# Quadratrix of Hippias

The **quadratrix** or **trisectrix of Hippias** (also **quadratrix of Dinostratus**) 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 in an attempt to solve the angle trisection problem (hence trisectrix). Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle (hence quadratrix).

## 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 angular velocity on the quarter circle arc from *D* to *B*. In addition the point *F* travels with a constant velocity from *D* to *A* on the line segment *AD*, in such a way that *E* and *F* start at the same time at *D* and arrive at the same time in *B* and *A*. Now the quadratrix is defined as the locus of the intersection of the parallel to *AB* through *F* and the line segment *AE*.[1][2]

If one places such a square *ABCD* with side length *a* in a (cartesian) coordinate system with the side *AB* on the *x*-axis and vertex *A* in the origin, then the quadratix is described by a planar curve with:

This description can also be used to give an analytical rather than a geometric definition of the quadratrix and to extend it beyond the interval. It does however remain undefined at the singularities of except for the case of , where due to the singularity is removable and hence yields a continuous planar curve on the interval [3][4]

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 *AB* on *y*-axis rather than on the *x*-axis. Then the quadratrix is given by the following function *f*(x):[5][6]

## 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 (*n* = 3) becomes possible. In practical terms the quadratrix can be drawn with the help of a template or a quadratrix compass (see drawing).[1][2]

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* ( ≤ 90°) construct a square *ABCD* over its leg *AB*. The other leg of the angle intersects the quadratrix of the square in a point *G* and the parallel line to the leg *AB* through *G* intersects the side *AD* of the square in *F*. Now the segment *AF* corresponds to the angle *BAE* and due to the definition of the quadratrix any division of the segment *AF* in *n* equidistant parts yields a corresponding division of the angle *BAE* into *n* parts of equal size. To divide the segment *AF* into n equidistant parts proceed as follows. Draw a ray a with origin in *A* and then draw n equidistant segments (of arbitrary length) on it. Connect the endpoint *O* of the last segment with *F* and draw lines parallel to *OF* through all the endpoints of the remaining *n* − 1 segments on *AO*, these parallel lines divide the segment *AF* on *AD* into *n* equidistant segments. Now draw parallel lines to *AB* through the endpoints of those segments on *AF*, 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.[5]

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.[2][3]

## 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 .[1] For a given quarter circle with radius *r* one constructs the associated square *ABCD* with side length *r*. The quadratrix intersect the side *AB* in *J* with . Now one constructs a line segment *JK* of length r being perpendicular to *AB*. Then the line through *A* and *K* intersects the extension of the side *BC* in *L* and from the intercept theorem follows . Extending AB to the right by a new line segment yields the rectangle *BLNO* with sides *BL* and *BO* 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 *ON* by a line segment and draws a half circle to right of *NQ*, which has *NQ* as its diameter. The extension of *BO* meets the half circle in *R* and due to Thales' theorem the line segment *OR* is the altitude of the right angle triangle *QNR*. Hence the geometric mean theorem can be applied, which means that *OR* forms the side of a square *OUSR* with the same area as the rectangle *BLNO* and hence as the quarter circle.[7]

Note that the point *J*, where the quadratrix meets the side *AB* 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 (see drawing). 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.[8][9]

## Historical sources

The quadratrix is mentioned in the works of Proclus (412–485), Pappus of Alexandria (3rd and 4th centuries) and Iamblichus (c. 240 – c. 325). 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.[10][11][12]

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.[11][12]

## Notes

- Horst Hischer:
*Klassische Probleme der Antike – Beispiele zur "Historischen Verankerung"*Archived 2012-03-28 at the Wayback Machine. In: Blankenagel, Jürgen & Spiegel, Wolfgang (Hrsg.):*Mathematikdidaktik aus Begeisterung für die Mathematik — Festschrift für Harald Scheid*. Stuttgart/Düsseldorf/Leipzig: Klett 2000, pp. 97 – 118 - Hans-Wolfgang Henn:
*Elementare Geometrie und Algebra*. Verlag Vieweg+Teubner 2003, pp. 45–48 "Die Quadratur des Kreises" (*excerpt*, p. 47, at Google Books) - Hans Niels Jahnke:
*A History of Analysis*. American Mathematical Society 2003, ISBN 0821826239, pp. 30–31 (*excerpt*, p. 30, at Google Books) - Weisstein, Eric W. "Quadratrix of Hippias".
*MathWorld*. - Dudley Underwood:
*The Trisectors*. Cambridge University Press 1994, ISBN 0883855143, pp. 6–8 (*excerpt*, p. 6, at Google Books) - O'Connor, John J.; Robertson, Edmund F., "Quadratrix of Hippias",
*MacTutor History of Mathematics archive*, University of St Andrews. - Audun Holme:
*Geometry: Our Cultural Heritage*. Springer 2010, ISBN 9783642144400, pp. 114–116 (*excerpt*, p. 114, at Google Books) - Jean-Paul Delahaye:
*Pi – Die Story*. Springer 1999, ISBN 3764360569, p. 71 (*excerpt*, p. 71, at Google Books) - O'Connor, John J.; Robertson, Edmund F., "Dinostratus",
*MacTutor History of Mathematics archive*, University of St Andrews. - Van der Waerden:
*Science Awakening*. Oxford University Press 1961, p. 146 - James Gow:
*A Short History of Greek Mathematics*. Cambridge University Press 2010, ISBN 9781108009034, pp. 162–164 (*excerpt*, p. 162, at Google Books) - Thomas Little Heath:
*A History of Greek Mathematics. Volume 1. From Thales to Euclid*. Clarendon Press 1921 (Nachdruck Elibron Classics 2006), pp. 182, 225–230 (online copy at archive.org)

## References

- Claudi Alsina, Roger B. Nelsen:
*Charming Proofs: A Journey Into Elegant Mathematics*. MAA 2010, ISBN 9780883853481, pp. 146–147 (*excerpt*, p. 146, at Google Books) - Felix Klein:
*Famous Problems of Elementary Geometry*. Cosimo 2007 (Nachdruck), ISBN 9781602064171, pp. 57–58 (*excerpt*, p. 57, at Google Books) (complete online copy at archive.org) - Audun Holme:
*Geometry: Our Cultural Heritage*. Springer, 2010, ISBN 9783642144400, pp. 114–116 (*excerpt*, p. 114, at Google Books) - Thomas Little Heath:
*A History of Greek Mathematics. Volume 1. From Thales to Euclid*. Clarendon Press, 1921 (Nachdruck Elibron Classics 2006), pp. 225–230 (online copy at archive.org) - Horst Hischer:
*Klassische Probleme der Antike – Beispiele zur "Historischen Verankerung"**Klassische Probleme der Antike – Beispiele zur "Historischen Verankerung"*. In: Blankenagel, Jürgen & Spiegel, Wolfgang (Hrsg.):*Mathematikdidaktik aus Begeisterung für die Mathematik — Festschrift für Harald Scheid*. Stuttgart/Düsseldorf/Leipzig: Klett 2000, pp. 97 – 118 (German) - Hans-Wolfgang Henn:
*Elementare Geometrie und Algebra*. Vieweg+Teubner, 2003, pp. 45–48 "Die Quadratur des Kreises" (*excerpt*, p. 45, at Google Books) (German)

## External links

Wikimedia Commons has media related to .Quadratrix of Hippias |

- Michael D. Huberty, Ko Hayashi, Chia Vang:
*Hippias' Quadratrix* - Weisstein, Eric W. "Quadratrix of Hippias".
*MathWorld*. - O'Connor, John J.; Robertson, Edmund F., "Quadratrix of Hippias",
*MacTutor History of Mathematics archive*, University of St Andrews.