Quadratrix of Hippias


The quadratrix or trisectrix of Hippias is a curve, which is created by a uniform motion. It is one of the oldest examples for a kinematic curve, that is a curve created through motion. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC to trisect an angle. Later around 350 BC Dinostratus used it to square the circle.

Definition

Consider a square ABCD with an inscribed quarter circle centered in A, such that the side of the square is the circle's radius. Let E be a point that travels with a constant speed on the quarter circle from B to D and let F be a point that travels with a constant speed from A to D on the line segment in such a way that F starts at "A" at the same time as E starts at B and E and F both arrive at D at the same time. Now the quadratrix is defined as the locus of the intersection of the parallel to through F and the line segment.
If one places such a square ABCD with side length a in the first quadrant of a coordinate system with the side on the x-axis and vertex A at the origin, then the quadratix is described by a planar curve with:
If the domain of and are extended to include the entire real line, these equations describe a family of curves. The domain of can be further extended to include t=0 because. If we define the curve is continuous for.
To describe the quadratrix as simple function rather than planar curve, it is advantageous to switch the y-axis and the x-axis, that is to place the side on y-axis rather than on the x-axis. Then the quadratrix is given by the following function f:

Angle trisection

The trisection of an arbitrary angle using only ruler and compasses is impossible. However, if the quadratrix is allowed as an additional tool, it is possible to divide an arbitrary angle into n equal segments and hence a trisection becomes possible. In practical terms the quadratrix can be drawn with the help of a template or a quadratrix compass.
Since by the definition of the quadratrix the traversed angle is proportional to the traversed segment of the associated squares' side dividing that segment on the side into n equal parts yields a partition of the associated angle as well. Dividing the line segment into n equal parts with ruler and compass is possible due to the intercept theorem.
For a given angle BAE construct a square ABCD over its leg. The other leg of the angle intersects the quadratrix of the square in a point G and the parallel line to the leg through G intersects the side of the square in F. Now the segment corresponds to the angle BAE and due to the definition of the quadratrix any division of the segment in n equidistant parts yields a corresponding division of the angle BAE into n parts of equal size. To divide the segment into n equidistant parts proceed as follows. Draw a ray a with origin in A and then draw n equidistant segments on it. Connect the endpoint O of the last segment with F and draw lines parallel to through all the endpoints of the remaining n − 1 segments on, these parallel lines divide the segment on into n equidistant segments. Now draw parallel lines to through the endpoints of those segments on, these parallel lines will intersects the trisectrix. Connecting those points of intersection with A yields a partition of angle BAE into n parts of equal size.
Since not all points of the trisectrix can be constructed with circle and compass alone, it is really required as an additional tool next to compass and circle. However it is possible to construct a dense subset of the trisectrix by circle and compass, so while you cannot assure an exact division of an angle into n parts without a given trisectrix, you can construct an arbitrarily close approximation by circle and compass alone.

Squaring of the circle

Squaring the circle with ruler and compass alone is impossible. However, if one allows the quadratrix of Hippias as an additional construction tool, the squaring of the circle becomes possible due to Dinostratus' theorem. It lets one turn a quarter circle into square of the same area, hence a square with twice the side length has the same area as the full circle.
According to Dinostratus' theorem the quadratrix divides one of the sides of the associated square in a ratio of. For a given quarter circle with radius r one constructs the associated square ABCD with side length r. The quadratrix intersect the side in J with. Now one constructs a line segment of length r being perpendicular to. Then the line through A and K intersects the extension of the side in L and from the intercept theorem follows. Extending to the right by a new line segment yields the rectangle BLNO with sides and the area of which matches the area of the quarter circle. This rectangle can be transformed into a square of the same area with the help of Euclid's geometric mean theorem. One extends the side by a line segment and draws a half circle to right of, which has as its diameter. The extension of meets the half circle in R and due to Thales' theorem the line segment is the altitude of the right angle triangle QNR. Hence the geometric mean theorem can be applied, which means that forms the side of a square OUSR with the same area as the rectangle BLNO and hence as the quarter circle.
Note that the point J, where the quadratrix meets the side of the associated square, is one of the points of the quadratrix that cannot be constructed with ruler and compass alone and not even with the help of the quadratrix compass based on the original geometric definition. This is due to the fact, that the 2 uniformly moving lines coincide and hence there exists no unique intersection point. However relying on the generalized definition of the quadratrix as a function or planar curve allows for J being a point on the quadratrix.

Historical sources

The quadratrix is mentioned in the works of Proclus, Pappus of Alexandria and Iamblichus. Proclus names Hippias as the inventor of a curve called quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappus only mentions how a curve named quadratrix was used by Dinostratus, Nicomedes and others to square the circle. He neither mentions Hippias nor attributes the invention of the quadratrix to a particular person. Iamblichus just writes in a single line, that a curve called a quadratrix was used by Nicomedes to square the circle.
Though based on Proclus' name for the curve it is conceivable that Hippias himself used it for squaring the circle or some other curvilinear figure, most historians of mathematics assume that Hippias invented the curve, but used it only for the trisection of angles. Its use for squaring the circle only occurred decades later and was due to mathematicians like Dinostratus and Nicomedes. This interpretation of the historical sources goes back to the German mathematician and historian Moritz Cantor.