Rhind Mathematical Papyrus: Mathematics in Ancient Egypt

The Rhind Mathematical Papyrus is an ancient Egyptian handbook of mathematics, copied by a scribe named Ahmes around 1550 BC. It contains tables and 84 worked problems on fractions, the division of food, the areas of fields, the volumes of granaries and the slopes of pyramids. It is the largest and best-preserved mathematical text from ancient Egypt and the main source for how Egyptian scribes calculated.

Background: scribes and numbers

In ancient Egypt, mathematics was a working tool of government. Scribes had to calculate rations for workers, taxes on harvests, the area of fields after the annual Nile flood, and the materials needed for building projects. They learned these skills in scribal training, partly by copying model problems.

Egyptians used a decimal system with separate signs for 1, 10, 100, 1,000 and higher powers of ten. Written in hieratic, the cursive script used on papyrus, the numbers had compact signs of their own. Multiplication and division were done by repeated doubling and halving, a method that needed no multiplication tables beyond the two times table.

The scribe Ahmes and the date

The papyrus opens with a title promising accurate reckoning and insight into all things, including obscure matters and secrets. The scribe then names himself as Ahmes and states that he copied the text in regnal year 33 of King Apepi, one of the Hyksos kings who ruled northern Egypt. He adds that his copy was made from an older original written under King Nimaatre, the throne name of Amenemhat III of the Twelfth Dynasty.

This makes the papyrus a copy, made around 1550 BC, of a work composed more than two centuries earlier, in the Middle Kingdom. Ahmes is the earliest mathematician-scribe known by name, though he was a copyist rather than the author of the methods.

The back of the scroll also carries later notes, including a brief record of military events in a “year 11”, which many historians link to Ahmose I’s campaign against the Hyksos.

What the papyrus contains

The papyrus is about 32 centimetres high and, in its two main pieces, a little over 5 metres long. Modern editors number its problems from 1 to 84, although the exact count differs between editions.

Tables of fractions

Egyptian scribes expressed fractions as sums of different unit fractions, that is, fractions with 1 on top, such as 1/2, 1/3 or 1/7. The only common exception was 2/3, which had its own sign. To work with this system, scribes needed ready-made tables.

The papyrus begins with a table giving 2 divided by every odd number from 3 to 101 as a sum of unit fractions. For example, 2/5 is written as 1/3 + 1/15. A second table covers the division of the numbers 1 to 9 by 10. These tables were the foundation for the problems that follow.

The choices in the 2/n table are not random. The scribes generally preferred decompositions with few terms, small denominators and even numbers that were easy to halve, and modern mathematicians have spent considerable effort reconstructing the rules that may lie behind them. Whatever the method, the table let a scribe divide almost any quantity without starting from scratch.

Arithmetic and algebra

Many problems deal with sharing loaves of bread or measures of beer among a number of men. Others, known as “aha” problems from the Egyptian word for a quantity, amount to simple linear equations. Problem 24, for instance, asks for a quantity which, when its seventh is added to it, makes 19. The scribe solves it by “false position”: guessing a convenient number, seeing what result it gives, and scaling the guess to fit.

Problem 40 divides 100 loaves among five men in an arithmetic progression, so that one-seventh of the three largest shares equals the two smallest. Problem 79 lists 7 houses, 49 cats, 343 mice, 2,401 ears of spelt and 16,807 measures of grain, powers of 7 added together to give 19,607. It recalls the much later English rhyme “As I was going to St Ives”.

Geometry

The geometric problems calculate the areas of rectangles, triangles and circles, and the volumes of cylindrical and rectangular granaries. To find the area of a circle, the scribe takes the diameter, removes one-ninth of it and squares the result. A circle 9 units across is thus given the area of a square 8 units on a side, 64. This rule is equivalent to a value of about 3.16 for pi, remarkably close to the true 3.14.

Problems 56 to 60 deal with pyramids. They use the seked, a measure of slope given as the number of palms a face runs horizontally for each cubit of height. A seked of 5 palms and 1 finger, used in several problems, gives a slope of about 53 degrees, close to that of Khafre’s pyramid at Giza; the slope of the Great Pyramid of Giza corresponds to a seked of 5 palms and 2 fingers.

Discovery and study

The papyrus is said to have been found at Thebes, possibly in or near the Ramesseum, the mortuary temple of Ramesses II, although its exact findspot is not recorded. The Scottish lawyer and antiquarian Alexander Henry Rhind bought it in Luxor in 1858. Rhind, who had gone to Egypt for his health, died in 1863, and the British Museum acquired the papyrus in 1865.

By then it was in two pieces with a missing section between them. Fragments from that gap were later identified among papyri in the collection of the New-York Historical Society, and are now in the Brooklyn Museum.

The first full English edition, by the Egyptologist T. Eric Peet, appeared in 1923. A second edition with photographs, led by the American mathematician Arnold Buffum Chace, was published by the Mathematical Association of America in 1927–1929. Recent scholars such as Annette Imhausen have studied the text in the context of Egyptian administration and scribal training.

Comparisons with other ancient mathematics

The Rhind Papyrus is not the oldest Egyptian mathematical text. The Moscow Mathematical Papyrus, dated to about 1850 BC, is older but shorter; it includes a correct method for the volume of a truncated pyramid. Mesopotamian scribes, working on clay tablets such as Plimpton 322, used a base-60 system and handled some problems, including square roots and quadratic equations, in a more advanced way.

Egyptian methods, with their unit fractions and practical problems, influenced later Greek arithmetic. Unit fraction calculations remained in use in the Mediterranean world for many centuries.

Myths and facts

The papyrus is sometimes described as a book of abstract theory. In fact it is a practical collection of examples and procedures, with no general proofs in the Greek sense. Another claim is that the Egyptians “knew pi”. They had an accurate rule for the area of a circle, but they did not treat pi as a constant. Nor does the papyrus reveal secret pyramid-building knowledge: its pyramid problems are classroom exercises on slope.

Timeline

• c. 1831–1786 BC: Reign of Amenemhat III, when the original text was written.
• c. 1550 BC: Ahmes copies the papyrus in year 33 of King Apepi.
• 1858: Alexander Henry Rhind buys the papyrus in Luxor.
• 1865: The British Museum acquires it.
• 1923: T. Eric Peet publishes the first full English edition.
• 1927–1929: Arnold Buffum Chace’s edition is published.

Why it matters

The Rhind Mathematical Papyrus is the clearest window into how ancient Egyptians counted, divided and measured. It shows a sophisticated mathematical culture that served farming, building and administration, and it remains one of the most important documents in the history of mathematics.