### Table of Contents

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# General overview

## Sources

It can be estimated to about 2000 the number ofmathematical tabletsknown to date, a large majority of which already published. The distribution of these sourcesaccording to time and spaceis extremely irregular, as shown by table 1 below. This table contains rough data, established with the help of the CDLI consulted September 13, 2015; note that most of the tablets from unknown or uncertain provenience according to the CDLI are very probably Old Babylonian.

Periods | Number of tablets | Provenience |
---|---|---|

Early Dynastic / Old Akkadian (2500-2100) | 27 | Southern Mesopotamia |

Ur III (2100-2000) | 8 | Southern Mesopotamia |

Old Babylonian (2000-1600) | 1613 | Entire Ancient Near East |

Medio-Assyrian /-Babylonian (1600-1100) | 51 | Northern Mesopotamia |

Neo- Assyrian /-Babylonian (911-539) | 44 | Northern Mesopotamia |

Late Babylonian (547-63) | 38 | Southern Mesopotamia |

Uncertain or unknown | Ca. 300 | |

Table 1: approximate distribution of mathematical tablets

The bulk of the mathematical sources (about 85%) are dated to the Old Babylonian period. Only small sets of mathematical tablets are dated to the 3^{rd}millennium, but they are of special interest as they document the earliest mathematical practices known to date (see articles on 3^{rd}millenium mathematics). The first millennium mathematical *corpus*, albeit modest too, is important for the history of exact sciences, especially the 38 texts dated to the Achaemenid and Hellenitic periods produced in the milieu of priests who developed mathematical astronomy (see article 1^{st}millenium mathematics).

As for the proveniences, the picture is also quite contrasted. The 3rd millennium sources come mainly from Adab, Šuruppak, Zabalam, Nippur, Girsu, and Umma, that is a small southern area of no more than one hundred kilometers diameter. Old Babylonian sources come from a huge area, which includes not only the low Mesopotamian plain, but also the Diyala valley, the eastern Syria, and the western Iran (Susa). Late 1st millennium sources again come from a reduced area: Achaemenid mathematical sources are attested in southern and central Mesopotamia, and Hellenistic ones only in Uruk and Babylon.

The communities of scribes who produced mathematical texts are very diverse. For example, the 3^{rd}millennium mathematical texts seem to have been written in connection with the control of the lands and the large scale works by governors and high ranked officials (see see article on “Early Dynastic mathematical texts”). The Old Babylonian mathematical production is closely connected with the activities of scribal schools (see article on “Learning mathematics in Old Babylonian scribal schools”). These schools produced mathematical traditions which appear quite different according to the places and the time. Sources from the Diyala Valley, Susa or southern Mesopotamia, for example, exhibit different local features. Early Old Babylonian southern texts can be distinguished from the late Old Babylonian mathematical tradition developed in Kiš or Sippar after the collapse of southern cities. The late Babylonian picture is completely different. The scholars who wrote mathematical texts in Achaemenid and Hellenistic periods belonged to the families in charge of the service in the great temples of Uruk, Babylon, Nippur or Sippar, and were involved in astronomy, astrology and medicine (see article on “Mathematical texts of the Achaemenid and Hellenistic periods”).

In conclusion, the general label "mathematical cuneiform texts" encompasses several mathematical traditions, developed in different contexts for different goals. The articles attached to this brief introduction shed light on this diversity.

## Numbers and metrology

Despite the diversity of mathematical methods invented in different communities of scribes in different periods, mathematical cuneiform texts exhibit striking common features: a standardized metrology, and the use of the sexagesimal place value notation, at least from the late third millennium.

The standardized metrology used in mathematical texts emerged probably in the context of the reforms undertaken by rulers of the first centralized empires, from the mid-third millennium under the Old Akkadian dynasties and subsequently under the Ur III rulers, as shown by Marvin Powell. This standardized metrology dominates in mathematical texts of all periods, even if the archaic texts use different systems, and the late texts show a competition with other systems used in Mesopotamia from mid-first millennium. The standardized metrology and earlier variants are presented with details is in the tool "Numbers and Metrology" of the wiki. See in particular the metrology that was taught in Old Babylonian scribal school.

The reform realized during the second part of 3^{rd} millennium produced not only a standardized and coherent metrology, but also a new numerical system, the sexagesimal place value notation, which impacted strongly the development of mathematics in the Ancient Near East. (see article on the “Sexagesimal Place Value Notation”).

## The content of the mathematical texts

Third millennium mathematical texts deal mainly with the evaluation of surfaces. This is the case of the five known early-dynastic tables, which contain a correspondence between lengths of squares or rectangles and their surface measurement. This is also the case of the field texts from Old Akkadian period. Other interesting texts, for example from Šuruppak, shed light on division algorithms used before the invention of sexagesimal place value notation. Ur III mathematical texts are reciprocal tables which provide the earliest known examples of the systematic use of the sexagesimal place value notation.

Most of the known Old Babylonian mathematical cuneiform texts are school exercises. These school texts appeared too simple and repetitive to deserve some interest, and only recently they began to be systematically published. However, they provide valuable information on education in Mesopotamia, an important facet of the social life. Moreover, they illuminate the mathematical methods that were used by learned scholars who wrote advanced mathematical texts.

Old Babylonian advanced mathematical texts were probably written by masters of the scribal schools. Most of them seem to treat concrete subjects (fields, canals, excavations, bricks, walls, buildings, economics…), and use realistic data. However, the mathematical treatment of such subjects can hardly be considered as realistic. The most often, the problems solved, including quadratic and higher degree problems, are of interest for mathematicians, not for practitioners. To illustrate the "supra-utilitarist" (to use Høyrup's expression) nature of some mathematical texts, here is problem 9 from catalogue YBC 4652, where a stone is weighed after removing improbable fractions of it:

`14. I found a stone. Its weight I don’t know. 1/7 I subtracted, then 1/11 I added,`

`15. then 1/13 I subtracted. I weighed: 1 `

*mana*. The initial weight how much?

`16. The initial weight 1 `

*mana* 9 1/2 *gin* 2 1/2 *še*.

The mathematical texts generally include compendia of problems provided with a detailed solution (BM 13901 and BM 85194 are famous examples). These "procedure texts" show a specific and ingenious practice of the mathematical proof (see article on “Reasoning”). In some instances, only the statements are provided on single column tablets (catalogue texts), or on series of multicolumn tablets (series texts). Sometimes, diagrams illustrate the problems. Old Babylonian mathematical *corpus* includes also numerical texts, which show the virtuosity of the ancient calculators (Plimpton 322 and CBS 1215 are fascinating examples).

Mathematical texts of Achaemenid period include metrological tables and procedure texts focused on the calculation of surfaces using different metrological systems. Hellenistic mathematical texts show a strong interest for the very large sexagesimal regular numbers and the calculation of reciprocals. Procedure texts witness the encyclopedic mathematical knowledge of the *kalû* of the temples of Uruk and Babylon. Another facet of mathematical practices in the Hellenistic period can be discovered in astronomical texts, as shown by Mathieu Ossendrijver (*Babylonian Mathematical Astronomy: Procedure Texts*: Springer, 2012). (see article on “Mathematical texts of the Achaemenid and Hellenistic periods”).

Bibliography

CP

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