The early Egyptians settled along the fertile Nile valley as early as about 6000 BC, and they began to record the patterns of lunar phases and the seasons, both for agricultural and religious reasons. The Pharaoh’s surveyors used measurements based on body parts to measure land and buildings very early in Egyptian history, and a decimal numeric system was developed based on our ten fingers. Like a palm was the width of the hand, a cubit the measurement from elbow to fingertips. The oldest mathematical text from ancient Egypt discovered so far, though, is the Moscow Papyrus, which dates from the Egyptian Middle Kingdom around 2000 – 1800 BC. Civilization reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land by the river Nile with a pleasing climate. It was also a country which was easily defended, having few natural neighbors to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly. They were possibly the first civilization to practice the scientific arts.
Within the long sweep of Egyptian history, certain events or epochs have been crucial to the development of Egyptian society and culture. One of these was the unification of Upper Egypt and Lower Egypt sometime in the 3000 B.C. The ancient Egyptians regarded this event as the most important in their history, comparable to the “First Time,” or the creation of the universe. With the unification of the “Two Lands” by the legendary, if not mythical, King Menes, the glorious Pharaonic Age began. Power was centralized in the hands of a god-king, and, thus, Egypt became the first organized society. Agriculture developed making heavy use of the regular wet and dry periods of the year. The Nile flooded during the rainy season providing fertile land which complex irrigation systems made fertile for growing crops. Knowing when the rainy season was about to arrive was vital and the study of astronomy developed to provide calendar information. The large area covered by the Egyptian nation required complex administration, a system of taxes, and armies had to be supported. As the society became more complex, records required to be kept, and computations done as the people bartered their goods. A need for counting arose, then writing and numerals were needed to record transactions. By then they also developed their Hieroglyphic writing. Hieroglyphs for writing and counting gave way to an useful and important script for both writing and numerals.
Early hieroglyphic numerals can be found on Egyptian temples, stone monuments and vases. They give very little knowledge about any mathematical calculations which might have been done with the number systems or with the numerals. While these hieroglyphs were being carved in stone there was no need to develop symbols which could be written more quickly. However, once the Egyptians began to use flattened sheets of the dried papyrus reed as ‘paper’ and the tip of a reed as a ‘pen’ there was reason to develop more rapid means of writing. This prompted the development of hieratic writing and numerals.
But Egyptian number systems were not well suited for arithmetical calculations. They had some drawbacks similar to that of Roman numerals. It was easy to understand the numerals and although addition of Egyptian numerals is quite satisfactory, multiplication and division are essentially impossible. However, the Egyptians were very practical in their approach to mathematics and their trade required that they could deal in fractions. Trade also required multiplication and division to be possible so they devised remarkable methods to overcome the deficiencies in the number systems with which they had to work. And it was quite interesting that they had to devise methods of multiplication and division which only involved addition.
There was a large number of papyri, many dealing with mathematics in one form or another, but unfortunately since the material was fragile almost all had destroyed. But the good news is two major mathematical documents survived probably due to the consequence of the dry climatic conditions in Egypt: Rhind Mathematical Papyrus and Moscow Mathematical Papyrus. They are now kept at The British Museum and the Pushkin State Museum of Fine Arts in Moscow respectively. The Moscow Mathematical Papyrus is older than the Rhind Mathematical Papyrus, while the latter is the larger of the two.
It is from these two documents that most of our knowledge of Egyptian mathematics comes and most of the mathematical information in this article is taken from these two ancient documents.
The Rhind papyrus, a scroll about 20 feet long and 13 inches wide, was written around 1650 BC by the scribe Ahmes who states that he is copying a document which is 200 years older. The original papyrus on which the Rhind papyrus is based therefore dates from about 1850 BC.
While based on the palaeography and orthography of the hieratic text, the text in the Moscow papyrus was most likely written down in the 13th dynasty and based on older material probably dating to the Twelfth dynasty of Egypt, roughly 1850 BC. Approximately 18 feet long and varying between 1½ and 3 inches wide, its format was divided into 25 problems with solutions by the Soviet Orientalist Vasily Vasilievich Struve in 1930.
The Rhind papyrus contains 87 problems while the Moscow papyrus contains 25. The problems are mostly practical but a few are posed to teach manipulation of the number system itself without a practical application in mind. For example the first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n = 1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6. Clearly fractions are involved here and, in fact, 81 of the 87 problems given involve operating with fractions. Historical mathematician G. R. Rising discusses these problems of fair division of loaves in his book “The Egyptian use of unit fractions for equitable distribution“, which was particularly important in the development of Egyptian mathematics.
Some problems ask for the solution of an equation. For example Problem 26: a quantity added to a quarter of that quantity become 15. What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets 1/8 of a hekat more than the one before. Some problems involve geometry. For example Problem 50: a round field has diameter 9 khet. What is its area? The Moscow papyrus also contains geometrical problems.
Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus.
G. G. Joseph, in his book “The crest of the peacock” and many other authors give some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind. The angle between the base and one of the faces is 51° 50′ 35″. The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034. Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base. On the other hand the cotangent of the slope angle of 51° 50′ 35″ is very close to π/4. Again of course nobody believes that the Egyptians had invented the cotangent, but again it is the ratio of the sides which it is believed was made to fit this number. Now the observant reader will have realised that there must be some sort of relationship between the golden ratio and π for these two claims to both be at least numerically accurate. In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4 (in fact 3.996168).
Historical mathematician G. Robins and C. C. D. Shute, in their book “Mathematical bases of ancient Egyptian architecture and graphic art” argue against both the golden ratio or π being deliberately involved in the construction of the pyramid. They claim that the ratio of the vertical rise to the horizontal distance was chosen to be 5 1/2 to 7 and the fact that (11/14) × 4 = 3.1428 and is close to π is nothing more than a coincidence. Similarly Robins claims the way that the golden ratio comes in is also simply a coincidence. Robins claims that certain constructions were made so that the triangle which was formed by the base, height and slope height of the pyramid was a 3, 4, 5 triangle. Certainly it would seem more likely that the engineers would use mathematical knowledge to construct right angles than that they would build in ratios connected with the golden ratio and π. Interesting point, right?
As we mentioned above, it was also important for the Egyptians to know when the Nile would flood and so this required calendar calculations. The beginning of the year was chosen as the heliacal rising of Sirius, the brightest star in the sky. The heliacal rising is the first appearance of the star after the period when it is too close to the sun to be seen. For Sirius this occurs in July and this was taken to be the start of the year. The Nile flooded shortly after this so it was a natural beginning for the year. The heliacal rising of Sirius would tell people to prepare for the floods. The year was computed to be 365 days long and this was certainly known by 2776 BC and this value was used for a civil calendar for recording dates. Later a more accurate value of 365 1/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account. In fact two calendars ran in parallel, the one which was used for practical purposes of sowing of crops, harvesting crops etc. being based on the lunar month. Eventually the civil year was divided into 12 months, with a 5 day extra period at the end of the year. The Egyptian calendar, although changed much over time, was the basis for the Julian and Gregorian calendars.
So the work continues. We will discuss more about these two papyri we have found in my next article. Till then stay tight and let me know your views in the comment section.