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Foundations of Mathematics Buried in School Garbage (Southern Mesopotamia, Early Second Millennium BCE)

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Interfaces between Mathematical Practices and Mathematical Education

Abstract

In this chapter, I suggest an analysis of mathematical texts by relying on notions, concepts, and tools that were instilled during their early education in scholars who were active in Southern Mesopotamia during the Old Babylonian period (early second millennium BCE). The sources considered are mathematical texts from scribal schools that flourished in the Ancient Land of Sumer. I propose to examine the content of the garbage from these schools to access the “internal meanings” of advanced mathematical texts. First, I outline some characteristics of the mathematics taught in elementary education, drawing mainly from sources found in Nippur. Then, I discuss two mathematical texts in detail and show how they can be interpreted using the tools that were taught to (or invented by) their authors or users. The first example (CBS 1215) deals with the extraction of reciprocals by factorization; the second (CBS 12648) deals with volume problems using coefficients. The conclusion underlines the mathematical work that some ancient erudite scribes accomplished in order to emancipate them from the world of elementary education.

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Notes

  1. 1.

    For example, Veldhuis (1997), Robson (2001), and Proust (2007).

  2. 2.

    Below is the translation of lines 36–38 of the text “Dialogue between two scribes,” reconstructed after several duplicates from Nippur and some other Southern cities (my translation after Civil and Vanstiphout).

  3. 3.

    Sumerian is a dead language at this time, while the mother tongue of the young scribes attending the schools was Akkadian, a Semitic language.

  4. 4.

    See Veldhuis (1997), Robson (2001), Proust (2007) and bibliography provided by these publications.

  5. 5.

    The same metrological and numerical tables are attested in many duplicates from Nippur and other proveniences. The variations from one source to another being insignificant, any copy can be used indifferently. The sources that I present in this section come from Nippur if possible. However, in some cases, I use sources from other proveniences that are in better condition than those from Nippur. All the intermediate level sources used in part 2.2 come from Nippur. The tablet possibly reflecting the advanced level quoted in part 2.3 comes from another Southern city, possibly Larsa. All the sources quoted in this chapter are dated to the Old Babylonian period.

  6. 6.

    The order adopted in the curriculum, metrological tables of capacity, weight, surface, length and height, numerical tables of reciprocal, multiplication, and so on, is not the most intuitive for a modern reader. For an approach closer to the curriculum, see Proust (2007, Forthcoming-a).

  7. 7.

    Tablet MS 3869/11 belongs to the Schøyen Collection and is of unknown provenience, probably from Southern Mesopotamia. This tablet was published in (Friberg 2007, p. 119); photo available at https://cdli.ucla.edu/P252942

  8. 8.

    Tablet HS 243 is now kept at the University of Jena, and was published in (Hilprecht 1906, pl. 27; Proust 2008, n°31). The photo is available at https://cdli.ucla.edu/P388162

  9. 9.

    Tablet MS 2186 of the Schøyen Collection (Friberg, 2007, p. 110) is of unknown provenience, probably from Southern Mesopotamia. The photo is available online (https://cdli.ucla.edu/P250902). About the use of the section for small values of tables of weights as tables of surfaces and volume, see Proust (Forthcoming-b).

  10. 10.

    Tablet HS 217a from Nippur is now kept at the University of Jena and was published in (Hilprecht 1906; Proust 2008); photo at https://cdli.ucla.edu/P254585

  11. 11.

    Tablet ERM 14645 is kept at the Hermitage Museum, St Petersburg (Friberg and Al-Rawi 2017, Sect. 3.5). Photo https://cdli.ucla.edu/P211991

  12. 12.

    Tablet Ist Ni 10241 from Nippur is now kept at the Archaeological Museums of Istanbul (Proust 2007); photo https://cdli.ucla.edu/P368962

  13. 13.

    Here and elsewhere in the chapter, the reader is invited to check the calculations using MesoCalc, the Mesopotamian calculator implemented by Baptiste Mélès at http://baptiste.meles.free.fr/site/mesocalc.html

  14. 14.

    For Neugebauer and Sachs, “marginal numbers” means numbers noted on the top left; they note “GAR” the length unit ninda.

  15. 15.

    A detailed discussion of this issue is developed in Proust (2014).

  16. 16.

    Tablet YBC 4663 is now kept at the Yale Babylonian Collection, Yale University. It was published in Neugebauer and Sachs (1945). The elements of interpretation proposed here are detailed in Proust (Forthcoming-a).

  17. 17.

    My translation. The line numbers were added by the modern editors.

  18. 18.

    A discussion on this issue is proposed in Proust (2013).

  19. 19.

    CBS 1215 is now kept at the Museum of Archaeology and Anthropology, University of Pennsylvania, Philadelphia. It was published by Abraham Sachs (1947), who was the first historian to understand this puzzling numerical text. A detailed analysis is proposed in Proust (2012b). Photo: https://cdli.ucla.edu/P254479

  20. 20.

    The term “trailing part” was coined by Friberg to designate the last digits of a sexagesimal number.

  21. 21.

    This interpretation originates in a tablet, probably of later date, VAT 6505, which explains the algorithm for some of the entries found in CBS 1215. The procedure includes an addition. In my view, this addition reflects an incursion into the multiplication algorithm, and is not a step in the procedure (more discussion in Proust 2012a).

  22. 22.

    For a complete study and bibliography, see Proust (2007), Chap. 7.

  23. 23.

    The first entry in the table is 1/2 še; the entry 1/12 še is not attested but the corresponding number in SPVN is easily obtained by dividing 20 (which corresponds to 1 še) by 12.

  24. 24.

    This tablet, now kept at the Yale Babylonian Collection, was published by Neugebauer and Sachs (1945). For an updated study and bibliography, see Proust (Forthcoming-b).

  25. 25.

    Such terms are attested in other mathematical texts. See for example Str. 368 (Thureau-Dangin 1938, p. 91–92); VAT 8391 #3 (Høyrup 2002, p. 83) the terms « truth » (gi-na = kînum) length and « false » (lul = sarrum) length. See a synthesis in Høyrup (2002, p. 100–101).

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Acknowledgements

I thank the people of La Filature du Mazel (Valleraugue, France) who offered me a wonderful co-working space to write this chapter.

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Authors and Affiliations

  1. CNRS and Université Paris Diderot, Paris, France

    Christine Proust

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  1. Christine Proust
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Correspondence to Christine Proust .

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  1. Department of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Gert Schubring

Appendix

Appendix

The diagrams below represent the metrological systems and correspondences with SPVN according to Old Babylonian sources from Nippur. The arrows represent the factors between different units (e.g., gin ←180− še means 1 gin is equal to 180 še); the numbers below the units are the numbers in SPVN that correspond to these units in the metrological tables (e.g., 20 below še means that 1 še corresponds to 20 in metrological tables).

Length and other horizontal dimensions (1 ninda represents about 6 m)

  • ninda ←12− kuš ←30− šu-si

  • 1 5 10

Heights and other vertical dimensions (1 ninda represents about 6 m)

  • ninda ←12− kuš ←30− šu-si

  • 5 1 2

Surface and volume (a surface of 1 sar is that of a square of 1 ninda-side; a volume of 1 sar is that of a rectangular cuboid of 1 sar-base and 1 kuš-high)

  • gan ←100− sar ←60− gin ←180− še

  • 1:40 1 1 20

Weight (1 mana represents about 500 g)

  • mana ←60− gin ←180− še

  • 1 1 20

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Proust, C. (2019). Foundations of Mathematics Buried in School Garbage (Southern Mesopotamia, Early Second Millennium BCE). In: Schubring, G. (eds) Interfaces between Mathematical Practices and Mathematical Education. International Studies in the History of Mathematics and its Teaching. Springer, Cham. https://doi.org/10.1007/978-3-030-01617-3_1

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