Abstract
This chapter analyses different ways of quantifying tri-dimensional entities in Old Babylonian mathematical cuneiform texts. In administrative documents, the metrology adopted by scribes depended on the nature of the things being quantified. In mathematical texts too, one finds different ways of quantifying tri-dimensional entities, but these different approaches to spatial extension are not completely independent from each other as they are in administrative texts. Some mathematical texts exhibit conversions between capacity, volume and brickage metrologies. It is the case of a set of eight ‘catalogue texts’ preserved at Yale University, which contain lists of statements of problems carefully organized according to thematic criteria, and some other related texts. This set of texts provides an interesting view of the mathematical approach to spatial problems attested in a southern city, and may reflect a specific mathematical culture for quantifying spatial entities (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.).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
An abundant literature was devoted to these different practices of quantification, as attested in administrative and economic documentation. An updated bibliography and new approaches can be found in the books edited by the SAW project. For containers and capacities in the third and second millennium, see Chap. 3 by Sallaberger in this volume. For volumes of earth in Ur III texts, see Rost’s chapter in Michel and Chemla (2020). For brick metrology in Ur III and Old Babylonian periods, see Chap. 2 by Heimpel’ and Chap. 6 by Middeke-Conlin’s in this volume, and Sauvage’s chapter in Michel and Chemla (2020). For the evaluation of quantities of metals by their weights, see the example of the Assyrian merchants’ practices in early second millennium in Michel’s chapter in Michel and Chemla (2020).
- 2.
Other approaches to spatiality, such as counting bundles or baskets, are attested. See for example the estimation of number of bundles in Ur III administrative texts in Heimpel, Chap. 2 in this volume. See also the estimation of number of baskets for carrying earth in Robson 1999: 78–82, and in Middeke-Conlin, Chap. 6 in this volume.
- 3.
Conversions between volumes and capacities can be found also in the Haddad 104 mathematical tablet, from Tell Haddad, a city located in the Diyāla Valley, in northern Mesopotamia. Thus, another approach to spatiality may emerge from a study of the mathematical texts from the Diyāla Valley; however, this study will not be undertaken in the present chapter.
- 4.
The tablet was classified by Jöran Friberg as belonging to a southern group as other catalogues (Friberg 2000: 164). Neugebauer considered this tablet to be strongly connected with other series texts dated from the late Old Babylonian period (Neugebauer 1935–1937: Vol. I, Chap. 7). On this point I followed Neugebauer (Proust 2012) as did J. Høyrup (2002: 349–352), who suggested Kiš, in central Mesopotamia, as a possible provenance.
- 5.
An extended bibliography and some new light on the history of positional notations in Mesopotamia can be found in Ouyang and Proust (Chap. 5 in this volume).
- 6.
Moreover, I use common Assyriological conventions for transliteration. In particular, Sumerian logograms (sumerograms) are transliterated in plain font, and Akkadian words are transliterated in italics.
- 7.
- 8.
YBC 4607, YBC 4612, YBC 4652, YBC 4657, YBC 4666, YBC 5037, YBC 6492, YBC 7164.
- 9.
See in Neugebauer and Sachs (1945) the edition and mathematical interpretation of catalogues and procedure texts from Yale University. See especially ibid: 66–76, texts G, H and J, for the analysis of the relation between catalogue YBC 4657 and procedure texts YBC 4663 and YBC 4662. Further discussion on function, provenance and date of catalogues and series texts is provided in Proust (2012).
- 10.
- 11.
In the following, I consider in somewhat improper fashion that sections of texts noted with only Sumerian logograms are written in Sumerian, even if the language underlying these sumerograms is not clear. See discussion of this problem in Proust (2012) and, for the case of mathematical series texts, in Proust (2009a: 170; 229–230).
- 12.
For the geometrical meaning of terms adopted for operations in mathematical texts, see Høyrup 2002: 3–49.
- 13.
See image of the tablet at CDLI (http://cdli.ucla.edu/P368708). For more information, see detailed analysis of this tablet in Proust (2007: 190–196).
- 14.
- 15.
Information on measurement units is provided in Annex A.1 at the end of this volume.
- 16.
I follow Høyrup (2002: 44) who translates this verb as ‘to make hold’.
- 17.
According to Høyrup, the term našum, and the derived grammatical forms, including the Sumerian counterpart il2, ‘designate the determination of a concrete magnitude by means of a multiplication. They are used for all multiplications by technical constants and metrological conversions; for the calculation of volumes from base and height; and for the determination of areas when this is not implied by the construction of a rectangle’ (Høyrup 2002: 22).
- 18.
For an overview of metrological tables for heights and depths, see Proust (2007: 107–111).
- 19.
The unit of volume 1 sar is defined as a unit of surface 1 sar with a 1 kuš-height, the unit of volume 1 iku gan is defined as a unit of surface 1 iku gan with a 1 kuš-height, and so on. This abstract concept of volume probably emerged from reforms of metrology undertaken by rulers of centralized states during the second part of the third millennium. According to Powell (1987–1990: 488), ‘The standard system of volume measures canonized in OB mathematical texts is an invention of the Akkad period [ca. 2340–2200]’.
- 20.
- 21.
Powell (1982: Appendix 2) labeled the types of brick in order of increasing volume. Other authors (Friberg, Robson) opted for a different system of naming the types of bricks.
- 22.
In known sources, this portion is included only in metrological tables for weights. However, the units of weight mana (SPVN 1), of capacity sila (SPVN 1), and of surface sar (SPVN 1), have the same sub-units, gin and še, defined with the same factors. As a consequence, the portion of the table for gin and še is common to the three systems. This portion is detailed in tables for weights, but omitted in tables for capacities and for surfaces.
- 23.
- 24.
- 25.
On the cuneiform text, we read actually ‘brick-volume’ (sig4) instead of ‘capacity’ (še), which is expected after a measurement of capacity (Appendix 1 Text 5).
- 26.
Here and thereafter, ‘table C’ means metrological table for capacities, ‘table S’ means metrological table for surfaces, and so on.
- 27.
The relation between volume and capacities was elucidated by Neugebauer and Sachs (1945: 96–97) using the same sources (YBC 4607 and BM 85194 #30). However, Neugebauer and Sachs’ explanations do not rely on metrological tables, and moreover, they consider the coefficient 5 as conveying an absolute value (5,0,0). Thus, their arguments are not exactly the same as those developed here.
- 28.
- 29.
See Middeke-Conlin 2020.
- 30.
Moreover, see Heimpel and Middeke-Conlin, respectively Chap. 2 and 6 in this volume for extensive examples where brick metrology is adopted in administrative contexts, and see Sauvage’s chapter in Michel and Chemla (2020) for different aspects, including archaeological, of the treatment of bricks in Mesopotamia. Mathematical aspects of brick metrology are examined in Proust (2007: 210–214).
- 31.
Two equivalent methods are possible: either dividing the area of the layer by the area of the base of each brick; or calculating the total volume of the layer, and converting this volume into the number of bricks. I opt here for the second method, which is simpler.
- 32.
- 33.
See Chap. 6 by Middeke-Conlin for other examples of such composite notations. The brickage could be easily obtained by dividing the total number of bricks (14:24 in SPVN) by 12, since 1 gin contains 12 bricks (14:24 / 12 = 14:24 × 5 = 1:12). The result 1:12 (in SPVN) corresponds to 1 sar 12 gin according to the metrological table for surfaces/volumes and a mental check for the orders of magnitude.
- 34.
- 35.
See the transliteration and translation of the first part of this text in Appendix 2, 6.
- 36.
- 37.
‘One is tempted to view the values of the nalbanum as the number of SAR (= 12,0) of bricks of the different types in 1 volume-SAR, which may have been a unit of special significance in the molding of bricks’ (Neugebauer and Sachs 1945: 138). The nalbanum is defined equivalently as ‘The number of sarb in a sarv for each type of brick’ (Robson 1999: 59) or ‘The “molding number” nalbanum … is an expression for the number of bricks of a given type in a unit of volume’ (Friberg 2001: 71). Powell is more cautious as he recognizes only a calculation function to the nalbanum: ‘This is a key number which can be used to calculate the number of bricks of a given type in volume-šar. This is obtained by multiplying C1 (the nalbanum, which is the number of brick-šar of a given type in a volume-šar) by D1 (=12,0, the igigubbû which is the number of bricks in a brick-šar)’. (Powell 1982: 120).
- 38.
Metrological tables are not attested in the Ur III period, date of the earliest known use of brickage. However, SPVN is attested in Ur III mathematical texts, and traces of the use of correspondences similar to those provided by metrological tables can be detected in administrative texts (see Ouyang and Proust, Chap. 5 in this volume).
- 39.
For an updated presentation of standard vessels and containers used in Ur III and Old Babylonian Mesopotamia, and for an extensive bibliography of the subject, see Chap. 3 by Sallaberger in this volume.
- 40.
See Proust (2009b) for explanations on the different functions of cuneiform graphemes (arithmograms, metrograms and arithmo-metrograms).
- 41.
Friberg (2000: 129–131) supposes that the vessels are cylinders, and then offers a completely different interpretation of sections 1–9.
- 42.
See the thorough study of the different versions of this list by Niek Veldhuis (Veldhuis 1997).
- 43.
On norms, see Michel’s chapter in Michel and Chemla (2020).
- 44.
See Sallaberger, Chap. 3 in this volume.
- 45.
DCCLT, http://oracc.museum.upenn.edu/dcclt. Nippur sources: CBS 4827; CBS 6426; CBS 7139 + CBS 07,152 + N 0330; HS 1745 + HS 1797 + HS 2630 + HS 2902; N 1564; N 5201; N 6102. Ur source: UET 6/3–677 + 678 + UET 7–087 + 091.
- 46.
I reconstructed the Nippur composite text according to the photos of the sources listed by N. Veldhuis along with the composite text online at http://oracc.museum.upenn.edu/dcclt/corpus (03/05/2014), page on ‘OB Nippur Ura 01’, and, of course, with the help of his own composite text. I found some slight divergences: Veldhuis omits nig2 in the line for 2 sila3 (line 520 of the composite text); however, nig2 appears in N 5201; Veldhuis inverts 2/3 sila3 and 1/3 sila3 (line 521 and 523 of the composite text). I transliterated UET 6/3–677 + 678 + UET 7–087 + 091 according to the photo posted on CDLI (P346714).
- 47.
Proust 2009b: Sect. 9, http://www.cdli.ucla.edu/pubs/cdlj/2009/cdlj09_001.html, accessed December 2016.
Bibliography
Charpin, Dominique. 1986. Le clergé d’Ur au siècle d’Hammurabi. Geneva: Droz.
Friberg, Jöran. 2000. Mathematics at Ur in the Old Babylonian period. Revue d'Assyriologie 94: 98–188.
Friberg, Jöran. 2001. Bricks and mud in metro-mathematical cuneiform texts. In Changing Views on Ancient Near Eastern Mathematics, ed. J. Høyrup and P. Damerow, 61–154. Berlin: Dietrich Reimer Verlag.
Friberg, Jöran. 2011. Review of proust, tablettes mathématiques de Nippur, 2007. Archiv für Orientforschung 52: 259–263.
Goetze, Albrecht. 1945. The Akkadian dialects of the Old-Babylonian mathematical texts. In Mathematical Cuneiform Texts, O. Neugebauer, and A.J. Sachs, 146–151. New Haven: American Oriental Series and American Schools of Oriental Research.
Gurney, Olivier R. 1974. Middle Babylonian Legal Documents and Other Texts. Book Series: Ur Excavations. Texts 7. London/Philadelphia, British Museum/University of Pennsylvania.
Hilprecht, Herman V. 1906. Mathematical, Metrological and Chronological Tablets from the Temple Library of Nippur. Book Series: Babylonian Expedition 20/1. Philadelphia: University of Pennsylvania.
Høyrup, Jens. 2002. Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and its Kin. Book Series: Studies and Sources in the History of Mathematics and Physical Sciences. Berlin: Springer.
King, Leonard William. 1900. Cuneiform Texts from Babylonian Tablets in the British Museum. Book Series: Cuneiform Texts 9. London: British Museum.
Michel, Cécile, and Karine Chemla (eds.). 2020. Mathematics and Administration in the Ancient World. Book Series: Why the Sciences of the Ancient World Matter 5. Dordrecht: Springer.
Middeke-Conlin, Robert, and Christine Proust. 2014. Interest, price, and profit: an overview of mathematical economics in YBC 4698. Cuneiform Digital Library Journal 2014–3. http://cdli.ucla.edu/pubs/cdlj/2014/cdlj2014_003.html.
Middeke-Conlin, Robert. 2020. The Making of a Scribe: Errors, Mistakes, and Rounding Numbers in the Old Babylonian Kingdom of Larsa. Book Series: Why the Sciences of the Ancient World Matter 4. Dordrecht: Springer.
Muroi, Kazuo. 1990. Interest calculation in Babylonian mathematics: New interpretations of VAT 8521 and VAT 8528. Historia Scientiarum 39: 30–34.
Neugebauer, Otto, and Abraham J. Sachs. 1945. Mathematical Cuneiform Texts, American Oriental Studies. New Haven: American Oriental Series & American Schools of Oriental Research.
Neugebauer, Otto. 1935–1937. Mathematische Keilschrift-Texte I-III. Berlin: Springer.
Powell, Marvin A. 1982. Metrological notes on the Esagila tablet and related matters. Zeitschrift für Assyriologie und Vorderasiatische Archäologie 72: 116–123.
Powell, Marvin A. 1987–1990. Masse und Gewichte. In Reallexikon der Assyriologie, (ed.). D.O. Edzard, 457–517. Berlin: De Gruyter.
Proust, Christine. 2009a. Deux nouvelles tablettes mathématiques du Louvre: AO 9071 et AO 9072. Zeitschrift für Assyriologie und Vorderasiatische Archäologie 99 (2): 167–232.
Proust, Christine. 2009b. Numerical and metrological graphemes: From cuneiform to transliteration. Cuneiform Digital Library Journal 2009:1. https://cdli.ucla.edu/pubs/cdlj/2009b/cdlj2009b_001.html
Proust, Christine. 2012. Reading colophons from Mesopotamian clay-tablets dealing with mathematics. NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin 20 (3): 123–156.
Proust, Christine (with the collaboration of Antoine Cavigneaux). 2007. Tablettes Mathématiques de Nippur. Istanbul: Institut Français d'Etudes Anatoliennes/Paris: De Boccard.
Proust, Christine (with the collaboration of Manfred Krebernik and Joachim Oelsner). 2008. Tablettes Mathématiques de la Collection Hilprecht. Leipzig: Harrassowitz.
Robson, Eleanor. 2001. The tablet house: A scribal school in Old Babylonian Nippur. Revue D’assyriologie 95: 39–66.
Robson, Eleanor. 1999. Mesopotamian Mathematics, 2100–1600 BC. Technical Constants in Bureaucracy and Education. Book series: Oxford Editions of Cuneiform Texts 14. Oxford: Clarendon Press.
Scheil, Vincent. 1915. Le calcul des volumes dans un cas particulier à l’époque d’Ur. Revue D’assyriologie 12: 161–172.
Thureau-Dangin, François. 1937. La mesure du ‘qa.’ Revue D’assyriologie 34: 80–86.
Thureau-Dangin, François. 1938. Textes Mathématiques Babyloniens. Leiden: Brill.
Veldhuis, Niek. 1997. Elementary Education at Nippur, the Lists of Trees and Wooden Objects. Ph.D. diss.: University of Groningen.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendices
Appendix 1: Main Texts
I collated the YBC tablets provided in this section at Yale University in 2009 and 2010, with the kind permission of Ulla Kasten and Benjamin Foster. The transliteration and translation are mine, with the help of primary editions (mainly Neugebauer 1935–1937; Thureau-Dangin 1938: 20, and Neugebauer and Sachs 1945). See conventions and abbreviations at the end of the chapter, in the references section.
1.1 Text 1. Catalogue Text YBC 4657 #1–2 and Colophon
Publication: Neugebauer and Sachs 1945: text G.
CDLI number: P254982.
Provenance: unknown; Goetze’s Group 2 (southern group, perhaps from Ur).
Transliteration
Obverse
# | li | |
1 | 1. 2. 3. | [ki]-la2 [5(diš) ninda uš 1(diš) 1/2] ninda sag 1/2 ninda bur3-bi 1(u) gin2 sahar eš2-kar3 6(diš) še [a2-bi lu2-huğ-ga2] gagar sahar-[hi-a erim-hi-a] u3 ku3-babbar en-[nam] 7(diš) 1/2 gagar 45 sahar-hi-a 4(geš2) 3(u) [erim-hi-a] 9(diš) gin2 ku3-babbar |
2 | 4. 5. | ku3 ki-⌈la2⌉ [9(diš) gin2] 1(diš) 1/2 ninda sag 1/2 ninda < bur3-bi > 1(u) gin2 eš2-gar3 6(diš) še a2-bi lu2-huğ-[ga2] uš-bi [en-nam] 5(diš) ninda nindasic uš |
Upper edge
3(u) 1(diš) im-šu ki-la2
Translation
Obverse
# | li | |
1 | 1. 2. 3. | A trench. 5 ninda its length, 1 1/2 ninda its width, 1/2 ninda its depth. 10 gin the assigned volume (per worker). 6 še (silver) [the wage per worker] The base, the volume, the number of worker and the silver (of the wages) how much? 7 1/2 the base, 45 the volume 4 × 60 + 30 [workers], 9 gin the silver (of the wages) |
2 | 4. 5. | The silver (for) a trench (is) 9 gin. 1 1/2 ninda the width, 1/2 ninda < its depth > , a volume of 10 < gin > the work assignment, 6 še (of silver) the wages Its length how much? Its length is 5 ninda |
Upper edge
31 sections (on) excavations
3.1 Text 2. Procedure Text YBC 4663 #1–2
Publication: Neugebauer and Sachs 1945: text H.
CDLI number: P254984.
Provenance: unknown; Goetze’s Group 2 (southern group, perhaps from Ur).
Transliteration
Obverse
# | li | |
1 | 1. 2. 3. 4. 5. 6. | ki-la2 5(diš) ninda uš 1(diš) 1/2 ninda < sag > 1/2 ninda bur3-bi 1(u) < gin2 > sahar eš2-kar3 6(diš) ⌈še⌉ [a2-bi] gagar sahar-hi-a erim-hi-a u3 ku3-babbar en-nam za-e in-da-zu-de3 uš sag gu7-gu7-ta 7:30 i-na-ad-di-ik-ku 7:30 a-na bur3-bi i-ši 45 i-na-ad-di-ku igi eš2-gar3 du8 6 i-na-ad-di-ku a-na 45 ⌈i⌉-ši 4:30 i-na-di-ku 4:30 a-na i-di i-ši 9 i-na-di-ku ki-a-am ⌈ne2-pe-šu⌉ |
2 | 7. 8. 9. 10. 11. 12. 13. | 9(diš) gin2 ku3-babbar ki-la2 1(diš) 1/2 ninda < sag > 1/2 ninda bur3-bi 1(u) < gin2 > sahar eš2-gar3 6(diš) še [a2]-bi uš-bi en-nam za-e in-da-zu- < de3 > sag u3 bur3-bi gu7-gu7-ta 9 i-na-ad-di-ku-um igi eš2-gar3 pu-ṭu3-ur a-na 9 i-ši 54 i-na-ad-di-ik-ku-um 54 a-na i-di i-ši 1:48 i-na-ad-di-ku-um igi 1:48 33:20 i-na-ad-di-ik-ku 33:20 a-na 9 ku3 i-ši ⌈5⌉ i-na-ad-di-ku-um 5(diš) ninda uš-bi ki-a-am ne2-pe-šu |
Translation
Obverse
# | li | |
1 | 1. 2. 3. 4. 5. 6. | A trench. 5 ninda the length, 1 1/2 ninda < the width > , 1/2 ninda its depth, a volume of 10 < gin > the work assignment, 6 še [of silver the wages] The base, the volume, the (number of) workers and the silver how much? You, in your procedure, the length and the width make hold together, 7:30 it will give you 7:30 to its depth raise, 45 it will give you The reciprocal of the work assignment loosen, 6 it will give you. To 45 raise, 4:30 it will give you 4:30 to the wages raise, 9 it will give you. Such is the procedure |
2 | 7. 8. 9. 10. 11. 12. 13. | 9 gin silver for a trench. 1 1/2 ninda < the width > , 1/2 ninda its depth, a volume of 10 < gin > the work assignment, 6 še [of silver the] wages Its length how much? You, in your procedure, the width and the depth make hold together, 9 it will give you. The reciprocal of the work assignment loosen, (and) to 9 raise, 54 it will give you 54 to the wages raise, 1:48 it will give you The reciprocal of 1:48 < loosen > , 33:20 it will give you. 33:20 to 9, the silver, raise, 5 it will give you. 5 ninda is its length. Such is the procedure |
Notes
In #1, obv. 2 and #2, obv. 8, Neugebauer and Sachs read ‘kid9-da-zu-de3’; however, the reading ‘in-da-zu-de3’ seems clear to me.
In #1, obv. 3 and #2, obv. 8, ‘gu7-gu7’ is transliterated as ‘UR.UR’ by Neugebauer and Sachs (1945: 69).
5.1 Text 3. Catalogue YBC 4607
Publication: Neugebauer and Sachs 1945: text O.
CDLI number: P235642.
Provenance: unknown; Goetze’s Group 2 (southern group, perhaps from Ur) (Fig. 4.8).
YBC 4607 (courtesy of the Yale Babylonian Collection, photo C. Proust)
Obverse
# | li | |
1 | 1 2 3 4 5 | sig4 1/2 kuš3 uš-bi 1/3 kuš3 sag-bi 5(diš) šu-si sukud-bi gagar sahar-bi u3 i3-šam2* sahar-bi en-nam 1(u) 2(diš) še šu-ri-a še gagar-bi 2(diš) še u3 < igi > -12-gal2 še sahar-bi 3(diš) 1/3 sila3 8(diš) 1/3 gin2 i3-šam2 sahar-bi |
2 | 6 7 8 9 | sig4 1(u) 8(diš) šu-si uš-bi 1(u) 2(diš) šu-si sag-bi 5(diš) šu-si sukud-bi gagar sahar-bi u3 i3-šam2 sahar-bi en-nam 1(u) 8(diš) še gagar 3(diš) še sahar-bi [5(diš)] ⌈sila3 i3-šam2⌉ sahar-bi |
3 | 10 11 12 13 14 15 | sig4-ab2 2/3 kuš3 uš-bi 1/3 kuš3 sag-bi 5(diš) šu-si sukud-bi gagar sahar-bi u3 i3-šam2 sahar-bi en-nam 1(u) 6(diš) še šu-ri-a še u3 igi-6-gal2 ⌈še še-bi⌉ 2(diš) še šu-ri-a še igi-4-gal2 [še u3 igi-9 igi]-4-gal2 sahar 4(diš) 1/2 sila3 6sic(diš) 2/3 < gin2 > 2(u) še i3-[šam2 sahar-bi] |
4 | 16 17 18 19 20 | sig4-al-ur3-[ra] 2/3 kuš3-ta-am3 ib2-sa2 5(diš) šu-si [sukud-bi] gagar sahar-bi u3 i3-šam2 sahar-bi en-nam igi-6-gal2 < gin2 > 3(diš) še u3 igi-3-gal2 še gagar 5(diš) še šu-ri-a še u3 igi-9-gal2 šu-ri-a še sahar |
Reverse
5 | 1 2 3 | sig4-al-ur3-ra 1 kuš3-ta-am3 ib2-sa2 5(diš) šu-si sukud-bi gagar sahar-bi u3 i3-šam2 sahar-bi en-nam 1/3 gin2 u3 1(u) 5(diš) še gagar 1(u) ⌈2(diš)⌉ še šu-ri-a še sahar-bi |
6 | 4 5 6 | sig4 1/2 kuš3 uš-bi 1/3 kuš3 sag-bi 5(diš) šu-si sukud-du 1 sar gagar ta-ad-di-ta-[am en-nam i3-dib2] 1(diš) sar 2(geš2) 2(u) 4(diš) i3-dib2 |
7 | 7 8 9 | sig4 18 šu-si uš-bi 1(u) 2(diš) šu-si sag-bi 5(diš) šu-si sukud-bi 1(diš) sar gagar ta-ad-di-ta-am en-nam i3-dib2 1(geš’u) sig4 i3-dib2 |
8 | 10 11 12 | sig4-ab2 2/3 kuš3 uš-bi 1/3 kuš3 sag-bi 5(diš) šu-si sukud-bi 1(diš) sar gagar ta-ad-di-ta-am en-nam i2-dib2 1(geš’u) 4(u) 8(diš) sig4 i3-dib2 |
9 | 13 14 15 16 | sig4-al-ur3-ra 2/3 kuš3-ta-am3 ib2-sa2 5(diš) šu-si sukud-bi 1(diš) sar gagar ta-ad-di-⌈ta⌉-am en-nam i2-dib2 5(geš2) 2(u) 4(diš) sig4-al-ur3-ra i3-dib2 |
10 | 17 18 19 | sig4-al-ur3-ra 1(diš) kuš3-ta-am3 < ib2-sa2 > 5 šu-si sukud-bi 1(diš) sar gagar ta-ad-di-ta- < am > en-⌈nam⌉ i3-dib2 2(geš2) 2(u) 4(diš) sig4-al-[ur3-ra] i3-dib2 |
Left edge
1(u) im-šu-meš.
Translation
Obverse
# | li | |
1 | 1 2 3 4 5 | A brick (sig4). 1/2 kuš its length, 1/3 kuš its width, 5 šusi its height Its base, its volume and its equivalent capacity how much? 12 še and one-half a še its base, 2 še and one-12th še its volume, 3 1/3 sila 8 1/3 gin its equivalent capacity |
2 | 6 7 8 9 | A brick (sig4). 18 šusi its length, 12 šusi its width, 5 šusi its height Its base, its volume and its equivalent capacity how much? 18 še its base, 3 še its volume, [5] sila its equivalent capacity |
3 | 10 11 12 13 14 15 | A half-brick (sig4-ab2). 2/3 kuš its length, 1/3 kuš its width, 5 šusi its height Its base, its volume and its equivalent capacity how much? 16 še and one-half a še and one-6th še its base! (capacitysic), 2 še and one-half a še and one-4th [še and one-9th of] one-4th its volume, 4 1/2 sila 7! 2/3 (6sic 2/3) < gin > 20 še its equivalent [capacity] |
4 | 16 17 18 19 20 | A square baked brick (sig4-al-ur3-ra) 2/3 kuš each equal-side, 5 šusi [its height] Its base, its volume and its equivalent capacity how much? One-6th < gin > 3 še and one-3rd še [its] base 5 še and one-half a še and one-half of one-9th še sahar |
Reverse
5 | 1 2 3 | A square baked brick (sig4-al-ur3-ra). 1 kuš each equal-side, 5 šusi its height. Its base, its volumes and its equivalent capacity how much? 1/3 gin and 15 še [its] base, 12 še one-half a še its volume |
6 | 4 5 6 | A brick (sig4). 1/2 kuš its length, 1/3 kuš its width, 5 šusi its height. A layer of bricks (taddītum) which base is 1 sar, how much (bricks) takes? (The layer) takes 1 sar 2 × 60 + 24 (bricks) |
7 | 7 8 9 | A brick (sig4). 18 šusi its length, 12 šusi its width, 5 šusi its height A layer of bricks (taddītum) which base is 1 sar, how much [bricks] takes? [The layer)] takes 600 bricks |
8 | 10 11 12 | A half-brick (sig4-ab2). 2/3 kuš its length, 1/3 kuš its width, 5 šusi its height A layer of bricks (taddītum) which base is 1 sar, how much [half-bricks] takes? [The layer] takes 600 + 48 half-bricks |
9 | 13 14 15 16 | A square baked brick (sig4-al-ur3-ra). 2/3 kuš each equal-side, 5 šusi its height. A layer of bricks (taddītum) which base is 1 sar, how much [square baked bricks] takes? [The layer] takes 5 × 60 + 24 square baked bricks |
10 | 17 18 19 | A square baked brick (sig4-al-ur3-ra). 1 kuš each equal-side, 5 šusi its height. A layer of bricks (taddītum) which base is 1 sar, how much [square baked bricks] takes? [The layer] takes 2 × 60 + 24 square baked bricks |
Left edge
10 sections
Notes
In #3 obv. 13, Neugebauer and Sachs (1945) read ‘⌈še še-bi⌉’, while ‘še gagar-bi’ is expected. While the cuneiform sign ‘še’ is not clear, I did not identify the expected sign ‘gagar’ with certainty when I collated the tablet. 2/3 še is noted ‘šu-ri-a še u3 igi-6-gal2’, that is, ‘half a še and one-6th še’ (1/2 še + 1/6 še).
In #4, obv. 20, I translate ‘igi-9-gal2 šu-ri-a še’ as ‘one-half of one-9th še’ because ‘one-half’ (šu-ri-a) acts here as an operator and, as such, is postponed with regard to the argument ‘one-9th’. While here ‘one-half’ (šu-ri-a) has a multiplicative function, elsewhere in the text, ‘one-half’ (šu-ri-a) is an additive element (e.g. in #1, obv. 4, ‘12 še šu-ri-a še’ means ‘12 še and half a še’).
6.1 Text 4. YBC 4669 #1–9
Publication: Neugebauer 1935–1937: I, 514–516 (transliteration of #1–9; transliteration and translation of B1and B11); III, 27 ss. (transliteration and commentary of whole text); Thureau-Dangin 1937: 80–81 (translation of #1–9); Thureau-Dangin 1938: 208 ss. (#B1-B4; B7; B10-B11). Neugebauer and Sachs 1945: 103 (#B4), 138–139 (#B6).
Secondary publications: Friberg (2000): 129–131 (#2–4); Friberg (2001): 88–90 (#10–12); 93 (#13), 112–3 (#B10); Muroi (1990): 30 (#B11); 2007, 11 (#B7-B8); Robson (1999): 66 (#12); 89 (#B10); 126–7 (#B6).
CDLI number: P254987.
Provenance: unknown; Goetze’s Group 2 (southern group, perhaps from Ur) according to Friberg; Goetze’s Group 6 (perhaps from Kiš) according to Neugebauer and Høyrup.
Transliteration and Translation
Obverse
Col. i
# | li | ||
1 | 1 2 3 4 | gišba-ri2-ga 2/3 kuš3 4(diš) šu-si dal sukud-bi en-nam 2/3 kuš3 2(diš) 1/2 šu-si sukud | A bariga-vessel of one bariga 2/3 kuš 4 šu-si is the transversal Its height, how much is it? 2/3 kuš 2 1/2 šu-si is its height |
2 | 5 6 7 8 | gišba-an 3(ban2) 1/2 kuš3 3(diš) šu-si dal sukud-bi en-nam 2/3 kuš3 sukud-bi | A ban-vessel of 3 ban 1/2 kuš 3 šu-si is the transversal Its height, how much is it? 2/3 kuš is its height |
3 | 9 10 11 12 | gišba-an 10 1/3 kuš3 2(diš) šu-si dal sukud-bi en-nam 1/2 kuš3 sukud | A ban-vessel, 10 1/3 kuš 2 šu-si is the transversal Its height, how much is it? 1/2 kuš is its height |
4 | 13 14 15 16 | gišnig2 1(diš) sila3 6(diš) šu-si dal sukud-bi en-nam 6(diš) šu-si sukud | A vessel of 1 sila 6 šu-si is the transversal Its height, how much is it? 6 šu-si is its height |
5 | 17 18 19 20 | gišnig2 1/2 sila3 4(diš) 1/2 šu-si dal sukud-bi en-nam 5(diš) 1/3 šu-si sukud | A vessel of 1/2 sila 4 1/2 šu-si is the transversal Its height, how much is it? 5 1/3 šu-si is its height |
6 | 21 22 23 24 | gišnig2 1/3 sila3 4(diš) šu-si dal sukud-bi en-nam 4(diš) 1/2 šu-si sukud | A vessel of 1/3 sila 4 šu-si is the transversal Its height, how much is it? 4 1/2 šu-si is its height |
Col. ii
7 | 1 2 3 4 | gišnig2 1(u) gin2 3(diš) šu-si dal sukud-bi en-nam 4(diš) šu-si sukud | A vessel of 10 gin 3 šu-si is the transversal Its height, how much is it? 4 šu-si is its height |
8 | 5 6 7 8 | gišnig2 5(diš) gin2 2(diš) šu-si dal sukud-bi en-nam 4(diš) 1/2 šu-si sukud | A vessel of 5 gin 2 šu-si is the transversal Its height, how much is it? 4 1/2 šu-si is its height |
9 | 9 10 11 12 | gišnig2 1 ⌈gin2⌉ 1(diš) šu-si [dal] sukud-bi en-nam 3(diš) 1/2 šu-si sukud | A vessel of 1 gin 1 šu-si [is the transversal] Its height, how much is it? 3 1/2 šu-si is its height |
Related Lexical Lists
The list of vessels provided in YBC 4669 #1–9 has striking parallels in lexical lists, more precisely in the lexical list of wooden objects. According to the database developed by Niek Veldhuis, the Digital Corpus of Cuneiform Lexical Texts, most of the sources containing a section on standard vessels, or a part of it, come from Nippur, and one comes from Ur.Footnote 45 Figure 4.9 below shows a comparison between the entries found in mathematical text YBC 4669 #1–9, the composite text of Nippur sources containing items of the section on vessels (lines 516–526), and the transliteration of the source from Ur containing this section.Footnote 46
Comparison of YBC 4669 #1–9, ‘OB Nippur Ura 01’ lines 516–526, and UET 6/3–677 + 678 + UET 7–087 + 091
7.1 Text 5. BM 85194 #30
Publication: King (1900): 8–13 (copy); Neugebauer 1935–1937: I, 142–193; Thureau-Dangin 1938: 21–39.
This tablet belongs to a set of five tablets which exhibit a strong common feature: they end with a colophon providing the number of procedures (kibsum). They were classified by Goetze (1945: 150–151) and Høyrup (2002: 329–332) into groups belonging to northern traditions, and dated from the late Old Babylonian period. According to Høyrup, some of these tablets (BM 85194, BM 85196, BM 85200 + VAT 6599 and BM 85210) belong to a very homogeneous sub-group which he calls 6A: ‘The uniformity of group 6A is evidence of an effort to create and observe a particular canon. There is little doubt that all 6A-texts come from the same school—or that this school was located in Sippar’ (Høyrup 2002: 332).
Tablet BM 85194 contains 35 problems, including the detailed procedures, on volumes, walls, military constructions, circles, and work. The colophon indicates ‘total 35 procedures of calculation’ (šu-nigin 35 ki-bi-is minûti). Only problem 30 is provided here.
Transliteration of #30
Reverse
col. iii
7 | ⌈šum⌉-ma gišma2 1(diš) sar sig4 i-na-aš-ši-i |
8 | ⌈še⌉-a-am en-nam i-na-aš-ši za-e 41:40 sahar-hi-a |
9 | […] sig4-hi-a [41]:40 a-na 5 < igi > -gub i-ši 3:28:20 |
10 | sahar-hi-a [3(diš) 1/3] sila3 8(diš) 1/3 gin2 sig4sic (še!) 1(diš) sig4 |
11 | 3:28:[20] a-na 1(u) 2(diš) šu-sisic (šu-ši!) i-ši 41:40 ta-mar |
12 | 8(aš) ⌈1(bariga) 4(ban2)⌉ še gur ta-mar ne-pe2-šum |
Translation
7 | If a boat transports 1 sar of bricks, |
8 | what is the capacity that it transports? You. 41:40 is the volume |
9 | of a brick. [41]:40 to 5, the coefficient, raise. 3:28:20 is |
10 | the volume. [3 1/3] sila 8 1/3 gin is the capacity! (bricks-volumesic) of 1 brick |
11 | 3:28:[20] to 12 sixties raise, 41:40 you see |
12 | 8 gur 1 bariga 4 ban you see. That is the procedure |
Notes
In rev. col. iii, 9, the sign after ‘5’ may be ‘GUB’, that I interpret as an abbreviation of ‘igi-gub’, the technical term for coefficient which appears in the heading of the coefficient list YBC 5022 (Table 6 in Appendix 2, 6) However, Neugebauer and Thureau-Dangin read gan2, and Robson (1999: 115) follows them, even if she admits (1999: 118) that ‘the reading of GAN2 in BM 85194 is uncertain (cf. Powell 1987–90: 491)’. According to the copy and to the photos posted in the British Museum website, the reading ‘GAN2’ seems to me barely possible.
In rev. col. iii, 12, Neugebauer and Thureau-Dangin read 1.40, which does not make sense after the signs 8(aš). The reading is not certain because there is a line break at this place, but the remaining traces are compatible with the reading ‘1(bariga) 4(ban2)’ which is expected.
Appendix 2: Tables
Conventions and abbreviations: see the end of the chapter in the references section.
10.1 Table 1-UET 7–114
Publication: Gurney (1974): No. 114.
Secondary publication: Friberg (2000): 154–156.
CDLI number: P347073.
Provenance: Ur (Fig. 4.10).
UET 7–114 (copy Gurney 1974: No. 114)
Obverse
Col. i
[1(diš)] | šu-si | 10 |
[1(diš) 1/2] | šu-si | 15 |
[2(diš)] | šu-si | 20 |
[2(diš) 1/2] | šu-si | 25 |
[3(diš) | šu-si] | 30 |
[4(diš) | šu-si] | 40 |
[5(diš) | šu-si] | 50 |
[6(diš) | šu-si] | 1 |
[7(diš) | šu-si] | 1:10 |
8(diš) | šu-si | 1:20 |
9(diš) | šu-si | 1:30 |
1/3 | kuš3 | 1:40 |
1/3 | kuš3 1(diš) šu-si | 1:50 |
1/3 | kuš3 2(diš) šu-si | 2 |
1/3 | kuš3 3(diš) šu-si | 2:10 |
1/3 | kuš3 4(diš) šu-si | 2:20 |
1/2 | kuš3 | 2:30 |
1/2 | kuš3 1(diš) šu-si | 2:40 |
1/2 | kuš3 2(diš) šu-si | 2:50 |
1/2 | kuš3 3(diš) šu-si | 3 |
1/2 | kuš3 4(diš) šu-si | 3:10 |
2/3 | kuš3 | 3:20 |
2/3 | kuš3 1(diš) šu-si | 3:30 |
2/3 | kuš3 2(diš) šu-si | 3:40 |
[2/3] | kuš3 3(diš) šu-si | 3:50 |
2/3 | kuš3 4(diš) šu-si | 4 |
[5/6 | kuš3 3(diš) šu-si] | 4:40 |
Col. ii
5/6 | kuš3 4(diš) šu-si | 4:50 |
1(diš) | kuš3 | 5 |
1(diš) 1/3 | kuš3 | 6:40 |
1(diš) 1/2 | kuš3 | 7:30 |
1(diš) 2/3 | kuš3 | 8:20 |
2(diš) | kuš3 | 10 |
3(diš) | kuš3 | 15 |
4(diš) | kuš3 | 20 |
5(diš) | kuš3 | 25 |
1/2 | ninda | 30 |
1/2 | ninda 1(diš) kuš3 | 35 |
1/2 | ninda 2(diš) kuš3 | 40 |
1/2 | ninda 3(diš) kuš3 | 45 |
1/2 | ninda 4(diš) kuš3 | 50 |
1/2 | ninda 5(diš) kuš3 | 55 |
1(diš) | ninda | 1 |
1(diš) 1/2 | ninda | 1:30 |
2(diš) | ninda | 2 |
2(diš) 1/2 | ninda | 2:30 |
3(diš) | ninda | 3 |
3(diš) 1/2 | ninda | 3:30 |
4(diš) | ninda | 4 |
4(diš) 1/2 | ninda | 4:30 |
5(diš) | ninda | 5 |
5(diš) 1/2 | ninda | 5:30 |
6(diš) | ninda | [6] |
6(diš) 1/2 | ninda | [6:30] |
7(diš) | ninda | [7] |
7(diš) 1/2 | ninda | [7:30] |
8(diš) | ninda | [8] |
Reverse
Col. i
[…] | |||
[3(u) | ninda | 30] | |
[3(u) 5(diš) | ninda] | 35 | |
[4(u)] | ninda | 40 | |
[4(u)] 5(diš) | ninda | 45 | |
5(u) | ninda | 50 | |
5(u) 5(diš) | ninda | 55 | |
1(diš) | UŠ | 1 | |
1(diš) | UŠ 1(u) ninda | 1:10 | |
1(diš) | UŠ 2(u) ninda | 1:20 | |
1(diš) | UŠ 3(u) ninda | 1:30 | |
1(diš) | UŠ 4(u) ninda | 1:40 | |
1(diš) | UŠ 5(u) ninda | 1:50 | |
2(diš) | UŠ | 2 | |
3(diš) | UŠ | 3 | |
4(diš) | UŠ | 4 | |
5(diš) | UŠ | 5 | |
6(diš) | UŠ | 6 | |
7(diš) | UŠ | 7 | |
8(diš) | UŠ | 8 | |
9(diš) | UŠ | 9 | |
1/3 | danna | 10 | |
[1/2] | danna | 15 | |
[2/3] | danna | 20 | |
[1(diš)] | danna | 30 | |
[1(diš) 1/3] | danna | 40 | |
[1(diš) 2/3] | danna | 50 | |
[2(diš)] | danna | 1 | |
[nam-uš]-dagal-la-še3 | for the lengths and the widths | ||
Col. ii (Colophon)
ti-la dnisaba | Live Nisaba, |
dha-ia3 | Haya, |
den-ki | Enki |
šu e2-a-šar-i3-li | The hand of Ea-šar-ili, |
dub-sar tu-ra | the young scribe, |
in-sar | wrote (the tablet) |
Table 2-UET 7–115
Publication: Gurney (1974): No. 115.
Secondary publication: Friberg (2000): 156.
CDLI number: P347074.
Provenance: Ur (Fig. 4.11).
UET 7–115 (copy Gurney 1974: No. 115)
Obverse
Col. i
[…] | ||
[5/6 | kuš3 | 4]0.10 |
[1(diš) | kuš3 | 5] |
[1(diš) 1/3 | kuš3 | 6]0.40 |
[1(diš) 1/2 | kuš3] | 7.30 |
[1(diš) 2/3 | kuš3] | 8.20 |
[2(diš)] | kuš3 | 10 |
[3(diš)] | kuš3 | 15 |
[4(diš)] | kuš3 | 20 |
[5(diš)] | kuš3 | 25 |
1/2 | ninda | 30 |
1/2 ninda | 1(diš) kuš3 | 35 |
1/2 ninda | 2(diš) kuš3 | [40] |
[1/2 ninda] | 3(diš) [kuš3 | 45] |
[…] |
Col. ii
5(u) | [ninda | 50] |
1(diš) | [UŠ | 1] |
1(diš) 1(u) (ninda) | [UŠ | 1:10] |
1(diš) 2(u) (ninda) | [UŠ | 1:20] |
1(diš) 3(u) (ninda) | [UŠ | 1:30] |
1(diš) 4(u) (ninda) | [UŠ | 1:40] |
1(diš) 5(u) (ninda) | [UŠ | 1:50] |
2(diš) | [UŠ | 2] |
3(diš) | [UŠ | 3] |
4(diš) | [UŠ | 4] |
5(diš) | [UŠ | 5] |
[…] |
Reverse (columns right to left).
Col. i
[…] | ||
nam-uš#-[dagal-la-še3] | ||
= = = = = = = = = = = = = = = = = = | ||
1(diš) | šu-[si | 2] |
1 (diš) 1/2 | šu-[si | 3] |
2(diš) | šu-[si | 4] |
2(diš) 1/2 | [šu-si | 5] |
3(diš) | [šu-si | 6] |
4(diš) | [šu-si | 8] |
[…] | ||
Col. ii
5(diš) | ninda | [1] |
5(diš) 1/2 | ninda | 1.6 |
6(diš) | ninda | 1.12 |
6(diš) 1/2 | ninda | 1.18 |
7(diš) | ninda | 1.24 |
7(diš) 1/2 | ninda | 1.30 |
8(diš) | ninda | 1.36 |
8(diš) 1/2 | ninda | 1.42 |
9(diš) | ninda | 1.48 |
9(diš) 1/2 | ninda | 1.54 |
1(u) | ninda | 2 |
1(u) 5(diš) | ninda | 3 |
2(u) | ninda | 4 |
2(u) 5(diš) | ninda | 5 |
3(u) | ninda | 6 |
3(u) 5(diš) | ninda | 7 |
4(u) | ninda | 8 |
4(u) 5(diš) | ninda | 9 |
5(u) | ninda | 10 |
5(u) 5(diš) | ninda | 11 |
1(diš) | UŠ | 12 |
1(diš) 1(u) (ninda) | UŠ | 14 |
1(diš) 2(u) (ninda) | UŠ | 16 |
1(diš) 3(u) (ninda) | UŠ | 18 |
1(diš) 4(u) (ninda) | UŠ | 20 |
1(diš) 5(u) (ninda) | UŠ | 22 |
[…] |
Col. iii
5(diš) | UŠ | 1 | |
5(diš) 1/2 | UŠ | 1.6 | |
6(diš) | UŠ | 1.12 | |
6(diš) 1/2 | UŠ | 1.18 | |
7(diš) | UŠ | 1.24 | |
7(diš) 1/2 | UŠ | 1.30 | |
8(diš) | UŠ | 1.36 | |
8(diš) 1/2 | UŠ | 1.42 | |
9(diš) | UŠ | 1.48 | |
9(diš) 1/2 | UŠ | 1.54 | |
2(u) | danna | 2 | |
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = | |||
nam-sukud-bur3-še3 | For the heights and the depths | ||
Table 3-Ist Ni 3703 + N 3901 + UM 29–15-483
Published in Proust (2007): 107; joint made in November 2003.
Provenience: Nippur (Fig. 4.12).
Ist Ni 3703 + N 3901 + UM 29–15-483 (copy Proust (2007): pl. XVI)
Obverse
[2(diš)] | kuš3 | 2 |
[3(diš)] | kuš3 | 3 |
4(diš) | kuš3 | 4 |
[5(diš)] | kuš3 | 5 |
[1/2] | ninda | 6 |
[1/2] ninda | 1(diš) kuš3 | 7 |
[1/2] ninda | 2(diš) kuš3 | 8 |
1/2 ninda | 3(diš) kuš3 | 9 |
1/2 ninda | 4(diš) kuš3 | 10 |
Reverse
1/2 ninda | 5(diš) kuš3 | 11 |
[1(diš)] | ninda | 12 |
[1(diš) 1/2] | ninda | 18 |
[2(diš)] | ninda | 24 |
[2(diš) 1/2] | ninda | 30 |
[3(diš)] | ninda | 36 |
[3(diš) 1/2] | ninda | 42 |
[4(diš)] | ninda | 48 |
[4(diš)] 1/2 | ninda | 54 |
Upper edge
5(diš) | ninda | 1 | |
dnisaba za3-mi2 | Praise to Nisaba | ||
Table 4-HS 243
Publication: Hilprecht (1906): No. 41.
Secondary publication: Proust (2008): No. 31.
CDLI number: P388162.
Provenience: Nippur (Fig. 4.13).
HS 243 (courtesy of the Jena University, photo C. Proust)
Obverse
1(diš) | šu-si | 2 |
2(diš) | šu-si | ⌈4⌉ |
3(diš) | šu-si | ⌈6⌉ |
4(diš) | šu-si | ⌈8⌉ |
5(diš) | šu-si | ⌈10⌉ |
6(diš) | šu-si | ⌈12⌉ |
7(diš) | šu-si | ⌈14⌉ |
8(diš) | šu-si | 16 |
9(diš) | šu-si | 18 |
1/3 | kuš3 | 20 |
1/2 | kuš3 | 30 |
2/3 | kuš3 | 40 |
1(diš) | kuš3 | 1 |
Reverse
2(diš) | kuš3 | 2 |
3(diš) | kuš3 | 3 |
4(diš) | kuš3 | 4 |
5(diš) | [kuš3] | 5 |
[1/2 | ninda] | 6 |
[1/2] ninda | [1(diš) kuš3] | 7 |
1/2 [ninda | 2(diš) kuš3] | ⌈8⌉ |
1/2 [ninda | 3(diš) kuš3 | 9] |
1/2 [ninda | 4(diš) kuš3 | 10] |
[1/2 ninda | 5(diš) kuš3 | 11] |
1 | [ninda | 12] |
Upper edge
šu da-mi-⌈iq⌉-[…] | hand of xxx |
Left edge
šu ⌈im?⌉-di-3(u) | hand of xxx |
Right edge
ni/ir-N[E?…] |
Table 5-MS 3874
Publication: Friberg (2007: 69).
CDLI number: P252955.
Provenance: unknown; may come from a southern city according to (Friberg 2007: 69).
Obverse
1-da-am3 2/3-bi | 40-am3 | 2/3 of 1 is 40 |
šu-ri-a-bi | 30-am3 | its half is 30 |
igi-3 gal2-bi | 20 | the reciprocal of 3 is 20 |
igi-4 gal2-bi | 15 | the reciprocal of 4 is 15 |
igi-5 gal2-bi | 12 | the reciprocal of 5is 12 |
igi-6 gal2-bi | 10 | and so on |
igi-8 gal2-bi | 7:30 | |
igi-9 gal2-bi | 6:40 | |
igi-10 gal2-bi | 6 | |
igi-12 gal2-bi | 5 | |
igi-15 gal2-bi | 4 | |
igi-16 gal2-bi | 3:45 | |
igi-18 gal2-bi | 3:20 | |
igi-20 gal2-bi | 3 | |
igi-24 gal2-bi | 2:30 | |
igi-25 gal2-bi | 2:24 | |
igi-27 gal2-bi | 2:13:20 | |
igi-30 gal2-bi | 2 | |
igi-32 gal2-bi | 1:52:30 | |
igi-36 gal2-bi | 1:40 | |
igi-40 gal2-bi | 1:30 | |
igi-45 gal2-bi | 1:20 |
Reverse
igi-48 gal2-bi | 1:15 |
igi-50 gal2-bi | 1:12 |
igi-54 gal2-bi | 1:6:40 |
igi-1 gal2-bi | 1 |
igi-1:4 gal2-bi | 56:15 |
igi-1:21 gal2-bi | 44:26:40 |
= = = = = = = = = = = = = = = = = = = | |
im-gid2-da ši-ip-suen | Elongated tablet of ši-ip-suen |
u4 1(diš) […] | Day 1 [rest of the date lost] |
Table 6-YBC 5022
Publication: Neugebauer and Sachs (1945): text Ud.
Secondary publication: Robson (1999): text F.
CDLI number: P255026.
Provenance: unknown; Goetze’s Group 1 (southern group, probably from Larsa) (Fig. 4.14).
YBC 5022 (Neugebauer and Sachs 1945: pl. 18)
Transliteration obv. #1–16
# | |||
1 | igi-gub-ba | ša ne 2 -pi-iš-tum | |
2 | 4:30 | na-az-ba-al | sig4 |
3 | 7:12 | na-al-ba-an | sahar |
4 | 5:24 | na-al-ba-an | sig4-ab2 |
5 | 3:22:30 | na-az-ba-al | -ša |
6 | ⌈2⌉:42 | na-al-ba-an | sig4-al-ur3-ra |
7 | 1:41:15 | na-az-ba-al | -ša |
8 | 1:40 | na-az-ba-al | sahar |
9 | 1:12 | na-al-ba-an | 1(diš) kuš3 sig4 |
10 | 45 | na-az-ba-al | -ša |
11 | 4:48 | na-al-ba-an | 1/2 kuš3 sig4 |
12 | 4:23:36 | na-az-ba-al | -ša |
13 | 9 | na-al-ba-an | 1/3 kuš3 sig4 |
14 | 8:20 | na-al-ba-an | im-dugud |
15 | 5:38 | na-az-ba-al | -ša |
16 | 7:12 | ša | sig4-anše |
Translation obv. #1–16
# | ||
1 | coefficient | of the procedure |
2 | 4:30 | nazbalum of the brick |
3 | 7:12 | nalbanum of the (brick-)volume |
4 | 5:24 | nalbanum of the brick sig-ab |
5 | 3:22:30 | its nazbalum |
6 | 2:42 | nalbanum of the brick sig-al-ur-ra |
7 | 1:41:15 | its nazbalum |
8 | 1:40 | nazbalum of the volume |
9 | 1:12 | nalbanum of the brick 1 kuš |
10 | 45 | its nazbalum |
11 | 4:48 | nalbanum of the brick 1/2 kuš |
12 | 4:23:36 | its nazbalum |
13 | 9 | nalbanum of the brick 1/3 kuš |
14 | 8:20 | nalbanum of the clod of clay |
15 | 5:38 | its nazbalum |
16 | 7:12 | (coefficient) of the pile of bricks |
Appendix 3: Composite Text of Metrological Tables
This section offers a composite text made from Old Babylonian Nippur sources. This composite text does not exist as such in any preserved source, but is attested only by portions in many different tablets and fragments. However, due to the high standardization of Nippur’s curriculum, very little variation from one version of the tables to another can be detected. These variations essentially lie in the granularity of the tables (introduction or not of sub-units in the enumeration of measurement values). Thus, the composite text below may be considered as a relatively good reflection of the metrological tables memorized by students educated in Nippur’s scribal schools. Moreover, the metrological tables found in other southern schools (mainly in Ur, Uruk and Larsa) deviate very little from Nippur sources. In conclusion, the complete composite text can be considered as quite close to the set of results that was memorized by all of the students and scholars involved in mathematical activities in Old Babylonian southern Mesopotamia before the collapse of the southern cities, that is, before about 1740 BCE.
The following tables are extracts of the complete composite text. The complete composite text is available on CDLJ website.Footnote 47 To make the use of the tables easier for the modern reader, I added subtitles and indications of orders of magnitudes, referring to concrete realia, noted in parenthesis.
1-Capacities (še).
1 gin2 še | 1 |
1 1/3 gin2 | 1:20 |
1 1/2 gin2 | 1:30 |
1 2/3 gin2 | 1:40 |
1 5/6 gin2 | 1:50 |
2 gin2 | 2 |
3 gin2 | 3 |
4 gin2 | 4 |
5 gin2 | 5 |
6 gin2 | 6 |
7 gin2 | 7 |
8 gin2 | 8 |
9 gin2 | 9 |
1gin2 | 10 |
11 gin2 | 11 |
12 gin2 | 12 |
13 gin2 | 13 |
14 gin2 | 14 |
15 gin2 | 15 |
16 gin2 | 16 |
17 gin2 | 17 |
18 gin2 | 18 |
19 gin2 | 19 |
1/3 sila3 | 20 |
1/2 sila3 | 30 |
2/3 sila3 | 40 |
5/6 sila3 | 50 |
1 sila3 | 1 |
1 1/3 sila3 | 1:20 |
1 1/2 sila3 | 1:30 |
1 2/3 sila3 | 1:40 |
1 5/6 sila3 | 1:50 |
2 sila3 | 2 |
3 sila3 | 3 |
4 sila3 | 4 |
5 sila3 | 5 |
6 sila3 | 6 |
7 sila3 | 7 |
8 sila3 | 8 |
9 sila3 | 9 |
1(ban2) še | 10 |
2(ban2) še | 20 |
3(ban2) še | 30 |
4(ban2) še | 40 |
5(ban2) še | 50 |
1(barig) še | 1 |
2(barig) še | 2 |
3(barig) še | 3 |
4(barig) še | 4 |
1(aš) gur | 5 |
2 gur | 10 |
3 gur | 15 |
4 gur | 20 |
5 gur | 25 |
6 gur | 30 |
7 gur | 35 |
8 gur | 40 |
9 gur | 45 |
1gur | 50 |
11 gur | 55 |
12 gur | 1 |
13 gur | 1:5 |
14 gur | 1:10 |
15 gur | 1:15 |
16 gur | 1:20 |
17 gur | 1:25 |
18 gur | 1:30 |
19 gur | 1:35 |
2gur | 1:40 |
3gur | 2:30 |
4gur | 3:20 |
5gur | 4:10 |
1(geš2) gur | 5 |
1(geš2) 1gur | 5:50 |
1(geš2) 2gur | 6:40 |
1(geš2) 3gur | 7:30 |
1(geš2) 4gur | 8:20 |
1(geš2) 5gur | 9:10 |
2(geš2) gur | 10 |
3(geš2) gur | 15 |
4(geš2) gur | 20 |
5(geš2) gur | 25 |
6(geš2) gur | 30 |
7(geš2) gur | 35 |
8(geš2) gur | 40 |
9(geš2) gur | 45 |
1(geš’u) gur | 50 |
2(geš’u) gur | 1:40 |
3(geš’u) gur | 2:30 |
4(geš’u) gur | 3:20 |
5(geš’u) gur | 4:10 |
1(šar2) gur | 5 |
2(šar2) gur | 10 |
3(šar2) gur | 15 |
4(šar2) gur | 20 |
5(šar2) gur | 25 |
6(šar2) gur | 30 |
7(šar2) gur | 35 |
8(šar2) gur | 40 |
9(šar2) gur | 45 |
1(šar’u) gur | 50 |
2(šar’u) gur | 1:40 |
3(šar’u) gur | 2:30 |
4(šar’u) gur | 3:20 |
5(šar’u) gur | 4:10 |
1(šargal)gal gur | 5 |
1(šargal)gal šu-nu-tag gur | 5 |
2-Weights.
1/2 še ku3-babbar | 10 | (grain) |
1 še | 20 | |
2 še | 40 | |
3 še | 1 | |
6 še | 2 | |
9 še | 3 | |
12 še | 4 | |
15 še | 5 | |
18 še | 6 | |
21 še | 7 | |
22 še | 7:20 | |
23 še | 7:40 | |
24 še | 8 | |
25 še | 8:20 | |
26 še | 8:40 | |
27 še | 9 | |
28 še | 9:20 | |
29 še | 9:40 | |
igi 6-gal2 gin2 | 10 | |
igi 4-gal2 gin2 | 15 | |
1/3 gin2 | 20 | |
1 gin2 | 1 | (nugget) |
2 gin2 | 2 | |
3 gin2 | 3 | |
4 gin2 | 4 | |
5 gin2 | 5 | |
6 gin2 | 6 | |
7 gin2 | 7 | |
8 gin2 | 8 | |
9 gin2 | 9 | |
1gin2 | 10 | |
11 gin2 | 11 | |
12 gin2 | 12 | |
13 gin2 | 13 | |
14 gin2 | 14 | |
15 gin2 | 15 | |
16 gin2 | 16 | |
17 gin2 | 17 | |
18 gin2 | 18 | |
19 gin2 | 19 | |
1/3 ma-na | 20 | |
1/2 ma-na | 30 | |
2/3 ma-na | 40 | |
5/6 ma-na | 50 | |
1 ma-na | 1 | (stone) |
2 ma-na | 2 | |
3 ma-na | 3 | |
4 ma-na | 4 | |
5 ma-na | 5 | |
6 ma-na | 6 | |
7 ma-na | 7 | |
8 ma-na | 8 | |
9 ma-na | 9 | |
10 ma-na | 10 | |
11 ma-na | 11 | |
12 ma-na | 12 | |
13 ma-na | 13 | |
14 ma-na | 14 | |
15 ma-na | 15 | |
16 ma-na | 16 | |
17 ma-na | 17 | |
18 ma-na | 18 | |
19 ma-na | 19 | |
20 ma-na | 20 | |
30 ma-na | 30 | |
40 ma-na | 40 | |
50 ma-na | 50 | |
1 gu2 ku3-babbar | 1 | (load of a donkey) |
2 gu2 | 2 | |
3 gu2 | 3 | |
4 gu2 | 4 | |
5 gu2 | 5 | |
6 gu2 | 6 | |
7 gu2 | 7 | |
8 gu2 | 8 | |
9 gu2 | 9 | |
10 gu2 | 10 | |
… |
3- Surfaces
1/3 sar a-ša3 | 20 | (small garden) |
1/2 sar | 30 | |
2/3 sar | 40 | |
5/6 sar | 50 | |
1 sar | 1 | |
2 sar | 2 | |
3 sar | 3 | |
4 sar | 4 | |
5 sar | 5 | |
6 sar | 6 | |
7 sar | 7 | |
8 sar | 8 | |
9 sar | 9 | |
10 sar | 10 | (orchard) |
20 sar | 20 | |
30 sar | 30 | |
40 sar | 40 | |
1(ubu) GAN2 | 50 | |
1(ubu) GAN2 1sar | 1 | |
3(iku) GAN2 | 5 | |
1(eše3) GAN2 | 10 | |
1(eše3) 3(iku) GAN2 | 15 | |
2(eše3) GAN2 | 20 | |
2(eše3) 3(iku) GAN2 | 25 | |
1(bur3) GAN2 | 30 | (domain) |
1(bur3) 1(eše3) GAN2 | 40 | |
1(bur3) 2(eše3) GAN2 | 50 | |
2(bur3) GAN2 | 1 | |
4(bur3) GAN2 | 2 | |
6(bur3) GAN2 | 3 | |
8(bur3) GAN2 | 4 | |
1(bur’u) GAN2 | 5 | |
2(bur’u) GAN2 | 10 | |
3(bur’u) GAN2 | 15 | |
4(bur’u) GAN2 | 20 | |
5(bur’u) GAN2 | 25 | |
1(šar2) GAN2 | 30 | (district) |
2(šar2) GAN2 | 1 |
4-Lengths.
1 šu-si | 10 | (finger) |
2 šu-si | 20 | |
3 šu-si | 30 | |
4 šu-si | 40 | |
5 šu-si | 50 | |
6 šu-si | 1 | (tablet) |
7 šu-si | 1:10 | |
8 šu-si | 1:20 | |
9 šu-si | 1:30 | |
1/3 kuš3 | 1:40 | |
1/2 kuš3 | 2:30 | |
2/3 kuš3 | 3:20 | |
5/6 kuš3 | 4:10 | |
1 kuš3 | 5 | |
2 kuš3 | 10 | |
3 kuš3 | 15 | |
4 kuš3 | 20 | |
5 kuš3 | 25 | |
1/2 ninda | 30 | (house) |
1 ninda | 1 | (small garden) |
2 ninda | 2 | |
3 ninda | 3 | |
4 ninda | 4 | |
… | ||
10 ninda | 10 | |
20 ninda | 20 | |
30 ninda | 30 | |
40 ninda | 40 | |
45 ninda | 45 | |
50 ninda | 50 | |
1 UŠ | 1 | (palace) |
2 UŠ | 2 | |
3 UŠ | 3 | |
4 UŠ | 4 | |
5 UŠ | 5 | |
… | ||
1UŠ | 10 | |
1/2 danna | 15 | |
2/3 danna | 20 | |
5/6 danna | 25 | |
1 danna | 30 | |
2 danna | 1 | (district) |
4 danna | 2 | |
6 danna | 3 | |
8 danna | 4 | |
9 danna | 4:30 | |
10 danna | 5 | |
20 danna | 10 | |
30 danna | 15 | |
40 danna | 20 | |
50 danna | 25 |
5-Heights, depths.
1 šu-si | 2 | |
2 šu-si | 4 | |
3 šu-si | 6 | |
4 šu-si | 8 | |
5 šu-si | 10 | |
6 šu-si | 12 | |
7 šu-si | 14 | |
8 šu-si | 16 | |
9 šu-si | 18 | |
1/3 kuš3 | 20 | |
1/2 kuš3 | 30 | |
2/3 kuš3 | 40 | |
5/6 kuš3 | 50 | |
1 kuš3 | 1 | (small canal) |
2 kuš3 | 2 | |
3 kuš3 | 3 | |
4 kuš3 | 4 | |
5 kuš3 | 5 | |
2 kuš3 | 2 | |
3 kuš3 | 3 | |
4 kuš3 | 4 | |
5 kuš3 | 5 | |
1/2 ninda | 6 | |
1 ninda | 12 | (canal) |
2 ninda | 24 | |
3 ninda | 36 | |
4 ninda | 48 | |
5 ninda | 1 | (tower) |
10 ninda | 2 | |
20 ninda | 4 | |
30 ninda | 6 | |
40 ninda | 8 | |
50 ninda | 10 | |
1 UŠ | 12 | |
2 UŠ | 24 | |
3 UŠ | 36 | |
4 UŠ | 48 | |
5 UŠ | 1 | |
6 UŠ | 1:12 | |
7 UŠ | 1:24 | |
8 UŠ | 1:36 | |
9 UŠ | 1:48 | |
1UŠ | 2 | |
1/2 danna | 3 | |
2/3 danna | 4 | |
5/6 danna | 5 | |
1 danna | 6 |
References
Conventions
The exponent ‘sic’ after a sign indicates that I interpret this sign as an error in the ancient text, and the exponent ‘!’ indicates a corrected sign. I indicate both: the written sign and the expected sign in the transliteration and in the translation. For example, the transliteration ‘sig4sic (še!)’ means that the sign ‘sig4’ is written on the tablet, but that the sign ‘še’ is expected).
For the notations related to numbers and quantities, see Annex A.1 at the end of the volume.
Abbreviations
- li.:
-
means line
- col.:
-
means column
- obv.:
-
means obverse
- rev.:
-
means reverse
- #:
-
means section
SPVN means Sexagesimal Place-Value Notation
‘Table C’ means metrological table for capacities (complete composite text in Appendix 3 Table 1).
‘Table W’ means metrological table for weights (complete composite text in Appendix 3 Table 2).
‘Table S’ means metrological table for surfaces (complete composite text in Appendix 3 Table 3).
‘Table L’ means metrological table for lengths and widths (complete composite text in Appendix 3 Table 4).
‘Table Lh’ means metrological table for heights and depths (complete composite text in Appendix 3 table 5).
‘□’ represents the ‘multiplication’ of two measurement units
Primary Sources
Museum number | Primary publication | CDLI no |
|---|---|---|
BM 85194 (Text 5) | Neugebauer (1935)–1937: I, 142–193 Thureau-Dangin (1938): 21–39 | P254438 |
HS 243 (Table 4) | P388162 | |
Ist Ni 3703+ (Table 3) | Proust (2007): 107 | P368775 |
MS 3874 (Table 5) | Friberg (2007): 69 | P252955 |
UET 7-114 (Table 1) | Gurney (1974): No.114 | P347073 |
UET 7-115 (Table 2) | Gurney (1974): No.115 | P347074 |
YBC 4607 (Text 3) | Neugebauer and Sachs (1945): text O | P235642 |
YBC 4657 (Text 1) | Neugebauer and Sachs (1945): text G | P254982 |
YBC 4663 (Text 2) | Neugebauer and Sachs (1945): text H | P254984 |
YBC 4669 (Text 4) | Neugebauer and Sachs (1945): 103 Neugebauer (1935–1937): Vol. I, 514; Vol. III, 26 Thureau-Dangin (1937): 80–81 Thureau-Dangin (1938): 208 | P254987 |
YBC 5022 (Table 6) | Neugebauer and Sachs (1945): text Du Robson (1999): text F | P255026 |
Websites
CDLI: Cuneiform Digital Library Initiative. University of California, Los Angeles, University of Oxford, and Max Planck Institute for the History of Science, Berlin. http://cdli.ucla.edu/. Accessed December 2016.
DCCLT: Digital Corpus of Cuneiform Lexical Texts. 2003. Niek Veldhuis, University of California Berkeley, National Endowment for the Humanities. http://oracc.museum.upenn.edu/dcclt. Accessed December 2016.
MesoCalc. A Mesopotamian calculator. 2013. Baptiste Mélès, http://baptiste.meles.free.fr/site/mesocalc.html. Accessed December 2016.
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Proust, C. (2022). Volume, Brickage and Capacity in Old Babylonian Mathematical Texts from Southern Mesopotamia. 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_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-98361-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98360-4
Online ISBN: 978-3-030-98361-1
eBook Packages: HistoryHistory (R0)