View source for Quadratrix

{{short description|Curve with axes measuring the area of another curve}}
In [[Geometry|geometry]], a '''quadratrix''' ({{ety|la|quadrator|squarer}}) is a [[Curve|curve]] having ordinates which are a measure of the [[Area|area]] (or quadrature) of another curve. The two most famous curves of this class are those of [[Biography:Dinostratus|Dinostratus]] and E. W. Tschirnhaus, which are both related to the [[Circle|circle]].

==Quadratrix of Dinostratus==
{{Main|Quadratrix of Hippias}}
The quadratrix of Dinostratus (also called the ''quadratrix of Hippias'') was well known to the [[Social:Ancient Greek|ancient Greek]] geometers, and is mentioned by [[Biography:Proclus|Proclus]], who ascribes the invention of the curve to a contemporary of [[Biography:Socrates|Socrates]], probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of [[Biography:Plato|Plato]], discussed the curve, and showed how it effected a mechanical solution of [[Squaring the circle|squaring the circle]]. [[Biography:Pappus of Alexandria|Pappus]], in his ''Collections'', treats its history, and gives two methods by which it can be generated. 
# Let a [[Helix|helix]] be drawn on a right circular cylinder; a screw surface is then obtained by drawing [[Line (geometry)|lines]] from every point of this spiral [[Perpendicular|perpendicular]] to its axis. The orthogonal projection of a section of this surface by a [[Plane (geometry)|plane]] containing one of the perpendiculars and inclined to the axis is the quadratrix. 
# A right cylinder having for its base an [[Archimedean spiral]] is intersected by a right circular [[Conical surface|cone]] which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.
[[File:Quadratrix_no_anim.svg|thumb|right|Quadratrix of Dinostratus (in red)]]
Another construction is as follows. {{mvar|DAB}} is a [[Circular sector|quadrant]] in which the line {{mvar|DA}} and the arc {{mvar|{{overarc|DB}}}} are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn [[Parallel (geometry)|parallel]] to {{mvar|AB}} and through the corresponding points on the radius {{mvar|{{overline|DA}}}}. The [[Locus (mathematics)|locus]] of these intersections is the quadratrix. 
[[File:Quadratrixfunktion.svg|thumb|left|Quadratrix of Dinostratus with a central portion flanked by infinite branches]]
Letting {{mvar|A}} be the origin of the [[Cartesian coordinate system]], {{mvar|D}} be the point {{math|(''a'', 0)}}, {{mvar|a}} units from the origin along the {{mvar|x}}-axis, and {{mvar|B}} be the point {{math|(0, ''a'')}}, {{mvar|a}} units from the origin along the {{mvar|y}}-axis, the curve itself can be expressed by the equation<ref>{{SpringerEOM|title=Dinostratus quadratrix|id=Dinostratus_quadratrix}}</ref>
:<math>y = x\cot\left(\frac{\pi x}{2a}\right).</math>
Because the [[Cotangent|cotangent]] [[Function (mathematics)|function]] is invariant under negation of its argument, and has a simple [[Zeros and poles|pole]] at each multiple of {{pi}}, the quadratrix has [[Reflection symmetry|reflection symmetry]] across the {{mvar|y}}-axis, and similarly has a pole for each value of {{mvar|x}} of the form  {{math|1=''x'' = 2''na''}}, for [[Integer|integer]] values of {{mvar|n}}, except at {{math|1=''x'' = 0}} where the pole in the cotangent is canceled by the factor of {{mvar|x}} in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the {{mvar|y}}-axis has {{math|1=''y'' = 2''a/π''}}; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a [[Rational number|rational]] multiple of {{math|1/''π''}}, leading to a solution of the classical problem of [[Squaring the circle|squaring the circle]]. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge.
An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and straightedge: [[Doubling the cube|doubling the cube]] and trisecting an angle.

==Quadratrix of Tschirnhaus==
[[File:Tschirnhaus.svg|thumb|Tschirnhaus' quadratrix (red),<br/> Hippias quadratrix (dotted)]]
The quadratrix of Tschirnhaus<ref>See definition and drawing in the following online source: {{cite book|author=Hutton C.|title=A Philosophical and Mathematical Dictionary Containing... Memoirs of the Lives and Writings of the Most Eminent Authors |volume=2|location=London|pages=271–272|year=1815|url=https://books.google.com/books?id=_xk2AAAAQAAJ&pg=PA272}}</ref> is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before. The mutual intersections of the lines drawn from the points of division of the arc parallel to ''DA'', and the lines drawn parallel to ''AB'' through the points of division of ''DA'', are points on the quadratrix. The Cartesian equation is <math>y = a \cos\!\big(\tfrac{\pi x}{2a}\big)</math>. The curve is periodic, and cuts the ''x''-axis at the points <math>x = (2n - 1)a</math>, <math>n</math> being an integer; the maximum values of <math>y</math> are <math>a</math>. Its properties are similar to those of the quadratrix of Dinostratus.

== Other quadratrices ==
Other curves that have historically been used to square the circle include the [[Archimedean spiral]] and the [[Cochleoid|cochleoid]].

==References==
*{{1911|wstitle=Quadratrix|volume=22|page=706}}
{{reflist}}

==External links==
* [http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Quadratrix.html Quadratrix of Hippias] {{Webarchive|url=https://web.archive.org/web/20120204120351/http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Quadratrix.html |date=2012-02-04 }} at the MacTutor archive.
*[http://www.maa.org/press/periodicals/convergence/an-investigation-of-historical-geometric-constructions Hippias' Quadratrix] at [http://www.maa.org/press/periodicals/convergence Convergence] (MAA periodical)

[[Category:Curves]]
[[Category:Area]]

{{Sourceattribution|Quadratrix}}