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1999 Brown, Walker & Fermor The Water Clock in Mesopotamia

1999, Archiv für Orientforschung 46/47: 130-148

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 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 REFERENCES Linked references are available on JSTOR for this article: https://www.jstor.org/stable/41668444?seq=1&cid=pdfreference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms is collaborating with JSTOR to digitize, preserve and extend access to Archiv für Orientforschung This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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." This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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, This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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.). This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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). This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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: This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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). This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms and - ba 148 David Brown - John Fermor - Christopher Walker BM 29371 obv. (Copyright British Museum) BM 29371 rev. (Copyright British Museum) This content downloaded from 81.100.74.116 on Thu, 11 Feb 2021 11:57:53 UTC All use subject to https://about.jstor.org/terms