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Über dieses Buch

This monograph presents in great detail a large number of both unpublished and previously published Babylonian mathematical texts in the cuneiform script. It is a continuation of the work A Remarkable Collection of Babylonian Mathematical Texts (Springer 2007) written by Jöran Friberg, the leading expert on Babylonian mathematics.
Focussing on the big picture, Friberg explores in this book several Late Babylonian arithmetical and metro-mathematical table texts from the sites of Babylon, Uruk and Sippar, collections of mathematical exercises from four Old Babylonian sites, as well as a new text from Early Dynastic/Early Sargonic Umma, which is the oldest known collection of mathematical exercises. A table of reciprocals from the end of the third millennium BC, differing radically from well-documented but younger tables of reciprocals from the Neo-Sumerian and Old-Babylonian periods, as well as a fragment of a Neo-Sumerian clay tablet showing a new type of a labyrinth are also discussed. The material is presented in the form of photos, hand copies, transliterations and translations, accompanied by exhaustive explanations. The previously unpublished mathematical cuneiform texts presented in this book were discovered by Farouk Al-Rawi, who also made numerous beautiful hand copies of most of the clay tablets.
Historians of mathematics and the Mesopotamian civilization, linguists and those interested in ancient labyrinths will find New Mathematical Cuneiform Texts particularly valuable. The book contains many texts of previously unknown types and material that is not available elsewhere.



1. Late Babylonian Tables of Many-Place Regular Sexagesimal Numbers, from Babylon, Sippar, and Uruk

For the notion of many-place regular sexagesimal numbers, and for many explicit examples, both Old and Late Babylonian, the reader is referred to Friberg, MSCT 1 (2007), Sec. 1.4 and App. 9. In particular, it is important to recall that a sexagesimal number n is called “regular” if another sexagesimal number n´ can be found such that n times n´ equals some power of 60. (In Babylonian “relative” place value notation, every power of 60 is written as ‘1’.) The number n´ is called the “reciprocal” of n. In the following, it is conveniently referred to as rec. n.
Jöran Friberg, Farouk N. H. Al-Rawi

2. Direct and Inverse Factorization Algorithms for Many-Place Regular Sexagesimal Numbers

BM 46550 is a small Neo-Babylonian clay tablet, published for the first time in Sec. 2.1 below. On the obverse of the tablet is a teacher’s model text, showing that the reciprocal of the 6-place regular sexagesimal number n= 1 01 02 06 33 45 is the 5-place sexagesimal number rec. n = 28 · 126 = 58 58 56 38 24.
Jöran Friberg, Farouk N. H. Al-Rawi

3. Metrological Table Texts from Achaemenid Uruk

The corpus of published mathematical and/or metrological cuneiform texts from the 1st millennium BC is only moderately extensive. In Sec. 1.1 above, 34 Late Babylonian (Neo-Babylonian, Achaemenid, or Seleucid) texts dealing with many-place regular sexagesimal numbers were listed, and 11 of them were discussed in Chs. 1-2.
Jöran Friberg, Farouk N. H. Al-Rawi

4. CBS 8539. A Mixed Metrological Table Text from Achaemenid Nippur

CBM 8539 is a fragment of a Large Combined mertological table, with sub-tables for lenth measure, of four kinds, for weight measure, and for capacity measure.
Jöran Friberg, Farouk N. H. Al-Rawi

5. Five Texts from Old Babylonian Mê-Turran (Tell Haddad), Ishchali and Shaduppûm (Tell Harmal) with Rectangular-Linear Problems for Figures of a Given Form

IM 121613 (see the hand copies in Figs. 5.1.20-21 below) is a large and fairly well preserved Old Babylonian clay tablet from ancient Mê-Turran (the site Tell Haddad, situated in the Himrin basin near Diyala). The various fragments of the text were gathered together by Farouk Al-Rawi, who also made the hand copies of the text. Thanks are due to the excavators Dr. Nail Hanoun and Mr. Burhan Shakir for their permission to publish and for their support during the copying of the text.
Jöran Friberg, Farouk N. H. Al-Rawi

6. Further Mathematical Texts from Old Babylonian Mê-Turran (Tell Haddad)

Only thelast theree lines of the solution algorithm in this exercies are (partly) preserved. Without the missing context, the object of the exercise cannot be decided, not even conjectured. Without further context, the translation of line 4´ is very uncertain.
Jöran Friberg, Farouk N. H. Al-Rawi

7. A Recombination Text from Old Babylonian Shaduppûm Concerned with Economic Transactions

This text was first Published by Bruins in Sumer 10 (1954), 57-61, in an inadequate transliteration without hand copies or photos, and with in most cases unsatisfactory mathemadical interpretations.
Jöran Friberg, Farouk N. H. Al-Rawi

8. Six Fragments of Problem Texts of Group 6, from Late Old Babylonian Sippar

BM 80078 is a relativelly large fragment originally forming the lower right cvorner of an Old Babyloinian clay tablet inscribed in two columns on the ovberse and two on ther reverse with a mathematical recombination text apparently with problems for bricks as a common theme.
Jöran Friberg, Farouk N. H. Al-Rawi

9. More Mathematical Cuneiform Texts of Group 6 from Late Old Babylonian Sippar

In Ch. IV of Neugebauer and Sachs, MCT (1945), A. Goetze divided published Old Babylonian mathematical texts without known provenance into 6 different groups with respect to their Akkadian orthography. Goetze’s classification was later refined and extended by J. Høyrup in LWS (2002), Ch. IX. Groups 5 and 6 were classified by Goetze as “northern”.
Jöran Friberg, Farouk N. H. Al-Rawi

10. Goetze’s Compendium from Old Babylonian Shaduppûm and Two Catalog Texts from Old Babylonian Susa

The three tablets IM 52916 (in the present chapter, Sec. 10.1), and IM 52685 + IM 52304 (in Sec. 10.2) were published and correctly interpreted by Goetze in Sumer 7 (1951). Goetze called them together “a mathematical compendium from Tell Harmal”. The three tablets are very poorly preserved.
Jöran Friberg, Farouk N. H. Al-Rawi

11. Three Old Babylonian Recombination Texts of Mathematical Problems without Solution Procedures, Making up Group 2b

YBC 4698 is a mathematical recombination text, a mixed bag of loosely related exercises, all concerned with commercial problems. See the hand copy and conform transliteration in Fig. 11.1.1 below. Since exercises belonging more closely together are sometimes separated from each other in this text, the exercises are numbered in two different ways in the conform transliteration, both in the order they are inscribed on the tablet (## 1-17) and according to their problem types (§§ 1-5).
Jöran Friberg, Farouk N. H. Al-Rawi

12. An Early Dynastic/Early Sargonic Metro-Mathematical Recombination Text from Umma with Commercial Exercises

CUNES 52-18-035 is a fairly well preserved Early Dynastic III/Early Sargonic clay tablet (2350-2300 BC), piblished by Vitali Bartash in CUSAS 23(2013), no. 77, as a text from the Umma region. It is a metro-mathematical recombination text with (at least) seven simple but closely related exercises, complete with questions and answers, but with no solution procedures.
Jöran Friberg, Farouk N. H. Al-Rawi

13. An Ur III Table of Reciprocals without Place Value Numbers

In Fig. 13.1.1 below is shown a hand copy and conform transliteration of SM 2685, a clay tablet from the Suleimaniyah Museum in the Kurdistan region in northeastern Iraq. The clay tablets in the Suleimaniyah Museum are acquired in the antiquities market and are therefore unprovenanced, but in most cases probably from Old Babylonian Larsa. However, the writing on SM 2685 is such that the text can be either from the Neo-Sumerian Ur III period or Early Old Babylonian, and, as will be shown below, the atypical table of reciprocals inscribed on the tablet is clearly older than all earlier known Ur III tables of reciprocals.
Jöran Friberg, Farouk N. H. Al-Rawi

14. Fragments of Three Tablets from Ur III Nippur with Drawings of Labyrinths

The two almost perfectly preserved clay tablets MS 3194 and MS 4515, with drawings of a rectangular labyrinth in the former case and of a square labyrinth in the latter, were published and thoroughly analyzed in Friberg, MSCT 1 (2007), Sec. 8.3. Since then, the two clay tablets have been baked and cleaned, and new, much improved photos of them have been published in George, CUSAS 18 (2013), nos. 39-40, and online at, nos. P253616 and P274587.
Jöran Friberg, Farouk N. H. Al-Rawi


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