The History of Mathematical Proof in Ancient Traditions

396 Christine proust

Even though the texts of the Tablets A and B refer to the same algorithm,

some features distinguish them. In the fi rst case, the text is two-dimensional:

the spatial arrangement of the numbers plays a critical role, referring to the

steps of calculation but not to the manner of carrying out the multiplica-

tions. In the second case, the text concerns a linear continuation of the

instructions, which refer not only to the algorithm, but also to the execution

of the multiplications. Another diff erence appears in Section 5. When the

numbers for which the reciprocal is determined reach a certain size, the

phenomenon of iteration appears in Tablet A, but not in Tablet B (so far as

the preserved portion allows us to judge).

Iteration

Let us consider Section 20, of which the transcription and the copy are

given in Table 12.3. (Th e bold type and underscoring have been added.)

First, I will explain the fi rst part of the section, concerning the reciprocal of

5.3.24.26.40 (lines 1 to 9).

Th e idea of determining the reciprocal through factorization is used

with more force here. Th e number for which the reciprocal is sought is

Table 12.3 Transcription and copy of Section 20

Line Transcription Copy Robson 2000 : 23

1 5.3.24.2 6.40 [9]

2 45.30. 40 1.30

3 1.8. 16 3.45

4 4. 16 3.45

5 1 6 3.45

6 14.3.45

7 5[2.44].3.45

8 1.19.6.5.37.30

9 11.51.54.50.37. 3 0 2

10 23.43.49.41. 15 4

11 1.34.55.18. 45 * 16

12 25.18. 45 * 16

1 3 6.45 1.20

14 9 6.[40]

15 8.53.20

16 2.22.13. 20

17 37.55.33.20

18 2.31.42.13.20

19 5.3.24.26.40