**MATHEMATICS (arithmetic, algebra and geometry)**

Mathematics is an important field of science in which the ancient Kemetyu worked. The accurate measurements of their enormous architectural and sculptural monuments are worthy proof of their preoccupation with precision. They would never have been able to reach this pitch of perfection without a minimum of mathematical capacity. Kemetyu mathematics may be considered under the three headings of arithmetic, algebra and geometry.

Two important mathematical papyri have come down to us from the Middle Kingdom (—2000 to —1750), those of Moscow and Rhind. The Kemetyu method of numeration, based on the decimal system, consisted of repeating the symbols for numbers (ones, tens, hundreds, thousands) as many times as necessary to obtain the desired figure. There was no zero. It is interesting to note that the Kemetyu symbols for the fractions 1/3, 1/2, 1/4 and so on originate in the myth of Horus and Seth, in which one of Horus’ falcon eyes was torn out and cut into pieces by Seth. It is these pieces that symbolize certain fractions.

Kemetyu administrative organization required a knowledge of arithmetic. The efficiency of the highly centralized administration depended on knowing exactly what was happening in each province, in all spheres of activity. It is not surprising, then, that the scribes spent an enormous amount of time keeping records of the area of land under cultivation, the

quantities of products available and their distribution, the size and quality of the staff, and so on.

The Kemetyu method of calculation was simple. They reduced all operations to a series of multiplications and divisions by two (duplication), a slow process which requires little memorization and makes multiplication tables unnecessary. In divisions, whenever the dividend was not exactly divisible by the divider, the scribe introduced fractions, but the system used only fractions whose numerator was the number 1. The operations on fractions were also done by systematic doubling. The texts contain numerous examples of proportional shares obtained in this way, with the scribe adding at the end of his calculations the formula ‘it is exactly that’, which is equivalent to our ‘QED’.

All the problems posed and solved in Kemetyu treatises on arithmetic have one trait in common: they are all material problems of the type that a scribe, isolated in some remote outpost, would have to solve daily, like the apportioning of seven loaves of bread among ten men in proportion to their rank in the hierarchy, or the calculation of the number of bricks required to build an inclined plane. It was, then, basically an empirical system, with little in it of an abstract nature. It is difficult to judge what elements of such a system might have passed into neighbouring cultures.

It is not exactly clear whether one may properly speak of a Kemetyu algebra and specialists in the history of science hold different views on this matter. Certain problems described in the Rhind Papyrus are formulated as follows: ‘ A quantity [ahâ in Egyptian] to which is added [or subtracted] this or that increment («) results in quantity (N). What is this quantity?’ Algebraically, this would be expressed as x ±- x/n = N, which has led some historians of science to conclude that the Egyptians used algebraic calculations. However, the solutions proposed by the scribe of the Rhind Papyrus to this type of problem are always reached by simple arithmetic, and the only instance in which algebra might have been used is a problem of division which implies the existence of a quadratic equation. The scribe solved this problem as a modern algebraist would do, but instead of taking an abstract symbol like x as the basis of calculation, he took the number 1. The question whether Kemetyu algebra existed or not depends therefore on whether one accepts or rejects the possibility of doing algebra without abstract symbols.

The Greek writers Herodotus and Strabo concur in the view that geometry was invented by the Kemetyu. The need to calculate the area of the land eroded or added each year by the flooding of the Nile apparently led them to its discovery. As a matter of fact, Kemetyu geometry, like mathematics, was empirical. In ancient treatises, the task was first and foremost to provide the scribe with a formula that would enable him to find rapidly the area of a field, the volume of grain in a silo or the number of bricks required for a building project. The scribe never applied abstract reasoning to the solution of a particular problem but just provided the practical means in the shape figures. None the less, the Kemetyu knew perfectly well how to calculate the area of a triangle or a circle, the volume of a cylinder, of a pyramid or a truncated pyramid, and probably that of a hemisphere. Their greatest success was the calculation of the area of a circle. They proceeded by reducing the diameter by one-ninth and squaring the result which was equivalent to assigning a value of 3.1605 to ** pi**, which is much more precise than the value 3 given to

*by other ancient peoples.*

**pi**Knowledge of geometry proved of considerable practical use in land surveying, which played a significant role in Kemet. There are many tombs with paintings showing teams of surveyors busy checking that the boundary-stones of fields have not been shifted and then measuring with a knotted cord, the forerunner of our surveyor’s chain, the area of the cultivated field. The surveyor’s cord or ** nouh** is mentioned in the earliest texts (c.—2800). The central government possessed a cadastral office, the records of which were ransacked during the Memphite revolution (c. —2150) but were restored to order during the Middle Kingdom (c. -199).

Note:

The terms: Black Egypt, Pharaonic Egypt and Ancient Egypt or Egypt have all been replaced with the more appropriate term ‘Kemet’.

Whereas the terms: Black Egyptians, Pharaonic Egyptians and Ancient Egyptians or Egyptians have all been replaced with the more appropriate term ‘Kemetyu’.

For a scholarly explanation on the above changes; see ‘The Egyptians as they saw themselves’.

Authors: **R. El Nadoury** with collaboration of **Vercoutter**. General History of Africa Vol.II. [Editor: G.Mokhtar] Ancient Civilisations of Africa. Chapter 5. Legacy of Pharaonic Egypt.

**R. El Nadoury** (Egypt); specialist in ancient history; author of numerous works and articles on the history of the Maghrib and of Egypt; Professor of Ancient History and Vice Chairman of the Faculty of Arts, University of Alexandria.