Cuneiform Digital Library Journal 2009:1 <http://www.cdli.ucla.edu/pubs/cdlj/2009/cdlj2009_001.html> © Cuneiform Digital Library Initiative ISSN 1540-8779 Version: 22 June 2009 Numerical and Metrological Graphemes: From Cuneiform to Transliteration1 Christine Proust CNRS, Paris §1. Introduction §1.3. From a methodological point of view, the paper §1.1. The aim of this paper is two-fold: fi rst, to analyze will for the most part depend on the visual properties of some normative aspects of metrological and numerical the tablets, and will examine closely the way in which notations in mathematical cuneiform texts; second, to the texts are displayed. This kind of analysis potentially examine issues raised by modern conventions of trans- yields a classifi cation of graphemes most similar to that literations. of ancient scribes. In another respect, this paper is based on the general principles and functional classifi cation of §1.2. The argument presented in this paper relies graphemes developed by CDLI collaborators.2 It con- mainly on Old Babylonian school tablets because these tains, moreover, an attempt to import the descriptive sources bear deep traces of normalization processes, system of graphemes used in the fi eld of Mycenaean and they serve as examples that elucidate the principles epigraphy.3 of notations used in mathematical texts. In the Old Babylonian period, metrology and place value notation §1.4. This paper will fi rst present a detailed analysis of were taught in scribal schools in which this knowledge texts used in scribal schools to teach metrological nota- made up the fi rst level of the mathematical curriculum. tions (§§2-3) and place value notation (§4). Problems School tablets provide us with valuable evidence of the raised by the distinction between positional and non- elements that the teachers considered essential. Thus, positional numbers will then be examined (§5). The last they constitute a good source for understanding the section (§6) advances some practical suggestions for a new concepts involved in numeration and metrology greater standardization of the transliteration of math- that emerged at the end of the 3rd millennium BC. ematical texts. §2. School Tablets §2.1. School tablets have been unearthed at almost all 1 I wrote the fi rst version of this paper for a meeting of great sites of the ancient Near East, but the bulk of the the Cuneiform Digital Library Initiative (CDLI) held extant corpus comes from Nippur.4 This city provided in Berlin in May of 2008, particularly to contribute to ongoing discussions about the transliteration of nu- merical and metrological signs used in the mathematical texts. The exchange of views during the meeting and in 2 See the “white paper” posted by S. Tinney (2004), subsequent e-mails were very fruitful, and I am grateful <http://cdl.museum.upenn.edu/doc/ATF/wnm.html>. to Jacob Dahl, Peter Damerow, Steve Tinney, Manfred 3 Bennett 1963; Bennett 1972; Olivier and Godart 1996: Krebernik and Bob Englund, as well as to Madeleine 12. My warmest thanks go to Françoise Rougemont and Fitzgerald and the referees of CDLJ, for their help, clari- Maurizio Del Freo, who provided me with the biblio- fi cations and comments. Abbreviations follow those of graphic references and numerous helpful ideas from the CDLI (<http://cdli.ucla.edu/wiki/doku.php/abbrevia- fi eld of Mycenology. tions_for_assyriology>), adding: MMT Neugebauer and Sachs 1984 4 The numbers of mathematical school tablets found at TMH 8 Proust 2008a some important sites are, for example, the following: ca. Needless to say, any remaining faults in this paper are 900 tablets at Nippur; ca. 150 at Mari; 64 at Ur; and 62 my own. at Kiš. Cuneiform digital Library Journal 2009:1 page 1 of 27 us with one of the few groups of school tablets that §2.3. These data show that the fi rst step of mathemati- permit qualitative as well as statistical analysis. How- cal education focused on the notation of measures. The ever, the striking homogeneity of content in school curriculum indicates quite clearly that metrology was mathematical tablets found in Mesopotamia, as well as not just an integral part of mathematics, but clearly in its neighboring regions, gives to the Nippur docu- an essential component, since metrological texts rep- mentation a relevance beyond the local scale.5 Recent resented approximately half of all Nippur mathemati- studies have allowed the reconstruction of the Nippur cal school texts. Where metrology constituted the fi rst curriculum.6 It has been established that the education part of mathematical education, writing and using place started with a fi rst level, called “elementary” by modern value notation made up the second part. For the scribes, scholars. The young scribes had to memorize huge lists, the memorization of an ordered set of elementary results and then reconstruct them in a given order as a written (reciprocals, multiplications, etc.) was essential for the form, and probably also as an oral recitation. In the fi eld mastery of algorithms for calculation. School tablets are of mathematics, these compositions include a coherent and strongly structured group of texts that • metrological lists (enumeration of an increasing pro- focuses on notation of measures and calculation. Con- gression of measures of capacity, weight, surface, and sequently, the extant corpus of such documents enables length, in that order); us to produce an exhaustive and methodical overview • metrological tables (enumeration of the same items of metrological and numerical cuneiform notations, as as in the metrological lists, but including, in front of well as an analysis of the broader systems into which each item, its correspondence with a number written these notations fi t. It is precisely this ancient presenta- in place value notation); tion made by the scribes themselves that I would like to • numerical tables (tables of reciprocals, multiplications, re-examine in order to gain a better understanding of squares, square roots, and cube roots). some basic principles applied by scribes in mathemati- cal texts. §2.2. After this fi rst level came a more advanced pro- gram dedicated to calculation, namely, algorithms for §3. Metrological Lists the calculation of multiplications, reciprocals, surfaces, §3.1. Introduction and probably also volumes. A rough idea of the propor- §3.1.1. Following the natural progression of the cur- tion of tablets containing these different texts can be riculum, let us begin with metrological lists and ex- 7 gathered from the following distribution : amine the organization of information in these texts. Metrological lists: 187 tablets Metrological lists are documented by a good number of Metrological tables: 161 tablets duplicates (see, for example, the last column of fi gure 1 Numerical tables: 417 tablets below). In the following, I will consider both the com- Calculation exercises: 38 tablets posite text and the individual tablets. The composite text given in §8 is the reconstructed list of items found at least once among the Nippur tablets. Individual tab- 5 In this regard, the comparison between Nippur and lets contain realizations of this “ideal” text. The com- Mari is enlightening. The contents of elementary math- ematical tablets from both sites are highly similar, prov- plete set of metrological lists was preserved at Nippur ing a strong uniformity of the knowledge transmitted. in the so-called “type I” tablets, which include four lists The differences in tablet typology merely indicate local (capacity, weight, surface, length).8 variations in pedagogical methods. 6 See the study of N. Veldhuis on the lexical texts from §3.1.2. The fi rst column of fi gure 1 (below) indicates Nippur (Veldhuis 1997); see also E. Robson (2001) on how the lists appear on the tablets. Though it is some- the tablets found in House F at Nippur, and my own what artifi cial (in particular, I have noted only the be- work (2007) on the complete corpus of Nippur math- ginning for each list), this presentation is a faithful re- ematical texts. The reconstruction of the curriculum is mainly based on the correlation of texts written on “type production of the visual properties of these lists. These II” tablets, as initiated by N. Veldhuis (1997, ch. 2). properties are particularly well illustrated by the tablets HS 249+1805 (=TMH 8, no. 3) and HS 1703, reverse 7 These data derive from tablets excavated in the course of archaeological campaigns funded by the University (=TMH 8, no. 8). of Pennsylvania towards the end of the 19th century (Babylonian Expedition), which provided the bulk of Nippur sources. For other statistical data concerning 8 Examples of metrological lists in Type I tablets: HS 249 mathematical tablets from Nippur, see Robson 2001; + (=TMH 8, no. 3); Ist Ni 3515; Ashm 1931-137 (Rob- Proust 2007: 268-275. son 2004: 33-34). page 2 of 27 Cuneiform Digital Library Journal 2009:1 Cuneiform text Transliteration Sources List of capacities 1(diš) gin2 še Ist Ni 3238 Ist Ni 3279 1 1(diš) /3 gin2 Ist Ni 3772 1 Ist Ni 3976+ 1(diš) /2 gin2 Ist Ni 4750 2 1(diš) /3 gin2 Ist Ni 5293 5 Ist Ni 5339 1(diš) /6 gin2 Ist Ni 10203 2(diš) gin2 etc. 1 2(diš) /3 gin2 1 2(diš) /2 gin2 ... ================= 1 List of weights /2 še ku3-babbar Ist Ni 3742 Ist Ni 3515 1(diš) še Ist Ni 5196 1(diš) 1/2 še 2(diš) še 2(diš) 1/2 še 3(diš) še 4(diš) še 5(diš) še ... ================= 1 List of surfaces/3 sar a-ša3 Ist Ni 5263 Ist Ni 3814 1/ sar 2 Ist Ni 5295 2/3 sar 5/6 sar 1(diš) sar 1(diš) 1/3 sar 1(diš) 1/2 sar 1(diš) 2/3 sar ..