# Mathematical computation

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1 Mathematical computation in the management of public works in Mesopotamia (End of the 3rd and beginning of the 2nd millennium BC). SAW seminar, 6 April Constructions and excavations Martin Sauvage CNRS, USR 3225, Société préhistorique française et UMR 7041, équipe «Histoire et ARchéologie de l Orient Cunéiforme»

2 Tomb of Rekhmirê, Thebes (Egypt, XVIIIst dynasty)

3 Mixing clay with water and straw

4 Brick moulding

5 ConstrucBon of religious buildings during the Third Dynasty of Ur Persian Gulf

6 The Ziggurat of Ur after excavations and before restauration

7 The four Ur III ziggurats

8 SUMERO-BABYLONIAN METROLOGICAL UNITS Mass 1 load (gún / biltu) = 60 pounds 30 kg 1 pound (ma-na / manû) = 60 shekels 500 g 1 shekel (gín / šiqlu) = 180 grains 8.33 g 1 grain (še / uṭṭetu) g Distance 1 league (dana / bêru) = 180 cords 10.8 km 1 cord (éše / ašlu) = 10 rods 60 m 1 rod (ninda / nindânu) = 12 cubits 6 m 1 reed (gi / qanû) = 6 cubit 3 m 1 cubit (kùš / ammatu) = 30 fingers 50 cm 1 finger (šu-si / ubånu) = 6 grains 1.66 cm 1 grain (še /uṭṭetu) 0.28 cm Surface 1 bùr (bur / buru) = 18 iku 6.48 ha 1 iku (iku / kû) = 100 sar 0.36 ha 1 sar (sar / mušaru) 36 m 2 ( ha) Capacity 1 gur (gur / kurru) = 5 bariga 300 l 1 bariga (bariga / pânu) = 6 bán 60 l 1 bán (bán / sûtu) = 10 silà 10 l 1 silà (silà / qû) = 60 shekels 1 l 1 shekel (gín / šiqlu) ml Bricks 1 bùr = 1800 sar = 1,296,000 bricks 1 ešè = 600 sar = 432,000 bricks 1 iku= 100 sar = 72,000 bricks 1 sar = 60 gín = 720 bricks 1 gín = 12 bricks 1 sar a = 1 ninda 1 ninda (areas, surfaces) 1 sar v = 1 ninda 1 ninda 1 cubit (volumes) 1 sar b = 720 bricks (when counting bricks, whatever the brick type) Volume 1 iku (iku / kû) = 100 sar 1,800 m 3 1 sar (sar /mušaru) 18 m 3 1 sar = 5,00,00 [= 18,000] sila ( 180 hl)

9 Iškarum: daily tasks for workers

10

11

12 total volume or surface daily task numbers of daysof work earth carryingcoefficient distance of each trip volume of earth carried in a day) going(ninda / day) basket(sar v ) daily rate of carrying(sar distance(ninda) v / day) Carrying building materials: tupšikkum (the basket ) = the worker load muttalliktum (the going ) = the daily distance

13 Brick types in YBC 4608 Brick types in OB mathemabcal texts

14 5 4 Brick types in YBC Lengths and widths of UR III bricks 1

15 5 4 Brick types in YBC Lengths and widths of OB bricks 1

16

17 TWO EXAMPLES OF OB MATHEMATICAL TEXTS YBC 4669 (xii) [Part of an OB mathemabcal text, unknown provenance. Robson 1999: 66]. CalculaBons on a brick- pile. The length is 3½ nindan, 3 cubits. The bricks are ⅓ cubit square. The height of the pile is 4 cubits. The width of the pile is 2 cubits. What is the volume of the brick- pile and how many bricks are there? The volume of the pile is 2;15 sar. There are 20 sar and 3 sixbes of bricks. * YBC 4673 (v) [Part of an OB mathemabcal text, unknown provenance. Robson 1999: 90]. CalculaBons on brick- making and construcbon of a brick- pile. One man carried earth over a distance of 30 ninda and he built a brick- pile. For what proporbon of the day he carry earth; for what proporbon of the day did he built the brick- pile; and how many bricks were there? 2,40 [i.e. total of bricks: 160].

18 TWO EXAMPLES OF OB MATHEMATICAL TEXTS YBC 4669 (vi) [Part of an OB mathemabcal text, unknown provenance. Robson 1999: 89]. Combined coefficient exercise. Let a man carry (earth) for a length of 20 nindan and let him demolish (a wall). For what proporbon of the day shall he demolish (the wall), and for what proporbon [of the day] shall he carry (the earth)? In a fihh of the day, or 0;12, he demolished 4 shekels (of earth); in 0;48, the remainder of the day, he carried 4 shekels (of earth). * YBC 4673 (v) [Part of an OB mathemabcal text, unknown provenance. Robson 1999: 90]. CalculaBons on brick- making and construcbon of a brick- pile. One man carried earth over a distance of 30 nindan and he built a brick- pile. For what proporbon of the day he carry earth; for what proporbon of the day did he built the brick- pile; and how many bricks were there? 2,40 [i.e. total of bricks: 160].

19 Robson 1999: men to make 8 sar of bricks [5,760 bricks] 16 men to carry 8 bricks each over 60 nindan 8 men to mix mortar 9 men to carry 90 reed-bundles for mats 3 men to mix mortar 18 men to carry 180 reed-bundles for putting into the brickwork 6 men to mix mortar Total : 84 man-days 240 bricks a day for moulding do not correspond to OB mathematical coefficients: one gives 0;25 sar v /day or 2,161 bricks/day of Type 2 bricks; one other gives 0;20 sar v /day or 1,729 bricks/day of type 2 bricks). The task probably consists here in both carrying and mixing earth, and moulding bricks. But in this case, we do not know the distance for carrying earth before mixing, or carrying the mixture before moulding. An average value of 240 bricks/day for brick making is often attested in both Ur III (e.g. Garshana, see Heimpel, 2009: 224) or OB administrative texts (e.g. NBC 5506, see Robson 1999: 153).

20 Brick carrier text I. 1. [?] sig 4 - hi- tum 2. [n] a- hu- wa- qar 3. [n] im- B- dam 4. [n] nu- ri- li 2 5. [n] um- mi- da- hi 6. [n] ba- ku- tum 7. [n] la- ma- za- t[um] 8. [n] za- zi- x- tum 9. [n....] 10. [n] x- ki- um- mi 11. [n] na- a- a? ša- IGI.DU- na? zi- ip- ri- tum ša- at- i 3 - li [i]m?- ta- tum ugula? / x- [....] [+n] 18. nam- 10 š[a- at-....] en- um- am?- na?- a? ša- at- er 3 - ra im- B- dam a- zi- num d.adad- nu- ri ša- at- i 3 - li x- ra- x- at a- ni- a- hi II du- ni- [....] x- x- tum [t]a?- B- la? 30. [2520+] [n] im- B- dam 32. [n] nu- ur 2 - i 3 - li [n] a- da- lal er 3 - ra- an- dul puzur 4 - a- bi im- B- dam a- hu- ni šu- er 3 - ra u- bar- um bi 2 - x- um [....] [...g]i- ba- ša 43. [n....]- ga 44. [n....]- hi- i 3 - li [n....]- x- x 46. [n+] nam- 10 [a?]- hu- ni B ba- ra- x- B simat- d.nin- šubur geme 2 - d.nin- lil ur- me- lam d.adad- [....] d.šamaš- [a- bi] III. 55. [n....] [....] [....] nam- 10 d.adad- [....] 60. [n] na- ab- [....] 61. [n] e 2 - a- [....] 62. [n] a- da- [lal 2 ] ? a- ni- [....] 64. [n] mi- ni- šu- da ša- at- i 3 - li ša- at- er 3 - ra 67. [n] a- ni- a- hi 68. [n] a- ni- a- bi 69. [n] d.adad- [nu]- ri na- ab- r[i- tu]m ša- hu- na- [x] a- ni- a- hi min la- ma- za- tum i- zi- na[- x?] ša- hi- B[- x?] nam- 10 a- ni- a- bi na- na- a d.šamaš- a- bi ga- [....] x[....] IV. 82. [....] 83. [....] 84. [n+] BLANK SPACE 86. šu+nigin 2 48[+n] 87. uš- bi a 2 - geme sig 4 - ta 89. geme 2 - bi u 4-1- še še- bi sila3 / gur 91. [....] sig4 il2- ga2 92. [....] e2- uš- bar- še3 93. [....] simat- e2- a ugula / lu2- hun- ga2 94. [gi]ri3 dad[ad- il]at šabra 95. [u3] puzur4- d.[nin- k]ar- ak / du[b- sa]r 96. [ša3 gar- š]a- an- na.[ki] 97. ib a2- k]i- B 98. [mu d.šu- d.en].zu lugal / [....]

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