4.6: BM 13901 #23

Rev. II

11 About a surface, the four widths and the surface I have heaped, \(41^{\prime} 40^{\prime \prime}\).

12 4, the four widths, you inscribe. igi 4 is \(15^{\prime}\).

13 \(15^{\prime}\) to \(41^{\prime} 40^{\prime \prime}\) you raise: \(10^{\prime} 25^{\prime \prime}\) you inscribe.

14 1, the projection, you join: by \(1^{\circ} 10^{\prime} 25^{\prime \prime}\), \(1^{\circ}5^{\prime}\) is equal.

15 1, the projection, which you have joined, you tear out: \(5^{\prime}\) to two

16 you repeat: \(10^{\prime}\), \(\mathrm{NINDAN}\), confronts itself.

Whereas the previous problem illustrates the “modern” aspect of Old Babylonian mathematics, the present one seems to illustrate its archaic side—even though they come from the same tablet.

This is no real contradiction. The present problem #23 is intentionally archaic. In other words, it is archaizing and not truly archaic, which explains its appearance together with the “modern” problems of the same collection. The author is not modern and archaic at the same time, he shows his virtuosity by playing with archaisms. In several ways, the formulations that are used here seem to imitate the parlance of Akkadian surveyors. The text speaks of the width of a square, not of a “confrontation”; further, this word appears in syllabic writing, which is quite exceptional (cf. note 4, page 16). The introductory phrase “About a surface”⁹ seems to be an abbreviated version of the characteristic formula introducing a mathematical riddle: “if somebody asks you thus about a surface …” (cf. pages 34, 110, 111 and 127). The expression “the four widths”¹⁰ reflects an interest in what is really there and for what is striking, an interest that characterizes riddles in general but also the mathematical riddles that circulated among the mathematical practitioners of the pre-Modern world (see page 106). Even the method that is used is typical of riddles: the use of an astonishing artifice that does not invite generalization.

The problem can thus be expressed in the following way:

\(4 c+\square(c)=41^{\prime} 40^{\prime \prime}\).

Figure 4.12 makes clear the procedure: 4c is represented by 4 rectangles \((1, c)\); the total \(41^{\prime} 40^{\prime \prime}\) thus corresponds to the cross-shaped configuration where a “projection” protrudes in each of the four principal directions.

Lines 12–13 prescribe to cut out \(\frac{1}{4}\) of the cross (demarcated by a dotted line) and the “joining” of a quadratic complement \(\square(1)\) to the gnomon that results. There is no need to “make hold,” the sides of the complement are already there in the right position. But it is worthwhile to notice that it is the “projection” itself that is “joined”: it is hence no mere number but a quadratic configuration identified by its side.

Figure \(4.12\): The procedure of BM 13901 #23.

The completion of the gnomon gives a square with area \(1^{\circ} 10^{\prime} 25^{\prime \prime}\) and thus side \(1^{\circ}5^{\prime}\). "Tearing out" the "projection"—now as a one dimensional entity—we find \(5^{\prime}\). Doubling the result, we get the side, which turns out to be \(10^{\prime}\). Here again, the text avoids the usual term and does not speak of a "confrontation" as do the "modern" problems of the collection; instead it says that \(10^{\prime} \mathrm{NINDAN}\) "confronts itself."

This method is so different from anything else in the total corpus that Neugebauer believed it to be the outcome of a copyist's mixing up of two problems that happens to make sense mathematically. As we shall see below (page 109), the explanation is quite different.

The archaizing aspect, it should be added, does not dominate completely. Line 12, asking first for the "inscription" of 4 and stating afterwards its igi, seems to describe the operation on a tablet for rough work that were taught in school (see note 5, page 65, and page 120).