View source for Quadratrix of Hippias

{{mergefrom|Dinostratus' theorem|date=February 2025}}
{{Short description|Curve where spinning and moving lines cross}}
{{Use list-defined references|date=January 2025}}
{{CS1 config|mode=cs2}}
[[File:Quadratrix no anim.svg|thumb|right|Quadratrix (red); snapshot of E and F having completed 60% of their motions]]
The '''quadratrix''' or '''trisectrix of Hippias''' (also called the '''quadratrix of Dinostratus''')<ref name="Klein"/> is a [[curve]] which is created by a uniform motion. It is traced out by the crossing point of two [[Line (geometry)|lines]], one moving by [[translation (geometry)|translation]] at a uniform speed, and the other moving by [[rotation (geometry)|rotation]] around one of its points at a uniform speed. An alternative definition as a [[parametric curve]] leads to an equivalence between the quadratrix and the graph of the [[Lambert W function]], or (after a 90° rotation) as the graph of the function <math>y=x\cot x</math>.

The discovery of this curve is attributed to the Greek sophist [[Hippias|Hippias of Elis]], who used it around 420 BC in an attempt to solve the [[Angle trisection|angle trisection problem]], hence its name as a [[trisectrix]]. Later around 350 BC [[Dinostratus]] used it in an attempt to solve the problem of [[squaring the circle]], hence its name as a [[quadratrix]]. '''Dinostratus's theorem''', used in this attempt, relates an endpoint of the curve to the value of {{pi}}. Both angle trisection and squaring the circle can be solved using a compass, a straightedge, and a given copy of this curve, but not by compass and straightedge alone. Although a [[dense set]] of points on the curve can be constructed by compass and straightedge, allowing these problems to be approximated, the whole curve cannot be constructed in this way.

The quadratrix of Hippias is a [[transcendental curve]]. It is one of several curves used in [[Greek mathematics]] for squaring the circle.

== Definitions ==
===By moving lines===
Consider a [[square]] <math>ABCD</math>, and an inscribed quarter circle arc centered at <math>A</math> with radius equal to the side of the square. Let <math>E</math> be a point that travels with a constant [[angular velocity]] along the arc from <math>D</math> to <math>B</math>, and let <math>F</math> be a point that travels simultaneously with a constant [[velocity]] from <math>D</math> to <math>A</math> along [[line segment]] <math>\overline{AD}</math>, so that <math>E</math> and <math>F</math> start at the same time at <math>D</math> and arrive at the same time at <math>B</math> and <math>A</math>. Then the quadratrix is defined as the locus of the intersection of line segment <math>\overline{AE}</math> with the [[parallel (geometry)|parallel line]] to <math>\overline{AB}</math> through <math>F</math>.<ref name="Hischer"/><ref name="Henn"/>

===Helicoid section===
If a line in three-dimensional space, perpendicular to and intersecting the {{nowrap|<math>z</math>-axis,}} rotates at a constant rate at the same time that its intersection with the {{nowrap|<math>z</math>-axis}} moves upward at a constant rate, it will trace out a [[helicoid]]. As [[Pappus of Alexandria]] observed, the curve formed by intersecting this helicoid with a non-vertical plane, when projected onto the {{nowrap|<math>xy</math>-plane,}} forms a quadratrix.<ref name="Koetsier"/>

===Parametric equation===
[[File:Quadratrixkurve.svg|thumb|right|Quadratrix as a plane curve for side length <math>a=1</math>, as given by the parametric formula for {{math|&minus;&infin; < ''t'' < &infin;}}, with singularities when {{mvar|t}} is a nonzero integer multiple of {{pi}}]]
If one places such a square <math>ABCD</math> with side length <math>a</math> in a [[Cartesian coordinate system|(Cartesian) coordinate system]] with the side <math>\overline{AB}</math> on the <math>x</math>-axis and with vertex <math>A</math> at the origin, then the quadratrix is described by a [[parametric equation]] that gives the coordinates of each point on the curve as a function of a time parameter <math>t</math>, as
<math display=block> \gamma(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}=\begin{pmatrix}\frac{2a}{\pi} t\cot(t)\\\frac{2a}{\pi} t\end{pmatrix}</math>
This description can also be used to give an analytical rather than a geometric definition of the quadratrix and to extend it beyond the <math>(0,\tfrac{\pi}{2}]</math> [[interval (mathematics)|interval]]. It does however remain undefined at the points where <math>\cot(t)</math> is [[singularity (mathematics)|singular]], except for the case of <math>t=0</math> where the singularity is removable by evaluating it using the [[limit (mathematics)|limit]] <math>\lim_{t\to 0} t \cot(t)=1</math>. Removing the singularity in this way and extending the parametric definition to negative values of <math>t</math> yields a continuous planar curve on the range of parameter values <math>-\pi<t<\pi</math>.<ref name="Jahnke"/>

===As the graph of a function===
[[File:Quadratrixfunktion.svg|thumb|right|Quadratrix as the graph of a function for <math>a=1</math>]]
When reflected left to right and scaled appropriately, the quadratrix forms the graph of the [[principal branch]] of the [[Lambert W function]].<ref name="CGHJK"/> However, this is a [[multivalued function]], whose graph has many branches above each point of the <math>x</math>-axis. To describe the quadratrix as the graph of an unbranched [[function (mathematics)|function]], it is advantageous to swap the <math>y</math>-axis and the <math>x</math>-axis, that is to place the side <math>\overline{AB}</math> on the <math>y</math>-axis rather than on the <math>x</math>-axis. Then the quadratrix forms the graph of the function<ref name="Underwood"/><ref name="MacTutor1"/>
<math display=block>f(x) = x \cdot \cot\left(\frac{\pi}{2a} \cdot x \right).</math>

== Angle trisection ==
[[File:Mechanical Quadratrix or Quadratrixzirkel.svg|thumb|right|upright=1.0|Quadratrix compass]]
[[File:Angle trisection quadratix hippias.svg|thumb|right|upright=1.25|Angle trisection]]
The trisection of an arbitrary [[angle]] using only [[compass and straightedge]] is impossible. However, if the quadratrix is allowed as an additional tool, it is possible to divide an arbitrary angle into <math>n</math> equal segments and hence a trisection (<math>n=3</math>) becomes possible. In practical terms the quadratrix can be drawn with the help of a [[stencil|template]] or a quadratrix compass (see drawing).<ref name="Hischer"/><ref name="Henn"/><ref name="Simoson"/>

By the definition of the quadratrix, the traversed angle is proportional to the traversed segment of the associated squares' side. Therefore, dividing that segment on the side into <math>n</math> equal parts yields a partition of the associated angle into <math>n</math> equal parts as well. Dividing the line segment into <math>n</math> equal parts with ruler and compass is possible due to the [[intercept theorem]].<ref name="Ostermann"/>

In more detail, to divide a given angle <math>\angle BAE</math> (at most 90°) into any desired number of equal parts, construct a square <math>ABCD</math> over its leg <math>\overline{AB}</math>. The other leg of the angle intersects the quadratrix of the square in a point <math>G</math> and the parallel line to the leg <math>\overline{AB}</math> through <math>G</math> intersects the side <math>\overline{AD}</math> of the square in <math>F</math>. Now the segment <math>\overline{AF}</math> corresponds to the angle <math>\angle BAE</math> and due to the definition of the quadratrix any division of the segment <math>\overline{AF}</math> into <math>n</math> equal segments yields a corresponding division of the angle <math>\angle BAE</math> into <math>n</math> equal angles. To divide the segment <math>\overline{AF}</math> into <math>n</math> equal segments, draw any ray starting at <math>A</math> with <math>n</math> equal segments (of arbitrary length) on it. Connect the endpoint <math>O</math> of the last segment to <math>F</math> and draw lines parallel to <math>\overline{OF}</math> through all the endpoints of the remaining <math>n-1</math> segments on <math>\overline{AO}</math>. These parallel lines divide the segment <math>\overline{AF}</math> into <math>n</math> equal segments. Now draw parallel lines to <math>\overline{AB}</math> through the endpoints of those segments on <math>\overline{AF}</math>, intersecting the trisectrix. Connecting their points of intersection to <math>A</math> yields a partition of angle <math>\angle BAE</math>  into <math>n</math> equal angles.<ref name="Underwood"/>

Since not all points of the trisectrix can be constructed with circle and compass alone, it is really required as an additional tool beyond the compass and straightedge. However it is possible to construct a [[dense set|dense subset]] of the trisectrix by compass and straightedge. In this way, while one cannot assure an exact division of an angle into <math>n</math> parts without a given trisectrix, one can construct an arbitrarily close approximation to the trisectrix and therefore also to the division of the angle by compass and straightedge alone.<ref name="Henn"/><ref name="Jahnke"/>

== Squaring the circle ==
[[File:Circle quadrature quadratix hippias2.svg|thumb|right|upright=1.25|Squaring of a quarter circle with radius 1]]
Squaring the circle with compass and straightedge 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's theorem relating an endpoint of this circle to the value of {{pi}}. One can use this theorem to construct a square with the same [[area]] as a quarter circle. Another square with twice the side length has the same area as the full circle.

===Dinostratus's theorem===
According to Dinostratus's theorem the quadratrix divides one of the sides of the associated square in a ratio of <math>\tfrac{2}{\pi}</math>. More precisely, for the square <math>ABCD</math> used to define the curve, let <math>J</math> be the endpoint of the curve on edge <math>AB</math>. Then<ref name="Hischer"/> 
<math display=block>\frac{\overline{AJ}}{\overline{AB}}=\frac{2}{\pi}.</math>

The point <math>J</math>, where the quadratrix meets the side <math>\overline{AB}</math> 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. This is due to the fact that (as [[Sporus of Nicaea]] already observed) the two uniformly moving lines coincide and hence there exists no unique intersection point.<ref name="MacTutor2"/> However relying on the generalized definition of the quadratrix as a function or planar curve allows for <math>J</math> being a point on the quadratrix.<ref name="Delahaye"/><ref name="MacTutor2"/>

===Construction===
For a given quarter circle with radius <math>r</math> one constructs the associated square <math>ABCD</math> with side length <math>r</math>. The quadratrix intersect the side <math>\overline{AB}</math> in <math>J</math> with <math>\left|\overline{AJ}\right|=\tfrac{2}{\pi}r</math>. Now one constructs a line segment <math>\overline{JK}</math> of length <math>r</math> being [[perpendicular]] to <math>\overline{AB}</math>. Then the line through <math>A</math> and <math>K</math> intersects the extension of the side <math>\overline{BC}</math> in <math>L</math> and from the [[intercept theorem]] follows <math>\left|\overline{BL}\right|=\tfrac{\pi}{2}r</math>. Extending <math>\overline{AB}</math> to the right by a new line segment <math>\left|\overline{BO}\right|=\tfrac{r}{2}</math> yields the rectangle  <math>BLNO</math> with sides <math>\overline{BL}</math> and <math>\overline{BO}</math> 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 [[Geometric mean theorem|Euclid's geometric mean theorem]]. One extends the side <math>\overline{ON}</math> by a line segment <math>\left|\overline{OQ}\right|=\left|\overline{BO}\right|=\tfrac{r}{2}</math> and draws a half circle to right of <math>\overline{NQ}</math>, which has <math>\overline{NQ}</math> as its diameter. The extension of <math>\overline{BO}</math> meets the half circle in <math>\overline{R}</math> and due to [[Thales' theorem]] the line segment <math>\overline{OR}</math> is the [[altitude (triangle)|altitude]] of the [[right-angled triangle]] <math>QNR</math>. Hence the geometric mean theorem can be applied, which means that <math>\overline{OR}</math> forms the side of a square <math>OUSR</math> with the same area as the rectangle <math>BLNO</math> and hence as the quarter circle.<ref name="Holme"/>

== Other properties ==
For a quadratrix constructed from a [[unit square]], the area under the quadratrix is<ref name="Simoson"/>
<math display=block>\frac{2\ln 2}{\pi}\approx 0.44127.</math>

[[Inversive geometry|Inverting]] the quadratrix by a circle centered at the axis of the rotating line that defines it produces a [[cochleoid]], and in the same way inverting the cochleoid produces a quadratrix.<ref name="Eves"/>

== History ==
The quadratrix of Hippias is one of several curves used in [[Greek mathematics]] for squaring the circle, the most well-known for this purpose.<ref name="Klein"/> Another is the [[Archimedean spiral]], used to square the circle by Archimedes.<ref name="Boyer"/>

It is mentioned in the works of [[Proclus]] (412–485), [[Pappus of Alexandria]] (3rd and 4th centuries) and [[Iamblichus]] (c.&nbsp;240&nbsp;–&nbsp;c.&nbsp;325). Proclus names Hippias as the inventor of a curve called a quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappus only mentions how a curve named a quadratrix was used by Dinostratus, [[Nicomedes (mathematician)|Nicomedes]] and others to square the circle. He relays the objections of [[Sporus of Nicaea]] to this construction, but 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.<ref name="Van der Waerden"/><ref name="Gow"/><ref name="Heath"/>

From Proclus' name for the curve, it is conceivable that Hippias himself used it for squaring the circle or some other curvilinear figure. However, most historians of mathematics assume that Hippias invented the curve, but used it only for the trisection of angles. According to this theory, 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]].<ref name="Gow"/><ref name="Heath"/>

[[Rüdiger Thiele]] claims that [[François Viète]] used the trisectrix to derive [[Viète's formula]], an [[infinite product]] of [[nested radicals]] published by Viète in 1593 that converges to <math>2/\pi</math>.<ref name="Jahnke"/> However, other sources instead view Viète's formula as an elaboration of a method of nested polygons used by Archimedes to approximate <math>\pi</math>.<ref name="Beckmann"/> In his 1637 book ''[[La Géométrie]]'', [[René Descartes]] classified curves either as "geometric", admitting a precise geometric construction, or if not as "mechanical"; he gave the quadratrix as an example of a mechanical curve. In modern terminology, roughly the same distinction may be expressed by saying that it is a [[transcendental curve]] rather than an [[algebraic curve]].<ref name="Bos"/> [[Isaac Newton]] used [[trigonometric series]] to determine the area enclosed by the quadratrix.<ref name="Jahnke"/>

== References ==
{{reflist|refs=

<ref name="Beckmann">{{citation
 | last = Beckmann | first = Petr | author-link = Petr Beckmann
 | edition = 2nd
 | isbn = 978-0-88029-418-8
 | mr = 0449960
 | pages = [https://books.google.com/books?id=XqqUUSyz138C&pg=PA94 94–95]
 | publisher = The Golem Press | location = Boulder, Colorado
 | title = A History of {{pi}}
 | title-link = A History of Pi
 | year = 1971}}</ref>

<ref name="Bos">{{citation
 | last = Bos | first = H. J. M. | author-link = Henk J. M. Bos
 | doi = 10.1007/BF00357312
 | issue = 4
 | journal = [[Archive for History of Exact Sciences]]
 | jstor = 41133624
 | mr = 0635811
 | pages = 295–338
 | title = On the representation of curves in Descartes' Géométrie
 | volume = 24
 | year = 1981}}</ref>

<ref name="Boyer">{{citation  | last = Boyer  | first = Carl B. | author-link = Carl Benjamin Boyer | title = A History of Mathematics  | publisher = Princeton University Press  | year = 1968 | location =Princeton, New Jersey  | pages = 140–142  | isbn = 0-691-02391-3}}</ref>

<ref name="CGHJK">{{citation
 | last1 = Corless | first1 = R. M.
 | last2 = Gonnet | first2 = G. H. | author2-link = Gaston Gonnet
 | last3 = Hare | first3 = D. E. G.
 | last4 = Jeffrey | first4 = D. J.
 | last5 = Knuth | first5 = D. E. | author5-link = Donald Knuth
 | doi = 10.1007/BF02124750
 | issue = 4
 | journal = Advances in Computational Mathematics
 | mr = 1414285
 | pages = 329–359
 | title = On the Lambert {{mvar|W}} function
 | url = https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf
 | volume = 5
 | year = 1996}}</ref>

<ref name="Delahaye">{{citation
 | last = Delahaye | first = Jean-Paul
 | isbn = 3764360569
 | page = 71
 | publisher = Springer
 | title = <math>\pi</math> – Die Story
 | year = 1999
 | url = {{Google books|RNb95MU9fUUC|plain-url=yes|page=71}}}}</ref>

<ref name="Eves">{{citation
 | last = Eves | first = Howard | author-link = Howard Eves
 | date = November 1948
 | doi = 10.5951/mt.41.7.0311
 | issue = 7
 | journal = [[The Mathematics Teacher]]
 | jstor = 27953353
 | pages = 311–313
 | title = A graphometer
 | volume = 41}}</ref>

<ref name="Gow">{{citation
 | last = Gow | first = James
 | isbn = 9781108009034
 | pages = 162–164
 | publisher = Cambridge University Press
 | title = A Short History of Greek Mathematics
 | year = 2010
 | url = {{Google books|KSe_ZEmHaXEC|plain-url=yes|page=162}}}}</ref>

<ref name="Heath">{{citation
 | last = Heath | first = Thomas Little | author-link = Thomas Little Heath
 | pages = 182, 225–230
 | publisher = Clarendon Press
 | title = A History of Greek Mathematics, Volume 1: From Thales to Euclid
 | year = 1921 | url = https://archive.org/details/historyofgreekma01heat}}</ref>

<ref name="Henn">{{citation
 | last = Henn | first = Hans-Wolfgang
 | contribution = Die Quadratur des Kreises
 | pages = 45–48
 | publisher = Verlag Vieweg+Teubner
 | title = Elementare Geometrie und Algebra
 | year = 2003
 | url = {{Google books|2caZW8KRMtAC|plain-url=yes|page=47}}}}</ref>

<ref name="Hischer">{{citation
 | last = Hischer | first = Horst
 | editor1-last = Blankenagel | editor1-first = Jürgen
 | editor2-last = Spiegel | editor2-first = Wolfgang
 | contribution = Klassische Probleme der Antike – Beispiele zur "Historischen Verankerung"
 | contribution-url = https://horst.hischer.de/publikationen/buch-beitraege/2000-Festschrift_Scheid/Festschrift_Scheid-60.pdf
 | location = Stuttgart/Düsseldorf/Leipzig
 | pages = 97–118
 | publisher = Klett
 | title = Mathematikdidaktik aus Begeisterung für die Mathematik – Festschrift für Harald Scheid
 | year = 2000}}</ref>

<ref name="Holme">{{citation
 | last = Holme | first = Audun
 | isbn = 9783642144400
 | pages = 114–116
 | publisher = Springer
 | title = Geometry: Our Cultural Heritage
 | year = 2010
 | url = {{Google books|zXwQGo8jyHUC|plain-url=yes|page=114}}}}</ref>

<ref name="Jahnke">{{citation
 | last = Jahnke | first = Hans Niels
 | isbn = 0821826239
 | pages = 30–32
 | publisher = [[American Mathematical Society]]
 | title = A History of Analysis
 | year = 2003}};  {{Google books|CVRZEXFVsZkC|excerpt|page=30}}</ref>

<ref name="Klein">{{citation
 | last = Klein | first = Felix | author-link = Felix Klein
 | location = Boston
 | pages = 57–58
 | publisher = Ginn
 | title = Famous Problems of Elementary Geometry
 | url = https://archive.org/details/famousproblemsof00kleiiala/page/56
 | year = 1897}}</ref>

<ref name="Koetsier">{{citation
 | last = Koetsier | first = Teun
 | doi = 10.1007/978-3-031-39872-8
 | isbn = 9783031398728
 | page = 60
 | publisher = Springer Nature Switzerland
 | series = History of Mechanism and Machine Science
 | title = A History of Kinematics from Zeno to Einstein: On the Role of Motion in the Development of Mathematics
 | year = 2024}}</ref>

<ref name="MacTutor1">{{MacTutor Biography|id=Quadratrix|title=Quadratrix of Hippias|class=Curves}}</ref>

<ref name="MacTutor2">{{MacTutor Biography|id=Dinostratus|title=Dinostratus}}</ref>

<ref name="Ostermann">{{citation |title=Geometry by Its History|first1=Alexander|last1=Ostermann|first2=Gerhard|last2=Wanner
|publisher=Springer|year=2012|isbn=978-3-642-29163-0|pages=[https://books.google.com/books?id=eOSqPHwWJX8C&pg=PA6 6–7]|doi=10.1007/978-3-642-29163-0}}</ref>

<ref name="Simoson">{{citation
 | last = Simoson | first = Andrew J.
 | editor-last = Takenouchi | editor-first = Kazuki
 | contribution = A quadratrix trammel
 | date = September 2024
 | doi = 10.1007/978-3-031-71013-1_25
 | isbn = 9783031710131
 | pages = 268–279
 | publisher = Springer Nature Switzerland
 | series = Lecture Notes on Data Engineering and Communications Technologies
 | title = ICGG 2024 – Proceedings of the 21st International Conference on Geometry and Graphics}}</ref>

<ref name="Underwood">{{citation
 | last = Dudley | first = Underwood | author-link = Underwood Dudley
 | isbn = 0883855143
 | pages = 6–8
 | publisher = Cambridge University Press
 | title = The Trisectors
 | year = 1994}}; {{Google books|tp3kHvbMjqUC|excerpt|page=6}}</ref>

<ref name="Van der Waerden">{{citation
 | last = van der Waerden | first = Bartel Leendert | author-link = Bartel Leendert van der Waerden
 | page = 146
 | publisher = Oxford University Press
 | title = Science Awakening
 | year = 1961}}</ref>

}}

== Further reading ==
*Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA 2010, {{ISBN|9780883853481}}, pp.&nbsp;146–147 ({{Google books|mIT5-BN_L0oC|excerpt|page=146}})

== External links ==
{{Commonscat|Quadratrix of Hippias}}
*Michael D. Huberty, Ko Hayashi, Chia Vang: [http://www.geom.uiuc.edu/~huberty/math5337/groupe/quadratrix.html  ''Hippias' Quadratrix'']
*{{MathWorld|urlname=QuadratrixofHippias|title=Quadratrix of Hippias|mode=cs2}}

{{Ancient Greek mathematics}}

[[Category:Euclidean plane geometry]]
[[Category:Curves]]
[[Category:Squaring the circle]]
[[Category:Area]]