The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus is a working document. It contains eighty-four problems, each with a worked solution, copied out by a scribe named Ahmes in the thirty-third regnal year of a king called Awserre. That places the copy around 1550 BC, in the Second Intermediate Period, a time when Egypt was divided and the Hyksos kings controlled the north. Ahmes tells us he is copying from an older manuscript made during the reign of Nimaatre, roughly two centuries before his own time. He is preserving knowledge, not creating it.

The problems are practical. Divide a hundred loaves among ten men. Calculate how much grain is needed to make a given quantity of beer. Work out the area of a circular field. Find the slope of a pyramid face. Distribute rations proportionally. These are the problems a scribe in an administrative role would need to solve: accounting, surveying, construction, resource allocation. There is no abstract theory here, no proofs, no exploration of why a method works. Each problem states a question, shows the steps to the answer, and moves on.

Problem 79 is typical. It lists a household with seven people, forty-nine cats, three hundred and forty-three mice, two thousand four hundred and one ears of grain, and sixteen thousand eight hundred and seven measures of grain. The problem asks for the total. The solution simply adds them up. The numbers follow a geometric progression with a ratio of seven, but the papyrus does not say so. It treats this as an accounting exercise, not as an illustration of a pattern.

The seked problems are more revealing. The seked is a measure of slope, expressed as the horizontal distance you move inward for every cubit you rise. Problem 56 gives a pyramid with a height of 250 cubits and a square base 360 cubits on each side. The solution takes half the base, 180 cubits, and divides it by the height. That gives a ratio, which is then converted into palms, the smaller unit. The answer is five palms and one finger per cubit of rise. This is a real calculation for pyramid construction, where maintaining a consistent slope was essential.

Ahmes makes mistakes. In some problems the arithmetic does not balance. In others a step is skipped or a method is applied inconsistently. These errors are informative. They show that Ahmes did not always understand the material he was copying, and that Egyptian mathematics was learned by example rather than by general principles. A scribe would memorise procedures for standard problems and adapt them as needed. When the problem changed in an unfamiliar way, the method might fail.

The papyrus also contains a long table at the beginning, converting fractions of the form two divided by an odd number into sums of unit fractions. Two divided by five becomes one-third plus one-fifteenth. Two divided by seven becomes one-quarter plus one-twenty-eighth. This table was essential because Egyptian mathematics used almost exclusively unit fractions: one-half, one-third, one-tenth, but not two-fifths or three-sevenths. The only exception was two-thirds, which had its own hieratic sign. Every other fraction had to be broken down into unit parts, and the table provided standard conversions the scribe could refer to.

This system makes some operations straightforward and others laborious. Adding one-half and one-quarter is simple. Multiplying fractions or dividing them becomes a series of expansions and recombinations. Modern readers sometimes assume the Egyptians must have had a more efficient method they kept hidden, but the papyrus shows no sign of it. The unit fraction system is what they used, and Ahmes works within it throughout.