Is It Impossible To Trisect An Arbitrary Angle The Problem Cannot Be Solved Through The Plane Or Euclidian Methods They Used?
Introduction
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...
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