**THE QUADRATRIX**
**Trisecting an angle - Squaring the circle
**

**Introduction**

Three famous geometrical construction problems, originating from ancient Greek mathematics occupied many mathematicians until modern times. These problems are

**the duplication of the cube:**

construct (the edge of) a cube whose volume is double the volume of a given cube,**angle trisection:**

construct an angle that equals one third of a given angle,**the squaring of a circle:**

given (the radius of) a circle, construct (the side of) a square whose area equals the area of the circle.

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 quadratrix**

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:

This expression doesn't have any sense for y = 0. The

Once we admit that we can use the limit point L, only ruler and compass are needed for the further construction of the square with side length sqrt(pi). The area of this square equals the area of the circle with radius OR.

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