I · THE OBJECT · BRITISH MUSEUM
The Rhind Mathematical Papyrus
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Ancient Egypt built in stone and calculated on papyrus. This roll shows how: eighty-four problems in arithmetic and measurement, worked step by step, copied from a text already ancient.
At a glance
- Material
- Papyrus
- Culture
- Second Intermediate Period Egypt
- Collection
- British Library
- Scribe
- Ahmes
Look closer
A copy of a copy
The scribe Ahmes states in his opening that he copied this text in regnal year thirty-three, month four of Akhet, from an ancient copy made in the time of an earlier king. The papyrus is therefore a copy of an earlier manuscript, itself already old when Ahmes worked. How much older is not recorded. The text presents itself as giving accurate reckoning for inquiring into things, and the knowledge of all things, mysteries and all secrets — a claim to comprehensive mathematical knowledge, though what follows is a practical handbook rather than a theoretical treatise.
The method for slopes
Several problems concern the seked — the slope of a pyramid, expressed as the horizontal distance moved for each cubit of vertical rise. The method treats slope as a ratio, calculated in palms and fingers. A pyramid with a base of a certain width and a height of a certain number of cubits has its seked worked out by dividing half the base by the height, then converting the result into palms. The calculation is concrete: it tells you how far out the stone must sit for each layer you build up.
Division without division
Egyptian arithmetic had no division operation as such. To divide by a number, the scribe multiplied by its reciprocal — its unit fraction. The papyrus includes a table at the front converting numbers of the form two divided by an odd number into sums of unit fractions: two divided by five becomes one-third plus one-fifteenth. These decompositions are used throughout the problems that follow. Why this method was preferred is not stated in the text, and remains a matter of scholarly discussion.
The story
The papyrus is a working manual. It contains eighty-four problems, each set out as a question followed by a worked solution. The problems cover practical matters: dividing loaves among men, calculating the volume of cylindrical granaries, finding the area of fields, determining the strength of beer, and working out the slope of pyramid faces. The arithmetic is concrete, not abstract. A problem does not ask for a general method; it asks how to divide nine loaves among ten men, and then shows the steps.
The method throughout is additive. Multiplication is done by doubling: to multiply by ten, you double, double again, then double once more, then add the results. Division, as noted, is handled by multiplying by a reciprocal fraction. The Egyptians wrote fractions almost exclusively as unit fractions — one over something — and the papyrus includes a reference table at the front that breaks two divided by each odd number from three to one hundred and one into sums of unit fractions. This table is used in the problems that follow.
One worked problem will serve. Problem forty-one asks: what is the volume of a cylindrical granary with a diameter of nine cubits and a height of ten cubits? The scribe first finds the area of the circular base. The method is to take away one-ninth of the diameter, then square the result. Nine minus one is eight; eight times eight is sixty-four. That is the area in square cubits. Multiply by the height of ten cubits and the volume is six hundred and forty cubic cubits. The answer is given, and the problem ends.
The method for the circle is an approximation. Taking away one-ninth of the diameter and squaring gives a value equivalent to using a number close to what we would call pi, though the papyrus does not discuss the approximation or state the principle behind it. It simply gives the procedure, and it works well enough for the purpose.
The pyramid problems are more involved. The seked is a measure of slope, defined as the horizontal offset in palms for every cubit of vertical rise. A cubit is seven palms, so the seked is a way of expressing gradient in whole numbers that a builder can use. The problems show how to calculate the seked from the base and height, and how to work backwards from a given seked to find the height. The method is consistent and the arithmetic is shown step by step, but the text does not explain why the procedure works. It demonstrates that it does.
This is characteristic of the papyrus as a whole. It is a manual of procedures, not a treatise on principles. The scribe does not prove that the method for the circle is correct, or derive the formula for a pyramid's seked from a general theory of slope. He shows you what to do, and the fact that the method is copied from an older text suggests it had been tested and found reliable. The knowledge is practical, handed down, and written out so that someone else can follow the steps.
Why it mattered then
The papyrus was copied during regnal year thirty-three, and the verso — the back of the roll — carries a note in a different hand recording events in regnal year eleven. The note mentions that a place called Heliopolis was entered in the second month of Shemu, and that in the first month of Akhet, on day twenty-three, someone referred to as he of the South broke into a place called Tjaru. Scholars identify this person as a Theban king and the events as part of a campaign against rulers in the north, but the note does not name the individuals or explain the context. It is a brief historical record on the back of a mathematical text, and it places the papyrus in a period of political division. The mathematics itself would have served administrators, scribes and builders. Calculating rations, measuring fields, assessing taxes, planning construction — all required arithmetic. The problems in the papyrus are not abstract exercises. They concern the quantities that mattered in a managed agrarian state: bread, beer, grain, land and stone. A scribe who could work through these problems could handle the calculations required to keep records, allocate resources and plan projects. The knowledge was professional, and the fact that it was copied from an older text suggests it was part of a scribal curriculum, passed from one generation to the next.
Why it matters now
The papyrus is evidence of mathematical practice rather than mathematical theory. It shows what methods were in use, how they were recorded, and what kinds of problems were considered worth solving. The procedures work, and they were refined enough to handle non-trivial calculations — volumes, slopes, divisions by fractions — but the text does not discuss why the methods are correct or how they were discovered. That makes it a different kind of mathematical document from the ones that later Greek and Islamic scholars produced, where proof and generalisation were central. The question of what the Egyptians knew is harder than it looks. The papyrus shows what they could calculate, but it does not show what they understood about the structure of the mathematics they were using. Did they know that their method for the area of a circle was an approximation, or did they treat it as exact? Did they have a concept of proof, or was reliability enough? The text does not say, and it is easy to read too much into the procedures or too little. The honest position is to describe what the papyrus records and to acknowledge where it is silent. What is clear is that the mathematics was practical, systematic and stable enough to be copied over time. The scribe Ahmes was working from a text already considered ancient, and the methods he recorded were still in use. That longevity suggests they were effective for the purposes they served, and that they were embedded in a scribal tradition that valued accuracy and continuity. The papyrus is a window into that tradition, and it remains a key source for understanding how calculation was done in a literate bureaucratic society that left most of its records on a material that does not survive.
The surprising detail
The verso of the papyrus — the back of the roll — carries a historical note that has nothing to do with mathematics. It records events in regnal year eleven: the entry into Heliopolis and a breakthrough at a place called Tjaru by someone referred to as he of the South. Scholars have linked this to a military campaign by a Theban king against rulers in the north, but the note itself does not name the individuals. It is a brief, factual entry, written in a different hand from the mathematical problems, and it suggests the papyrus was kept and used over time, with its blank back serving as a convenient surface for recording something else entirely.
What is disputed
The historical note on the verso has been interpreted as evidence for the date of the papyrus and the political context in which it was copied, but the note itself does not name the individuals involved. Scholars including Thomas Schneider and Irene Forstner-Mueller identify he of the South as the Theban king Ahmose I and the regnal year eleven as belonging to a northern ruler named Khamudi, but these identifications rest on broader historical reconstructions rather than statements in the papyrus itself.
Remember this
Eighty-four problems, each worked through step by step. Not a theory of mathematics, but a manual of methods that were old when the scribe copied them and still reliable enough to be written down.
Test yourself
The Egyptian method for finding the area of a circle was to take away one-ninth of the diameter and then square the result. Does this mean the Egyptians knew the value we call pi?
The papyrus does not say. The method gives a result close to the correct area, and it is consistent across the problems, which suggests it was a standard procedure. But the text does not discuss the approximation, does not state a principle behind it, and does not compare it to other possible methods. It simply gives the procedure and uses it. We can calculate that the method implies a value close to what we call pi, but the papyrus does not present it that way, and we do not know whether the scribes understood it as an approximation or treated it as exact. The lesson generalises: a working method is not the same as a stated principle, and a text that shows you what to do is not always a text that tells you why it works.
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