For many centuries the indecipherability of Egyptian hieroglyphs helped to perpetuate the Greek belief in Egypt as the source of higher knowledge and wisdom, in mathematics as well as other matters. However, with the decipherment of the trilingual Rosetta Stone (hieroglyphic, demotic and Greek) by Jean Champollion in the 1820s, the picture changed to reveal a civilisation much more pragmatic and down to earth. Although the pyramids and other monumental constructions provide us with substantial evidence that the Egyptians had a good knowledge of mechanics and astronomy, when it comes to mathematics the story is rather different; there is disappointingly little evidence of the Egyptians’ mathematical attainments. This is because most Egyptian documents were written on papyrus which is extremely fragile and deteriorates over time. Of the few papyri that survive only a tiny number (about a dozen) are concerned with mathematical calculation, of which the earliest dates from 1850 BCE and the most recent from 750 AD.
It is generally agreed that these mathematical texts were used to teach apprentice scribes basic numerical accounting and other techniques that they would need in their later professional life. They are written in a script called hieratic, a semi-cursive derivative of hieroglyphs. (Hieratic came into use for writing on papyri from about 2000 BCE, although hieroglyphic was retained for monumental stone-carving and more formal inscriptions.) Most of our knowledge of Egyptian mathematics is derived from two papyri, the Rhind papyrus (1650 BCE), the largest and the best preserved, and the Moscow papyrus (1850 BCE). The other mathematical papyri are mostly fragmentary in nature.
The Rhind papyrus, which (according to its scribe) is a copy of a text from 200 years earlier, was allegedly discovered in the ruins of a small building close to the temple of Ramesses II at Thebes. It is named after the man who bought it while on holiday in Luxor in 1858 and is now in the British Museum. It is approximately 18 feet long and 13 inches high, and contains extensive lists of divisions on one side and 87 mathematical problems on the other. The Moscow papyrus, now in the Pushkin Museum of Fine Arts, Moscow, is 15 feet long but only about 3 inches high. It contains 25 mathematical problems.
There are also a few Eygptian mathematical texts not on papyrus, the most significant of which are the Leather Roll (1650 BCE), which was so brittle that it remained unopened for 60 years, and the Two Thebes Wooden Tablets (2000 BCE). The Leather Roll contains an addition table, in duplicate, and is now in the British Museum, whilst the Thebes Wooden Tablets contain calculations relating measures of capacity and are now in the Cairo Museum.
The texts can be divided into two different types—problem texts and table texts. The problem texts pose mathematical problems and give their solution in the form of a step-by-step procedure, with each step representing a single mathematical command such as ‘add’ or ‘multiply’. These texts use plain language, not symbolism (apart from actual numbers themselves, of course) and the problems always involve particular numbers, not general formulae. The table texts are tables of numbers that are used in solving mathematical problems, e.g. tables of addition, tables used in fraction reckoning, and tables for conversions of measures.
The Egyptian number system is not too difficult to follow since integers were written according to a decimal system, with different symbols being used to represent the powers of ten: 1, 10, 100, … up to 1,000,000. In hieroglyphic notation these symbols were written additively with each symbol being repeated as often as necessary, eg the number 472 was expressed by writing the symbol for 100 four times, the symbol for 10 seven times, and the symbol for 1 twice. In hieratic, each number from 1 to 9 had a specific symbol, as did each multiple of 10, each multiple of 100, and so on. Thus in hieratic a given number, such as 472 was expressed by putting the symbol for two next to that for seventy and putting both of these symbols next to the symbol for four hundred. Although a zero is not necessary in such a system, the Egyptians did have a symbol for zero but it only occurs in papyri dealing with architecture and accounting.
Egyptian calculations were fundamentally additive. The most frequent operations were doubling and halving. Multiplication is reduced to repeated additions, and division, because it is the inverse of multiplication, is seen in terms of what one number must be multiplied by in order to get another, e.g. a problem such a 100 divided by 13 would be given as multiply 13 so as to get 100.
The most remarkable feature of Egyptian mathematics is its use of fractions. All fractions, with the lone exception of 2/3, are reduced to sums of what we call unit fractions, that is fractions with numerator 1, eg 1/2, 1/7, 1/34. Like integers, unit fractions are written additively, so that:
In hieroglyphic fractions are written with an elongated oval above the whole number, and in hieratic fractions are written with a dot over the whole number. The exception, 2/3, had its own special symbol. The reduction to sums of unit fractions was made possible by tables which gave the decompositions for fractions of the form 2/n, e.g:
The Rhind papyrus contains a 2/n table giving the decompositions for all odd n from 5 to 101.
Many of the problems are quite simple and do not go beyond a linear equation with one unknown. They deal with everyday concerns, such as the strength of bread and of different kinds of beer, the feeding of animals and the storage of grain, although often the numbers involved mean that the problems do not have any basis in reality. There are also geometrical problems, mostly related to measuring, and they too are conceived in a practical setting—finding the volume of a granary was particularly popular. From these problems we know that the Egyptians had formulae for the area of a triangle and of a circle, a value for the constant we call pi of 256/81 (3.1605), and formulae for solid volumes, such as the cube, the cylinder and, remarkably, the truncated pyramid. We are still waiting to find an Egyptian formula for an ordinary pyramid!