The ancient Egyptian civilisation developed on the banks of the Nile about four thousand years before Christ. We have been able to learn about their system for recording words and numbers by using hieroglyphic symbols from papyrus manuscripts. Although the Babylonian tablets survived for so long so well, this wasn't the case with the papyrus documents. However, understanding the system has been made much easier since the discovery of the Rosetta Stone in 1799. This large tablet is inscribed with text written in three different scripts: Greek, Demotic (the local language) and hieroglyphs. This allowed hieroglyphs to be translated into Greek and then into other languages. The most important mathematic document to have been discovered is the Rhind Papyrus (named after the  Scottish antiquary who found it in 1858) or the Ahmes Papyrus (as it is also known, named after the scribe who copied it in about 1650 B.C. The material appears to have belonged to the Middle Kingdom (2000 to 1800 B.C.) and some of its knowledge may have come from Imhotep, who was the architect in charge of the building of Pharaoh Zoser's pyramid about 5,000 years ago. Other papyri of great importance are the Kahun Papyrus, the Berlin Papyrus and the Moscow Papyrus. The Rhind Papyrus and the Moscow Papyrus both contain examples of mathematical problems. The table below reflects the extent of mathematical knowledge during ancient Egyptian civilisation. It is important to bear in mind that the Egyptians used mathematics to solve everyday problems (e.g. to survey the land by the Nile which was often flooded when the river burst its banks and to construct pyramids). All the examples refer to specific cases, concrete shapes with fixed measurements, which are descriptive but do not include what we nowadays refer to as proof. These particular examples were probably used as teaching material at the time.

Hieroglyphic script		Left: The Rhind Papyrus inscribed with hieratic script 1650 B.C.      Right: The Moscow Papyrus
Number theory	Whole fractions and 2/3 Addition, multiplication and division using the duplication process and the "rule of three".
Algebra	Algebraic problems which are solved using the "method of false position". The unknown is referred to as a heap:  Problem 24: Work out the value of the heap if the heap and a seventh of the heap is equal to 19. Solution: Choose a possible value for the heap, e.g. 7, carry out the operations and you get 8 as an answer. Compare 8 to 19 and you see that you have to multiply 8 by (2+1/4+1/8) to get 19. In the same way multiply the brackets by 7 to get the value of the unknown heap. Finally show that 7·(2+1/4+1/8)=16+1/2+1/8, which is the value of the unknown heap. As you can see, only unit fractions are used.
Geometry	The area of an isosceles triangle The area of an isosceles trapezium The area of a circle: pi=3.16 The length of the circumference: the ratio of the area of a circle to the circumference is the same as the ratio of the area of the circumscribed square to its perimeter. "Seqt", which is a type of Trigonometric Ratio or quotient between the length, measured in "hands" and the height measured in "cubits". The volume of the frustum of a pyramid: V=h·(a2+ab+b2)/3, where h is the height and a and b are the sides of the square bases. The surface area of a basket or half-cylinder knowing the diameter and length: Multiply the length of the semicircle by the length of the cylinder.

The area of an isosceles triangle is equal to the area of a rectangle whose base is half that of the triangle and whose height is the same as that of the triangle.

The area of an isosceles trapezium whose larger base is 6, smaller base is 4 and the distance between them is 20, is equal to a rectangle with the same height but whose base is half the sum of the trapezium bases.

 The area of an isosceles trapezium is equal to half the area of a parallelogram whose base is the sum of the bases of the trapezium and whose height is the same as that of the trapezium. 

 1.- This diagram is more intuitive but requires the use of a parallelogram. However, the Egyptians didn't make use of these shapes.

This problem gives an idea of how the Egyptians used to work out the area of a circle by using a method of approximation.

1.- An octagon is used. The method has several stages:

Draw a circle

Draw a square whose side is equal to the diameter of the circle.

Divide the side into three equal parts and keep the central segment.

Cut off the four corner isosceles triangles to obtain an octagon whose area is close to that of the circle.

In this window you can see the calculation that are made and compare the approximate results given to the actual results.