The Water Clock in Mesopotamia
Author(s): David Brown, John Fermor and Christopher Walker
Source: Archiv für Orientforschung , 1999/2000, Bd. 46/47 (1999/2000), pp. 130-148
Published by: Archiv für Orientforschung (AfO)/Institut für Orientalistik
Stable URL: https://www.jstor.org/stable/41668444
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The Water Clock in Mesopotamia
By David Brown* (Oxford), John Fermor (Glasgow),
and Christopher Walker (London)
This paper discusses the evidence pertaining to water clocks in Mesopotamia, revealing the serious flaws
that exist in our current understanding of the devices and attempting to remedy this, while recognising the
limitations inherent in the exercise. We possess no recognised examples, however fragmentary, from ancient
Mesopotamia of outflowing water clocks. Any reconstruction of them relies on textual evidence and what is
known to be both physically possible and impossible. J. Fermor undertook the experimental work. This paper
also presents BM 29371, which was edited by C. B. F. Walker and published in a photograph in Astronomy
before the Telescope (1996, ed. C. B. F. Walker) p. 47. BM 29371 describes weights, times and the lengths
of shadows on various days through the year and was inscribed during the Late Babylonian period.
the ratio may have owed more to notions of symmetry
Contents
1. Timing in a Divinatory Context
1.1 The Ratio of the Longest to the Shortest Day
1.2 Names of the Device
1.3 The OB Mathematical dibdibbu Texts
1.4 The Neugebauer Clock and its Problems
1.5 A New Model for the Mesopotamian Water
Clock
1.6 Ideals as Opposed to Empirical Reality
1.7 The Mašqú - Apparent Precision versus Accuracy
2. Timing in an Astronomical Context
2.1 A 3:2 Ratio
2.2 The Accurate Measurement of Time after the
mid- 8th Century BC
3. BM 29371 (98-11-14, 4)
or numerical simplicity than it did to observation.
The earliest known explicit attestation of the ratio
is in an OB text (BM 17175+) copied by C. B. F.
Walker and published in an appendix by H. Hunger
and D. Pingree in their MuLApin - An Astronomical
Compendium in Cuneiform = AfO Beih. 24 (1989) p.
163. It is there interpreted as describing weights of
water in a clepsydra corresponding to watches of the
night and day at the solstices and equinoxes. This is
a reconstruction, for the values are without units and
water is nowhere mentioned. BM 17175+ assigns the
values 2, 3, 4 & 3 to the watches of the night on the
15th of months III, VI, IX and XII respectively. It
does, therefore, describe the ratio of 2:1, also a year
of 12 months and evenly distributed equinoxes and
solstices with the vernal equinox taking place on the
15th of month XII. Amongst several examples from
the OB period, IM 80213 (and copy IM 80214)
published by L. de Meyer in Zikir šumim. Assyrio-
1. Timing in a Divinatory Context
logical Studies presented to F. R. Kraus on the
occasion of his 70th Birthday, ed. van Driel et al.
(1982), pp. 271-8, describes the year as being 360
1.1 The Ratio of the Longest to the Shortest Day days long2 and undoubtedly this is also meant in BM
A qualitative feeling that winter days are shorter solar azimuth at the two solstices. This would imply the use
and nights longer than in summer has a history of some device to measure or compare angles - perhaps a
gnomon - but we have no evidence for such devices as early
perhaps as long as humanity has had memory sufficient
for the comparison. There is, however, no simpleas the OB period. Indeed there is no evidence of any Meso-
potamian attempt to measure angles in an astronomical context
route to the correct ratio of the longest to the shortest
before the 7th century BC, and even then the units used
day. The 2: 1 ratio, long held to apply in Mesopotamia,describe fractions of a circumference rather than angles subonly poorly corresponds to reality in that part of the
tended from a centre. To our minds Bremner's ideas are
world. Empirical origins have been suggested1, butanachronistic. O. Neugebauer ("The Water Clock in Ancient
Astronomy," Isis 37 (1947) pp. 37-43) p. 39 first suggested
*) David Brown's work for this paper was made possiblethat the ratios in times may have derived from the corresponding
ratios in weights of water pouring from a cylindrical outflow
by the award of a postdoctoral fellowship of the British
Academy. I would like to thank H. Hunger for his help on clock. This idea has gained popularity over the years and is
discussed at length below.
§ 2.1.
2) R. Englund in "Administrative Timekeeping in Ancient
') R. W. Bremner ("The shadow length table in Mul.Apin,"
Mesopotamia," JESHO 31 (1988) pp. 121-85 suggests that
in Die Rolle der Astronomie in den Kulturen Mesopotamiens
the 360-day year may have had its origins in administrative
(1993), ed. H. D. Galter, pp. 367-82) p. 370 speculates that
practices and may be much older than the OB period. See
it may have arisen from a comparison of the daily range of
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The Water Clock in Mesopotamia 131
iii 15) this is
explicit
171 75+. The ideal year there described
ismade
made
upwith the line: 40 nindan
of 360 days, with 12 30-day months.
The
months
are
riappalti
ümi
u múši "40
nindan, the difference for
daytime
time" which closely mirrors the
also ideal, since on average about half
ofand
allnight
lunations
description
in the of
OB coefficient
lists. 40 nindan
are not 30 days in length. The even
spacing
the
times 180,
the number
days between solstices in
equinoxes and solstices is again ideal,
since
they of
are
not so spaced in reality due to refraction.
the ideal year (which is always used in Mul.Apin),
Further evidence of this 2:1 ratio in OB times
comes from the OB coefficient lists3. There we find
is 7200 nindan, which is 120 or 2,0; UŠ7, which
would be written "2." "2" is, thus, the implied
three unitless coefficients that pertain to the visibility difference in the length of daylight between the
of the moon. One of them, "40," is described as the summer and winter solstices. Since one nychthemeron
nappaltum "the decrease" from day to night. Because lasts 360 UŠ (n. 7), it follows that the coefficient 40
of the absence of a marker for zero in this notation, nindan indicates that at the summer solstice the
the coefficient "40" could refer to 40,0; that is 60 daylight is supposed to last 240 UŠ (written "4" for
times 40, or to 0;40, that is 60"1 times 40 and so on.
4,0;) and at winter 120 UŠ (written "2" for 2,0;), and
Two possible meanings of this coefficient can be
so forth. These are, of course, the numbers we see
in BM 171 75+, and imply a longest to shortest day
discerned from later sources, both of which imply a
ratio of 2:1 in time . A third reference to the coefficient
ratio of 2:1 for the longest to the shortest day:
In the text known by its incipit as Mul.Apin, "40" in Mul.Apin II iii 41 does not specify the units.
attested only in NA and NB copies but composed Despite this Pingree writes op. cit. p. 151 that the
much earlier4, a coefficient "40" is attested in II i 11- "40 nindan" mentioned in this part of Mul.Apin
12 (and repeated in II i 17-18), where the sun is said refer to units of weight, namely to 0;0,40 minas8.
to move south after the summer solstice at 40
Why?
nindan per day. Nindan, Akkadian nindãnu or
Firstly, when values are given corresponding to
"rod," is a unit of length and, at least by the first the lengths of the "watches of the day and night," in
Mul.Apin they are given in terms of weights in
Instead the 40 nindan refer to the daily changing minas and shekels (I ii 42 - iii 7, II i 9-21, II ii 21
length of daylight (or night). Later in Mul.Apin (II - iii 12). The maximum day length is always given
as "4" and the minimum as "2," figures and a ratio
that
again correspond with those in BM 17175+. A
also Archaic Bookkeeping: Writing and Techniques of Economic Administration in the Ancient Near East (1993), ed. daily change in minas corresponding to a change in
Nissen et al ., p. 28 for the proto-cuneiform numerical sign day length between solstices of 2 minas would be
millennium BC, of time5. Length is not meant here6.
systems including one for describing what may be 12 30-day
months in a year.
3) See now E. Robson (1999) Mesopotamian Mathematics
2100-1600 BC : Technical Constants in Education and Bu-
2/i8o or 0;0,40 minas, which would indeed be written
"40." It seems plausible at first sight to interpret the
coefficient "40" in Mul.Apin, and also in the OB
coefficient lists, as meaning 0;0,40 minas, but since
reaucracy - OECT 14, Ch. 8.2.
on
4) Elements of the text may go as far back as the OB two out of three occasions the units are specified
period, cf. A. George, ZA 81 (1991) pp. 301-6, especially p. as nindan we must think again. As we will explain
304. The series was perhaps brought into its final form at the
below the values pertaining to day and night lengths
end of the second millennium BC, as proposed by Hunger have often been interpreted as weights of water
and Pingree, op. cit .
5) M. Powell RIA "Masse und Gewichte" p. 463f.
6) The description in Mul.Apin II i 11-12 might appear to 7) UŠ are firstly units of length, with one measuring about
imply that what is being referred to is the changing orientation
360 metres. They become units of time at least by the second
of the rising sun, that is a change in azimuth (the angle from
millennium BC, with 360 UŠ corresponding more or less to
North) of the rising sun. If a length corresponding to an arc
the time between successive sunsets. 1 UŠ is thus, fairly
were meant, then 40 nindan per day would correspond to an
accurately, both 4 minutes and Io of right ascension. During
arc of 2h° per day (see n. 7, 1 UŠ = 60 nindan). 40 nindan the final seven centuries BC, 1 UŠ came to mean Io of
per day would mean that over the 180 days between solstices
celestial arc more generally. O. Neugebauer referred to the
an arc of 120° would be swept out by the rising sun on the
UŠ as a time-degree, and we shall continue this practice. See
eastern horizon. This is vastly more than the maximal change
D. Brown, "The Cuneiform Conception of Celestial Space
in azimuth of the rising sun from mid-summer to mid-winter
and Time," Cambridge Archaeological Journal (CA J) forthat the latitude of Mesopotamia, where at Babylon, say, it coming,
is
for details.
about 56°. However, if the 40 nindan referred to an actual
8) Although a theoretical relationship between nindan and
length (of 40 times 6 metres, the approximate length of 1
units of weight does exist (see Powell, op. cit. p. 509 and
nindan) along the eastern horizon corresponding to the daily
table XVI), it is of the form 0;0,0,1 nindan = 2 minas, and
change in the position of the rising sun, this would imply thenot of the form 0;0,40 minas = 40 nindan. Pingree is wrong
existence of a sighting apparatus of gargantuan proportions
to equate these saying op. cit. p. 153: "0;0,40 minas, which
and so must be excluded.
were called 40 nindan in II i 12."
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132 David Brown - John Fermor - Christopher Walker
flowing from a small outflowing clepsydra of constant 1.2 Names of the Device
cross section, which has also meant that the woefully
inaccurate ratio of 2:1 for the longest: shortest day
could be accounted for. However, although Mul.Apin
specifies weights when describing day and night
lengths, the fact is that it gives the daily change in
day length in terms of time units, which also means
that the weight and time units were thought of as
being in direct proportion. Projecting back to BM
17175+, this would mean that the figures "2," "3,"
and "4" there describe periods of time of 2,0 UŠ, 3,0
UŠ, and 4,0 UŠ whether or not 2, 3 or 4 minas9 are
actually being referred to, and that the coefficient
"40" described 40 nindan of time, even if this were
to be measured by 0;0,40 minas in a water clock.
This tells us a great deal about the type of water
clock that was used.
In the OB coefficient lists we also find the
statements "12 maltaktum" and "12 maltaktum ša
mušitim ," "12 (is the coefficient of) the maltaktum "
and "12 (is the coefficient of) the maltaktum of the
night." Maltaktum is a word that has been interpreted
to mean "water clock." As we will explain below,
the coefficient "12" denotes an interval of time
which suggests strongly that the device to which it
pertains did indeed measure time. Also, we know
from the lexical text OB Lu A 171 10 that ša maltaktim
"he of the maltaktum" is the equivalent of Sumerian
lú.a.lá "the one who weighs water," and in the OB
epic Atra-hasïs III i 36-37 we find the lines ipte
maltakta Suãti umalli (37) ba-a-a1 abübi 7 můšišu
iqbišu; "He opened the maltaktum and filled it. (37)
Further evidence to support this perceived direct
He announced to him the coming of the flood for the
proportion between time and weight will be adduced
7th night."11 This again implies that the device was
from other texts below, but more that backs it used
up to measure time.
exists in Mul.Apin itself. In II ii 43 - iii 15, not only Maltaktum is probably the instrumental, nominal
are night lengths given in minas and shekels, but the
form maprast from latãkum "to test, check (of
times between sunset and moonset at the beginning
instruments)" meaning "tested/testing instrument." It
of each month and between sunset and moonrise in
seems likely to have involved the medium of water,
the middle of each month are given in time units.
though perhaps not exclusively. Von Soden in AHw
These times are constructed mathematically on the
596 maltaktu 3, considers that it might mean a sand
basis that the period between sunset and moonset on
clock, based on his reading of line III i 37 of Atrathe 1st of any month is Vis"1 of the total night length
hasïs (see n. 11). Either way, the Atra-hasïs quote
since (in the presupposed ideal system) on the 15th
suggests that the maltaktum required filling in some
of the month the moon is visible all night. The other
way.
lengths (expressed in UŠ and nindan) are thus in In the first millennium SB lexical series Ur5-ra IV
direct proportion to the values for the night lengths
6-10 maštak/qtum is one of the equivalents to gBdibdib12, in Akkadian dibdibbu - words with a possible
(expressed in minas etc.) The minas expressing the
night lengths, and so the day lengths, were thought,
onomatopoeic origin13. CAD M/l pp. 171, 392 &
in this instance at least, to describe lengths of time393 derives maštaktum from maltaktum and only
in direct proportion to their magnitude. Pingree
accepts an SB reading for maštaqtu "a cutting off
writes op. cit. p. 154: "Clearly some Babylonian
astronomers simply took the ratio of weights of
l0) See CAD M/l, p. 172, but see George and Al-Rawi
water, 2:1, which had been used since the Old
Babylonian period at least, to be the ratio of times."
The fact is, there is no evidence to suppose that that
ratio of times was ever thought to be anything other
than 2:1. It is only a construct on the part of modern
scholars wishing to find an ancient sympathy to
contemporary notions of accuracy to presume
that the 2:1 ratio in the texts hid a more
(AfO 38/9, 1991/2) n. 15 for the correct reading.
") After Lambert-Millard Atra-hasïs. They take ba-a-a '
from bâ'u "to come forth." Von Soden reads ba-a-as' from
baffu "sand" and reads line III i 37 "den Sand für die sieben
Nächte der Flut (einzufüllen) trug er ihm auf." See idem in
(TUAT III/2 1994) p. 638.
u) Ur5-ra = frubullu IV 6ff.; MSL V pl. 51 f.
giš.dib-dib ŠU-bu dibdibbu
giš.dib-dib mu-zib-bu "that which drips" < zâbu
accurate ratio (of say V2:l, see below) in
giš.dib-dib mu-ši-ib-bu "that which grows/waxes" < šáhu
reality. Mul.Apin is explicit. The 2:1 ratio giš.dib-dib
is in
mu-kan-zib-tum see CAD D, p. 134
terms of time, whether the units are UŠ
or
giš.dib-dib
maš-tak-tum "tested measure" see CAD M/l, p. 171
minas, and BM 17175+ and the timing giš.ki.lá
device maš-tak-tum see discussion in main text.
it implies should be interpreted accordingly. Text cited in Al-Rawi and George (1991/2) n. 15.
") Also suggested in N. Veldhuis, Elementary Education
in Nippur
- The lists of trees and wooden objects Diss.
9) We now know that the length of the "watch of
the
Groningen (1997) Ch. 5 note to line 144 of
night" in this context applies to the whole night andRijksuniversiteit
not to a
thenight
OB Nippur and south Mesopotamian "giš list" which
third. It is also clear that the length of the equinoctial
giš.dib-dib, with variants giš.dab5-dab5 and giš.dubwas measured by 3 minas, and not by 3,0 or 0;3 reads:
etc. See
below.
dub, all of which could be renderings of dripping water.
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The Water Clock in Mesopotamia 133
and takes
as before,
both meanings should be understood
(derived from ãatãqu)" AHw p. 630
mašlltaqtu
to be a water clock and maltaktutoto
a sand
clock,
be be
referring
to a period
of time. Firstly, in EAE
as mentioned. Robson (1999 § 8.2.3) argues that
14 table A15, 12 UŠ is the daily change in time the
maltaktum and maštaqtum derive from latãku and
moon remains visible after sunset in an "equinoctial"
satãqu respectively and that both refer to an "obsermonth, by which we mean a month, the middle of
vational device which functions by separating out its
which occurs on the equinox. "Equinoctial month" is
contents uniformly over time." However, there seems
an anachronism, and probably should be thought of
to be no evidence for an OB maštak/qtu that has
as referring simply to the months containing the
anything to do with a gBdib-dib14. All the known
equinoxes. The term suffices for our purposes, howreferences can be explained as later hypercorrections
ever. As in the model used in Mul.Apin, on the 15th
of a month the moon was considered to remain
for maltaktu, as proposed by CAD M/l p. 171. This
theory better explains the variants and also avoids
visible all night, so on the 7lh of the month, say, it
having to understand a dibdibbu as a "cutting off
would be visible for Vis"1 of the night and so forth.
(device)."
Since the increment between nights is there regarded
A giSdib-dib is known from OB mathematical
as 12 UŠ, then 12 times 15 UŠ will be the corre-
texts (see below) to be a device, probably of constant sponding measure of the length of the entire equicross-section, out of which water, probably, poured. noctial night. 12 times 15 is 180 UŠ, or 3,0 UŠ16.
In Ur5-ra IV: 7 one of the equivalents to gBdib-dibThis matches the value "3" found in the OB text BM
is muzibbu (note 12), which perhaps refers to the 17175+ for night lengths of the entire "equinoctial"
conduit leading into or out of the device, and if it month. Alternatively, in table B of EAE 1417, "12"
derives from the D participle of zâbu, means some-represents the daily change in time the moon remains
thing like "the oozer." This suggests that the flow visible after sunset in an equinoctial month, this time
rate into or out from the device was very slow.expressed in terms of weight in units of shekels. It
Another of the equivalents in Ur5-ra IV: 8 is mušihhu was F. X. Kugler18 who first suggested that the
(n. 12) "that which grows/lengthens/waxes" which weights in question in this text might refer to the
perhaps indicates that part of the dibdibbu includesweights of a substance within a water clock. If this
something which increased in size, which suggests is indeed the case, then EAE table B makes clear
that part of the device may have involved thethat 12 times 15 shekels of this substance, which
flowing in, rather than out, of a substance. This has equal 180 shekels or 3 minas19, corresponds to the
to be reconciled with the Atra-frasïs quote and what length of the equinoctial night - that is to 12 hours
can be gleaned from the OB mathematical texts and to that figure in BM 1 7 1 75+. The coefficient "of
(below). In Ur5-ra IV: 11 (n. 12) maštaktum is the maltaktum" is thus either 12 shekels, or 12 UŠ,
equated with gBki.lá, a word which is loaned into and whether or not this refers to a weight it means
Akkadian as kalakku, sometimes meaning "vessel." 12 time-degrees, or 48 minutes.
Ki.lá refers to "weight" or "the act of weighing,"
In Mul.Apin we have the equivalent. For example
suggesting that a maštaktum may have been under-in II iii 13-15 we find the lines: "concerning the
stood as a wooden (giš) weighing device. Since itcoefficients of the visibility of the moon; 3 minas
seems also to have measured time, this was perhaps(are) the watch of the night. Multiply by 4 and you
done by weighing the fluid involved, hence theget 12, the visibility of the moon."
minas and shekels we see in Mul.Apin, EAE 14 and
the like.
In summary, the philological evidence from the
1.3 The OB Mathematical dibdibbu Texts
OB period on suggests that a wooden giïdib-dib,
known in Akkadian as a dibdibbu and sometimes as
a maltaktum, was a time-measuring device which
could be filled and which involved a cylinder/prism
out of which dripped water, the quantity of which
may have been assessed by weight. It may also have
included a part which filled slowly. Perhaps an
alternative existed which used sand.
Two OB texts, probably from the same archive,
describing four interrelated mathematical problems
which concern the gi8dib-dib were translated and
ls) Enûma Anu Ellil "When Anu and Ellil ..." the opening
lines of the great celestial divination series comprising some
70 tablets. Tablet 14 is published now in Al-Rawi and
George, op. cit. p. 55f. The deviations from a straight line of
As with the coefficient "40," two possible meanthe values in UŠ at the beginning and end of Table A are not
ings of the OB coefficient "12" of the maltaktum
of consequence here. They are referred to by the third of the
seem possible based on the evidence of later texts,OB coefficients "3;45."
14) CAD M/l, p. 392 - the only OB attestation of
maštaq/ktum is in a lexical context different from that where
the ("dib-dib is found.
I6) As made explicit in EAE 14 Table A: 1. 15.
") Al-Rawi and George, op. cit. p. 56f.
,8) SSB Erg 1 (1913) 96.
") As made explicit in EAE 14 Table B: 1. 15.
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134 David Brown - John Fermor - Christopher Walker
discussed by F. Thureau-Dangin in RA 29example
(1932) 1:
pp.
height = 133 cm, volume when full = a/b
133-620. The variables in the problems are the= height
32 qû I litres
(let this be called "a") of the device when
full and
examples
2 & 3: height =167 cm, volume when full
the amount of fluid which leaves the devices in each
case. For each gBdib-dib a constant "b" is established
which designates the amount by which the height
changes when 1 sila ( qû ) (of water?) "falls." The
verb used in BM 85210 IV: 11 in this context is
= 40 litres
example 4: height =167 cm, volume when full =135
litres.
The devices described in the OB mathematical
texts are large. It would appear from the name gišdib-
maqãtu "to fall" in the G perfect "it fell."
dibThe
that wooden drip-timing devices were meant, but
substance in the gBdib-dib is not mentioned in either
we have no means of knowing if the dimensions
text. In BM 85194 II 27 and 34 we find the phrase,
suggested by these two texts corresponded to those
gi5dib-dib epte lA sila, "I opened the gUdib-dib (of)
V2 devices implied by BM 17175+, Mul.Apin,
of the
qû." In BM 85194 II 34 we find g8dib-dib 3,20
epte,
EAE
14 etc. There is no obvious relationship between
"I opened the gSdib-dib of 3,20 (qû)." As far
as
the minas
in the latter texts and the qû in the former,
these texts are concerned the "opening" of the but
device
1 qû of water weighs approximately 2 minas.
made the level, of what must have been the fluid
The fractions "f ' suggest that quite small differences
inside, fall. In three cases (examples 1, 2 & 3) "b"
is 2V2 fingers (1 šu.si = 1 ubãnum « 12A cm), in the
fourth it is 20/27 fingers. In all four problems "b" is
expressed as a fraction of 10 fingers, which was
presumably an important marker of the difference
between levels in the devices. In three cases (examples 2, 3 & 4) "a" measured 100 fingers and in the
other it measured 80. The four problems revolve
were measured. In the last example the "b" value of
20/27 fingers is given as '"/9th of 2Ards of 10 fingers."
Perhaps the scale envisaged on that device was made
up of intervals corresponding to 10 fingers, broken
down into thirds with each third broken down into
ninths (or finer). However, since it is likely that the
numbers were chosen deliberately in order to lead
towards the number 0;44,26,40, the final number in
around a fraction "f" of the initial level (ki) by
the Standard Table of Reciprocals (Neugebauer and
Sachs MCT (1945) p. 11), the sizes of this and the
f = height dropped/height when full ("a")
other devices may have only broadly corresponded
where the height dropped is determined by the
to those of real dibdibbus. Nevertheless, it seems
amount of fluid that has been released. In the first
reasonable to suggest that a high level of precision
three examples the outflow is V2 qû in each case in measurement was apparently thought to be possible
(assumed but not stipulated in example 1), which with these devices. Perceived precision must be
would result in a height drop of bh. Thus, in thesedistinguished from true accuracy, however, for a
which the final level is lower:
three cases f = b/2a. In the last example the outflow finely divided scale does not guarantee a timing
is 3 'A qû (gBdib-dib 3;20 sila epte) so f4 = (b4.3V3)/a4. device free from inconsistencies of flow brought
Thus we have for:
about by temperature changes, impurities in the
Ex. 1 = BM 85210 IV 10-16: a = 80 fingers, b = 2'/2
fluid, or even fundamental errors in design.
fingers when 1 qû flows out, f, = Vm (0;0,56,15) It is also worth noting that three different devices
Ex. 2 = BM 85194 II 27-33: a= 100 fingers, b = 2'/2,
are described in these mathematical texts. Apparently
f2-V«o (0;0,45)
no norm had been established, though this again may
Ex. 3 = BM 85194 II 34-40: a= 100 fingers, b = 2V4,
only be a reflection of three different mathematical
f3 = '/so (0;0,45)
problems. The constants "b" suggest that for these
Ex. 4 = BM 85194 II 41-48: a= 100 fingers, b = 20/27
dibdibbus an outflow of 1 qû may have been asso(0;44,26,40), f4 = 2/si (0;1,28,53,20).
ciated with a given interval of time. That is, they
imply that the sizes of the outflow orifices were of
1 qû is about 1 litre in size (Powell, op. cit. p.a fixed diameter in the devices, since the heads ("a")
503). Knowing a, and b we can calculate the dimenwere pretty much identical in all three cases21. All
sions of the three sizes of dibdibbu discussed in
that was seemingly varied was their cross-sectional
these OB texts if we assume them to be of constant
areas, and thus by how far the levels would fall for
cross-section:
each qû or mina outflowing. A consistency in the
outflow rate between devices is implied by the
weights quoted in later texts. If the OB mathematical
20) The texts are BM 85194 = 99-4-15;] = CT 9 Pis. 813 col. II 27-48 and BM 85210 = 99-4-15;17 = CT 9 Pis. 14- texts at least broadly reflect reality, then the devices
15 rev. col. II 10-16. See also Neugebauer, MKT I (1935)described therein appear to have had fairly similar
pp. 145, 155, 223 & 227 and p. 173f. where the prismatic/outflow rates in so far as near-constant heads were
cylindrical shape is justified, and Thureau-Dangin, TMB
2I) The one variant is in the only example on BM 85210.
(1938) pp. 25f. & 52f.
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The Water Clock in Mesopotamia 135
That is, in Neugebauer's
used and the volume outflowing, lation.
as opposed
to the model, it was observed
in Babylon
that 4 minas (or a multiple thereof, see
height dropped, seems to have been
important.
below) of water flowed out during the course of the
night of the winter solstice, and 2 minas (or the same
multiple
thereof) sufficed for the night of the summer
1.4 The Neugebauer Clock and its
Problems
solstice. All the weights corresponding to the other
night lengths
the year were then determined
One of the main reasons why Pingree
and through
Hunger
by describing
calculation - by interpolation
- and not by
chose to interpret BM 17175+ as
the
observation.
Thatintervals
is, the ratio of the equinoctial day
weights of water in a water clock rather
than
length
to
the
shortest
day length would be given by
of time can be traced, we propose, to an influential
'/2(V4+V2):V2
not by V3:V2, that is 1.207:1 as
article by Neugebauer (1947 - see
n. 1). and
There
to 1.225:1, and
so forth.
Neugebauer pointed to the parts compared
of Mul.Apin
and
Neugebauer's model is superficially very attractive,
EAE 14 discussed above, and followed Kugler's
a timing device, even though there is no explicit
but it has some serious flaws. One we have just
noted - that it requires some empirical observations
mention of a gBdib-dib or of a maltaktu anywhere in
to explain away the woefully inaccurate ratio of 2:1,
earlier suggestion that they referred to substances in
the two series. Neugebauer then proposed a model but not so many that linear interpolation would be
for the timing devices, incorporating some of what seen only very roughly to provide the values correcan be learnt from the OB mathematical texts just sponding to the lengths of the days and nights away
discussed, which not only accounted for the seemingly
dreadful inaccuracy of the ratio 2:1, but also described
the workings of the clocks. This model we will now
discuss:
Neugebauer observed (op. cit. p. 39) that the time
it takes for a cylindrical or prismatic vessel to empty
of fluid by means of a hole in its bottom was
governed by the equation t = cVh, where c is a
constant, t is time and h is the original height of the
fluid in the vessel22. Since the weight of the fluid in
the device was proportional to h, then the time it
would take to empty would be in proportion to the
square root of the weight of fluid originally placed
therein. Thus if the weights were in a ratio of 2:1,
then the emptying times would be in a ratio of
1.414:1. This ratio is much closer to that observed
from the solstices. Two much more significant problems exist, however.
Firstly, as we have discussed, even though weights
are made explicit in this context in some parts of
both EAE 14 and Mul.Apin, the ratio of the longest
to the shortest day was still conceived of in terms of
time in other parts of those texts. Rather than argue
that the examples which use time units are a corruption
of a system originally based on weights of water in
a cylindrical outflow clock, we suggest that perhaps
the current reconstruction of the timing device is
incorrect. When weights are mentioned, they describe
amounts of water (or possibly sand) in the devices
used in such a way that they correspond to a measure
of time in proportion to their value and not to the
square root of their value. We will return below to
the question of what purpose such an inaccurate ratio
for the lengths of the longest to the shortest days for
the latitude of Babylon - the supposed place of
composition of Mul.Apin and EAE 14. Thus, Neugebauer's hypothesis at once describes the basic
could have had.
structure of the gi5dib-dib as a cylindrical/prismatic,
The second problem pertains to the amount of
fluid in the device. It was unclear to Neugebauer in
1947 how many minas were supposed to constitute
outflowing clepsydra and asserted a high level of
an equinoctial night. We remarked above that in the
empirical correctness in the ratio. He finally proposed
that only those values determined for the solstices
were used, all the others were obtained by interpo22) In fact this is by no means general. Two flow types are
exhibited in a vessel freely draining under gravity - laminar
and turbulent. Only fast turbulent flow (obviously not applicable here) or flow through sharp edged holes leads to the
(Torricelli) equation used by Neugebauer. Some sort of spout
being possible, the longer it is the more fully will laminar
flow be established, and this flow type has a quite different
relationship between head and discharge rate. For details on
and references to the fluid dynamics of water clocks see
Fermor, Burgess and Przybylinski, "The timekeeping of Egyptian outflow clocks," Endeavour, New Series Vol. 7, No. 3
(1983).
texts it was measured by 3 minas, which is now
confirmed in the publication by George and Al-Rawi
of EAE 14 table B (see also ibid. pp. 59-60 and n.
23). The inscription "three minas," and not 3,0 etc.,
is absolutely clear from table B because of the use
there of the signs for fractions and the values in
shekels. A mina of water weighs about 0.5 kg23 and
its outflow from the device was supposed to last 4
hours. Whether this is physically possible in the case
where the device empties is open to serious doubt.
Such a low discharge rate could only be obtained
by using an outlet with a minuscule bore. However
in an outflow device the force driving water through
23) Powell, op. cit. Section V.5 p. 510.
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136 David Brown - John Fermor - Christopher Walker
such an outlet diminishes as the level falls. Two
different lengths cut from the same stainless steel
impediments to the flow might then become manifest.
tube. Lengths of 3 cm, 2 cm and 1 cm promised full
drainage in about 2.5 hours - 2 hours and 1.5 hours
The first of these concerns any fine particles
suspended in the water which at low heads might
using the calculation previously described. Thus all
settle in an outlet conduit halting the flow. These
the spouts seemed set to empty the vessel in less
might include transparent algae such as are found
in otìe third of a shortest day or night (a little over
than
standing tanks on high rise buildings today and
no at Babylon). However, in actual fact their
3 hours
doubt were common in Mesopotamian water sources.
drainage was protracted beyond the watch in all
Tiny bubbles released from long standing water
cases. Moreover outflow ceased entirely at heads
well above the outlet (due to the effects of surface
might also become entrained in the outflow causing
airlocks. Chinese commentators speak of blockage
as
tension),
for the 3 cm length with 44% of the water
a common problem in their water clocks24. They
still undrained. The proportions undrained for the
speak of flow from the same clocks as being "as 2fine
cm and 1 cm spouts were 25% and 31% respecas hair" indicating a continuous stream. Intively.
the The fact that the shortest spout did not prove
Mesopotamian case, the name dibdibbu suggests that
the best at draining suggests that even the omission
the flow was reduced to drips indicating a yet slower
of a spout would not solve the problem, a point
discharge that would exacerbate the problemconfirmed
of
by piercing small holes (of somewhat
blockage.
indeterminate bore) through the floors of similar
The second possible impediment arises from surface
tension. A thin film of water will stick to surfaces
vessels25.
We do not claim that a functioning Neugebauer
and the thinner the film the more energy is requiredclock is impossible to construct, for the design
to move it. It appears in the present case as if thevariables are legion, including different materials
restriction on the outlet bore necessary to achieve the and spouts of greater slope. We even note that it is
low discharge rate implied by the texts consideredof possible relevance that Egyptian water clocks of
might be such as to prevent the egress of the water the New Kingdom through the Ptolemaic ages were
at low heads due to the opposition of capillary water. scaled internally, but always with the scales ending
We can report on the following experiments, 4 cm or so above the outlet, as if in recognition of
which were designed to try and produce a workingthe draining problem made clear by our experiments.
Neugebauer clock. Hypodermic needles, set horizon-Conceivably a similar solution was adopted in Mesotally, were used as the discharge conduits of a potamia, though we argue for an alternative reconcylindrical clock holding 333 ml (c. 2A mina) abovestruction. Enough has been done at this stage, howthe outlet giving an initial head of 8 cm. This
ever, to put the onus on the supporters of the
represents the amounts used per watch during the Neugebauer clock to demonstrate a working model.
solstice months for either the day or night watches
according to the texts. The water was kept at around
20° C. Initial trials with tap water supplied by roof
storage demonstrated repeated stoppages which, when
cleared by tilting the spout more steeply, were
followed by prolonged further flow. Trials were
continued using distilled water and such blockages
did not recur.
The discharge rate for each of the spouts used
was measured at the full 8 cm head while this was
1.5 A New Model for the Mesopotamian Water
Clock
Our present concern is to seek a new best-fit
model of the timing device based on the textual and
experimental evidence collected above26. The theo2S) Incidentally, the escape holes of water clocks in some
kept constant. This allowed an estimate of the empSanskrit texts are described as round tubes of gold four digits
tying time to be made, for if unimpeded, the average
in length. For details see JHA 4 (1973) pp. 3-4 where Pingree
discharge rate from the sinking level should be half
also argues that these timing devices were dependent on
Mesopotamian scientific influence into India during the Achaethat from the initial head. After initial trials a spout
menid period.
of approximately 0.7 mm in diameter was selected
(the manufacturer quotes a nominal internal diameter2<s) Many other texts are known from a time after the OB
period which include figures plainly referring to weights or
of 0.027 inches or 0.685 mm, perhaps better consid-
ered as about 0.7 mm). Spouts of wider bores
emptied the vessel too quickly and vice versa for
those of a narrower bore. The spouts then used were
24) J. Needham & F. Wang, Science and Civilisation in
China, (CUP 1959), Vol. 3 p. 316f.
times throughout the year which match those discussed.
Examples include the "Zwölfmaldrei," or "astrolabes," some
of which may well have been first composed in the OB
period, but which are now known only from later copies. See
C. B. F. Walker and H. Hunger, "Zwölfmaldrei," MDOG
109 (1977) pp. 27-34. None of these texts add to our
knowledge of the water clock, however.
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The Water Clock in Mesopotamia 137
retical device should be of constant
cross-section,
quoted are
understood to be amounts collected in an
and if anything like the 8i5dib-dib
ofvessel,
the OB
matheinflow
however,
it becomes clear that the
matical texts, involve the dripping
orofoozing
out
of
smallness
the volumes
collected
is the counterpoint
water (or conceivably sand). It would
need filling
toin the supply vessel.
to the considerable
amounts
With
this construction,
such small amounts could
work ( Atra-hasïs III i 36), measure
time
in proportion
to weight and (if water-based) involve
water
have been enough
collected even
over those long periods of
to work without capillary interference,
should
time. This water and
clock with
a near constant head, a
large supply vessel combined with a narrow conduit
A first model would be one in of
which
a large
store
egress and
a weighed
inflow vessel, was a true
of water was coupled with a small
rate
of
egress,
clock and not a mere watch and
timer. In a Neugebauer
an amount of water was collected
in an
a even
smaller
vessel,
clock
scale (suggested
by the OB mathe-
not empty.
whose filling marked the end ofmatical
the texts)
watch,
and then
would produce
gross errors, the fourth
(possibly) returned to the main store.
filling
isduration as the first
scale hour,This
say, having
the same
perhaps implied by mušihhu (n.
12).
The Direct
amount
three
together.
evidenceof
for the timing of
intervals
smaller than a watchby
is limited
before the
water having flowed out would
be designated
a
weight or by a volume. The main store would
mid-S01 century BC. The short intervals described in
contain a lot of water (just as we find in the OB
EAE 14, Mul.Apin and the astrolabes were all
mathematical texts) and would require some effort to mathematically constructed. Even the lengths of
fill (perhaps hence the reference in Atra-hasïs). Very
"seasonal hours,"29 which were perhaps developed
little would actually flow out during a watch, during
which time the flow rate would be virtually constant.
before the 8th century, were calculated in this way.
It would seem probable, however, that these shorter
In this new model, the three watches of any given
night would each be measured by the same amount
of water collected in the smaller vessel, and would
be of equal length.
and fractions of the time between sunsets, were still
assessed with a clock, in the same way as we believe
At first glance it might seem, since the specified
flow rate remains unchanged, that we face the same
(though, see below). In Mul.Apin II i 23 it states, for
example, that "you observe the visibility times30 of
problem as the Neugebauer clock. However, our
the moon ... and you will find how many days are
in excess" for the purposes of intercalation. This
proposal is for a supply vessel of such great capacity
that the head will barely change during the course of
a watch and that the rate of egress, though slow, will
be unaffected by the opposition of capillary water at
least. We suggest that the Mesopotamians proceeded
by first producing a spout of roughly the right
dimensions27 and then fine tuning the clock by
adjusting the near constant head to that producing a
sufficient fit to the natural checks. Thus, supply
vessels would vary in capacity, just as in the OB
intervals, despite being mathematical elaborations
the length of a watch of the night was determined
indicates that these times, of the order of 40 minutes,
were measured, or at least were thought to be
measurable. We develop this further, below, in the
context of the masqû.
There is also evidence that the timing of the
intervals of a number of phenomena played a part in
celestial divination, a discipline that was established
at least by the OB period. See for example the
mathematical texts, since it would probably be beyond
omens "if the day reaches its normal length; a reign
of long days" and the antithesis "if the day is short
the competence of the makers to produce spouts of
compared to its normal length; a reign of short
uniform bore at these small scales. As to the near
days."31 That correspondence with an ideal (see
constant heads, if the devices were like those described
was scaled down from the top and was never fully drained.
in the first three OB mathematical text examples,
Also, in
see below on the ziqpu text AO 6478 and BE 13918.
then an outflow of 4 minas or 2 qû would result
29) Reiner & Pingree, "A Neo-Babylonian Report on
a drop of only 5 fingers, a small fraction of the total.
Seasonal Hours," AJO 25 (1975) pp. 50-55.
We may note that if the Mesopotamians used30) NA.MEŠ or manzãzu. These are most probably the
simple emptying outflow clocks, then they could
same as the na recorded in the Diaries (see beiow, also Sachs
have avoided the problems we have cited by increas& Hunger, 1988, 21), which are both the periods of time
ing their discharge rate28. Once the weights of between
water sunset and moonset on the first of the months, and
the times between sunrise and moonset in the middle of the
27) A. Gwinnett & L. Gorelick, "Beadmaking in Iran
in
months.
These are also probably the intervals presented schethe early Bronze Age," Expedition 23/4 (1981) pp.matically
10-24 in EAE 14 table D, in which case those on the 15th
report stone beads bored with holes down to 0.5 mm.
of each month are not between sunset and moonrise (as stated
28) Egyptian practice showed an appreciation of these
by George and Al-Rawi, op. cit. p. 58) but between sunrise
and moonset on the 15th.
problems. An outflow clock of the "flower-pot" design known
from the Oxyrhynchus papyri has a mean discharge rate31)
17H. Hunger, Astrological Reports to Assyrian Kings =
times that implied by the cuneiform texts. Moreover this
clock
SAA
8 (1992) 7:3 and 457:4.
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138 David Brown - John Fermor - Christopher Walker
if boded
we insist
below) boded well and non-correspondence
ill on establishing an empirical basis for
texts dealing with what has casually been called
appears to have been a general truth of cuneiform
celestial divination. For details see D. Brown Meso"early Mesopotamian Astronomy,"32 we must argue
that the OB and later users of those same texts did
potamian Planetary Astronomy-Astrology - Styx,
not notice that the year was not 360 days long, or
forthcoming. After c. 750 BC, as part of a new spirit
abroad in cuneiform astronomy, water clocks were
that months were not all 30 days long. The latter two
regularly used to measure shorter time periods, asfacts
we were plainly known, as the references to intershall see. In that case accuracy rather than merely
calation in Mul.Apin, or to the ominous significance
apparent precision characterised the endeavour. of 29-day months in EAE make clear. The 360-day
1.6 Ideals as Opposed to Empirical Reality
year and the 30-day months were not the early,
rather poor results of empiricism, but were ideals.
Without doubt the 2:1 ratio was the same.
Once it is recognised that the purpose of the texts
The OB mathematical texts implied a varyingwe
in have at our disposal concerning Mesopotamian
celestial interests prior to the 8th century might not
the initial heights of water "a" within a similar
have been "astronomical," but instead "divinatory"
storage cylinder (compare examples 1 and 2). Neugebauer's 1947 idea could be resurrected in a modified
in a broad sense, it becomes easier to accept that the
form with the suggestion that the total amount2:1
ofratio may not have come about as a consequence
of the fortunate effect of an outflow clock. If one
water placed in the storage vessel was for a midthinks of the texts we have as expressing how the
winter night twice that for a summer night. This
would approximately increase the outflow rate ideal
by universe might run, then we can understand
sections in Mul.Apin and in EAE 14 on the
about 1.4. Since in reality the midwinter nightthose
is
approximately 1.4 times the length of the midsummer
visibility of the moon, say, to be examples of
night at the latitude of Babylon, 1.4 times 1.4,mathematical
or
exegesis33 on the preconceived idea of
about twice as many minas of water would flow out
what ought to happen, and not the result of precise
of the storage container and be collected. Thatempirical
is,
observations. The texts we possess conthere may have been a fortuitous empirical vindication
cerning the water clock during that time may in fact
have had as much a role in divination as in the
of the pre-conceived notion of a 2:1 ratio of maximum: minimum night lengths, as his 1947 article
practical needs for dividing up the nights into thirds
suggests.
for sentry duty or whatever. They tell us, perhaps,
that for the diviners, since it appears to have been
Naturally, many variants on the clock hypothesised
they who used them, the ideal universe had a longest
above could be proposed. It all depends on the
strength of one's desire to fit the attested figuresday
to twice that of the shortest, and the consequences
ofof
that notion34.
empirical reality (see n. 1). However, regardless
whether or not the storage vessel had more water in We certainly know that the diviners noted the
watch during which an event occurred, but the extent
it in winter than in summer, we argued that the
to which their use of these intervals was affected by
preconceived notion of a 2:1 ratio was already
any failure in the schemes to accord with the natural
understood in terms of time. Night lengths were
is a moot point. Linear interpolation between
referred to by the amount of water collected checks
per
seasonal extremes is contrary to reality, but not
watch, but this amount was synonymous with the
excessively so, and of the same order as expected
length of the night in the time units UŠ. 4 minas
clock variability. However, if the device we propose
meant a night was considered to last 4,0 UŠ. This
makes it difficult to satisfy the desire to have were
the used without seasonal adjustment of the initial
Mesopotamian users of the g>idib-dib/maltaktum care
as much about time keeping and accuracy as do we.32) E. g. B. L. van der Waerden, Birth of Astronomy
We should be more literal and accept that the
(1974) Chs. 2 & 3 "Old-Babylonian Astronomy," O. Neuge-
device was of the constant head variety where twice
bauer, A History of Ancient Mathematical Astronomy (=
the amount of water flowing out measured twice the
HAMA, 1976) p. 54 If. "Early Babylonian Astronomy" and
the titles of Hunger and Pingree, op. cit. and George and AItime and not 1.4 times the time. If not, we are forced
Rawi, op. cit. That any of the texts described under these
into arguing that the outflow clepsydra was the cause
headings
could be called "astronomical" is seriously to be
of a misunderstanding concerning the relative lengths
doubted.
of the midwinter and midsummer nights - a misun33) Termed arû, perhaps. See S. Lieberman, "A Mesopo-
derstanding that lasted from the OB period until
tamian Background for the so-called Aggadic "Measures" of
perhaps as late as the 8th century (see below), and
Biblical Hermeneutics," HUCA 58 (1987) p. 188.
which filtered into every aspect of celestial divinationM) See D. Brown, op. cit. Chs. 3.2 and 5.1.3 in Styx
forthcoming.
from EAE to Mul.Apin and the Astrolabes. Equally,
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The Water Clock in Mesopotamia 139
course also
mean
"a vessel which fills (with water),"
level then large discrepancies between
the
calculated
hence the water clock connection. The quoted context
end of the third watch and sunrise or sunset must
have followed near the solstices. These could have
in which it is found suggests that the mašqú was
used to establish the period of time for which the
been much reduced by changing the initial level, but
moon remained visible after sunset at the beginning
equally they may have been ignored, with the natural
of the month, and the time for which it could be seen
check perhaps simply replacing the device indication
for the end of the third watch. It comes down to a
before dawn towards the end of the month, when
observations could not be made due to cloudy
level of concern for empirical confirmation, and those
in
conditions. The text indicates that ordinarily those
this they were perhaps less exercised than are we. In
time intervals were measured. It is not clear, however,
conclusion, we suggest that any attempt to account
for the inaccurate 2:1 ratio in terms of the construction
how a mašqú could be used to determine them if the
of the water clock may be unfounded. We await
phenomena to which they pertained could not be
further discoveries, however. Terms in Ur5-ra=Aw¿w//M
seen. Possibly, they were calculated from the mašqú measured lengths of lunar visibility on days 2, 3 and
IV 6f. (n. 12) like mukanzibtu "pendulum?" remain
to be incorporated into the model.
so forth of the month.
It appears that the purpose of knowing these time
intervals was to determine when intercalation should
1.7 The Masqû - Apparent Precision versus Accuracy
take place39. It seems probable, then, that it was
considered possible (by the author of this text, at
least) to see if the lunar year was falling behind the
In a commentary text on the eclipse section ofsolar by measuring the period of time the moon took
EAE35 and in a text now known as "a Babylonian to set on its first day and comparing that with the
Diviner's Manual,"36 both written in the late NA ideal values for this determined in texts such as
period (though possibly older), we find the lines: Mul.Apin and in EAE 14. That is, the length of the
e-nu-ma ina igi.du8.a ( tãmarti ) d30 u4-mu er-pu gál-moon's first visibility period was considered to be,
ka li-ti-ik-šú d[ugmaš-qu-u]
say, V 15th of the length of the night40. A measure of
e-nu-ma ina bi-ib-lu u4-mu er-pu gál-Ara li-ti-ik-šúthat would, according to this theory, provide a
measure of the length of the night. If this did not
dugmaš-qu-u
correspond with what was expected for that date of
"If at the (first) appearance of the moon you have a
the
year then the lunar year would be known to be
cloudy day, its checking device is a mašqú vessel.
falling
behind the solar, and a new month would be
If on the day of the moon's disappearance you have
known to be needed. Potentially, this would assist
a cloudy day, its checking device is a masqû
vessel."37
the diviners in averting evil omens, the prognostications for which vary by month, which appears to
"Checking device" or litku is a simple nominal
have
been one of the main purposes of the Diviner's
derivation from latäku , much as maltaktum (see
Manual41. This namburbû (line 56) or method to
above) is38. This in itself suggests that a mašqú was
dispel the evil prognostications, could not be thwarted
perhaps a form of clock. Mašqu itself is a mapras,
by the mere presence of clouds. By using the mašqú ,
place or instrumental form (like maltaktum ) from
however, the diviner would still have been able to
šaqú "to make drink, fill (with water)" and means "a
establish that the month in which the evil omen
drinking place" or "a drinking vessel." It could of
apparently took place had not in fact taken place.
If a mašqú truly was a form of water clock, the
35) C. Virolleaud, ACh 2 Supp. Sin 19 = K 3123:7f. ....
Babylonian Diviner's Manual shows us a little more
] LU li-ti-ik-šú dug (8) ...li]-ti-ik-šú maš-qu-ú kimin na 4.ašhow such a device may have been used in Mesopopu-ú ša u4-sakar ša šá d30, "its checking device is a vessel
the crescent of the moon." K 3123 parallels
Klunations
250 in
parts
39) Because 12
fall
some 11 days short of the
(ref. Borger, HKL).
equinoctial year, every three years or so an additional month
needs 197-220.
to be intercalated The
into the calendar
the lunar and solar
36) A. L. Oppenheim, JNES 30, pp.
linesif in
question are nos. 64 and 65.
years are to remain synchronised. The methods described in
37) Oppenheim, op. cit . translates:
"Should
it
happen were
to necessary are very
this text
to determine
if intercalation
you that at the first visibility/disappearance
of the
moon the
similar to those outlined
in Mul.Apin.
weather should be cloudy, the water
be
40)clock?
Or possiblyshould
the values in
the the
Nippur variant parts of
means of computing it" and suggests
38
"at and
periods
EAEin
14, n.
table
A -that
see George
Al-Rawi, op. cit.
of poor visibility a water clock device
masqû)
was
41) Lines (called
49-52: "(when)
they ask you
to save the city, the
used to establish the exact length of
day."
kingthe
and his
subjects from enemy, pestilence and famine
(predicted)
whatequated
will you say?in
... how
will you make (the
38) Indeed litiktu (fem.) and maltaktu
are
the
lexical series. See sub maltaktu in the CAD.
evil) bypass?" (after Oppenheim, op. cit.).
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140 David Brown - John Fermor - Christopher Walker
tamia, both in the NA period and earlier, for what
is described in this text was no doubt meant in
2. Astronomical Timing
Mul.Apin II i 23 (above). The theory in Mul.Apin,
2.1 A 3:2 Ratio
EAE 14, and in the Diviner's Manual, underlying the
length of lunar visibility on the first of the month 3:2
is is a good ratio for the longest to the shortes
incredibly inaccurate. There is absolutely no day
wayat the latitude of Babylon (when atmosphe
that these lengths even approximate '/15th of therefraction
night
is taken into account42) and is probably t
length. Many factors determine the length of lunar
best ratio that can be obtained if small numbers
first visibility, including the separation of the moon
(below 10, say) are to be used. It was assumed by
and sun on the ecliptic, the angle the ecliptic makes
F. X. Kugler43 that I.NAM.giš. hur.an.ki.a, a learned
with the horizon, the latitude of the moon, horizon
explanatory text whose earliest exemplar can be
and atmospheric effects. It is not a simple function
dated to the reigns of Sargon II or Sennacherib (c.
of night length. Its measurement could not tell 700
you BC) but which is probably much older, contains
if the lunar year were falling behind the solar
in one of its pirsu "divisions" (K 2164+) the earliest
without a profound understanding of all the mechapositive statement of the ratio of the longest daylight
nisms at work. A measurement of the length ofto
the
the whole day as 1 : 1 ;40 or l:l2/3. He read K
night, say, could tell you this immediately, however.
2164+ 26: 1;40 ud.da zal-e u4-mu "1;40 (times the
So what was the point in measuring the length
of of the longest day(light) is a (whole) day"
length)
lunar first visibility? It appears from the Manual(For
thatan edition of the text which maintains this
its measurement provided the diviner with the excuse
translation see A. Livingstone MMEW (1986) pp.
he needed to avert portended evil. This, perhaps
we If we assume that the addition of the longest
27-9).
might call it traditional, theory provided valuesand
for the shortest daylights equals the length of the
these intervals, based on ideal suppositions and
mathematics. If a measurement of the interval produced a result that corresponded with the ideal value,
then this would provide one piece of evidence that
no intercalation was necessary. Other phenomena
would be measured and their values compared with
other ideals. Only the sum of the conclusions would
determine whether or not an intercalated month was
required. Simply because we know that some such
correspondences would have provided better assessments than others, does not mean that the ancient
Mesopotamians knew. For such measurements any
timing device used would not have needed to be
accurate, but we argue that it would have been the
case that an apparent precision of measurement
would have added credibility to the diviner's actions.
Had the water clock been genuinely accurate it
would rapidly have shown up the differences between
the observed and predicted intervals for lunar first
and last visibility, which are equal in the traditional
theory. Perhaps, we should think of the mašqú, and
likewise the dibdibbu or maltaktum, not as timers in
this early period but as divinatory devices used to
"show up anomalies" and thereby serve the purposes
of the diviner.
whole day, then their ratio according to this text is
1:0;40 or 3:2. However, Kugler's translation of the
passage is incorrect, as was pointed out by Neugebauer
and Sachs in JCS 10 p. 135 n. 4 (see the parallel in
Mul.Apin, Hunger-Pingree, op. cit. p. 153 on II ii
13-17). I.NAM.giš.bur.an.ki.a, therefore, does not
contain the 3:2 ratio. It has been also argued that the
ratio 3:2 can be found in Mul.Apin44, but this is by
no means clear45. In theory, however, the 3:2 ratio
in Mul.Apin could have been a later addition. As
yet, then, we have no firm evidence that the 3:2 ratio
was known in the 7th century BC, and have ample
evidence that the 2:1 ratio was used at and before
this time.
In the later Persian and Seleucid periods, however,
we have many examples of the scribes at that time
employing a 3:2 ratio in the course of trying to
predict planetary phenomena46. They used two arithmetical schemes to represent the changing length of
daylight through the year47. In both, the 360-day
ideal year was replaced by a 360 UŠ zodiac. The
schemes were still schematic. There is no evidence,
«) O. Neugebauer, (1947) p. 38.
43) F. X. Kugler, SSB Erg. I-II, p. 89 n. 1.
44) See Hunger-Pingree, op. cit. p. 153 on II ii 21-42 of
We have little idea what a mašqú was like, short
of what can be gleaned from its etymology and the the text.
45) See J. Fermor "Timing the Sun in Egypt and MesopoDiviner's Manual. Its context suggests that it was
tamia," Vistas in Astronomy 41 p. 165.
perhaps a measurer of fixed time intervals, marked
46) By which we mean the phenomena of all seven
by its filling with water. In CAJ forthcoming D. wandering celestial bodies visible to the naked eye.
Brown tentatively suggests that it may refer to a
47) Now known as Systems A and B. Cf. O. Neugebauer,
sinking-type water clock, a possible example of
"The Rising Times in Babylonian Astronomy," JCS 7 (1953)
which he discusses.
pp. 100-2, which includes references to earlier literature.
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The Water Clock in Mesopotamia 141
for example, of the plotting of empirically
determined
so-called water clock
texts of the period before the
day lengths against solar longitude.
The
question
7th century
BC and
the mathematical astronomical
naturally arises as to the origin
of
the
ratio
texts,
in that
the 3:2
argument
is still the 360 days of the
employed in the last centuries BC.
Was
result
ideal year
and it
yetthe
the ratio
is 3:248.
of the interaction of observation and
theory,
eveninif
The 3:2
ratio by weight
BM 29371 certainly
once determined the ratio was used in mathematical
implies a clock in which time and weight of water
idealisations of reality?
were in direct proportion. Although the construction
Firstly, a 3:2 ratio is so simple a ratio, particularly
of the timing devices could have changed over
in base 60, that it may have been derived as muchtime49, it seems to us that the evidence of this late
through numerical play as empiricism. The ratio is,
text favours our proposal that the water clocks from
in a sense, the next simplest after 2:1 and we should earlier times also operated with a constant, or near
probably not put much store by its possible appearance
constant head. Also, by continuing to employ the
in Mul.Apin. Nevertheless, the period in which it
astronomically inadequate descriptions of the universe
would then be attested for the first time is charac-
(the 360-day year50 and 30-day months) BM 29371
terised by a development in cuneiform celestial reveals itself to be a divinatory text in the manner
concerns of singular importance in the history of of the OB text BM 171 75+. It is as if the astronomiscience. These included the appearance for the firstcally useful 3:2 ratio had by then become de rigueur
time of "accurately" recorded data and the attempt toeven for the purposes of divination. BM 29371 also
predict celestial phenomena to an accuracy of less implies that precise timing was possible, just as in
than a day. This revolution in science was much the OB mathematical texts. It includes a column in
discussed by D. Brown (op. cit. forthcoming) and which the increments are in 0;01 units (of perhaps
the details need not concern us here, though the minas, or perhaps of time directly, see below). On
question of the "accuracy" of the recorded observa-the one hand this is an example of rather clever
tions dating from the 7th century BC or so will bemathematical play, on the other it appears to suggest
discussed shortly. Conceivably, the presence of the the existence of devices which at that time could
3:2 ratio in a 7th century exemplar of Mul.Apin couldrecord units to this level of precision. Such precision
be put down to this revolution. Whatever, it is thewas not meant for astronomical measurement though,
use and not the discovery or invention of the 3:2 for the true lengths of any of the days quoted in the
ratio, at a time when the celestial diviners were
text would have varied from year to year by amounts
much greater than the smallest increments. So although this text utilises a relatively accurate day
divinatory prediction, which is the key fact. What length ratio of 3:2 and suggests the fineness of the
was descriptively more apposite may earlier have scales perhaps used on contemporary water clocks,
been considered less useful or even incorrect from a
its purpose was no doubt similar to that outlined in
attempting to provide the basis (in terms of theory
and data) for astronomical prediction over and above
divinatory point of view. The application of the ratio the Diviner's Manual. For evidence as to the real
in the centuries following the 7th century BC suggests accuracy of water clocks used after the 8th century
that in the endeavour to predict certain celestial BC, we must look elsewhere.
phenomena in advance, the "traditional" 2:1 ratio
One further text, also from the LB period and as
had needed to be superseded. To this extent the 3:2 yet unpublished51, BM 29440 also writes of what are
ratio was a result both of empiricism and of a probably weights pertaining to the 15th of each month
changing divinatory climate in which the prediction
of phenomena was playing a more and more signifi-
48) Interestingly, F. Rochberg, "The Rising times of the
Zodiac and the Length of Daylight in Babylonian Astronomy,"
The text BM 29371, identified by C. B. F. Walker
(JHA forthcoming) argues that some ziqpu texts of the postand edited here below, is important in this context.zodiac period were comparably intermediate, in that they
It dates from the Late Babylonian period and gives, implied a ratio of 2:1 in night lengths, but used the 360° of
at five-day intervals, the weights (indicated by ki.lá) the zodiac as the argument of seasonal change.
of water (probably) corresponding to some fraction 49) Indeed the weights referred to in BM 29371 do not
of the night. The text presupposes the ideal year ofcorrespond with those in Mul.Apin, EAE 14 etc, suggesting
360 days and undoubtedly applies to some form of a slightly different outflow rate.
!0) A more accurate value for the length of the year is
water clock, though no name is given. It describes
implied in BM 36731 which describes the period from 616-
cant role.
itself as an arû "a mathematical table," but see also
588 BC. See Neugebauer & Sachs, JCS 21 (1967) pp. 183f.
n. 33. Most significantly, the text explicitly describes
5I) But see E. Leichty and C. B. F. Walker forthcoming.
a ratio of the longest to the shortest night as 3:2 in The text assigns the value 1;0 to the 15,h of month Su (IV)
terms of weight. It is in a sense intermediate to the and 1;30 to the 15lh of month ab (X).
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142 David Brown - John Fermor - Christopher Walker
the question of the accuracy of both
in the ideal year. It too describes the ratio approached
3:2 for the
prediction and observational measurement made by
longest to the shortest nights.
the LB scribes in the case of eclipses and recently
of the so-called "lunar six." This has been made
2.2 Accurate Measurement of Time after the mid-
possible through the determination of what is known
as the clock error (AT). This is the difference
between the terrestrial time (TT), defined by the
motion of the planets, and universal time (UT)
NA period examples of Mul.Apin and EAE, and
8th Century BC
extracts from the series sent in Letters and Reportsdefined by the rotation of the earth. That is AT = UT
written by scholars to their kings, attest to the- TT. This difference is caused by the slowing down
continued use of the ideal year and the 2:1 ratio of
in the earth's rotation and when accumulated over
the million or so days between now and the LB
divinatory contexts. At the same time, however,
period amounts to several hours. A value for AT is
these same Reports and Letters sometimes included
needed before any comparison can be made between
accurate descriptions of both the times and locations52
of celestial phenomena. In an Assyrian Report foundthe time of celestial events as calculated from today
in Nineveh and dating to 657 BC53, the time ofand
a their time according to the scribes of old57. This
done, Stephenson et al. have been able to establish
solar eclipse is given as 2Vi béru u4-mu "5 hours of
the day." How this was measured is unspecified, buta number of facts about the accuracy with which
a water clock may well have been used. Eclipse
Reports that include data from the mid-8th century
BC on, assigned times to the nearest 10 UŠ, or 40
minutes. In one dating to the 7th century BC the time
interval is given to the nearest UŠ, or 4 minutes.
From 560 BC on they often give times to the nearest
UŠ54. In the Babylonian Diary from 568 BC, line 4,
a time interval between sunrise and moonset was
observations were made by the LB scribes. In all
cases they assume, though without offering justifica-
tion, that the times recorded were done so with a
water clock (letters a through f refer to n. 55)58:
In (a) p. 26 If. they show that the mean discrepancy
in the times recorded for the phases of a lunar
eclipse (between -561 and -119) is as much as 7 UŠ,
or 28 minutes for intervals of mostly between 1 and
4 hours. They suggest that if the discrepancy inrecorded to the nearest UŠ. Accuracy at this apparent
creases with time it does so at the rate of about 13%
level is found in the later examples55.
To what extent were these statements the result of
observation using an accurate timing device, rather
than examples of apparent precision brought about
by the use of a finely graduated cylinder, say, that
had not been well calibrated against reality? In a
series of articles in the Journal for the History of
Astronomy (JHA), F. R. Stephenson et al.56 have
52) S. Parpóla, Letters from Assyrian and Babylonian
Scholars (1993) 47:r.7; 148:7; 149:r,5 and H. Hunger, Astrological Reports to Assyrian Kings (1992) 82:8; 104:4; 489:r.7.
53) Hunger, op. cit. 104:1.
M) The Eclipse Reports are published in Pinches, Sachs &
Strassmaier, Late Babylonian Astronomical and Related Texts
(= LBAT, 1955) Nos. 1414f. and are translated in P. Huber,
Babylonian Eclipse Observations 750 BC to 0 (1973), a
privately circulated manuscript.
") The Astronomical Diaries, 652 to 61 BC, have now
been published in 3 volumes by A. Sachs & H. Hunger,
Astronomical Diaries and Related Texts from Babylonia
(1988, 1989 & 1996).
56) (a) JHA 25 (1993) pp. 255-67, F. R. Stephenson and
L. J. Fatoohi, "Lunar Eclipse Times Recorded in Babylonian
History."
(b) JHA 26 (1994) pp. 99-1 10, F. R. Stephenson and L. J.
Fatoohi, "The Babylonian Unit of Time."
(c) JHA 28 (1997) pp. 119-31, F. R. Stephenson and
J. M. Steele, "Lunar Eclipse Times Predicted by the Babylo-
(a, p. 262). Clearly, this represents a very poor
performance and the authors compare this with the
accuracy achieved by pre-telescopic Arab astronomers
in (a) p. 261 of about 1 UŠ using altitude determinations. Their table 1, columns 1 and 3 and table 2,
columns 1 and 2 (pp. 260-1) also seem to show that
there is no evidence that the device used to measure
these intervals slowed down. This provides good
evidence that if a water clock was used it had a
constant head.
In (b) they show that the UŠ has no seasonal
variation or change over the centuries and was equal
to 4 minutes. They argue pp. 109-110 that their
evidence suggests that a water clock with a constant
inflow or outflow of water must have been used.
In (c) p. 123 the authors argue that since the
scatter in the AT values for larger time intervals
(e) JHA 28 (1997) pp. 337-45, F. R. Stephenson, J. M.
Steele and L. V. Morrison, "The Accuracy of Eclipse Times
Measured by the Babylonians."
(f) Forthcoming (perhaps in Astronomy & Geophysics)
J. M. Steele, "The accuracy of Babylonian observations of
lunar sixes."
57) See now F. R. Stephenson and L. V. Morrison, "Longterm fluctuations in the Earth's rotation 700 BC to AD
nians."
1990," in Phil. Trans. R. S. Lon. A cccli pp. 165-202, and
(d) JHA 28 (1997) pp. 133-9, J. M. Steele, "Solar Eclipse
Times Predicted by the Babylonians."
op. cit. for references to earlier literature.
58) (a) p. 258, (b) 100, (e) 338.
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The Water Clock in Mesopotamia 143
that the observers were unaware of the limitations of
measured by the Babylonians is much greater, then
the "clepsydras used ... were subject to considerable
the apparatus they were using. Much of the apparent
drift." They also show that the accuracy of the times
accuracy continued to be perceived precision, and
for predicted eclipses was of the order of an hour,
since there is no obvious way to determine the ratio
without any appreciable improvement over time.
between the longest and the shortest nights without
This is twice as inaccurate as the times recorded for
direct measurement of their duration, it is no wonder
that no refinement on the 3:2 ratio was achieved
observed eclipses, demonstrating that the phases of
observed eclipses were indeed measured (or if preafter 700 BC. However, the inability to devise mor
dicted after first contact, done so by methods different accurate timing devices did little to impede the
from those that predicted first contact).
development of the mathematical astronomy that
In (d) Steele shows that between -357 and +37 flowered in the last few centuries BC. It appears, for
there was no improvement in the accuracy of solar
example, that many of the fundamental parameter
eclipse prediction, which was half as accurate again
that underpinned this astronomy were derived merely
as lunar eclipse prediction.
from the dates of the observations of phenomena59.
In (e) the authors demonstrate that there was no
However, it also appears that one of the key functions
improvement in the accuracy of measurement over
in the lunar mathematical astronomical texts may
the 500 years or so of the records they investigate
have been in part derived from the records of the
(p. 342), nor any seasonal variation of accuracy (p.
lunar six phenomena60. It is perhaps significant, then,
343). In the latter case they suggest that this is
that the accuracy of their recording was substantially
surprising given the importance of temperature-debetter (f). Perhaps more accurate devices were empendent viscosity on the flow-rate of water. They
ployed in order to realise this part of cuneiform
suggest that the Babylonians' design of clepsydra
astronomical theory. Were these the descendants o
may deliberately have circumvented this problem.
the mašqúl
On pp. 343-4 the authors determine that the clock
The periods of time recorded in the Eclipse
drift (the increase of inaccuracy with time) is of the
order of 9% (reduced from 13% in a).
In (f) Steele argues that there was no improvement
in the accuracy of the observations of the intervals
around full and new moon between 400 and 78 BC,
with errors in the region of 2 UŠ for new moon and
4 UŠ for full moon sightings, and large clock drifts.
The conclusion from these interesting studies is
that little changed to improve the accuracy of observation in Babylonia between the 6th century BC and
1st century AD. The device(s) used appear(s) to have
been quite inaccurate - good only to the nearest 5 or
10 UŠ at best. However, it appears not to have
varied with temperature, and not to have slowed
down over time. These facts imply that if the device
were indeed a water clock, it was of the constant
head variety and may in some way have been
isolated from the extremes of temperature, perhaps
inside a building. The device's inaccuracy appears to
have increased as the period of its use lengthened,
but not in a way consistent with its slowing down.
This might suggest a variable flow rate, the erratic
topping-up of the device or any number of things
that might both underestimate or overestimate the
Records and the Diaries, and also in the NA Letters
and Reports used equinoctial units, i. e. UŠ and
béru. Seasonal hours, however, appear not to have
been used to record astronomical data. This is perhaps
because it would have been difficult to calibrate a
water clock for them, although it would have been
straightforward in the case of a sundial. The celestial
phenomena of interest were, of course, mainly noc-
turnal. Also, equinoctial units would have made
comparison of results simpler, although times were
generally measured from sunset, itself a seasonal
epoch61. Seasonal hours continued to play a divinatory
role until well into the Hellenistic period. It is
perhaps of note that the length of the midsummer
night seasonal hour in the text cited in n. 29 was
idealised to 10 UŠ, commonly the limit of perceived
precision at that time.
We remarked that there was evidence to suggest
that little or no improvement in the accuracy of what
were probably constant head outflowing water clocks
was made in the centuries after 750 BC. The clock
implied by the LB text considered below appears to
have used a flow rate comparable to the very earliest
devices. In the text AO 647862, however, a Seleucid
time measured. All the above serves only to increase
the likelihood that the device used in the late period
59) N. Swerdlow, The Babylonian Theory of the Planets
in the service of predictive astronomy, if a water
(1998) including earlier literature.
clock, was similar to the dibdibbu or maltaktum or
**) L. Brack-Bernsen, Die Babylonische Mondtheorie
even the mašqú described above.
(1997) including earlier literature.
However, since after c. 560 BC values in the
6I) For details see D. Brown, CAJ forthcoming.
Eclipse Reports and in the Diaries were recorded to
62) F. Thureau-Dangin, "Distances entre étoiles fixes,"
better than the nearest 5 or 10 UŠ, it also appears
RA 10 pp. 215-25.
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144 David Brown - John Fermor - Christopher Walker
leads to our final point, which concerns the
period list of stars and the times betweenThis
their
individuals who used the water clocks. There was no
culminations, a flow rate for a water clock of 10
distinction between diviner and astronomer in ancient
times that found in Mul.Apin is implied. 60 minas
correspond to one whole day. The same is probably Mesopotamia. Scholars performed both functions,
meant in the late period text BE 1391863, which
and there is good evidence to indicate that the
assigns the value "30" to the equinoctial day and prediction of celestial phenomena both assisted the
night. Presumably, variant water clocks were used in
the final few centuries BC, although explicit evidence
art of divination and enabled the complementary art
characterised this period.
to determine in which hour of the night a child was
born for the purposes of determining horoscopes64,
of zodiacal astrology to develop. Undoubtedly the
is still wanting. We still lack proof, however, that same devices were used by the astrologer-astronomers
any improvement in the accuracy of recording data to time visibility periods for astronomical prediction,
To conclude, this paper has presented textual and
and to see whether or not the universe was corre-
physical evidence to suggest that the water clock
sponding to the ideal for the purposes of divination.
used by the celestial diviners and those who produced
The inaccuracies in the water clock did not present
the astronomical data after c. 750 BC was of the
difficulties to them in any of these activities, for the
outflow variety. It was probably cylindrical or prisdivination aimed at throwing up anomalies regardless
matic, and combined a large reserve of water with
of areality, and the predictive models were based
small outflow conduit. This resulted in a near-
largely on the dates of celestial happenings and
constant head of fluid in the device that measured
celestial locations in space. At best the mathematically
equal times by equal amounts. The amounts outflow- derived predictions provided a date and a vague time
ing were weighed and convention initially establishedduring which a phenomenon would occur. This was
that 1 mina corresponded to 60 UŠ or 4 hours ofsufficient so far as the diviner was concerned, for the
time. We discussed a possible sand clock and the phenomenon still had to be seen in order to be
mašqú. A 2:1 ratio of the longest to the shortestinterpreted fully. If the water clock were truly the
night in terms of time, though inaccurate, played andevice the Scholars continued to use, even to the end
important role in divination, and we suggested that
of the cuneiform tradition, then the accuracy of the
attempting to account for it otherwise revealed more
device in its OB form would have been adequate for
about the aspirations of our society rather than the all the purposes stated. In a cultural milieu in which
Mesopotamian. Once the predicting of celestial phethe credibility of a discipline was in proportion to its
nomena became the subject of sustained activity in
purported antiquity, it would not be remotely surpristhe period after 750 BC, the keeping of accurateing if the last authors of the mathematical astronomical
records of heavenly events was combined with the
texts continued to employ the inaccurate dibdibbu or
repeated use of another, perhaps also ancient, ratio
maltaktum, a design by then perhaps two millennia
of 3:2. This ratio came to be used in both divinatory old.
and astronomical contexts thereafter.
♦
3.
BM
29371
(98-11
-14,
4)65
Obv.
1 ina itu.šu (4) u4-15-kam 1 ki.lá 1,12 1 ina itu.šu (4) u4-15-kam 1 ki.lá 1,12
u4-20-kam 1;0,50 1,13 u4-10-kam 1;0,50 1,13
u4-25-kam 1;1,40 1,14 u4-5-kam 1;1,40 1,14
63) F. H. Weissbach, Babylonische Miscellen (1903) pp.
50-1 and Tf. 15 No. 4. See also Kugler,
Erg
I-II,
mas. It SSB
includes
some .150
tabletsp.
of 90.
the N/LB period. Those
F. Rochberg, "Babylonian Seasonal
Hours,"
Centaurus
tablets
which record
their place of writing and date come
32 (1989) pp. 146-70.
from Babylon or Borsippa and from the mid-7!h to the early
65) Our thanks go to the Trustees5th
of
the British
Museum
centuries
BC: earliest date
Šamaš-šumu-ukin year 13 (BM
for their kind permission to publish
thisdate
tablet.
C. 35
B.(BM
F. 29569). Note also
29415), latest
Darius I year
Walker's "edition" of BM 29371 was first circulated at a
that BM 29440 mentioned above (n. 50) was acquired as part
conference in Graz in 1991. This tablet has no provenance, of the same collection. In the absence of other texts mentioning
having been purchased from a dealer. The 98-1 1-14 collection, our scribe we cannot be more precise about the date of the
comprising 247 tablets, was purchased from Mrs F. A. Sha-tablet.
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The Water Clock in Mesopotamia 145
u4-30-kam 1 ;2,30 1,15 1 ina itu.sig4 (3) u4-30-kam 1;2,30 1,15
1 ina itu.ne (5) u4-5-kam 1;3,20 1,16 u4-25-kam 1;3,20 1,16
u4-10-kam 1;4,10 1,17 u4-20-kam 1;4,10 1,17
u4-15-kam 1;5 1,18 u4-15-kam 1;5 1,18
u4-20-kam 1;5,50 1,19 u4-10-kam 1;5,50 1,19
u4-25-kam 1;6,40 1,20 u4-5-kam 1;6,40 1,20
u4-30-kam 1;7,30 1,21 1 ina itu.gu4 (2) u4-30-kam 1;7,30 1,21
1 ina itu.kin (6) u4-5-kam 1;8,20 1,22 u4-25-kam 1;8,20 1,22
u4-10-kam 1;9,10 1,23 u4-20-kam 1;9,10 1,23
u4-15-kam 1 ; 10 1,24 u4-15-kam 1 ; 10 1,24
u4-20-kam 1; 10,50 1,25 u4-10-kam 1; 10,50 1,25
u4-25-kam 1 ; 1 1 ,40 1,26 u4-5-kam 1 ; 1 1 ,40 1,26
u4-30-kam 1 ; 12,30 1,27 1 ina itu.bar (1) u4-30-kam 1 ; 12,30 1,27
1 ina itu.du« (7) u4-5-kam 1 ; 1 3,20 1,28 u4-25-kam 1 ; 1 3,20 1,28
u4-10-kam 1 ; 14, 1 0 1,29 u4-20-kam 1 ; 14, 10 1,29
u4-15-kam 1;15 1,30 u4-15-kam 1;15 1,30
u4-20-kam 1 ; 1 5,50 1,31 u4-10-kam 1 ; 1 5,50 1,31
u4-25-kam 1;16,40 1,32 u4-5-kam 1;16,40 1,32
u4-30-kam 1;17,30 1,33 1 ina itu.Se (12) u4-30-kam 1;17,30 1,33
Rev.
1 ina itu.apin (8) u4-5-kam 1 ; 1 8,20 1,34 u4-25-kam 1; 18,20 1,34
u4-10-kam 1 ; 19, 10 1,35 u4-20-kam 1 ; 1 9, 10 1,35
u4-15-kam 1;20 1,36 u4-15-kam 1;20 1,36
u4-20-kam 1;20,50 1,37 u4-10-kam 1;20,50 1,37
u4-25-kam 1;21,40 1,38 u4-5-kam 1;21,40 1,38
u4-30-kam 1;22,30 1,39 1 ina itu.áš (11) u4-30-kam 1;22,30 1,39
1 ina itu.gan (9) u4-5-kam 1;23,20 1,40 u4-25-kam 1;23,20 1,40
u4-10-kam 1;24,10 1,41 u4-20-kam 1;24,10 1,41
u4-15-kam 1;25 1,42 u4-15-kam 1;25 1,42
u4-20-kam 1;25,50 1,43 u4-10-kam 1;25,50 1,43
u4-25-kam 1;26,40 1,44 u4-5-kam 1;26,40 1,44
u4-30-kam 1;27,30 1,45 1 ina itu.ab (10) u4-30-kam 1;27,30 1,45
1 ina itu.ab (10) u4-5-kam 1;28,20 1,46 u4-25-kam 1;28,20; 1,46
u4-10-kam 1;29,10 1,47 u4-20-kam 1;29,10 1,47
u4-15-kam 1;30 ki.lá 1,48 u4-15-kam 1;30 1,48
arw(a.rá-ú) «ê/ne^(nam.kù.zu) dNabů(ng) g///i(im.gíd.da)
mdAfa6«(ag)-a/>/a(ibila)-Wí//«(sum.na)
mãri(z)-sú šá mdNabü(skg)-nädin(mu)-§umi(mu ) mãr( a) méš-g
ana tówar/;'(igi.du8)-šú «/Mr(in.sar)
Every line of the table ends with
statement:
1
"Inthe
the month
Du'üzu (month
4 = itu.šu), 15th day,
ammatÇkM) ?illu( giš.mi) 12A bëru(
u4-mu
"1
1 danna)
is the weight
(in minas
which corresponds to?)
cubit shadow, 1 2 h double-hours of
day."
1,12
(UŠ?). 1 cubit shadow, 1 2h double-hours of
The tablet is somewhat damaged,day."
but because it can
be restored with complete confidence
orfor zero which
- In line 2 damaged
a symbol is used
lost signs are not noted here. The elsewhere
table is
be read astronomical and
in to
LB mathematical,
texts is used
9.
down the left side and up then upeconomic
the right.
It for
gives
in sexagesimal numbers two amounts
(thereads:
first by
The colophon
weight = ki.lá = šuqultu) probably
"Mathematical
associated
table,
with
the wisdom
a
of Nabû, one column
water clock which measure some fraction of the
tablet (that) Nabû-apla-iddin, son of Nabû-nâdinnight, the amounts varying at five day intervals
šumi, descendant of Ešguzi-mansum, priest, copied
in order to read it."
throughout the year. The basic pattern is:
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146 David Brown - John Fermor - Christopher Walker
- Nabú-nadin-šumi can alternatively be read
nightNabûrequiring a still slower rate of egress of water
in the device.
šuma-iddin.
- Ešguzi-mansum as an ancestral name is otherwise Whatever the units implied in either column it is
unknown at present, but cf. the scribal name Ešguzi- clear that this text describes a water clock slightly at
gin-a (5 R 44 iii 44) explained as Esagil-kïn-apli,
and Lambert, JCS 11 (1957) 6.
variance with those described above. As we have
already seen, a variant with a flow rate 10 times that
- arû : CAD offers the following translations: productused in the older models may also have been used
(in multiplication), mathematical table, ephemeris. in the latest periods of cuneiform writing. These
The present tablet is a table of coefficients rathervariations are entirely consistent with the new interests
in accurate recording and prediction at this time,
than products. See also n. 33 above.
though in this case the developments in water clock
The text presupposes the ideal year of 360 days, technology have been applied once again to a dividivided into twelve 30-day months. The vernal equi- natory scheme.
nox was assumed to fall on the 15th of Nisannu
Most striking in BM 29371 is the 3:2 ratio in
terms
of weight. This makes more than likely the
(month 1 = itu.bar). The ratio of values assigned
to
case for direct proportionality between the weight of
the longest and the shortest nights in both columns
water
is equal to 3:2 (1 ;30: 1 or 1,48:1,12). The numbers
in outflowing and the time measured thereby. If
our 1
surmise that the second column contains values
the second column increase and then decrease by
in time units is correct, then the proportionality is
every five days. This suggests the use of a very
made explicit in this text at 1:72 (minas to UŠ). The
practical timing device, but in reality the numbers
use of the more accurate ratio in what is a traditional,
may have been the result of little more than simple
mathematical elegance. The ratio between the ideal
two form may indicate that this text was first
written long before the LB period. However, the
prevalence of the 2:1 ratio in both EAE and Mul.Apin
No units are attested, but weights are specified
(even as late as the Hellenistic period) suggests that
for the first column of numbers. From older parallels
the divinatory use of the 3:2 ratio may only have
it would seem at first sight probable that the weight
units were minas, with the values correspondingtaken
to place after the NA period. In which case it was
"watches," or thirds of the night. 1;15 minas would
perhaps a direct result of the improved accuracy in
columns is 1:72.
timing initiated then, and the development of more
thus have measured a period equivalent to one-third
accurate, rather than simply apparently precise, water
of the equinoctial night or 4 hours, or 1,0;0 (= 60)
clocks to that end.
UŠ in the equinoctial time units of the era. Were this
to have been the case, the ancient water clock
The statement concerning the one cubit shadow at
equation between 1 mina and 60 UŠ would not
l2/3 double-hours of the day, apparently unvarying
apply. The units implied in the second column mightthroughout the year, seems at first sight to be
also have been minas, but it seems probable that they
decidedly odd. It was perhaps the result of unthinking
were time units, and that the intervals described
copying. However, it may have had some basis in
corresponded to the weights in the first column in reality. If we assume that the shadow of one cubit
the ratio 1:72. C. B. F. Walker noticed that the
was cast from a vertical gnomon of one cubit in
largest and smallest numbers in the second column
length, then the sun at that moment must have been
45° above
the horizon. The question is then posed:
- 1,48 and 1,12 - are precisely half those used
to
at 12/î
describe the longest and shortest nights in system
A béru, or 3 hours and 20 minutes, after sunrise
of the mathematical astronomical texts in UŠ
units
how
high is the sun on any given day of the year?
- 3,36 and 2,24s6. If the time units in the second
Ignoring atmospheric and horizon effects, spherical
trigonometry
gives the following result for the solar
column were also UŠ, then at the equinoxes
1;15
altitude
minas (of water) flowed out, we believe, in 1;30
UŠ"a" at hour angle "H" for terrestrial latitude
solar declination "d"67:
(= 6 minutes) or in 1,30;0 UŠ (= 6 hours). "L"
Inand
the
sin(a) = sin(d)sin(L) + cos(d)cos(L)cos(H) (1)
transliteration we have opted for the latter possibility.
The solar declination d varies from approximately
+23
Vi" to -2314° over the course of the year. It is of
equation between 1 mina and 60 UŠ would again
not
course zero at the equinoxes. Babylon's terrestrial
apply and also suggest that the seasonally varying
latitude L is about 32'/2°. We can determine an hour
intervals described in minas and in UŠ were not
We note that in this case the ancient water clock
thirds of a night, but halves. In this eventuality angle
2;30 H0 when the altitude is zero (i. e. when the sun
is
just rising) for any declination. The hour angle 3
minas, and not the more frequently attested 3 minas,
would correspond to the length of an equinoctial
w) Neugebauer, ACT p. 47.
67) See for example figures 2.2 and 2.3 and equation 2.8
in R. M. Green, Spherical Astronomy (CUP 1985).
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The Water Clock in Mesopotamia 147
hours and 20 minutes after sunrise, H, will be
autumn. Only in win
and
20 minutes after s
H0 -50°, since that period of time constitutes 5%6oths
of 24 hours. Thus,
longer than one unit.
H0 = arccos (-tan(d)tan(L)) and (2)
H = H0 -50° (3)
large.
If
we
compare
hours after sunrise (F
in shadow lengths is m
after
sunrise,
fo
solarbéru
altitude
"a"
against
d
Determining
Babylon
will
cast a
s
(knowing "H") using
(1) will
provide
th
the question of how
length
high of
theabout
sun is
one
3 h
minutes after sunrise
on any
day of
the
rather
cryptic
statem
29371
was
based
on so
this, we can plot the
length
of
a shadow
in the
style
these
gnomon using l/tan(a).
This
is of
done
in
idealised
"one
As can be seen, the
lengthto
of
the cubi
shad
but
to
little
from
early
year.
mid-summer
spring
(declination
(declination
23°)
Fig. 1. Shadow length of a Gnomon of unit height for all
declinations, 3 hours and 20 minutes after sunrise at Babylon.
Fig. 2. Shadow length of a Gnomon of unit height for all declinations,
5 hours after sunrise at Babylon.
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and
-
ba
148 David Brown - John Fermor - Christopher Walker
BM 29371 obv. (Copyright British Museum)
BM 29371 rev. (Copyright British Museum)
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