394 Christine proust
- the product of 2.13.20 by 18 is 40; 40 is therefore a second factor and
it is regular; 40 is set out on the left and its reciprocal 1.30 is set out on
the right; - the number 2.13.20 is therefore factored into the product of two
elementary regular factors: 3.20 and 40; - the reciprocal of 2.13.20 is the product of the reciprocals of these two
factors, namely the numbers set out on the right: 1.30 and 18; - the product of 1.30 by 18 is 27
- 27 is the desired reciprocal.
Th en, the reciprocal of this result is found, leading back to 2.13.20, the same
number as the initial data. For the time being, let us put aside this last step
in order to comment on the reciprocal algorithm, as I have reconstituted it
in the steps above.
Essentially, the algorithm is based on two rules. On the one hand, a
regular number can always be decomposed into the product of elemen-
tary regular factors – that is, into the product of numbers appearing in the
standard reciprocal table. 26 On the other hand, the reciprocal of a product
is the product of reciprocals. Th ese rules correspond to the spatial arrange-
ment of the numbers into two columns.
Th e factorization of 2.13.20 appears in the left column:
2.13.20 = 3.20 × 40
Th e factorization of the reciprocal appears in the right column:
18 × 1.30 = 27
Let us note an interesting diff erence between Tablets A and B in their
manner of executing the procedure. No addition appears in Tablet A, but
one instance appears in Tablet B (line 5). Th is addition may be interpreted
as being a step in the multiplication of 2.13.20 by 18. Th e number 2.13.20 is
decomposed into the summation of 2.10 and 3.20. Th en each term is mul-
tiplied separately by 18, and fi nally the two partial products are added. Th is
method of multiplication is economical. With one of the partial products
being obvious (3.20 × 18 is equal to 1 by construction), the multiplication is
reduced to 2.10 × 18. Th is decomposition of multiplication may draw on the
practices of mental calculation or the use of an abacus. It therefore seems
that the instructions of text B refer not only to the steps of the algorithm,
but also to the execution of multiplications. Text A, on the contrary, makes
reference only to the steps of the algorithm. Th e execution of multiplication
26 Naturally, this decomposition is not unique. Th e choices made by the scribes will be analysed
later.