1.5: Footnotes

1

However, around 1930 one had to begin with texts that were much more complex than the one we consider here, which was only discovered in 1936. But the principles were the same. The most important contributions in the early years were due to Otto Neugebauer, historian of ancient mathematics and astronomy, and the Assyriologist François Thureau-Dangin.

2

A literal retranslation of François Thureau-Dangin’s French translation. Otto Neugebauer’s German translation is equivalent except on one point: where Thureau-Dangin translated “\(1^{\circ}\), the unit” Neugebauer proposed “1, the coefficient.” He also transcribed place-value numbers differently.

3

Nobody, except perhaps Neugebauer, who on one occasion observes (correctly) that a text makes use of a wrong multiplication. In any case it must be noticed that neither he nor Thureau-Dangin ever chooses a wrong operation when restituting the missing part of a broken text.

4

More precisely, the word translated “length” signifies “distance”/“extension”/“length” while that which is translated “width” means “front”/“forehead”/“head.” They refer to the idea of a long and narrow irrigated field. The word for the area (eqlum/a.šà) originally means “field” but in order to reserve it for technical use the texts use other (less adequate) words when speaking of genuine fields to be divided. In what follows, the term will be translated “surface,” which has undergone a similar shift of meaning, and which stands both for the spatial entity and its area.

A similar distinction is created by other means for lengths and widths. If these stand for “algebraic” variables they are invariably written with the logograms uš and sag̃; if used for general purposes (the length of a wall, a walking distance) they may be provided with phonetic complements or written syllabically as šiddum and pūtum .

5

In the absence of a sexagesimal point it is in principle impossible to know whether the basic unit was 1 \(\mathrm{NINDAN}\), 60 \(\mathrm{NINDAN}\) or \(\frac{1}{60} \mathrm{NINDAN}\). The choice of 1 \(\mathrm{NINDAN}\) represents what (for us, at least) seems most natural for an Old Babylonian calculator, since it already exists as a unit (which is also true for 60 \(\mathrm{NINDAN}\) but not for \(\frac{1}{60} \mathrm{NINDAN}\)) and because distances measured in \(\mathrm{NINDAN}\) had been written without explicit reference to the unit for centuries before the introduction of the place-value system.

6

It is not to be excluded that the Babylonians thought of the mina as standard unit, or that they kept both possibilities open.

7

The verbal form used would normally be causative-reciprocative. However, at times the phrase used is “make \(p\) together with \(q\) hold” which seems to exclude the reciprocative interpretation.

8

When speaking of a “school” in the Old Babylonian context we should be aware that we only know it from textual evidence. No schoolroom has been identified by archaeologists (what was once believed to be school rooms has turned out to be for instance store rooms). We therefore do not know whether the scribes were taught in palace or temple schools or in the private homes of a master scribe instructing a handful of students; most likely, many were taught by private masters. The great number of quasi-identical copies of the table of reciprocals that were prepared in order to be learned by heart show, however, that future scribes were not (or not solely) taught as apprentices of a working scribe but according to a precisely defined curriculum; this is also shown by other sources.

9

It may seem strange that the multiplication of igi 4 by 30 is done by “raising.” Is this not a multiplication of a number by a number? Not necessarily, according the expression used in the texts when igi 4 has to be found: they “detach” it. The idea is thus a splitting into 4 equal parts, one of which is detached. It seems that what was originally split (when the place-value system was constructed) was a length—namely 1‵ [\(\mathrm{NINDAN}\)], not 1 [\(\mathrm{NINDAN}\)]. This Ur-III understanding had certainly been left behind; but the terminological habit had survived.

10

And, tacitly understood, that n itself can be written in this way. It is not difficult to show that all “regular numbers” can be written \(2^{p} \cdot 3^{q} \cdot 5^{r}\), where \(p\), \(q\) and \(r\)are positive or negative integers or zero. \(2\), \(3\) and \(5\) are indeed the only prime numbers that divide 60. Similarly, the “regular numbers” in our decimal system are those that can be written \(2^{p} \cdot 5^{q}\), 2 and 5 being the only prime divisors of 10.

11

The expression “posit to” refers to the way simple multiplication exercises were written in school: the two factors were written one above the other (the second being “posited to” the first), and the result below both.

12

More precisely, the Babylonian word stands for “a situation characterized by the confrontation of equals.”

13

On the problem of the “school” see note 8, page 20, and page 101.

14

In Akkadian, the verb comes in the end of the phrase. This structure allows a number to be written a single time, first as the outcome of one calculation and next as the object of another one. In order to conserve this architecture of the text (“number(s)/operation: resulting number/new operation”), this final position of the verb is respected in the translations, ungrammatical though it is. The reader will need to get accustomed (but non-English readers should not learn it so well as to use the construction independently!).