Cultural Essay 4
Archimedes is known to be the third greatest mathematician of all time, and there is no secret as to why this is true. Between his major works and accomplishments The Method, the Measurement of the Circle, the Quadrature of the Parabola and his oddly invented shapes, Archimedes is nothing short of brilliant, and very accurate in his calculations. When developing “The Measurement of the Circle”, in proposition one he takes the area of a triangle A=(1/2)bh, sets the base of a triangle equal to the radius b=r of a circle and the height equal to the circumference of a circle h=2πr. From this he uses the area of a triangle to solve for the area of a circle we still use today.
A=(1/2)(2πr)r → A=πr^2
In proposition two and three of “The Measurement of the Circle”, he uses the interesting techniques of utilizing shapes to find values of π such dividing the area by the diameter of a circle to solve for π values.
((πr^2)/((2〖r)〗^2 ))→(π/4)
Archimedes has the same idea when using a 96 sided polygon to better estimate the value of π by dividing now the circumference by the diameter.
Another famous concepts of Archimedes is that of “The Quadrature of the Parabola” which in modern days something we use quite often in Calculus I and II. In this contribution, he experiments with different shapes by not using a straight edge or a compass to draw. He was drawing these shapes of a graph and trying to solve for the area underneath them by, yet again, breaking up the shape using triangles to get the most accurate answer. From this, he developed the formula, sum of triangles = 1 + 2next + 4next + 8next and so on. What he found to be the answer was (4a/3) which is incredibly accurate when we use today’s methods of integration and get the same solution. Archimedes continues playing with different shapes and came up with “On Spirals” which is another method he used to derive the circumference and area of circles which again proved to be extremely accurate when using the integral method of today. The “On the Sphere and the Cylinder” was a huge contribution to mathematics as it began to focus now on volume and surface area rather than just areas of shapes. He did so by placing a sphere inside a cylinder and using ratios of volumes and ratios of surface areas to simplify and find the comparisons. For example,
Volume of sphere=(4/3)πr^2 Volume of cylinder= πr^2 h=2πr^3
Vc/Vs=(2πr^3)/((4/3)πr^3 )=2/3
Surface area of sphere=4πr^2 Surface area of cylinder=6πr^2
SAc/SAs=(6πr^2)/(4πr^2 )=3/2

Reflective Tag
It amazes me how accurate all of Archimedes theories and methods were, so much so that they are still used very often today. Although, it does make me wonder about the notation that was used when finding these discoveries and how over time they were translated into the notation we use today. And if anything was lost in translation or if the person translating was really the one that discovered these ideas. It was clear Archimedes like to play around and create new shapes with makes me wonder about the architecture and how that was changing overtime unless it was some epiphany that these shapes came out of. It is surprising also that since he was so accurate on his ways of finding out things about these unusual shapes that he had to have done it more than once. But if no tools were used to draw them in the first place, then how did he recreate them so accurately? On the other hand, like many other mathematicians did, using triangles to cut existing shapes into pieces to discover more about this is shown to be a common practice that continues to help the development of mathematics.
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