Mathematics In Ancient Egypt

the Ancient Egyptians possess an ingenious skill for calculation? Assem Deif* works out an ancient problem

The Greeks developed mathematics as a deductive science that reached its climax with Euclid of Alexandria in his masterpiece The Elements. Before that, during the ancient Egyptian era, mathematics was an inductive discipline of a utilitarian nature used to perform practical tasks such as flood control or land measurement using rope. It has been suggested that mathematics then amounted to no more than the two-times table and the ability to find two-thirds of any number. The whole structure of Egyptian mathematics was said to be based on these two simple rules, and indeed no evidence exists of a textual geometry with constructions and proofs.

Yet, looking at the Egyptians' stunning monuments, as well as a civilisation that spanned three millennia, one might expect to find a similar element of grandeur in their sciences -- especially in mathematics and astronomy. How did they configure the manpower and materials needed to build more than 90 pyramids? It is obvious that to calculate the vast amount of computations they needed, the ancient Egyptians reached a fairly advanced mathematical knowledge.

Several eminent Greek mathematicians -- Pythagoras, Thales and Archimedes, to name just a few -- worked in Egypt, and it is likely that Egyptian mathematics was absorbed into the body of Greek mathematics. The Giza pyramids offer definitive evidence of the ancient accuracy of measuring. Built in the middle of the third millennium BC, shortly after the first known evidence of Egyptian writing, they predate by 600 years any early mathematical tools. The Great Pyramid of Khufu was built of 2,300,000 limestone blocks each averaging 2.5 tons. Simple calculations reveal that, since it took 20 years to complete, and assuming that work lasted eight hours per day, it was possible to fit 2,300,000/20 x 365 x 8 x 60 = 0.7 blocks per minute. In other words it took about 10 minutes to fit seven such huge blocks neatly into place at such an elevation. This does not account for the time taken to construct or demolish the ramp using to pull up the stones.

One engineer reckons that such a ramp would require 18,000,000 m of material -- seven times the amount used for the pyramid itself, and necessitating a work force of 240,000 during Khufu's reign and more than 300,000 to dismantle it for at least eight years afterwards. Neither does it account for the time taken to position the nine blocks each weighing 50 tons for the inside of the royal chambers, or the time to clad the monument with casing stones. Astonishingly, an experiment by Japanese researchers 15 years ago to build a pyramid using new technology was abandoned after six months when their calculations showed it would take more than 1,000 years to complete their task.

No two Egyptologists agree on the exact dimensions of the Great Pyramid, yet all accept that the sides agree in length within 0.01 per cent, and that the right angles are equally accurate. The pyramid's 350-foot-long descending passage is so straight that it deviates from a central axis by less than a quarter of an inch from side to side and only one tenth of an inch up and down. This compares only with the best laser-controlled drilling of today.

Another perplexing feature of the Great Pyramid are the four so-called "air shafts", two in the King's Chamber and two in the Queen's. In each chamber, one is directed precisely to the North while the other is set precisely to the South. Whether these shafts were intended for ventilation or to serve a religious purpose is a mystery. The alignment of the shafts was difficult to attain, especially since they were made during construction. The builders appear to have selected a "target star", visible to the naked eye and rising high enough so as not to be disturbed by the earth's atmosphere. This would be viewed through the shaft during each phase of construction. The pyramid builders were able to insert these almost perfectly straight shafts directly North and South hundreds of feet from inside the pyramid and with almost a laser-beam precision. The shafts' alignments to the star's culmination points are so precise that they point exactly to the three stars of Orion's Belt, which the Egyptians relied heavily on in their astronomical observations.

The casing stones covering the monument are also so perfectly shaped that the mortar-filled joint is just 1/15th of an inch. Egyptologist Flinders Petrie compared such phenomenal precision with that of the finest optician, saying it was beyond the capabilities of modern technology. Again, these stones show no tool marksn and the corners are not even slightly chipped.

Monuments elsewhere show equal feats of engineering. The Karnak temple complex has 134 carved granite pillars, each 22m in height and 3.5m in diameter. Some obelisks are 42m high and weigh 1,100 tons. How did these early engineers raise them upright?

They had no electronic calculators, only ropes and rods. Yet they knew accurate values for both pie and sigma . They were aware of Pythagoras's theorem -- and not just as having sides with the ratio 3:4:5. Pythagoras himself called it the "Sacred Triangle". In our view, he might have given this name not only to the triangle, but also to the Great Pyramid with its dimensions 220c, 280c and 356c. History records that Pythagoras announced his theorem as he departed from Egypt in 600 BC after living there for 22 years.

The Golden Ratio, also called Divine Proportion, is what artists reckon to be the ratio controlling the dimensions of any beautiful figure and which applies to monuments from the Parthenon and the domes of Persia, to the art of the Renaissance.

It is beyond doubt that the Great Pyramid is a testament to the builders' remarkable ability precisely to measure directions, angles and lengths on the earth's surface. The pyramid exhibits such a high degree of precision in construction and orientation that it is little wonder ill- founded legends have grown up around it. It is said to be the most accurately aligned structure in existence, facing true North with only 3/ 60th of a degree of error (the misalignment in the telescope's sensor axis of the Paris observatory is 7min of arc, or twice the pyramid's error, while the Meridian Building at Greenwich Observatory in London has an inclination of 9min). Moreover, the pyramid's site was selected so as to allow for astronomical observations. It was determined as a site that would be suitable for a building with 61/2 million tons of stone, whose height was 147m and base area 53000 m . So, whereas Egyptologists adopt the view that the ancient Egyptians built the Great Pyramid as a tomb for Khufu, others suggest that their intention was to build a geodesic monument that would demonstrate their knowledge of the earth's shape and size, or perhaps an astronomical observatory.

In any event, what knowledge did the ancient Egyptians possess in order to construct such colossal structure and with such outstanding precision? We are forced to conclude that the pyramid builders were capable of making precise geodesic and astronomical calculations.

Another reason for believing in these skills is their accurate calendar. The Egyptians could not have devised a calendar with such remarkable sophistication unless they were well-versed in astronomy, a science we cannot dissociate from either mathematics or religion in ancient Egypt. A nation capable of mastering astronomy must have possessed advanced mathematical know-how.

One of the most astounding pieces of Egyptian architecture is Abu Simbel. A marvel of engineering, the temple construction depends on precise astronomical calculations. Thanks to the orientation of the temple, twice a year on 22 February and 22 October -- the anniversaries of Rameses's birthday and his coronation day -- the statues of the gods Amun-Ra and Re-Horakhte and of the pharaoh in the inner temple are struck at dawn by a shaft of sunlight. This spectacle continued for more than 3,200 years until the 1960s when the temple was dismantled and relocated to make way for the High Dam. After that the illumination shifted by one day.

Two major mathematical documents have survived; the Rhind and the Moscow papyri. Also still in existence are the Egyptian Mathematical Leather Roll, a table of 26 decompositions of unit fractions, a well as the Berlin Papyrus which contains two problems on simultaneous equations, one of second degree, and the Reisner Papyrus demonstrating the practical application of mathematics in construction and commerce. It is from the first two documents that we have obtained most of our information on Egyptian mathematics.

The papyrus, purchased by A Henry Rhind in Luxor in 1858 was written about 1650 BC by the scribe Ahmes, who stated that he was copying a document 200 years older. The papyrus contains multiplication tables, along with 87 problems involving a variety of mathematical processes.

The Moscow Papyrus which dates from 1890 BC contains some 25 problems. Number 14 shows a figure resembling an isosceles trapezoid: the calculations associated with it indicate that it is the frustum of a square pyramid. The formula was not written on the papyrus, but it was evidently known to the Egyptians.

"Squaring the circle" is the most fascinating problem that the Egyptians tackled, and, by far, the most famous and intricate mathematical problem ever posed in antiquity. By using simple geometrical instruments such as a compass and ruler, it seeks to find a square of an area equal to that of a given circle. Only after three and a half millennia (in the late 19th century) was it shown that such a square could not be constructed. The reason is that it is not an algebraic number. The Egyptians were the first to pose this problem, by stating in problem number 50 of the Rhind Papyrus, that a circle of nine units in diameter is equal in area to a square with a side of eight units.

By far the most intriguing is problem 14 of the Moscow Papyrus. It asks for the volume of a truncated pyramid (frustum), stating: "Given a truncated pyramid of height 6, base 4, and top 2".

An important find at Saqqara was a Third-Dynasty limestone ostracon dating from about 2700 BC. Egyptologists believe this architect's plan of a curved section of a roof is an example of the use of rectangular coordinates. For horizontal coordinates spaced one cubit apart, the vertical height is given for points which define a curve. The curve in the sketch exactly matches the curve of a nearby temple roof. This appears to be the earliest use of rectangular coordinates, and is another example of sophisticated mathematical concepts found in practical applications outside of the surviving mathematical papyri.

Instead of numbers, the Egyptians used symbols which started at one and went up to a million. Number one was a papyrus leaf, 10 a tied leaf, 100 a piece of rope, 1000 a lotus flower, 10,000 a snake, 100,000 a tadpole and 1,000,000 a scribe with raised arms. One major disadvantage was its lack of the zero, but neither the Babylonians nor the Greeks had zero either, although the Hindus, Greeks and Mayans knew of it as a symbol. It was the Arabs near the end of the first millennium AD who introduced it in numbers and later used it to solve algebraic equations.

Hieroglyphic numerals did not remain constant, but changed continuously over time. A New Kingdom script differs from the Middle Kingdom, and so on. When hieroglyphs were carved on stone, there was no need to develop forms which were quick to write. However, once the Egyptians began to use dried papyrus reed as paper and its tip as a pen, they needed to develop a more rapid means of writing. This prompted the development of fast hieratic writing. Later, a system of hieratic numerals was introduced, allowing numbers to be written in a more compact form: the number 9999 had just four hieratic symbols instead of 36 hieroglyphs. Examples of hieratic writing are the Rhind and Moscow papyri; meanwhile the carving on stone remained in hieroglyphs.

Today's scientists are searching desperately to fill the many blanks in the history of the Egyptian civilisation. There are very few sources on Egyptian mathematics, but these still give plenty of information about the level of mathematics. In fact, what current knowledge the West considers as originating mostly -- if not all -- from Babylon or Greece is beyond any doubt inherited from the ancient Egyptians. Such early historians as Solon, Hecataeus of Melitus, Herodotus, Diodorus and Strabo agreed that all the prominent Greek scientists, without a single exception, visited Egypt. Some historians, physicians and even philosophers stayed for more than 10 years in Waset, or Thebes. Further, All historians agree that one science in which the Greeks borrowed heavily from the Egyptians was medicine, so it seems plausible that they also borrowed in the other sciences.

If this is the case, then it would be legitimate to ask why most of the ancient written heritage was lost but the Greek was preserved to reach European Renaissance in the form we know today. The answer probably lies in that sciences in the Hellenistic era were written in Greek, a language that was understood and thus translated into Latin or Arabic. Hieroglyphs and hieratic, unidentified and written on fragile papyrus or parchment, did not survive. Thus it was left to the Greeks to reap the acclaim.

Very far-fetched claim that the Egyptians were the ones who came up with the mathematics ordinarily credited to the Greeks. The writer obviously has an agenda.

The big innovation that the Greek mathematicians (geometers and philosophers) came up with was to prove theorems logically, deductively -- they transformed mathematics from a bunch of empirical rules (some exact, some approximate) to the first modern science. If you read Euclid's Elements you will be struck at the sophistication of its contents and its thoroughly modern tone. (This text dates to two or three centuries before Christ.) This is what is interesting about Greek mathematics. Not its precision, a simple matter of arithmetic. (The Greeks, by the way, used the Babylonian system of arithmetic, which was base 60, from which we get our degrees-minutes-seconds. This was a completely modern form of arithmetic, only superceded by the Indian base 10 system in the middle ages. The arithmetic of the ancient Egyptians was a horror show by comparison. They did not use a base system, they used a system resembling Roman numerals, with pictograms representing numbers such as 1, 5, 10, 100, etc. This was tough to do arithmetic with. Worse, except for the fraction 2/3, they only allowed fraction with numerator 1. So they would allow 1/5 but they wouldn't allow 2/5, 3/5, etc. They needed complicated rules for adding fractions to achieve answers with only numerator 1, because strangely they did not allow repeated fractions of the same denominator.)

This is not to denigrate Egyptian mathematics, some of which was quite impressive for premodern times. But it wasn't the Egyptians who conceived of the notion of a rational number -- or who proved that the square root of two or similar numbers are irrational. That was the Greeks. The fact that much of this happened in Alexandria -- a Greek colony in Egypt -- is beside the point.

While not especially relevant to the question of what the ancient Egyptians knew about mathematics, it's interesting to recall that in his Timaeus, Plato—who was in a position to know the nature and quality of Egyptian learning since he spent time in Egypt after the execution of Socrates—attributes his famous Atlantis tale to a wise Egyptian priest, who tells Solon, the great Athenian lawmaker and poet (one of the Seven Wise Men of Greece), that he and his fellow Greeks are like children because they don't remember their past:

Listen then, Socrates, to a tale of Solon's, who, being the friend of Dropidas my great-grandfather, told it to my grandfather Critias, and he told me. The narrative related to ancient famous actions of the Athenian people, and to one especially, which I will rehearse in honour of you and of the goddess. Critias when he told this tale of the olden time, was ninety years old, I being not more than ten.
The occasion of the rehearsal was the day of the Apaturia called the Registration of Youth, at which our parents gave prizes for recitation. Some poems of Solon were recited by the boys. They had not at that time gone out of fashion, and the recital of them led some one to say, perhaps in compliment to Critias, that Solon was not only the wisest of men but also the best of poets. The old man brightened up at hearing this, and said: Had Solon only had the leisure which was required to complete the famous legend which he brought with him from Egypt he would have been as distinguished as Homer and Hesiod. 'And what was the subject of the poem?' said the person who made the remark. The subject was a very noble one; he described the most famous action in which the Athenian people were ever engaged. But the memory of their exploits has passed away owing to the lapse of time and the extinction of the actors. 'Tell us,' said the other, 'the whole story, and where Solon heard the story.'
He replied-- There is at the head of the Egyptian Delta, where the river Nile divides, a city and district called Sais; the city was the birthplace of King Amasis, and is under the protection of the goddess Neith or Athene. The citizens have a friendly feeling towards the Athenians, believing themselves to be related to them. Hither came Solon, and was received with honour; and here he first learnt, by conversing with the Egyptian priests, how ignorant he and his countrymen were of antiquity. Perceiving this, and with the view of eliciting information from them, he told them the tales of Phoroneus and Niobe, and also of Deucalion and Pyrrha, and he endeavoured to count the generations which had since passed.
Thereupon an aged priest said to him: 'O Solon, Solon, you Hellenes are ever young, and there is no old man who is a Hellene.'
'What do you mean?' he asked.
'In mind,' replied the priest, 'I mean to say that you are children; there is no opinion or tradition of knowledge among you which is white with age; and I will tell you why. Like the rest of mankind you have suffered from convulsions of nature, which are chiefly brought about by the two great agencies of fire and water. The former is symbolized in the Hellenic tale of young Phaethon who drove his father's horses the wrong way, and having burnt up the earth was himself burnt up by a thunderbolt. For there occurs at long intervals a derangement of the heavenly bodies, and then the earth is destroyed by fire. At such times, and when fire is the agent, those who dwell by rivers or on the seashore are safer than those who dwell upon high and dry places, who in their turn are safer when the danger is from water. Now the Nile is our saviour from fire, and as there is little rain in Egypt, we are not harmed by water; whereas in other countries, when a deluge comes, the inhabitants are swept by the rivers into the sea. The memorials which your own and other nations have once had of the famous actions of mankind perish in the waters at certain periods; and the rude survivors in the mountains begin again, knowing nothing of the world before the flood. But in Egypt the traditions of our own and other lands are by us registered for ever in our temples. The genealogies which you have recited to us out of your own annals, Solon, are a mere children's story. For in the first place, you remember one deluge only, and there were many of them, and you know nothing of that fairest and noblest race of which you are a seed or remnant. The memory of them was lost, because there was no written voice among you. For in the times before the great flood Athens was the greatest and best of cities and did the noblest deeds and had the best constitution of any under the face of heaven.'
Solon marvelled, and desired to be informed of the particulars.
'You are welcome to hear them,' said the priest, 'both for your own sake and for that of the city, and above all for the sake of the goddess who is the common foundress of both our cities. Nine thousand years have elapsed since she founded yours, and eight thousand since she founded ours, as our annals record. Many laws exist among us which are the counterpart of yours as they were in the olden time. I will briefly describe them to you, and you shall read the account of them at your leisure in the sacred registers.
In the first place, there was a caste of priests among the ancient Athenians, and another of artisans; also castes of shepherds, hunters, and husbandmen, and lastly of warriors, who, like the warriors of Egypt, were separated from the rest, and carried shields and spears, a custom which the goddess first taught you, and then the Asiatics, and we among Asiatics first received from her. Observe again, what care the law took in the pursuit of wisdom, searching out the deep things of the world, and applying them to the use of man. The spot of earth which the goddess chose had the best of climates, and produced the wisest men; in no other was she herself, the philosopher and warrior goddess, so likely to have votaries.
And there you dwelt as became the children of the gods, excelling all men in virtue, and many famous actions are recorded of you. The most famous of them all was the overthrow of the island of Atlantis. This great island lay over against the Pillars of Heracles, in extent greater than Libya and Asia put together, and was the passage to other islands and to a great ocean of which the Mediterranean sea was only the harbour; and within the Pillars the empire of Atlantis reached in Europe to Tyrrhenia and in Libya to Egypt. This mighty power was arrayed against Egypt and Hellas and all the countries bordering on the Mediterranean. Then your city did bravely, and won renown over the whole earth. For at the peril of her own existence, and when the other Hellenes had deserted her, she repelled the invader, and of her own accord gave liberty to all the nations within the Pillars. A little while afterwards there were great earthquakes and floods, and your warrior race all sank into the earth; and the great island of Atlantis also disappeared in the sea. This is the explanation of the shallows which are found in that part of the Atlantic ocean.'

Of course, Plato does say that the Athenians had an even more ancient lineage than the Egyptians of Sais, saving a modicum of face for his city.

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