Reverse algorithms in several Mesopotamian texts 393
parallels) as well as in verbal version (Tablet B) (see Tables 12.6 , 12.7 and
12.8 below). However, in my estimation, this formula does not permit us to
explain the diff erences between the Tablets A and B, nor to grasp specifi c
objectives pursued by them in referring to the algorithm. Th e principal
shift s that I note between the ‘Sachs formula’ and the texts that it supposedly
describes are the following:
(1) Th e tools employed by Sachs in his interpretation (algebraic notation,
using semicolons and zeros) are not those used by the Old Babylonian
scribes. Th e ‘Sachs formula’ leaves unclear the actual practices of calcu-
lation to which the texts of Tablets A and B make reference.
(2) Th e text of Tablet B, just like the remains of Tablet A, does not refer to
the algorithm in an abstract manner but in a precise manner, with a
series of particular numbers, namely 2.5 and its successive doublings.
Th e algebraic formula does not explain the choice of these particular
numbers.
(3) None of the properties of Tablet A (spatial arrangement, iteration and
reciprocity) are found in Tablet B. Th e ‘Sachs formula’ does not allow
the stylistic diff erences that separate Tablets A and B to be described or
interpreted.
I would like to draw attention to the fact that Tablet A tells us much more
than an algebraic formula in modern language can convey. What informa-
tion is conveyed by the text of Tablet A but not contained by the ‘Sachs
formula?’ Answering this question will help us understand the original
process of the ancient scribes and their methods of reasoning. In that
attempt, I will concentrate for now on the particular properties of the text
of Tablet A, then on the particular numbers found therein.
Spatial arrangement
Using Sachs’ interpretation as a starting point, I am ready to detail the
algorithm of determining a reciprocal to which Tablet A refers. I rely on
the numeric data in Tablet A Section 7, which are presented above and in
Appendix 1 :
- the number 2.13.20 terminates with 3.20, which appears in the
reciprocal table, thus 3.20 is an elementary regular factor 25 of 2.13.20; - the reciprocal of 3.20 is 18; 18 is set out on the right;
25 As indicated above, I call any factor which appears in the standard reciprocal table (that is,
Table 12.2 ) an ‘elementary regular factor’.