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The Quadratrix of Hippias

Given a line segment , construct the square on and a quarter circle of radius   with center at .  The length of  an arc of a circle is proportional to the central angle corresponding to the arc.  Let be curve having the property that the ratio of the lengths of arcs of and on the quarter circle is equal to the ratio of the length of line segments and , where is the intersection of radius vector of with the curve.  In other words, the length is proportional to the length of the arc . 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 .  

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

• the first one   rotates (counterclockwise) about until it coincides with
• its second copy moves uniformly and parallel to along the positive -axis until reaches the final position at the same time.

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

(1)

and

(2)

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

(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 and the right angle into 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

• draw the line segment through parallel to which meets in
• trisect the line segment
• find intersection points of the quadratrix with lines drawn through these trisection points parallel to    
• the semi lines going through these intersection points and trisect the given angle .

If the equation of the quadratrix in  Cartesian coordinates is

(4)

or parametrically

(5)

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

In polar coordinate system we have the equation

(6)

Notes

References

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