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# Historiography of Cuneiform Mathematics

The first mathematical cuneiform text to have been brought to public knowledge was a copy of tablet BM 92698 published by Henry C. Rawlinson and Georges Smith in 1875 in "Cuneiform inscriptions of Western Asia" vol. 4. This important document is a large multi-column tablet containing a set of metrological and numerical tables linked to the calculation of surfaces; this text was really understood only in 1930 by Thureau-Dangin (Thureau-Dangin, François. 1930. "La table de Senkereh." *Revue d'Assyriologie* 27: 115-116). The other important early publication was offered in 1906 by Herman V. Hilprecht. With this book "*Mathematical, Metrological and Chronological Tablets from the Temple Library of Nippur*" Hilprecht revealed the existence of scribal schools and mathematical teaching in Nippur.

The pioneering works of the French assyriologist François Thureau-Dangin (1872-1944) were, in a first period, focused on metrological matters. Between 1896 and 1930, he published numerous papers, mainly in the Revue d'Assyriologie, where he elucidated most of the metrological systems used in cuneiform texts – a prerequisite for a full understanding of mathematical texts – and the nature of the sexagesimal place value notation used in mathematical texts. From 1930 until his death, he published a lot of mathematical texts and comments on them, including his landmark book *Textes Mathématiques Babyloniens*, 1938.

Otto E. Neugebauer (1899-1990), an Austro-American mathematician, chose to devote himself to the "exact sciences in Antiquity" (mathematics and astronomy) in the early stages of his career in the context of the very open and innovative mathematical school of the University of Göttingen, Germany. As soon as the early 1930, he undertook at Göttingen the systematic publication of the cuneiform mathematical texts preserved in European Museums, and finished a first book in Copenhagen, after fleeing Nazi Germany (*Mathematische Keilschrifttexte* vol. I-III, 1935-1937). He emigrated to the United States in 1939, and he completed his work in Brown University, Providence. He published with Abraham Sachs and the assistance of Albrecht Götze a second book focused on the American collections (*Mathematical Cuneiform Texts*, 1945).

The most notable contributions of the subsequent generation are those of Evert Bruins, who published the mathematical texts from Susa with Marguerite Rutten (*Textes Mathématiques de Suse*, 1961) and of Aizik A. Vaiman who published mathematical tablets preserved in the Hermitage Museum, Saint Petersburg, Russia (*Shumero-vavivonskaya matematika III-I tysyacheletiya do n.e.* = *Sumero-Babylonian Mathematics of the Third to First Millennia BC*, 1961). In close connection with the study of mathematical texts, the contribution of Marvin A. Powell on metrology turned to be essential for the understanding of mathematical practices (*Sumerian numeration and metrology*, PhD thesis 1971, and the landmark article "Masse und Gewichte" in *Reallexikon der Assyriologie*, vol. 7, pp. 457-517, 1987-1990).

By the end of the 1970, Jöran Friberg revived the field of cuneiform mathematics. His elucidation of the metrologies used in proto-cuneiform texts opened new avenues of research. The role played by mathematicians in Assyriology was already illustrated by the works of Neugebauer. This role deserves to be emphasized again here, by quoting Robert K. Englund:

"It may surprise some that the most important recent advances in the decipherment of the proto-cuneiform documents have been made by and in collaboration with mathematicians with no formal training in Assyriology, J. Friberg and P. Damerow. But remembering that the great majority of archaic texts are administrative records of the collection and distribution of grain, inventories of dairy fats stored in jars of specific sizes, and so on, that is, documents above all made to record in time quantifiable objects, it is reasonable to expect that such documents would contain, no less than the accounts of current institutions, evidence of mathematical procedures used in the archaic period and that they would thus contain the seeds of the mathematical thinking which developed during the third millennium. (Englund 1998. "Texts from the Late Uruk Period." Pp. 15-233 inMesopotamien.Späturuk-Zeit und Frühdynastische Zeit, vol. 160,OBO, edited by J. Bauer, R. K. Englund, and M. Krebernik. Freibourg, Göttingen, p. 111).

Jöran Friberg recently published important books, including the edition of mathematical texts of the Schøyen collection (*A Remarkable Collection of Babylonian Mathematical Texts*, 2007).

The approach of cuneiform mathematics has been revolutionized in the last decades by the works of Jens Hørup, who elucidated the methods of reasoning used by the ancient scribes to solve quadratic problems (*Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and its Kin*, 2002) – see [9]. The social sciences entered in the field with the work of Eleanor Robson (*Mesopotamian Mathematics, 2100-1600 BC. Technical Constants in Bureaucracy and Education*, 1999, and *Mathematics in Ancient Iraq: A Social History*, 2008).

## Bibliography

- Jones, Alexander, Christine Proust, and John Steele. 2016. "A Mathematician's Journeys: Otto Neugebauer and Modern Transformations of Ancient Science" in
*Archimedes. New Studies in the History and Philosophy of Science and Technology*, Springer. - Høyrup, Jens. 1996. "Changing trends in the historiography of Mesopotamian mathematics: an insider's view."
*History of science*34:1-32. - Melville, Duncan J. forthcoming. "After Neugebauer: Recent developments in Mesopotamian mathematics." Pp. *** in
*A Mathematician's Journeys: Otto Neugebauer and Modern Transformations of Ancient Science*,*Archimedes*, edited by A. Jones, C. Proust, and J. M. Steele: Springer.

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