Vistas in Astronomy Vol. 41, No. I, pp. 151-167, 1991 @ 1997 Elsevier Science Ltd

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TIMING THE SUN IN EGYPT AND MESOPOTAMIA

JOHN FERMOR Department of Language and Media, Glasgow Caledonian University, Cowcaddens

Road, Glasgow, G4 OBA, U.K.

Abstract- By the 5th Century B.C. three great cultures were in contact in the Eastern Mediterranean area. The newest of these, the Greek, be- gan by borrowing and improving on the discoveries of Babylonians and Egyptians. The knowledge of the gnomon is attested as one such borrow- ing. An overview is attempted of the state of timekeeping, with particular emphasis on the measure of shadows, in these two older cultures, reveal- ing the uncertainties as to just what they had to offer the Greeks, and how they had come to their conclusions. 01997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

As the centres of high culture around the Mediterranean and Middle East began to inter- penetrate from the 7th Century B.C. onward, it was an intellectual sport to argue who had first invented each advance, a contest that even extended to the invention of language itself, and one reminiscent of the cold war claims of a later age. The Greeks, as parvenus, could hardly compete directly but they soon developed a convincing riposte. Herodotus, writing around 450 B.C., candidly admitted that Greek science had borrowed liberally from others at its inception, but added that it had everywhere improved on what was taken. Discussing some borrowings from Egypt, he insisted, in an aside, that the Greeks had learnt the use of the gnomon and of the twelve hours of the day from Babylonia rather than from Egypt.* The implication is that knowledge of these matters was available in either of the senior cultures. Accordingly the state of time measurement by day is examined in each of these cultures down to the 6th Century B.C. to see what the Greeks could have taken from them. It will be shown that our knowledge of this is far from exhaustive. Nevertheless examina- tion reveals a number of contrasts between these cultures in the manner of measuring time from shadows cast, and in the balance between empirical and theoretical approaches which cast further light on Herodotus’ bare statement,

*Herodotus II, 109.

157

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Fig. 1. The Egyptian shadow stick: (a) diagram on roof of sarcophagus chamber; (b) device with shadow casting T-bar.

2. THE EGYPTIAN SHADOW STICK

Egypt has the best claim to be the senior culture in the matter of time measurement. A system for measuring the hours of the night based on stellar observations is known to have existed from at least the first intermediate period (c. 2000 B.C.). In the Middle Kingdom it continued to be actively used and further developed, the risings of stars being replaced by their culminations by 1870 B.C. This last development compares with the earliest known evidence of the observation of stellar culminations in Babylonia occurring c. 1000 B.C. As to daytime measure the Egyptians employed both the sundial, the first known example being from the 13th Century B.C., and a modification of the gnomon which measured shadow length transformed to a preset direction. This latter instrument is known from the reign of Thutmose III (1490-1436 B.C.)* but other evidence suggests a much older use.

In the cenotaph of Seti I at Abydos one of the funerary texts covers the whole subject of time measurement in Egypt at a particular date. The date was not however that of King Seti’s death in 1290 B.C., but can be dated astronomically to the 19th Century B.C. In these texts time is measured during the day using a shadow stick, a device with a shadow-casting upright connected to a horizontal scale arm, illustrated in Fig. l(a), with accompanying instructions.+ These latter show gaps suggesting that the passage of six centuries had taken their toll, but the final copy still presents an understandable procedure, the salient point being that the stick is to be aligned east-west and is only moved at noon so that the upright is due east of its scale arm in the morning and due west in the afternoon. Since shadows alter during the day not only in length but in angle, the upright would need to have been widened in some way if its shadow is always to fall on the scale. Borchardt (1920) suggested an arrangement as in Fig. l(b) featuring a T-bar. Neither the sarcophagus diagram nor surviving shadow sticks include this feature, although two holes in the top of the upright in one specimen of the latter suggest an addition of some sort.

The Seti diagram features a “clock” which directly measures only 8 h. Yet the text is clear that from sunset to sunset there are 24 h. The 12 h of the night are found from the decan stars (Fermor, 1993) and the full count is made up by the 8 h when the shadow end falls on the scale bar, plus 2 h each in the early forenoon and late afternoon when the device is

*All regnal dates used are from Finegan (1979) Archaeological history of the Ancient Middle East, Dawson, Folkestone.

t It will be seen that this diagram is not to scale, and the text makes clear that the markers begin three units from the upright.

Timing the Sun

Table I, Shadow stick scales and equal hours

159

Hour 5 Hour 4 Hour 3 Hour 2 Hour 1 (a) The early scheme: 0.38 0.77 I .63 3.76

Distance from upright of Hgt=l (3) (6) (13) (30) to nearest integer

(b) The later scheme 0.31 0.68 1.18 2.08 4.69

(3) (6.5) (11) (20) (45)

(4.5) (IO) (30) (68)

(c) The Egyptian scale (3) (9) (18) (30) (45)

inoperative. Neugebauer and Parker (1960) make the plausible suggestion that 1 h of each pair represents twilight and the other the time from sunrise or sunset to the outermost scale marker on the stick.

The Thutmosid and later shadow sticks measured 10 h directly, so that a 12 h day from sunrise to sunset seems indicated. This development was foreshadowed by the invention of a waterclock capable of dividing the night into 12 seasonal hours in the reign of Amunhotep I (1527-l 506 B.C.). Such a clock could operate as well in twilight as in full darkness and it is likely that the hours of the day and of the night were equalised after this date. There is one important continuity between the earlier and later shadow sticks however, for the underlying formula for the spacing of hour markers on the scale is an arithmetical progression of the second order viz.: 3,9, 18, 30 units from the upright, and in later instruments this is simply extended by adding a marker 45 units from the upright.

If we ask how equally such devices divided the day we face the difficulty that we do not know how much any T-bar added to the height of the shadow casting upright. (Mac- naughton, 1944) produced a result by supposing that a T-bar is unnecessary since the stick was kept aligned directly toward the Sun. He concluded that the scale hours would only be close to reality for the summer solstice and thought it had some ceremonial use at that date being otherwise of little utility. His approach is however at variance with the text. Bor- chardt also gave durations on the unlikely assumption that the T-bar added nothing to the height of the device. On this premise the eight scale hours about noon total 10 actual hours and are quite varied in length (Sloley, 1931).

An alternative approach explored here is to follow Neugebauer and Parker’s hypothesis and to suppose that initially the Egyptians aimed to divide the period from sunrise to noon into 5 equal hours. The solar altitudes and azimuths for the first 4 h have been calculated for the equinoxes in Table 1, and the relative scale lengths of the shadow cast by any T-bar can then be compared. These prove to conform poorly with the actual lengths used, and a similar result is found if we test by dividing the period between sunrise and noon into six in accordance with Thutmosid usage.

Bruins (1965) considered this latter case, producing essentially the same results, but with a very different conclusion, as he claimed them “close to the values of the drawing in the cenotaph”. In his presentation the value for the second hour marker instead of the fifth is made to fit the formula and the third and fourth hours come close to the expected values, but the fifth and first hours are far removed from them. Hence we dismiss Bruin’s assertion. Consequently it is not necessary to ask how the Egyptians came by the correct timings to which they then found a close mathematical expression, since their timings are not correct. Perhaps the formula is not empirically based but stems from some false conception of

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celestial geometry and the path of the sun.* Herodotus may have known of the shadow stick, but he would be quite clear that it was

not a source of Greek knowledge of the gnomon, since, unless employed in the manner of Macnaughton’s suggestion, it was indeed not a gnomon but an independent instrument with a more complex theoretical background, albeit one of which the Egyptians may have been largely ignorant. The Egyptians also employed sundials from at least the reign of Seti’s grandson Merneptah (1224-1204 B.C.), and these were hung vertically to catch the shadow of a horizontal gnomon projecting from the centre of the top of the semicircular dial. The shadow would sweep round such a dial more rapidly in the early morning and late afternoon than around midday, but the Egyptians simply divided the dial into 12 1 S’ sectors or “hours”. This is perhaps the crudest order of gnomon use and provides little of either theoretical or empirical interest for the Greeks. One gets a distinct impression that after the 13th Century B.C. the interest in time measurement in Egypt became stultified until the invasion by the Assyrians with its possibilities of cultural interchange in the 7th Century. If true this would likely take the form of preserving the construction rules for instruments, but forgetting the original contribution of empirical checks or any underlying theory.

3. THE BABYLONIAN SHADOW LENGTH TABLE

The evidence currently available to us concerning the use of the gnomon in Mesopotamia is almost totally from a single source, the second tablet of the MUL.APIN series, a com- pendium of astronomical knowledge including inter alia a description of the common path of the Sun, moon, and planets, the systematic observation of the risings and culminations of selected stars, and the periods of visibility of the moon through the seasons. This com- pendium is known in many copies, the oldest from Assur dated to 687 B.C., but bearing the remark “Copy from Babylon”. Some of the stellar material has been analysed by Pingreet and is said to fit an origin around 1000 B.C. This restricts the date of compilation to the period 700-1000 B.C. with the first production of the shadow table itself possibly before the earlier date.

Before evaluating the shadow table we need to see it in the wider context of Mesopotamian time measurement. From at least Old Babylonian times the day began at sunset and was divided into six watches, three for the night and three for the daylight. Thus the duration of a watch varied seasonally with night watches taking or giving time to day watches. Where the Babylonians differed from the Egyptians was that parallel to these seasonal watches they also reckoned time in units which were intended to be seasonally invariant. The two systems are most clearly integrated at the equinox, when night and day watches are equal, and where the conversion is 1 watch=2 beru, the etymology of the latter term being a half (watch). However the beru was also a standard distance (c. 11 km), linked with time via the notion of a standard pace. In our terms the beru was intended to be 2 h and the pace about 3.5 miles per hour. As to its duration this intent was not uniformly reahsed because of a curious failing in Babylonian science whereby the ratio of the longest day to the shortest night or vice versa was reckoned to be 2:1, a difference far in excess of reality. Bremner (1993) has hypothesised that the origin of this error comes from the angular distances “swept” by the sun around the horizon at the two solstices, which are indeed close to such a ratio at these latitudes. Be that as it may, such a ratio once believed gives the beru a variable

*A similar conclusion was reached about older Babylonian conceptions by Toulmin (1967) ‘For a full discussion of dating see Hunger and Pingree (1989).

Timing the Sun 161

time span, not in the same sense as the seasonal watches but to a quite opposite effect whereby the midwinter night or midsummer day have more beru than they should, each with shorter durations than their equinoctial equivalents, the converse holding for winter days and summer nights. All this apparently, without the Babylonians being aware of this variation.

A curious coincidence helped to keep the Babylonians in thrall to this error. From some- time in the second millennium at latest they used water clocks to time their watches, equat- ing time with weight of water discharged. If Bremner is correct to think the erroneous seasonal ratio already in existence, then a challenge to it might have been expected. How- ever new instruments need calibration and just as in Egypt where an erroneous seasonal ratio was designed into the first waterclock on the authority of a previous measure, so too in Babylon. Thus the weights of water put in the water clocks on a midsummer day or midwinter night were double those added on midwinter days and midsummer nights. To operate in this way and yet to conform with nature insofar that the emptying times are congruent at least roughly with the actual durations of night and day, might be thought a difficult problem of clock design. Yet it proved deceptively simple, as a uniform cross sec- tion, be it cylindrical or prismatic, plus an outlet that is a sharp edged hole rather than a long spout, is all that is required. As Neugebauer (1947) was the first to realise such a de- vice discharges in proportion to the square root of the head of water, so that a head ratio of 2: 1 leads to a time ratio of 1.414: 1, close to the true seasonal swing in these latitudes. A set of mathematical problems of Old Babylonian date (Thureau-Dangin, 1932) illustrates a conclusion stemming from this, that it was believed that the discharge rate was indepen- dent of the head. It was not to be the last time in the history of science that astronomical concerns were bedevilled by false physics.

Eventually the Babylonians corrected the false ratio, substituting a value of 3:2. As this is still a trifle too large, its use by Ptolemy a millennium later to determine the latitude of Babylon led to a northward skew for the Middle East on his famous world map, but as a simple ratio it is much nearer the mark than 2: 1. It is here that we return to a consideration of the gnomon table, for in it we discover just such a 3:2 ratio. This has led Assyriologists to see the table as a proof that the better ratio for the seasons was already known at the time of compilation of MUL.APIN. It even led some to hope that a full understanding might show that the sundial was the engine producing the new ratio .* However the presentation proved sparse in detail. Entries have the form “2 cubits of shadow 1 beru 7 USH 30 GAR”, where the conversions are 60 GAR= 1 USH, 30 USH= 1 beru, 12 beru=24 h. Altogether timings for shadows l-3 cubits long are given for each equinox, and for shadows of 1-6 and 8-10 cubits for each solstice, these year points being placed on the 15th day of months 1,4,7 and 10 in a schematic lunar calendar of 360 days. The timings presented are clearly calculated rather than observed, conforming exactly to the law that the product of the shadow length in cubits and its timing in beru is a constant on any 1 day, having the value three for the winter solstice, 2.5 for the equinoxes, and two for the summer solstice. The absence of timings for 7 cubit shadows reflects the impossibility of exact expression in the given units.

*Neugebauer (1941). Neugebauer says here that “the methods of determining time and latitude from sundials are well known from Greek and Roman sources. Their existence in Babylon has been proved by E. F. Weidner” In his Isis paper (op. cit. 38) he further remarks “it (the shadow table) is based on the assumption of a ratio of 3:2 for the lengths of the longest and the shortest daylight”. More recently Al- Rawi and George (op. cit. 59) repeat “the ratio 3:2 computed for longest to shortest day from data given in MUL.APIN’s gnomon table”. The words here italicised have no warrant in the source document.

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The bareness of the tabulation leaves some matters open to surmise. It is generally agreed that the shadows are cast by a vertical gnomon 1 cubit in height upon a horizontal surface. Weidner (1924) supposed the times to be counted from 6 a.m., but all later commentators assume sunrise, however defined and calculated. The timings are thought to be by water clock, as weights of water per day are included in the text. The consensus has been that the time units are seasonally invariant, despite the range of these weights being from 4 to 2 minas. The Babylonians may well have considered 4 to 2 as a time ratio in this case, as they did in all other sections of the MUL.APIN, (Hunger and Pingree, 1989). This would lead us to expect a longer duration for the beru in winter than in summer.

One line of the table claims a shadow of 1 cubit on the winter solstice, 3 beru after sunrise. This is impossible on the assumptions above, for the time indicated would be 1 p.m. if uniform beru are meant, and mid afternoon with longer winter measure, while the shadow would be longer than this even at noon. This led Van der Waerden (1951) to dismiss the winter timings as very bad though he described the summer ones as quite good approximations. Bremner (1993) overcame the difficulty of a solar altitude of 45 in winter by supposing the record a fabrication “added for reasons of symmetry and to show the value of the constant for that solstice”. He concluded from his own calculations that his assumptions, viz. a 1 cubit gnomon, a latitude of 32.5N (Babylon), and an ecliptic obliquity of 23.8 (700-1000 B.C.), cannot be far wrong. These calculations show an interesting feature. The times he infers were used for sunrise require a solar altitude at the summer solstice two degrees higher than for other seasons, such as might obtain with a mountainous northeastern horizon. As this does not apply to Babylon it lead to the tentative suggestion of an Assyrian locale. We shall see that other explanations are possible.

4. A NEW CALCULATION

The calculations that follow define sunrise as first gleam and the location as Babylon. The extremely flat easterly aspect from this city suggests an artificially elevated site to avoid obstructed views, and this is assumed to be the rooftop of the Temple of Heaven. Bernheimer (1988) gives the height of the ziggurat in Babylon as 270 feet. Observation from the roof edge to the horizon has an angle of dip of 16 min. Hence apparent sunrise of the upper rim will occur when the centre of the Sun is 1“ 6’ below the ground level horizon.* The employment of a limited roof area may explain the otherwise surprising shortness of the gnomon (1 KUS (cubit)=20.9 in. or 531 mm).

In Table 2, three different timepieces are envisaged. Section A employs a simple cylindrical outflow clock with the initial head varied by a factor of two between the solstices and interpolated linearly. Section B uses an inflow clock supplied by a constant head device, but with that head still varied seasonally as above. Section C keeps the head in the supply vessel the same throughout the year. All these clocks are to be read so that unit rise or fall of water connotes the passing of unit time.

It is at once apparent why all previous reports have assumed the equivalent of Section C. Yet even here the agreement of the reconstructed measure with that claimed by the Babylonian law is distinctly patchy-it is close for all shadow lengths in winter and for the longer shadows at the equinox, but deteriorates for the three shortest shadows, just those listed in MUL.APIN. In summer there is almost total disagreement except for the 1 cubit shadow. van der Waerden’s more positive description (Van der Waerden, 1951) probably

*This is an estimate using average values for atmospheric refraction, a highly variable phenomenon

Timing rhe Sun

Table 2. Values for Beru since sunrise X Cubits of shadow

A B C

163

Sh Su Eq Wi Su Eq Wi Su Eq Wi 10 6.1 4.8 4.5 3.3 2.6 2.4 3.0 2 6 3.0 9 5.9 4.1 4.4 3.2 2.6 2.4 2.9 2.6 3.0 8 5.9 4.6 4.2 3 2 2.6 2.4 2.9 2 6 3.0 I 5.1 4.5 4.1 3.1 2.5 2.4 2.8 2.5 3.0 6 5.5 4.4 4.0 3.1 2.5 2.4 2.8 2 5 2.9 5 5.4 4.3 3.9 3.0 2.5 2.3 2.1 2s 2.9 4 5.0 4.0 3.6 2.9 2.4 2.3 2.6 2.4 2.9 3 4.6 3.7 3.3 2.8 2.3 2.4 2.5 23 2.9 2 3.9 3.2 2.6 2.6 2.2 2.4 2.3 2 2 3.0 1 2.5 2.0 - 2.1 1.9 - 1.9 1.9 -

arises from using a rounded value of 5 a.m. for sunrise.] If we wish to present Table 1, Section C, as the observational base from which the Babylonians induced their simple law, we need to know how they could measure in standard time while seeming to conserve older exaggerated ideas of the seasonal swing, and on what basis they selected observations which allowed such a partial summary.

5. THE CALENDAR CONNECTION

Both before and after the compilation of MUL.APIN the Babylonians described time keeping in terms of weights. The weight of water put into cylindrical vessels and allowed to empty as a measure of a watch, was varied month by month or even day by day in a linear fashion between the solstices. The Babylonians kept strict lunar months, 12 of which made a normal calendar year, setting the calendar adrift against the solar year. The effect on time keeping was to make the calendar seasons with their assigned weights of water progressively out of phase with the actual lengthening and shortening of days and nights. Over any 24 h the scheme is self correcting since overly long night watches would be followed by overly short day watches or vice versa. The Babylonian day began at sunset, which probably meant that watches and sunset were kept synchronised. This results in any departure from the natural checks due to calendar drift being loaded onto sunrise. Over the 3 yr or so before intercalation the last watch of the night would progressively extend beyond sunrise on autumn and winter mornings, but would end before sunrise in spring and summer.

When to add an extra month to the year, was a problem that long exercised the Baby- lonians. From the use of erratic weather indications in the Old Babylonian period (c. 1800 B.C.) the matter was finally systematised in cyclic formulae in the Persian period (c. 500 B.C.). Between these times various schemes using astronomical observations were mooted, including two using stellar risings in the MUL.APIN itself. The disjunction of the official and actual ending of the night described above provided another possibility, which was outlined in a prescient article by Smith (1969), subsequently supported by the publication of a text, dated to the 7th Century B.C., in which an official writes to the king of such a method, said to have been long preserved in the oral tradition (Pingree and Reiner, 1977). In essence when the degree of departure each morning reaches 20 USH the calendar is 1 month behind the solar year. In practice it will have been less clear-cut because of various false assumptions such as the linearity of the change in daylight, and the belief that a lin- ear change in weight leads to a linear change in the length of the watch. Moreover in this period the priests proposed and kings disposed, so that the ground was probably prepared by measuring and reporting the gradual increase in the departure.

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All this calls for an instrument capable of measuring reliably periods of time possibly as short as 1 USH (4 min). An initial calibration would be necessary and here the fact that the time units doubled as measures of distance is highly suggestive. The two usages are linked by the notion of a constant pace. Providing the distance is short this seems capable of considerable consistency, recent research suggesting that we carry the equivalent of a metronome in our heads (Meek, 1966). It is therefore proposed that a device akin to a crude egg timer was calibrated against a carefully paced distance of l/30 beru. (The term USH, which is given to the corresponding time interval, means “length”.) Such a timer will have the following characteristics: it gives constant measure in all seasons, and it is sufficiently laborious to use for extended periods as to deter its employment as a check on total daylength, especially when this is not at issue.

On spring and summer nights the priests timed the gap between watch end and sunrise, and on autumn and winter mornings between sunrise and watch end. In these latter cases they did so to the accompaniment of shortening shadows. Hypothetically the following ensued. First correlation was made between shadow lengths and times elapsed up to about 80 min after sunrise for autumn and winter mornings. This discovered a family of inverse linear relationships between shadow length and time after sunrise for shadows down to 4 or 5 cubits which provided an easier way to time watch discrepancies thereafter. At some later date an individual of more philosophical bent wished to extend the shadow clock formula to cover shorter shadows and all seasons. He may have simply extrapolated the two trends already discovered, or he may have asked the shadow lengths at known times, as these were believed. Thus thinking the midwinter morning to be 2 beru in duration his formula suggested a noonday north pointing shadow of 1.5 cubits, confirmed by enquiry (actual value of 1.48.) At midsummer a like prediction is that 4 beru after sunrise the shadow should be 0.5 cubits, but it is only a third of that. But this is well outside the range of the MUL.APIN table, and he may have preferred to ask the shadow at the end of the first watch, believed to be 2 beru 40 USH after sunrise with a predicted shadow of 3/4 cubits. We can calculate the time as 9.32 and the shadow of the gnomon as 2/3 cubits. He may have settled for the answer received and looked no further, ignoring contraindications.

To anyone unacquainted with the field the generalising claimed to produce the final table may seem overbold. It is not untypical however. In a recent commentary on K90, a tablet describing the durations of lunar appearances through a winter month, Al-Rawi and George (199 1) observe that the scribal calculations are at odds with plainly observable fact and add “that he could do this suggests an interest in mathematics and theory rather than practical observation”.

6. THE SEASONAL SWING CLAIMED IN MUL.APIN

This account of the matter holds that two instruments coexisted and their results were thought compatible, though in truth the one measured constant beru and the other had unsuspected seasonal variation in this time unit.

The fact that the shadow table shows some evidence of measure in seasonally constant units has, heretofore, been the most cogent reason for believing the better 3:2 ratio of day and nightlength, known from later Babylonian documents, to be already demonstrated in MUL.APIN. If the force of this argument is reduced, there is little further to support that notion other than a mere coincidence of ratios, which is without evidential value.

Timing the Sun 165

Neugebauer long felt otherwise. In the Isis paper cited he refers to the shadow table as being based on a 3:2 seasonal daylength ratio. On closer examination he realised that from his perspective the constants 3 and 2 are applied to the wrong seasons, and he proposed a scribal transposition (Neugebauer, 1975). Following this train of thought we can say that the shadow table supposes the elapsed time from sunrise to any solar altitude to have a seasonal ratio of 3:2, whilst later Babylonian science held that the elapsed time from sunrise to noon had the same seasonal ratio. The two statements can only be linked if one accepts a constant solar altitude at noon, and since no one supposes the Babylonians believed that, perhaps the scribe was correct in his seasonal attribution after all.

There is in the shadow table itself a summary section that may hold the clue as to which seasonal swing ratio was held at this time. It reads as follows, using sexagesimal arithmetic.. “If you are to find the difference for 1 cubit of shadow, you multiply 40, the difference for daytime and night-time, by 7,30, and you find 5, the difference for 1 cubit of shadow”. (Tablet II ii 4142).

We can interpret the numbers 40 and 5 as rates of change expressed directly in time units. * The difference for daytime and night-time is the daily increment of the seasonal swing according to the schematic calendar. A value of 40 GAR equates with the older idea that the swing in day and night length is 4 beru in 6 months. The difference for 1 cubit of shadow refers to the time from sunrise until such a shadow length is attained. Its daily increment or decrement is only 10 GAR or 1 beru in 6 months, a quarter of the seasonal swing rate. The multiplier of 7,30 (7.5 decimal) applied to the 40 GAR can be partitioned into x0.25 to allow for this, and x30, which converts the result to the rate of change per schematic month. The product of 5 USH is in perfect agreement with the shadow table. In other words lines 41-42 are an instruction for interpolating the shadow time tables for months other than the solstices and equinoxes. This is done, simply and exactly, by applying a linear zigzag function based on a 2:1 seasonal ratio of day and night lengths. Neither the shadow table, nor any other section of the MUL.APIN, need be read as witnessing to the knowledge of a better value than this.

7. SOME WIDER IMPLICATIONS

If the Greeks learnt about the gnomon from the MUL.APIN they would be introduced to it in a theoretical rather than an empirical context. Moreover they would rapidly discover the need to reinvestigate the matter in other latitudes and would thus be stimulated to improve the theory, thus justifying Herodotus’ description of the matter. The transmission is likely to have occurred in the 6th Century B.C. It cannot be later because of the report that Anaximander (floreat 560-570 B.C.) raised a gnomon in Sparta, but it cannot have been much earlier because of the bracketing of the gnomon with the 12 hours of the day. While the Egyptians long used such a division, its employment in Babylon is only attested from the 7th Century B.C., and may well stem from increased contact with the Egyptians as the Assyrian empire expanded. One is left to ponder which of the seasonal swing ratios

*Hunger and Pingree op. cit. pp. 153-154, prefer to interpret 40 as the change in the weight of water+O;0,40 minas-to be put in waterclocks each day or night. This is to be multiplied by 0;7,30 (the equivalent of division by 8) to produce O;O,S minas as the difference for 1 cubit of shadow. They take this to mean that the shadow decreases by 1 cubit for each eighth of the watch. This last step can be criticised on many counts, not least because the table itself shows that the decrease is far from linear, and in any event is not related to watches.

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for Babylon was transmitted to the Greeks at this time, and whether the shadow formula had by then been improved upon in its country of origin.

If the 3 to 2 seasonal swing ratio is not attested in MUL.APIN then its earliest known appearance is in the series i.NAM.gis.hur.an.ki.a. (Livingstone, 1986). Lines 25-29 contain unequivocal statements that 1 2/3 longest days is a day of 24 h, and 600 longest days is a year, though the total passage is far from fully understood. These lines are embedded, apparently unconformably, in material concerning the moon. This might indicate a late addition to the main work, but it cannot be later than the reign of Sennacherib (704-681 B.C.). On the other hand the persistence of a continued belief in the 2 to 1 ratio throughout the 7th Century B.C., not merely in weights of water but in durations, is shown by tablets K.2077 and BE. 13918. The latter tabulates daylight and darkness as sixtieths of the total 24 h day. Duzu 15th (the schematic date of the summer solstice) records 20 night, 40 daylight. Van der Waerden* considers this document to date from the time of the Chaldean kings and thus post 626 B.C. The former contains the statement in lines l-2 obverse: “(When the Sun) stands in Cancer and the Moon stands in Capricorn, then the day is 8 beru, the night 4 beru”. It is dated as around 650 B.C.

The 7th Century B.C. in Babylonia saw not only new ideas on the seasonal swing but new types of clock. Stephenson and Fatoohi (1994) have argued that the near constancy of the value of time units across the seasons that their analysis of astronomical diaries from 650 B.C. has demonstrated, suggests the use of a constant head inflow clock. Walker (Private communication) has published a table-BM 2937 1, which he interprets as describing the weights of water to be put into a waterclock every fifth night of the schematic 360 day year. The annual regime changes linearly between midwinter and midsummer extremes with respective weights in the ratio of 3 to 2. Presumably a 3 to 2 time ratio is intended, so that the cylindrical watchclocks previously discussed cannot be involved, and these weights might be from an outflow clock of Egyptian style (a truncated cone) or from a constant head inflow clock, the former seeming more likely as this gives the table a function. In addition every line of Walker’s table ends with the statement-“one cubit shadow, 1 2/3 beru day”. As he remarks, it makes no sense to quote this value for all times of the year and the scribe must have copied the phrase from his source and unthinkingly repeated it. There is however another possibility, for as Table 1 column C shows, a value not so different to this actually obtains at both the summer solstice and the equinoxes. Perhaps the original theorist rather than the scribe generalised from his values for the same three times. We can even go further in reconstructing this new gnomon table (for new it must be as the value cited is not found in MUL.APIN) as there is an archaic formula embedded amongst more advanced analysis in a 5th Century A.D. Indian treatise and considered to be Babylonian in origin (Neugebauer and Pingree, 1971). This gives S - b = a/T where S =shadow length, T =time since sunrise, and b and a are constants of unstated value. Such a formula is capable of a closer approximation to reality than the S = a/T of MUL.APIN where u is constant in any one season and takes values from 2 to 3.

We see therefore that in the 7th Century B.C. the Babylonians were engaged in a debate about all aspects of time measurement, with rival schools of thought. This chimes in with the contentious spirit of early Greek science in the next century. Perhaps the Babylonians

*Van der Waerden, B. Wissenschaft. Veroff. d. Deutsche Orient-Gesell. No. 4. It is his general practice to use the ambiguous term Neo-Babylonian, as here, as a synonym for Chaldean. The tablet concerned is published by Weissbach, E. (1903) Eubylonische Miscellen, pp. 50-51, Leipzig. Associated finds were Chaldean or early Persian.

Timing the Sun 167

had first acquired this greater openness through increased contact with Egyptian ideas when both they and the Egyptians came under fitful Assyrian hegemony from 670 B.C. The Greeks were to improve on the degree of autonomy allowed to scientific enquiry but in this, as in its substance, they did so by standing on the shoulders of giants.

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