Reflective Essay
Nick Boline
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In this essay, I will discuss western civilization’s approach to solving geometric
problems, including a discussion of how the cultural lenses through which they viewed
mathematics affected the way they approached mathematics in general. I will conclude that in
the case of the Egyptians and Babylonians, the use of geometry was a practical matter based on
the types of applications they used for math, but the Greeks had a cultural predisposition for
geometric solutions, even to problems which would later be seen as non-geometric. I will then
discuss the way in which these conclusions affected my perceptions of each culture. In terms of
artifacts, I have selected the first four cultural essays as indicative of my progress towards
understanding geometry in these cultures.
At the very beginning of the story of Geometry, we have the ancient Egyptians. With no
documented academic heritage to inherit, they became the cornerstone of the history of math in
the West. As I stated in Cultural Essay I, the Egyptians were among the first to develop formulae
for determining the area of geometric shapes. These problems were framed exclusively in word
problems, since notation was so primitive that it was more effective simply to write the problem
out. Despite their lack of inherited knowledge, they were able to determine the area of more noncircular shapes, including the triangle, rectangle, trapezoid and pyramid. They had a unique
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formula for finding the area of a circle: 𝐴 = (9 𝑑) 2 . With this formula, we find that the value of
π is 3.16046. This is remarkably accurate considering that their civilization existed more than
5,000 years ago. The inaccuracy that stems from this incorrect value of π was often
inconsequential, given that this formula would have been known only to a very small elite
employed in the administration of the nobility’s property. Since they would be calculating the
area of circular fields for tax purposes, the difference between the calculated size and the actual
size would have been unimportant.
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This understanding of the nature of Egyptian math in general and geometry in particular
(which were often one and the same) has more or less determined my entire outlook on Egyptian
society. Prior to this class I had very little knowledge of the technical capabilities of the
Egyptians. Now I have a much more robust understanding of the incredible capabilities and the
limitations of their mathematics. This is the kind of perspective I took this class to obtain. The
ability of the Egyptians to intuit basic geometric truths from only their senses and reasoning has
given me a new respect for Egyptian culture. It has also reinforced my belief that there is
something intrinsic about mankind which allows us to overcome nature and thrive.
The Babylonian, whose recorded history began after the Egyptians, but with whom they
existed contemporaneously, had a remarkably similar approach to geometric problems. By and
large, the Babylonians also used mathematics for administrative purposes. Because of this, they
focused primarily on arithmetic and geometry, particularly the calculation of area. They had
accurate formulae for non-circular shapes just as the Egyptians did, as well as some formulae for
three dimensional shapes which the Egyptians lacked. In terms of circular geometry though, they
were not quite as advanced as the Egyptians. The Babylonians considered the ratio of a circle’s
circumference to diameter to be 1/3, meaning that their value of π was 3.00. This is clearly not as
accurate as the Egyptian value, but given that it was used primarily for assessing taxes on
circular fields, the lack of accuracy was not a major limitation. The Babylonians also framed
their mathematics in terms of word problems due to the lack of notation.
All things considered, the approach to mathematics by the Babylonians is not
significantly different than the Egyptians. A study of Babylonian and Egyptian mathematics
offers a unique chance to compare and contrast two cultures which developed along similar lines
and with similar concerns. What we see is that both civilizations come to largely the same
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conclusions about mathematics in general and geometry in particular. They did not consider it an
academic discipline; rather, they saw it as a tool to be used in the process of administration, a
new cultural development which would unlock the boundless potential of civilization and
mankind. I draw these conclusions from my examination of each culture over the course of the
class, as well as from an analysis of cultural essays one and two.
The Greeks represent an entirely different civilization, which evolved with limited access
to the accomplishments of the Egyptians and Babylonians until later in their civilization. The
conditions under which they developed were markedly different from the other civilizations I
discuss. The Greeks, along their development, saw huge advances in geometry. In my cultural
essays three and four, I discuss how their focus on geometric solutions to general math problems
could be the result of the special place the circle holds in Greek culture. Indeed, we can see the
attention which the Greeks devoted to circular geometry by examining proofs such as the Lune o
Hippocrates. This was the first example of a mathematician being able to equate the area of a
non-circular shape with that of a circular shape. Archimedes, a Greek mathematician was also
able to determine the value of π to seven decimal places using an ingenious geometric solution in
which we inscribed and superscribes an n-sided polygon on a circle and increases n to 96 to find
the difference in area of the two. This turns out to be 3.14159, the correct value for π. The list of
important proofs and solutions goes on far beyond the scope of this paper, but the contributions
of the Greeks to geometry are countless. One needs look no further than the names of many
famous theories and postulates in geometry to see the mark the Greeks left upon history.
Prior to this class, I was familiar only with the historical and literary aspect of ancient
Greek society. Now, having studied them thoroughly, I have a much fuller understanding of
Greek culture. I have a much greater appreciation for the value of the Greek contribution to
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western society in all fields of study. It was interesting for me to equate Greek ideas of the
perfection of the circle with their vigorous efforts to understand the geometry of the circle. They
began to delve in to mathematics purely for its own sake, something neither the Egyptians nor
the Babylonians did. They improved notation and set the state for future civilizations to advance,
securing then their place as one of the most historically significant cultures in the west. Because
of this, they are still studied to this day.
The process of learning about these different civilizations’ approaches and attitudes
toward geometric problems was a rather comfortable one. The class was structured much like
other history classes with which I am familiar. The lecture being derived from the reading, I very
much enjoyed the early readings on these civilizations and was able to apply what I’d read to the
lecture notes and come away with a more complete understanding. Solving some example
problems and deriving some of the formulas for myself really helped me appreciate what was
going on in the minds of the mathematicians of the era.
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