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A New Geometric School Text
Hagan Brunke
<habrunke@gmail.com>




The school text CUNES 52243 gives the solution to a geometric exercise which is to compute the circumferences and the areas of the annuli defined by a system of eight concentric circles. It therefore probably shows a pupil’s sketchy solution of an assignment as it is known from a number of problem texts.
Figure 1: CUNES 52243 obv. Photograph by Laura JohnsonKelly.
Figure 2: Schematic transliteration of the drawing on CUNES 52243 obv.
The obverse of the lentilshaped school tablet CUNES 52243 (90×92×32mm; reverse empty, provenience unknown) shows the solution to a geometric exercise dealing with a system of eight concentric circles (figs. 1 and 2). It consists of a scetchy drawing with inscribed numbers representing various length and area values. The drawing depicts only a sectorshaped section of the figure involved and is highly out of scale. In fact, though the annuli bounded by the subsequent circles increase in width by one at each step (see fig. 3), the circles are drawn equidistantly. In the following I will use the symbol π̂ (with value 3) to denote the Mesopotamian equivalent to our π.^{[1]} The numbers given are one possible interpretation. Statements remain valid when at the same time numbers representing lenghts are multiplied by 60^{k} , and those representing areas by 60^{2k} for arbitrary integer k.
The interpretation is the following. The innermost circle (“circle 0” in the following) has the area A_{0} = 08; 20 from which its circumference c_{0} and radius r_{0} result as
The circumference c_{0} seems to be written twice at the left side inside the inner circle. The seven outer circles will be referred to as “circle 1, circle 2, . . .”, in the order of increasing diameter. As indicated by the numbers in the central “column” of the drawing, the widths of the annuli bounded by the subsequent circles (i.e., the increments of the circles’ radii) are ∆_{i} = i (1 ≤ i ≤ 7), i.e. increase arithmetically from inside to outside, whence radius r_{n} and circumference c_{n} of circle n (1 ≤ n ≤ 7) are^{[2]}
and
The area Ã_{n} of the nth annulus (bounded by the nth and (n − 1)th circle) can either be found by computing the areas A_{n} and A_{n−1} of the bounding circles as
and then Ã_{n} = A_{n} − A_{n−1}, or directly from the circumferences c_{n} and c_{n−1} as^{[3]}
The numerical values for the circumferences c_{n} of the circles, the radial increments ∆_{i}, and the areas Ã_{n} of the annuli are collected in the following table. These are the respective values written in the left, central, and right column of the drawing. The situation is depicted to scale in fig. 3.
Figure 3: The situation of CUNES 52243 obv. drawn to scale
The underlying problem text would most likely have given the central circle’s area 08 20 and the arithmetically increasing sequence of radius increments (∆_{i})_{1≤i≤7} and asked for the outer circles’ circumferences and the areas of the annuli.^{[4]} The solution would then start by computing the innermost circle’s circumference and radius (see (1)) and proceed by computing the radii of the outer circles (either by successive addition of the radius increments ∆_{i} or by means of (2)) and their circumferences (3). Finally, the areas of the annuli are determined, probably by means of (4) as it is done in W 23291x, § 2 (Friberg, Hunger & alRawi 1990: 494496), see below.
There are three problems (all accompanied by drawings) that are sort of “complementary” to this one, in that given and asked data are partially exchanged:
There are some more signs on the tablet below the drawing which may or may not be connected to the problem of the concentric circles. On the right side immediately below the drawing there is clearly discernible 41 20 next to damaged surface. The area A_{6} of the sixth circle (not annulus) is 25 41;20, but this need not mean anything.
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Version: 15 September 2018 