The problems presented in Chapters 2–5 were so different one from the others that it was necessary to accompany each of them with a copious commentary. In order to allow the reader who may like to explore some Old Babylonian texts without being held firmly by the hand, this appendix contains problems in translation only, or at most accompanied by the most necessary clarifications. Some are counterparts of problems that were presented above and come from the same tablets.
This problem belongs to one of two twin tablets, containing a total of ten problems about the rent paid for two parcels of a field. On one parcel the rent is 4 gur
As explained on page
A modern reader may find it strange that the two rents per bùr, which in lines I.1–2 are given in gur (per bùr), are translated into sìla in lines I.6–7 without multiplication; in general, as we see, the text skips no intermediate step. The explanation is that the conversion is made by means of a “metrological table”
The modern reader may also wonder that the text does not indicate once for all the value of the bùr in sìla and its igi. Once more the reason is that the text describes the Old Babylonian calculational technique
A small explanation may be necessary in order to facilitate understanding of the procedure: first the text determines what the difference between the two rents would be if the two parcels had been equal in area, that is, 15‵ sar each. This difference is not large enough—it is 2‵30 sìla, 5‵50 sìla too small—and therefore the first parcel must be enlarged. Each time a sar is transferred from the second to the first parcel, the difference grows by 40′+30′ sìla (the two “changes”
As support for the interpretation, a diagram may serve (Figure 1). Then the text almost explains itself, in particular if one keeps in mind BM 13901 #10
One should take note of the use of the multiplicative operations “make hold,” “raise”
This problem deals with the same mutilated rectangle as #4 of the same tablet (see page
Exceptionally in this type, the “joining”
This is the third problem
This problem comes from the same tablet as the “excavation problem”
This is one of the texts from the Eshnunna region, and thus belongs to the earliest phase (and as we see, it uses the phrase “to one join, from one cut off,” not respecting the “norm of concreteness”). With fair precision it can be dated to c. 1775 bce. The problem is one of the riddles which the Old Babylonian school
Lines 1–9 find the difference between the length and the width of the rectangle; the method is shown in the upper part of Figure 3. Afterwards, the sides are found from this difference and the area by the procedure which we already know perfectly well, for instance from YBC 6967
[1]
The tablet is damaged at this point, but the traces of signs that remain could well come from the word takkirtum, which means “change” or “modification” but does not occur in other mathematical texts. In any case, this philological doubt does not touch the interpretation of the mathematical procedure.
Table of Contents
1 Introduction: The Issue – and Some Necessary Tools
2 Techniques for the First Degree
3 The Fundamental Techniques for the Second Degree
4 Complex Seconddegree Problems
5 Application of QuasialgebraicTechniques to Geometry
9 A Moral
Appendix A: Problems for the Reader
Appendix B: Transliterated Texts
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