A scribe’s handbook of worked problems in arithmetic and geometry (fractions, rations, granaries, field areas) and the only ancient text that shows how Egyptian builders specified the slope of a pyramid: by its seked, the horizontal run in palms for each cubit of rise.
The papyrus is a roll a little over five metres long, now in two main pieces in the British Museum (with small fragments in the Brooklyn Museum). Its scribe, Ahmes, dates his copy to the thirty-third year of the Hyksos king Apepi, around 1550 BC, and says he is copying an older book of the time of Amenemhat III of the Twelfth Dynasty, some three centuries earlier. The Scottish antiquarian Alexander Henry Rhind bought it at Luxor in 1858, and it passed to the British Museum after his death.
It is a teaching text. After a long table for splitting fractions of the form 2⁄n into sums of unit fractions (the only fractions Egyptian arithmetic used, apart from ⅔), it sets out some eighty-four problems, each with its answer and a worked check. They are the problems of an administrator: how to divide loaves among men, how much grain a round or square granary holds, how large a field is, how strong a batch of beer or bread will be.
For architecture the precious pages are Problems 56 to 60. They ask for the seked of a pyramid from its base and height, or its height from its base and seked. The seked is given in palms and fingers per cubit of rise (the royal cubit being seven palms of four fingers), which is exactly how a mason would set out a sloping face with a square and a plumb-line. Problem 56, for instance, takes a pyramid 360 cubits across and 250 high and finds a seked of 5 1⁄25 palms; Problems 57 to 59 use 5 palms 1 finger, the 3 : 4 : 5 slope of about 53° that is very close to the measured slope of Khafre’s pyramid at Giza.
Elsewhere the papyrus finds the area of a circle by squaring eight-ninths of its diameter (equivalent to taking π as about 3.16), computes the volumes of granaries, and handles slopes, areas and volumes with the same practical good sense. It does not explain why its rules work; it shows how to use them, which is how most architectural knowledge was handed down before the Greeks.
The book, part by part
I The 2⁄n table
A reference table decomposing 2 divided by every odd number from 3 to 101 into sums of distinct unit fractions, the working tool for every later division.
II Problems 1–40: arithmetic
Dividing loaves among men, “completion” problems, and the aha problems: linear equations solved by false position (guess, check, and scale the guess).
III Problems 41–47: granaries
Volumes of cylindrical and rectangular granaries in cubic cubits, converted into the corn measure (the khar and the hekat).
IV Problems 48–55: areas
Areas of rectangles, triangles and trapezia; the area of a round field found by squaring eight-ninths of its diameter, illustrated by an octagon drawn in a square.
V Problems 56–60: the seked of pyramids
The slope of a pyramid given as a seked, so many palms (and fingers) of horizontal run for one cubit of vertical rise, found from the base and height, or used to find the height. Problem 60 applies the same reasoning to a steeper monument whose seked is a quarter of a cubit.
Rules: The seked of a pyramid · The Great Pyramid: 440 cubits by 280 · Measure in cubits, palms and fingers
VI Problems 61–84: practical reckoning
Multiplying fractions, the “strength” (pefsu) of bread and beer, feed for fowl and cattle, and a geometric progression of houses, cats, mice and grain.