The History of Mathematical Proof in Ancient Traditions

402 Christine proust

32 Th is property is the result of a more general rule: for a given base, the divisibility of an integer by

the divisors of the base is seen in the last digits of the number. For a discussion of the particular

problems resulting from divisibility in ‘fl oating’ base-60 cuneiform notation, a system in which

there is no diff erence between whole numbers and sexagesimal fractions, see Proust 2007 : §6.2.

33 Th e word ‘large’ has nothing to do with the magnitude of the abstract numbers, since

magnitude is not defi ned, but with their size. A two-place number is ‘larger’ than a single-

place number; for numbers with the same number of digits, the ‘larger’ number is the last

in the lexicographical order. Th e speed of the algorithm depends on the size of the numbers

thus defi ned: the ‘larger’ the factors are, the fewer factors there will be and thus fewer

iterations. Let us specify that the order according to the size of the numbers is diff erent from

the lexicographical order mentioned above. Th e two orders appear in cuneiform sources.

Th e order according to the size appears in the Old Babylonian reciprocal tables, and the

lexicographical order occurs in the Seleucid reciprocal tables such as AO 6456, as well as in the

arrangement of the multiplication tables in the Old Babylonian numerical tables.

a sequence of digits which form a regular number. 32 All that is needed is to

adjust for a suitable sequence. (In the case of 2.13.20, we may take 20, or

3.20, or even 13.20.) In practice, the fi nal part , insofar as it is an elementary

regular number, is likely to be a factor. (For 2.13.20, the factor might be 20,

or 3.20.) In the majority of cases, the scribe chose, from among the pos-

sible factors, the ‘largest’ (3.20 rather than 20), in order to render the algo-

rithm faster. 33 Th us, in general, the selected factor is the largest elementary

Table 12.5 Selection of factors in the reverse sequences

Number

to factor Section

F a c t o r

chosen

Reciprocal

of factor

Largest

elementary

regular factor

7. 12 3 1 2 5

1.41. 15 11 15 4 1.15 (2)

23.43.49.41. 15 19 15 4 1.15

2. 15 5 1 5 4 ( 2′)

14. 24 2 24 2.30
    3.22. 30 10 30 2 2.30
    12.39.22. 30 14 30 2 2.30
    47.27.39.22. 30 18 30 2 2.30 (3)
    50.37. 30 12 30 2 7.30
    11.51.54.50.37. 30 20 30 2 7.30


15. 30 8 3 0 2 ( 3′)


16. 36 4 36 1.40


17. 45 9 45 1.20


18. 48 5 48 1.15


19. 48 1 48 1.15
    25.18. 45 13 3.45 16 45
    1.34.55.18. 45 17 3.45 16 45 (4)
    5.55.57.25.18. 45 21 3.45 16 45