# The Rhind Papyrus or Advanced Ancient Egyptian Mathematics

1300 years before Thalès was born, Ancient Egyptians solved the famous theorem which now bears his name, Theorème de Thales in French, or Intercept theorem in English. Back then, it was called problem Number 53, and was part of the Rhind Papyrus. The value for π was already approximated as 3.16 (a 0.6% margin error, extremely good even by modern standards), 4000 years before that value was fixed at 3.14. So why are these theorems called after Pythagoras or Thales, when they had already existed thousands of years prior to their living?

The Rhind Papyrus is a famous papyrus written by the scribe Ahmes (Ahmose) around 1650 BC. It was copied from a now lost text from the reign of king Amenemhat III (12th dynasty) 1500 years prior to Ahmose’s birth. His papyrus is one of the best known examples of advanced Egyptian mathematics; mathematician-priests of the Nile valley knew no peers. It was found during illegal excavations in or near the Ramesseum. It has been housed in the British Museum since 1865 along with the Egyptian Mathematical Leather Roll. Originally, this papyrus was 5 m long and 33 cm high. This is the most famous mathematical papyrus to have survived from Ancient Egypt. Pyramid of Giza in the 19th century

This papyrus shows that Ancient Egyptians were very advanced mathematicians and were familiar with both roots and square roots. They could plot an arch by using offsets that were measured at regular intervals from a base line, and they could also find out areas. To find the area of a circle, the Egyptians used an area of a square on an 8/9 of the diameter, or (7/8) squared. They could also figure out the area of a triangle. They knew that the volume of a frustum of a square pyramid equaled (1/3) h (a2 + ab + b2)modern mathematicians, 4000 years later, have still not found a better approximation. They also knew that to make right angled triangles, they had to use the ratio of 3:4:5. The Great Pyramid of Khufu, Great Pyramid of Giza, from the 4th Dynasty is a mathematical wonder: It is laid out with geometric precision – a near-perfect square base, with sides of 230 m that differ from each other by less than 20 cm, and faces that sloped upwards at an angle of 51 to reach an apex nearly 150 m above the desert floor. Khufu’s pyramid was built long before the Ahmose papyrus was written, indicating the beginning of this mathematical theory was about 1,000 years old by the year 1650 B.C.E.

The Rhind Papyrus is divided in 3 books. Book 1 includes problems 1 – 40, and is about algebra and arithmetics. Book 2 focuses on Geometry and spans problems 41 – 59, while Book 3 focuses on miscellaneous problems from number 60 – 87.

The first part of the Rhind papyrus, book 1, consists of reference tables and a collection of 21 arithmetic and 20 algebraic problems. The first part of the papyrus is taken up by the 2/n table. The fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. Problems 41 – 46 show how to find the volume of both cylindrical and rectangular granaries. In problem 41, Ahmose computes the volume of a cylindrical granary. In modern mathematical notation (and using d = 2r) this gives V = (8/9)2 d2h = (256/81)r2h. The fractional term 256/81 approximates the value of π as being 3.1605.

Problem 47 is a table with fractional equalities which represent the ten situations where the physical volume quantity of “100 quadruple heqats” is divided by each of the multiples of ten, from ten through one hundred. The quotients are expressed in terms of Horus eye fractions, sometimes also using a much smaller unit of volume known as a “quadruple ro”. Egyptian numerals were based on 10, a precursor to our decimal system.