# 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.

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.

Problems 48–55 show how to compute an assortment of areas. Problem 48 is notable in that it succinctly computes the area of a circle by approximating π. Specifically, problem 48 explicitly reinforces the convention (used throughout the geometry section) that “a circle’s area stands to that of its circumscribing square in the ratio 64/81.” Problem number 53 is the famous Thales’s theorem, 1300 years before he was born!

Other problems show how to find the area of rectangles, triangles and trapezoids. The final six problems are related to the slopes of pyramids.

The third part of the Rhind papyrus consists of the remainder of the 91 problems, being 61, 61B, 62-82, 82B, 83-84, and “numbers” 85-87, which are items that are not mathematical in nature. This final section contains more complicated tables of data, several pefsu problems which are algebraic problems concerning food preparation, and even an amusing problem (number 79) which is suggestive of geometric progressions, geometric series, and certain later problems and riddles in history.

So, it is truly no surprise that over several thousand years later, modern-day architects still cannot reproduce the great pyramids of Egypt… well, simply because these were quite advanced mathematicians and scientists. In all ancient Egyptian mathematics, there is not a single mathematical error, not in trigonometry, algebra, geometry, arithmetic, or mechanics. No wonder we still are in awe at the Egyptian Pharaonic era. It is a pity that some of their work was “stolen” by the likes of Thales and Pythagoras and renamed after them. It is a pity that so many of us have lived in ignorance of such great science on our continent. It is sad that most African kids have been taught in their classrooms the Thales theorem or Pythagorean one or Archimede (problem 10 of the Moscow Papyrus solves Archimede’s at least 1700 years earlier) without ever being told about their famous intelligent ancestors who had the answers to all these over 3000 years earlier. So today, after looking at the Rhind Papyrus and at problem number 53, I no longer call this the Theorem of Thales or Intercept Theorem, but rather the Egyptian theorem, and Pythagoras triangle, the Egyptian triangle.

For more information, check out the works of Cheikh Anta Diop (Apport de l’Afrique à la Civilisation Universelle, 1985 – published in Presence Africaine 1987), Beatrice Lumpkin (African and African-American contributions to mathematics, PPS Geocultural Essaysbseries, 1987), Ivan van Sertima, and Elikia M’Bokolo.

# The Ishango Bone: Craddle of Ancient Mathematics

Today, I would like to talk about the Ishango bone, or rather the first evidence of a calculator in the world.  Named after the place where it was found in the Democratic Republic of Congo (DRC), the Ishango bone is what is called a bone tool or the craddle of mathematics.  Dating as far back as 22000 years ago, in the Upper Paleolitic era, the Ishango bone is a dark brown bone which happens to be the fibula of a baboon, with a sharp piece of quartz affixed to one end for engraving It is the oldest attestation of the practice of arithmetic in human history.

The Belgian geologist Jean de Heinzelin de Braucourt uncovered the bone buried in layers of volcanic ashes on the shores of Lake Edward in the Ishango region in DRC, near the border with Uganda.  The Ishango bones are actually two (2) bones of baboon, 10 to 14 cm long, with several incisions on each faces.  The smallest of the two bones was the first to be exposed at the Royal Belgian Institute of Natural Sciences in Brussels; it carries several incisions organized in groups of three columns.

The left column can be divided in 4 groups, with each group possessing 19, 17, 13, and 11 notches.  The sum of these being 60.  Those are the 4 successive prime numbers between 10 and 20.  This constitutes a quad of prime numbers.

The central column is divided in groups of 8.  By an approximate count, one can find (in the parenthesis, is the maximum number): 7 (8), 5 (7), 5 (9), 10, 8 (14), 4 (6), 6, 3.  The minimal sum is 48, while the maximal sum is 63.

The right column is divided into 4 groups, where each group has 9, 19, 21, and 11 notches.  The sum of these 4 numbers is 60.

The second bone has not been well-studied.  However, we know that it is composed of 6 groups of 20, 6, 18, 6, 20, and 8 notches.