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The Quadratrix of Hippias

Given a line segment typeset structure, construct the square typeset structure on typeset structure and a quarter circle of radius  typeset structure with center at typeset structure.  The length of  an arc of a circle is proportional to the central angle corresponding to the arc.  Let typeset structure be curve having the property that the ratio of the lengths of arcs of typeset structure and typeset structure on the quarter circle is equal to the ratio of the length of line segments typeset structure and typeset structure, where typeset structure is the intersection of radius vector of typeset structure with the curve.  In other words, the length typeset structure is proportional to the length of the arc typeset structure. If such a curve exits then it can be used to divide an acute angle to any given ratio, simply using the corresponding division of typeset structure.  

The curve described above can be “dynamically” 1 constructed  as the locus of intersection points of two line segments moving with constant velocity:

The point typeset structure on the segment typeset structure is obtained by a limiting process. If the length typeset structure and typeset structure is the angle typeset structure then typeset structure, i.e. typeset structure. Further

| A H | = (| A E |)/(tan φ) = (2 | A C |)/π · φ/(tan φ),(1)


| A G | = lim _ ( φ -> o) (2 | A C |)/π · φ/(tan φ) = (2 | A C |)/π ·(2)

Clearly, Greeks would not proceed in this way, since there was no specific number typeset structure, but Deinostratos using a double reductio absurdum argument showed that

(arc length B D C)/(| A C |) = (| A C |)/(| A G |) .(3)

This shows that the length of the circumference of the circle can be expressed in terms of the lengths of straight lines. This leads to the construction of a square equal to the circle (cf. below), and clearly rectifies the quarter circle.


The curve was conceived by the sophist Hippias of Elis in about 430 B.C. and was used by him in his work to trisect the angle ( [1] , Book IV, XXX, p. 252 ). About 350 BCE the curve was used by Deinostratos for squaring the circle. Because of the later application it is called quadratrix of Hippias, but due to the original application it is also called the trisectrix of Hippias.

Quadratrix of Hippias is the first named curve other than circle and line. It is also the first curve requiring more than a straightedge and compass and the first one that must be plotted point by point. Following a construction described by jesuit monk Christoph Clavius (1537-1612) we can construct a dense set of points on quadratrix in such a way that we divide the segment typeset structure and the right angle typeset structure into typeset structure equal parts by consecutive halving and construct the corresponding intersection points as described above. Clavius believed that in such a way we can construct every point of the curve, but Descartes (correctly) opposed this conclusion.   

To trisect an acute angle typeset structure


If typeset structure the equation of the quadratrix in  Cartesian coordinates is

x = y cot (π y)/(2 a) .(4)

or parametrically

x = (2 a t cot t)/π,      y = (2 a t)/π .(5)

Since typeset structure is a many-valued function the graph of quadratrix has infinitely many branches.


In polar coordinate system we have the equation

r(φ) = (2 a φ )/(π sin φ) .(6)


1 This curves is historically the first curve constructed kinematically (the second one was the Archimedes spiral). For instance, Descartes classification of curves divides curves into two classes algebraic and mechanically generated.


[1]  Pappi Alexandrini. (1875-1878). Collectiones quae supersunt (ed. F. Hultsch) 3 Vols.. Berlin (reprint Amsterdam: Hakkert 1965).

Cite this web-page as:

Štefan Porubský: The Quadratrix of Hippias.

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