Throughout history, Geometry has been widely considered as one of the seven liberal arts.
The liberal arts that make up the trivium are: grammar, rhetoric, dialectic and the liberal arts that make up the quadrivium are: arithmetic, geometry, astronomy and music.
In fact, until fairly recently, geometry was highly regarded as an essential part of a liberal education; that is, not specifically leading to a particular profession.
The birthplace of Geometry was with the ancient Greeks, who regarded the geometric constructions performed with only a simple compass and straightedge, as the more elegant than with any other device or instrument.
The simplest geometric construction is probably the bisection of a line segment.
Consider the line segment AB.
First place the compass point at A and draw can arc that crosses the line AB.
Repeat this procedure at point B so that the two arcs intersect.
The line drawn through where the arcs cross bisects the line segment AB.
In fact, for over 2,000 years there were continual efforts using a compass and straightedge to prove the three great geometric construction problems: trisecting an angle, squaring the circle and duplicating the cube. These problems have since been proven to be futile.
In the tenth century a Persian mathematician by the name of Abul Wefa decided to impose an even greater restriction on compass and straightedge geometric constructions. He proposed that all geometric constructions should use a fixed compass, a compass that never alters its radius. This was dubbed the “rusty compass”.
The set of drawings below outline how to bisect a line segment more than twice of a “rusty” compass.
Another application of the “rusty” compass is the geometric construction of an equilateral triangle, (all sides are equal).
Consider the line segment AB below. If the “rusty” compass is set at the same length as AB, we can very simply construct an equilateral triangle by making the two arc crossings formed when the compass point is placed at both A and B.
Lines drawn through where the arcs cross to both A and B completes the equilateral triangle geometric construction.
Geometric constructions using a compass and straightedge were largely employed using a circle to construct regular polygons. This is the focus of the next post!