Cultural Essay 3
The Greeks such as Hippocrates, Aristotle and Euclid made major advancements particularly in the field of geometry through the Elements, Construction Problem and the Method of Exhaustion. It was important when working with shapes that there be a way to denote the different sides and so forth of the shapes for communication purposes. This was Hippocrates contribution in that he designated the figures with letters and present material in a chain of ideas using a Lune. The Construction Problem was also created around this time, mid 400 BCE. In this idea was the logic that straight edges and compasses should be used to make the shapes more accurate. Using these tools, the Pythagorean Theorem was able to be proven yet again with the combination of the new use of straight edges and denoting shapes with letters. It was now possible that given a triangle, one could draw a segment line perpendicular to a side such that the triangle became a right triangle and could prove the Pythagorean Theorem to be true.
The Elements consist of thirteen books with the first six discussing plane geometry and eleven through thirteen covering solid geometry. In the first book, the controversy of the 5th Postulate is brought about stating if one has a line and a point, that there is only one possible line that can both intersect that point and be parallel to the line. Later, in book four, inscribed and circumscribed polygons with circles were introduced. In other words, circles were placed in squares and vice versa and corners of the square were cut off into triangles getting the square more and more like the circle to better estimate the value of pi. Although there are many things throughout the Elements we still use today such as the shapes created in book eight which included the cube, tetrahedron, dodecahedron, octahedron, and icosahedron, one of the most common creations of this time was that of the Method of Exhaustion. The Method of Exhaustion was created by Democritus and is so significant because it is believed to be the connection between Euclid and Egyptian math. The method uses rectangles to find the area under a curve on a graph. Their thought process for this method was to use ‘nice’ shapes to ‘exhaust’ the area of interest.

Reflective Tag
To me, these developments have such an impact on how math continues to work today through logic. During this time period, mathematicians began using more logic to figure out problems such as The Method of Exhaustion we still use today. Simply by thinking, the smaller the widths of the rectangles, the more rectangles there will be, therefore having less of an error in the space unaccounted for and being more accurate is an example of this logic thinking. Another example of this logic thinking is when a square was drawn and having the edges continue to be cut off to find the closest value to pi possible was, in all honesty, probably how I would have gone about trying to solve the mysterious number myself.