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Ancient Egyptian mathematics: New perspectives on old sources

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Although Egyptian mathematics will probably never have the vast number of sources that still can be found in other cultures like India or Mesopotamia, there is more available than has been used so far.33 The analysis of all the available mathematical texts, taken along with the additional material from administrative economic and literary contexts related to Egyptian mathematics, is certain to provide a better foundation for understanding its role within Egyptian culture. This integrated approach represents an important advance beyond the early studies that relied exclusively on an internal analysis of a small corpus of mathematical texts, which served for several decades as the sole basis for assessing nearly three millennia of mathematical life in ancient Egypt. By carefully rereading these classical mathematical texts while according the new sources a serious first reading, we may anticipate that the fate of Egyptian mathematics faces an exciting future.

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Notes

  1. I thank David Rowe for his comments on previous versions of this article and for his corrections of my English. I also thank Richard Parkinson of the British Museum and Stephen Quirke of the Petrie Museum for permission to include photographs of sources.

  2. See Annette Imhausen and Jim Ritter, “Mathematical Papyri,” in: Mark Collier and Stephen Quirke (eds.),The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford: Arcaheo press 2004.

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  3. Among the early editions, the most noteworthy are still Thomas E. Peet,The Rhind Mathematical Papyrus. British Museum 10057 and 10058, London: Hodder and Stoughton 1923, and Wasili W. Struve,Mathematischer Papyrus des Staatlichen Museums der Schonen Kunste in Moskau(Quellen und Studien zur Geschichte der Mathematik, Abteilung A: Quellen, Vol. 1), Heidelberg: Springer 1930.

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  4. Otto Neugebauer,Die Grundlagen der agyptischen Bruchrechnung, Berlin: Julius Springer 1926.

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  5. Otto Neugebauer,A History of Ancient Mathematical Astronomy (Part Two). Berlin, Heidelberg, New York: Springer 1975: 559.

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  6. See for example the interpretations of Plimpton 322, e.g., compare Joran Friberg, “Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples and the Babylonian triangle parameter equations,”Historia Mathematica 8 (1981): 277–318 and the recent reassessment by Eleanor Robson (Eleanor Robson,“Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322,” Historia Mathematica 28 (2001); 167-206 and Eleanor Robson, “Words and pictures: new light on Plimpton 322,”American Mathematical Monthly 109 (2002): 105-120).

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  7. An extreme example of this is Richard Gillings, “The Volume of a Truncated Pyramid in Ancient Egyptian Papyri,”The Mathematics Teacher 57 (1964): 552–555.

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  8. For Egyptian mathematics, see for example James Ritter, “Chacun sa vérité: les mathématiques en Égypte et en Mésopotamie,” in: Michel Serres (ed.),Ele ments d’histoire des sciences: 39–61, Paris: Bordas 1989; James Ritter, “Egyptian Mathematics,” in: Helaine Selin (ed.),Mathematics Across Cultures: The History of Non-Western Mathematics: 115-136, Dordrecht, Boston, London: Kluwer 2000, as well as Annette Imhausen,Ägyptische Algorithmen: Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden: Otto Harrassowitz 2003. For Greek Mathematics, cf. Serafina Cuomo,Ancient Mathematics, London, New York: Routledge 2001, Michael N. Fried and Sabetai Unguru,Apollonius of Perga’s Conica. Text, Context, Subtext, Leiden: Brill 2001, as well as David Fowler,The Mathematics of Plato’s Academy: A New Reconstruction (Second Edition), Oxford: Clarendon Press 1999, and Reviel Netz,The Shaping of Deduction in Greek Mathematics: A Study of Cognitive History (Ideas in Context 51), Cambridge: Cambridge University Press 1999. For Mesopotamian mathematics, see most recently Jens Høyrup,Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and its Kin, New York: Springer 2002, and Eleanor Robson,Mesopotamian Mathematics, 2100-1600 BC: Technical Constants in Bureaucracy and Education (Oxford Editions of Cuneiform Texts XIV), Oxford: Clarendon Press 1999.

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  9. See Gary Urton,The Social Life of Numbers. A Quechua Ontology of Numbers and Philosophy of Arithmetic, Austin, Texas: University of Texas Press 1997, and Marcia Ascher,Mathematics Elsewhere. An Exploration of Ideas across Cultures, Princeton, N.J.: Princeton University Press 2002.

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  10. See, for example, for Mesopotamia Jens Høyrup,Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and its Kin, New York: Springer 2002.

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  11. See note 8.

  12. See Annette Imhausen and Jim Ritter, “Mathematical Papyri,” in: Mark Collier and Stephen Quirke (eds.),The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford: Arcaheopress 2004. Another mathematical fragment will be published in the next volume of that series.

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  13. See Oleg Berlev, “Review of William Kelly Simpson: Papyrus Reisner III: The Records of a Building Project in the Early Twelfth Dynasty, Boston: Museum of Fine Arts 1969,”Bibliotheca Orientalis 28 (1971): 324–326, esp. p. 325.

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  14. Richard Parker, “A Demotic Mathematical Papyrus Fragment,”Journal of Near Eastern Studies 18 (1959): 275–279; Richard Parker,Demotic Mathematical Papyri, Providence, R.I.: Brown University Press 1972; Richard Parker, “A Mathematical Exercise—P. Dem. Heidelberg 663,”Journal of Egyptian Archaeology 61 (1975):189-196. A list of Demotic mathematical ostraca can be found in Jim Ritter, “Egyptian Mathematics,” in: Helaine Selin (ed.),Mathematics across Cultures. The History of Non-Western Mathematics, Dordrecht: Kluwer 2000: 134, note 27.

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  15. See Gunter Dreyer,Umm el-Qaab I. Das prädynastische Köcnigsgrab U-j und seine frühen Schriftzeugnisse, Mainz: Von Zabern 1998.

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  16. For a discussion of the inscriptions on these tags, see Günter Dreyer,Umm el-Qaab I. Das prädynastische Königsgrab U-j und seine frühen Schriftzeugnisse, Mainz: Von Zabern 1998, pp. 137–145, and John Baines, “The Earliest Egyptian Writing: Development, Context, Purpose,” in: Stephen D. Houston,The First Writing. Script Invention as History and Process, Cambridge: Cambridge University Press 2004:150-189.

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  17. See problems 56–60 of the Rhind Mathematical Papyrus.

  18. Jim Ritter, “Mathematics in Egypt,” in: Helaine Selin (ed.),Encyclopedia of the History of Science, Technology and Medicine in Non-Western Cultures, Dordrecht, Boston, London: Kluwer 1997, p. 631.

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  19. For the prehistory of Egyptian fractions and their development see Jim Ritter, “Metrology and the Prehistory of Fractions,” in: Paul Benoit, Karine Chemla, Jim Ritter (eds.),Histoire de fractions, fractions d’histoire: 19–34, Basel, Boston, Berlin: Birkhäuser 1992.

  20. See, for example, the description of Couchoud: “ . . . il ne semble avoir connu que les fractions unitaires, c’est à dire celles dans lesquelles le numérateur est toujours équivalent à I’unité,...” (Sylvia Couchoud,Mathématiques Égyptiennes. Recherches sur les connaissances mathématiques de I’Égypte pharaonique, Paris: Le Léopard d’Or 1993, p. 21) or that of Gillings: “When the Egyptian scribe needed to compute with fractions he was confronted with many difficulties arising from the restriction of his notation. His method of writing numbers did not allow him to write such simple fractions as -3/4 or 5/9 because all fractions 3/5 or5/9 had to have unity for their numerators (with one exception).” (Richard J. Gillings,Mathematics in the Time of the Pharaohs, Cambridge, Mass.: MIT Press 1972, p. 20).

  21. See Jim Ritter, “Egyptian Mathematics,” in: Helaine Selin (ed.),Mathematics across Cultures. The History of Non-Western Mathematics, Dordrecht, Boston, London: Kluwer 2000, p. 120.

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  22. For a discussion of the use of an abstract number system and conversions into metrological systems, see Jim Ritter, “Egyptian Mathematics,” in: Helaine Selin (ed.),Mathematics across Cultures. The History of Non-Western Mathematics, Dordrecht, Boston, London: Kluwer 2000, pp. 121–122.

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  23. The cubit was the Egyptian standard measure of length. 1 cubit consisted of 7 palms; each palm, of 4 digits.

  24. An English example for its use would be the statement “If I have a stone in my hand, and let it drop, then the stone falls to the ground.” The last part of this statement “then the stone falls to the ground” is where the sdm. ((h))r=f is used in an Egyptian text.

  25. Examples can be found in Annette Imhausen and Jim Ritter, “Mathematical Fragments: UC32114, UC32118, UC32134, UC32159—UC32162,” in: Mark Collier and Stephen Quirke,The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical (British Archaeological Reports International Series 1209): 71/2-96, Oxford; Archaeopress 2004, esp. pp. 85–86.

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  26. The New Kingdom Ostracon Senmut 153 may be interpreted as a table of 1/7 in this way, see David Fowler,The Mathematics of Plato’s Academy: A New Reconstruction (Second Edition), Oxford: Clarendon Press 1999, p. 269.

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  27. Jim Ritter, “Chacun sa verite: les mathémathiques en Égypte et en Mésopotamie,” in: Michel Serres (ed.),Élements d’histoire des sciences: 39-61, Paris: Bordas 1989 (English edition: Jim Ritter, “Measure for Measure: Mathematics in Egypt and Mesopotamia,” in: Michel Serres (ed.),A History of Scientific Thought. Elements of a History of Science: 44-72, Oxford: Blackwell 1995).

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  28. Annette Imhausen,Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten. (Agyptologische Abhandlungen 65). Wiesbaden: Otto Harrassowitz 2003.

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  29. For a detailed discussion of these problems see Annette Imhausen, “Egyptian Mathematical Texts and their Contexts,”Science in Context 16, 2003: 367–389.

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  30. Michel Guillemot, “Les notations et les pratiques opératoires permettent-elles de parler de ‘fractions egyptiennes’?”, in: Paul Benoit, Karine Chemla, Jim Ritter (eds.),Histoire de fractions, fractions d’histoire, Basel, Boston, Berlin: Birkhäuser 1992: 53–69.

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  31. Annette Imhausen, “Calculating the Daily Bread: Rations in Theory and Practice,”Historia Mathematica 30 (2003): 3–16.

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  32. A first attempt to analyze the mathematical content of some parts of the Reisner Papyri has been made by Richard J. Gillings,Mathematics in the Time of the Pharaohs, Cambridge, Mass. MIT Press 1972, pp. 218–231.

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  33. A variety of architectural sources (with athematical implications) can be found in Corinna Rossi,Architecture and Mathematics in Ancient Egypt, Cambridge: Cambridge University Press 2004.

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    Annette Imhausen

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Imhausen, A. Ancient Egyptian mathematics: New perspectives on old sources. The Mathematical Intelligencer 28, 19–27 (2006). https://doi.org/10.1007/BF02986998

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