Quadratrix from Tschirnhaus

Tschirnhaus-Qudratrix (red), Quadratrix des Hippias (dashed)

The Quadratrix von Tschirnhaus , named after Ehrenfried Walther von Tschirnhaus (1651–1708), is a compressed sine curve , with the help of which a square of the circle , i.e. H. the conversion of a circle into an exactly equal square using a construction with compass and ruler . Because of this property, it is called a quadratrix . Since it can also be used to divide the angle into three , it is also a trisectrix .

definition

Tschirnhaus's quadratrix is ​​defined similarly to Hippias's quadratrix by two uniform movements and it is therefore assumed that Tschirnhaus was inspired by that of Hippias in its construction.

In a square ABCD , a point F moves from D to A at a constant speed on the AD side . At the same time, a further point moves E also at a constant speed on the quarter circle around A from D to B . The two constant speeds are selected so that F and E arrive at their destinations A and B at the same time . Through the point F a parallel to the side is now AB drawn and the point E , the solder on the side AB like. The parallel and the perpendicular intersect at a point S , which moves from D to B as E and F move . The quadratrix of Tschirnhaus is now defined as the locus of the point S .

If you place the square ABCD in the coordinate system in such a way that the point D is at the origin and the side AD is on the x-axis, then the quadratrix can be described as follows using a compressed sine function:

${\displaystyle f(x)=r\sin \left({\frac {\pi x}{2r}}\right)}$

Here r is the radius of the quarter circle or the side length of the square.

Angular division

Trisection of an angle, ${\displaystyle \alpha =\beta =\gamma }$

Due to the construction of the quadratrix via uniform movements, the ratio of the length of a route section on the square side AD to the length of the total side corresponds to the ratio of the length of the corresponding section on the circular arc to the length of the quarter circle. The latter ratio, however, corresponds to the angle belonging to the circular arcs, so using the quadratrix for a division of a segment on the square side, a corresponding division of the associated angle is also obtained. This enables the following method for dividing an angle into n equal parts.

For a given angle BAE , the square ABCD and the associated square rix are built over its leg AB . The quarter circle around A with radius | AB | cuts the other side of the angle in E . Of E can now coat the solder (vertical) to the angle legs AB and the solder intersects the quadratrix in U . Draw a parallel through U to leg AB , which intersects the side of the square AD in F. The subdivision of the segment AF into n equal parts then provides a subdivision of the angle into n equal parts via the quadratrix by drawing further parallels to the leg AB through the partial points of the segment AF . At the points of intersection with the quadratrix, perpendiculars to AB are set up , the points of intersection with the quadrant finally provide the division of the angle. The subdivision of the distance AF even with compass and ruler is possible due to the theorem of rays (see drawing and division of a distance with compass and ruler in a given ratio ).

Circle quadrature

Circle quadrature using the quadratrix and its tangent

Assuming that in addition to the quadratrix itself, you also have its tangent available, i.e. the construction device for the quadratrix also supplies the associated tangent at least in its end points B and D , then the tangent through D intersects the extension of the side AB in E. with . A rectangle can now be constructed with AE , the area of ​​which corresponds to a quarter circle, and this rectangle can be converted into a square with the same area using Euclid's theorem of heights or the cathetus set (see drawing). ${\displaystyle |AE|={\tfrac {\pi }{2}}|AB|}$

literature

• John Martin Frederick Wright: An Algebraic System of Conic Sections, and other Curves . Black and Armstrong, 1836, pp. 206-208
• Nikolaus Fialkowski: Division of the angle and the circle, or: Bi-, tri-, quadri- and polysection of any angle in 72 new methods. Gerold, Vienna 1860, pp. 208-211
• AB Ivanov: Quadratrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).