The Mathematics of Uruk and Susa (c.3500-3000 BCE)

This is Part 2 in the Ancient Mathematics series. (Part 1: The Prehistoric Origins of Mathematics, Part 3: Exploring Cuneiform Culture 8500-2500 BCE)

Summary The written mathematics of ancient Iraq and Iran (Mesopotamia, Khuzistan) developed out of an administrative/bureaucratic program to control the surplus raw and manufactured goods of the settled societies of the late neolithic/early bronze age: grains & grain products, sheep & other herded animals, jugs of dairy fats & beer, rope & textiles. It evolved through a sequence of literary and mathematical innovations, each making more efficient the ability to record quantitative/metrological information and use it for planning and control. Initially, impressed tokens and pictographs were used whose meaning was clear by association. Subsequently, this repertoire was written signs was expanded in a consious effort to provide a standard, all-encompassing collection of signs/symbols (ideographs/logograms) that could represent all aspects of importance in early thought (professions, animals, foods, containers, textiles, etc.). The standard sign lists were spread through scribal schools to produce the scribes that administered the temple economies of the early city-states.

Uruk was the hegemonic centre of this innovation in mathematics and writing, starting from 3500 BCE. The increased administrative control generated economic efficiencies accelerating Uruk’s growth and which supported greater military effectiveness and the ability to dominate neighboring polities and support longer distance trading missions [Adams/2005], [Algaze/2013]. The success of Uruk’s structures had the effect of radiating the new inventions outward throughout the Greater Mesopotamian region (evidence in Aratta/Susa adoption of writing/adminstrative control), even reaching Anatolia (Turkey) in the far north (Uruk expansion phenomenon).

The gains in economic power and increased resilience to subsistence unpredictability conferred by the new planning and control capabilities, set in motion the development of a bureaucratic administrative culture in the southern Mesopotamian city states that, over the next 1000 years would reach its hypertrophic apex in the ambitious Ur III program under King Shulgi to plan, manage, and control all economic/productive assets in his vast empire through mathematics (c.2050 BCE). This required an army of scribes which in turn led to the standardization and systematization of the scribal school institution responsible for producing them.

Two examples of mathematical innovation are from the cattle redistribution center Puzrish-Dagan outside Nippur during the Ur III empire. One shows perfection of the form of tabular accounting (world’s earliest normalized two-dimensional table with rows and columns and sums in both dimensions) [Robson/2003]. The other shows the population growth modeling of a cattle-herd over 10 years with projected economic yields in dairy and cheese, solving, in modern terms, population difference equations in table form (see illustrated explanation of cuneiform tablet TCL 2, no.5499, [Nissen/1993: 97-102])

In this paper, we will look in more detail at mathematical development during the archaic period of writing (3500-3000 BCE) which gave rise to a new literate and quantitative layer in society in the main urban centres of Mesopotamia. Our thesis (which we have seen play out already in Part 1) is that technology (in this case mathematics/writing) and culture (in this case the impulse to plan/control) are inextricably linked. Their development influences the trajectory of the surrounding societies.¹

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Mathematically, quantity was managed through distinct metrological systems for distinct commodities (one for sheep and other discrete countable things, one for frain and rationed goods, one for area, one for weight, one for time, etc.). Each system has its unique factor relations. Sums were calculated using grouping and replacement operations to ensure all quantities were written in canonical form (like a change making algorithm), a precursor of the place value system. Unit fractions (1/2, 1/3, …, 1/10) and scaling were used to record production inputs. By the end of the fourth millennium BCE (3000 BCE), economic control through writing and mathematics had become a standard part of how city-states were run, touching off a 500-year period (Early Dynastic) in which city-states with centralized temple run economies would joust for dominance.

The innovations in writing used for planning and control included the use of:

1. administrative seals for economic control (from 6000 BCE in the Ubaid period),
2. clay tokens to keep the count (in use between 6000 and 3500 BCE),
3. clay envelopes for storing tokens from a single transaction (from 3500 BCE),
4. numerical signs recording quantity, initially made on the surface of the clay envelopes, then afterward on their own tablets, separately from the token counters (Uruk V period, 3500-3350 BCE),
5. numero-ideographic tablets combining records of quantity as well as commodity (the latter was known from context before) (Uruk IV period, 3350-3200 BCE), and
6. proto-cuneiform signs (word lists) allowing highly structured administrative records to capture origin, recipient, and intent (Uruk III/Jemdet Nasr period, 3200-3000 BCE). These developments in economic control through advances in archaic bookkeeping, are visible in archaeological layers in Uruk, Susa and other sites of comparable size.

1. The urban mathematics of Uruk and Susa

As we have seen in Part One paper Prehistoric Origins of Mathematics, written mathematics originated with scribal record-keeping (bookkeeping) associated with the redistributive temple-economy of the largest Neolithic cities of Sumerian Mesopotamia (southern Iraq) and Elamite Khuzistan (western Iraq). Temple bookkeepers accounted for products of staple finance and surplus goods arising from irrigation agriculture supplemented by herding, fishing, and hunting. This was the time of the ascendancy of Uruk in the southern Mesopotamia, its excursions to the east (Aratta, Awan, Susa), the time before the Flood (c.2900 BCE), and before these economic and adminstrative innovations became standard through the region.

An impetus toward an accounting function was the use of communal labor to increase the productivity of the community through large-scale efforts such as irrigation canals, cultivation of broader plots of land, care of larger sized herds, the manufacture of goods (baskets, pottery jars, etc.) and the construction of progressively elaborate temples. This required coordination, the pooling of results of this labor (farmers expected to deliver raw and processed grains, herdsmen expected to deliver milk, dairy fats, butters and cheeses, etc.), storage and annotation of surpluses, and distribution of the resulting goods from the central store (rations). This is evident in the changing construction of houses and settlement plans, with community storage in the center.

Around this redistributive function arose chiefs that combined leadership, authority, and stewardship of resources with justice, unity, and the resulting power to mobilize and direct labor (often displayed through constructivion of prestige buildings, temples, ziggurates, etc.) as well as temples and temple-workers that included priests, scribes, as well as the specialized crafts needed to build and finish buildings, storage containers, and symbolic objects.

What is clear is that development of writing and mathematics in Sumeria around 3500 BCE was the culmination of a long period of increasing social and cultural complexity that accompanied the material prosperity of increasing large settled population centers at the end of the Ubaid period. The resulting mathematics was a reflection of this complexity and long heritage, as can be seen from the documented evidence of at least half a dozen metrological systems, each with its own factor list and signs. See [Englund/1998, pp.30-44] and [Hoyrup/1991] for cultural context behind these developments.

2. Signs and Tablets

2.1. Tokens for early accounting.

The early quantitative notation grew out of a practice of accounting using clay tokens of different shapes to designate fixed measures of designated commodities using fixed metrology tables, possibly using counting boards. How extensive this token system was is not known, but archaeological and anthropological evidence shows use of pebbles (stones, abzu) by shepherds to keep track of herds, and the use of such in separate containers to track the gender and status of herded animals (male/female lamb, male/female adult, new birth, milk producing etc.)

2.2. Clay Envelope

We know the context of token use with greater certainty when the tokens were contained in a clay envelope and kept as a unit, and when their contents were impressed on the surface of the envelope, using the same signs that were later explained with meaning when writing became more expressive. We know this because the same form ( abzu, pebbles or small stones) are documented in the same usage 1500 years later with cuneiform writing (see Oppenheim’s find of the clay envelope at Nuzi). We know this anthropologically because it is still used in much the same form by shepherds in Middle East.

A special find at Nuzi by Leo Oppenheim dating from 1500 BCE, when cuneiform script was already advanced, confirms the usage hypothesis. On this particular clay envelope is inscribed a detailed cuneiform description of the meaning of the 48 abzu (small stones) inside representing 48 individual small cattle (sheep and goats).

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2.3. Early Numerical tablets (Uruk V, 3500-3350 BCE)

The next three stages can be observed in tablets dating from the Uruk V period from 3500 BCE to 3350 BC. Although many of these tablets come from Uruk, they cannot be dated into precise periods as they were discovered in large rubbish heaps where they had been cast aside as detritus. Some were “recycling” as building filler (and discovered in building remains). The dating (in many cases sequencing) comes from finding similar symbolism in Susa and other sites in situ amidst distinct archaeological layers (cf. Englund/1998, p.56). (See Appendix 3 for timelines).

At this stage, canonical representation had not yet become standardized, i.e. the collection of quantity grouped into the largest units, i.e. the equivalent of the remainder theorem, as became standard numerical representation in later tablets. This occurs in the late numerical tablets.

Examples:

• 

  Early numerical tablets from Jebel Aruda, c.3500-3350 BCE (Uruk V period). In no. 6 (top), the use of a grouping operation to represent quantitites in with the fewest possible signs (canonical form) is not yet standard. In no. 2 (bottom) the kind of rule breaking depends on which metrology system is presumed: if this is SEG (sexigesimal) system, a clear grouping rule is broken (since 6x N14 = 1x N34), but in the SZE system, this may be an acceptable exception since 10x N14 = 1x N45, so this usage may have been to avoid a 0 in the lowest two values. In any case, the total value is very different in SZE (900 + 300 + 3 = 1203) than in SEG (60 + 100 + 3 = 973). Source: Englund/1998, p.51, Fig.13. Tablets Fs Kraus 012-025, no. 6 (top) and no. 2 (bottom)

• Thirteen additional attestations from Jebel Aruda described by G. van Driel (1982).

2.4 Early Geometric Calculation

From Susa, we have what appears to be early geometric calculations, dated from Uruk V period, i.e. 3500-3350 BCE.

We also see in this period triangular tablets (MW 0188/107) and circular tablets (MW 0188/112, the shape which may be indicative of area or circumference records.

2.5 Simple Numero-Ideographic tablets, single information cell

These tablets were found as late as Uruk V period, i.e. 3500 – 3350 BCE, counting sheep (UDU) and jars (DUG) of liquid.

From Godin Tepe in Iran, c.3500-3350 (Uruk V period), we have the following.

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2.6. Late stage simple numerical tablets

The next phase is during the Uruk IV period from 3350 BCE to 3200 BCE, there is a rise in complexity of information captured in tablets, and what commodities are recorded.

What is significant is that by now the canonical representation of large numbers was standard, i.e. a sort of remainder theorem (making change type algorithm) was applied, so that repetition of smaller signs never exceeded the limit that would allow grouping and replacement/substitution with the next larger unit in the metrological sequence. Example: 6 x 10 (u) = 1 x 60 (ges), so the maximum number of tens (u) units is 5. Similarly 10 x 60 (ges) = 1 x 600 (gesu) so the maximum number of 60s (ges) units is 9. See above, where this is the case for both the 1s (dis) and the 60s (ges).

Using the Sumerian words for numbers, this might have been said: “gesu (600) dis (1) ges (60) ilimmu (9) u (10) limmu (4) dis (1) ia (5)”, or if the places were assumed, “dis (1) ilimmu (9) limmu (4) ia (5)”, with whatever object was being counted.

2.7. Simple tablets, but expanding ideographic repertoire

Texts began to capture more than just quantity, but also other details through additional signs: commodity, ownership, use function. Example: 127 finds from Uruk (Uruk IV period 3350BCE to 3200 BCE). Tags were solely ideographic. Tablets combined number with attributes, not all of which have been deciphered, sometimes with what appears to be signature of an individual connected with the transaction.

Examples:

• 

  Tag W15662 from Uruk (c.3350-3200 BCE, Uruk IV period), ideographic only, Source: Englund/1998, p.60, Fig. 18

• 

  Tablet W15658 from Uruk (3350-3200BCE, Uruk IV period), ideographic, Source: Englund/1998, p.60, Fig. 18

• 

  Tablet w9478,o from Uruk (Uruk IV period, 3350-3200 BCE). This is an example of an ideographic tablet conveying a great deal more information about the context of the transaction (conj. transl. “Three [from/to?] Elam (NIM) Uruk (UNUG) create (TU) herald (SUKKAL) document (DUB)”) than simply the quantity three. Unfortunately, we don’t know what the three refers to.

• 

  Tablet w15897, c24 from Uruk (Uruk IV period, 3350-3200 BCE). This tablet describes a quantity of prepared food intended for a house. The reverse (Uruk) might be the origin or destination, source, or recipient.

  Of course, any attempt at translation is hypothetical, pending similar sign groupings or corroboration with later texts or other contextual implements. But this serves to illustrate the change that started to occur with the broader use of ideograms.

See Appendix 1 for proto-cuneiform (archaic) sign lists and Appendix 4 for primary source research tools to decipher/verify transliteration/translation of proto-cuneiform primary source texts.

2.8. Complex Numero-ideographic tablets, with multiple information cells

This is the scribal equivalent of a spreadsheet, containing lists of multiple items with quantity and commodity, and sometimes attributes. Proto-cuneiform from the Uruk influence listed the number first. Proto-elamite from the Susa influence listed the commodity first.

The primary administrative activity in archaic Mesopotamia was of grain storage and distribution, and these by far have the greatest number of accounts in Uruk (Englund/2001,p.3)

There are accounts of:

• grain and grain products (emmer wheat, barley, breads and other baked goods, cereal products, rations),
• beer of various strengths and its primary ingredients (barley, hops, malt),
• other liquid grain products (e.g. mixed with dairy fats)
• herded animals (sheep, goats, cows, pigs) and their production (dairy fats, milk, cheese)
• land usage
• labor management, wages, and the distribution of rationed goods

These activities and their signs are found on archaic tablets from Uruk and surrounding cities in the periods Uruk IV (3350 BCE – 3100 BCE) and Uruk III/Jemdet Nasr (3100 BCE – 3000 BCE), and indeed appeared in the period Uruk V (3500-3350 BCE) in early attestations.

2.9. Double-sided tablets
The first double-sided complex numero-ideographic tablets were seen during Uruk IV (3350 BCE-3200 BCE). It is unclear whether and how the information on the reverse was in every case related to that on the obverse (front), and how much was fixed by convention vs. varied by tablet/context. Could the double-sided documents show the two trades on separate sides? Was the reverse in some cases a remainder after settling a transaction (i.e. input/output)? What we know is that in many cases the reverse was a high-level summation (grand total) operations over all quantities provided with detailed accounting on the front. (cf. Englund/1998, p.61.ff)

Examples

• Tablet W7227,a which contains 54 cows (AB2) and bulls (GU4), the largest attested herd of cattle in Uruk IV period.
• Tablet IM 074343, which looks like what might be traded on one side (complex products: 10x jars of beer, 25x fruits, 40x apples, 4x special fruits, 3x foreign or exotic fruits, 5x luxurious fruits or almonds, 2x apple fruit, 15x wool, 71x ?, 2x onion/garlic, 3x perfume) with the counter trade on the reverse (simple raw materials: 20x onions/garlic, 20x blocks/slabs of stone, 16x boxes of fish.
• Tablet W6966,b looks like a wage distribution receipt for 20 male laborers (GURUSZ) receiving 31 wage rations (BA).
• Tablet MSVO 1,185 (from Jemdet Nasr during Uruk III period) is a 4-column account of total rope (DUR) production over three years from two production inputs (column 4): pre-existing rope pieces (BA.DUR) and fresh reed (GI). A yearly total is given in column 3, an subtotal of each individual input (just rope pieces and just reed separately) over all three years (column 2), and then finally the grand total of new ropes is given in column 1, obtainable by summing either column 2 or column 3.
  
  

  Tablet MSVO 1,185 during Uruk III period (3200-3000 BCE) shows detailed accounting of the production of ropes (DUR) from inputs pre-existing pieces (BA.DUR) and new reed (GI). Source: Englund/1998, p.63

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3. Mathematical Capability

In what follows, we will look at some of the mathematics evident from the early archaic tablets.

3.1. Simultaneous Metrology Systems
One of the complexities of the Sumerian measurement system was a set of conventional measures that had different units based on what commodity was being measured. This meant almost a dozen parallel metrology systems were in simultaneous use, some using the same signs, but with different values, sequences, and factors between them.

The following tablet illustrates the commodity specific context behind the use of metrology systems even on the same tablet.

Example: Uruk IV use of SZE and B systems on same tablet

Metrology systems
Looking at the metrology sequences in use, the most common systems were the sexagesimal (S system) for counting discrete objects, bisexagesimal (B system) for counting rationed goods, SZE system for counting grain capacity, and the GAN2 (G) system for measuring area. The S system progresses by factors of 10 and 6, the B system appends a factor of 2 after the 10 and 6, the SZE system has a completely different sequence 5, 6, 10, 3, and the G system reverses the last two with 6, 3, 10. This means the same symbol means 1 unit (dis) in the S and B systems but 5 sila (bowls) in the SZE system, and 1 iku in the G system. A small circle is then worth 10 units (u) in the S system, e.g. when counting sheep, 6 when counting barley, and 18 when measuring the area of a field. [Nissen/1993, p.28 and 132].

The factors of the above systems are:

• sexagesimal series (for general counting) with alternating factors of 10x and 6x e.g. 1, 10, 60, 600, 3600, 36000, …,
• bisexagesimal series (for counting rationed products) with factors 10x, 6x, 2x e.g. 1, 10, 60, 120, 1200, 7200, …,
• sze series (for grain capacity) with factors 5x, 6x, 10x, 3x, and all unit fractional designations from 1/2, 1/3, …, to 1/10, e.g. 1, 5, 30, 300, 900, 9000, …
• dug series (for liquid capacity) with factors 5x, 2x, e.g. 1, 5, 10, 50, 100, …
• gan series (for area masures) with factors 10x, 6x, 3x, 10x, 6x, 3x, … e.g. 1, 10, 60, 180, 1800, 10800, 32400, …
• en series (for weight measures) with factors 4x, 2x, 2x, 10x, e.g. 1, 4, 8, 16, 160, …
• u series (for time and calendar) with factors 10x, 3x, and 12x, corresponding to 1, 10, 30, 360 (day, 10-day week, month, and year).

Number words and language:

Later cuneiform representation of numbers:

3.2. Arithmetic – Sums with Grouping and Replacement

From Uruk III, complex tablets had detailed accounting on the front (obverse) and a simple sum tally of the higher level related items, on the back (reverse). As we have seen above, this involved grouping and replacement using the appropriate metrological factors depending on what was being counted.

Tablets from Jemdet Nasr (MSVO 1) in N. Mesopotamia (an economic outpost perhaps of Uruk) cover broader aspects of the archaic provincial economy for which accounting was used, including herding, land management, utilization, and yield planning, worker rationing, and other distributive mechanisms. These show the more detailed accounting practice.

Example: MSVO 1,216 (Nissen/1993, p.133)

It is unclear whether the early scribes used a wooden counting board to perform the arithmetic/groupings (see Christine Proust’s reconstructions).

3.3. Arithmetic – Fractions and Multiples

The SZE system for measuring grain capacity is where we see the use of fractions.

The example below is from the Uruk III period (3200-3000 BCE) from the Erlenmayer collection of archaic tablets (MSVO 3), which appear to have been an administrative archive of a production unit concerned with the distribution of beer and the ingredients used in beer brewing (unprocessed grain emmer and barley, malt, coarse-ground barley groats). The tablets in this collection document production processes, e.g. how much grain and malt was needed to produce a certain type, size, and strength of beer.

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Appendices

Appendix 1. Proto-Cuneiform (Archaic) Word Signs (Vocabulary)

Table 1. Animals

Origin of animal signs. Most are clear just by looking. What’s unique about a goat? It’s horns sticking out from it’s face. So we have a goat depicted as a cross. What is a sheep but a fat/fluffy goat? So a sheep is a cross surrounded by a circle. (Imagine looking at a sheep head on. Plump wooly head (circle), with the vertical axis defined by broad nose ridge and horizontal axis defined by the extended ears.) With fat-tailed sheep (GUKKAL, a distinct type of sheep, 25% of world sheep population), the fat tail (up to 16% of sheep’s weight, concentrated in the tail and therefore easy to harvest as a source of cooking fat/tallow) is pinned to the back of the symbol.

Table 2. Grain and Grain products, Beverages, Jugs of Liquid

Table 3. People and Professions

Etymology for People & Professions

• The origin of signs for male and female should be pretty obvious just by looking.
• Farmer (ENGAR) is from lord (EN) and a grain product (GAR) and uses the sign of a plow (APIN).
• Male laborourer (GURSZ) is from GUR (referring to various forms of labor and the largest form of capacity measure e.g. for grain, 1 gur = 300 sila (approx 300 litres) [Nissen/1993, p.142], or 480 sila, approx. 480 litres, [Robson/2007, p.70]) and USZ (male).
• Shepherd (SIPA) uses the double-sign for PAP (to check, verify, count) and UDU (sheep). Note, interestingly the word UDU is from UD (day) and U (pasture).
• Potter (DUB.NAGAR) is from clay (DUB) and carpenter (NAGAR)
• Scribe (DUBSAR) is from clay (DUB) and ‘to write’ (SAR). The sign for SAR (to write) is, interestingly, grain (SZE) writing on a tablet, perhaps referencing the use of a stalk of grain to scratch out the early signs on the clay.
• Intelligent person (LU2xGESZTU) is from LU2 (person) and GESZTU (ears, intelligence). Note that the sign for GESZTU is the same for ears, suggesting intelligence associated with listening.
• King (LUGAL) is from LU2 (person) and GAL (large or great).
• Chief Administrator (EN) is common first part of names of rulers, e.g. En-mer-kar, En-men-bara-ges-i, En-men-lu-an-na, En-men-gal-an-na, En-sipa-zi-anna, En-men-nun-na, En-nun-dara-an-na, En-shakush-an-na, En-bi-esztar. MER is crown, BARA2 is pedestal, LU2 is person, AN is god, GAL is great, SIPA is shepherd, NUN is prince or god.
• Herald (NIMGIR) is from NIM (high) and GIR (fish)
• Child (DUMU) uses the sign TUR (small)

Table 4. Places and Geographical Names

Etymology for Place Names

• Eridu has the the sign NUN (prince or god).
• KI is place/earth. According to Sumerian mythology, the god ENKI from EN (administrator) and KI (earth) is the god that cared for mankind enough to teach civilizations, and first taught these arts in Eridu.
• Uruk (UNUG) has the sign sea (AB) with many people
• Ur has the signs SZESZ (brother) and Uruk (UNUG)
• Kish, interestingly, has the sign donkey (ANSZE) with many people. Kish is in N. Mesopotamia (Akkad) and as a major trading area of its own would have been closely associated with donkeys.
• Babylon is the signs KA2 (gate) and AN (god, sky)
• Susa has the signs ERIN (cedar) and Inanna. Background, in the Gilgamesh cycle, it is clear that cedar was the sought-after wood from the mountains used to build the great temples, and Inanna was the patron diety (goddess) of Uruk. Susa is in the Zagros mountains, and is likely one of the main trading centers of Uruk in the Uruk Expansion period (Uruk IV).
• Aratta is from the signs LAM (abundance), KUR (foreign, i.e. over the mountains, these would be the Zagros mountains to the East of Mesopotamia), and RU (to cast down, i.e. dominate). See literature Uruk Cycle: Enmerkar (founder of Uruk) and the Lord of Aratta
• Elam has the sign HIGH
• Storehouse is a vessel from which flows out.
• Animal shed (TUR3) has the signs NUN (prince) and cover (SZU2, DU6)

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Appendix 2. Sumerian Language (Emegir)

See [Hoyrup, 1992] for a fascinating discussion of the Sumerian question, in particular the evidence that the Sumerian language may have arisen as a CREOLE emerging from the melting pot of Uruk V and Uruk IV society between 5000-4000 BCE.

See [Jagersma, 2010] for a presentation of the language and grammar.

Sumerian Nouns – Table 1

Sumerian Nouns – Table 2

Sumerian Verbs – Table 3

Sumerian Verbs – Table 4

Sumerian Adjectives – Table 5

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Appendix 3. Timelines

Simplified Mesopotamian Chronology (Main Periods)

Mesopotamian Chronology (Key Periods and Notable Rulers)

Development of Mesopotamian Mathematics (4000 BCE onwards)

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Appendix 4. Primary Sources and Research Aids

Where does one find the primary sources and research aids?

1. CDLI (Cuneiform Digital Library Initiative), a joint project of UCLA, University of Oxford, and the Max Planck Institute for the History of Science (Berlin), aims to store digitally high resolution images, line drawings, and transliterations of all known cuneiform texts. It is a fantastic resource!
   You can put any of the tablet names in the publication box, or specify one of approx 100 search attributes in Full Search.
   

   Cuneiform Digital Library Initiative (CDLI) aims to store digitally high resolution images, line drawings, and transliterations of all known cuneiform texts.

2. For proto-cuneiform, you will need access to the latest archaic sign list, hosted by CDLI.
3. What do the signs mean? You need sign readings list, hosted by CDLI.
4. Pre-Uruk (8500-3500 BCE) and Uruk V (3500-3350 BCE) periods: 632 texts.
5. Uruk IV (3350-3200 BCE) period: 1861 texts
6. Cornell’s Cuneiform Library with 219 texts from their archaic collection
7. Sumerian/Akkadian and English dictionary, hosted by Penn State.
8. DCCLT (Digital Corpus of Cuneiform Lexical Texts), e.g. Lexical List LU2 A (standard professions list), from ORACC, with links to the attested tables. Example:
   LU2 A Lexical List of Standard Professions, from 3200 BCE (Uruk IV) through to the Fara schooltexts.
   Source: Englund/1998, p.104, Fig 32.
   Transliteration: ORACC
   Tablet attestation: MS 2429 (from Umma, c.3200-3000 Uruk III period)
9. Old books (with expired copyrights) from Archive.org, e.g. The Sumerian Kinglist by Thorkild Jacobsen
10. Abbreviations for Assyriology
11. Academia.edu, a central hub for papers on Assyriology by leading researchers, e.g. Jens Hoyrup’s papers (x238), Eleanor Robson’s papers (x81), Joran Friberg’s papers, Christine Proust’s papers
12. Homepages of key researchers with their publications: Robert Englund’s publications at CDLI, Joran Friberg’s publications at Chalmers U. and his staff page, Jens Hoyrup’s page, Hans Nissen’s page
13. Google Scholar for citations, cross-references, and PDF papers online, e.g. Robert Englund’s work
14. Proto-cuneiform short history and bibliography (on CDLI)
15. Intro to Sumerian language and culture, primary sources collection from course at Masaryk University (Czech). Other courses: Art and iconography, Neolithic Pottery of Near East, The Chalcolithic Near East, Course 49
16. Christie’s auction of the Erlenmeyer Collection (most of which was bought by the Government of Berlin)
17. ORACC (Open Richly Annotated Cuneiform Collection) and List of dozens of collaborative projects
18. ETCSL (Electronic Text Corpus of Sumerian Literature), hosted by Oxford University, containing over 400 items
19. DCCMT (Digital Corpus of Cuneiform Mathematical Texts), by Eleanor Robson of Oxford University
20. Chicago Assyriological Dictionary (CAD), at University of Chicago
21. MSVO 1, 241 tablets
22. MSVO 2, 175 texts
23. MSVO 3, 86 texts
24. MSVO 4, 80 tablets

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Appendix 5: Cultural History of Sumer from Uruk period onward

For period before the Ubaid, see Prehistoric Origins of Mathematics, Appendix 3

• Ubaid (6500-3800 BCE)/Chalcolithic (4500 BCE) – transition to permanent unwalled settlements with specialized craftspeople (potters, weavers, metalworkers), cultivation of grain under arid conditions through the use of irrigation canals (some up to 5km long) requiring large collective labor efforts, the growth of an extensive trade netowrk, and the building of temples. First known settlement in S.Mesopotamia is Tell el-Ouelli (Ubaid 0) (6500 BCE-5400 BCE), 4km SE of Larsa, 25km SE of Uruk. Next is Eridu (Ubaid 1) up to 4,000 residents in 20-25 hectares, irrigation agriculture, limited use of copper metal tools, expansion of art and aesthetics, and the beginnings of stratification of society, professional specialization, and the clustering of villages around centers


Artists conception of Ubaid life (unwalled settlements, communal labor, irrigation agriculture, copper supplementing stone and wood tools)



Ubaid period cultures, c.6000 BCE onwards.

• City-State Period (4,000 BCE-2,900 BCE) – early bronze age, expansion of settlement size to large cities with walls (Uruk, Ur, Susa), with up to 50,000 residents in 6 km2 (Uruk c.2900BCE), hierarchical society with an established elite (temples and lords), warrior class, slavery, long distance trade, large surpluses and the controlled use of labor for prestige buildings – emergence of writing, the state, arithmetic, ancient book-keeping. Uruk city (founded in Eridu Ubaid 1 period 5,000 BCE onwards) originated as two separate temple sites to Innana and An (Kullaba district).See FAQ1 for discussion of middle-chronology dating of Mesopotamian events from Early Dynastic onward


Uruk City, founded in Ubaid 1 (Eridu) period 5,000 BCE as two temple sites, Eanna (to Inanna) and An, consolidated into a single walled urban site, eventually with an intricate inter-city canal system allowing heavy goods (wood, stone, etc.) to be brought into the city by boat from outposts, colonies, and distant trading partners (Venice in the desert)

• Early Dynastic Sumer (c.2900 BCE) – the establishment of royal lineages and cementing of power and authority, economic and military rivalry between city-states.(cf. The Sumerian King List [Jacobsen/1939]). It was a tumultuous time, with overlapping leaderships, tribute relations, alliances, and hegemonies. Not a clean linear order, despite the attempt by the Sumerian King List to establish such a chronology. In effect, the chronology shows where the hegemonic center was, but then starts the king list before the time of the hegemonic shift to provide the ancestry of the hegemonic leader.
  

  Cover illustration from The Sumerian King List (Source: Jacobsen/1939)



The Weld-Blundell Prism is the most complete version of the Sumerian King List. Currently housed at the Ashmolean Museum in London. (Source: Ashmolean Museum)

• Eridu (Alulim, Alalgar, and the abgallu 7 sages), Bad-Tibira, Larak, Sippar, Shurrupak (modern Fara, city of Utnapishtim/Ataharsis/Noah, and later location of a flourishing scribal school of mathematics and literature), The Flood (c.2900BCE),
• Kish I (with 23 kings who ruled after the Flood, including 12th king Etana of the Etana myth, and the 21st king who was the first archaeologically attested ruler
• Enmenbaragesi (c.2600 BCE), who built the first temple at Nippur to Enlil establishing Nippur as the holy city of Sumer, controlled Eshnunna, and conquered Elam), controlled Uruk (capturing Dumuzid the Fisherman, but his son Aga was defeated by Gilgamesh of Uruk, ending the Kish dynasty and transferring hegemony to Uruk I.
• Eanna/Uruk I founded by Enmerkar who conquered Aratta
• Lugalbanda of Uruk who fought with Enmerkar
• Gilgamesh, son of Lugalbanda, who defeated Aga of Kish and escaped the supremacy of Kish
• Ur I, (First Dynasty)
• Awan (Elam), Kish II, Hamazi (Guti), Uruk II
  

  First 11 cities of the Sumerian King List – from Eridu to Hamazi (Source: Jacobsen/1939)

• At this juncture, we have a fork:
  • The king list continues: Ur II, Adab, Maeri, Kish III, Akshak, Kish IV, Uruk III (Lugal-za-gesi), Akkad.
  • Outside the kinglist, we have contemporaneously the rise of a pre-empire before Sargon centred around Lagash and Umma: Lagash, whose rulers (First Dynasty) are Ur-Nanshe, Eannatum (c.2500 BCE), Entemena, ending with Urukagina (c.2350 BCE), king of Lagash and Girsu, who overthrew the corrupt Lugalanda and was viewed as a reformer, with the Code of Urukagina being the first written law code we know of in history (AO 03278, also here). Urukagina was defeated by Lugal-za-gesi of Umma/Uruk III who first unified Sumeria, but then was overthrown by Sargon of Akkad, his cup-bearer. The Lagash kings are speculated not to be in the Sumerian King List due to the political rivalry at the end of Ur III (when the Sumerian King List was written)between Ur/Isin and Larsa, which was founded by Eannatum of Lagash.
  • Uruk III/Umma: Lugal-za-gesi
  • Akkad: Sargon
• Age of Empire – unification of the entire region under a single hegemonic ruler:
• Sargon of Akkad (cf. [Jacobsen/1939], p.111),
• Lagash, c. 2050BCE, Gudea of the Gudea Cylinder (texts here)
• Ur-Nammu and Shulgi of Ur III (cf. ibid. p.123) which was the last great Sumerian empire, followed by the secondary Sumerian city-states Isin and Larsa, and the rise of the Amorites (Martu) and Hammurabi of Babylon (c.1763). The Sumerian King List chronology is: Akkad, Uruk IV, Gutium, Uruk V, Ur III, Isin, where it ends. Subsequently we know it continued:
• Isin (100 years)
• Larsa (100 years)
• Babylon.


Near East c. 1700 BCE. This is approx. the time of Hammurapi’s unification of Mesopotamia centered at Babylon, following the fall of Ur III, the rise of Isin and its subsequent fall to Larsa. Ultimately, the Hittites would sack Babylon and retreat allowing the Kassites (from the Mittani Empire) to rule.



The Cuneiform Empires of the Near East – from Uruk (3700 BCE) through to the fall of Assyria and the start of the Achaemenid (Persian) empire of Darius the Great.

Appendix 6: Sumerian Language and Literature

Language: Sumerian is agglutinative, like Kiswahili. It seems these languages are older, based on small morphemes strung together, with the meaning accumulating through suffixes or endings. You get more chattering like sounds in such languages: Tamil, Hungarian, Finnish, Malay, Basque, Japanese, Arabic, etc.

• Sumerian King List from the Weld-Blundell Prism
• The Tummal Inscription
• Enmerkar cycle, 4 narrative poems about Enmerkar of Uruk and his commander Lugalbanda.
  1. Enmerkar (of Uruk I) and the Lord of Aratta
  2. Enmerkar and En-suhgir-anna
  3. Lugalbanda in the mountain cave
  4. Lugalbanda and the Anzu bird
• Gilgamesh cycle: 5 poems about Gilgamesh of Uruk and Aga of Kish, son of Enmenbaragesi
  1. Epic of Gilgamesh: Gilgamesh & Humbaba (Huwawa) (Version A, Version B),
  2. Gilgamesh and the Bull of Heaven,
  3. Gilgamesh and Aga (of Kish I),
  4. Gilgamesh, Enkidu, and the nether world,
  5. The death of Gilgamesh
• Sumerian Precedence Debates (Disputation Literature): these provide insight into the ways of life of the Sumerians
  1. Debate between Sheep and Grain
  2. Debate between Winter and Summer,
  3. Debate between Hoe and Plough,
  4. Dumuzid and Enkimdu (Herdsman and Farmer) (or Innana Chooses Farmer)
  5. Debate between Bird and Fish,
  6. Debate between Tree and Reed,
  7. Debate between Silver and Copper,
  8. Song of the Hoe,
  9. Date Palm and the Tamarisk
  10. Three Ox Drivers from Adab
• Wisdom Literature:
  1. Proverbs Collection;
  2. Instructions of Shurrupak (apparently to Zinsudda, aka Utnapishtim, Atrahasis, Noah, before the flood),
  3. Farmer’s Instructions,
• Epic of Atrahasis (aka Noah, Utnapishtim, Zinsuddu)

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Appendix 7. Assigning dates to historical Mesopotamian events

Mesopotamian events are typically dated in relative terms referencing the “year events” of respective kings within the respective city (e.g. conquest of Larsa by Hammurapi of Babylon in year 30 of his reign). These are associated with each other via synchronism using historical evidence and year event lists from related cities/rulers (e.g. Hammurapi of Babylon year 30 = Rim-Sin of Larsa year 60). They are finally assigned an absolute chronology using one of are 5 competing chronologies for dating Mesopotamian events: high chronology (HC), middle chronology (MC), middle-low chronology (L-MC), low chronology (LC), and new chronology (NC). 152 years lie between High and New chronologies, providing an uncertainty of almost a century for key events such as the Fall of Babylon to the Hittites under Mursuli (direct descendent of Hattusa).

A new technique using carbon 14 dating of tree rings (dendrochronology) has been able to narrow this uncertainty down to +/- 8 years by demonstrating that the only viable chronologies are the middle (MC) and middle-low chronologies (L-MC), and that both of these are currently compatible with known astronomical evidences. The fall of Babylon is now established as between 1587-1595 BCE. [Manning/2016]

Other key dates in the middle chronology (Ref: [Fitzgerald/2002])

• Shulgi of Ur III, 2091 BCE
• Rise of Larsa, 1941 BCE (Zabaia, fourth named king of Larsa).
• Fall of Isin to Larsa, 1794 BCE (Rim-Sin I of Larsa conquering Damiq-ilisu of ISin in Year 29 (rim-sin) and Year 23 (damiq-ilisu0
• Hammurapi takes throne of Babylon, 1792 BCE (year 31 of rim-sin of Larsa) (Old Babylonian period)
• Fall of Larsa to Babylon, 1763 BCE (year 60 of rim-sin of Larsa, year 30 of Hammurapi)
• Fall of Babylon to Hittites, 1595 BCE. Start of Kassite dynasty (from Zagros mountains) (End of Old Babylonian period, Start of Middle Babylonian period)
• Fall of Kassite Babylon, 1155 BCE.


C14 dating of tree rings has improved absolute dating to within +/- 8 years, and validated Middle (MC) and Middle-Low (L-MC) chronologies. Source: [Manning/2016]

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Bibliography

Mesopotamian Mathematics
1. Robert Englund, 1998, Texts from Late Uruk, published in J. Bauer, R. Englund and M. Krebernik, Mesopotamien: Späturuk-Zeit und Frühdynastische Zeit (transl. Mesopotamia: Late Uruk Time and Early Dynastic Time) OBO 160/1, Freiburg Switzerland 1998, 275pp. [Englund/1998]
2. Jens Hoyrup, 1991, Mathematics and Early State Formation [Hoyrup/1991]
3. Eleanor Robson, 2007, Mesopotamian Mathematics, pp.57-186 [Robson/2007]
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5. Denise Schmandt-Besserat, 1977, An Archaic Recording System and the Origin of Writing,”; Syro-Mesopotamian Studies I., 1977, pp.31-70; [Besserat/1977]
   This first publication of her findings builds on prior work of Amiet (1966) on Susa findings, on Oppenheim (1959) on Nuzi findings including an inscribed bulla from 2000-1500BCE, and on Falkenstein (1936) on archaic signs (proto-writing). Subsequent detailed investigations of Besserat’s hypothesis have supported the following points (1) sealed bullae containing tokens provide the evidence of the use of tokens for accounting commercial transactions, (2) that this transition from tokens to inscribed bullae provides a key missing link between pre-writing numerical practice, proto-writing, and the proto-cuneiform that followed, (3) that this critical transition happened c.3200 BCE in Uruk (aka Warka) in southern Mesopotamia. The rest of her many claims in subsequent publications have been demolished, in particular the claim that clay tokens were an accounting system in wide use across the Near East. See critical reviews by Zimansky/1993, Englund/1993, Englund/1998, and the use of contextual archaeology to close the case on Besserat’s speculations, see masters thesis Niemi/2016, and Bennison/2018
6. Tonje Niemi, 2016, Near Eastern tokens. A contextual analysis of near eastern tokens from the 7th to the 4th millenium BC, Master’s thesis, The University of Bergen [Niemi/2016]
   Based heavily on the work of Charvat/2002, Niemi reviews the claims of Besserat using contextual archaeological analysis. She finds, as have Damerow, Englund, Nissen, and others, that while the evidence for token use for book-keeping is convincing in the 4th millenium site layers, there is insufficient contextual evidence for mathematical use of tokens in any other strata due to (1) insufficient quantity of token finds across time and location to be draw significant conclusion, and (2) contradictory micro-local finds of the tokens suggesting use of tokens for other purposes (e.g. funerary rites, game pieces, etc.)
7. A. Leo Oppenheim, April 1959, Journal of Near Eastern Studies, 18:121-128, “An Operational Device in Mesopotamian Bureaucracy“. [Oppenheim/1959]
   Oppenheim describes a bulla containing 48 tokens dated from 1500 BCE that also contains a cuneiform description of the reading of these tokens as itemizing types of sheep and goats (male, female, young, old ,etc.). Unfortunately, between cataloging the tokens and analysis in the museum, the tokens got separated from the bulla, so the opportunity to assign token type to animal type is lost.
8. Joran Friberg, 1984, Numbers and Measures in the Earliest Written Records, Scientific American, Feb 1984, Volume 250, Number 2, pages 110-118 [Friberg/1984]
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10. Robert Englund, 2004, Proto-Cuneiform Account-Books and Journals, in Hudson/Wunsch Creating Economic Order, CDL Press, pp.23-46 [Englund/2004]
11. Marvin Powell, 1971, Sumerian Numeration and Metrology, PhD Dissertation, University of Minnesota. [Powell/1971]
12. Robert Englund, 2001, Grain Accounting Practices in Archaic Mesopotamia [Englund/2001]
13. Peter Damerow, 1999, The Origins of Writing as a Problem of Historical Epistemology, Max Planck University Preprint P114, Invited Lecture at Multi-Origins of Writing Workshop, March 1999 [Damerow/1999w]
14. Hans Nissen, 1986, Archaic Texts from Uruk (ATU2), World Archaeology, Vol 17, Issue 3 [Nissen/1986]
    Outstanding discussion of what we know about the evolution of writing and how we have been able to decipher it.
15. Eleanor Robson, 2000, The Uses of Mathematics in Ancient Iraq: 6000 BCE-600BCE, in Selin’s Mathematics Across Cultures [RobsonSelin/2000]
16. Mesopotamian Mathematics: Some Historical Background, Eleanor Robson, 2000. [Robson/2000]
17. Mathematics in Ancient Iraq: A Social History, Eleanor Robson, 2008, Princeton University Press, Download Chapter 1 (Academia.eu) [Robson/2008]

Mesopotamian History

18. Hans Nissen, 1995, Western Asia before the Age of Empires [Nissen/1995]
    Succinct, 8-page summary of Mesopotamian history.
19. Land, History, and Geography, 2011, Notes from course on Sumerian at Masaryk University (Czech)
20. Petr Charvat, 2002, Mesopotamia Before History, Taylor & Francis (Revised edition of Ancient Mesopotamia 1993), [Charvat/2002]
    Detailed description, based on archaelogical finds, of how the Near East went from Paleolithic to Mesolithic to Neolithic to Chalcolithic, before arriving at the Uruk period of city states. Each find site is reviewed in detail, and an interpretation is given covering all aspects of the associated culture (material conditions, social practice, art and ritual, modes of sustenance, food and commensality, individual work profiles, housing conditions, etc.)
21. L.W. King and R.C. Thompson, 1907, The sculptures and inscription of Darius the Great on the Rock of Behistûn in Persia : a new collation of the Persian, Susian and Babylonian texts, The British Museum [Behistun/1907]
22. Thorkild Jacobsen, 1939, The Sumerian King List, University of Chicago Press [Jacobsen/1939]
    Provides an account, written toward the end of the Sumerian period, and before the conquest by Babylon, of the Sumerian lineages, from Eridu to the flood, to Kish and Uruk (Gilgames), to Ur, to the Akkadian conquest (Sargon), the Sumerian reconquest Ur III, and finally to Isin. Here the King List stops c.1753 BCE. What we know is that within 50 years (and one more transition to Larsa), the dissolution of the Sumerian dynastic lineage would occur with the conquest by Babylon under Hammurabi, a brother of the next to last regent of Larsa (Warad-Sin). See Prehistoric Origins, Appendix 3 for details.
23. Madeleine A. Fitzgerald, 2002, The Rulers of Larsa, PhD Dissertation, Yale University [Fitzgerald/2002]
    Gives a detailed history of Larsa and its environs in the aftermath of Ur III (early 2nd millenium), when Isin was hegemonic. Discusses evidence for the gradual growing in strength of Larsa until its pre-eminence, the waning of Isin, the rise of Babylon, and ultimately the defeat of Larsa (see Prehistoric Origins, Appendix 7 on establishing chronology for these events). Shows the relative insecurity in these cities and the way in which fortunates waxed and waned in the human timescales of a generation. Shows that rulers were succeeded quite rapidly in times of conflict (probably death in battle), and that militarily successful rulers had long reigns. Detailed discussion of the year name system on which synchronist approach to relative chronologies are based.
24. Staurt Manning, et.al, 2016, Resolving Mesopotamian Chronology: Integrated Tree-Ring Radiocarbon High-Resolution Timeframe to resolve Earlier Second Millenium BCE Mesopotamian Chronology, PLOS Journal, July 2016 [Manning/2016]
    Summary: Carbon-14 dating of tree rings shows that absolute dating of Mesopotamian events can be accurate to +/- 8 years. Of the 5 major chronologies, only the Middle (MC) and Middle-Low (L-MC) chronologies are compatible with the data. The fall of Babylon is now established as between 1587-1595 BCE.

    Language

25. [Hoyrup, 1992], Sumerian: The Descendant of a Proto-Historical Creole?: Alternative Approach to the Sumerian Question, by Jens Hoyrup, 1992,
26. [Jagersma, 2010], A Descriptive Grammar of Sumerian, Bram Jagersma, PhD Thesis, University of Leiden, [Jagersma/2010]
1. Also linked, but out of scope for this paper, is the impact of institutional values in enhancing/suppressing innovation. Laws limiting exploitation by the powerful were put in place by Sargon of Akkad, Gudea and Entemena of Lagash, and Hammurapi of Babylon. The military policies of King Shulgi of Ur III stimulated massive state investment, drove institutional innovation but suppressed individual innovation. In the freedoms of the Old Babylonian period we see indiviual innovation thrive. See (Hoyrup/1991) and (Hoyrup/2009: 31-32) for a survey and further reading. ↩