This document is one of the main sources of our knowledge of Egyptian mathematics. It dates from around 1650 BC, but the scribe Ahmes states that he copied it from an earlier document dating from the XII-th dynasty - around 1800 BC. It begins as follows:

*Correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets.*

The papyrus consists of a famous table of the numbers 2/n, where n=3, 5, 7, ..., 101, all expressed as a sum of fractions with a numerator 1, and about 85 assorted mathematical exercises together with their solutions. Exercises 56-60 deal with pyramids. Here are exercises 56 and 57. Note that the term "ukha thebt" refers to the length of one side and the term "peremus" refers to the height of the pyramid.

56. *Example of reckoning a pyramid 360 in its ukha-thebet and 250 in its peremus. Cause thou that I know the seked of it.
You are to take half of 360; It becomes 180.
You are to reckon with 250 to find 180.
Result:
1/2 + 1/5 + 1/50.
A cubit being 7 palms, you are to multiply by 7.
1 7
1/2 3 + 1/2
1/5 1 + 1/3 + 1/15
1/50 1/10 + 1/25
Its seked is 5 1/25 palms.*

57. *A pyramid 140 in its ukha thebt, and 5 palms, 1 finger in its seked. What is the peremus thereof?
You are to divide 1 cubit by the seked doubled, which amounts to 10 1/2.
You are to reckon with 10 1/2 to find 7 for this one cubit.
Reckon with 10 1/2.
Two-thirds of 10 1/2 is 7.
You are to reckon with 140, for this is the ukha thebt.
Make two-thirds of 140, namely 93 1/3.
This is the peremus thereof.*

Here is a brief summary of the pyramid exercises from the Rhind Papyrus:

Exercise 56 concerns determining the seked, given the length of one side and the height. The seked turns out to be 5 1/25 palms (per cubit of height).

Exercise 57 concerns determining the height, given the length of one side and the seked. The seked is five palms, one finger (per cubit).

Exercise 58 concerns determining the seked, given the length of one side and the height. The seked is again five palms, one finger (per cubit).

Exercise 59 seems to concern two pyramids - one with a seked of five palms, two fingers and the other with a seked of five palms, one finger.

Exercise 60 seems to concern a pillar instead of a pyramid. It is rather small and steep, with a seked of four palms (per cubit).

**SOME LINKS**

THE RHIND PAPYRUS

The Egyptian Mathematics Papyri

The Rhind Mathematical Papyrus

Picture of the Rhind Papyrus

Ahmes: What were you thinking?

The Rhind (Ahmed) Papyrus

Egyptian Mathematics and the Rhind Papyrus

The History of Geometry in Egypt

THE SEKED

The Seked and the Geometry of the Egyptian Pyramids

The angle of the slope of the pyramid is accidental, and determined by its Seked.

The Slope of an Angle

Back to *PI* and the Great Pyramid