3.7: Footnotes

1

Léon Rodet, Journal asiatique, septième série 18, p. 205.

2

The inverse of the “heaping” operation, on the other hand, is no subtraction at all but a separation into constitutive elements. See note 3, page 99.

3

The verb in question (nadûm) has a broad spectrum of meanings. Among these are “to draw” or “to write” (on a tablet) (by the way, the word lapātum, translated “to inscribe,” has the same two meanings). Since what is “laid down” is a numerical value, the latter interpretation could seem to be preferable—but since geometrical entities were regularly identified by means of their numerical measure, this conclusion is not compulsory.

4

Here we see one of the stylistic reasons that would lead to a formulation in terms of falling-short instead of excess. It might as well have been said that one side exceeds the other by one sixth, but in the “multiplicative-partitive” domain the Babylonians gave special status to the numbers 4, 7, 11, 13, 14 and 17. In the next problem on the tablet, one “confrontation” is stated to exceed the other by one seventh, while it would be just as possible to say that the second falls short of the first by one eighth.

5

One might believe the underlying idea to be slightly different, and suppose that the original squares are subdivided into 7 7 respectively 6 6 smaller squares, of which the total number would be 1‵25, each thus having an area equal to \(\frac{21^{\circ} 15^{\prime}}{1^{1} 25}=15^{\prime}\) and a side of \(30^{\prime}\). However, this interpretation is ruled out by the use of the operation “to make hold”: Indeed, the initial squares are already there, and there is thus no need to construct them (in TMS VIII #1 we shall encounter a subdivision into smaller squares, and there their number is indeed found by “raising”—see page 78).

6

This part of the tablet is heavily damaged. However, #24 of the same tablet, dealing with three squares but otherwise strictly parallel, allows an unquestionable reconstruction.

7

In a simple false position, indeed, the provisionally assumed number has to be reduced by a factor corresponding to the error that is found; but if we reduce values assumed for \(c_{1}\) and \(c_{2}\) with a certain factor—say, \(\frac{1}{5}\)—then the additional \(5^{\prime}\) would be reduced by the same factor, that is, to \(1^{\prime}\). After reduction we would therefore have \(c_{2}=\frac{2}{3} c_{1}+1^{\prime}\).

8

This meticulous calculation shows that the author thinks of a new square, and does not express \(\square\left(c_{2}\right)\) in terms of \(\square\left(c_{1}\right)\) and \(c_{1}\).

9

This device was used constantly in the solution of non-normalized problems, and there is no reason to suppose that the Babylonians needed a specific representation similar to Figure 3.7. They might imagine that the measuring scale was changed in one direction—we know from other texts that their diagrams could be very rough, mere structure diagrams—nothing more than was required in order to guide thought. All they needed was thus to multiply the sum \(\Sigma\) by \(\alpha\), and that they could (and like here, would) do before calculating \(\beta\).

10

The quotient is called ba.an.da. This Sumerian term could mean “that which is put at the side,” which would correspond to way multiplications were performed on a tablet for rough work, cf. note 11, page 21.

11

That the value of \(c_{1}\) is calculated as \(1 \cdot c\) and not directly identified with \(c\) confirms that we have been working with a new side \(c\).

12

The tablet is rather damaged; as we remember, passages in ^¿…^? are reconstructions that render the meaning (which can be derived from the context) but not necessarily the exact words of the original.

13

The word ki.gub.gub is a composite Sumerian term that is not known from elsewhere and which could be an ad hoc construction. It appears to designate something stably placed on the ground.