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[[Old Babylonian Diyala – Mathematical Tablets]]
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 ===== Old Babylonian Diyala – Mathematical Tablets ===== ===== Old Babylonian Diyala – Mathematical Tablets =====
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 **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. **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=|and]][[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.+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 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.
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   * Bruins, E. M. (1953b) ‘Three geometrical problems’, //Sumer// 9: 255-259.   * Bruins, E. M. (1953b) ‘Three geometrical problems’, //Sumer// 9: 255-259.
  
-  * Bruins, E. M.  (1954) ‘Some mathematical texts’, //Sumer// 10: 55-61.+  * 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. (2000) ‘Mathematics at Ur in the Old Babylonian period’, //Revue d'Assyriologie// 94: 98-188.
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   * Neugebauer, Otto and Abraham Sachs. (1945) //Mathematical Cuneiform Texts//. American Oriental Series XXIX. New Haven, Connecticut: American Oriental Society.   * Neugebauer, Otto and Abraham Sachs. (1945) //Mathematical Cuneiform Texts//. American Oriental Series XXIX. New Haven, Connecticut: American Oriental Society.
  
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 ==== Metrological Systems Present in Mathematical Tablets from the Old Babylonian Diyala ==== ==== Metrological Systems Present in Mathematical Tablets from the Old Babylonian Diyala ====
  
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 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. 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.
  
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 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. 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 === === Units of capacity ===
  
-                                         +                                         
  
-| sila<sub>3</sub>//, qûm// | {{Old%20Babylonian%20Diyala%20v5_fichiers:image002.jpg?26x28}} | +| sila<sub>3</sub>//, qûm// | {{:Dyala002.jpg?26x28}} | 
-| **↓ × 10** || +|  **↓ × 10**  || 
-| (ban<sub>2</sub>)//, sūtum// | {{Old%20Babylonian%20Diyala%20v5_fichiers:image004.jpg?23x26}} | +| (ban<sub>2</sub>)//, sūtum// | {{:Dyala004.jpg?23x26}} | 
-| **↓ × 6** || +|  **↓ × 6**  || 
-| (barig), //parsiktum// | {{Old%20Babylonian%20Diyala%20v5_fichiers:image006.jpg?9x39}} | +| (barig), //parsiktum// | {{:Dyala006.jpg?9x39}} | 
-| **↓ × 5** || +|  **↓ × 5**  || 
-| gur | {{Old%20Babylonian%20Diyala%20v5_fichiers:image008.jpg?33x32}} |+| gur | {{:Dyala008.jpg?33x32}} |
  
- + 
  
 === Units of weight === === Units of weight ===
  
- + 
  
-| še | {{Old%20Babylonian%20Diyala%20v5_fichiers:image010.jpg?19x28}} | +| še | {{:Dyala010.jpg?19x28}} | 
-| **↓ × 180** || +|  **↓ × 180**  || 
-| gin<sub>2</sub>//, šiqlum// | {{Old%20Babylonian%20Diyala%20v5_fichiers:image012.jpg?33x28}} | +| gin<sub>2</sub>//, šiqlum// | {{:Dyala012.jpg?33x28}} | 
-| **↓ × 60** || +|  **↓ × 60**  || 
-| ma-na | {{Old%20Babylonian%20Diyala%20v5_fichiers:image014.jpg?33x31}}{{Old%20Babylonian%20Diyala%20v5_fichiers:image016.jpg?29x30}} | +| ma-na | {{:Dyala014.jpg?33x31}}{{:Dyala016.jpg?29x30}} | 
-| **↓ × 60** || +|  **↓ × 60**  || 
-| gu<sub>2</sub> | {{Old%20Babylonian%20Diyala%20v5_fichiers:image018.jpg?41x29}} |+| 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. \\ 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 -System S
  
- + 
  
 | Transliteration | Sign | Value | | Transliteration | Sign | Value |
-| 1(aš) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image020.jpg?21x38}} | 1 | +| 1(aš) | {{:Dyala020.jpg?21x38}} | 1 | 
-| 1(u) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image022.jpg?12x35}} | 10 |+| 1(u) | {{:Dyala022.jpg?12x35}} | 10 |
  
- + 
  
 Example: Example:
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 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]] 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 === === Units of surface and volume ===
  
- + 
  
-| sar//, mušarum// | {{Old%20Babylonian%20Diyala%20v5_fichiers:image024.jpg?42x37}} | +| sar//, mušarum// | {{:Dyala024.jpg?42x37}} | 
-| **↓ × 100** || +|  **↓ × 100**  || 
-| gan<sub>2</sub> | {{Old%20Babylonian%20Diyala%20v5_fichiers:image026.jpg?38x36}} |+| 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. 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 -System G
  
- + 
  
-| Transliteration | Signs | Value | +| Transliteration         | Signs                                           | Value      | 
-|   | {{Old%20Babylonian%20Diyala%20v5_fichiers:image028.jpg?26x36}} | 1/4 iku | +|                         | {{:Dyala028.jpg?26x36}}                         | 1/4 iku    | 
-| **↓ × 2** ||   | +|  **↓ × 2**                                                                           ||| 
-| 1(ubu) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image030.jpg?19x40}} | ½ iku | +| 1(ubu)                  | {{:Dyala030.jpg?19x40}}                         | ½ iku      | 
-| **↓ × 2** || ** ** | +|  **↓ × 2**                                                                           ||| 
-| 1(iku) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image020.jpg?21x38}} | 1 iku | +| 1(iku)                  | {{:Dyala020.jpg?21x38}}                         | 1 iku      | 
-| **↓ × 6** || ** ** | +|  **↓ × 6**                                                                           ||| 
-| 1(eše<sub>3</sub>) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image032.jpg?30x44}} | 6 iku | +| 1(eše<sub>3</sub>)      | {{Dyala032.jpg?30x44}}                          | 6 iku      | 
-| **↓ × 3** || ** ** | +|  **↓ × 3**                                                                           ||| 
-| 1(bur<sub>3</sub>) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image033.jpg?12x35}} | 18 iku | +| 1(bur<sub>3</sub>)      | {{:Dyala033.jpg?12x35}}                         | 18 iku     | 
-| **↓ × 10** || ** ** | +|  **↓ × 10**                                                                          ||| 
-| 1(bur’u) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image035.jpg?14x35}} | 180 iku | +| 1(bur’u)                | {{:Dyala035.jpg?14x35}}                         | 180 iku    | 
-| **↓ × 6** || ** ** | +|  **↓ × 6**                                                                           ||| 
-| 1(šar<sub>2</sub>) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image037.jpg?22x35}} | 1080 iku | +| 1(šar<sub>2</sub>)      | {{:Dyala037.jpg?22x35}}                         | 1080 iku   | 
-| **↓ × 10** || ** ** | +|  **↓ × 10**                                                                          ||| 
-| 1(šar’u) | {{Old%20Babylonian%20Diyala%20v5_fichiers:image039.jpg?22x35}} | 10800 iku | +| 1(šar’u)                | {{:Dyala039.jpg?22x35}}                         | 10800 iku  | 
-| **↓ × 6** || ** ** | +|  **↓ × 6**                                                                           ||| 
-| 1(šar<sub>2</sub>) gal | {{Old%20Babylonian%20Diyala%20v5_fichiers:image040.jpg?22x35}}{{Old%20Babylonian%20Diyala%20v5_fichiers:image042.jpg?26x34}} | 34800 iku |+| 1(šar<sub>2</sub>) gal  | {{:Dyala040.jpg?22x35}}{{:Dyala042.jpg?26x34}}  | 34800 iku  |
  
- + 
  
 Examples: Examples:
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 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]] 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 === === Units of length ===
  
-// // 
  
-| //ubānum// || 
-| **↓ × 30** || 
-| kuš<sub>3</sub>//, ammatum// | {{Old%20Babylonian%20Diyala%20v5_fichiers:image044.jpg?23x34}} | 
-| **↓ × 3** || 
-| //nikkassum// || 
-| **↓ × 3** || 
-| //qanûm// || 
-| **↓ × 2** || 
-| ninda | {{Old%20Babylonian%20Diyala%20v5_fichiers:image046.jpg?18x34}} | 
-| **↓ × 5** || 
-| //ṣuppum// || 
-| **↓ × 2** || 
-| //ašlum// || 
  
- +|  //ubānum//                                                || 
 +|  **↓ × 30**                                                || 
 +|  kuš<sub>3</sub>//, ammatum// {{:Dyala044.jpg?23x34}}      || 
 +|  **↓ × 3**                                                 || 
 +|  //nikkassum//                                             || 
 +|  **↓ × 3**                                                 || 
 +|  //qanûm//                                                 || 
 +|  **↓ × 2**                                                 || 
 +|  ninda {{:Dyala046.jpg?18x34}}                             || 
 +|  **↓ × 5**                                                 || 
 +|  //ṣuppum//                                                || 
 +|  **↓ × 2**                                                 || 
 +|  //ašlum//                                                 ||
  
- +  
 + 
 + 
  
 === Bridges === === Bridges ===
- 
-  
  
 The relation between units of length and units and surface is that: The relation between units of length and units and surface is that:
- 
 a square of sides 1 //ninda// has area 1 //sar//. a square of sides 1 //ninda// has area 1 //sar//.
- 
-  
  
 The relation between units of height, surface and volume is that: 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 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 ==== ==== Metrological Volume and Capacities ====
- 
-  
  
 In the evidence from the Diyala, one finds two different equivalencies between volume given in sar and capacities. In the evidence from the Diyala, one finds two different equivalencies between volume given in sar and capacities.
  
- + 
  
 For clay: For clay:
Line 229: Line 212:
 1 sar of volume is equivalent to 60 gur of capacity (Haddad 104, problems #7-#8). 1 sar of volume is equivalent to 60 gur of capacity (Haddad 104, problems #7-#8).
  
- + 
  
 For grain: For grain:
Line 235: Line 218:
 1 sar of volume is equivalent to 72 gur of capacity (Haddad 104, problems #1-#5). 1 sar of volume is equivalent to 72 gur of capacity (Haddad 104, problems #1-#5).
  
- + 
  
-=== Fractions ===+==== Fractions ====
  
- +|  1/3  | {{:Dyala048.jpg?26x40}}  | 
 +|  1/2  | {{:Dyala050.jpg?18x34}}  | 
 +|  2/3  | {{:Dyala052.jpg?23x36}}  | 
 +|  5/6  | {{:Dyala054.jpg?23x35}}  |
  
-| 1/3 | {{Old%20Babylonian%20Diyala%20v5_fichiers:image048.jpg?26x40}} | + 
-| 1/2 | {{Old%20Babylonian%20Diyala%20v5_fichiers:image050.jpg?18x34}} | +
-| 2/3 | {{Old%20Babylonian%20Diyala%20v5_fichiers:image052.jpg?23x36}} | +
-| 5/6 | {{Old%20Babylonian%20Diyala%20v5_fichiers:image054.jpg?23x35}} | +
- +
- +
  
 All of these fractions occur in [[http://www.cdli.ucla.edu/P368229|A 21948]] 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]] See also, for 1/3, 1/2 and 5/6, [[http://www.cdli.ucla.edu/P225105|IM 063106]]
 +
 +
 +
 +//
 +//
 +Page prepared by Carlos Gonçalves (digital version prepared by BG)
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