http://weekly.ahram.org.eg/2007/829/hr1.htm
25 - 31 January 2007
Issue No. 829
Published in Cairo by AL-AHRAM established in 1875
Mathematics in Ancient Egypt
Did 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.
Click to view caption
Khafre's pyramid with the Sphinx
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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.
* The writer is professor of Mathematics at Cairo University
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