Quadratrix
In geometry, a quadratrix is a curve that can be used for quadrature, constructing the area under another curve.
For instance, in integral calculus as developed by Gottfried Wilhelm Leibniz, the quadratrix of a curve was another curve, the graph of its indefinite integral: the area under the first curve could be constructed from the -coordinates of points on the quadratrix. The property of being an indefinite integral was expressed geometrically, as an equality between the -coordinates on the first curve and the subnormals of the second, the difference between the -coordinate of a point on the curve and the -coordinate of the point where a perpendicular line to the curve crosses the -axis.
Certain specific curves are called a quadratrix. The two most famous curves of this class are the quadratrix of Hippias and the quadratrix of E. W. Tschirnhaus, which can be used for squaring the circle, the construction of a square with the area of a given circle.
Quadratrix of Dinostratus
The quadratrix of Hippias was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus, in his Collections, treats its history, and gives two methods by which it can be generated.One construction is as follows. is a quadrant in which the line and the arc are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to and through the corresponding points on the radius. The locus of these intersections is the quadratrix.
The point where the curve crosses the -axis has ; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of, leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge.
An accurate construction of the quadratrix would also allow the solution of another classical problem known to be impossible with compass and straightedge: trisecting an angle.