There were three problems that the ancient Greeks tried unsuccessfully to solve by Euclidean methods. They were the doubling of a cube, trisecting an angle and squaring a circle. These problems became the interest of mathematicians for tens of centuries after their proposal, all of which were proven unsolvable by these means as much as around two thousand years later, as a result of progress in algebra, and the idea of analytic geometry in the sense of Descartes. In this essay I mean to discuss only one of these problems, the trisection of an angle. What methods did the ancient Greeks apply to solve this problem and why is it impossible to trisect an arbitrary angle the problem cannot be solved through the plane or Euclidian methods they used? I aim to break the problem down to an easily understandable level for anyone with a minor level of understanding of mathematics to comprehend, and to show why the problem cannot be solved through construction History
The problem of trisecting an angle differs from the two other problems mentioned above in the sense that it has no specific history about where it was first developed. What makes this seem odd is the fact that the problem still came to the attention of the greatest mathematicians and logical thinkers in ancient Greece. The problem cannot be dated exactly, but the first writings found about it appeared around two thousand years ago. After this numerous mathematicians attempted to solve the problem, until great progress was made for the first time by Carl Friedrich Gauss (1777-1855) and Pierre Wantzel (1814-1848). Mathematicians managed to find numerous solutions for trisecting an arbitrary angle using other methods than plane geometry. Finally Wantzel proved the impossibility of the construction in 1837. Introducing the Euclidian principles and Constructible lengths I will first introduce the rules of construction and Euclidian principles in relation to the problem discussed in this essay. The rules are the methods that are allowed to be used rather than any specific formula like we would see in algebra. In construction we are allowed to use a compass and an unmarked straight edge to draw with. There are some individual operations with this equipment which may be conducted and they are called the fundamental constructions which were formulated by an ancient Greek mathematician and geometrician Euclid. The fundamental constructions
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Given 2 points, we may draw a line trough them, extending it indefinitely in each direction.
Given 2 points we may draw the line segment connecting them.
Given a point and a line segment, we may draw a circle with center at this point and radius equal to the length of the line segment. I would also like to add note that a point is only constructible as two lines intersect, or as a line intersects with a circle or two circles intersect. I would also like to state that the diagrams are no specific proof for any trisections made but are only illustrations and guidance to understand and visualize the problems. To be exact and sure the problems must be proven mathematically, because in an ideal picture lines for example would have no thickness and thus no error.
Constructing Real lengths trough rational operations
With the fundamental constructions and the tools mentioned above we can, given a length a multiply this length by any rational, and divided it by any rational. We can also add and subtract from this length. We say that given two lengths a and b we can add them, subtract them, multiply them or divide one by the other. Adding two lines one after another and subtracting them from one another
To prove the construction through multiplication and division we chose one line to be the unit length of 1 and we name the other line a. We can prove both the product and the quotient by drawing two non-collinear rays (a and 1) emanating from the same point.. After this the two...