Abstract
The use of sexagesimal place-value notation (hereafter SPVN) is one of the most striking features of cuneiform mathematics. The earliest attestations of a systematic use of SPVN in cuneiform sources are found in a small set of mathematical texts dated to the Ur III period (ca. 2112 to 2004 BCE). Besides this mathematical corpus, traces of numbers written in positional notations have been found in some Ur III administrative texts. Just a few dozen tablets, among tens of thousands of known Ur III administrative documents, exhibit such numbers noted in positional notation. Moreover, such numbers noted in positional notation appear almost only as a kind of graffiti in the margins. This chapter focuses on these scanty ‘marginal numbers’. Marginal numbers in Ur III administrative texts testify to diverse practices with positional notations and sexagesimal factors in an administrative context, and, in this way, exemplify different ‘cultures of computation and quantification’ in the Ur III period. Through a close analysis of the positional notations found in Ur III administrative texts, we detect a diversity of graphical systems for what was considered until now as a uniform notion of SPVN. We show that these graphical systems vary according to the operations (multiplication, reciprocal, addition, subtraction), the contexts (administrative or mathematical), and the archaeological sites considered in this study (Umma, Girsu, Puzriš-Dagan, Nippur). Our goal is to show that this diversity of notations reflects different computational methods.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804. On the part of **aoli Ouyang, this article was finished during the period when she received general support from the ‘Eastern Scholar’ talent grant (2013–02) awarded by the Shanghai municipal government in China. This work deals with notations which are often omitted in publications and copies, and would not have been possible without the help of curators and colleagues working in different museums and collections. Our warmest thanks go to Irina Antonova, Boris Perlov and Yuriy Saveliev from the Pushkin State Museum of Fine Arts, Benjamin Foster, Ulla Kasten and Elizabeth Payne from the Yale Babylonian Collection, Béatrice André Salvini from the Musée du Louvre, Jean-Luc Chappaz from the Musée d’Art et d’Histoire of Geneva and Kenneth Woodrow Henke from the Special Collections of Princeton Theological Seminary for their kind authorization to work on tablets or use photos or both. We are also very grateful to numerous colleagues who facilitated our work and provided photos or access to archives, including Alassi Mahmoud and Laure-Cassandre Devic who helped with working on tablets and consulting archives at the Department of Near Eastern Antiquities of the Louvre, and Emmert Clevenstine who helped us with the Geneva tablets.
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Notes
- 1.
- 2.
Ist L 7375 is an Ur III mathematical tablet kept at the Archaeological Museum in Istanbul, Delaporte (1911); YBC 1793 (text 10 in the present study) is an Ur III administrative tablet kept at Yale, Keiser (1919). Another mathematical text from Ur III Girsu, AO 2728 (= AOT 304 = RTC 413), was published by Thureau-Dangin in 1903 (Thureau-Dangin 1903: 149), but was recognized as a mathematical text only in 1987 by Friberg (1987–1990: 451). See more on this text in Sect. 5.1.4 and edition in Appendix 4.
- 3.
Delaporte (1912).
- 4.
Delaporte (1911: 132–133).
- 5.
Neugebauer (1935: 10).
- 6.
Powell (1976a: 417). While this point is now widely shared among historians, some doubts persist. For Damerow and Englund, the supposed invention of SPVN as early as the Ur III period is ‘based primarily on the alleged dating to this period of several numerical tables,’ suggesting that for them this dating is far from proven (Nissen et al. 1993: 142).
- 7.
‘I have rather cautiously suggested that Sumero-Akkadian scribes of the Sargonic period were using a mental construct analogous to Old Babylonian place notation. Having reflected on the matter at length, I see that I have been overly hesitant: place notation of some type, if not the classical type that appears in Old Babylonian mathematical texts, must have been in use by the Sargonic period’ (Powell 1976b: 13).
- 8.
Powell (1976b).
- 9.
- 10.
After studying the table of surfaces CUNES 50-08-01, dated from the Early Dynastic period (mid-third millennium), Friberg (2007: 426) concluded that the ‘way of counting with small sexagesimal fractions could easily have led to the invention of sexagesimal place-value notation already in the middle of the third millennium!’.
- 11.
- 12.
‘Conceptions about the nature and origin of Babylonian place notation have always been closely linked or inextricably entangled with notions about the sexagesimal system of counting. The two phenomena are not, however, identical’ (Powell 1976a: 418).
- 13.
- 14.
- 15.
- 16.
Another system is also used for the evaluation of large surfaces (system G). See examples in Text 9 below.
- 17.
We have provided a copy instead of a photo of the marginal numbers, because the numbers appear faint in a photo and would be hard for a non-specialist to recognize.
- 18.
The Ur III mathematical texts are not the same if we compare for example the list obtained by entering the criteria ‘mathematical’ and ‘Ur III’ in CDLI, the list provided in Robson (2008: 306, Table B7) or Robson (1999, 169–71), and the examples quoted in Friberg (2009: §4). Even the two lists by Robson differ from each other as she explains in (2008: 349, note 9).
- 19.
Ist Ni 374 is published in Proust (2007) and HS 201 in Oelsner (2001); except Ist L 7375, the Girsu tablets are unpublished and were kindly mentioned to C. Proust by B. Lafont. Other tables of unknown provenience were tentatively considered to date from the Ur III period, even if they do not share these features, see Robson (2003–2004: 356–360); Proust (2008b: Chap. 2); Friberg (2009: Sect. 4.2.1–4.25).
- 20.
M(2,2) means a multicolumn tablet, where the text is divided into two columns on the reverse and two columns on the obverse. Old Babylonian reciprocal tables are single column tablets (type S).
- 21.
- 22.
However, according to the archives kept at the Musée du Louvre’s Antiquités Orientales library, the tablet was bought in 1898 from the dealers Morel and Géjou.
- 23.
- 24.
See the complete set of metrological tables in Proust (2009: Chap 9).
- 25.
1 gin-volume is a volume with a 1/60 sar base and a 1 kuš height. 1/60 sar corresponds to 1 (in floating SPVN) and 1 kuš corresponds to 5, thus 1 gin-volume corresponds to 5 and, as a result, 1 gur also corresponds to 5.
- 26.
Personal communications in the context of CDLI technical meetings. Probably, more examples can be found in different collections, for example at Yale, as noted by Powell (1976a: 435, note 6): ‘When I discussed YBC 1793 with W. W. Hallo, curator of the Yale Babylonian Collection, he told me that he remembered seeing sexagesimal notations rather frequently in balanced accounts from the Ur III period and referred me to YBC 4179, published by Ellis [1970]’.
- 27.
See respectively, http://bdts.filol.csic.es/ and http://cdli.ucla.edu/.
- 28.
Molina (2008: 52–53). The statistics concerning Umma and Girsu take into account a small number of texts, 260 and 126 respectively, with an uncertain provenience. Moreover, texts from two sites, Girsu and the much smaller Lagaš, are both counted as Girsu texts because they were sister sites located in the same administrative province we call Girsu or Girsu-Lagaš.
- 29.
For a brief overview of this issue and further references, see Ouyang (2013: 28–29).
- 30.
Although text 7 does not contain the phrase that we translate as ‘balanced account’ (níg-kas7-ak), it adopts the general structure of such an account and features, as discussed below, two other key terms (sag-nig2-gur11-ra-kam and ša3-bi-ta) characteristic of a balanced account.
- 31.
Such as the two texts AO 5674 (AS 3 xii) and AO 5676 (ŠS 2) mentioned in Sect. 5.5.1.
- 32.
See Ouyang (2013: chapter 5) for a systematic study of the Umma merchants.
- 33.
It has thus far been published only in a hand-copy, so we provide a full edition in Appendix 4.
- 34.
In fact, the name Lugal-hegal is attested in all the major corpora of the Ur III period. Despite that, the genre of this text (i.e. the silver account—see discussion above) provides the critical evidence for deducing the provenience of the tablet. In the Umma text, somebody named Lugal-hegal appears identified as a son of Šeš-kala in VAT 7042 (dated to AS 8; OrSP 47 382 53), as a son of Ur-Sin in Erm 14,994 (dated to IS 2; Santag 6 340), and as a pig farmer in HMA 9–02,824 (dated to AS 4; UCP 9–2-1 100); see Ouyang (2013: p. 84, note 178; p. 145, note 687); Snell (1982: 101).
- 35.
The transformation of the entire amount into SPVN would yield 12:57:50, according to the correspondence attested in Old Babylonian metrological tables.
- 36.
Powell (1976a: 435, note 6) first noted this kind of strange notation, found on the tablet BM 19027: ‘There is an instructive example of mixed notation (sexagesimal + standard metrological notation) written on the edge of a tablet dated to the last year of Shulgi [King 1898, pl. 30 No. 19027]’. See also Brunke (2011) and Ouyang (2016). Note that the notation of both numbers, 12:50 and 23:30, cannot be confused with system S (see Appendix 2).
- 37.
The quantity in-kind 1 barig 3 ban 6 1/2 sila 2 gin (SPVN 1:36:32) of lard was equivalent to 5 1/2 gin 23 1/2 še (SPVN 5:37:50) of silver. Both numbers and their quotient are irregular, thus the rate is irregular, and the calculation would involve approximation (see similar situation in Text 4). From the relation of 1:36:32 × 3:30 = 5:37:52, and the fact that 5:37:52 is very close to 5:37:50, we deduce that the rate in-silver was approximately 3:30 (10 1/2 še per sila). The rate in-kind used by the scribes in this evaluation was probably 17 (17 sila per gin) because 3:30 is an excellent approximation of the reciprocal of 17. By contrast, it is easy to see that the rate in-silver of the copper used here is 1 1/2 gin (of cooper) per gin of silver. The value in-silver is obtained by dividing 11 by the rate (1:30 in SPVN), that is multiplying 11 by 40, the reciprocal of 1:30, which gives 7:20 in SPVN, or 7 1/3 gin.
- 38.
It has thus far been published in a hand-copy by Snell (1982: No. 5) and in transliteration with a study by Englund (1992: 85–86). Scattered references to it appear in other studies concerning the Umma merchants. See, for example, Snell (1982: 25–26) cited as AS5PdB; Ouyang (2013: 125, note 452) cited as Ledgers pl. 8 5. We provide a full edition of this tablet in Appendix 4.
- 39.
Pada appeared to be one of the three best-documented merchants in Umma. For his documentation, see Ouyang (2013: 220–2); for the most recent discussion of his business activities, see ibid., Chap. 5.
- 40.
Englund (1992: 85) has pointed out that this carried-over balance is recorded as the deficit (la2-ia3) of Pada in the balanced account Ashm. 1924–667 (AS 5 xi) compiled earlier in the same year.
- 41.
As Snell (1982: 107) and Englund (1992: 85) have noted, this deficit appears almost the same as (only 10 še less than) the carried-over balance (2/3 mana 7 1/2 gin 27 še) in a balanced account of the next year, PUL Ex 662 (AS 6 xi). Images of that tablet show that between the two signs denoting the number 20 in ‘27 še’, the first U sign appears smaller and fainter than the second U sign, but does not look like an erasure.
- 42.
Englund (1992: 96, note 23) has noted the correspondence between the marginal number and the total of the expenditures: ‘48.40.10 appears to be a sexagesimal reconstruction of the total 48;50!,10 expressed in shekel. The final ‘10’ may have represented the 10 grains of the total, graphically separated from the notation itself since not a consequent part of a position value notation. A parallel situation seems to be attested in the text Nik. 2 402’.
- 43.
The quantity in-kind, 4 barig 1 ban 9 sila (SPVN 4:19) of bitumen, was equivalent to 2 5/6 gin (SPVN 2:50) of silver. Both numbers and their quotient are irregular. The rate in-kind hovers around 1:30, and the rate in-silver around 40 (reciprocal of 1:30). We cannot say anything about the other expenditure as its value in-kind is partially lost.
- 44.
- 45.
Actually, the amount appears with a mistake: a sign 5(diš) is inserted, but it cannot appear at this place according to the syntax of the measures of capacity. Perhaps the sign 5(diš) is an older sign that was badly erased.
- 46.
The correspondence between metrological notations in the main text and marginal numbers is well analyzed by Ellis (1970: 267).
- 47.
- 48.
Texts 4, 5 and 10 provide the rates explicitly. This situation is rare, as noted by Englund (2012: 440): ‘Texts such as MVN 11, 101, with multiple instances of explicit equivalency values in the form of 1(aš) 4(barig) 4(ban2) 6 sila3 mungur 3(aš) gur-ta / ku3-bi 2/3 (gin2) la2 3(diš) še (obverse 19–reverse 1) … are very rare’.
- 49.
An edition of the text appears in Ouyang and Brookman (2012: Sect. 3.4).
- 50.
- 51.
As cited by Englund (2012: 441), the quantity of animals, including bandicoot rats (peš2-giš-gi), two kinds of birds (amar-sag niga maš2 and amar-sag u2-ga maš2) and fish, and their respective value in-silver are recorded on the Umma tablet MAH 19353 (Š 39). Additional evidence can be found in Ouyang (2013: 291–2, Table 5.3.M-1), which lists quantities of fish, donkeys, cows, sheep and goats alongside their value in-silver.
- 52.
- 53.
The amount of silver corresponds to SPVN 7:57:16. Divide it by 10, that is, multiply it by 6 (the reciprocal of 10), we would get the result 47:43:36. The scribe might have rounded it down to 47:40 and then transformed it into 2/3 mana 7 2/3 gin.
- 54.
- 55.
The product of 1:27:50 and 5:54:42:8 approximates to 8:39:14:40. Thus the rate in-silver may fall between 5:54 and 5:55 (i.e. the reciprocal of the rate in-kind may fall between 10:7 and 10:9).
- 56.
Explanations on the reconstruction. For section A, the text is too damaged to be completely reconstructed. For section B, the calculations can be completely reconstructed, as the text is perfectly preserved. In section C, the first entry (obv. col. ii 3) containing the surface and the seeding rate for plot 8 is partially destroyed. However, the seeding rate is probably the same as in the first entry of section B for plot 4 (1 gur 2 barig 3 ban, SPVN 7:30), and the missing surface can be reconstructed. Indeed, we know the total quantity of seed for the four plots of field C (obv. ii 7) and we can calculate the quantity of seed for plots 9, 10 and 11 thanks to the information given in obv. col. ii 4–6. We obtained the quantity of seed for plot 8 by subtracting the seed for the other plots from the total, and finally, we obtained the surface of plot 8. In SPVN, the surface of plot 8 must be 9.29.57.29.40 (col. I of Table 5.22), which corresponds to 1 šaru 8 šar 5 buru 9 bur 2 eše 4 1/4 iku gan 24 sar 2/3 gin. Thus, line obv. ii 3 for plot 8 can be restored as follows: [1(šar’u) 8(šar2) 5(bur’u) 9(bur3) 2(eše)3] 4 1/4 (iku) GAN2 [24 sar 2/3 gin2 1(aš) 2(barig) 3(ban2)-ta]. In this case, 4 1/2 (iku) GAN2 must be read ‘4 1/4 (iku) GAN2’. For section D, the text is too damaged to be reconstructed. Another possibility is that this method, namely the multiplication by 2 and the use of a metrological table based on the correspondence of 1(bur3) GAN2 to 30 as in the Old Babylonian period, was replaced by a unique step, using a non-standardized table of capacity based on the correspondence of 1(bur3) GAN2 to 1 (instead of 30). In Old Babylonian B period, a unique set of standardized metrological tables served for all the calculations. A set of coefficients allowed the use of these tables. Here, this coefficient is 2.
- 57.
After its partial publication and interpretation by Thureau-Dangin (1897: 26–27; pl. 30 No. 79) at the end of the nineteenth century, this text has received little attention. It was mentioned by Powell (1976a: 435 note 6) who notes: ‘Two undated texts from this period which indicate the use of place notation in making calculations are Thureau-Dangin [1903, nos. 408 and 413]’. Lafont (1985) catalogued it under No. 61. We found neither the reference of the text nor its study elsewhere. The transliteration, translation (in Appendix 4) and analysis offered here complete Thureau-Dangin’s publication.
- 58.
For this reason, Thureau-Dangin (1897: 26) argued that ‘thus, it appears to be certain that it was considered as a unit’ (Il paraît donc certain que c’était considéré comme unité). The sign is now transliterated as 1(bur3).
- 59.
Perhaps, the trace […]2:[…]:40 (second line of marginal numbers) may correspond to the number 12:21:40 possibly involved in the calculation of the seeds for plot 7 of field B (obv. i 9). The number 45 (last line of marginal numbers – see rev. col. ii 8’ of the transliteration, Appendix 4) may correspond to one of the numbers ending with the digits 45 in column III of Table 5.22 (seeds for plot 4 or 5 of field B, or for plot 9 of field C).
- 60.
- 61.
In theory, the six numbers could also be partial-SPVN numbers, as we cannot distinguish the two systems when a number has no digits correspondent to the weight unit še. But in view of the parallelism with the four SPVN numbers at the start of the tablet, these six numbers are more likely to be SPVN numbers.
- 62.
The following hypothesis relies on Middeke-Conlin (2020).
- 63.
Such as the two 4s in obv. col. i 1. For normalized paleography, see the second number 4 in line obv. col. i 1, number 7 in obv. col. i 3, and the first number 40 in obv. col. i 3; for non-normalized paleography, see the first 4 in obv. col. i 1 and the second 40 in obv. col. i 3.
- 64.
As 1 gu 28 mana minus? 10 gin in rev. 7.
- 65.
- 66.
http://cdli.ox.ac.uk/wiki/doku.php?id=adopted_periodisation_in_cdli, accessed October 2015.
- 67.
For counting gan, the largest unit of surface, another system, the so-called ‘system G’ was used (see Text 9). The graphical repertoire of signs used in system G exhibits some similarities with system S, but the factors are different (see Annex A.1). Two units of capacity (ban and bariga) used different principles, with no graphical separation between numerical value and measurement unit. See more on this in Annex A.1.
- 68.
Note again that the term’sexagesimal’ is confusing because in fact system S is not purely sexagesimal, as underlined in Sect. 5.1.2. In particular, the first factor is ten, thus for low values, the system looks like a decimal numeration.
- 69.
The fact that SPVN always consists of isolated numbers (number without specification) is widely attested in Ur III and Old Babylonian mathematical texts, more details in Proust (2008a). The same rule applies in the texts examined in this chapter. Numbers in system S can appear isolated only in margins (see Sect. 5.1.2).
- 70.
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Thureau-Dangin, François. 1897. Un cadastre chaldéen. Revue d'Assyriologie 4: 13–27.
Thureau-Dangin, François. 1903. Recueil de Tablettes Chaldéennes. Paris: Leroux.
Thureau-Dangin, François. 1932. Esquisse d’une Histoire du Système Sexagésimal. Paris: Geuthner.
Whiting, Robert M. 1984. More evidence for sexagesimal calculations in the third millennium B. C. Zeitschrift für Assyriologie und Vorderasiatische Archäologie 74: 59–66.
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Appendices
Appendix 1: Notations and Chronology
(A map of Southern Mesopotamia in Ur III period is provided at the end of the book, Annex B, map 1).
1.1 Transliterations and Translations
-
For the transliterations and translations of numbers and measurement values, we follow the conventions provided in Annex A.1 at the end ofthis volume. However, given the peculiarity of the sources studied in this chapter, we add the following adjustment: when a semi-positional number uses horizontal wedges, we specify ‘aš’ in the transliteration, as in the example of 5(aš) 1:30 in Text 3.
-
In commentaries, we use capital letters to refer to individual cuneiform signs isolated from their context, e.g. AŠ versus aš, such as in our discussion of how the number of gur is written in the marginal numbers of Text 3.
-
The symbol ‘/’ is used to indicate layout features (e.g. /space/).
-
The notation < < x > > means that the sign ‘x’ is not expected.
-
In the tables summarizing the data from the texts (Sect. 5.2), we use parentheses to mark quantities and numbers in our calculation that are not attested in the texts.
1.2 Location and Orientation
The location of a certain line of a text on a tablet is specified in the following format: ‘side, column, line’. For example, ‘obv. col. ii 3’ means ‘obverse, column ii, line 3’; ‘rev. col. i 2’ means ‘reverse, column i, line 2’.
Upper and lower edges are defined in relation to the obverse. As tablets are read by rotation around the lower edge, notations on the upper edge are a continuation of the text of the reverse.
1.3 Chronology
General chronology, following the so-called middle chronology, after CDLI.Footnote 66
Period | Approximate dates |
---|---|
Early Dynastic I-II | ca. 2900–2700 BCE |
Early Dynastic IIIa | ca. 2700–2500 BCE |
Early Dynastic IIIb | ca. 2500–2340 BCE |
Old Akkadian (or Sargonic) | ca. 2340–2200 BCE |
Lagash II | ca. 2200–2100 BCE |
Ur III (or Third Dynasty of Ur) | ca. 2100–2000 BCE |
Old Babylonian | ca. 2000–1600 BCE |
Old Assyrian | ca. 1950–1850 BCE |
Ur III chronology
Ur III administrative texts are almost always dated, and historians refer to the dates provided by tablets using the year-month-day format in the following way: abbreviated name of the ruler, year of his reign, month in roman numerals (i, ii, …, xii and ‘diri’ which means ‘additional’ for intercalary months) and day.
The Ur III dynasty includes five rulers (the dates follow the middle chronology):
Ruler | Abbreviation |
---|---|
Ur-Namma (2112–2095) | UN |
Šulgi (2094–2047) | Š |
Amar-Suen (2046–2038) | AS |
Šu-Suen (2037–2029) | ŠS |
Ibbi-Suen (2028–2004) | IS |
For example, the date labelled in modern publications as ‘AS 5 ii 17’ represents the seventeenth day of the second month of the fifth year of Amar-Suen’s reign. Of course, the Sumerian system for naming years and months in cuneiform texts has nothing to do with this modern representation.
Appendix 2: System S
2.1 A Broad Definition of System S
System S, a sexagesimal additive system, is attested since the beginning of writing in Mesopotamia, and was used primarily for counting items such as animals, workers, days, years and other items of discrete collections. Diagram a of Fig. 5.25 represents the shape of the graphemes of system S used in the Ur III period, as well as the factors which define the value attached to each grapheme.
The numerical system used for expressing most of the measurement values exhibits the same features as system S (see diagram c, c’ and d in Fig. 5.25). The only variant is that the number 1 is represented by a horizontal wedge (aš) for counting the highest measurement units of capacity and weight (see diagram b of Fig. 5.25).Footnote 67
Small measurement units, namely, all of the units except the largest of each metrological system, were generally counted with numerical values less than sixty, noted with graphemes one ( ) and ten ( ) repeated as many times as necessary (see diagram c in Fig. 5.25). However, in the Ur III period, the notations were quite flexible, and, for example, the units ninda and sar may have been counted with numbers larger than sixty (see see diagram c’ in Fig. 5.25 and examples in Text 5).
Figure 5.25 shows that the systems represented by diagrams a, b, c, c’ and d share the same graphical repertory, the same base (alternation of factors ten and six), and the same additive principle. The only variation is the sign for 1, which is a horizontal wedge in b, but vertical in a, c, c’ and d. For the sake of simplicity, we term all of them as ‘system S’.Footnote 68
2.2 Distinguishing System S from SPVN
In diagrams a, c and c’, the cuneiform graphemes representing 1 and 60 are the same, namely, a vertical wedge ( ). In early texts, the wedge for sixty is bigger than the wedge for one. However, the sizes of both signs tended to become the same and confusion between one and sixty became inevitable. Subsequently, in the Old Babylonian period, the scribes differentiated one from sixty in ambiguous cases by specifying ‘sixty’ (šu-ši) when necessary (see Proust 2009: Sect. 5.5 for details), or by introducing new units (for example the length unit UŠ for sixty ninda), or by using fractions of superior units (for example, 1/2 gan for fifty sar).
In some instances, when numbers are less than 600 and use vertical wedges as units (see diagrams a and c), there is no graphical difference between SPVN and system S. However, some criteria can be used to recognize system S:
-
The function: system S is used for counting discrete items or measuring units.
-
The textual context: in main texts, a number in system S is always followed by the name of the items counted or a measurement unit; moreover, fractions never appear just after a number in SPVN.Footnote 69
-
The paleography: in the exemple we analyse in this article, the non-normalized paleographyPaleography is adopted in sytem S, while the normalized paleography is adopted for positional systems (SPVN and different forms of partial-SPVN).
Of course, for numbers more than 600, there is no ambiguity. For example, in Text 1, the marginal number (transliteration 23:30) cannot be interpreted as a number noted in system S, the paleography of which should be (transliteration 2(geš’u) 3(ges2) 3(u)). Another example: the positional system adopted for the number 48:30 found in Text 6 cannot be confused with system S .
Appendix 3: Catalogue of Ur III Administrative Texts with Positional Notations
Text numbers given in the first column of the catalogue below are the numbers used in the present chapter. Documents without a text number are not studied in detail here.
Typology of tablets
M(n,n’) means a multi-column tablet, on which the text is divided into n columns on the obverse, and n’ columns on the reverse.
S means single-column tablet.
Date: See chronology in Appendix 1.
Numerical systems:
SPVN means sexagesimal place-value notation.
Partial-SPVN means that the notation is partially sexagesimal or partially positional or both.
MN means marginal number.
Ø means not inscribed.
Text No. | Museum No | Date | Type of tablet | Type of text | Quantity in relation to MN | Location of MN | Numerical system of MN | ||
---|---|---|---|---|---|---|---|---|---|
MM 396 | AS 1? | Frag | Deliveryof precious metals and objects | Weight of silverand gold | Left edge | SPVN? | |||
1 | Nik. 2.402 | AS 4 | S | Balanced accountfor silver in-value | Weight of silver | Obv. bottom | Partial-SPVN | ||
2 | YBC 16487 | AS 5 | S | Balanced accountfor silver in-value | Weight of silver | Rev. bottom | Partial-SPVN | ||
3 | YBC 4179 | AS 6 | M(2,2) | Balanced accountfor grain | Capacity of grain | Rev. col. ii middle section | Partial-SPVN | ||
Rev. upper Balanced accounton aromatics edge | Partial-SPVN | ||||||||
FMB 39.2 | AS 7 | S | Weight and capacity of aromatics | Left edge | SPVN? | ||||
Nik. 2.403 | AS 9 | S | Balanced accountfor silver in-value | Capacity of cash crops or weight of silver | Rev. middle section | Partial-SPVN?or SPVN? | |||
Erlenmeyer 152 | ŠS 2 | M(5,5) | Balanced accountfor workdays | Number of workdays | Obv. col. iii upper section | SPVN | |||
Lo. edgeof obv. col. iii–v, rev. col. i–iv | System S | ||||||||
4 | YBC 16607 | ŠS 5 | S | Balanced accountfor silver | Weight of silver | Lower edge | Partial-SPVN | ||
YBC 3883 | ŠS 5 | M(2,2) sealed | Agricultural work measured in workdays | Number of workdays | Obv ii upper section | SPVN? (normalized) | |||
5 | E 15550 | IS 2 | S | Earthwork for different sections of a dike | Volume of earth | Left edge | SPVN | ||
Puzriš-Dagan | |||||||||
6 | PTS 473 | Š 48 vii | S | Two batches of livestock as booty | Number of animals | Rev. lower section | SPVN | ||
WCMA 20.1.07 | ŠS 6 diri | S(1, Ø) | Distribution of barleyas rations | Capacity of grain | Obv. lower section | Partial-SPVN | |||
7 | NBC 6641 | ŠS 8 ix | S | Distribution of silver for purchase of gold | Weight of silver | Lower edge | SPVN | ||
Left edge | SPVN | ||||||||
8 | YBC 13418 | No date | S | Deliveriesof gold | Weight of silver | Upper edge | Partial-SPVN? | ||
Rev. bottom | Unknown | ||||||||
YBC 1778 | Date damaged | S | Balanced accountfor sheep and goats | Numbers of animals | Obv. middle section | SPVN? (normalized) | |||
Girsu | |||||||||
BM 19027 | Š 48 | M(4,4) | Grain balanced account | Capacity of grain | Left edge | Partial-SPVN?or SPVN? | |||
SM 1909.05.224 | No date? | S | Survey of two fields | Surfaceof fields | Obv. lower section | SPVN | |||
9 | AO 27307 | Date lost | M(2,2) | Seed for given surfaces | Capacity of seeds | Rev. col. ii lower section | SPVN | ||
Nippur | |||||||||
CBS 11661 | IS 2 xi 30 | M(2,2) | Capacity of grain | SPVN? (damaged) | |||||
Uncertain | |||||||||
10 | YBC 1793 | AS 5 ix | M(2,Ø) | Value in-silverof different deliveries, or deliveries of silver | Weight of silver | Obv. col. i upper section (main text) | SPVN | ||
Obv. col. ii lower section | SPVN |
Appendix 4: Texts
4.1 Mathematical Text
AO 2728 (=AOT 304 = RTC 413).
Location of marginal number: reverse.
Provenience: Girsu
Date: Ur III (no date written on the tablet).
Location of tablet: Musée du Louvre, Paris, France.
Reference No.: BDTNS 000853; CDLI P128566.
Ed. Thureau-Dangin (1903): No. 413.
Photo: http://www.cdli.ucla.edu/dl/photo/P128566.jpg
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | 6(diš) ninda 4(diš) 1/3 kuš3⌈gid2⌉ | 6 ninda4 1/3 kuš the length, |
2 | 1/2 ninda sukud | 1/2 nindathe height, |
3 | 2(diš) kuš35(diš) šu-si dagal | 2 kuš5 šusi the width |
4 | sahar-bi 3(diš) 1/2 sar 2(diš) 1/2 gin2 | Its volume 3 1/2 sar2 1/2 gin, |
5 | sig4-bi 2(u) 5(diš) 1/2 sar | its brick-volume 25 1/2 sar |
Lower edge | 30? 6 | |
Reverse | ||
1 | 6:31:50 | 6:31:50 |
2 | 3 10:50 | 3 10:50 |
Commentary:
Our transliteration follows the works of Thureau-Dangin (1903: No. 413) and Robson (1999: 66), and the collation in the Musée du Louvre on February 6, 2014, by C. Proust.
Lower edge: The numerical signs are visible, but they do not appear in Thureau-Dangin’s copy.
4.2 Administrative Texts
The following editions of texts rely on previous publications, on transliterations provided by CDLI and BDTNS, on digital photos, and, in most cases, on our own examination of the tablets. Both databases were last accessed in February 2014 unless otherwise stated. Only photos of the tablets (obverse and reverse) without images in CDLI are provided here. Photos of the edges with marginal numbers in Texts 1–10 appear in Sect. 5.2.
Text 1-Nik. 2 402
Location of marginal number: Obverse, bottom.
Provenience: Umma
Date: AS 4.
Location of tablet: Pushkin State Museum of Fine Arts, Moscow, Russia.
Museum No.: Unavailable.
Reference No.: BDTNS 004641; CDLI P122085.
Copy: Nikol'skij (1915): No. 402 (Fig. 5.26).
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | 1(u) 3(diš) 1/3 gin2ku3 | 13 1/3 gin of silver, |
2 | ša3-bi-ta | From its within: |
3 | 1(barig) 3(ban2) 6(diš) 1/2 sila32(diš) gin2 i3-šah2 | 1 barig3 ban 6 1/2 sila 2 gin of lard |
4 | ku3-bi 5(diš) 1/2 gin22(u) 3(diš) 1/2 še | Its value in-silver 5 1/2 gin 23 1/2 še; |
5 | 1(u) 1(diš) ma-na [uruda?] | 11 mana of … |
6 | ku3-bi 7(diš) 1/3 gin2 | Its value in-silver7 1/3 gin; |
7 | 12:50 /space/ 23:30 | 12:50 /space/ 23:30 |
Reverse | ||
1 | šu-nigin21(u) 2(diš) 5/6 gin2 2(u) 3(diš) 1/2 še | In total: 12 5/6 gin 23 1/2 še |
2 | zi-ga-am3 | Expended; |
3 | la2-NI 1/3 gin2 6(diš) 1/2 še | Deficit 1/3 gin 6 1/2 še |
4 | Blank line | |
5 | nig2-kas7-ak lugal-he2-gal2 | A balanced accountof Lugal-hegal |
6 | mu en-mah-gal-an-na en dnanna ba-hun | Year when Enmahgal-ana, the en-priestess of Nanna, was installed |
Commentary:
Obv. 5: The name of the product appears lost in BDTNS but is restored as [uruda] in CDLI. We find the rate in-kind of this product to be 1 1/2 mana per gin of silver, which is close to the rates in-kind of copper calculated by Snell (1982: 150) based on the Ur III evidence.
Obv. 7: The copy of the tablet indicates an erasure of three vertical wedges right above the second sign, 2(diš), of the marginal number. The images of the tablet show only an erasure, probably from the tip of a finger, in the same place, but we cannot tell how many wedges may have been erased.
Text 2-YBC 16487
Location of marginal number: Reverse, bottom.
Provenience: Umma
Date: AS 5.
Dimensions (cm): 4.9 × 4.5 × 2.
Location of tablet: Yale Babylonian Collection, New Haven, Connecticut, USA.
Reference No.: BDTNS 024111; CDLI P112497.
Copy: Snell (1982): No. 5 (Fig. 5.27).
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | 1(diš) 1/2 ma-na 6(diš) 1/3 gin22(u) 7(diš) še ku3-babbar | 1 1/2 mana 6 1/3 gin 27 še of silver, |
2 | si-i3-tum | Carried-over balance, |
3 | ša3-bi-ta | From its within: |
4 | 1/2(diš) ma-na 8(diš) gin2ku3-babbar | 1/2 mana 8 gin of silver |
5 | mu-kux(DU) | As delivery; |
6 | 4(barig) 1(ban2) 9(diš) sila3esir2 e2-a | 4 barig 1 ban 9 sila of bitumen |
7 | ku3-bi 2(diš) 5/6 gin2 | Its value in-silver 2 5/6 gin; |
8 | […] ┌1┐(barig) ┌5┐(ban2) 6(diš) sila3naga si-e3 gur | […] + 1 barig 5 ban 6 sila sprouted alkaline plants, |
9 | [ku3-bi x] ┌gin2┐[x] ┌še┐ | Its value in-silverx gin x še; |
Rest broken | ||
Reverse | ||
Beginning broken | ||
Blank line | ||
1’ | ┌šu-nigin22/3┐ ma-na 8(diš) 5/6 gin2 1(u) še ku3 | In total: 2/3 mana 8 5/6 gin 10 še of silver |
2’ | zi-ga-am3 | Expended; |
3’ | la2-NI 2/3 ma-na 7(diš) 1/2 gin21(u) 7(diš) še ku3 | Deficit 2/3 mana 7 1/2 gin 17 še of silver |
4’ | nig2-kas7-ak pad3-da dam-gar3 | A balanced accountconcerning Pada the merchant |
5’ | mu en-unu6-gal dinanna ba-hun | Year when En-unugal of Inanna was installed |
6’ | 48:40sic?/space/ 10 | 48:40 /space/ 10 |
Text 3-YBC 4179
Location of marginal numbers: Upper edge and middle section of rev. col. ii.
Provenience: Umma
Date: AS 6.
Dimensions (cm): 15.2 × 8.8 × 2.5
Location of tablet: Yale Babylonian Collection, New Haven, Connecticut, USA.
Reference No.: BDTNS 015950; CDLI P111807.
Copy: Ellis (1970): 268–269 (Fig. 5.28).
Transliteration | Translation | |
---|---|---|
Obverse | ||
Col. i | ||
1 | 8(aš) 1(ban2) 8(diš) sila3 še gur | 8 gur 1 ban 8 sila of barley |
2 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
3 | Erased line | |
4 | mu en dnanna | Year of Š 43; |
5 | 8(aš) 1(ban2) 8(diš) sila3gur | 8 gur 1 ban 8 sila |
6 | iti 1(u) 2(diš)-kam | For twelve months, |
7 | 3(barig) 2(ban2) 1(diš) 1/2 sila3iti diri | 3 barig 2 ban 1 1/2 sila for the additional month, |
8 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
9 | (erasureat beginning) še kaš mu-kux(DU) | Delivery of barley for beer |
10 | mu si-mu-ru-um lu-lu-bum2ki | For the year of Š 44; |
11 | 8(aš) 1(ban2) 8(diš) sila3gur | 8 gur 1 ban 8 sila |
12 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
13 | (erasureat beginning) še kaš mu-kux(DU) | Delivery of barley for beer |
14 | mu ur-bi2-lumki | For the year of Š 45; |
15 | 8(aš) 1(ban2) 8(diš) sila3gur | 8 gur 1 ban 8 sila |
16 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
17 | (erasureat beginning) še kaš mu-kux(DU) | Delivery of barley for beer |
18 | mu ki-maškiba-hul | For the year of Š 46; |
19 | [8(aš)] 1(ban2) 8(diš) sila3iti 1(u) 2(diš)-kam | 8 gur 1 ban 8 sila for twelve months |
20 | [3(barig) 1(ban2)] ⌈6⌉(diš) 1/2 sila3iti diri | 3 barig 1 ban 6 1/2 sila for the additional month |
21 | [še kaš] ⌈u4⌉-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
22 | (erasureat beginning) ⌈sila3⌉ še kaš mu-kux(DU) | Delivery of barley for beer |
23 | [mu us2]-⌈sa⌉ ki-maški | For the year of Š 47; |
24 | [8(aš) 1(ban2) 8(diš) sila3] gur | 8 gur 1 ban 8 sila for twelve months |
25 | [še kaš u4-sakar gišgigir]-⌈ra⌉ | Barley for beer for the chariot’s crescent |
26 | (erasureat beginning) še kaš [mu]- ⌈kux(DU)⌉ | Delivery of barley for beer |
Col. ii | ||
1 | mu ha-ar-šiki | For the year of Š 48; |
2 | 8(aš) ⌈1(ban2)⌉ [8(diš) sila3] ⌈gur⌉ | 8 gur 1 ban 8 sila |
3 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
4 | (erasureat beginning) še kaš mu-kux(DU) | Delivery of barley for beer |
5 | mu damar-dsuen lugal | For the year of AS 1; |
6 | 8(aš) 1(ban2) 8(diš) sila3gur | 8 gur 1 ban 8 sila |
7 | iti 1(u) 2(diš)-kam | For twelve months, |
8 | 3(barig) 1(ban2) 6(diš) 1/2(diš) sila3iti diri | 3 barig 1 ban 6 1/2 sila for the additional month, |
9 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
10 | (erasureat beginning) še kaš mu-kux(DU) | Delivery of barley for beer |
11 | mu damar-dsuen lugal-e | For the year of AS 2 |
12 | ur-bi2-lumkimu-kux(DU) | Delivery; |
13 | 8(aš) 1(ban2) 8(diš) sila3gur | 8 gur 1 ban 8 sila |
14 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
15 | (erasureat beginning) sila3 še kaš mu- < kux(DU) > | Delivery of barley for beer |
16 | mu gu-za den-lil2-la2 | For the year of AS 3; |
17 | 8(aš) 1(ban2) 8(diš) sila3iti 1(u) 2(diš)-kam | 8 gur 1 ban 8 sila for twelve months |
18 | 3(barig) 1(ban2) 6(diš) 1/2(diš) sila3iti diri | 3 barig 1 ban 6 1/2 sila for the additional month, |
19 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
20 | (erasureat beginning) še kaš mu-kux(DU) | Delivery of barley for beer |
21 | mu en-mah-⌈gal⌉ […] | For the year of AS 4; |
22 | 8(aš) 1(ban2) 8(diš) sila3gur | 8 gur 1 ban 8 sila |
23 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
24 | 3(barig) 3(ban2) še kaš mu-kux(DU) | 3 barig 1 ban, delivery of barley for beer |
25 | mu en-unu6-gal | For the year of AS 5; |
26 | 8(aš) 1(ban2) 8(diš) sila3⌈gur⌉ | 8 gur 1 ban 8 sila |
27 | iti 1(u) 2(diš)- ⌈kam⌉ | For twelve months |
Reverse | ||
Col. i | ||
1 | 3(barig) 1(ban2) 6(diš) 1/2(diš) sila3iti ⌈diri⌉ | 3 barig 1 ban 6 1/2 sila for the additional month |
2 | še kaš u4-sakar gišgigir-ra | Barley for beer for the chariot’s crescent |
3 | 4(barig) 3(ban2) še kaš mu-kux(DU) | 4 barig 1 ban, delivery of barley for beer |
4 | mu ša-aš-rukiba-⌈hul⌉ | for the year of AS 6; |
5 | 7(aš) 1(barig) 4(ban2) 6(diš) 1/2 sila3še kaš mu-kux(DU) | 7 gur 1 barig 4 ban 6 1/2 sila, delivery of barley for beer |
6 | mu en dnanna-ta mu en-mah-gal-še3 | From the year of Š 43 to AS 4; |
7 | šu-nigin21(geš2) 4(u) 8(aš) 4(barig) 5(ban2) še gur | In total: (60 + 48) gur 4 barig 5 ban |
8 | ša3-bi-ta | From its within: |
9 | 8(aš) gur mu en dnanna maš2-e i3-pa3 | 8 gur in the year of Š 43, |
10 | 8(aš) gur mu si-mu-ru-um lu-lu-bu-umki | 8 gur in the year of Š 44, |
11 | 9(aš) 1(ban2) 5(diš) sìla gur | 9 gur 1 ban 5 sila |
12 | mu ur-bí-lumki | In the year of Š 45, |
13 | 9(aš) 1(ban2) 5(diš) sila3gur | 9 gur 1 ban 5 sila |
14 | mu ki-maški | In the year of Š 46, |
15 | 9(aš) 1(ban2) 5(diš) sila3gur | 9 gur 1 ban 5 sila |
16 | mu us2-sa ki-maški | In the year of Š 47, |
17 | 3(u) 9(aš) 1(barig) 6(diš) 1/2 sila3gur | 39 gur 1 barig 6 1/2 sila |
18 | mu ha-ar-šiki-ta | From the year of Š 48 |
19 | mu en-mah-gal-še3 | To the year of AS 4, |
20 | 9(aš) 3(barig) gur | 9 gur 3 barig |
21 | mu gu-za ba-dim2 | In the year of AS 3, |
22 | 2(u) 2(aš) še gur | 22 gur of barley |
23 | mu en-unu6-gal u3mu ša-aš-ruki | In the years of AS 5 and AS 6, |
24 | ⌈kišib?⌉ lu2-dnin-šubur | Sealed tablet of Lu-Ninšubur, |
Col. ii | ||
1 | šu-nigin21(geš2) 5(u) 3(aš) 4(barig) 5(ban2)1(diš) 1/2 sila3 še gur | In total: (60 + 53) gur 4 barig 5 ban 1 1/2 sila of barley |
2 | zi-ga-am3 | Expended |
3 | diri 5(aš) 1(diš) 1/2 < < 5(diš) > > sila3še gur | Surplus: 5 gur 1 1/2 sila of barley |
5(aš) /erasure/ 1:30 | 5(aš) 1:30 | |
4 | nig2-ka9-ak še kaš u4-sakar gišgigir-ra | The balanced accountconcerning the barley for beer for the chariot’s crescent |
5 | lú-dnin-šubur | In the charge of Lu-Ninšubur |
6 | mu1(u) 2(diš)-kam mu en dnanna maš2-e i3-pa3-ta | Twelve years from the year of Š 43 |
7 | mu ša-aš-rukiba-hul-šè | To AS 6 |
Upper edge | ||
7(aš) 1:46:30 | 7(aš) 1:46:30 |
Commentary:
Obv. col. i 7: The amount for the additional month happens to equal the annual total in obv. col. i 5 divided by twelve. However, the amount for the additional month in other years (Š 47, AS 2, AS 4, AS 6) turns out to be 5 sila less than the annual total divided by twelve (see summary of data in Sect. 5.2).
Obv. col. i 19: Restoration of [8(aš)] based on parallel amounts attested for other years.
Obv. col. i 20: Restoration of [3(barig) 1(ban2)] ⌈6⌉(diš) 1/2 sila3 based on the numerical relationship between quantities of different years (see summary of data in Sect. 5.2).
Obv. col. i 24: Restoration of [8(aš) 1(ban2) 8(diš) sila3] based on parallel amounts attested for other years.
Obv. ii 2: Restoration of [8(diš) sila3] based on parallel amounts attested for other years.
Rev. col. i 5: Although there appears to be no ruling between this line and the following line 6, the latter does not have an indenture at the beginning. So we separate these two lines in our numbering.
Text 4-YBC 16607
Location of marginal number : Lower edge
Provenience: Umma
Date: ŠS 5.
Dimensions (cm): 8.8 × 4.7 × 2.2
Location of tablet: Yale Babylonian Collection, New Haven, Connecticut, USA.
Reference No.: BDTNS 024119; CDLI P112505.
Copy: Snell (1982): No. 13.
Photo: http://www.cdli.ucla.edu/dl/photo/P112502.jpg
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | 1(geš2) 1(u) še gur 1(aš) 5(ban2) gur-ta! | 60 + 10 gur of barley (at the rate of one gin of silver) per 1 gur 5 ban, |
2 | 1(geš2) 5(u) gur 1(aš) 4(ban2) gur-ta | 60 + 50 gur (at the rate of one gin of silver) per 1 gur 4 ban, |
3 | ku3-bi 2(diš) 1/2 ma-na 7(diš) gin2 9(diš) 1/2 še | Their value in-silver 2 1/2 mana 7 gin 9 1/2 še, |
4 | mu dšu-dsuen lugal | Year when Šu-Sin (became) king (ŠS 1); |
5 | 1(geš2) 1(u) gur | 60 + 10 gur, |
6 | ku3-bi 1(diš) ma-na 1(u) gin2 | Its value in-silver 1 mana 10 gin, |
7 | mu ma2 dara3 ab-zu den-ki ba-ab-du8 | Year when the boat ‘ibex of the abzu’ was caulked for Enki (ŠS 2); |
8 | 2(geš2) 1(u) ┌4┐(aš) gur | 60 × 2 + 14 gur, |
9 | ku3-bi 2(diš) 1/2┌ma┐- < na > 7(diš) 1/2 < gin2 > | Its value in-silver 2 1/2 mana 7 1/2 gin, |
10 | mu si-ma-num2ki ba-hul | Year when Simanum was destroyed (ŠS 3); |
11 | Blank line | |
12 | še ku3-še3 sa10-a | Barley sold for silver; |
Lower edge | ||
6:57:20 /space/ 15 | 6:57:20 /space/ 15 | |
Reverse | ||
1 | 1/2 ma-na 2(diš) 5/6 gin2 5(diš) še ku3-babbar | 1/2 mana 2 5/6 gin 5 še of silver, |
2 | la2-NI su-ga ugula kikken2 sa2-du11 ensi2-ka | Repaid arrears of the overseer of a milling house, as the sadu-offering of the governor; |
3 | Blank line | |
4 | [6(diš)] ┌5/6┐ ma-na 7(diš) 1/3! gin2 1(u) 5(diš) še | 6 5/6 mana 7 1/3 gin 15 še |
5 | ┌ki?┐ […]-ta | From … |
6 | 1(u) 4(diš) 2/3(diš) gin2 sám x x x zabar uruda šu-nir-ra x | 14 2/3 gin as price … for bronze and copper for the emblem …; |
7 | 4(diš) 1/2 ma-na ku3-babbar | 4 1/2 mana of silver, |
8 | ┌kišib┐┌ensi2┐-ka | Sealed receipt of the governor; |
9 | la2-NI 2(diš) ma-na 1(u) 3(diš) gin2 ku3-babbar | Deficit: 2 mana 13 gin of silver |
10 | nig2-ka9-ak ku3ensi2-ka | A balanced account on the silver of the governor, |
11 | gir3 lu2-kal-la | Conveyed by Lu-kala |
12 | mu dšu-dsuen lugal-ta | From the year when Šu-Sin became king (ŠS 1) |
13 | mu us2-sa bad3 mar-tu ba-du3-še3 | To the year after the western wall was built (ŠS 5) |
Commentary:
Rev. 4: Restoration of [6(diš)] based on the relationship that the quantities in obv. 3, 6, 9 and rev. 1 add up to the total in rev. 4 (see summary of data in Sect. 5.2).
Regarding the fraction before gin2, both BDTNS and CDLI read it as 1/2, but we expect it to be 1/3 instead. Our collation of the tablet reveals two short vertical strokes below and above the horizontal stroke but to the right of the vertical stroke in the MAŠ sign representing 1/2. These two may indicate a long but half-erased vertical stroke. If so, then we may restore the fraction as 1/3.
Text 5-E 15550
Location of marginal number : left edge
Provenience: Umma
Date: IS 2.
Dimensions (cm): 10.6 × 5.5 × 2.5
Location of tablet: Peabody Essex Museum, Salem, Massachusetts, USA.
Reference No.: BDTNS 191620; CDLI P416398.
For photos, transliteration and translation, see Ouyang and Brookman (2012): 3.4.
Text 6-PTS 473
Location of marginal number: Reverse, middle section.
Provenience: Puzriš-Dagan
Date: Š 48 vii.
Location of tablet: Princeton Theological Seminary, Princeton, New Jersey, USA.
Dimensions (cm): 7.5 × 4.5 × 1.5
Reference No.: BDTNS 032993; CDLI P126749.
Copy: Sigrist (1990): No. 60.
Photo: http://www.cdli.ucla.edu/dl/photo/P126749.jpg
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | [2(geš2)] 4(u) ┌5(diš)┐ ab2 | 2 × 60 + 45 cows |
2 | […] [6 + ]2(geš2) [3(u)] 4(diš) udu | […] + 8 × 60 + 34 sheep |
3 | […] [5 + ]2(geš2) 2(u) 1(diš) maš2 | […] + 7 × 60 + 21 sheep |
4 | 2(geš2’u) 2(geš2) ┌3(u)┐ 4(diš) ud5 gun3 | 2 × 600 + 2 × 60 + 34 speckled female goats |
5 | [2 + ]1(geš2’u) ┌2(u)┐ 7(diš) ud5 | 3 × 600 + 27 female goats, |
6 | (Blank space) 2(geš2) 4(u) 5(diš) ab2 2(šar2) 8(geš2) 5(u) 6(diš) udu | (In total) 2 × 60 + 45 cows 2 × 3600 + 8 × 60 + 56 sheep |
7 | ┌gir3┐ bu-bu | Conveyed by Bubu; |
8 | 1(geš2) 6(diš) ab2 | 60 + 6 cows |
9 | 5(geš2’u) udu maš2 hi-a | 5 × 600 sheep and goats of various kinds |
10 | gir3 šu-den-lil2 | Conveyed by Šu-Enlil; |
11 | nam-┌ra┐-ak | Booty |
12 | ki-maški┌ha┐-ar-šiki | Of Kimaš and Harši |
Reverse | ||
1 | […]-x┌ki┐ | […] |
2 | šušin x […]-x-ra | Susa … |
3 | kišib sà-┌sí┐ […] x | Sealed receipt of … |
4 | šu-┌d┐[…] ┌TI?┐.A […] di-ku5-┌x┐ | … |
5 | ┌tum3┐-dam | To bring |
2/space/ 8:56 1 | 2:8:56 1 | |
6 | iti [ezem]-┌d┐šul-gi | The month of the ‘Festival of Šulgi’ |
7 | mu ┌ha-ar┐-šiki u3 ki-maš┌ki┐ ba-hul | The year when Harši and Kimaš were destroyed |
Commentary:
Obv. 1: Restored according to the first part of obv. 6.
Obv. 2: Partially restored based on the numerical relationship of this line with Obv. 3–6 (see summary of data in Sect. 5.2).
Obv. 3: Partially restored based on the numerical relationship of this line with Obv. 2, 4–6 (see summary of data in Sect. 5.2).
Obv. 4: In contrast to the hand-copy by Sigrist and the reading of BDTNS and CDLI, our collation of the tablet shows no A sign at the end.
Rev. middle section: A DIŠ sign appears in a second line and right below the 8(diš) sign of the marginal number. We do not understand the meaning of this sign.
Text 7-NBC 6641
Location of marginal numbers: Lower edge and left edge.
Provenience: Puzriš-Dagan
Date: ŠS 8 ix.
Dimensions (cm): 11.1 × 6.2 × 2.7
Location of tablet: Yale Babylonian Collection, New Haven, Connecticut, USA.
Reference No.: BDTNS 077468; CDLI P298493.
Copy: Garfinkle (2008)
Edition: Garfinkle (2008), Paoletti (2008) and Paoletti (2012): 448–9 (Fig. 5.29).
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | 1(aš) gu21(u) ma-na ku3-babbar | 1 gu 10 mana of silver |
2 | nig2-sa10-ma ku3-sig171(u)-ta-še3 | As purchase price of gold at the rate of 10:1 |
3 | sag-nig2-gur11-ra-kam ša3-bi-ta | This is the total of the receipts. From its within: |
4 | 9(diš) 1/2 ma-na 2(diš) 2/3 gin21(u) 1(diš) še | 9 1/2 mana 2 2/3 gin 11 še |
5 | kur-bi-la-ak | For Kurbilak; |
6 | 9(diš) 1/2 ma-na 2(diš) 2/3 gin21(u) 1(diš) še | 9 1/2 mana 2 2/3 gin 11 še |
7 | lu2-dašašgigi4 | For Lu-Ašgi; |
8 | 9(diš) 1/2 ma-na 2(diš) 2/3 gin21(u) 1(diš) še | 9 1/2 mana 2 2/3 gin 11 še |
9 | lu2-zabala4ki | For Lu-Zabala; |
10 | 7(diš) 5/6 ma-na 7(diš) gin2igi-4(diš)-gal2 3(diš)Footnote 70 še | 7 5/6 mana 7 1/4 gin 3 še |
11 | i3-li2-an-dul3 | For Ili-andul; |
12 | 7(diš) ma-na 9(diš) 1/2 gin28(diš) 1/2 še 1/2 še | 7 mana 9 1/2 gin 9 še |
13 | dsuen-kal | For Sin-kal; |
14 | 6(diš) 1/3 ma-na 1(diš) 5/6 gin2 | 6 1/3 mana 1 5/6 gin |
15 | nu-ur2-dutu | For Nur-Utu; |
Reverse | ||
1 | 4(diš) 2/3 ma-na 6(diš) 1/3 gin25(diš) 1/2 še | 4 2/3 mana 6 1/3 gin 5 1/2 še |
2 | lu2zimbirki-ke4-ne | For people of Zimbir; |
3 | 3(diš) ma-na 1(u) 5/6 gin21(u) 5(diš) še | 3 mana 10 5/6 gin 15 še |
4 | dingir-ba-ni | For Dingir-bani; |
5 | 3(diš) ma-na 1(u) 5/6(diš) gin21(u) 5(diš) še | 3 mana 10 5/6 gin 15 še |
6 | bu3-su2-num2 | for Busunum; |
7 | 2(diš) 2/3 ma-na 2(diš) gin2igi-6(diš)-gal2 1(u) 8(diš) 1/2 ┌še┐ | 2 2/3 mana 2 1/6 gin 18 1/2 še |
8 | ku3lu2 uri5ki-ke4-ne | For people of Ur; |
9 | 2(diš) [1/3 ma]-na 3(diš) gin2igi-6(diš)-gal2 3(diš) še | 2 1/3 mana 3 1/6 gin 3 še |
10 | ┌ku3┐[…]-ruki-ke4-ne | For […]; |
11 | […]-la | […] |
12 | 2(diš) [1/3 ma-na 3(diš) gin2igi-6(diš)]-gal2 3(diš) še | 2 [1/3 mana 3 1/6 gin] 3 še |
13 | [ku3lu2 …ki]-ke4-ne | For people of […]; |
14 | […] 2(u) 2(diš) 1/2(diš) še | […] + 22 1/2 še of silver |
15 | […]-um | […] |
Blank line | ||
17 | [šu-nigin2] 1(aš) ┌gu2┐1(u) ma-na ku3-babbar | In total: 1 gu 10 mana of silver |
18 | gaba-ri kišib dam-gar3-e-ne | Copy of the sealed receipt of the merchants |
19 | iti ezem-dšu-dsuen | The month of the ‘Festival of Šu-Sin’ |
20 | mu dšu-dsuen lugal uri5ki-ma-ke4ma2-gur8-mah den-lil2 dnin-lil2-ra mu-ne-dim2 | Year when Šu-Sin, king of Ur, constructed a grand boat for the gods Enlil and Ninlil |
Bottom edge | 5(u) / space / 6(diš) 5(u) | 50 / space / 6:50 |
Left edge | 2(u) / space / 1(u) 2(diš) 1(u) 7(diš) 3(u) | 20 / space / 12:17:30 |
Commentary:
Obv. 12: Last two signs read as << 1/2 (diš) še >> sic in CDLI, but as 1/2 (diš) še in the two editions above and BDTNS.
Rev. 9: The fraction before ma-na read as ⌈2/3⌉ in BDTNS, Paoletti (2008) and Paoletti (2012): 448–9, but as [1/3] in Garfinkle (2008) and CDLI.
Text 8-YBC 13418
Location of marginal numbers: Upper edge and bottom of reverse.
Provenience: Puzriš-Dagan
No date.
Dimensions (cm): 9.6 × 5.5 × 2.5
Location of tablet: Yale Babylonian Section, New Haven, Connecticut, USA.
Reference No.: BDTNS 061651; CDLI P210091.
No copy of this tablet is available (Fig. 5.30).
Transliteration | Translation | |
---|---|---|
Obverse | ||
1 | 1(diš) 1/3 ma-na ku3-sig17huš-a | 1 1/3 mana of ‘red’ gold |
2 | lu2-zabalam4ki | (From) Lu-Zabalam, |
3 | 1(diš) ma-na 6(diš) 2/3 gin2 | 1 mana 6 2/3 gin |
4 | puzur4-er3-ra | (From) Puzur-Erra, |
5 | 5/6 ma-na 3(diš) 1/3 gin2 | 5/6 mana 3 1/3 gin |
6 | nu-ur2-dutu | (From) Nūr-Utu, |
7 | 1/3šaba-sa6-ga | 1/3 mana(from) Basaga, |
8 | 1/3ša6(diš) 2/3 gin2 | 1/3 mana 6 2/3 gin |
9 | dingir-ba-ni | (From) Ilum-bani, |
10 | 1(u) 3(diš) 1/3 gin2i-re-eb | 13 1/3 gin (from) Ireb, |
11 | 1/2 ma-na 3(diš) gin2igi-6-gal2 8(diš) še | 1/2 mana 3 1/6 gin 8 še |
Reverse | ||
1 | lu2-dašaš7-gi4 | (From) Lu-Ašgi, |
2 | 2(diš) 5/6 ma-na 6(diš) še | 2 5/6 mana 6 še |
3 | nu-ur2-eš4-tar2 | (From) Nūr-Eštar, |
4 | 5/6 ma-na 6(diš) gin2 | 5/6 mana 6 gin |
5 | er3-ra-dan | (From) Erra-dan, |
6 | šu-nigin25(diš) ma-na 1(u) 1(diš) ┌gin2?┐1(u) 4(diš) še | In total: 5 mana 11 gin 10 še |
7 | ku3-bi 1(aš) gu22(u) 8(diš) ma-na ┌la2┐ 1(u) gin2 | Its value in-silver1 gu 28 mana minus 10 gin |
8 | 2(u) /space/ 4(u) 5(diš) | 20 /space/ 45 |
Upper edge | ||
1(aš) 2(u) 7(diš) 5(u) | 1(aš) 27:50 |
Commentary:
Obv. 7–8: The ša serves as the phonetic complement for the fraction of 1/3 (read as šušana). In the absence of the weight unit ma-na, this ša combined with the fraction implies that 1/3 mana is meant (Powell 1971: 133).
Rev. 6: This total turns out to be 3 mana 28 1/6 gin less than the expected total 8 1/2 mana 9 1/6 gin 14 še, which is calculated by adding all the numbers from the preceding individual entries. Our collation of the tablet reveals that a fraction before the MA sign might have been erased.
Rev. 7: Restoration of ┌lá┐ based on our collation of the tablet and the correspondence between the metrological notation here and the marginal number on the upper edge.
Rev. 8: The first part of the marginal number, 2(u), appears slightly above the second part, 4(u) 5(diš). For the digit 4(u), the two U signs on the top row appear much fainter and smaller than the two on the bottom.
Text 9-AO 27307 (=AOT c-61 = RTC 408)
Location of marginal number: Reverse, end of col. ii.
Provenience: Girsu
Date: Ur III (date probably written on the tablet, but lost).
Location of tablet: Musée du Louvre, Paris, France.
Reference No.: BDTNS 000848; CDLI P128561.
Edition: Thureau-Dangin (1897): 26–27; pl. 30 No. 79; Thureau-Dangin (1903): No. 408.
Photo: http://www.cdli.ucla.edu/dl/photo/P128561.jpg, accessed October 2015.
Transliteration | Translation | |
---|---|---|
Obverse Col. i | ||
1 | […] 3(bur’u) 6(bur3) [1(eše3)] 3(iku) GAN21(aš) 2(barig) ⌈3(ban2)-ta⌉ | […] 3 buru 6 bur [1 eše] 3 iku gan; (the rate is) 1 gur 2 barig 3 ban [per (bur)] |
2 | 1(bur’u) 4(bur3) 1(iku) GAN21(aš) 4(barig)-ta | 1 buru 4 bur 1 iku gan; (the rate is) 1 gur 4 barig per (bur) |
3 | 2(bur’u) 8(bur3) 1(eše3) 2(iku) GAN21(aš) 1(barig)-ta | 2 buru 8 bur 1 eše 2 iku gan; (the rate is) 1gur 1 barig per (bur) |
4 | ⌈še⌉-bi 4(geš’u) 7(geš2) 4(u) 1(aš) 2(barig) 2(ban2) 5 sila3gur | Its grain 4 × 600 + 7 × 60 + 41 gur 2 barig 2 ban 5 sila |
5 | GAN2uru4-a | The field for seeding |
6 | 2(šar2) 7(bur3) 2(eše3) GAN21(aš) 2(barig) 3(ban)-ta | 2 šar 7 bur 2 eše gan; (the rate is) 1 gur 2 barig 3 ban per (bur) |
7 | 1(šar2) 2(bur’u) 4(bur3) 2(eše3) 1(iku) 1/4 GAN21(aš) 4(barig)-ta | 1 šar 2 buru 4 bur 2 eše 1 iku 1/4 gan 1 gur 4 barig per (bur) |
8 | 4(bur’u) 1(bur3) GAN21(aš) 1(barig)-ta | 4 buru 1 bur gan; (the rate is) 1 gur 1 barig per (bur) |
9 | 4(bur3) 2(eše3) 5(iku) GAN21(aš)-ta | 4 bur 2 eše 5 iku gan; (the rate is) 1 gur per (bur) |
10 | še-bi 6(geš2) 3(u) 8(aš) 5(ban2) 5/6/ sila3gur | Its grain is 6 × 60 + 38 gur 5 ban 5/6 sila |
11 | GAN2bala-a | The field of bala |
12 | [šu-nigin2] 3(u) 8(aš) ⌈3(barig) 1(ban)?3 sila3 gur⌉ […] | [Total] 38 gur 3 barig? 1 ban 3 sila […] |
Col. ii | ||
[…] | […] | |
3 | [1(šar’u) 8(šar2) 5(bur’u) 9(bur3) 2(ese3)] 4(iku) 1(ubu)sicGAN2 [24 sar 2/3 gin2 1(aš) 2(barig) 3(ban2)-ta] | [1 šaru 8 šar 5 buru 9 bur 2 ešê] 4 iku 1/4! gan [24 sar 2/3 gin (the rate is) 1 gur 2 barig 3 ban] |
4 | 3(šar2) 8(bur3) 2(eše3) 1(aš) 1/4 GAN2 1(aš) 4(barig)-ta | 3 šar 8 bur 2 eše 1 iku 1/4 gan; (the rate is) 1 gur 4 barig per (bur) |
5 | 3(šar2) 4(bur’u) 2(bur3) 2(eše3) 3(iku) 1(ubu) 1/4 GAN2/ 1(aš) 1(barig)-ta | 3 šar 4 buru 2 bur 2 eše 3 iku 1/2 1/4 gan; (the rate is) 1 gur 1 barig per (bur) |
6 | 1(bur)32(eše3) GAN2 1(aš)-ta | 1 bur 2 eše gan; (the rate is) 1 gur per (bur) |
7 | še-bi 3(geš’u) 8(geš2) 3(u)/ 8(aš) 3(barig) 3(ban) 4 5/6 sila35 gin2 | Its grain 3 × 600 + 8 × 60 + 38 gur3 barig 3 ban 4 5/6 sila 5 gin |
8 | GAN2⌈uru4-a⌉ | The field for seeding |
9 | 2( šar2) 4(bur’u) 4(bur3) 1(eše3) 2(iku) [GAN2] 1(aš) 2(barig) ⌈3(ban2)⌉-[ta] | 2 šar 4 buru 4 bur 1 eše 2 iku gan; (the rate is)1 gur 2 barig 3 ban [per (bur)] |
10 | ⌈2(bur’u) 2 bur3⌉ […] | 2 buru 2 bur [gan …] |
Reverse Col. i | ||
[…] | […] | |
[…] | […] | |
1' | šu-nigin2[…] | Total […] |
2' | šu-nigin2[…] | Total |
3' | še-bi […] | Its grain |
4' | GAN2bala-a | The field for bala |
5' | šu-nigin2 | Total |
6' | šu-nigin2 | Total |
7' | šu-nigin2 | Total |
8' | šu-nigin2 | Total |
9' | še-bi | Its grain |
10' | GAN2bala-[a] | The bala of the field |
Col. ii | ||
1' | […] | […] |
2' | x x ŠU x […] /space/ | In Girsu? […] /space/ |
3’ | 20?2? 4? […] 5:41:56?:55 ⌈3⌉ /space/ 10? 3? 1:20 1:20 45 | 20?2? 4? […] 5:41:56?:55 ⌈3⌉ /space/ 10? 3? 1:20 1:20 45 |
Commentary:
Note on fractions of iku:
Cuneiform | Transliteration | Translation |
---|---|---|
| 1(iku) GAN2 | 1 gan |
| 1(ubu) GAN2 | 1/2 gan |
| 1/4 GAN2 | ¼ gan |
Obv. col. i 2: Contra Thureau-Dangin’s copy, the first sign of the line is visible; it is 1(bur’u).
Obv. col. ii 3: For explanations on the reconstruction of the damaged part, see Text 9 in Sect. 5.2. ‘4(iku) 1/2 GAN2’ is noted on the tablet, but, according to the reconstitution, ‘4(iku) 1/4 GAN2’ is expected.
Rev. col. ii 2’: CDLI and BDTNS restore ⌈ša3?⌉ ⌈gir2?-šu?ki⌉ ⌈x⌉ (in Girsu).
Text 10-YBC 1793
Location of marginal numbers: Obverse, beginning of col. i and lower section of col. ii.
Provenience: Unknown.
Date: AS 5 ix.
Dimensions (cm): 9.3 × 7 × 2.2
Location of tablet: Yale Babylonian Collection, New Haven, Connecticut, USA.
Reference No.: BDTNS 005335; CDLI P142357.
Copy: Keiser (1919): No. 293 (Fig. 5.31).
Transliteration | Translation | |
---|---|---|
Obverse | ||
Col. i | ||
1 | 14:54 | 14:54 |
2 | 29:56:50 | 29:56:50 |
3 | 17:43:40 | 17:43:40 |
4 | 30:53:20 | 30:53:20 |
5 | Blank line | |
6 | šu-nigin2 1(diš) 1/2 ma-na 3(diš) 1/2 gin2 la2 7(diš) še ku3-a | In total: 1 1/2 mana 3 1/2 gin minus 7 še of silver, |
7 | mu-kux(DU) didli | various deliveries; |
8 | 7(diš) ma-<na> 1(u) 9(diš) gin2 ku3-a mu-kux(DU) a-tu5-a lugal | 7 mana 19 gin of silver, delivery for the lustration of the king; |
9 | Blank line | |
10 | šu-nigin2 8(diš) 5/6 ma-na 2(diš) 1/2 gin2 la2 7(diš) še ku3-a | In (grand-)total: 8 5/6 mana 2 1/2 gin minus 7 še of silver, |
11 | ša3 im UD | Within the daily tablets |
12 | Blank line | |
13 | [mu]-kux(DU) iti ezem-mah | Delivery in the month of ‘Grand Festival’ |
14 | ┌mu┐ en dinanna | During the year when the en-priest of Inanna (was installed) |
Col. ii | ||
1 | 8(diš) 5/6 ma-na 4(diš) 1/2 gin2 la2┌x┐ 2(diš) še | 8 5/6 mana 4 1/2 gin minus x + 2 še |
2 | ša3 […]-a | […] |
2:┌54┐ ┌45┐ ┌28┐ 17 2:28 27 | 2:┌54┐ ┌45┐ ┌28┐ 17 2:28 27 | |
Reverse | ||
Not inscribed |
Commentary:
We follow CDLI for the unknown provenience of this tablet. BDTNS identifies it as a tablet from Puzriš-Dagan.
Obv. col. i 6, 8, 10: Our search in both BDTNS and CDLI (accessed 2013 August) did not find the expression ‘kù-a’ in any other Ur III document. The exact meaning of this phrase remains unclear to us.
Obv. col. i 11: Read as ‘šà im-babbar’ in CDLI. Our search in both BDTNS and CDLI (accessed 2013 August) found the phrase ‘šà im UD' only in this text. We follow Friberg (2005: 9) for the translation here.
Obv. col. ii 1: Friberg (ibid.) restored the amount in this line as the same in obv. col. i 10, but our collation does not support his restoration. Instead of the 2 1/2 gin2 as read by him, both the copy and our collation of the tablet show that the amount involved should be 4 1/2 gin2. As for the amount subtracted, since the two DIŠ signs do not appear under the horizontal stroke of the LA2 sign (cf. ‘la2 7 še' in obv. col. i 6, 10) but immediately after this sign, the amount subtracted may well exceed two še.
Obv. col. ii 2: Friberg (ibid.) restored the lost sign as bala without producing any evidence.
Primary sources
Museum number | Primary publication | CDLI number |
---|---|---|
AO 2728 = AOT 304 = RTC 413 | Thureau-Dangin(1903: 149, No. 413) | P128566 |
AO 27307 = AOT c-61 = RTC 408 (Text 9) | Thureau-Dangin (1897: 26–27; pl. 30 No. 79); Thureau-Dangin (1903: No. 408) | P128561 |
AO 5674 | Genouillac (1922) | P131745 |
AO 5676 | Genouillac (1922) | P131747 |
AO 6038 | Genouillac (1922) | P131752 |
Ashm 1924–0667 | Grégoire (1996) | P142827 |
BM 19027 | King (1898): pl. 27 | P108470 |
CBS 11661 | Myhrman (1910): 56 | P105608 |
E 15550 (Text 5) | Ouyang and Brookman (2012): Sect. 3.4 | P416398 |
Erlenmeyer 152 | Englund (2003): No. 1 | P109319 |
Erm 14994 | Koslova (2000): No. 340 | P212246 |
FLP 972 | Owen (1975): No. 290 | P113850 |
FMB 39.2 | Clevenstine, forthcoming | P427641 |
HMA 9–02824 | Lutz (1928): No. 100 | P136004 |
HS 201 | P254568 | |
Ist L 7375 | Delaporte (1911) | P254700 |
Ist L 9236 | Genouillac (1921) | P227599 |
Ist Ni 374 | Proust (2007) | P257557 |
MAH 16323 | Sauren (1974): No. 168 | P113467 |
MAH 16573 | Sauren (1974): No. 105 | P113404 |
MAH 16605 | Sauren (1974): No. 104 | P113403 |
MAH 19353 | Sauren (1974): No. 24 | P113323 |
MAH 19472 | Sauren (1974): No. 175 | P113474 |
MM 0396 | Schneider (1932) | P101631 |
MSR 1 | Goetze and Foster (2010): No. 158 | P142394 |
MVN 3.290 | Garfinkle (2008): 68 | P215676 |
NBC 6501 | Paoletti (2012): 447 | P298413 |
NBC 6641 (Text 7) | Garfinkle (2008) | P298493 |
Nik. 2.402 (Text 1) | Nikol’skij (1915): No. 402 | P122085 |
Nik. 2.403 | Nikol’skij (1915): No. 403 | P122086 |
PTS 473 (Text 6) | Sigrist (1990): No. 60 | P126749 |
PUL Ex 662 | Chiera (1922) | P130373 |
SM 1909.05.224 | Owen (1982): 91 | P116105 |
VAT 07042 | Schneider (1930): No. 382 | P125272 |
WCMA 20.1.07 | unpublished according to CDLI | P424374 |
YBC 01778 | Keiser (1919): No. 304 | P142368 |
YBC 01793 (Text 10) | Keiser (1919): No. 293 | P142357 |
YBC 03883 | Keiser (1919): No. 225 | P142289 |
YBC 04179 (Text 3) | Ellis (1970): 268 | P111807 |
YBC 13418 (Text 8) | Sigrist and Ozaki (2009): No. 20 | P210091 |
YBC 16487 (Text 2) | Snell (1982): No. 5 | P112497 |
YBC 16607 (Text 4) | Snell (1982): No. 13 | P112505 |
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Ouyang, X., Proust, C. (2022). Place-Value Notations in the Ur III Period: Marginal Numbers in Administrative Texts. In: Chemla, K., Keller, A., Proust, C. (eds) Cultures of Computation and Quantification in the Ancient World. Why the Sciences of the Ancient World Matter, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-98361-1_5
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