412 Christine proust
Th e number 16, its reciprocal, is set out on the right; on the same line, the
number 15, its square root, is set out on the left ; the product of 1.7.44.3.45
by 16 (which gives a second factor) is placed on the centre of the follow-
ing line. Th e process is repeated until a number for which the square root
is given by the standard tables is found. 45 Th e desired square root is the
product of the numbers recorded on the left.
It should be noted that this small text, like those found in the sections
of Tablet A, begins and ends with the same number, and as before, the cal-
culation forms a loop. It starts with an arithmetical operation (the square
of 1.3.45), then it proceeds by a sequence which carries out the reverse
operation (the square root of the resulting number, 1.7.44.3.45). Here, the
direct sequence and the reverse sequence rely on algorithms of a diff erent
nature, even though in the cases involving reciprocals, they rely on the same
algorithm. Could it be said that the calculation of the square of 1.3.45 is a
simple verifi cation of the result of the calculation of the square root? In this
case, it would be logical that the verifi cation should come at the end of the
calculation (as is the case in the verbal Tablet D) and not at the beginning.
Th e text thus illustrates something else, which seems to relate to the fact
that the square and the square root are reciprocal operations. Th is ‘some-
thing else’ is perhaps akin to what the author of Tablet A illustrated with the
reverse sequences.
Th e algorithm for calculating square roots is based on the same mecha-
nism of factorization as that for determining the reciprocal. In the numeric
versions, the rules concerning the layout are analogous: the factors areTable 12.9 Tablet C
Transcription Calculations Copy
1.3.45 1.3.45 × 1.3.45 = 1.7.44.3.45
1.3.45
15 1.7.44.3.45 16 inv(3.45) = 16; sq.rt.(3.45) = 15
15 18.3.45 16 inv(3.45) = 16; sq.rt.(3.45) = 15
17 4.49 sq.rt.(4.49) = 17
3.45 15 × 15 = 3.45
1.3.45 3.45 × 17 = 1.3.45
45 As in the case of the reciprocals, the calculations of the squares and square roots rely on a small
stock of basic results memorized by the scribes during their elementary education. Th e tables
of squares and square roots are largely found in the school archives. See, notably, Neugebauer
1935 –7: i ch. I.