414 Christine proust
seen above, it is based principally on the doublings of 2.5). Moreover, the
group of tablets containing the calculations of square roots is small, whereas
the group of exercises of the calculation of reciprocals is numerically impor-
tant. Th e great frequency of calculations of reciprocals is undoubtedly
explained by the importance of this technique in calculation, but another
reason may be postulated. In the reciprocal, the two components (factoriza-
tion and the determination of reciprocals) are superimposed. Th e algorithm
for the determination of a reciprocal by factorization puts the mechanism
of factorization fi rst. Th e determination of a reciprocal by factorization is
thus a fundamental procedure, 46 essential to other algorithms, even though
it is applied in a less general way for the roots than for the reciprocals.
Consequently, the reciprocal exercises probably occupy a more elementary
educational level than those that contain square roots. Th e calculations of
square roots may be situated between the work of beginning scribes and
works of scholars, in a grey area that has left us few traces.
What, then, of the cube roots? Th ey appear in two verbal texts, wherein
they are treated in a manner identical to the square roots, except for the
verifi cations, which do not appear in either case. 47 No numeric version is
known for these calculations. It cannot be excluded that the absence of a
numeric version of the calculation of a cube root is due to the chances of
preservation but other explanations are possible. Indeed, tables of squares,
square roots and cube roots are known to us from the preserved numeric
tablets, but tables of cubes are unknown. Th e absence of a table of cubes is
undoubtedly linked to the fact already mentioned that multiplication is an
operation with two arguments. Consequently, the cube root has no reverse
operation in the Mesopotamian mathematical tradition. Th is fact would
explain why it has not been found in a numeric format, which is founded
on the notion of reciprocity.
Th is analysis of the calculation of square roots also emphasizes by
contrast the fact that the reciprocal algorithm is a combination of two
diff erent components (factorization and the determination of a reciprocal).
In addition, it may be seen that the numeric texts have an approach46 Th e Akkadian term maks. arum probably has some link with the process of factorization. It
appears in two texts, in slightly diff erent senses: it appears in the incipit of tablet YBC 6295
cited in Table 12.6 ([ ma ] -ak-s. a-ru-um ša ba-si = the maks. arum of the cube root); it designates
an enlargement in tablet YBC 8633.
47 Note also the following curious detail: in VAT 8547, all the entries appear in the standard
tables of cube roots, and the application of the reciprocal algorithm to these numbers leads
to a complication of the situation. Th us, 27 is decomposed according to a somewhat artifi cial
manner as the product of 7.30 and 3.36. It is clear that in this case, as in that of Tablet A, the
purpose is not to obtain a new result.