Abstract
Most of the mathematical texts from Late Babylonian Uruk were found in the ‘House of the āšipus ’, altogether with other scholarly texts, including astronomical, astrological, medical, lexical texts, rituals, prayers and others. This chapter provides a close examination of the content of a set of mathematical tablets discovered in a room of the Achaemenid level of this house. These tablets deal mainly with diverse methods to evaluate surfaces. One of the more striking aspects of these methods is the way in which they confront ancient and new metrological systems. Indeed, the metrological systems for lengths, surface and capacity differ in many respects in Old-Babylonian and Late-Babylonian sources. Mathematical texts from the Achaemenid period echo a cross-fertilization between various metrological cultures, the ones resulting from a long transmission of mathematical knowledge from generations to generations during several centuries, the others developed in Mesopotamia in late periods for administrative or juridical purposes. This contribution analyses how different mathematical and metrological cultures were combined by scholars linked to the milieu of the āšipus of Achaemenid Uruk. The conclusion discusses the reasons of the interest shown by these scholars for the art of evaluating surfaces.
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Notes
- 1.
This corpus was recently extended with the discovery and publication of several Late Babylonian metrological tables from Babylon by Steele (2015).
- 2.
Clancier (2009: 404) mentions also that eleven mathematical tablets were found in this house. However, on the one hand, W22656/2 (SpTU 4, 178) is not mathematical (H. Hunger, personal communication April 2017, indicates that this text contains ‘some kind of recipe, maybe magical’). On the other hand, W 22260a (SpTU 1, 101), which was identified as a table of fractions (Rechentabelle (u. a. Namen von Bruchzahlen angebend)) in UVB 26–27: 97, is considered here as mathematical . Moreover, the attributions of tablets to the scholarly archives of the ‘House of the āšipus ’ are not exactly the same here and in Clancier (2009). Clancier supposes that W 23021 and W 22715/2 (SpTU 4, 176 and 177) are to be attributed to the scholarly archives of the family Šangî-Ninurta, but I am not sure of this attribution because the archaeological context is unclear (filling of level IV and level III), and these tablets do not have a colophon . Some of the mathematical texts were published along with the other tablets found in the ‘House of the āšipus ’ in the SpTU volumes (SpTU 1 = Hunger 1976; SpTU 4 = Von Weiher 1993; SpTU 5 = Von Weiher 1998; see for more details the introduction to this volume), and complementary editions focused on mathematical texts were published by Jöran Friberg, Hermann Hunger and Farouk Al-Rawi (BagM 21 = Friberg et al. 1990; BagM 28 = Friberg 1997—see list of abbreviations).
- 3.
This was the methodology developed by Clancier (2009) who restored the lost information by studying the colophons.
- 4.
- 5.
‘The latter tablet [W 23 291] was found in a clay jar together with two other mathematically important Late-Babylonian clay tablets, the large metrological table text W 23 273, and the large many-place table of reciprocals W 23 283. ‘(Friberg 1997: 252). ‘Three of them (W 23273 , W 23283, and W 23291 ) were even found together, stored with other tablets in a clay jar (Von Weiher, personal communication)’ (Friberg 1999: 139).
- 6.
According to Friberg (1997: 252), this tablet was ‘confiscated’.
- 7.
- 8.
Fincke and Ossendrijver (2016: 185, 196) observed that SpTU 4, 176 and SpTU 5, 316 are similar to BM 46550, probably to be dated to the Achaemenid period and possibly originated from Borsippa . However, tablet W 23021 (SpTu 4, 176) exhibits a paleographic difference with the other Achaemenid texts: the shape of the digit 9 () is different from those used in other tablets found in level IV. This feature may point to a later date, which is not impossible since the filling rubble of level IV contained tablets dated to different periods. For example, the astronomical tablet SpTU 4, 168 (W 22925), also found in a filling rubble of level IV, belongs to the so-called ‘Ekur-zakir Library’ (see John Steele’s chapter in this volume).
- 9.
- 10.
Neugebauer and Sachs (1945: 143–5).
- 11.
- 12.
- 13.
- 14.
- 15.
- 16.
The sexagesimal place value notation (SPVN thereafter) was used, mainly in mathematical texts, from the endof the third millennium to the end of the first millennium. As indicated by its name, this notation is sexagesimal (base sixty) and positional (the value of a sign derives from its place in the number). This notation uses 59 digits, written with ones () and tens () repeated as many times as necessary. According to the place value principle, each sign represents sixty times the same sign occupying the previous place (at the right hand). For example, in (1.21), the left hand wedge represent sixty times the right hand wedge. Moreover, the place of the unit in the number is not indicated in cuneiform writing, at least as far as mathematical texts are concerned (other features appear in some Seleucid astronomical texts). For example, the number represents at the same time 1, and 60, and 1/60, and any power of 60; the number represents at the same time 3, and 3 × 60, and 3/60, and 3 multiplied by any power of 60.
- 17.
This order appears for example in tablet HS 249, which contains all of the metrological lists (Proust 2008: 23).
- 18.
Proust (2007).
- 19.
See synthesis and references in Colonna d’Istria (2015).
- 20.
- 21.
See publication information and translation in Appendix 3.1
- 22.
Friberg (1993: 400) labelled the sections as follows: for Friberg, my section A is section G (gods); B is Ln(length in ninda), B′ is Ln′, C is Lc (lengths in cubits), C′ is Lc′, D is L&A (lengths and areas), E is A (areas), E′ is M (mass), F is C (catch line), G is also C, H is L-T a and L-T b (cubits and month names).
- 23.
Steele (2013).
- 24.
The table in section E′ is interpreted as a table of weights by Friberg (1993: 400) and Robson (GKAB). However, this table is similar to the previous one, a table for surfaces (E), with inversion of the order of the signs. This would fit with the general structure of the text, even if the traces of sign ‘ma-[na]’ and ‘gun 2′ at the end of the table argue in favor of Friberg’s interpretation.
- 25.
John Steele (personal communication) notes that: ‘It could be argued that the shadow scheme is not part of the metrological material because it appears after a catchline and the remark “finished”. In fact we seem to have two catchlines, one before and one after the shadow length scheme. Perhaps we should see this tablet as containing a copy of a metrological tablet, complete with the catchline etc., followed by an additional (unrelated?) text and a colophon .’
- 26.
Jens Høyrup suggested that the Late Babylonian metrological tables may result from copies of Aramaic texts written from right to left (personal communication).
- 27.
Transliteration and translation by Hunger (personal communication, May 2016).
- 28.
Note that in line 2 the place where we expect a number, namely 5, before the unit of surface še is blank according to the photo, and not damaged as suggested by Von Weiher’s copy.
- 29.
- 30.
- 31.
John Steele (personal communication) notes: ‘An exception to this rule is found in the first line of the shadow scheme which gives the Akkadian equivalent of kuš3. This adds further support to the idea that the shadow length material was copied onto this tablet from a different source text.’
- 32.
Text 2 contains lists of length, surface and capacity units. The system of length units have the same general structure as their Old Babylonian counterpart, even if some units are added, and the surface units are the same as in Old Babylonian period. However, the capacity system is the one used in Late-Babylonian period. In addition, text 2 contains a table of reciprocals.
- 33.
In other known tables of reciprocals from Achaemenid or Hellenistic period, the entries are (almost) all of the regular numbers beginning with 1, by 2, and sometimes by 3. See Ossendrijver’s chapter, this volume, for more details on Hellenistic reciprocal tables.
- 34.
See publication information on these tablets in the list of sources.
- 35.
‘Hilprecht (1906) place la rédaction de ce texte vers 1350, soit à l’époque Kassite; je ne sais sur quel fondement. L’écriture fait plutôt penser à l’époque néo babylonienne. De plus, l’échelle ici employée pour les mesures de capacité est, comme à l’époque néobabylonienne, 1/30 (de gur) = 6 qa …’ (Thureau-Dangin 1909: 84). Actually, there is no ‘qa’ in this table; Thureau-Dangin grounds the equivalence 1/30 gur = 6 qa on a reconstruction of the calculation of a surface expressed as capacity of grain (the so-called ‘seed-system’—see below).
- 36.
Powell (1999: 483).
- 37.
Sachs (1947: 67).
- 38.
The number 5, which corresponds to 1 gur Old Bablylonian metrology , is based on the equivalence of 60 gur with 1 sar-volume (1 ninda × 1 ninda × 1 kuš), that is about 18 m3. However, the Late-Babylonian gur, equivalent to 180 sila (about 180 L), should corresponds to 3, not to 5. For more details on the relation between volume and capacity in the Old Babylonian metrology , see Powell (1987–1990); for a detailed analysis of these relations in Old Babylonian mathematical texts, see Proust (forthcoming b).
- 39.
Translation by Thureau-Dangin (1909: 85): ‘Tel est le doigt, dont 30 font une coudée, coudée (qui sert à mesurer les superficies évaluées) en quantités de semence ou en cannes carrées’.
- 40.
The unit of surface square -kuš, termed as “small kuš” (kuš3 tur) in problem 19 of text 5, is used mainly in mathematical texts. However, Heather Baker has shown that this “small kuš” is attested in economic texts, for example in a text from Borsippa dated to 517 BCE and in a text from Babylon dated to the same year (Baker 2004, 2011: 310). It is striking that, in this latter text, the unit is termed literally as a “square -kus” (labag-kuš3) (Baker 2011: 310).
- 41.
Powell (1984: 35).
- 42.
In Table 3.9 and in the following, I represent the correspondence between measurement values and SPVN by an arrow, for example ‘1 kuš → 5’ means ‘1 kuš corresponds to 5 in SPVN’, ‘1 kuš → 1’ means ‘1 kuš corresponds to 1 in SPVN’, etc.
- 43.
For checking the calculations, the reader is invited to use MesoCalc, the Mesopotamian Calculator developed by Baptiste Mélès (http://baptiste.meles.free.fr/site/mesocalc.html).
- 44.
The standard surface is obtained by multiplying the length by the width, these magnitudes being represented by sexagesimal place value numbers according to tables of text 1 (see Sect. 3 and Appendix 3.1). The standard surface is termed “sar” surface in text 5, problem 5, obv. col. ii, line 11. Here, the scribe works with the correspondences 1 kuš → 5, so that the standard surface of 1 kuš-side square corresponds to 25 in SPVN.
- 45.
In fact, in the cuneiform text, the order of multiplication is recip(c) × recip(w) × S′ (read from left to right). It is striking that this order does not allow the reader to make sense to the intermediary steps recip(c) × recip(w). Indeed, the reciprocal of the product of the length by the coefficient of the seed (9) does not correspond to any actual magnitude. At the opposite, the calculations performed in the order S′ × recip(c) × recip(w) would have made more sense since the intermediate step S′ × recip(c) represents the standard surface . The fact that a formal calculation was preferred to a calculation whose steps make sense probably reflects the way in which the procedure prescribed in problem 1 was conceptualized in order to be reversed. But as problem 1 is destroyed, we cannot go further in this speculation. For a discussion of such phenomena, see Proust (forthcoming a).
- 46.
Note that in the Hellenistic text from Uruk AO 6484, calculations of volumes revive the old signification of these correspondences, with explicit references to horizontal and vertical lines.
- 47.
Powell (1984: 34), Baker (2011: 312). The system for seed-surfaces based on this coefficient is termed as ‘common seed measure ’ by Friberg (1997: 273). This coefficient is evaluated as 1/3 bariga for 60 × 60 square -kuš by Friberg (1997: 273), and as 36 gi 1 nikkas 1 kuš 15 3/7 šu-si for 1 bariga by Baker (2011: 312) (I use my own conventions for representing the quantities). These evaluations are of course equivalent to the ratio 30 square -kuš per GAR mentioned above.
- 48.
Notice 2 of CBS 8539 (obv. col. iii) provides the coefficient of the seed as follows: “(The rectangle) whose length is 1 hundred kuš and width is 1 hundred kuš has (a surface of) 5 ban 3 sila 3 1/3 GAR 39 of še-numun”. This surface represents 10000 square -kuš for 333 1/3 GAR (in LB capacity metrology ), that is, 30,000 square -kuš for 1000 GAR, that is, 30 square -kuš per GAR.
- 49.
As explained previously, the coefficient 25 comes from the fact that, in the second option, the SPVN corresponding to 1 kuš is divided by 5 (1 kuš corresponds to 1 instead of 5), thus, the surface is divided by 25. In order to use the metrological table for surfaces such as section E of text 1, the right value must be restored by multiplying by 25.
- 50.
Friberg et al. (1990: 508–509).
- 51.
- 52.
Steele (2013).
- 53.
Hunger (1994).
- 54.
- 55.
Among the exceptions is the use of the surface of a trapezoid to model the displacement of a celestial body moving with a variable speed attested in texts dated to the Hellenistic period (Ossendrijver 2016). However, in these texts, the metrological facet of the evaluation of the surface is absent.
- 56.
Baker (2011: 313–317).
- 57.
Powell (1984: 36)
- 58.
- 59.
Steele 2013 identifies this section as a catch line. Hunger reads the last two signs egir-šu 2 (‘comes after it’) and thus recognize also a catch line (personal communication, June 1st 2016).
- 60.
Steele (2013).
- 61.
Transliteration and translation by Hunger (personal communication April 2015).
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Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804. I was helped for transliteration and translation of non-mathematical sections and colophons by Hermann Hunger and I am deeply grateful toward him. The calculations were performed with MesoCalc, the Mesopotamian Calculator developed by Baptiste Mélès (http://baptiste.meles.free.fr/site/mesocalc.html accessed 05/09/2018).
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Appendices
Appendix 3.1: Translation of Text 1 (W 23273 = SpTU 4, 172)
Only the translation , necessary for the understanding of the chapter, is provided here. It is based upon Von Weiher’s copy (SpTU 4, 172), the black and white photograph taken by the excavators, the partial edition (detailed description of the content and structure of the metrological systems) published in Friberg (1993), and the edition of section H by Steele (2013). This translation benefited from the kind collaboration of Hermann Hunger. A complete edition is published in Friberg and Al-Rawi (2017). The conventions adopted for translating numbers and measurement values are detailed in Appendix 3.4.
Section A. table of gods
Obverse i
[1] | Anum |
[2] | Enlil |
[3] | Ea |
[4] | Sîn |
[5] | […] |
[6] | Enki |
[7] | Its Seven Gods |
8 | Igigi |
9 | Annunaki |
10 | Bel |
20 | Šamaš |
30 | Sîn |
40 | Ea |
50 | Enlil |
Section B. table for lengths, ‘if 5 is your ammatum ’ (1 kuš corresponds to 5)
[10] | 1 šu-si |
20 | 2 šu-si |
30 | 3 šu-si |
40 | 4 šu-si |
50 | 5 šu-si |
1 | 6 šu-si |
1.10 | 7 šu-si |
1.20 | 8 šu-si |
1.30 | 9 šu-si |
1.40 | 1/3 kuš |
2.30 | 1/2 kuš |
3.20 | 2/3 kuš |
5 | 1 kuš |
6.40 | 1 1/3 kuš |
7.30 | 1 1/2 kuš |
8.20 | 1 2/3 kuš |
10 | 2 kuš |
15 | 3 kuš |
20 | 4 kuš |
25 | 5 kuš |
30 | 1/2 ninda |
35 | 1/2 ninda 1 kuš |
40 | 1/2 ninda 2 kuš |
45 | 1/2 ninda 3 kuš |
50 | 1/2 ninda 4 kuš |
55 | 1/2 ninda 5 kuš |
1 | 1 ninda |
1.30 | 1 1/2 ninda |
2 | 2 ninda |
2.30 | 2 1/2 ninda |
3 | 3 ninda |
3.30 | 3 1/2 ninda |
4 | 4 ninda |
4.30 | 4 1/2 ninda |
5 | 5 ninda |
5.30 | 5 1/2 ninda |
6 | 6 ninda |
6.30 | 6 1/2 ninda |
7 | 7 ninda |
7.30 | 7 1/2 ninda |
8 | 8 ninda |
8.30 | 8 1/2 ninda |
9 | 9 ninda |
9.30 | 9 1/2 ninda |
Obverse ii
10 | [10 ninda] |
15 | [15 ninda] |
20 | [20 ninda] |
25 | [25 ninda] |
30 | [30 ninda] |
35 | [35] ninda |
40 | [40] ninda |
45 | [45] ninda |
50 | [50] ninda |
55 | [5(u)]5 ninda |
1 | 1 UŠ |
1.10 | 1 × 60 + 10 ninda |
1.20 | 1 × 60 + 20 ninda |
1.30 | 1 × 60 + 30 ninda |
1.40 | 1 × 60 + 40 ninda |
1.50 | 1 × 60 + 50 ninda |
2 | 2 UŠ |
3 | 3 UŠ |
4 | 4 UŠ |
5 | 5 UŠ |
6 | 6 UŠ |
7 | 7 UŠ |
8 | 8 UŠ |
9 | 9 UŠ |
10 | 10 UŠ |
11 | 11 UŠ |
12 | 12 UŠ |
13 | 13 UŠ |
14 | 14 UŠ |
15 | 15 UŠ |
16 | 16 UŠ |
17 | 17 UŠ |
18 | 18 UŠ |
19 | 19 UŠ |
20 | 1/3 danna |
25 | 1/3 danna 5 UŠ |
30 | 1 danna |
35 | 1 danna 5 UŠ |
40 | 2 danna |
45 | 1 1/2 danna |
50 | 1 2/3 danna |
55 | 1 2/3 danna 5 UŠ |
1 | 2 danna |
1.30 | 3 danna |
2 | 4 danna |
2.30 | 5 danna |
3 | 6 danna |
3.30 | 7 danna |
4 | 8 danna |
4.30 | 9 danna |
5 | 10 danna |
5.30 | 11 danna |
6 | 12 danna |
6.30 | 13 danna |
7 | 14 danna |
7.30 | 15 danna |
8 | 16 danna |
8.30 | 17 danna |
9 | 18 danna |
9.30 | 19 danna |
10 | 20 danna |
15 | 30 danna |
20 | 40 danna |
25 | 50 danna |
30 | 1 sixty danna |
Obverse iii
35 | 1 × 60 + 10 danna |
40 | 1 × 60 + 20 danna |
45 | 1 × 60 + 30 danna |
50 | 1 hundred danna |
Section B′. similar to section B, with inversion of the order of the signs
šu-si | 1 | 10 |
šu-si | 2 | 20 |
[šu-si | 3] | 30 |
šu-si | 4 | 40 |
šu-si | 5 | 50 |
šu-si | 6 | 1 |
šu-si | 7 | 1.10 |
šu-si | 8 | 1.20 |
šu-si | 9 | 1.30 |
kuš | 1/3 | 1.40 |
kuš | 1/2 | 2.30 |
kuš | 2/3 | 3.20 |
kuš | 1 | 5 |
kuš | 1 1/3 | 6.40 |
kuš | 1 1/2 | [7.30] |
kuš | 1 [2/3 | 8.20] |
kuš | [2 | 10] |
[kuš | 3 | 15] |
[kuš | 4] | 20 |
[kuš] | 5 | 25 |
ninda | 1/2 | 30 |
ninda | 1/2 ninda 1 kuš | 35 |
ninda | 1/2 ninda 2 kuš | 40 |
ninda | 1/2 ninda 3 kuš | 45 |
ninda | 1/2 ninda 4 kuš | 50 |
ninda | 1/2 ninda 5 kuš | 55 |
ninda | 1 | 1 |
ninda | 1 1/2 | 1.[30] |
ninda | 2 | [2] |
ninda | [2 1/2] | 2.30 |
ninda | [3] | 3 |
ninda | 3 1/2 | 3.30 |
ninda | 4 | 4 |
ninda | 4 1/2 | 4.30 |
ninda | 5 | 5 |
ninda | 5 1/2 | 5.30 |
ninda | 6 | 6 |
ninda | 6 1/2 | 6.30 |
ninda | 7 | 7 |
ninda | 7 1/2 | 7.30 |
ninda | 8 | 8 |
ninda | 8 1/2 | 8.30 |
ninda | 9 | 9 |
ninda | 9 1/2 | 9.30 |
ninda | 10 | 10 |
ninda | 15 | 15 |
ninda | 20 | 20 |
ninda | 25 | 25 |
[ninda] | 30 | 30 |
[ninda] | 35 | 35 |
[ninda] | 40 | 40 |
[ninda] | 45 | 45 |
[ninda] | 50 | 50 |
[ninda] | 5(u)5 | 55 |
UŠ | 1 | 1 |
ninda | 1 × 60 + 10 | 1.10 |
ninda | 1 × 60 + 20 | 1.20 |
ninda | 1 × 60 + 30 | 1.30 |
ninda | 1 × 60 + 40 | 1.40 |
Obverse iv
ninda | 1 × 60 + 50 | 1.50 |
UŠ | 2 | 2 |
UŠ | 3 | 3 |
UŠ | 4 | 4 |
UŠ | 5 | 5 |
UŠ | 6 | 6 |
UŠ | 7 | 7 |
UŠ | 8 | 8 |
UŠ | 9 | 9 |
UŠ | 10 | 10 |
UŠ | 11 | 11 |
UŠ | 12 | 12 |
UŠ | 13 | 13 |
UŠ | 14 | 14 |
UŠ | 15 | 15 |
UŠ | 16 | 16 |
UŠ | 17 | 17 |
UŠ | 18 | 18 |
[UŠ | 19 | 19] |
[danna | 2/3 | 20] |
[danna | 1 | 30] |
Section C. table for lengths, ‘if 1 is your ammatum ’ (1 kuš corresponds to 1)
2 | [10 šu-si]a | |
4 | [20] šu-si | |
6 | 30 šu-si | |
8 | 40 šu-si | |
10 | 50 šu-si | |
12 | [1 sixty šu-si] | |
14 | [1 × 60 + 10 šu-si] | |
[16 | 1 × 60 + 20 šu]-si | |
18 | 1 × 60 + 30 šu-si | |
20 | 1 × 60 + 40 (=) 1/3 kuš | |
30 | 2 × 60 + 30 (=) 1/2 kuš | |
40 | 3 × 60 + 20 (=) 2/3 kuš | |
1 | 5 | 1 kuš |
1.20 | 6.40 | 1 1/3 kuš |
1.30 | 7.30 | 1 1/2 kuš |
1.40 | 8.20 | 1 2/3 kuš |
2 | 10 | <2 > kuš |
3 | 15 | <3 > kuš |
4 | 20 | <4 > kuš |
5 | 25 | <5 > kuš |
6 | 30 | 1/2 ninda |
7 | 35 | <7 > kuš |
8 | 40 | <8 > kuš |
9 | 45 | <9 > kuš |
10 | 50 | <1(u) > kuš |
11 | 55 | <11 > kuš |
12 | 1 ninda | |
18 | 1 1/2 ninda | |
24 | 2 ninda | |
30 | 2 1/2 ninda | |
36 | 3 ninda | |
42 | 3 1/2 ninda | |
48 | 4 ninda | |
54 | 4 1/2 ninda | |
1 | 5 ninda | |
1.6 | 5 1/2 ninda | |
1.12 | 6 ninda | |
1.18 | 6 1/2 ninda | |
1.24 | 7 ninda | |
1.30 | 7 1/2 ninda | |
1.36 | 8 ninda | |
Obverse v
1.42 | 8 1/2 ninda |
1.48 | 9 ninda |
1.54 | 9 1/2 ninda |
2 | 10 ninda |
3 | 15 ninda |
4 | 20 ninda |
5 | 25 ninda |
6 | 30 ninda |
7 | 35 ninda |
8 | 40 ninda |
9 | 45 ninda |
10 | 50 ninda |
11 | 55 ninda |
12 | 1 [UŠ] |
Section C′. similar to section C, with inversion of the order of the signs
šu-si | [10 | 2] |
šu-si | 20 | 4] |
šu-si | 30 | 6] |
[šu-si | 40 | 8] |
[šu-si | 50 | 10] |
[šu-si | 1 šu | 12] |
[šu-si | 1 × 60 + 10 | 1]4 |
[šu-si | 1 × 60 + 20 | 1]6 |
šu-si | [1 × 60 + 30 | 1]8 |
kuš | [1 × 60 + 40 1/3] | 20 |
[kuš | 2 × 60 + 30 1/2] | 30 |
[kuš | 3 × 60 + 20 2/3] | 40 |
[kuš | 1 | 1] |
[kuš | 1 1/3 | 1].20 |
[kuš | 1 1/2] | 1.30 |
[kuš | 1 2/3] | 1.40 |
kuš | 2 | 2 |
kuš | 3 | 3 |
kuš | 4 | 4 |
kuš | 5 | 5 |
ninda | 1/2 | 6 |
ninda | 1/2 1 kuš | 7 |
ninda | 1/2 2 kuš | 8 |
ninda | 1/2 3 kuš | 9 |
ninda | 1/2 4 kuš | 10 |
ninda | 1/2 5 kuš | 11 |
ninda | 1 | 12 |
ninda | 1 1/2 | 18 |
ninda | 2 | 24 |
ninda | 2 1/2 | 30 |
ninda | 3 | 36 |
ninda | 3 1/2 | 42 |
ninda | 4 | 48 |
ninda | 4 1/2 | 54 |
ninda | 5 | 1 |
ninda | 5 1/2 | 1.6 |
ninda | 6 | 1.12 |
ninda | 6 1/2 | 1.18 |
ninda | 7 | 1.24 |
ninda | 7 1/2 | 1.30 |
ninda | 8 | 1.36 |
ninda | 8 1/2 | 1.42 |
ninda | 9 | 1.48 |
ninda | 9 1/2 | 1.54 |
ninda | 10 | 2a |
ninda | 15 | 3 |
ninda | 20 | 4 |
ninda | 25 | 5 |
ninda | 30 | 6 |
ninda | 35 | 7 |
ninda | 40 | 8 |
ninda | 45 | 9 |
ninda | 50 | 10 |
ninda | 55 | 11 |
UŠ | 1 | 12 |
Section D. orders of magnitudes of squares
Reverse i
Kingship, destiny
šu-si to še
1 kuš to [x] gin
gi to [x] sar
10 ninda to 1 iku GAN
1 UŠ to buru GAN
1 danna to 1 šar GAN
From 6 UŠ to 1 šar [GAN]
You do […]
Their multiplication (will be) correct
Their calculation will not be chopped (?).
Section E. table for ‘sar-surfaces’
10 | 1/2 še |
20 | 1 še |
30 | 1 1/2 še |
40 | 2 še |
50 | 2 1/2 še |
1 | 3 še |
1.20 | 4 še |
1.40 | 5 še |
2 | 6 še |
2.20 | 7 še |
2.30 | 7 1/2 še |
2.40 | 8 še |
3 | 9 še |
3.20 | 10 še |
3.40 | 11 še |
4 | 12 še |
4.20 | 13 še |
4.40 | 14 še |
5 | 15 še |
5.20 | 16 še |
5.40 | 17 še |
6 | 18 še |
6.20 | 19 še |
6.40 | 20 še |
7 | 21 še |
7.20 | 22 še |
[7].30 | 22 1/2 še 1/8 <gin> |
[7].40 | 23 še |
8 | 24 še |
8.20 | 25 še |
8.40 | 26 še |
9 | 27 še |
9.20 | 28 še |
9.40 | 29 še |
10 | 1/6 <gin> |
11.40 | <1/6> gin 5 še |
13.20 | <1/6> gin 10 še |
15 | <1/4> gin |
16.40 | <1/4> gin 5 še |
18.20 | <1/4> gin 10 še |
20 | 1/3 <gin> še |
30 | 1/2 <gin> še |
40 | 2/3 <gin> še |
50 | 5/6 <gin> še |
1 | 1 gin |
1.10 | 1 gin 1/6 |
1.15 | 1 gin 1/4 |
1.20 | 1 gin 1/3 še |
1.30 | 1 gin 1/2 še |
1.40 | 1 gin 2/3 še |
1.50 | 1 gin 5/6 še |
2 | 2 gin |
3 | 3 gin |
4 | 4 gin |
5 | 5 gin |
6 | 6 gin |
7 | 7 gin |
Reverse ii
8 | 8 gin |
9 | 9 gin |
10 | 10 gin |
11 | 11 gin |
12 | 12 gin |
13 | 13 gin |
14 | 14 gin |
15 | 1 gin 1/4 sar |
16 | 16 gin |
17 | 17 gin |
18 | 18 gin |
19 | 19 gin |
20 | 1/3 sar |
30 | 1/2 sar |
40 | 2/3 sar |
50 | 5/6 sar |
1 | 1 sar |
1.10 | 1 sar 10 gin |
1.15 | 1 sar gin 1/4 sar |
1.20 | 1 sar 1/3 sar |
1.30 | 1 sar 1/2 sar |
1.40 | 1 sar 2/3 sar |
1.50 | 1 sar 5/6 sar |
2 | 2 sar |
3 | 3 sar |
4 | 4 sar |
5 | 5 sar |
6 | 6 sar |
7 | 7 sar |
8 | 8 sar |
9 | 9 sar |
10 | 10 sar |
11 | 11 sar |
12 | 12 sar |
13 | 13 sar |
14 | 14 sar |
15 | 15 sar |
16 | 16 sar |
17 | 17 sar |
18 | 18 sar |
19 | 19 sar |
20 | 20 sar |
30 | 30 sar |
40 | 40 sar |
50 | 1/2 iku GAN |
1 | 1 × 60 sar* |
1.10 | 1 × 60 + 10 sar |
1.20 | 1 × 60 + 20 sar |
1.30 | 1 × 60 + 30 sar |
1.40 | 1 iku GAN |
2.30 | 1 1/2 iku GAN |
3.20 | 2 iku GAN |
4.10 | 2 1/2 iku GAN |
5 | 3 iku GAN |
5.50 | 3 1/2 iku GAN |
6.40 | 4 iku GAN |
7.30 | 4 1/2 iku GAN |
8.20 | 5 iku GAN |
9.10 | 5 1/2 iku GAN |
10 | 1 eše GAN |
11.40 | 1 eše 1 iku GAN |
13.20 | 1 eše 2 iku GAN |
15 | 1 eše 3 iku GAN |
16.40 | 1 eše 4 iku GAN |
18.20 | 1 eše 5 iku GAN |
20 | 2 eše GAN |
Reverse iii
21.40 | 2 eše [1] GAN |
23.20 | 2 eše 2 iku GAN |
25 | 2 eše 3 iku GAN |
26.40 | 2 eše 4 iku GAN |
28.20 | 2 eše 5 iku GAN |
30 | 1 bur GAN |
40 | 1 bur 1 eše GAN |
50 | 1 bur 2 eše GAN |
1 | 2 bur GAN |
1.30 | 3 bur GAN |
2 | 4 bur GAN |
2.30 | 5 bur GAN |
3 | 6 bur GAN |
3.30 | 7 bur GAN |
4 | 8 bur GAN |
4.30 | 9 bur GAN |
5 | 1 bur’u GAN |
5.30 | 1 bur’u 1 bur GAN |
6 | 1 bur’u 2 bur GAN |
6.30 | 1 bur’u 3 bur GAN |
7 | 1 bur’u 4 bur GAN |
7.30 | 1 bur’u 5 bur GAN |
8 | 1 bur’u 6 bur GAN |
8.30 | 1 bur’u 7 bur GAN |
9 | 1 bur’u 8 bur GAN |
9.30 | 1 bur’u 9 bur GAN |
10 | 2 bur’u GAN |
15 | 3 bur’u GAN |
20 | 4 bur’u GAN |
25 | 5 bur’u GAN |
30 | 1 šar GAN |
35 | 1 šar 1 bur’u GAN |
40 | 1 šar 2 bur’u GAN |
45 | 1 šar 3 bur’u GAN |
50 | 1 šar 4 bur’u GAN |
55 | 1 šar 5 bur’u GAN |
1 | 2 šar GAN |
1.30 | 3 šar GAN |
2 | 4 šar GAN |
2.30 | 5 šar GAN |
3 | 6 šar GAN |
3.30 | 7 šar GAN |
4 | 8 šar GAN |
4.30 | 9 šar GAN |
5 | 1 šar’u GAN |
5.30 | 1 šar’u 1 šar GAN |
6 | 1 šar’u 2 šar GAN |
6.30 | 1 šar’u 3 šar GAN |
7 | 1 šar’u 4 šar GAN |
7.30 | 1 šar’u 5 šar GAN |
8 | 1 šar’u 6 šar GAN |
8.30 | 1 šar’u 7 šar GAN |
9 | 1 šar’u 8 šar GAN |
9.30 | 1 šar’u 9 šar GAN |
Section E′. similar to section E, with inversion of the order of the signs and an additional section for weights (last two lines), or table of weights
še | 1/2 | 10 |
še | 1 | 20 |
še | 1 1/2 | 30 |
še | 2 | 40 |
še | 2 1/2 | 50 |
še | 3 | 1 |
Reverse iv
še | 4 | 1.20 |
še | 5 | 1.40 |
še | 6 | 2 |
še | 7 | 2.20 |
še | 7 1/2 | 2.30 |
še | 8 | 2.40 |
še | 9 | 3 |
še | 10 | 3.20 |
še | 11 | 3.40 |
še | 12 | 4 |
še | 13 | 4.20 |
še | 14 | 4.40 |
še | 15 | 5 |
še | 16 | 5.20 |
še | 17 | 5.40 |
še | 18 | 6 |
še | 19 | 6.20 |
še | 20 | 6.40 |
še | 21 | 7 |
še | 22 | 7.20 |
še | 22 1/2 = 1/8 (gin) | [7].30 |
še | 23 | [7].40 |
še | 24 | 8 |
še | 25 | 8.20 |
še | 26 | 8.40 |
še | [27 | 9] |
še | [28 | 9.20] |
še | [29 | 9.40] |
še | [30 = 1/6 gin | 10] |
še | 4[5 = 1/4 gin | 15] |
še | [60 = 1/3 gin | 20] |
še | 60 + [30 = 1/2 gin | 30 |
še | 2 × 60 = 2/3 gin | [40] |
še | 2 × 60 + 30 [= 5/6 gin | 50] |
še | 3 × 60 = 1 gin | [1] |
še | 3 × 3600 = 1 ma-[na | 1] |
še | 3 × 60 × 3600 = 1 gun | [1] |
Section F. catch Line
1 gin grain comes after itFootnote 59
Section G
Finished
Section H. shadow scheme Footnote 60
-
1 ammatum month IV
-
[…] kuš month V and month III the same
-
[…kuš] month VI and month II the same
-
[…kuš month VII and] month I the same
-
[…kuš month VII and month XII] the same
-
[…]
-
[…]
-
Month V 15 the shadow is delayed
-
Month VI 30 the shadow is delayed
-
Month VII 45 the shadow is delayed
-
Month VIII 1 the shadow is delayed
-
Month IX 1.15 the shadow is delayed
-
Month X 1.30 the shadow is delayed
-
I 1.12 shadow 1.40 danna a day after it
-
Colophon Footnote 61According to a tablet, original of
-
Babylon , Rīmūt-Ani,
-
[son of] Šamaš-iddin , descendant of Šangi-Ninurta,
-
[wrote and] checked it.
Appendix 3.2: Extracts of Text 4 (W 23291 = SpTU 4, 175)
The following transliterations and translations rely on the copy by Von Weiher (1993) and the edition of the text by Friberg (1997). I did not have access to either the photograph or to the tablet itself, which is held at the Baghdad Museum.
Only the problems discussed in Sect. 3.4.1 are provided here. Some explanations on problem 3, which are not detailed in Sect. 3.4.1, are added.
Problem 2
Obv. i
2′. | [1 me kuš3 sag uš en] ḫe2 KURsic (gid2!)-ma ḫe2 1(aš) gur še-numun |
3′. | [mu nu-zu-u2 igi-u2] ša sag a-ša3 il2-ma |
4′. | [u3 a-ra2 igi-u2] ⌈ša⌉ igi-gub-be2-e še-numun du-ak |
5′. | ⌈u3⌉ še-numun ša e-kaa du-ma sag igi-mar |
6′. | [šum]-ma 5 am-mat-ka 8.20 1 me kuš3 igi 8.20 7.12 |
7′. | [7.12] a-ra2 1.12sic (1.15!) du-ma 9 9 a-ra2 5 du-ma 45 |
8′. | ⌈45⌉ a-na šid-du a-ša3-ka ta-šak-kan |
9′. | šum-ma 1 am-mat-ka 1.40 1 me kuš3 igi 1.40 36 |
10′. | 36 a-ra2 3 du-ma 1.48 1.⌈48⌉ a-ra2 5 du-ma 9 |
11′. | ⌈5⌉ meb 40 ⌈kuš3⌉ a-⌈na⌉ šid-du ta-šak-kan |
Translation
2′. | [1 hundred kuš is the width. How much is the length] such as (the surface) is 1 gur of seed? |
3′. | [As you do not know: the reciprocal] of the width to the surface you raise |
4′. | [and you multiply by the reciprocal] of the coefficient of the seed. |
5′. | And by the seed, what was said to you, multiply: the length you see. |
6′. | If 5 is your ammatum , 8.20 is 1 hundred kuš. The reciprocal of 8.20 is 7.12. |
7′. | [7.12] by 1.15! multiply: it is 9. 9 by 5 multiply: it is 45. |
8′. | 45 as the length of your surface you set. |
9′. | If 1 is your ammatum , 1.40 is 1 hundred kuš. The reciprocal of 1.40 is 36. |
10′. | 36 by 3 multiply: it is 1.48. 1.48 by 5 multiply, it is 9. |
11′. | 540 (corresponding to the width in) kuš, (this) length you put down. |
Explanations: see Sect. 3.4.1
Problem 3
Obverse i
12′. | a-ša3 en am3 lu-[maḫ]-ḫir-ma ḫe2 1(aš) gur 2(ban2) še-numun |
13′. | mu nu-zu-u2 še-numun ša e-kaa e (?)14’⌈ša⌉ igi-gub-be2-e še-⌈numun⌉ du-ak x il ?2 -ma ši-id-du⌉ |
15′. | šum-ma 5 am-mat-ka 5.20 a-ra2 1.15b du-⌈ma⌉ 6.⌈40⌉-e16’20-am3 ti-qe2 20 ninda-am3 tu-⌈maḫ⌉-ḫar |
17′. | šum-ma 1 am-mat-ka 5.20 a-ra2 3 du-ma 16-e |
18′. | ⌈4⌉-am3 ti-qe2 2 me 40 kuš3-am3 tu-⌈maḫ⌉-ḫar |
Translation
12′. | A surface. How much is each (side) such as I make it square and the seed is 1 gur 2 ban? |
13′. | As you do not know: the seed, what I said to you, |
14′. | by the coefficient of the seed multiply. […] the side. |
15′. | If 5 is your ammatum , 5.20 by 1.15 multiply: it is 6.40. |
16′. | 20 each (side) you take. 20 ninda each (side) is the square. |
17′. | If 1 is your ammatum , 5.20 by 3 multiply: it is 16. |
18′. | 4 each (side) is what you take. 2 hundred 40 kuš each (side) is the square. |
Problem 3 states that the surface of a square field is 1 gur 2 ban. The side of the square is asked for. The problems has the same structure as problem 2: statement (line 12′), conventional formula which opens the procedure (line 13′), general procedure (lines 13′–14′), specific procedures with numerical values. As in problem 2, two options are offered to solve the problem: 1 ammatum (Sum. kuš3) corresponds to 5 and the coefficient of the seed is 48 (lines 15′–16′); 1 ammatum corresponds to 1 and the coefficient of the seed is 20 (lines 17′–18′).
Here is a synthetic explanation of the specific procedures with numerical values, according to the two options.
First option: ‘If 5 is your ammatum ’ (1 kuš corresponds to 5)
In this case, the coefficient of the seed is 48, reciprocal 1.15.
-
1 gur 2 ban corresponds to 5.20 (according to a metrological table for capacities similar to the Nippur metrological text CBS 8539 , section I, rev. ii)
-
5.20 × 1.15 produces 6.40
-
6.40 is the square of 20.
The side 20 corresponds to 20 ninda according to a metrological table for lengths such as Text 1, section B, obv. ii (broken but easy to restore) or section B′, obv. iii, and a control of the orders of magnitude.
Second option: ‘If 1 is your ammatum ’ (1 kuš corresponds to 1)
In this case, the coefficient of the seed is 20, reciprocal 3.
-
1 gur 2 ban → 5.20
-
5.20 × 3 produces 16
-
16 is the square of 4.
The side 4 corresponds to 20 ninda (240 kuš) according to a metrological table for lengths such as Text 1, section C or C′, obv. v, and a control of the orders of magnitude. Interestingly, in the second option, the result is expressed as a number of kuš in decimal system in a Late Babylonian fashion, while in the first option, it is expressed as a number of ninda in a traditional fashion.
Appendix 3.3: Extracts of Text 5 (W 23291x = BagM 21, 554–557)
The following transliteration and translation rely on the copy by Von Weiher (1993) and the edition of the text by Friberg et al. (1990). I did not have access to either the photograph nor to the tablet itself.
Only the problems discussed in Sect. 3.4.2 are provided here. I added some comments on problems 6–7 and 19–20 insofar as the explanations are not detailed in Sect. 3.4.2.
Problem 4
Obverse ii
8. | gi-meš ša2 gi 1 ninda uš gi 1 ninda sag 1 sar šum-⌈ma⌉ |
9. | 5 am-mat-ka mi-ḫi-il-tu4 a-ra2 ki-min u3 a-ra2 |
10. | 25 du-ak a-ra2 2.24-am3 |
Translation
8. | Reeds which the length is a reed of 1 ninda and the width is a reed of 1 ninda (have a surface of) 1 sar. If |
9. | 5 is your ammatum , the line times the same (line) and (if 1 is your ammatum , the line times the same line), by |
10. | 25 you multiply. (Reciprocally) times 2.24, each side (i.e. the square root) (you take). |
Explanations: see Sect. 3.4.2.
Problem 5
Obverse ii
11. | ⌈ṣu⌉-up-pan uš u3 ṣu-up-pan sag en sar-me šum-ma |
12. | ⌈5⌉ am-mat-ka 5 ṣu-up-pan 5 a-ra2 5 du-ma 25 25 sar sar |
13. | ⌈šum⌉-ma 1 am-mat-ka 1 ṣu-up-pan 1 a-ra2 1 du-lak-ma |
14. | ⌈1⌉ a-ra2 25 du-ma 25 |
Translation
11. | A ṣuppān the length, a ṣuppān the width. How much is the sar-(surface)? If |
12. | 5 is your ammatum , 5 is the ṣuppān. 5 by 5 multiply: it is 25. The sar-(surface) is 25 sar. |
13. | If 1 is your ammatum , 1 is the ṣuppān. 1 by 1 multiply: it is |
14. | 1. By 25 multiply: it is 25. |
Explanations: see Sect. 3.4.2
Problem 6
Obverse ii
15. | ⌈e2⌉ 25 sar-meš ur-a ḫe2-en šum-ma 5 am-mat-ka |
16. | ⌈am3 25 ti-qe2 šum-ma 1 am-mat-ka 25 a-na |
17. | ⌈2⌉.24 du-ma 1-e am3 ti-qe2 ṣu-up-pan ur-⌈a⌉ |
Translation
15. | A house of 25 sar. How much is the equal-side? If 5 is your ammatum , |
16. | each side (the square root) of 25 you take. If 1 is your ammatum , 25 by |
17. | ⌈2⌉.24 you multiply, it is 1. Each side (the square root of 1) you take. A ṣuppān is the equal-side. |
This problem is the reciprocal of the previous one: the surface of a square is given, find its side. The square has the same dimensions as in problem 5. The procedure is applied in the quantities specified in the statement. As usual, the measurement values are transformed into SPVN, and for that, as in the other problems of texts 3 and 4, two options are offered.
First option: ‘If 5 is your ammatum ’ (1 kuš corresponds to 5)
25 sar corresponds to 25 according to the metrological table for surfaces (see for example Text 1, section E, rev. ii). The square root of 25 is 5, the side. The number 5 is transformed into measurement of length (1 ṣuppān) only at the end of the problem (see below).
Second option: ‘If 1 is your ammatum ’ (1 kuš corresponds to 1)
The coefficient 2.24 (reciprocal of 25) is to be applied to the surface 25, as stated in problem 4.
-
25 × 2.24 produces 1
-
The square root of 1 is 1, the side is 1.
The last step, not detailed in the text, is the transformation of 1 into measurement value. 1 corresponds to 5 ninda according to the metrological table for lengths in the second option (see for example Text 1, table C), or, equivalently, to 1 ṣuppān. The equivalency between 5 ninda (that is, 10 gi) and 1 ṣuppān is stated in Text 2 (W 23281 = SpTU 173), obv. i, line 13: ‘10 gi are 1 ṣuppān’. This observation reveals a thread more that binds together the texts of room 4 . Here again, to control the order of magnitude, the author or reader of the text may have been guided by a notice such as section D of Text 1, line 4, which indicates that the surface of a square of some gi side has a surface of some sar, and that, by reverse reading, that a square of some some sar surface has a side of some gi.
Problem 7
Obverse ii
18. | [sag] 4 uš en ḫe2-gid2-da ḫe2 20 sar sar šum-ma [5] |
19. | [am]-mat-ka igi 4-gal2-la 15 15 a-ra2 20 du-[ma 5] |
20. | [1 ṣu]-up-pan gid ad2 šum-ma 1 am-mat-ka igi ⌈48⌉-[gal2-la] 1.15 |
21. | 1.15 a-ra2 2.24 du-ma 3 3 a-ra2 [20 du-ma 1] |
Translation
18. | [The width is] 4. How much is the long side so that the sar-surface is 20 sar? If [5]. |
19. | is your ammatum , (4 is the width). The reciprocal of 4 is 15. 15 by 20 multiply: it is [5]. |
20. | [1] ṣuppān is the long side. If 1 is your ammatum , (48 is the width). The reciprocal of 48 is 1.15. |
21. | 1.15 by 2.24 multiply: it is 3. 3 by [20 multiply: it is 1]. |
This problem is a variant of the previous ones: the surface and the width of a rectangle is given, find its length (the ‘long side’). As usual, the measurement values are transformed into SPVN. Here again, two options are offered.
First option: ‘If 5 is your ammatum ’ (1 kuš corresponds to 5)
-
The width is given directly in SPVN, 4.
-
20 sar corresponds to 20 according to the metrological table for surfaces (see for example Text 1, section E, rev. ii).
-
The length is obtained by dividing the surface (20) by the width (4), that is, multiplying 20 by 15, the reciprocal of 4:
20 × 15 produces 5
-
As in the previous problem, the length corresponding to 5 is 1 ṣuppān, using the data of texts 1 and 2 and the control of the order of magnitude according to section D of Text 1 (see details above).
Second option: “If 1 is your ammatum ” (1 kuš corresponds to 1)
-
The width 4 ninda correspond to 48 according to the metrological table for lengths with the correspondence 1 kuš → 1 (see Text 1, section C, obv. iv).
-
The coefficient 2.24 (reciprocal of 25) is to be applied to the surface 20, as stated in problem 4.
-
The length is then obtained by dividing the surface (20 × 2.24) by the width (48), that is, multiplying 20 × 2.24 by 1.15, the reciprocal of 48:
(20 × 2.24)/48 produces 20 × 2.24 × 1.15 which produces 20 × 3, that is, 1.
-
The length which corresponds to the number 1 is not given in this problem, but can be easily be found as 5 ninda, that is 1 ṣuppān using tables in texts 1 and 2 and section D of Text 1.
Problem 19
Reverse ii
14. | [gi-meš ša] 7 kuš3 uš 7 kuš3 sag 1 gi kuš3-meš ša tur qe2-ea |
15. | [mi-ḫi]-⌈il⌉-tu4 a-ra2 ki-min u3 1.12 du-ma ša2 a-na igi-ka e11-a |
16. | x mi-nu-ti gi-meš kuš3-meš tur-meš te-eṣ-ṣip 1.10 kuš3 ur-a |
17. | ⌈ḫe2⌉-en gi-meš 1.10 a-ra2 1.10 1.21.40 1.21.40 a-ra2 1.12 |
18. | 1.38 gi-meš 1.38 kuš3-meš tur-meš 2 gi-meš 2 gi-meš a-na |
19. | 1.38 te-eṣ-ṣip-ma 1 me gi-meš |
Translation
14. | [Reeds which] the length is 7 kuš is and the width is 7 kuš (have) the interior (surface equal to) 1 gi according to the small (system). |
15. | The line by the same (itself) and by 1.12 you multiply: then, to what comes up before you, |
16. | a number of gi (according to the system) of small kuš, you add (to adjust). 1.10, the kuš, is the equal-side. |
17. | How much is the gi-surface ? 1.10 times 1.10 is 1.21.40. (Then), 1.21.40 times 1.12 is |
18. | 1.38, the (number) of gi. (To) 1.38 gi!, according to the small (system), 2 gi (is to be added). 2 gi to |
19. | 1.38 you add (to adjust). The surface is 1 hundred gi. |
Problem 19 applies the traditional method to calculate the surface : transform the measurement values into SPVN, calculate with SPVN, and come back to the measurement values. However, there is a difficulty: this method was conceived for a metrology which uses only regular factors. But it is applied here to calculation of surface with Late Babylonian metrology , where the factor 7 appears (1 gi = 7 kuš). The traditional method applied on an inappropriate metrology leads to the thorny problem of reciprocal of non-regular numbers . Problem 19 deals with this issue.
The line 14 defines a new unit of surface : 1 gi is the surface of a 7 kuš-side square . It seems that the corresponding system is labelled as the ‘small’ one (tur). Lines 15–16 provide the general procedure to calculate the surface in the new ‘small system’: multiply the side by itself. Then, multiply the result by a coefficient 1.12, which is the reciprocal of 50. Thus, the square of the side is divided by 50 instead to be divided by 49, as expected, with the correspondence 1 kuš → 1 (implicitly assumed). Indeed:
-
The surface of the 7 kuš-side square with the correspondence 1 kuš → 1 is
7 × 7 = 49
-
The surface of the 7 kuš-side square in the small system is 1 gi. Thus, the number of gi is obtain by dividing by 49 the surface with the correspondence 1 kuš → 1.
Dividing by 50 instead of 49 is easier because 50 is regular, while 49 is not. However, this division introduces an error. Line 16, this error is corrected by adding an adjustment (eṣēpu). It is not explained how to calculate this adjustment.
‘1.10 kuš3’, that I translate ‘1.10, kuš’ must represent 10 gi (70 kuš) for sake of consistency of the problem. The text provides the SPVN counterpart of the measurement value, which is given as 1.10, and an indication on the metrological system : the calculations are performed with the correspondence 1 kuš → 1. Other interpretation should be to translate ‘1.10 kuš3’ as ‘1(geš2) 10 kuš’. With these elements in mind, the problem can be re-formulated as follows: if the surface of a 7 kuš-side square is 1 gi, the surface of a 70 kuš-side square is how many gi? A straightforward solution appears to the modern reader (and would have appeared to the ancient reader): the sought surface is 100 gi, as stated in line 19. The very interesting feature of the problem is not the solution, but the procedure leading to the solution.
In application of the procedure (lines 15–16), we expect that the number of gi is obtained by:
-
Converting the side in SPVN with the correspondence 1 kuš → 1.
-
Multiplying the side by the side in SPVN
-
Dividing the obtained surface by 49 (see Table 3.15).
Table 3.15 Number of gi in problem 19
Problem 20
Reverse ii
20. | 1 gi a-ra2 1 gi 1 gi 1 gi a-ra2 1 kus3 1 kuš3 |
21. | 1 gi a-ra2 1 šu-si 1 šu-si 1 kuš3 a-ra2 1 gi 1 kuš3 |
22. | [1] kuš3 a-ra2 1 kuš3 1 kuš3-tur-tu2 1 kuš3 a-ra2 1 šu-si 1 ⌈še⌉ |
23. | [1] šu-si < a-ra2 > 1 gi 1 šu-si 1 šu-si a-ra2 1 kuš3 1 [še] |
24. | [1 šu]-si a-ra2 1 šu-si turtu2 24 šu-si-⌈meš⌉ tur-meš |
25. | [1] še 7 še-meš 1 šu-si 24 šu-si-meš 1 kuš3 7 kuš3-meš 1 gi |
26. | 3 su-si 3 še 1 kuš3-turtu2 7 kuš3 tur-meš 1 kuš3 |
This section contains two tables (here again, I follow Friberg et al. 1990: 538–539). Lines 20–24 contain a table of metrological ‘multiplication’ in the ‘small’ system, that is, with surface expressed as numbers of gi. It is a kind of generalization of problem 19. Lines 24–26 contains relations between surface units in the ‘small’ system. I present the two tables in tabular format, with explanations on the metrology in the last columns (Table 3.16).
This table is based on the following length system:
gi ← 7– kuš3 ← 24– šu-si
Three surface systems are defined from this, based successively on the square -gi (=1 gi × 1 gi), the square -kuš (= 1 kuš × 1 kuš), and the square -šu-si (= 1 šu-si × 1 šu-si) (Table 3.17).
This second table is represented by a unique diagram in Friberg, Hunger, and Al-Rawi (1990: 540) as follows (Fig. 3.3):
Metrological system in the second table of #20, after Friberg, Hunger, and Al-Rawi (1990: 540)
Appendix 3.4
Convention: In this chapter, the measurement units are not translated in English, but represented in italics by their Akkadian or Sumerian name according to the original text. Examples:
Sumerogram | Akkadian | Translation |
|---|---|---|
kuš3 | ammatum | kuš or ammatum |
ninda | ninda | |
sila3 | qa | sila or qa |
barig | pi | barig or pi |
List of sources
(IM– = National Museum of Iraq, Baghdad, unknown number)
Museum number | Excavation number | Publication number | CDLI number | Text in this chapter | Secondary publication |
|---|---|---|---|---|---|
IM– | W 23273 | SpTU 4, 172 | P348765 | 1 | Friberg (1993: 400) |
IM– | W 23281 | SpTU 4, 173 | P348766 | 2 | Friberg (1993: 401–2) |
IM– | W 23283 + W 22905 | SpTU 4, 174 | P348767 | 3 | |
IM– | W 23291 | SpTU 4, 175 | P348768 | 4 | Friberg (1997) |
IM 75985 | W 23291x | BagM 21, 554–557 | P430090 | 5 | Friberg et al. (1990) |
IM– | W 23021 | SpTU 4, 176 | P348769 | ||
IM– | W 22715-2 | SpTU 4, 177 | P348770 | ||
IM– | W 22260a | SpTU 1, 101 | P348522 | Friberg (1993: 395–6) | |
IM– | W 22309a + b | SpTU 1, 102 | P348523 | Friberg 1993: 392–3 | |
IM– | W 22656-1 | SpTU 4, 178 | P348771 | ||
IM– | W 22661-3a + b | SpTU 5, 317 | P348899 | ||
IM– | W 23016 | SpTU 5, 316 | P348898 | ||
AO 6484 | TU, 33 | P254387 | |||
CBS 8539 | BE 20/1, 30 | P230041 | Friberg (1993: 299) | ||
CBS 11019 | Sachs 1947, 69–70 | P266196 | |||
CBS 11032 | Sachs (1947, 68) | P266208 | |||
HS 241 | BE 20/1, 42 | P388160 | |||
HS 249 | Proust (2008, 3) | P388149 | |||
N 2873 | Robson (2000: n°20) | P277942 | |||
N 2694 | Unpublished? | P277762 |
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Proust, C. (2019). A Mathematical Collection Found in the ‘House of the āšipus’. The Art of Metrology in Achaemenid Uruk. In: Proust, C., Steele, J. (eds) Scholars and Scholarship in Late Babylonian Uruk. Why the Sciences of the Ancient World Matter, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04176-2_3
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