One of the mathematical topics examined in the Old Babylonian period consisted of calculating the size of a reed which was used to measure either a longitude or the perimeter of a rectangle or trapezium. These subjects were solved, probably, applying the geometric construction called completing the square. In this paper, we analyse the problem texts on the tablets AO 6770 (5), Str 368, VAT 7532, and VAT 7535.

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure (...) texts – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure (...) representing the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics. Some implications of the use of geometric methods by Babylonian astronomers are also explored. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure (...) texts – mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

The Mesopotamian system of sexagesimal counting numbers was based on the progressive series of units 1, 10, 1·60, 10·60, …. It may have been in use already before the invention of writing, with the mentioned units represented by various kinds of small clay tokens. After the invention of proto-cuneiform writing, c. 3300 BC, it continued to be used, with the successive units of the system represented by distinctive impressed cup- and disk-shaped number signs. Other kinds of “metrological” number systems in (...) the proto-cuneiform script, with similar number signs, were used to denote area numbers, capacity numbers, etc. In a handful of known mathematical cuneiform texts from the latter half of the third millennium BC, the ancient systems of sexagesimal counting numbers and area numbers were still in use, alongside new kinds of systems of capacity numbers and weight numbers. Large area numbers, capacity numbers, and weight numbers were counted sexagesimally, while each metrological number system had its own kind of fractional units. In the system of counting numbers itself, fractions could be expressed as sixtieths, sixtieths of sixtieths, and so on, but also in terms of small units borrowed from the system of weight numbers. Among them were the “basic fractions” which we would understand as 1/3, 1/2, and 2/3. In a very early series of metro-mathematical division exercises and an equally early metro-mathematical table of squares, “quasi-integers” of the form “integer plus basic fraction” play a prominent role. Quasi-integers play an essential role also in a recently found atypical cuneiform table of reciprocals. The invention of sexagesimal numbers in place-value notation, in the Neo-Sumerian period c. 2000 BC, was based on a series of innovations. The range of the system of sexagesimal counting numbers was extended indefinitely both upward and downward, and the use of quasi-integers was abolished. Sexagesimal place-value numbers were used for all kinds of calculations in Old Babylonian mathematical cuneiform texts, c. 1700 BC, while traditional metrological numbers were retained in both questions and answers of the exercises. Examples of impressive computations of reciprocals of many-place regular sexagesimal place-value numbers, with no practical applications whatsoever, are known from the Old Babylonian period. In the Late Babylonian period, such computations were still popular, performed by the same persons who constructed the many-place sexagesimal tables that make up the corpus of Late Babylonian mathematical astronomy. (shrink)

a re-analysis of Old Babylonian letters reveals the construction of class identity for men called “gentlemen” through their use of sympathetic expressions positioning correspondents as brothers, friends, colleagues, etc. While this observation is not new, this article makes two further points. First, I argue that class consciousness was created through the policing of failures to enact the social relations expressed in the letters, rather than superficial claims that such relations existed in the first place. This reading requires that we (...) engage seriously with the contingent nature of class identity—that fears and anxieties about falling out of status were more in evidence as the motor for and incentive towards class membership than simple claims of inclusion or group solidarity. Second, I argue that the sympathy enjoined by the letters simulated the affective-spatial cognitive states necessary for group identity. Group problems of geographic and physical distance and even loneliness were solved by the letters’ production of sympathy. Not only does a sincere consideration of the subjective experience of class formation require an understanding of individual and group emotional states, the letters themselves, filled with expressions of pathos and worry, invite it. (shrink)

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, (...) 1996, and Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1). (shrink)

Most Probably: Epistemic Modality in Old Babylonian by Nathan Wasserman is the first attempt to provide a description of the domain of epistemic modality in Old Babylonian Akkadian. This attempt is not entirely successful, for several reasons. Methodological inconsistency often impairs the author’s ability to convince the reader of his solutions. Both primary data and existing secondary literature are used only selectively, which renders the proposed description problematic. Finally, the lack of cross-referencing between the chapters evokes the feeling (...) that the different particles do not quite constitute a system. (shrink)

Conditional Structures in Mesopotamian Old Babylonian. By Eran Cohen. Languages of the Ancient Near East, vol. 4. Winona Lake, Ind.: Eisenbrauns, 2012. Pp. x + 198. $44.50.

Texts from the Late Old Babylonian Period. By Seth F. C. Richardson. The Journal of Cuneiform Studies Supplemental Series, vol. 2. Boston: American Schools of Oriental Research, 2010. Pp. ix + 221.

Gods, Kings, and Merchants in Old Babylonian Mesopotamia. By Dominique Charpin. Publications de l’Institute de Proche-Orient Ancient du Collège de France. Leuven: Peeters, 2015. Pp. 223, illus, €41.

The Material and Ideological Base of the Old Babylonian State: History, Economy, and Politics. By Lukáš Pecha. Lanham, MD: Lexington Books, 2018. Pp. xvii + 316. $115.

Sumerian Model Contracts from the Old Babylonian Period in the Hilprecht Collection Jena. By Gabriella Spada. Texte und Materialien der Hilprecht Collection, vol. 11. Wiesbaden: Harrassowitz Verlag, 2018. Pp. x + 126, 32 pls. €78.