As the centuries passed, among the various units of measurement used by the Egyptians (Hannig 1995; Schlott-Schwab 1981) some survived almost unchanged, some appeared and disappeared (e. g. ZLaba, Verner et al. 1976: 83), and some changed their value (e. g. Imhausen 2003: 156). The basic unit of linear measurement was the cubit (‘‘royal’’ and ‘‘small’’) and its subunits, the palm and the finger. The small cubit corresponded to the length of the human forearm and was equal to 6 palms (c. 45 cm), whereas the royal cubit corresponded to 7 palms (c. 52.3 cm); each palm (c. 7.4 cm) was divided, in turn, into 4 fingers (c. 1.8 cm). The royal cubit was the basic unit of measurement used in architecture, as many standing monuments clearly attest (Arnold 1974a: 29-31; Arnold 1991b: 10-1; Rossi 2004: 96-147); cubic cubits appear in building records reporting on the foundations of standing buildings and on the works done in rock-cut tombs (Simpson 1963: 124-6; Koenig 1997: 9). Agricultural fields were also measured on the basis of the cubit: their dimensions were generally expressed in khet (equal to 100 cubits) and their areas in setat (also indicated with the Greek term aroura), equal to one square khet (100 x 100 cubits; Clagett 1999: 12-3; Imhausen 2003: 66-7; see also Arnold 1991: 252).
The mathematical sources contain problems of calculation of rectangular, triangular and trapezoidal areas, that may well reflect the actual practice of landsurveying. The calculation of the area of a circle as large as a field, on the other hand, is likely to represent a theoretical case (Gillings 1972: 139), most probably meant to complete the series of examples on how to calculate the area of the most common geometrical figures.
The main unit of measurement for capacity was the heqat (corresponding, at least in the Middle Kingdom, to c. 4.8 litres), with its multiples and submultiples ranging from the tiny ro, equal to 320 of heqat (c. 0,015 litres), to the khar, corresponding to 20 heqat (that is, c. 96 litres; Clagett 1999: 14-5; Imhausen 2003: 58). Beside the intrinsic commensurability of these units and subunits, an important aspect was the relationship existing between the khar and the cubic cubit: the former was known to be equal to | of the latter, and this proportion allowed scribes to shift easily from capacity to volume, and the opposite. A common problem assigned to a scribe, in fact, was calculating the volume of a granary and establishing how much grain could be stored inside, or the opposite.
In general, the subjects of the surviving mathematical problems clearly reflect the type of tasks that a Middle Kingdom scribe was likely to be assigned in his working life. They also reflect the various uses of mathematics which played an important role in several contexts: the administration of the community was based on an equitable distribution of goods; large-scale production of food had to be strictly regulated from the initial step of land-surveying to the final act of storing grain; and the construction of important monuments required a constant control of the workforce and of the geometry of the building (cf. Rossi 2009). The respected scribe, well fed and well taken care of by his community (cf. Lichtheim 1976: 167-8), owed part of the success of his profession to the mathematical documents at his disposal. As the first lines of the papyrus Rhind state, they contained ‘‘accurate reckoning for inquiring into things, and the knowledge of all things, mysteries [...] all secrets’’ (Clagett 1999: 122).
World History
