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[[Numbers & Metrology in the 2nd millennium]]
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numbers_metrology_2nd_millennium [2014/11/10 16:04] – [Old Babylonian scribal schools] gombertnumbers_metrology_2nd_millennium [2016/01/26 21:12] (current) – [Numbers & Metrology in the 2nd millennium] gombert
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 ====== Numbers & Metrology in the 2nd millennium ===== ====== Numbers & Metrology in the 2nd millennium =====
 +(Return to [[numbers_metrology]])\\
  
- +  * [[Old Assyrian private trade]] 
-(Return to [[numbers_metrology]]) +  * [[Old Babylonian scribal schools]] 
-==== Old Assyrian private trade ==== +  * [[Nuzi]] 
- +  * [[Old Babylonian Dyala]]
-**Sources:** 22 500 old Assyrian private tablets have been excavated for the greater part in the houses of the Assyrian merchants in the lower town of Kültepe, ancient Kaneš. There are also some 180 old Assyrian tablets excavated in two other Assyrian trade settlements in Anatolia: Boğazköy, ancient Hattuš, and Alişar, ancient Amkuwa. The twenty old Assyrian computation school exercises excavated at Aššur use the same weight system. All these sources date for the most part to the 19th century BC. A substantial part of them are already available on CDLI [[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&order=PrimaryPublication&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=&ExcavationNumber=&Period=old+assyrian&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&Language=&Genre=&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|(here)]].\\ +
- +
-**Scope:** The old Assyrian metrology is attested thus both in Aššur and in the Assyrian trade settlements in Anatolia during the 19th and 18th centuries BC. Since the Assyrians were interested mainly by the trade in metals, the weight system is the most used among the merchants. Capacities, length and surfaces are rarely attested.\\ +
- +
-**Bibliography:**\\ +
-  * Lewy, J. 1959: ‘The Old Assyrian Surface Measure šubtum’ //Analacta Biblica// 12: 216-26\\ +
-  * Michel, Cécile. 1998. Les marchands et les nombres: l’exemple des Assyriens à Kaniš, in J. Prosecky (ed.), //Intellectual Life of the Ancient Near East, Compte-rendu de la 43e Rencontre Assyriologique Internationale//, Prague, p. 249-267\\ +
-  * Michel, Cécile. 2006. [[http://culturemath.ens.fr/content/calculer-chez-les-marchands-assyriens-2096|Calculer chez les marchands assyriens du début du IIe millénaire av. J.-C.]], site Internet CultureMaths (site expert ENS/DESCO)-Juin 2006.\\  +
-  * Veenhof, Klaas, R. 2007. On Some Old Assyrian Linear Measures, //Nouvelles Assyriologiques Brèves et Utilitaires// 2007/50.\\ +
- +
-=== Metrological systems: === +
-== Units of weight == +
- +
-| še | {{ :oakk_se.jpg?nolink |}} | ca. 0,05 g | +
-|   **↓ × 180 **   ||| +
-| gin<sub>2</sub> | {{ :oass_gin2.jpg?nolink |}} | ca. 8,3 g | +
-|   **↓ × 60 **   ||| +
-| ma-na | {{ :oass_ma-na.jpg?nolink |}} | ca. 500 g | +
-|   **↓ × 60 **   ||| +
-| gun<sub>2</sub> | {{ :oass_gun.jpg?nolink |}} | ca. 30 kg | +
- +
-When computing copper or wool in Anatolia, the unit gun2 was not always used ; instead quantities could be given in hundreds of ma-na.\\ +
- +
- +
-== Units of capacity == +
- +
-The capacity system was used to quantify any liquid (oil, beer) or some solids linked to food (grain, nuts, etc.). It is built on measuring vessels of standard capacity. Measuring units are written syllabically, except when specified. +
- +
-| sila<sub>3</sub>, //qûm// | {{:oass_sila3.jpg?nolink|}} | - | 1 l | +
-|   **↓ × 10 **   |||| +
-| ban<sub>2</sub>(( 1 (ban<sub>2</sub>): {{:1_ban2_.png?nolink|}}, 2 (ban<sub>2</sub>): {{:2_ban2_.png?nolink|}}, 3 (ban<sub>2</sub>): {{:3_ban2_.png?nolink|}}, 4 (ban<sub>2</sub>): {{:4_ban2_.png?nolink|}}, 5 (ban<sub>2</sub>): {{:5_ban2_.png?nolink|}})) | {{:oass_ban2.jpg?nolink|}} | A vessel | 10 l| +
-|   **↓ × 3 **   |||| +
-| aš<sub>2</sub>, //ṣimdum// | {{ :oass_ǎs2.jpg?nolink |}} | - | 30 l| +
-|   **↓ × 4 **   |||| +
-| //naruqqum// | - | A bag | 120 l | +
- +
-Note: the unit aš<sub>2</sub>, //ṣimdum// may alternate with the unit dug, //karpatum// (see below) +
- +
-There was a variant: +
- +
-| sila<sub>3</sub>, //qûm// | {{:oass_sila3.jpg?nolink|}} | - | ca. 1 l | +
-|   **↓ × 15 **   |||| +
-| // šaršarānu// (15 sila<sub>3</sub>) | - | A vessel | ca. 15 l | +
-|   **↓× 2 **   |||| +
-| dug, //karpatum// (2 // šaršarānu// ) | {{:oass_dug.jpg?nolink|}} | A jar | ca. 30 l | +
-|   **↓ × 4 **   |||| +
-| // naruqqum// (4 dug) | - | A bag | ca. 120 l | +
- +
-There were other measuring vessels of standard capacity, as the //kirrum, “a pitcher of beer”//, but their capacities remain unknown. +
- +
- +
-== Units of length == +
-Measuring units are written syllabically.\\ +
-| //ubānum// |Finger| +
-|   **↓ × 15 **   || +
-|// ūṭum// | ½ cubit | +
-|   **↓ × 2 **   || +
-| // ammatum // | 1 cubit, ca. 50 cm| +
-|   **↓ × 21600 **   || +
-| // bērum // | - | +
- +
-Presumably in a parallel system we find the following measures:\\ +
- +
-|//idum// | “arm” | approximately the same as the cubit? | +
-|//kabistum//| foot | 3/4 cubit or 22 1/2 fingers| +
- +
-== Units of length == +
-Only the unit //šubtum// is attested; it most probably equals the Babylonian sar = //mušarum//, measuring ca. 36m<sup>2</sup> +
- +
-=== Numerical system: === +
- +
-Assyrian merchants use, to count object, a decimal additive system and not a sexagesimal system. Numbers 100 and 1000 are expressed with their nouns: //meat// and //lim//. +
- +
-| //ištēn// | {{:oass_isten.jpg?nolink|}} | 1| +
-|   **↓ × 10 **   ||| +
-| //ešer// | {{:oass_eser.jpg?nolink|}} | 10| +
-|   **↓ × 10 **   ||| +
-| //me-at// | {{ :oass_me-at.jpg?nolink&100 |}} | 100 | +
-|   **↓ × 10 **   ||| +
-| //li-im// | {{ :oass_li-im.jpg?nolink&100 |}} | 1000 |\\ +
-\\ +
-Note that, in this decimal system, 60 {{:oass_60.jpg?nolink|}} +
- +
-== Fractions == +
-| 1/6 | 1/4 | 1/3 | 1/2 | 2/3 | 5/6 | +
-| {{:oass_1-6.jpg?nolink|}} | {{:oass_1-4.jpg?nolink|}} | {{:oass_shushana.jpg?nolink|}} | {{:oass_mash.jpg?nolink|}} | {{:oass_shanabi.jpg?nolink|}} | {{:oass_kingusila.jpg?nolink|}} | +
- +
- +
-Page prepared by CM  +
- +
-------- +
-==== Old Babylonian scribal schools ==== +
- +
-**Sources**: the diagrams below represent data provided by metrological lists and tables from Nippur  +
- scribal schools. These sources are available on CDLI [[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&order=PrimaryPublication&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Nippur&ExcavationNumber=&Period=&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&Language=&Genre=&SubGenre=Metrological&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|(here)]].\\ +
- +
-**Scope**: Similar metrology is attested in other Old Babylonian scribal schools. The metrology taught in scribal schools was adopted in a large part of Mesopotamia. However, some minor variants may be observed in some administrative or economic archives from Southern Mesopotamia (see metrology in Old Babylonian Larsa, etc.), and important differences were cultivated in Northern regions (see metrology for example in Mari, Diyala Valley, Assur). \\ +
- +
-**Bibliography**:\\ +
-  * Proust, Christine. 2007. //Tablettes mathématiques de Nippur//. Istanbul: Institut Français d'Etudes Anatoliennes, De Boccard.\\ +
-  * Proust, Christine. 2009. [[http://cdli.ucla.edu/pubs/cdlj/2009/cdlj2009_001.html|"Numerical and metrological graphemes: from cuneiform to transliteration."]] //Cuneiform Digital Library Journal//, 2009:1.\\ +
-  * Robson, Eleanor. 2002. "More than metrology: mathematics education in an Old Babylonian scribal school." Pp. 325-365 in //Under One Sky. Astronomy and Mathematics in the Ancient Near East, vol. 297, Alter Orient und Altes Testament (AOAT)//, edited by J. M. Steele and A. Imhausen. Münster: Ugarit-Verlag.\\ +
- +
-=== Metrological systems, in the order in which they appear in metrological lists and tables === +
-== Units of capacity == +
- +
-| gin<sub>2</sub> | {{ :obab_gin2.jpg?nolink |}} | ca. 17 cm<sup>3</sup> | +
-|   **↓ × 60 **   ||| +
-| sila<sub>3</sub> | {{ :obab_sila3.jpg?nolink |}} | ca. 1 l | +
-|   **↓ × 10 **   ||| +
-| ban<sub>2</sub>(( 1 (ban<sub>2</sub>): {{:1_ban2_.png?nolink|}}, 2 (ban<sub>2</sub>): {{:2_ban2_.png?nolink|}}, 3 (ban<sub>2</sub>): {{:3_ban2_.png?nolink|}}, 4 (ban<sub>2</sub>): {{:4_ban2_.png?nolink|}}, 5 (ban<sub>2</sub>): {{:5_ban2_.png?nolink|}})) | {{ :obab_ban2.jpg?nolink |}} | ca. 10 l | +
-|   **↓ × 6 **   ||| +
-| barig ((1 (barig): {{:obab_barig.jpg?nolink|}}, 2 (barig): {{:2_barig_.png?nolink|}}, 3(barig): {{:3_barig_.png?nolink|}}, 4 (barig): {{:4_barig_.jpg?nolink|}}))| {{ :obab_barig.jpg?nolink |}} | ca. 60 l | +
-|   **↓ × 5 **   ||| +
-| gur | {{ :obab_gur.jpg?nolink |}} | ca. 300 l| +
- +
-== Units of weight == +
- +
-| še | {{ :obab_se.jpg?nolink |}} | ca. 0.04 g | +
-|   **↓ × 180 **   ||| +
-| gin<sub>2</sub> | {{ :obab_gin2.jpg?nolink |}} | ca. 8 g | +
-|   **↓ × 60 **   ||| +
-| ma-na | {{:obab_ma.jpg?nolink|}}{{:obab_na.jpg?nolink|}} | ca. 500 g | +
-|   **↓ × 60 **   ||| +
-| gun<sub>2</sub>| {{:obab_gun2.jpg?nolink|}} | ca. 30 kg | +
- +
-== Units of surface and volume == +
- +
-| sar  | {{:obab_sar.jpg?nolink|}} | ca. 36 m<sup>2</sup>  |  +
-|   **↓ × 100 **   ||| +
-| GAN<sub>2</sub> | {{:obab_gan2.jpg?nolink|}} | ca. 3600 m<sup>2</sup> | +
- +
-== Units of length == +
- +
-| šu-si | {{:obab_shu.jpg?nolink|}}{{:obab_si.jpg?nolink|}} | ca. 16 mm | +
-|   **↓ × 30 **   ||| +
-| kuš<sub>3</sub> | {{:obab_kush3.jpg?nolink|}} | ca. 50 cm | +
-|   **↓ × 12 **   ||| +
-| ninda | {{:obab_ninda.jpg?nolink|}} | ca. 6 m | +
-|   **↓ × 60 **   ||| +
-| UŠ | {{:obab_ush.jpg?nolink|}} | ca. 360 m | +
-|   **↓ × 30 **   ||| +
-| danna | {{:obab_danna.jpg?nolink|}} | ca. 10,5 km | +
- +
-Note: in mathematical texts, the unit gin<sub>2</sub> is also used for sub-dividing the surface unit sar into 60 parts, and the unit še is also used in capacity and surface systems for sub-dividing the unit gin<sub>2</sub> into 180 parts. In other words, if we consider all of the OB mathematical texts, the sequence   ←×60− gin2←×180−še can be grafted in capacity and surface systems.  +
- +
-== Bridges == +
-Bridge between length and surface units: 1 ninda × 1 ninda = 1 sar\\ +
-Bridge between surface and volume units: 1 volume-unit = 1 surface-unit × 1 kuš<sub>3</sub>  (example: 1 sar-volume = 1 sar-surface × 1 kuš<sub>3</sub>).\\ +
-Bridge between volume and capacity units: 1 sar-volume is equivalent to 60 gur (ca. 18 m<sup>3</sup> or 18 000 liters).\\ +
- +
-===  Numerical systems === +
-== System S, used for counting discrete items == +
- +
-| {{:obab_barig.jpg?nolink|}} |   **× 10\\ ←**   | {{:obab_u.png?nolink|}} |   **× 6\\ ←**   | {{:obab_barig.jpg?nolink|}} |   **× 10\\ ←**   | {{:obab_gesh_u.png?nolink|}} |   **× 6\\ ←**   | {{:obab_shar2.png?nolink|}} |   **× 10\\ ←**   | {{:obab_shar_u.png?nolink|}} | +
-|  diš | ::: |  u  | ::: | geš<sub>2</sub> | ::: | geš’u | ::: | šar<sub>2</sub> | ::: | šar’u | +
-|  1 | ::: | 10  | ::: | 60 | ::: | 600| ::: | 3 600 | ::: | 36 000 | +
- +
- +
-| {{:obab_shar_u.png?nolink|}} |   **× 10\\ ←**   | {{:obab_shar2.png?nolink|}} |   **× 6\\ ←**   | {{:obab_gesh_u.png?nolink|}} |   **× 10\\ ←**   |  {{:obab_barig.jpg?nolink|}} |   **× 6\\ ←**   |{{:obab_u.png?nolink|}} |      **× 10\\ ←**   | {{:obab_barig.jpg?nolink|}} |    +
-| šar’u | ::: | šar<sub>2</sub> | ::: | geš’u | ::: | geš<sub>2</sub> | ::: |  u  | ::: | diš | +
-| 36 000 | ::: | 3 600 | ::: | 600 | ::: | 60 | ::: | 10  | ::: |  1 | +
- +
-== Variants of System S == +
- +
-The highest units of capacity (gur) and weight (gun2) are counted with a variant of System S, where the number 1 is represented by the sign aš <sub>{{:obab_ash.png?10*5|}}</sub> (and not the sign diš <sub>{{:obab_barig.jpg?4*5|}}</sub> as in the System S used for discrete item).\\ +
- +
-The other measuring units (sila3, gin2, še, sar, danna, UŠ, ninda, kuš and šu-si) are counted with  <sub>{{:obab_barig.jpg?4*5|}}</sub> (1) and  <sub>{{:obab_u.png?4*5|}}</sub> (10) repeated as many times as necessary.\\ +
- +
-== System G, used for counting the highest unit of surface == +
- +
-| {{:obab_ubu.png?nolink|}} |   **× 2\\ ←**   | {{:obab_ash.png?nolink|}} |   **× 6\\ ←**   | {{:obab_eshe3.png?nolink|}} |   **× 3\\ ←**   | {{:obab_u.png?nolink|}} |   **× 10\\ ←**   | {{:obab_bur_u.png?nolink|}} |   **× 6\\ ←**   | {{:obab_shar2.png?nolink|}} |   **× 2\\ ←**   | {{:obab_shar_u.png?nolink|}} | +
-| ubu | ::: | iku  | ::: | eše<sub>3</sub> | ::: | bur<sub>3</sub>  | ::: | bur’u | ::: | šar<sub>2</sub> | ::: | šar’u | +
-| ½ | ::: | 1 | ::: | 6 | ::: | 18 | ::: | 180 | ::: | 1 080 | ::: | 10 800 | +
- +
-=== Fractions === +
- +
-| 1/6 | {{:obab_igi.png?nolink|}}{{:obab_6.png?nolink|}}{{:obab_gal2.png?nolink|}} (igi-6-gal<sub>2</sub>) | +
-| 1/3 | {{:obab_1.3.png?nolink|}} | +
-| 1/2 | {{:obab_1.2.png?nolink|}} | +
-| 2/3 | {{:obab_2.3.png?nolink|}} | +
-| 5/6 | {{:obab_5.6.png?nolink|}} | +
- +
- +
-Page prepared by CP\\ +
-(digital version prepared by BG)+
numbers_metrology_2nd_millennium.1415635466.txt.gz · Last modified: 2014/11/10 16:04 by gombert
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