===== Old Babylonian Diyala – Mathematical Tablets =====

**Sources:**more than 70 mathematical tablets produced in the valley of the river Diyala during the Old Babylonian period have been published. A substantial part of these tablets corresponds to school material: tables of reciprocals, tables of square and cube roots, metrological tables and extracts from arithmetical tables and metrological lists. Besides, advanced problem texts are also present in the Diyala corpus.

These tablets come specifically from the following sites: [[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Nerebtum&ExcavationNumber=&Period=Old+Babylonian&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&TranslationSearch=&CommentSearch=&StructureSearch=&Language=&Genre=mathematical&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|Ishchali (Nerebtum)]],[[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Tell+Dhiba%27i&ExcavationNumber=&Period=Old+Babylonian&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&TranslationSearch=&CommentSearch=&StructureSearch=&Language=&Genre=mathematical&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|Tell edh-Dhiba’i (Zaralulu)]], [[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Turran&ExcavationNumber=&Period=Old+Babylonian&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&TranslationSearch=&CommentSearch=&StructureSearch=&Language=&Genre=mathematical&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|Tell es-Seeb]], [[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Turran&ExcavationNumber=&Period=Old+Babylonian&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&TranslationSearch=&CommentSearch=&StructureSearch=&Language=&Genre=mathematical&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|Tell Haddad (Me-Turān)]], [[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Harmal&ExcavationNumber=&Period=Old+Babylonian&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&TranslationSearch=&CommentSearch=&StructureSearch=&Language=&Genre=mathematical&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|Tell Harmal (Šaduppûm)]] and TululKhattab.

The important site of Tell Asmar (Ešnunna) has at the moment only one attested mathematical tablet in the CDLI database ([[http://cdli.ucla.edu/search/search_results.php?SearchMode=Text&requestFrom=Search&PrimaryPublication=&Author=&PublicationDate=&SecondaryPublication=&Collection=&AccessionNumber=&MuseumNumber=&Provenience=Asmar&ExcavationNumber=&Period=&DatesReferenced=&ObjectType=&ObjectRemarks=&Material=&TextSearch=&TranslationSearch=&CommentSearch=&StructureSearch=&Language=&Genre=mathematical&SubGenre=&CompositeNumber=&SealID=&ObjectID=&ATFSource=&CatalogueSource=&TranslationSource=|OIC 19, 2-3]]), but this tablet is dated to the Old Akkadian period and so it is not considered here.

The valley of the Diyala becomes unified in the Old Babylonian period under Ipiq-Adad II, a king from Ešnunna. One can speak then of a Kingdom of Ešnunna, centered on the city of the same name. As a consequence, the tablets studied here correspond both to a geographical delimitation (the so-called Diyala) and to a political one (the Kingdom of Ešnunna).


**Scope:**the measurement units attested in mathematical tablets from the Old Babylonian Diyala are for the most part the same as those found in Southern Old Babylonian Schools. However, additional units are attested in the Diyala, namely //nikkassum// and //ṣuppum//.


**Bibliography:**
  * al-Fouadi, Abdul-Hadi. (1979) //Lenticular Exercise School Texts.// Texts in the Iraq Museum 10 (TIM 10). Baghdad: Republic of Iraq --- Ministry of Culture and Arts, The State Organization of Antiquities.

  * al-Rawi, F. N. H. and Michael Roaf. (1984) ‘Ten Old Babylonian mathematical problems from Tell Haddad, Himrin’, //Sumer// 43: 175-218.

  * Baqir, Taha. (1948) ‘Excavations at Harmal’, //Sumer// 4: 137-139.

  * Baqir, Taha. (1950a) ‘An important mathematical problem text from Tell Harmal (on a Euclidean theorem)’, //Sumer// 6: 39-54, plus 2 planches.

  * Baqir, Taha. (1950b) ‘Another important mathematical text from Tell Harmal’, //Sumer// 6, 130-148: plus 3 planches.

  * Baqir, Taha. (1951) ‘Some more mathematical texts from Tell Harmal’, //Sumer// 7: 28-45, plus 5 planches.

  * Baqir, Taha. (1962) ‘Foreword’, //Sumer// 18: 5-14, plus one planche.

  * Bruins, E. M. (1953a) ‘Revision of the mathematical texts from Tell Harmal’, //Sumer// 9: 241-253.

  * Bruins, E. M. (1953b) ‘Three geometrical problems’, //Sumer// 9: 255-259.

  * Bruins, E. M.  (1954) ‘Some mathematical texts’, //Sumer// 10: 55-61.

  * Friberg, Jöran. (2000) ‘Mathematics at Ur in the Old Babylonian period’, //Revue d'Assyriologie// 94: 98-188.

  * Friberg, Jöran and Farouk N. H. al-Rawi. (1994) ‘Equations for Rectangles and Methods of False Value in Old Babylonian Geometric-Algebraic Problem Texts’, with introduction by Jens Høyrup. Not published.

  * Goetze, Albrecht. (1951) ‘A mathematical compendium from Tell Harmal’, //Sumer// 7: 126-155, plus 8 planches.

  * Gonçalves, Carlos. (2015) //Mathematical Tablets from Tell Harmal.//New York: Springer.

  * Gonçalves, Carlos. (forthcoming) ‘Quantification and computation in the mathematical texts of the Old Babylonian Diyala’, in //Cultures of Computation and Quantification//, eds. Karine Chemla, Agathe Keller, Christine Proust.

  * Greengus, Samuel. (1979) Old Babylonian Tablets from Ishchali and Vicinity. Leiden: Nederlands Historisch-Archaeologisch Instituutte Istanbul.

  * Greengus, Samuel. (1986) //Studies in Ishchali Documents.// Bibliotheca Mesopotamica XIX. Malibu: Undena Press.

  * Isma'el, Khalid Salim. (1999a) ‘A new table of square roots’, //Akkadica// 112: 18-26.

  * Isma'el, Khalid Salim. (1999b). ‘A New Mathematical Text in the Iraq Museum’, //Akkadica// 113: 6-12.

  * Isma'el, Khalid Salim. (2007) //Old Babylonian Cuneiform Texts from The Lower Diyala Region. Tulul Khatab.//Edubba IX. London: Nabu Publications.

  * Isma'el, Khalid Salim and Eleanor Robson. (2010) ‘Arithmetical tablets from Iraqi excavations in the Diyala’, in //Your Praise is Sweet. A Memorial Volume for Jeremy Black from Students, Colleagues and Friends//, eds. Heather D. Baker, Eleanor Robson and Gábor Zólyomi. London, British Institute for the Study of Iraq: 151-164.

  * Neugebauer, Otto and Abraham Sachs. (1945) //Mathematical Cuneiform Texts//. American Oriental Series XXIX. New Haven, Connecticut: American Oriental Society.

 

==== Metrological Systems Present in Mathematical Tablets from the Old Babylonian Diyala ====

 

The following tables give the attested units and the numerical relationship between neighbouring units in each system of measurement: capacities, weights, surfaces and volumes and finally length.

 

Measurement units are presented below the way they are attested in the mathematical tablets, that is to say, phonetically or with logograms, sometimes both, according to the way they are written in the examined tablets.

 

=== Units of capacity ===

                                         

| sila<sub>3</sub>//, qûm// | {{:Dyala002.jpg?26x28}} |
|  **↓ × 10**  ||
| (ban<sub>2</sub>)//, sūtum// | {{:Dyala004.jpg?23x26}} |
|  **↓ × 6**  ||
| (barig), //parsiktum// | {{:Dyala006.jpg?9x39}} |
|  **↓ × 5**  ||
| gur | {{:Dyala008.jpg?33x32}} |

 

=== Units of weight ===

 

| še | {{:Dyala010.jpg?19x28}} |
|  **↓ × 180**  ||
| gin<sub>2</sub>//, šiqlum// | {{:Dyala012.jpg?33x28}} |
|  **↓ × 60**  ||
| ma-na | {{:Dyala014.jpg?33x31}}{{:Dyala016.jpg?29x30}} |
|  **↓ × 60**  ||
| gu<sub>2</sub> | {{:Dyala018.jpg?41x29}} |

\\ In order to express multiples of the weight measurement unit gu<sub>2</sub>, the so-called System S is employed. This is consistent with the use of System S in Southern Old Babylonian Schools, where it is additionally used to express multiples of the capacity measurement unit gur.

 

-System S

 

| Transliteration | Sign | Value |
| 1(aš) | {{:Dyala020.jpg?21x38}} | 1 |
| 1(u) | {{:Dyala022.jpg?12x35}} | 10 |

 

Example:

1(u) 2(aš) gu<sub>2</sub> of silver, meaning 12 gu<sub>2</sub> of silver, as in [[http://www.cdli.ucla.edu/P225113|IM 067330]]

 

 

=== Units of surface and volume ===

 

| sar//, mušarum// | {{:Dyala024.jpg?42x37}} |
|  **↓ × 100**  ||
| gan<sub>2</sub> | {{:Dyala026.jpg?38x36}} |

 

In order to express multiples of the surface (and volume) measurement unit gan<sub>2</sub>, the so-called System G is employed.

 

-System G

 

| Transliteration         | Signs                                           | Value      |
|                         | {{:Dyala028.jpg?26x36}}                         | 1/4 iku    |
|  **↓ × 2**                                                                           |||
| 1(ubu)                  | {{:Dyala030.jpg?19x40}}                         | ½ iku      |
|  **↓ × 2**                                                                           |||
| 1(iku)                  | {{:Dyala020.jpg?21x38}}                         | 1 iku      |
|  **↓ × 6**                                                                           |||
| 1(eše<sub>3</sub>)      | {{Dyala032.jpg?30x44}}                          | 6 iku      |
|  **↓ × 3**                                                                           |||
| 1(bur<sub>3</sub>)      | {{:Dyala033.jpg?12x35}}                         | 18 iku     |
|  **↓ × 10**                                                                          |||
| 1(bur’u)                | {{:Dyala035.jpg?14x35}}                         | 180 iku    |
|  **↓ × 6**                                                                           |||
| 1(šar<sub>2</sub>)      | {{:Dyala037.jpg?22x35}}                         | 1080 iku   |
|  **↓ × 10**                                                                          |||
| 1(šar’u)                | {{:Dyala039.jpg?22x35}}                         | 10800 iku  |
|  **↓ × 6**                                                                           |||
| 1(šar<sub>2</sub>) gal  | {{:Dyala040.jpg?22x35}}{{:Dyala042.jpg?26x34}}  | 34800 iku  |

 

Examples:

1(ubu) gan<sub>2</sub>, meaning ½ gan<sub>2</sub>, as in [[http://www.cdli.ucla.edu/P368229|A21948]]

2(eše<sub>3</sub>) gan<sub>2</sub>, meaning 2 x 6 gan<sub>2</sub> = 12 gan<sub>2</sub>, as in [[http://www.cdli.ucla.edu/P225012|IM 051750]]

1(bur<sub>3</sub>) 2(eše<sub>3</sub>) gan<sub>2</sub>, meaning 18 + 2 x 6 gan<sub>2</sub> = 30 gan<sub>2</sub>, as in [[http://www.cdli.ucla.edu/P368229|A21948]]

 

 

=== Units of length ===



|  //ubānum//                                                ||
|  **↓ × 30**                                                ||
|  kuš<sub>3</sub>//, ammatum// {{:Dyala044.jpg?23x34}}      ||
|  **↓ × 3**                                                 ||
|  //nikkassum//                                             ||
|  **↓ × 3**                                                 ||
|  //qanûm//                                                 ||
|  **↓ × 2**                                                 ||
|  ninda {{:Dyala046.jpg?18x34}}                             ||
|  **↓ × 5**                                                 ||
|  //ṣuppum//                                                ||
|  **↓ × 2**                                                 ||
|  //ašlum//                                                 ||

 

 

=== Bridges ===

The relation between units of length and units and surface is that:
a square of sides 1 //ninda// has area 1 //sar//.

The relation between units of height, surface and volume is that:
a right prism of square base, with height equal to 1 //ammatum// and area of the base equal to 1 sar has volume 1 sar


==== Metrological Volume and Capacities ====

In the evidence from the Diyala, one finds two different equivalencies between volume given in sar and capacities.

 

For clay:

1 sar of volume is equivalent to 60 gur of capacity (Haddad 104, problems #7-#8).

 

For grain:

1 sar of volume is equivalent to 72 gur of capacity (Haddad 104, problems #1-#5).

 

==== Fractions ====

|  1/3  | {{:Dyala048.jpg?26x40}}  |
|  1/2  | {{:Dyala050.jpg?18x34}}  |
|  2/3  | {{:Dyala052.jpg?23x36}}  |
|  5/6  | {{:Dyala054.jpg?23x35}}  |

 

All of these fractions occur in [[http://www.cdli.ucla.edu/P368229|A 21948]]

See also, for 1/3, 1/2 and 5/6, [[http://www.cdli.ucla.edu/P225105|IM 063106]]



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Page prepared by Carlos Gonçalves (digital version prepared by BG)