392 Christine proust
According to Sachs, whose notations I have reproduced, 23 the algorithm
is based on the decomposition of the initial number c as the sum a + b ,
this decomposition is summarized by the following formula (in which the
reciprocal of a number n is denoted by nˉ ) :
(^) caba=+=⋅+() 1 ba
Applied to the data in B Section 7, this formula leads to the following
reconstruction:^24
c
ca
a
ab
ab
==
=×=
+=+
2,13;20
=+= +b 3;20 2,10
3;20 0;18
0;18 2,10 39
1139940
1400;1,30
10;180;1,300;0,27
+==
=×+ = × =
ab
ca ab
On the one hand, the ‘Sachs formula’ allows us to follow the sequence of
calculations by the scribe and on the other hand it establishes for us the
validity of the algorithm according to modern algebra. Moreover, it pro-
vides historians with a key to understanding Tablet A and its numerous
parallels. In fact, as indicated above, the fi rst twelve sections of Tablet A
contain the same numeric data as their analogues in Tablet B. For example,
the transcription of Section 7 of Tablet A is as follows:
[2.]13.20 18
40 1.30
[27] 2.13.20
In Tablet A Section 7 are found, in the same order, the numbers which
appear in the corresponding section of Tablet B. Clearly, the numeric Tablet
A refers to the same algorithm as the verbal text of Tablet B. Until now, the
‘Sachs formula’ has provided a suitable explanation of the reciprocal algo-
rithm. Th is formula is generally reproduced by specialists in order to explain
texts referring to this algorithm in numeric versions (Tablet A and its school
23 I n translations , like Neugebauer, Sachs used commas to separate sexagesimal digits, but unlike
Neugebauer, he did not use ‘zeros’ and semicolons to indicate the order of magnitude of the
numbers. He used these marks only in the mathematical commentaries and interpretations of
the sources.
24 Sachs 1947 : 227.