Overview of Metrological Systems

This is a short outline of the most important metrological systems attested in cuneiform mathematics, as used in the translations presented here. For a more comprehensive description, see M. A. Powell, "Masse und Gewichte" [in English], in D. O. Edzard (ed.), Reallexicon der Assyriologie, vol. 7 (1987-90), Berlin: Walter de Gruyter, pp. 457–517.

Archaic metrologies are not given here. They are summarised most conveniently in H. Nissen, P. Damerow, and R. K. Englund, Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East, Chicago: University of Chicago Press, 1993, pp. 28-29.

Some third-millennium metrologies

Area, volume, and weight metrologies are identical to the Ur III and Old Babylonian periods (below). Compound metrological signs, which denote both number and unit, are represented in translation by the numerical value followed by the sign in parentheses, e.g. 2 (ban) 3 (sila) for 23 litres.

Capacity measures in Early Dynastic Ebla

1 anzam

1 sila = 6 anzam

1 shekel = 4 sila

1 barizu = 2 1/2 shekels

1 gubar = 2 barizu

Early Dynastic and Sargonic lengths

Ancient units	Approx. modern values
1 finger	17 mm
1 double-hand = 10 fingers	17 cm
1 cubit = 3 double-hands	50 cm
1 seed-cubit = 2 cubits	1 m
1 reed = 3 seed-cubits	3 m
1 rod = 2 reeds	6 m

Early Dynastic to Old Babylonian discrete counting (in transliteration)

Ancient units	Modern values
1(diš)	One
1(u)	Ten
1(geš₂)	Sixty
1(geš'u)	Six hundred
1(šar₂)	Three thousand, six hundred
1(šar'u)	Thirty-six thousand
1(šargal)^gal	Two hundred and sixteen thousand

Classic Ur III and Old Babylonian metrologies

During the Ur III period, scribes began to calculate with the Sexagesimal Place Value System (SPVS), in which numbers are divorced from commodities and metrologies. Measurements and quantifications continued to be recorded in traditional metrologies, however. The SPVS is transliterated purely numerically, with spaces separating the sexagesimal places, thus: 1 41 15. In translation, a semi-colon marks the integer-fraction boundary, and leading/trailing zeroes are used where appropriate to indicate absolute value, thus: 1 41;15 = 101 1/4 but 1 41 15 00 = 364,500 and 0;01 41 15 = 0.028125.

Standard units of calculation (which are often implicit within Old Babylonian word problems) are shown below in bold.

Length

Ancient units	Approx. modern values
1 finger	17 mm
1 cubit = 30 fingers	0.5 m
1 half-reed (later: pace) = 3 cubits	1.5 m
1 reed = 6 cubits	3 m
1 rod = 12 cubits	6 m
1 chain = 1 00 cubits or 5 rods	30 m
1 cable = 1 00 rods	360 m
1 league = 30 00 rods or 30 cables	10.8 km

The cubit was the standard unit of height.

Area and volume

The sar could be divided into 60 shekels and the shekel into 180 grains.

Ancient units	Approx. modern values
1 area sar = 1 rod square	36 m²
1 volume sar = 1 area sar x 1 cubit	18 m³
1 (ubu) = 50 sar	1800 m² or 900 m³
1 (iku) = 2 ubu = 1 40 sar	3600 m² or 1800 m³
1 (eše) = 6 iku	2.16 ha or 108,000 m³
1 (bur) = 3 eše	6.48 ha or 324,000 m³
1 (bur'u) = 10 bur	64.8 ha or 3,240,000 m³
1 (šar) = 6 bur'u	388.8 ha
1 (šar'u) = 10 šar	3,888 ha
1 (šargal) = 6 šar'u	23,328 ha

Capacity

The sila could be divided into 60 shekels and the shekel into 180 grains. Multiples of the gur were written with the sexagesimal place value system.

Ancient units	Approx. modern values
1 *sila*	1 litre
1 (ban) = 10 sila	10 litres
1 (barig) = 6 ban	60 litres
1 gur = 5 barig	300 litres

Weight

Early Dynastic weight standards were much more variable, and heavier (with a mina of 0.55-0.68 kg), than later (Powell 1987-90, 508). Multiples of the talent were written with the sexagesimal place value system.

Ancient units	Approx. modern values
1 grain	0.05 g
1 shekel = 3 00 grains	8.3 g
1 mina = 1 00 shekels	0.5 kg
1 talent = 1 00 minas	30kg

Bricks

Ancient units	(Approx.) modern values
1 brick *sar* = 12 00 bricks	720 bricks
Small, unbaked brick = 15 x 10 x 5 fingers	25 x 17 x 8 cm
Number of small bricks in 1 volume sar = 1 26 24 = 7;12 brick sar	5184 bricks
Square, baked brick = 20 x 20 x 5 fingers	33 x 33 x 8 cm
Number of square bricks in 1 volume sar = 32 24 = 2;42 brick sar	1944 bricks

Multiples of the brick sar were written in the same way as large area and volume measures.

First-millennium metrologies

Arû measure for lengths and areas

Ancient length units	Approx. modern values	Ancient area units	Approx. modern values
1 big finger	3.1 cm
1 arû cubit = 24 big fingers	75 cm
1 rod = 12 cubits	9 m	1 rod x 1 rod = 1 mušaru	81 m²
		1 ubû = 50 mušaru	0.4 ha
		1 ikû = 2 ubû	0.81 ha
		1 eblu = 6 ikû	4.86 ha
		1 būru = 3 eblu = 18 ikû	14.6 ha
		1 šar = 60 būru	875 ha

Arû 'seed measure' for areas

Ancient units	Approx. modern values
1 qû	270 m²
1 sūtu = 10 qû	0.27 ha
1 pānu = 6 sūtu = 2 ikû	1.62 ha
1 kurru = 5 pānu = 10 ikû	8.1 ha

Cable 'reed measure' for lengths and areas

Ancient units	Approx. modern values	Ancient area units	Approx. modern values
1 finger	2 cm
1 cable-cubit = 24 fingers	0.5 m
1 rod = 12 cubits	6 m	1 rod x 1 rod = 1 mušaru	36 m²
1 chain = 5 rods = 50 cubits	30 m
1 cable = 2 chains = 10 rods	60 m	1 cable x 1 cable = 25 mušaru	900 m²
		1 ikû = 1 40 mušaru	0.36 ha

Cable 'seed measure' for area

Ancient units	Approx. modern values
1 grain	70 cm²
1 akalu = 18 00 grains	7.5 m²
1 qû = 10 akalu	75 m²
1 sūtu = 6 qû	450 m²
1 pānu = 6 sūtu	0.27 ha
1 kurru = 5 pānu	1.35 ha

'Reed measure' for lengths and areas

Ancient length units	Approx. modern values	Ancient area units	Approx. modern values
1 finger	2 cm	1 finger x 1 finger = 1 small finger	4 cm²
1 cubit = 24 fingers	0.5 m	1 cubit x 1 finger = 1 grain	100 cm²
		1 cubit x 1 cubit = 1 small cubit	0.25 m²
1 reed = 7 cubits	3.5 m	1 reed x 1 finger = 1 area finger	730 cm²
		1 reed x 1 cubit = 1 area cubit	1.75 m²
		1 reed x 1 reed = 1 area reed	12.25 m²
1 rod = 2 reeds = 14 cubits	7 m

10 Jun 2022

Eleanor Robson

Eleanor Robson, 'Overview of Metrological Systems', The Digital Corpus of Cuneiform Mathematical Texts, Eleanor Robson, 2022 [http://oracc.museum.upenn.edu/dccmt/metrology/]