6.1: Drawings?

All the texts that were discussed above were illustrated by geometric drawings. However, only two of the tablets carried geometric diagrams, and in both cases these illustrated the problem statement, not the procedure.

Many aspects of the procedures are inexplicable in the traditional arithmetical interpretation but naturally explained in a geometrical reading. in consequence, some kind of geometry must have particpated in the reasoning of the Babylonians. It is not very plausible, however, that the Babylonians made use of drawings quiete like ours. On the contrary, many texts give us reasons to believe that they were stisfied with rudimentary structure diagrams; see for example page 52 on the change of scale in one direction. The absence of particular names for \(L = 3\lambda\) and \(W = 21\phi\) in TMS IX #3 (see page 59) also suggests that no new diagram was created in which they could be identified, while \(\lambda\) and \(\phi\) could be identified as sides of the "surface 2."

After all, that is no wonder. Whoever is familiar with the Old Babylonian techniques will need nothing but a rough sketch in order to follow the reasoning; there is not even any need to perform the divisions and displacements, the drawing of the rectangle alone allows one to grasp the procedure to be used. in the same way as we may perform a mental computation, making at most notes for one or two intermediate results, we may also become familiar with "mental geometry," at most assisted by a rough diagram.

A fair number of field plans made by Mesopotamian scribes have survived; the left part of Figure 6.1 shows one of them. They have precisely the character of structure diagrams; they do not aim at being faithful in the rendering of linear proportions, as will be seen if we compare with the version in correct proportions to the right. In that respect they are similar to Figure 4.5, whose true proportions can be seen in Figure 4.6— pages 65 and 68, respectively. Nor are they interested in showing angles correctly, apart from the “practically right” angles that serve area calculations and therefore have a structural role.

Figure \(6.1\): A neo-Sumerian field plan (21st century bce), left as drawn on the tablet, right redrawn in correct proportions After F. Thureau-Dangin, "Un cadastre chaldéen”. Revue d’Assyriologie 4 (1897–98), 13–27.

Practicing "mental geometry" presupposes that one has first trained concrete geometry; real drawyings of some kind must thus have existed. However, cut-and-paste operations are not easily made on a clay tablet. The dust abacus, used by Phoenician calculators in the first millennium bce and then taken over by Greek geometers,¹ is much more convenient for this purpose. Here it is easy to cancel a part of a figure and to redraw it in a new position. A school-yard strewn with sand (cf. page 33) would also be convenient.

In the same way, dust or sand appears to have served in the first steps of learning the script. From this initial phase, we know the tablets on which are inscribed the models the students are supposed to have reproduced in order to learn the cuneiform characters. From the next phase we also have the clay tablets written by the students—but from the first phase the work of students has left no archaeological traces, which means that these will probably have been drawn in sand or dust. There is therefore no reason to be astonished that the geometrical drawings from the teaching of algebra and quasi-algebra have not been found.