Reverse algorithms in several Mesopotamian texts 391
no name in Sumerian, contrary to the determination of a reciprocal ( igi , in
Sumerian) and multiplication ( a - ra 2 , in Sumerian).
Th e determination of the reciprocal of a regular number is thus a funda-
mental objective of Babylonian positional calculation. Th e standard tables
furnish the reciprocals of the ordinary regular numbers. In what follows,
I call the numbers that appear in Table 12.2 ‘elementary regular factors’.
For the other regular numbers which do not appear in the standard table,
the scribes had recourse to a reciprocal algorithm, which is precisely what
Tablet A addresses.
Sachs identifi ed the reciprocal algorithm thanks to the verbal text of
Tablet B (VAT 6505). 21 First, I present the way in which Sachs understood
this algorithm and described it in an algebraic formula. Th en, I will analyse
the way in which Tablets A and B both refer to the same algorithm and the
ways in which they diff er. Th is contrast will indirectly permit some of the
particular objectives pursued in Tablet A to be clarifi ed.
Sachs’ formula
Th e colophon of Tablet B indicates that the text is composed of twelve sec-
tions. Th e entries are the fi rst twelve terms of a geometric progression for an
initial number 2.5 with a common ratio of 2 – the same terms which consti-
tute the beginning of Tablet A. In fact, only fi ve sections are even partially
preserved but these remains allowed Sachs to reconstitute the entirety of
the original text. Th e well-preserved entry of the seventh section is 2.13.20,
that is 2.5 aft er six doublings. Th e text may be translated as follows: 22
- 2,[13],20 is the igûm .[What is the igibûm ?]
- [As for you, when you] perform (the operations),
- take the reciprocal of 3,20; [you will fi nd 18]
- Multiply 18 by 2,10; [you will fi nd 39]
- Add 1; you will fi nd 40.
- Take the reciprocal of 40; [you will fi nd] 1,30.
- Multiply 1,30 by 18,
- you will fi nd 27. Th e igibûm is 27.
- Such is the procedure.
of a number (see, for example, the series of problems such as A 24194). Finally, in rare cases,
approximations for the reciprocals of irregular numbers are found (H2002: 29, n. 50).
21 Sachs 1947.
22 B Section 7, translation by Sachs 1947 : 226. Damaged portions of text are placed in square
brackets. igûm and igibûm are Akkadian words for pairs of reciprocals.