Trisecting an angle - Squaring the circle
Three famous geometrical construction problems, originating from ancient Greek mathematics occupied many mathematicians until modern times. These problems are
In the ancient Greek tradition the only tools that are available for these constructions are a ruler and
a compass. During the 19th century the French mathematician
Pierre Wantzel proved that
under these circumstances the first two of those constructions are impossible and for the squaring of the circle it
lasted until 1882 before a proof had been given by
Ferdinand von Lindemann!
If we extend the range of tools the problems can be solved. New tools can be material tools (ex. a "marked ruler", that's a ruler with two marks on it, a "double ruler", that's a ruler with two parallel sides,...), or mathematical tools (ex. special curves as conics, spirals,...).
The quadratrixThe quadratrix (of Hippias) is one of the curves that can be used to solve the problem of the trisection of an angle and in some sense the squaring of the circle. The curve already appears in ancient Greek geometry. It's named after Hippias of Elis and was used by Dinostratus and Nicomedes.
The curve is defined in a "dynamical" way. Consider the square ABDC. Suppose that AC rotates uniformly about A until it coincides with AB and that in the same time CD descends uniformly to AB. So AC and CD reach their final position AB at the same time. The quadratrix is the locus of the intersection points of both moving line segments.
This definition of the curve suggests that it isn't very difficult to construct a mechanical tool to obtain a drawing of a quadratrix. The quadratrix has a starting point but no end point! Indeed, when both moving segments reach their final position there is no longer an intersection point. However we see that there should exist a limit point L. Using a coordinate system with A in the origin, B(1,0), C(0,1) and D(1,1) we obtain the following equation for the quadratrix:
|Starting with the constructed angle of 60 degrees, we obtain the points P and Q. The point R is determined in such a way that AQ = 3 AR. This leads to the point S on the quadratrix. From the definition of the quadratrix it follows immediately that the angle BAS measures 20 degrees.|
If we choose the radius OR of the given circle as the unit to measure line segments, then we have to construct a
line segment with length sqrt(pi). We already know that the length of OL is 2/pi. If OM = ML then
we already have constructed a line segment with length 1/pi.
The construction on the right where AC=1, CD=1/pi, CE=1 and BE//AD leads to the point B and to the line segment BC with length pi. Using a further well known construction we obtain the line segment CF with length sqrt(pi).
All these constructions are possible using only ruler and compass! The area of the square constructed on CF equals the area of the circle with radius OR!