Reverse algorithms in several Mesopotamian texts 389
restored the orders of magnitude, in keeping with the indeterminacy of the
value in the cuneiform writing. However, in these circumstances, might it
be possible to establish ‘equalities’ between numbers, although their values
are not specifi ed? Even though the sign ‘=’ might be considered an abuse
of language (and an anachronism), I use it in the commentary. Th is con-
venience seems acceptable to me insofar as we bear in mind that the sign
‘=’ denotes not a relationship of equality between quantities, but rather an
equivalence between notations. For example, 2 × 30 = 1 signifi es that the
product of 2 and 30 is noted as 1.
How were these sexagesimal numbers used in calculations? Th e great
number of school tablets discovered in the refuse heaps of the scribal
schools present relatively accurate information about both the way in which
place-value notation was introduced in education in the Old Babylonian
period and also its use. Th e course of the scribes’ mathematical education
is particularly well documented at Nippur, the principal centre of teaching
in Mesopotamia. 15 At Nippur, and undoubtedly in the other schools, the
fi rst stage of mathematical apprenticeship consisted of memorizing many
lists and tables: metrological lists (enumerations of measures of capacities,
weights, areas and lengths), metrological tables (tables of correspondence
between diff erent measures and numbers in place-value notation) and
numerical tables (reciprocals, multiplications and squares). 16 A ft er having
memorized these lists, the apprentice scribes used these tables in calcula-
tion exercises which chiefl y concerned multiplication, the determination of
reciprocals and the calculation of areas. Documentation shows that place-
value notation came at precise moments in the educational curriculum.
Place-value notation does not occur among the expression of measure-
ments which appeal to other numerations, based on the additive principle.
Th ey appear in the metrological tables, where each measure (a value written
in additive numeration followed by a unit of measure) is placed in relation
to an abstract number (a number in place-value notation, not followed by
a unit of measure). Moreover, the abstract numbers are found exclusively
in the numeric tables and in exercises for multiplication and advanced cal-
culations of reciprocals. 17 Th e calculation of areas necessitates the transfor-
mation of measures into abstract numbers and back again, transformations
assured by the metrological tables. 18
15 Robson 2001b ; Robson 2002 ; Proust 2007.
16 Th ese tables are described in detail in Neugebauer 1935 –7: i ch. I.
17 In the following pages, ‘abstract numbers’ will refer to the numbers written in sexagesimal
place value notation.
18 For more details about these mechanisms, see Proust 2008.