Aaron Mattson
May 3rd, 2010
MAT 266
Dr. Naala Brewer
Building the Foundations of Both Structures and Calculus
In the second term of Calculus for Engineers, among the numerous topics
pics that were covered, a
technique wass taught to find the volume of various geometric objects using integration. Quite a few
shapes were covered,
ered, including cones and hemispheres, but the class was left alone to find the volume
of a pyramidal frustum as part of an extra credit problem on th
thee online homework system, WeBWorK.
While not exceptionally hard to solve with ‘modern’ mathematics, it is hard to imagine how the Egyptian
and Babylonian civilizations of past millennia could have constructed such buildings as their colossal
pyramids so structurally sound
und without any of those more recent calculations being done
done. An interesting
discovery of a papyrus showed exactly how skilled the Egyptians were at a form of mathematics that had
yet to be defined.
The oldest pyramids that are still in existence today
were built by the Egyptians as a way to honor the deceased
pharaohs and their associates. The most famous of these are
the pyramids in the Giza Necropolis (Fig. 1),
1) each requiring a
significant amount of architectural and mathematical
planning due to their size and complex inner network of
tunnels and tombs. One of those is even a wonder of the
ancient world, the Great Pyramid of Khufu, standing
approximately 140 meters high and estimated
ated to have a
mass of nearly six million tons. Its construction took twenty
t
years, ending in 2560 BC.
Figure 1: Pyramids of Giza
A lengthy strip of papyrus
papyrus, which was purchased by Vladimir Golenishchevv around 1900 AD
provides good evidencee that the Egyptians had a very good grasp of mathematics. The ‘Moscow
Mathematical Papyrus’ is inscribed with hieratic, a shorthand form of hieroglyphics, that is dated at
about 1850 BC,, seven dynasties after the creation of the Great Pyramid
Pyramid. Currently stored in the Pushkin
State Museum of Fine Arts,
rts, located in Moscow, the papyrus was split into its twenty-five
five separate
problems by Soviet Orientalist Vasily Vasilievich Struve in 1930 AD. The majority of tho
those
se problems
pertained to beer and bread recipes, including ingredient ratios and prices, while
le a couple dealt with
volumes, problems 10 and 14.
Problem 14 (Fig. 2) is by far the most complicated of the problems on the papyrus and perhaps
the most interesting, as well. Translated, it describes a pyramid that has been truncated so that the top
has an area that is equal to 2 (its side length) squared, the bottom has an area that is equal to 4 (its side
length) squared, and a height of 6 units. It then goes on
to describe how one would find the volume of the
described frustum – squaring the 4 to get 16, doubling
the 4 to get 8, and then squaring the 2 to get 4,
afterwards adding these values together to get 28. This is
then multiplied by a third of the height, 6, ending up as
56, thee correct answer. (Simplified: ) The first translator, Boris
Touraeff, committed an error
rror when he made his claim.
He believed that the papyrus not only proved that the
Egyptians knew the formula for a pyramidal frustum, but
the volume of any frustum. This equation, Figure 2:: Problem 14 of the MMP
,, was not defined on the papyrus, but may well have been known by the Egyptians.
The Egyptians’ formula can be proven true with rather simple geometry. For instance, the
volume of a pyramid is
.. The top part that has been removed by the
truncation is also a pyramid, but instead, uses the square top’s length as its base length. Thus the
volume would end up at this point as ! "#$%&%'()
! * + ),-&%'() !
.
With the knowledge that
h=h2-h1 and making use of the Heronian mean, . ,, the formula is turned into an
equation equivalent to what the Egyptians used .,
The problem that is depicted on the papyrus is o
one
ne instance, but the equation must work for all
instances in order to be accurate. Thus, the problem that WeBWorK provided for Section 7.2 was
attempted. In this case, h (height) = 10, a (base length) = 18, and b (top length) = 5. Putting those
/
numbers into
o the Egyptians’ equation, it ended up looking like 0 0 . Simplifying
this, it ends up as 1, or 2 444444
3 1.. This ends up as approximately
1463.333, the correct answer.
Nothing has been discovered that explains how the Egyptians came up with such an impressively
accurate formula, but the find itself gives a tremendous amount of insight into the development of
mathematics in other cultures, both deceased and living today. The papyrus gives a clue of how the
Egyptians figured out how to create such massive structures as they did, even if it does not explain how
exactly the Egyptians moved 800 tons of the massive stones used in the Giza pyramids every day. The
10th problem on the same strip of papy
papyrus involves an estimation of 5 being used to find the volume of a
bowl, so their math was not always exact, but the development of calculus could be seen even so.