4.9: Footnotes

1

As the “to-be-joined” of page 37, this noun (wuṣubbûm) is derived from the verb “to join.”

2

We should observe that the half that appears here is treated as any other fraction, on an equal footing with the subsequent third. It is not a “moiety,” and the text finds it through multiplication by \(30^{\prime}\), not by “breaking.”

Let us also take note that the half of the length and the third of the width are “joined” to the “surface,” not “heaped” together with it. A few other early texts share this characteristic. It seems that the surveyors thought in terms of “broad lines,” strips possessing a tacitly understood breadth of 1 length unit; this practice is known from many pre-Modern surveying traditions, and agrees well with the Babylonian understanding of areas as “thick,” provided with an implicit height of 1 kùš (as inherent in the metrology of volumes, which coincides with that for areas—see page 17). The “projection” and “base” of BM 13901 and TMS IX #1 are likely to be secondary innovations due to the school—different schools, indeed, and therefore different words. They allowed segments to be thought of as truly one-dimensional while still permitting their transformation into rectangles with width 1.

3

The absence of this notion from the text should not prevent it from using it as a technical term of general validity.

4

Alternatively, the trick used by the text could be a leftover from the ways of surveyors not too familiar with the place-value system; or (a third possibility) the floating-point character of this system might make it preferable to avoid it in contexts where normal procedures for keeping track of orders of magnitude (whatever these normal procedures were) were not at hand.

5

It is not quite to be excluded that the text does not directly describe the construction but refers to the inscription twice of \(3^{\circ} 25^{\prime}\) on a tablet for rough work, followed by the numerical product—cf. above, note 11, page 21; in that case, the construction itself will have been left implicit, as is the numerical calculation in other texts. Even the “inscription” of 2, followed by its igi (II.3 and 6) might refer to this type of tablet. Then, however, one would expect that the “detachment” of the igi should follow the inscription immediately; moreover, the inscription of 3 in line II.4 is not followed at all by “detachment” of its igi, which after all speaks against this reading of the lines II.3–6 and II.21–22.

6

The position of the “upper” width to the left is a consequence of the new orientation of the cuneiform script (a counterclockwise rotation of \(90^{\circ}\)) mentioned in the box “Cuneiform writing.” On tablets, this rotation took place well before the Old Babylonian epoch, as a consequence of which one then wrote from left to right. But Old Babylonian scribes knew perfectly well that the true direction was vertically downwards—solemn inscriptions on stone (for example Hammurabi’s law) were still written in that way. For reading, scribes may well have turned their tablets \(90^{\circ}\) clockwise.

7

This distinction between two halves of which one is “left” is worth noticing as another proof of the geometric interpretation—it makes absolutely no sense unless understood spatially.

8

By error, line 30 of the text has \(1^{\prime} 57^{\prime \prime} 46^{\prime \prime \prime} 40^{\prime \prime \prime}\) instead of \(1^{\prime} 57^{\prime \prime} 21^{\prime \prime \prime} 40^{\prime \prime \prime}\); a partial product 25 has been inserted an extra time, which shows that the computation was made on a separate device where partial products would disappear from view once they had been inserted. This excludes writing on a clay surface and suggests instead some kind of reckoning board.

The error is carried over in the following steps, but when the square root is taken it disappears. The root was thus known in advance.

9

In the original, the word is “surface” marked by a phonetic complement indicating the accusative. An accusative in this position is without parallel, and seems to allow no interpretation but the one given here.

10

For once, the determinate article corresponds to the Akkadian, namely to an expression which is only used to speak about an inseparable plurality (such as “the four quarters of the world” or “the seven mortal sins”).

11

The use of a “raising” multiplication shows that the calculator does not construct a new rectangle but bases his procedure on a subdivision of what is already at hand—see the discussion and dismissal of a possible alternative interpretation of the procedure of BM 13901 #10 in note 5, page 49.

12

Line 10 speaks of this as \(5^{\prime}\) the length—namely the side of the small square. Some other texts from Susa also speak of the side of a square as its “length.”