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--- general_overview [2015/10/25 20:36] – [Numbers and metrology] gombert
+++ general_overview [2016/04/11 21:16] (current) – gombert
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 It can be estimated to about 2000 the number ofmathematical tabletsknown to date, a large majority of which already published. The distribution of these sourcesaccording to time and spaceis extremely irregular, as shown by table 1 below. This table contains rough data, established with the help of the CDLI consulted September 13, 2015; note that most of the tablets from unknown or uncertain provenience according to the CDLI are very probably Old Babylonian.
-\\ | **Periods** | **Number of tablets** | **Provenience** |
+\\
+^ Periods ^ Number of tablets ^ Provenience ^
 | Early Dynastic / Old Akkadian (2500-2100) | 27 | Southern Mesopotamia |
 | Ur III (2100-2000) | 8 | Southern Mesopotamia |
@@ Line 32: / Line 33: @@
 Despite the diversity of mathematical methods invented in different communities of scribes in different periods, mathematical cuneiform texts exhibit striking common features: a standardized metrology, and the use of the sexagesimal place value notation, at least from the late third millennium.
-\\ The standardized metrology used in mathematical texts emerged probably in the context of the reforms undertaken by rulers of the first centralized empires, from the mid-third millennium under the Old Akkadian dynasties and subsequently under the Ur III rulers, as shown by Marvin Powell. This standardized metrology dominates in mathematical texts of all periods, even if the archaic texts use different systems, and the late texts show a competition with other systems used in Mesopotamia from mid-first millennium. The standardized metrology and earlier variants are presented with details is in the tool [[http://cdli.ox.ac.uk/wiki/doku.php?id=numbers_metrology|"Numbers and Metrology"]] of the wiki. See in particular the metrology that was taught in [[http://cdli.ox.ac.uk/wiki/doku.php?id=old_babylonian_scribal_schools Old Babylonian scribal school]].
+\\ The standardized metrology used in mathematical texts emerged probably in the context of the reforms undertaken by rulers of the first centralized empires, from the mid-third millennium under the Old Akkadian dynasties and subsequently under the Ur III rulers, as shown by Marvin Powell. This standardized metrology dominates in mathematical texts of all periods, even if the archaic texts use different systems, and the late texts show a competition with other systems used in Mesopotamia from mid-first millennium. The standardized metrology and earlier variants are presented with details is in the tool [[http://cdli.ox.ac.uk/wiki/doku.php?id=numbers_metrology|"Numbers and Metrology"]] of the wiki. See in particular the metrology that was taught in [[http://cdli.ox.ac.uk/wiki/doku.php?id=old_babylonian_scribal_schools|Old Babylonian scribal school]].
 \\ The reform realized during the second part of 3<sup>rd</sup> millennium produced not only a standardized and coherent metrology, but also a new numerical system, the sexagesimal place value notation, which impacted strongly the development of mathematics in the Ancient Near East. (see article on the “Sexagesimal Place Value Notation”).
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 \\ Most of the known Old Babylonian mathematical cuneiform texts are school exercises. These school texts appeared too simple and repetitive to deserve some interest, and only recently they began to be systematically published. However, they provide valuable information on education in Mesopotamia, an important facet of the social life. Moreover, they illuminate the mathematical methods that were used by learned scholars who wrote advanced mathematical texts.
-\\ Old Babylonian advanced mathematical texts were probably written by masters of the scribal schools. Most of them seem to treat concrete subjects (fields, canals, excavations, bricks, walls, buildings, economics…), and use realistic data. However, the mathematical treatment of such subjects can hardly be considered as realistic. The most often, the problems solved, including quadratic and higher degree problems, are of interest for mathematicians, not for practitioners. To illustrate the "supra-utilitarist" (to use Høyrup's expression) nature of some mathematical texts, here is problem 9 from catalogue YBC 4652 < [[http://www.cdli.ucla.edu/P254981|http://www.cdli.ucla.edu/P254981]]> , where a stone is weighed after removing improbable fractions of it:
+\\ Old Babylonian advanced mathematical texts were probably written by masters of the scribal schools. Most of them seem to treat concrete subjects (fields, canals, excavations, bricks, walls, buildings, economics…), and use realistic data. However, the mathematical treatment of such subjects can hardly be considered as realistic. The most often, the problems solved, including quadratic and higher degree problems, are of interest for mathematicians, not for practitioners. To illustrate the "supra-utilitarist" (to use Høyrup's expression) nature of some mathematical texts, here is problem 9 from catalogue [[http://www.cdli.ucla.edu/P254981|YBC 4652]], where a stone is weighed after removing improbable fractions of it:
-\\ 14. I found a stone. Its weight I don’t know. 1/7 I subtracted, then 1/11 I added,
+''14. I found a stone. Its weight I don’t know. 1/7 I subtracted, then 1/11 I added,''\\
+''15. then 1/13 I subtracted. I weighed: 1 //mana//. The initial weight how much?''\\
+''16. The initial weight 1 //mana// 9 1/2 //gin// 2 1/2 //še//.''\\
-. then 1/13 I subtracted. I weighed: 1 //mana//. The initial weight how much?
+\\ The mathematical texts generally include compendia of problems provided with a detailed solution ([[http://www.cdli.ucla.edu/P254406|BM 13901]] and [[http://www.cdli.ucla.edu/P274707|BM 85194]] are famous examples). These "procedure texts" show a specific and ingenious practice of the mathematical proof (see article on “Reasoning”). In some instances, only the statements are provided on single column tablets (catalogue texts), or on series of multicolumn tablets (series texts). Sometimes, diagrams illustrate the problems. Old Babylonian mathematical //corpus// includes also numerical texts, which show the virtuosity of the ancient calculators ([[http://cdli.ucla.edu/P254790|Plimpton 322]] and [[http://www.cdli.ucla.edu/P254479|CBS 1215]] are fascinating examples).
-. The initial weight 1 //mana// 9 1/2 //gin// 2 1/2 //še//.
-\\ The mathematical texts generally include compendia of problems provided with a detailed solution (BM 13901 [[http://www.cdli.ucla.edu/P254406|http://www.cdli.ucla.edu/P254406]] and BM 85194 [[http://www.cdli.ucla.edu/P274707|http://www.cdli.ucla.edu/P274707]] are famous examples). These "procedure texts" show a specific and ingenious practice of the mathematical proof (see article on “Reasoning”). In some instances, only the statements are provided on single column tablets (catalogue texts), or on series of multicolumn tablets (series texts). Sometimes, diagrams illustrate the problems. Old Babylonian mathematical //corpus//includes also numerical texts, which show the virtuosity of the ancient calculators (Plimpton 322 <http://cdli.ucla.edu/P254790> and CBS 1215 <http://www.cdli.ucla.edu/P254479> are fascinating examples).
 \\ Mathematical texts of Achaemenid period include metrological tables and procedure texts focused on the calculation of surfaces using different metrological systems. Hellenistic mathematical texts show a strong interest for the very large sexagesimal regular numbers and the calculation of reciprocals. Procedure texts witness the encyclopedic mathematical knowledge of the //kalû// of the temples of Uruk and Babylon. Another facet of mathematical practices in the Hellenistic period can be discovered in astronomical texts, as shown by Mathieu Ossendrijver (//Babylonian Mathematical Astronomy: Procedure Texts//: Springer, 2012). (see article on “Mathematical texts of the Achaemenid and Hellenistic periods”).
-===== Bibliography =====
-A tentative complete bibliography by Duncan Melville is available here: [[http://it.stlawu.edu/~dmelvill/mesomath/biblio/bigbib.html|http://it.stlawu.edu/~dmelvill/mesomath/biblio/bigbib.html]].
-\\ A basic tool indispensable for any people interested by the subject is Friberg's commented bibliography, alas unpublished, //A Survey of Publications on Sumero-Akkadian Mathematics, Metrology and Related Matters (1854-1982)//, Göteborg 1982.
-\\   * Bruins, Evert M. and Marguerite Rutten. 1961. //Textes mathématiques de Suse//. Paris: Geuthner.
-  * Damerow, Peter (2001) "Kannten die Babylonier den Satz des Pythagoras? Epistemologische Anmerkungen zur Natur der babylonischen Mathematik" in Changing Views on Ancient Near Eastern Mathematics (Jens Høyrup and Peter Damerow, eds.), Berlin: 219-310.
-  * Englund, Robert K. (1988) "Administrative timekeeping in Ancient Mesopotamia" JESHO 31: 121-185.
-  * Englund, Robert K. and Jean-Pierre Grégoire (1991) The Proto-Cuneiform Texts from Jemdet Nasr (MSVO 1), Berlin.
-  * Friberg, Jöran (1978) "Early Roots of Babylonian Mathematics 1. A method for the decipherment … of proto- Sumerian and proto-Elamite semi-pictographic inscriptions" Department of Mathematics CTH-GU 1978- 9, Gothenburg.
-  * Friberg, Jöran (1990) "Mathematik" (in English) in RlA 7 (Dietz Otto Edzard, ed.), Berlin, New York: 531-585.
-  * Friberg, Jöran. 2000. Mathematics at Ur in the Old Babylonian period. //Revue d'Assyriologie// 94:98-188.
-  * Friberg, Jöran. 2005. //Unexpected Links Between Egyptian and Babylonian Mathematics//. Singapour: World Scientific.
-  * Friberg, Jöran. 2007. //Amazing Traces of Babylonian Origin in Greek Mathematics//. Singapour: World Scientific.
-  * Friberg, Jöran. 2007. //A Remarkable Collection of Babylonian Mathematical Texts//, vol. I. New York: Springer.
-  * Friberg, Jöran. 2005. On the Alleged Counting with Sexagesimal Place Value Numbers in Mathematical Cuneiform Texts from the Third Millennium B. C. //CDLJ// 2005:2.
-  * Friberg, Jöran. 2009. A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma. //CDLJ// 2009-3.
-  * Høyrup, Jens. 2002. //Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and its Kin//. Berlin & Londres: 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.
-  * Neugebauer, Otto. 1935-1937. //Mathematische Keilschrifttexte I-III//. Berlin: Springer.
-  * Neugebauer, Otto and Abraham J. Sachs. 1945. //Mathematical Cuneiform Texts//. New Haven: American Oriental Series & American Schools of Oriental Research
-  * Neugebauer, Otto. 1957. //The exact sciences in antiquity//. Providence: Brown University Press.
-  * Nissen, Hans J., Peter Damerow, and Robert K. Englund (1993) Archaic Bookkeeping, Early Writing and Techniques of Economic Administration in the Ancient Near East, Chicago, London Nougayrol, Jean, et al. (1968) Ugaritica V, Paris.
-  * Powell, M. A., Jr. (1976) "The antecedents of Old Babylonian place notation and the early history of Babylonian mathematics" HM 3: 417-439.
-  * Powell, Marvin A. 1987-1990. Masse und Gewichte. In //Reallexikon der Assyriologie//.
-  * Proust, Christine (with the collaboration of A. Cavigneaux). 2007. //Tablettes mathématiques de Nippur//. Istanbul: Institut Français d'Etudes Anatoliennes, De Boccard.
-  * Proust, Christine (with the collaboration of M. Krebernik and J. Oelsner). 2008. //Tablettes mathématiques de la collection Hilprecht//. Leipzig: Harrassowitz.
-  * Proust, Christine. 2009. Numerical and metrological graphemes: from cuneiform to transliteration. //Cuneiform Digital Library Journal// 2009:1.
-  * Robson, Eleanor. 1999. //Mesopotamian Mathematics, 2100-1600 BC. Technical Constants in Bureaucracy and Education//, vol. XIV. Oxford: Clarendon Press.
-  * Robson, Eleanor. 2008. //Mathematics in Ancient Iraq: A Social History//. Princeton: Princeton University Press.
-  * Thureau-Dangin, François. 1938. //Textes Mathématiques Babyloniens//. Leiden: Ex Oriente Lux.
+[[http://cdli.ox.ac.uk/wiki/doku.php?id=mathematics#bibliography|Bibliography ]]
 \\ CP