Abstract
The written culture of the Ancient Near East, whose history covers more than three millennia (from the beginning of the third millennium to the end of the first millennium BCE), underwent profound transformations over the centuries and showed many faces according to the region of the vast territory in which it developed. Yet despite the diversity of contexts in which they worked, the scholars of Mesopotamia and neighboring regions maintained and consistently cultivated a true ‘art of lists’, in the fields of mathematics, lexicography, astrology, astronomy, medicine, law and accounting. The study of the writing techniques particular to lists represents therefore an important issue for the understanding of the intellectual history of the Ancient Near East. In this chapter, I consider extreme cases of list structures, and to do this I have chosen very long lists, most items of which are not semantically autonomous. More specifically, I shall study one of the most abstract and concise lists that have come down to us. It belongs to a series, of which one tablet is kept in the Oriental Institute in Chicago (no. A 24194). The study of this case will allow to set forth some of the writing techniques that were particularly developed in the series. Such a study of the structures of the mathematical texts could benefit other areas in Assyriology.
Translation Theodora Seal
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Notes
- 1.
The categories of mathematical texts referred to here ( procedure texts, catalogue texts, series texts) are those defined by J. Friberg and J. Høyrup; these authors completed the categories previously defined by Neugebauer (Høyrup 2002, p. 9). For more discussion on these classifications, see (Proust 2012.)
- 2.
Neugebauer named them Serientexte in his first publications in German (Neugebauer 1934–1936), then series texts in his following publications. Thureau-Dangin used the name textes de séries (Thureau-Dangin 1938, p. 214).
- 3.
A colophon is a small additional text usually written at the end of the text on the reverse of the tablet, sometimes on the edges, that gives information on the tablet or its context (number of lines or sections, author, date, name of the text, praise to a good, etc.). In the Old Babylonian period, mathematical texts rarely have colophons; when they do, the colophons are brief (giving one or two pieces of information, not more).
- 4.
A detailed discussion on the use of Sumerian logograms in mathematical texts is given in (Høyrup 2002). Further, it might be useful to recall some pieces of information concerning the written languages in the Ancient Near East. The Sumerian language was written and spoken in southern Mesopotamia during the entire third millennium BCE. Later, Sumerian was supplanted by Semitic languages, in particular Akkadian in the Old Babylonian period. Nevertheless, Sumerian continued to be used in the scribal schools and for scholarly activities during a major part of the second millennium, and was maintained within certain erudite circles until the disappearance of cuneiform writing at the beginning of our era. In their great majority, the mathematical texts are written in Akkadian. Akkadian writing is syllabic, therefore the cuneiform signs represent sounds. However, Sumerian logograms are frequently inserted into this phonetic writing. Although they originally represented words of the Sumerian language, they were probably read in Akkadian. With respect to the mathematical series, the connection between writing and language is more complex (see below). Let us end this note with some points concerning Sumerian. Sumerian words are formed of an invariable root, usually monosyllabic, to which grammatical particles are added: suffixes (which, for example, give the cases for the nouns), prefixes and infixes (for the verbs). For example, in the text examined in this article, the root ‘zi’ (to subtract) is found alone or in a conjugated form (‘ba-zi’, I have subtracted).
- 5.
Yale: tablets YBC 4668, YBC 4669, YBC 4673, YBC 4695, YBC 4696, YBC 4697, YBC 4698, YBC 4708, YBC 4709, YBC 4710, YBC 4711, YBC 4712, YBC 4713, YBC 4714, YBC 4715 (Neugebauer 1935, Chap. 7 and Neugebauer 1935–1937) ; Berlin: tablets VAT 7528 et VAT 7537 (Neugebauer 1935, Chap. 7); Louvre: tablets AO 9071 and AO 9071 (Proust 2009); Chicago: tablets A 24194 and A 24195 (Neugebauer and Sachs 1945, texts T and U).
- 6.
Neugebauer (1934–1936).
- 7.
Neugebauer (1934–1936, p. 107).
- 8.
Reign of Ammi-ṣaduqa (1711–1684); Sippar lies north of the Mesopotamian plain. For a more detailed presentation of the different hypotheses concerning the date and provenance, along with the corresponding bibliography, see (Proust 2009).
- 9.
The interested reader will find the list of numerical graphemes and the systems to which they belong on the CDLI site (http://cdli.ucla.edu/).
- 10.
An area of 1(eše3) GAN2(or 6 GAN2) approximately represents 21 600 m². Indeed, 1 GAN2 is equal to 100 sar; a sar is the area of a square of side 1 ninda; a ninda is a unit of length approximately equal to 6 m.
- 11.
This relation can be verified by a very simple calculation: as will be seen below, the length uš corresponds to 30, the width sag to 20, therefore the area corresponds to: 30 × 20 = 600 = 60 × 10, which is written 10 in the cuneiform ‘floating value’ system. For more details on the relations between measures and place value notations, see (Proust 2008).
- 12.
This term is inspired from Neugebauer’s work; Neugebauer designated the content of the long sections by “main problems” and the content of the others by “variants”. Nevertheless, I noted the name of these expressions as “P” ( principal) and “S” ( secondaire) in order to be coherent with the notations I used in my French publications, as (Proust 2009).
- 13.
In section 5, for example, if uš had been omitted, the sentence would end with ‘-ma’; this seems impossible within the syntax of the text. This is quite comprehensible, since it is difficult to imagine a sentence in modern languages that would end with ‘:’. The problem is the same in sections 12, 18 and 39. Let us note that in other series texts the difficulty is circumvented by using the verb ‘sa2’, which means ‘is equal’ (Proust 2009).
- 14.
The first case is attested to in tablet AO 9071 ( Ibid.), the second case in tablet YBC 4712 (Neugebauer 1935, p. 433, n. 12a) and in the present text (as indicated above, section 52 begins at the end of column 2 of the obverse and ends at the beginning of column 3).
- 15.
This double meaning can be compared to that of a word such as ‘book’: it can denote an object such as ‘a library of 300 books’, or a text such as ‘300 copies of the book were printed’.
- 16.
On this subject, see the typology of the Old Babylonian divination texts developed by J.J. Glassner, in particular that of the series from Sippar, which date from the end of Hammurabi’s dynasty (seventeenth century BCE) (Glassner 2009). Let us note that a ‘line’ is sometimes more a theoretical entity than a practical one: when a sentence is long, it can be written on two lines
- 17.
This comparison is also of historical interest. Indeed, careful observation of the colophons suggests that the tradition of the mathematical series is not so different than that of the divination texts studied by J.-J. Glassner, see note above (for more details, see Proust 2009).
- 18.
The word equation is understood here in a very broad sense: a numerical relation between two unknown magnitudes.
- 19.
See the example of the list of food lacking in a house which, in the hands of the person shopping, can be used to carry out the act of giving instructions (Chap. 6, Sect. 6.6). Other examples have been suggested by J. Virbel: ‘a very large number of texts have both the status of an assertion and that of an order (and often also of commitment): the agenda of a meeting, the menu of a restaurant’ (REHSEIS seminar, September 2008; see Chap. 6, Sect. 6.4).
- 20.
It does arise however in other series, in which the statements give sums of lengths and areas, in particular in the twin tablet A 24195 and in one of the Louvre tablets. On this subject, see (Proust 2009).
- 21.
In the cuneiform text, this concerns the juxtaposition of the signs uš and sag ([Image]). I represent this sequence by ‘uš/sag’ and not ‘uš-sag’ such as the usual transliteration norms would require; this, in order to avoid the graphical similarity with the subtraction uš—sag.
- 22.
See for example tablets YBC 4673, VAT 7528, YBC 4698, only to mention the tablets belonging to series (Neugebauer 1935–1937).
- 23.
In the following, the ordinary arguments are denoted by the letters A, B etc. In the particular cases where they represent numbers, they are denoted N, as mentioned above.
- 24.
Let us remark that the complete grammatical form would be A B-ta ba-zi, but the suffix –ta (ablative, translated by ‘from’) never appears in this text and the verbal prefix ba- is often omitted.
- 25.
I have only found this exact form in tablet Str 366. It is found in slightly different forms in some tablets dating from the end of the Old Babylonian period (for example: A a-ra2N tab-ba in tablet BM 85194), and in some tablets of the Schøyen Collection (for example: A a-na N-e tab in Tablets MS 2792, MS 3052, MS 5112– see (Friberg 2007)). Let us note that according to J. Friberg, Tablet MS 5112 could date back to the Kassite period (sixteenth–twelfth century BCE)
- 26.
This expression designates a number whose inverse cannot be written in base 60 with a finite number of digits. A number is regular in a given base if it can be written as the product of divisors of the base. In base 60, this means that its decomposition into prime factors only contains the factors 2, 3 and 5.
- 27.
It is rare for a statement (or a cycle of statements) to begin on one tablet and end on another. Nevertheless, this case is documented (see tablet AO 9072, Proust 2009).
- 28.
For example, the thematic groupings of statements may not correspond to the groupings by tablets ( Ibid).
- 29.
As mentioned above, it is generally believed that the series date back to the end of the Old Babylonian period (seventeenth century BCE).
- 30.
See in this regard the comparative study of a list of solved problems (BM 13901) and a list of statements from a series text (YBC 4714), in which J. Høyrup shows a connection between the problems in terms of the topics (concentric squares) and the resolution methods considered, so far as one can reconstruct them as regards the tablet YBC 4714. He notes however that the variants found in the series are much more sophisticated than those found in the list of solved problems. He suggests that one of the first tablets of the series to which YBC 4714 belongs might contain statements similar to those of tablet BM 13901 (Høyrup 2001, p. 199).
References
Friberg, Jöran. 2007. A remarkable collection of Babylonian mathematical texts. Manuscripts in the Schøyen collection: Cuneiform Texts Vol. I. New York: Springer.
Glassner, Jean-Jacques. 2009. Ecrire des livres à l’époque paléo-babylonienne: le traité d’extispicine. Zeitschrift für Assyriologie und Vorderasiatische Archäologie 99:1–81.
Høyrup, Jens. 2001. The Old Babylonian square texts—BM 13901 and YBC 4714. Retranslation and analysis. In Changing Views on Ancient Near Eastern Mathematics, ed. J. Høyrup and P. Damerow, 155–218. Berlin: Dietrich Reimer Verlag.
Høyrup, Jens. 2002. Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and its Kin. Berlin & Londres: Springer.
Høyrup, Jens. 2006. Artificial language in ancient mesopotamia—a dubious and less dubious case. Journal of Indian Philosophy 34:57–88.
Neugebauer, Otto. 1935. Mathematische Keilschrifttexte I. Berlin: Springer.
Neugebauer, Otto. 1935–1937. Mathematische Keilschrifttexte II–III. Berlin: Springer.
Neugebauer, Otto, and Abraham J. Sachs. 1945. Mathematical Cuneiform texts. American Oriental Studies Vol. 29. New Haven: AOS & ASOR.
Proust, Christine. 2008. Quantifier et calculer: usages des nombres à Nippur. Revue d’histoire des mathematiques 14:143–209.
Proust, Christine. 2009. Deux nouvelles tablettes mathématiques du Louvre: AO 9071 et AO 9072. Zeitschrift für Assyriologie 99:1–67.
Proust, Christine. 2012. Reading colophons from Mesopotamian clay-tablets dealing with mathematics. NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin 20:123–56.
Thureau-Dangin, François. 1938. Textes Mathématiques Babyloniens. Leiden: Ex Oriente Lux.
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Appendices
Appendix 1: Copy of Tablet A 24194 (Fig. 8.1)
Copy Neugebauer and Sachs 1945, pl. 15
Appendix 2: Structure and Distribution of the Information in Tablet A 24194 (Fig. 8.2)
Tree structure in tablet A 24194, obverse, columns 1 and 2. The branches of the tree correspond to the different levels of information: root = level 4; main nodes = level 3; secondary nodes = level 2; extremities = level 1. The column located to the right gives the number and content of the sections that correspond to each path in the tree
Appendix 3: Glossary
This glossary is a list of forms attested in the extract analyzed in this article. The notations are the same as before: N denotes specified numbers and the letters A, B and C denote any type of expression (specified numbers, uš, sag, simple or complex combinations of uš and sag, explicit or implicit). The grammatical suffixes and prefixes frequently omitted in the cuneiform text are given in parentheses.
3.1 Arguments
3.2 Relations
3.3 Operations
- A:
-
Tablet of the Oriental Institute, Chicago
- AO:
-
Tablet from Antiquités Orientales, the Louvre, Paris
- BM:
-
Tablet from British Museum, London
- MS:
-
Manuscript of the Schøyen Collection
- VAT:
-
Tablet of the Vorderasiatisches Museum, Berlin
- YBC:
-
Tablet from Yale Babylonian Collection, New Haven
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Proust, C. (2015). A Tree-Structured List in a Mathematical Series Text from Mesopotamia. In: Chemla, K., Virbel, J. (eds) Texts, Textual Acts and the History of Science. Archimedes, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-319-16444-1_8
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