Abstract
Chapter 4 represents the most important part of the book. In it, the 12 selected tablets are presented and analysed. Each tablet receives initially a transliteration and a transcription:
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• A transliteration of the cuneiform signs identifies the signs by their conventional names and tries to respect their isolated pronunciation.
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• A transcription renders morphological and syntactical aspects of the original text (see also Chap. 3 for the details and conventions governing each of these parts).
The tablets are then translated and commented. The only exception is IM54010, whose bad state of conservation prevents a complete understanding.
Transliteration and transcription are directed to specialists in cuneiform mathematics or to readers who possess at least a basic command of the Akkadian and a desire to improve their understanding of the original texts. Translation and commentary, on the other hand, were designed to satisfy the readers’ appetite for the mathematical ideas and techniques that these texts bear.
Besides, in this chapter, I also offer some philological remarks and a mathematical analysis for each tablet.
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Notes
- 1.
I am indebted to Hermann Hunger for allowing me to have access to these photos.
- 2.
For the benefit of the mathematically minded reader, it should be said that these equivalence classes would not be the usual ones obtained in the modulus-60 arithmetics. Instead, two numbers are equivalent here if one is the other multiplied by a power of 60. So it is not true that 62 and 2 are equivalent in this floating point system. But 2,0 (that is to say, 120) and 2 are indeed equivalent (Proust 2013).
- 3.
Here too, I am indebted to Hermann Hunger for the access to the photos.
- 4.
The first three lines on the edge seem to have been written in two columns. This produces, for each line, an initial and a final segment. The separation is marked by double slashes.
- 5.
The concept of unorthographic writing serves to explain deviations from expected writings. In the present example, dug4 is the expected way a scribe writes the verb to say, but as the phonetic values of the sign tug include the pronunciation “dug”, this sign can be used instead, constituting an unexpected or unorthographic writing.
- 6.
See mathematical commentary for the order of magnitude.
- 7.
Following Friberg’s (2000, 118) interpretation.
- 8.
The second segments of lines E1, E2 and E3 on the edge are marked with double slashes: E1//, E2// and E3//.
- 9.
That is to say, Text 14 of Bruins and Rutten’s TMS.
- 10.
The metrological table of surfaces was used for volumes too.
- 11.
See also the commentary to IM54478, where the strategy of positing a second figure, similar to that of the problem, might have been in action too.
- 12.
See again IM54478, where a similar computation of a linear ratio from the ratio of volumes might have been present.
- 13.
- 14.
However, Hussein (2009, 92) states that this tablet also comes from room 252.
- 15.
Which contained perhaps a mistyping. I propose “On two times the length (3 × the width) add 10; on the width, add 10”.
- 16.
The occurrences of this word are registered under ellum in the vocabulary.
References
al-Rawi, F.N.H., Roaf, M.: Ten old Babylonian mathematical problems from Tell Haddad, Himrin. Sumer XLIII, 175–218 (1984)
Attinger, P.: À propos de AK “faire” (I). Zeitschrift für Assyriologie 95, 46–64 (2005)
Baqir, T.: An important mathematical problem text from Tell Harmal (on a Euclidean theorem). Sumer VI, 39–54 (1950a). Plus 2 planches (IM55357)
Baqir, T.: Another important mathematical text from Tell Harmal. Sumer VI, 130–148 (1950b). Plus 3 planches (IM52301)
Baqir, T.: Some more mathematical texts from Tell Harmal. Sumer VII, 28–45 (1951). Plus 5 planches (IM54478, IM53953, IM54538, IM53961, IM53957, IM54010, IM53965, IM54559, IM54464, IM54011)
Bruins, E.M.: Comments on the mathematical tablets of Tell Harmal. Sumer VII, 179–185 (1951) (includes a letter to the editor and a “note added during correction”)
Bruins, E.M.: Revision of the mathematical texts from Tell Harmal. Sumer IX, 241–253 (1953a)
Bruins, E.M.: On the system of Babylonian geometry. Sumer XI, 44–49 (1955)
Drenckhahn, F.: A geometrical contribution to the study of the mathematical problem-text from Tell Harmal (IM55357) in the Iraq Museum, Baghdad. Sumer VII, 22–27 (1951)
Drenckhahn, F.: Ein geometrischer Beitrag zu dem mathematischen Problem-Text von Tell Harmal IM 55357 des Iraq Museums in Baghdad. Zeitschrift für Assyriologie und Vorderasiatische Archäologie, Neue Folge, Band 16 50, 151–162 (1952a)
Drenckhahn, F.: Letter to the editor. Sumer VIII, 234–235 (1952b)
Friberg, J.: Mathematics at Ur in the old Babylonian period. Revue d’Assyriologie 94, 98–188 (2000)
Gundlach, K.-B., von Soden, W.: Einige altbabylonische Texte zur Lösung ‘quadratischer Gleichungen’. Abhandlungen aus dem mathematischen Seminar der Universität Hamburg 26, 248–263 (1963)
Høyrup, J.: Algebra and naive geometry. An investigation of some basic aspects of old Babylonian mathematical thought II. Altorientalische Forschungen 17, 262–354 (1990)
Høyrup, J.: Lengths, widths, surfaces. A portrait of old-Babylonian algebra. Springer, New York (2002)
Høyrup, J.: A note about the notion of exp10(log10(modulo 1))(x) Concise observations of a former teacher of engineering students on the use of the slide rule. In: Contribution au Séminaire SAW: Histoire des mathématiques, histoire des pratiques économiques et financières. Séance du 6 janvier 2012: “Usage de la position—pratiques mathématiques, pratiques comptables” (2012)
Hussein, L.M.: Tell Harmal. Die Texte aus dem Hauptverwaltungsgebäude “Serai”. Inaugural-Dissertation zur Erlangung der Doktorwürde dem Fachbereich Fremdsprachliche Philologien der Philipps-Universität Marburg (2009)
Proust, C.: La multiplication babylonienne: la part non écrite du calcul. Revue d’histoire des mathématiques 6, 293–303 (2000)
Proust, C.: Tablettes mathématiques de Nippur. Varia Anatolica, vol. XVIII. Institut Français George Dumézil, De Brocard, Paris (2007)
Proust, C.: Du calcul flottant en Mésopotamie. La Gazette des Mathématiciens 138, 23–48 (2013)
Robson, E.: Mesopotamian Mathematics, 2100–1600 BC. Technical Constants in Bureaucracy and Education. Oxford Edition of Cuneiform Texts, vol. XIV. Clarendon, Oxford (1999)
Robson, E.: Mathematical cuneiform tablets in Philadelphia. Part 1: problems and calculations. SCIAMVS 1, 11–48 (2000)
von Soden, W.: Zu den mathematischen Aufgabentexten vom Tell Harmal. Sumer VIII, 49–56 (1952)
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Gonçalves, C. (2015). Mathematical Tablets. In: Mathematical Tablets from Tell Harmal. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-22524-1_4
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