Search Loci: Convergence:
Music is the pleasure the human soul experiences from counting without being aware that it is counting.
In N. Rose, Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
An Investigation of Historical Geometric Constructions
Construction of the Quadratrix I
Hippias’ quadratrix was described as follows: “ABCD is a square and BED is part of a circle, centre A radius AB. As the radius AB rotates about A to move to the position AD then the line BC moves at the same rate parallel to itself [i.e., B′C′] to end at AD. Then the locus of the point of intersection F of the rotating radius AB [i.e., AE] and the moving line BC is the quadratrix” (from Book IV of Pappus' Synagoge circa 340) (see Figure 1).
Figure 1: Quadratrix of Hippias
To construct the quadratrix using Geometer’s Sketchpad, you will first need to construct a square, ABCD, in your sketch window. Select any point on CD and call it C'. Construct a parallel line to BC, through the point C'. Name the point of intersection of AB and the parallel line point B'. Construct the segment B'C' and hide the line B'C' (see Figure 2).
Figure 2: Construction of the Quadratrix
Construct the circle with center A and radius AB. Next, construct arc BXD by identifying a point on the circle inside the square (point X). Then select, in order, points B, X, D and choose Arc Through 3 Points under the Construct menu (see Figure 3).
Figure 3: Continuing the Construction of the Quadratrix by Constructing Arc BXD.