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Computational Techniques and Computational Aids in Ancient Mesopotamia

  • Jens Høyrup
Chapter
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 11)

Abstract

Any history of mathematics that deals with Mesopotamian mathematics will mention the use of tables of reciprocals and multiplication in sexagesimal place-value notation—perhaps also of tables of squares and other higher arithmetical tables. Less likely there is a description of metrological lists and tables and of tables of technical constants. All of these belong to a complex of aids for accounting that was created during the “Ur III” period (twenty-first c. bce). Students’ exercises from the Old Babylonian period (2000–1600 bce) teach us something about their use. First metrological lists then metrological tables were learned by heart. These allowed the translation of real measures into place-value measures in a tacitly assumed basic unit. At an advanced level, we see multiplications, where first two factors and then the product are written in sequence on a clay tablet for rough work. Problem texts show us more about the use of the metrological tables and the tables of technical constants. Neither genre allows us to see directly how additions and subtractions were made, nor how multiplications of multi-digit numbers were performed. A few errors in Old Babylonian problem texts confirm, however, that multiplications were performed on a support where partial products would disappear once they had been inserted—in a general sense, some kind of abacus. Other errors, some from Old Babylonian period and some others from the Seleucid period (third and second c. bce), show that the “abacus” in question had four or five sexagesimal levels, and textual evidence reveals that it was called “the hand”. This name was in use at least from the twenty-sixth c. bce until c. 500 bce. This regards addition and subtraction from early times onward, and multiplication and division in Ur III and later. A couple of problem texts from the third millennium deals with complicated divisions, namely divisions of large round numbers by 7 and by 33. They use different but related procedures, suggesting that no standard routine was at hand.

Keywords

Sexagesimal place-value system Mathematical tables (Mesopotamia) Abacus (Mesopotamia) Scribe school curriculum (Mesopotamia) Centennial system (Mari) 

References

  1. Brack-Bernsen, Lis, and Hermann Hunger. 2008. BM 42484+42294 and the Goal-Year method. SCIAMUS, 9: 3–23.Google Scholar
  2. Bruins, Evert M., and Marguerite Rutten. 1961. Textes mathématiques de Suse. Paris: Paul Geuthner.Google Scholar
  3. Chambon, Grégory. 2012. Notations de nombres et pratiques de calcul en Mésopotamie; Réflexions sur le système centésimal de position. Revue d’Histoire des Mathématiques, 18: 5–36.Google Scholar
  4. Edzard, Dietz Otto. 1969. Eine altsumerische Rechentafel (OIP 14, 70). In Lišān mitḫurti. Festschrift Wolfram Freiherr von Soden zum 19.VI.1968 gewidmet, ed. W. Röllig, 101–104. Kevelaer: Butzon & Bercker/Neukirchen-Vluyn: Neukirchener Verlag des Erziehungsvereins.Google Scholar
  5. Feliu, Lluís. 2012. A new early dynastic IIIb metro-mathematical table tablet of area measures from Zabalam. Altorientalische Forschungen, 39: 218–225.CrossRefGoogle Scholar
  6. Friberg, Jöran. 1986. The early roots of Babylonian mathematics. III: Three remarkable texts from ancient Ebla. Vicino Oriente, 6: 3–25.Google Scholar
  7. Friberg, Jöran. (2007). A Remarkable Collection of Babylonian Mathematical Texts. New York: Springer.Google Scholar
  8. Høyrup, Jens. (1982). Investigations of an early Sumerian division problem, c. 2500 B.C. Historia Mathematica, 9, 19–36.CrossRefGoogle Scholar
  9. Høyrup, Jens. 2002a. Lengths, widths, surfaces: A portrait of Old Babylonian algebra and its kin. New York: Springer.CrossRefGoogle Scholar
  10. Høyrup, Jens. 2002b. A note on Old Babylonian computational techniques. Historia Mathematica, 29: 193–198.CrossRefGoogle Scholar
  11. Hunger, Hermann, and Teije de Jong. 2014. Almanac W22340a from Uruk: The latest datable cuneiform tablet. Zeitschrift für Assyriologie und Vorderasiatische Archäologie, 104: 182–194.CrossRefGoogle Scholar
  12. Neugebauer, Otto. 1934. Vorlesungen über Geschichte der antiken mathematischen Wissenschaften. I: Vorgriechische Mathematik. Berlin: Julius Springer.Google Scholar
  13. Neugebauer, Otto. 1935. Mathematische Keilschrift-Texte. I–III. Berlin: Julius Springer (1935–1937).CrossRefGoogle Scholar
  14. Neugebauer, Otto. 1955. Astronomical cuneiform texts: Babylonian ephemerides of the Seleucid period for the motion of the sun, the moon, and the planets. London: Lund Humphries.CrossRefGoogle Scholar
  15. Neugebauer, Otto. 1957. The exact sciences in antiquity, 2nd ed. Providence, Rh.I.: Brown University Press.Google Scholar
  16. Neugebauer, Otto, and Abraham Joseph Sachs. 1945. Mathematical cuneiform texts. New Haven, Connecticut: American Oriental Society.Google Scholar
  17. Powell, Marvin A. 1976. The antecedents of Old Babylonian place notation and the early history of Babylonian mathematics. Historia Mathematica, 3: 417–439.CrossRefGoogle Scholar
  18. Proust, Christine. 2000. La multiplication babylonienne: la part non écrite du calcul. Revue d’Histoire des Mathématiques, 6: 293–303.Google Scholar
  19. Proust, Christine. 2008. Avec la collaboration de Manfred Krebernik et Joachim Oelsner. Tablettes mathématiques de la collection Hilprecht. Wiesbaden: Harrassowitz.Google Scholar
  20. Robson, Eleanor. 1999. Mesopotamian mathematics 2100–1600 BC. Technical constants in bureaucracy and education. Oxford: Clarendon Press.Google Scholar
  21. Whiting, Robert M. 1984. More evidence for sexagesimal calculations in the third millennium B.C. Zeitschrift für Assyriologie und Vorderasiatische Archäologie, 74: 59–66.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Roskilde UniversityRoskildeDenmark
  2. 2.Institute for the History of Natural SciencesChinese Academy of SciencesBeijingChina
  3. 3.Max-Planck-Institut Für WissenschaftsgeschichteBerlinGermany

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