Ancient Egyptian
Mathematics
The use of organized mathematics in Egypt
has been dated back to the third millennium BC. Egyptian mathematics
was dominated by arithmetic, with an emphasis on measurement and calculation
in geometry. With their vast knowledge of geometry, they were able
to correctly calculate the areas of triangles, rectangles, and trapezoids
and the volumes of figures such as bricks, cylinders, and pyramids.
They were also able to build the Great Pyramid with extreme accuracy.
Early surveyors found that the maximum error in fixing the length of the
sides was only 0.63 of an inch, or less than 1/14000 of the total length.
They also found that the error of the angles at the corners to be only
12″, or about 1/27000 of a right angle (Smith 43). Three theories
from mathematics were found to have been used in building the Great Pyramid.
The first theory states that four equilateral triangles were placed together
to build the pyramidal surface. The second theory states that the
ratio of one of the sides to half of the height is the approximate value
of P, or that the ratio of the perimeter to the height is 2P. It
has been discovered that early pyramid builders may have conceived the
idea that P equaled about 3.14. The third theory states that
the angle of elevation of the passage leading to the principal chamber
determines the latitude of the pyramid, about 30o N, or that the passage
itself points to what was then known as the pole star (Smith 44).
Ancient Egyptian mathematics was based
on two very elementary concepts. The first concept was that the Egyptians
had a thorough knowledge of the twice-times table. The second concept
was that they had the ability to find two-thirds of any number (Gillings
3). This number could be either integral or fractional. The Egyptians
used the fraction 2/3 used with sums of unit fractions (1/n) to express
all other fractions. Using this system, they were able to solve all
problems of arithmetic that involved fractions, as well as some elementary
problems in algebra (Berggren).
The science of mathematics was further
advanced in Egypt in the fourth millennium BC than it was anywhere else
in the world at this time. The Egyptian calendar was introduced about
4241 BC. Their year consisted of 12 months of 30 days each with 5
festival days at the end of the year. These festival days were dedicated
to the gods Osiris, Horus, Seth, Isis, and Nephthys (Gillings 235).
Osiris was the god of nature and vegetation and was instrumental in civilizing
the world. Isis was Osiris’s wife and their son was Horus.
Seth was Osiris’s evil brother and Nephthys was Seth’s sister (Weigel 19).
The Egyptians divided their year into 3 seasons that were 4 months each.
These seasons included inundation, coming-forth, and summer. Inundation
was the sowing period, coming-forth was the growing period, and summer
was the harvest period. They also determined a year to be 365 days
so they were very close to the actual year of 365 ? days (Gillings
235).
When studying the history of algebra, you
find that it started back in Egypt and Babylon. The Egyptians knew
how to solve linear (ax=b) and quadratic (ax2+bx=c) equations, as well
as indeterminate equations such as x2+y2=z2 where several unknowns are
involved (Dauben).
The earliest Egyptian texts were written
around 1800 BC. They consisted of a decimal numeration system with
separate symbols for the successive powers of 10 (1, 10, 100, and so forth),
just like the Romans (Berggren). These symbols were known as hieroglyphics.
Numbers were represented by writing down the symbol for 1, 10, 100, and
so on as many times as the unit was in the given number. For example,
the number 365 would be represented by the symbol for 1 written five times,
the symbol for 10 written six times, and the symbol for 100 written three
times. Addition was done by totaling separately the units-1s, 10s,
100s, and so forth-in the numbers to be added. Multiplication was
based on successive doublings, and division was based on the inverse of
this process (Berggren).
The original of the oldest elaborate manuscript
on mathematics was written in Egypt about 1825 BC. It was called
the Ahmes treatise. The Ahmes manuscript was not written to be a
textbook, but for use as a practical handbook. It contained material
on linear equations of such types as x+1/7x=19 and dealt extensively on
unit fractions. It also had a considerable amount of work on mensuration,
the act, process, or art of measuring, and includes problems in elementary
series (Smith 45-48).
The Egyptians discovered hundreds of rules
for the determination of areas and volumes, but they never showed how they
established these rules or formulas. They also never showed how they
arrived at their methods in dealing with specific values of the variable,
but they nearly always proved that the numerical solution to the problem
at hand was indeed correct for the particular value or values they had
chosen. This constituted both method and proof. The Egyptians
never stated formulas, but used examples to explain what they were talking
about. If they found some exact method on how to do something, they
never asked why it worked. They never sought to establish its universal
truth by an argument that would show clearly and logically their thought
processes. Instead, what they did was explain and define in an ordered
sequence the steps necessary to do it again, and at the conclusion they
added a verification or proof that the steps outlined did lead to a correct
solution of the problem (Gillings 232-234). Maybe this is why the
Egyptians were able to discover so many mathematical formulas.
They never argued why something worked, they just believed it did.
BIBLIOGRAPHY
Berggren, J. Lennart. “Mathematics.”
Computer Software. Microsoft, Encarta 97 Encyclopedia.
1993-1996. CD- ROM.
Dauben, Joseph Warren and Berggren,
J. Lennart. “Algebra.” Computer Software.
Microsoft, Encarta 97 Encyclopedia. 1993-1996. CD- ROM.
Gillings, Richard J. Mathematics
in the Time of the Pharaohs. New York: Dover Publications,
Inc., 1972.
Smith, D. E. History of Mathematics.
Vol. 1. New York: Dover Publications, Inc., 1951.
Weigel Jr., James. Cliff Notes
on Mythology. Lincoln, Nebraska: Cliffs Notes, Inc., 1991.
