und die somatischen relationen der Thebanischen Studentafeln Aegyptologische Studien 2, 57-128 Leipzig.

Neugebauer, O &: Parker, R.A.(1964) EAT II, The Ramesside star clocks. Brown Uni- versity Press. 21. Censorinus.(238) De die natali liber. 22. Ingham, M.F.(1969) The length of the Sothic cycle,lEA 36-40 23. Fermor, J.H.,Burgess, A.,& Przybylinski. V.(1983) The timekeeping of Egyptian out- flow clocks. Endeavour, New Se~'ies 7(3). 133-136 24. Lucas, A.(1962) A ocieT~t Egyptia, materials and industries. Edward Arnold.