View source for Witch of Agnesi

{{short description|mathematical curve}}
{{Use dmy dates|date=July 2018}}
{{good article}}
[[File:Witch of Agnesi, a 1, 2, 4, 8.svg|thumb|upright=1.3|right|The witch of Agnesi with parameters {{math|1=''a'' = 1}}, {{math|1=''a'' = 2}}, {{math|1=''a'' = 4}}, and {{math|1=''a'' = 8}}]]
In [[mathematics]], the '''witch of Agnesi''' ({{IPA-it|a.ˈɲe.zi}}) is a [[cubic plane curve]] defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician [[Maria Gaetana Agnesi]], and from a mistranslation of an Italian word for a [[Sheet (sailing)|sailing sheet]]. Before Agnesi, the same curve was studied by [[Pierre de Fermat|Fermat]], [[Luigi Guido Grandi|Grandi]], and [[Isaac Newton|Newton]].

The [[Graph of a function|graph]] of the derivative of the [[arctangent]] function forms an example of the witch of Agnesi.
As the [[probability density function]] of the [[Cauchy distribution]], the witch of Agnesi has applications in [[probability theory]]. It also gives rise to [[Runge's phenomenon]],
has been used to approximate the energy distribution of [[spectral line]]s, and models the shape of hills.

The witch is tangent to its defining circle at one of the two defining points, and [[asymptotic]] to the [[Tangent lines to circles|tangent line to the circle]] at the other point. It has a unique [[vertex (curve)|vertex]] (a point of extreme curvature) at the point of tangency with its defining circle, which is also its [[osculating circle]] at that point. It also has two finite [[inflection point]]s and one infinite inflection point. The area between the witch and its asymptotic line is four times the area of the defining circle, and the volume of revolution of the curve around its defining line is twice the volume of the [[torus]] of revolution of its defining circle.

==Construction==
[[File:Witch of Agnesi, construction.svg|thumb|upright=1.3|right|The witch of Agnesi (curve ''MP'') with labeled points]]
[[Image:Agnesi.gif|thumb|right|An animation showing the construction of the witch of Agnesi]]
 
To construct this curve, start with any two points ''O'' and ''M'', and draw a circle with ''OM'' as diameter. For any other point ''A'' on the circle, let ''N'' be the point of intersection of the [[secant line]] ''OA'' and the tangent line at ''M''.
Let ''P'' be the point of intersection of a line perpendicular to ''OM'' through ''A'', and a line parallel to ''OM'' through ''N''. Then ''P'' lies on the witch of Agnesi. The witch consists of all the points ''P'' that can be constructed in this way from the same choice of ''O'' and ''M''.{{r|eagles}} It includes, as a limiting case, the point ''M'' itself.

==Equations==
Suppose that point ''O'' is at the [[Origin (mathematics)|origin]] and point ''M'' lies on the positive ''y''-axis, and that the circle with diameter ''OM'' has radius ''a''.
Then the witch constructed from ''O'' and ''M'' has the Cartesian equation{{r|lawrence|yates}}
: <math>y = \frac{8a^3}{x^2+4a^2}.</math>
This equation can be simplified, by choosing ''a'' = 1/2, to the form
: <math>y = \frac{1}{x^2+1}.</math>
In its simplified form, this curve is the [[Graph of a function|graph]] of the [[derivative]] of the [[arctangent]] function.{{r|calc}}

The witch of Agnesi can also be described by [[parametric equation]]s whose parameter {{mvar|θ}} is the angle between OM and OA, measured clockwise:{{r|lawrence|yates}}
: <math>x = 2a \tan \theta,</math>
: <math>y = 2a \cos ^2 \theta.</math>

==Properties==
The main properties of this curve can be derived from [[integral calculus]].
The area between the witch and its asymptotic line is four times the area of the fixed circle, <math>4\pi a^2</math>.{{r|lawrence|yates|larsen}}
The [[volume of revolution]] of the witch of Agnesi about its asymptote is <math>4\pi^2a^3</math>.{{r|lawrence}} This is two times the volume of the [[torus]] formed by revolving the defining circle of the witch around the same line.{{r|larsen}}

The curve has a unique [[Vertex (curve)|vertex]] at the point of tangency with its defining circle. That is, this point is the only point where the [[curvature]] reaches a local minimum or local maximum.{{r|egdc}} The defining circle of the witch is also its [[osculating circle]] at the vertex,{{r|haftendorn}} the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature.{{r|kiss}} Because this is an osculating circle at the vertex of the curve, it has [[Contact (mathematics)|third-order contact]] with the curve.{{r|omnibus}}

The curve has two [[inflection point]]s, at the points
:<math>\left( \pm\frac{2a}{\sqrt{3}}, \frac{3a}{2}\right)</math>
corresponding to the angles <math>\theta=\pm\pi/6</math>.{{r|lawrence|yates}} When considered as a curve in the [[projective plane]] there is also a third infinite inflection point, at the point where the [[line at infinity]] is crossed by the asymptotic line. Because one of its inflection points is infinite, the witch has the minimum possible number of finite real inflection points of any non-singular cubic curve.{{r|arnold}}

The largest area of a [[rectangle]] that can be inscribed between the witch and its asymptote is <math>4a^2</math>, for a rectangle whose height is the radius of the defining circle and whose width is twice the diameter of the circle.{{r|larsen}}

==History==
===Early studies===
[[File:Witch of Agnesi (Agnesi, 1748).jpg|thumb|Agnesi's 1748 illustration of the curve and its construction{{r|agnesi}}]]
The curve was studied by [[Pierre de Fermat]] in his 1659 treatise on [[Quadrature (mathematics)|quadrature]]. In it, Fermat computes the area under the curve and (without details) claims that the same method extends as well to the [[cissoid of Diocles]]. Fermat writes that the curve was suggested to him "''ab erudito geometra''" [by a learned geometer].{{r|fermat}} {{harvtxt|Paradís|Pla|Viader|2008}} speculate that the geometer who suggested this curve to Fermat might have been [[Antoine de Laloubère]].{{r|fmq}}

The construction given above for this curve was found by {{harvtxt|Grandi|1718}}; the same construction was also found earlier by [[Isaac Newton]], but only published posthumously later, in 1779.{{r|stigler}}
{{harvtxt|Grandi|1718}} also suggested the name  ''versiera'' (in Italian) or ''versoria'' (in Latin) for the curve. The Latin term is also used for a [[Sheet (sailing)|sheet]], the rope which turns the sail, but Grandi may have instead intended merely to refer to the [[versine]] function that appeared in his construction.{{r|larsen|stigler|truesdell|grandi}}

In 1748, [[Maria Gaetana Agnesi]] published ''Instituzioni analitiche ad uso della gioventù italiana'', an early textbook on [[calculus]].{{r|agnesi}}
In it, after first considering two other curves, she includes a study of this curve. She defines the curve geometrically as the locus of points satisfying a certain proportion, determines its algebraic equation, and finds its vertex, asymptotic line, and inflection points.{{r|struik}}

===Etymology===
[[Maria Gaetana Agnesi]] named the curve according to Grandi, ''versiera''.{{r|truesdell|struik}} Coincidentally, at that time in Italy it was common to speak of the [[Devil]], the adversary of God, through other words like ''aversiero'' or ''versiera'', derived from Latin ''adversarius''. ''Versiera'', in particular, was used to indicate the wife of the devil, or "witch".{{r|fanfani}} Because of this, Cambridge professor [[John Colson]] mistranslated the name of the curve as "witch".{{r|mulcrone}} Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened.{{r|singh|darling}} [[Dirk Jan Struik|Struik]] mentions that:{{r|struik}}
{{quote|The word [''versiera''] is derived from Latin ''vertere'', to turn, but is also an abbreviation of Italian ''avversiera'', female devil. Some wit in England once translated it 'witch', and the silly pun is still lovingly preserved in most of our textbooks in English language. ... The curve had already appeared in the writings of [[Pierre de Fermat|Fermat]] (''Oeuvres'', I, 279–280; III, 233–234) and of others; the name ''versiera'' is from Guido Grandi (''Quadratura circuli et hyperbolae'', Pisa, 1703). The curve is type 63 in [[Isaac Newton|Newton]]'s classification. ... The first to use the term 'witch' in this sense may have been B. Williamson, ''Integral calculus'', 7 (1875), 173;{{r|oed}} see ''Oxford English Dictionary''.}}
On the other hand, [[Stephen Stigler]] suggests that  Grandi himself "may have been indulging in a play on words", a double pun connecting the devil to the versine and the sine function to the shape of the female breast (both of which can be written as "seno" in Italian).{{r|stigler}}

==Applications==
A scaled version of the curve is the [[probability density function]] of the [[Cauchy distribution]]. This is the probability distribution on the [[random variable]] <math>x</math> determined by the following [[Experiment (probability theory)|random experiment]]: for a fixed point <math>p</math> above the <math>x</math>-axis, choose uniformly at random a line through <math>p</math>, and let <math>x</math> be the coordinate of the point where this random line crosses the axis. The Cauchy distribution has a peaked distribution visually resembling the [[normal distribution]], but its [[Heavy-tailed distribution|heavy tails]] prevent it from having an [[expected value]] by the usual definitions, despite its symmetry. In terms of the witch itself, this means that the <math>x</math>-coordinate of the [[centroid]] of the region between the curve and its asymptotic line is not well-defined, despite this region's symmetry and finite area.{{r|stigler|alexander}}

In [[numerical analysis]], when approximating functions using [[polynomial interpolation]] with equally spaced interpolation points, it may be the case for some functions that using more points creates worse approximations, so that the interpolation diverges from the function it is trying to approximate rather than converging to it. This paradoxical behavior is called [[Runge's phenomenon]]. It was first discovered by [[Carl David Tolmé Runge]] for Runge's function <math>y=1/(1+25x^2)</math>, another scaled version of the witch of Agnesi, when interpolating this function over the interval <math>[-1,1]</math>. The same phenomenon occurs for the witch <math>y=1/(1+x^2)</math> itself over the wider interval <math>[-5,5]</math>.{{r|runge}}

The witch of Agnesi approximates the [[spectral energy distribution]] of [[spectral lines]], particularly [[X-ray]] lines.{{r|spencer}}

The cross-section of a smooth [[hill]] has a similar shape to the witch.{{r|coppin}} Curves with this shape have been used as the generic topographic obstacle in a flow in mathematical modeling.{{r|snyder|lamb}}
[[Soliton|Solitary wave]]s in deep water can also take this shape.{{r|benjamin|noonan}}

A version of this curve was used by [[Gottfried Wilhelm Leibniz]] to derive the [[Leibniz formula for π|Leibniz formula for {{pi}}]]. This formula, the [[infinite series]]
:<math>\frac{\pi}{4} = 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots,</math>
can be derived by equating the area under the curve with the integral of the function <math>1/(1+x^2)</math>,
using the [[Taylor series]] expansion of this function as the infinite [[geometric series]] <math>1-x^2+x^4-x^6+\cdots</math>, and integrating term-by-term.{{r|yates}}

==In popular culture== 
''The Witch of Agnesi'' is the title of a novel by Robert Spiller. It includes a scene in which a teacher gives a version of the history of the term.{{r|gazette}}
 	
''Witch of Agnesi'' is also the title of a music album by jazz quartet Radius. The cover of the album features an image of the construction of the witch.{{r|discogs}}

==Notes==
{{reflist|refs=

<ref name=agnesi>{{citation
 | last = Agnesi | first = Maria Gaetana | author-link = Maria Gaetana Agnesi
 | title = Instituzioni analitiche ad uso della gioventú italiana
 | url = https://archive.org/details/A298183
 | year = 1748}} See in particular Problem 3, [https://archive.org/details/A298183/page/n403 pp. 380–382], and [https://archive.org/details/A298183/page/n507 Fig. 135].</ref>

<ref name=alexander>{{citation
 | last = Alexander | first = J. McKenzie
 | doi = 10.5840/jphil20121091233
 | issue = 12
 | journal = Journal of Philosophy
 | pages = 712–727
 | title = Decision theory meets the Witch of Agnesi
 | volume = 109
 | year = 2012}}</ref>

<ref name=arnold>{{citation
 | last = Arnold | first = V. I. | authorlink = Vladimir Arnold
 | contribution = The principle of topological economy in algebraic geometry
 | doi = 10.1017/CBO9780511614156.003
 | location = Cambridge
 | mr = 2166922
 | pages = 13–23
 | publisher = Cambridge University Press
 | series = London Mathematical Society Lecture Note Series
 | title = Surveys in modern mathematics
 | volume = 321
 | year = 2005}}. See in particular [https://books.google.com/books?id=dpJo33o6InEC&pg=PA15 pp. 15–16].</ref>

<ref name=benjamin>{{citation
 | last = Benjamin | first = T. Brooke
 | date = September 1967
 | doi = 10.1017/s002211206700103x
 | issue = 3
 | journal = Journal of Fluid Mechanics
 | page = 559
 | title = Internal waves of permanent form in fluids of great depth
 | volume = 29| bibcode = 1967JFM....29..559B
 }}</ref>

<ref name=calc>{{citation
 | last1 = Cohen | first1 = David W.
 | last2 = Henle | first2 = James M.
 | isbn = 9780763729479
 | page = 351
 | publisher = Jones & Bartlett Learning
 | title = Calculus: The Language of Change
 | url = https://books.google.com/books?id=QNfZls4urMoC&pg=PA351
 | year = 2005}}</ref>

<ref name=coppin>{{citation
 | last1 = Coppin | first1 = P. A.
 | last2 = Bradley | first2 = E. F.
 | last3 = Finnigan | first3 = J. J.
 | date = April 1994
 | doi = 10.1007/bf00713302
 | issue = 1–2
 | journal = Boundary-Layer Meteorology
 | pages = 173–199
 | title = Measurements of flow over an elongated ridge and its thermal stability dependence: The mean field
 | volume = 69
 | quote = A useful general form for the hill shape is the so-called 'Witch of Agnesi' profile| bibcode = 1994BoLMe..69..173C
 }}</ref>

<ref name=darling>{{citation
 | last = Darling | first = David | authorlink = David J. Darling
 | isbn = 0-471-27047-4
 | location = Hoboken, NJ
 | mr = 2078978
 | page = 8
 | publisher = John Wiley & Sons
 | title = The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
 | year = 2004}}</ref>

<ref name=discogs>{{citation|url=https://www.discogs.com/Radius-Witch-Of-Agnesi/release/1335001|publisher=Discogs|title=Radius – Witch Of Agnesi (Plutonium Records, 2002)|accessdate=28 May 2018}}</ref>

<ref name=eagles>{{citation|title=Constructive Geometry of Plane Curves: With Numerous Examples|first=Thomas Henry|last=Eagles|publisher=Macmillan and Company|year=1885|contribution=The Witch of Agnesi|pages=313–314|contribution-url=https://archive.org/stream/constructivegeom00eagluoft#page/312/mode/2up}}</ref>

<ref name=egdc>{{citation
 | last = Gibson | first = C. G.
 | at = [https://books.google.com/books?id=_G4cYUQpFucC&pg=PA131 Exercise 9.1.9, p. 131]
 | doi = 10.1017/CBO9781139173377
 | isbn = 0-521-80453-1
 | location = Cambridge
 | mr = 1855907
 | publisher = Cambridge University Press
 | title = Elementary Geometry of Differentiable Curves: An Undergraduate Introduction
 | year = 2001}}</ref>

<ref name=fanfani>[[Pietro Fanfani]], ''Vocabolario dell'uso toscano'', [https://books.google.com/books?id=SOczmy2F2y0C&pg=PA334 p. 334]</ref>

<ref name=fermat>{{citation|first=Pierre|last=de Fermat|authorlink=Pierre de Fermat|title=Oevres|volume=1|publisher=Gauthier-Villars et fils|year=1891|url=https://archive.org/stream/uvresdefermat00natigoog#page/n330/mode/2up|language=Latin|pages=280–285}}</ref>

<ref name=fmq>{{citation
 | last1 = Paradís | first1 = Jaume
 | last2 = Pla | first2 = Josep
 | last3 = Viader | first3 = Pelegrí
 | issue = 1
 | journal = Revue d'Histoire des Mathématiques
 | mr = 2493381
 | pages = 5–51
 | title = Fermat's method of quadrature
 | url = https://www.numdam.org/item/RHM_2008__14_1_5_0
 | volume = 14
 | year = 2008}}</ref>

<ref name=gazette>{{citation|newspaper=[[The Gazette (Colorado Springs)|The Gazette]]|title=Local teacher, author figures math into books|first=Dave|last=Phillips|date=12 September 2006|url=http://gazette.com/local-teacher-author-figures-math-into-books/article/6514}}</ref>

<ref name=grandi>{{citation|first=G.|last=Grandi|authorlink=Luigi Guido Grandi|year=1718|contribution=Note al trattato del Galileo del moto naturale accellerato|title=Opera Di Galileo Galilei|volume=III|page=393|location=Florence|language=Italian}}. As cited by {{harvtxt|Stigler|1974}}.</ref>

<ref name=haftendorn>{{citation
 | last = Haftendorn | first = Dörte | authorlink = Dörte Haftendorn
 | contribution = 4.1 Versiera, die Hexenkurve
 | doi = 10.1007/978-3-658-14749-5
 | language = German
 | pages = 79–91
 | publisher = Springer
 | title = Kurven erkunden und verstehen
 | year = 2017| isbn = 978-3-658-14748-8 }}. For the osculating circle, see in particular p. 81: "Der erzeugende Kreis ist der Krümmungskreis der weiten Versiera in ihrem Scheitel."</ref>

<ref name=kiss><!-- ridiculous dictionary definition of "kiss" reference requested by Gatoclass for DYK -->{{citation|title=Multivariable Calculus with MATLAB®: With Applications to Geometry and Physics|first1=Ronald L.|last1=Lipsman|first2=Jonathan M.|last2=Rosenberg|publisher=Springer|year=2017|isbn=9783319650708|page=42|url=https://books.google.com/books?id=PABCDwAAQBAJ&pg=PA42|quote=The circle "kisses" the curve accurately to second order, thus is given the name osculating circle (from the Latin word for "kissing").}}</ref>

<ref name=lamb>{{citation
 | last = Lamb | first = Kevin G.
 | date = February 1994
 | doi = 10.1017/s0022112094003411
 | issue = –1
 | journal = Journal of Fluid Mechanics
 | page = 1
 | title = Numerical simulations of stratified inviscid flow over a smooth obstacle
 | url = https://web.archive.org/web/20140106224752/https://mseas.mit.edu/download/evheubel/LambJFM1994.pdf
 | volume = 260| bibcode = 1994JFM...260....1L
 }}</ref>

<ref name=larsen>{{citation
 | last = Larsen | first = Harold D.
 | date = January 1946
 | doi = 10.1111/j.1949-8594.1946.tb04418.x
 | issue = 1
 | journal = School Science and Mathematics
 | pages = 57–62
 | title = The Witch of Agnesi
 | volume = 46}}</ref>

<ref name=lawrence>{{citation|title=A Catalog of Special Plane Curves|series=Dover Books on Mathematics|first=J. Dennis|last=Lawrence|publisher=Courier Corporation|year=2013|isbn=9780486167664|contribution-url=https://books.google.com/books?id=9rrFAgAAQBAJ&pg=PA90|contribution=4.3 Witch of Agnesi (Fermat, 1666; Agnesi, 1748)|pages=90–93}}</ref>

<ref name=mulcrone>{{citation
 | last = Mulcrone | first = T. F.
 | doi = 10.2307/2309605
 | journal = [[American Mathematical Monthly]]
 | mr = 0085163
 | pages = 359–361
 | title = The names of the curve of Agnesi
 | volume = 64
 | issue = 5
 | year = 1957| jstor = 2309605
 }}</ref>

<ref name=noonan>{{citation
 | last1 = Noonan | first1 = Julie A.
 | last2 = Smith | first2 = Roger K.
 | date = September 1985
 | doi = 10.1080/03091928508245426
 | issue = 1–4
 | journal = Geophysical & Astrophysical Fluid Dynamics
 | pages = 123–143
 | title = Linear and weakly nonlinear internal wave theories applied to 'morning glory' waves
 | volume = 33| bibcode = 1985GApFD..33..123N
 }}</ref>

<ref name=oed>{{citation|title=Oxford English Dictionary|url=http://www.oed.com/view/Entry/229575|publisher=Oxford University Press|year=2018|at=witch, ''n''.2, 4(e)|accessdate=3 July 2018|quote=1875 B. Williamson ''Elem. Treat. Integral Calculus'' vii. 173   Find the area between the witch of Agnesi <math>xy^2 = 4a^2(2a-x)</math> and its asymptote.}}</ref>

<ref name=omnibus>{{citation
 | last1 = Fuchs | first1 = Dmitry | author1-link = Dmitry Fuchs
 | last2 = Tabachnikov | first2 = Serge | author2-link = Sergei Tabachnikov
 | doi = 10.1090/mbk/046
 | isbn = 978-0-8218-4316-1
 | location = Providence, RI
 | mr = 2350979
 | page = 142
 | publisher = American Mathematical Society
 | title = Mathematical Omnibus: Thirty Lectures on Classic Mathematics
 | url = https://books.google.com/books?id=IiG9AwAAQBAJ&pg=PA142
 | year = 2007}}</ref>

<ref name=runge>{{citation
 | last1 = Cupillari | first1 = Antonella
 | last2 = DeThomas | first2 = Elizabeth
 | date = Spring 2007
 | issue = 2
 | journal = Mathematics and Computer Education
 | pages = 143–156
 | title = Unmasking the witchy behavior of the Runge function
 | url = https://search.proquest.com/docview/235858817
 | volume = 41}}</ref>

<ref name=singh>{{citation
 | last = Singh | first = Simon | authorlink = Simon Singh
 | isbn = 0-8027-1331-9
 | location = New York
 | mr = 1491363
 | page = 100
 | publisher = Walker and Company
 | title = Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem
 | year = 1997| title-link = Fermat's Last Theorem (book) }}</ref>

<ref name=snyder>{{citation
 | last1 = Snyder | first1 = William H.
 | last2 = Thompson | first2 = Roger S.
 | last3 = Eskridge | first3 = Robert E.
 | last4 = Lawson | first4 = Robert E.
 | last5 = Castro | first5 = Ian P.
 | last6 = Lee | first6 = J. T.
 | last7 = Hunt | first7 = Julian C. R.
 | last8 = Ogawa | first8 = Yasushi
 | date = March 1985
 | doi = 10.1017/s0022112085000684
 | issue = –1
 | journal = Journal of Fluid Mechanics
 | page = 249
 | title = The structure of strongly stratified flow over hills: dividing-streamline concept
 | volume = 152| bibcode = 1985JFM...152..249S
 }}</ref>

<ref name=spencer>{{citation
 | last = Spencer | first = Roy C.
 | date = September 1940
 | doi = 10.1364/josa.30.000415
 | issue = 9
 | journal = Journal of the Optical Society of America
 | page = 415
 | title = Properties of the Witch of Agnesi—Application to Fitting the Shapes of Spectral Lines
 | volume = 30| bibcode = 1940JOSA...30..415S
 }}</ref>

<ref name=stigler>{{citation
 | last = Stigler | first = Stephen M. | author-link = Stephen Stigler
 | date = August 1974
 | doi = 10.1093/biomet/61.2.375
 | issue = 2
 | journal = [[Biometrika]]
 | jstor = 2334368
 | mr = 0370838
 | pages = 375–380
 | title = Studies in the History of Probability and Statistics. XXXIII. Cauchy and the Witch of Agnesi: An Historical Note on the Cauchy Distribution
 | volume = 61}}</ref>

<ref name=struik>A translation of Agnesi's work on this curve can be found in: {{citation|title=A Source Book in Mathematics, 1200–1800|last=Struik|first=Dirk J.|authorlink=Dirk Jan Struik|year=1969|publisher=Harvard University Press|location=Cambridge, Massachusetts|pages=178–180|url=https://archive.org/stream/B-001-001-112#page/n199/mode/2up}}</ref>

<ref name=truesdell>{{citation |first=C. |last=Truesdell |authorlink=Clifford Truesdell|title=Correction and Additions for "Maria Gaetana Agnesi" |journal=Archive for History of Exact Sciences |volume=43 |issue=4 |year=1991 |pages=385–386 |doi=10.1007/BF00374764 |quote=[…] nata da' seni versi, che da me suole chiamarsi la ''Versiera'' in latino però ''Versoria'' […]}}</ref>

<ref name=yates>{{citation|first=Robert C.|last=Yates|title=Curves and their Properties|publisher=National Council of Teachers of Mathematics|series=Classics in Mathematics Education|volume=4|year=1954|url=https://files.eric.ed.gov/fulltext/ED100648.pdf|contribution=Witch of Agnesi|pages=237–238}}</ref>

}}

==External links==
{{Wikisource1911Enc|Witch of Agnesi}}
{{commons|Witch of Agnesi}}
*[http://www-groups.dcs.st-and.ac.uk/~history/Curves/Witch.html "Witch of Agnesi" at MacTutor's Famous Curves Index]
*{{MathWorld | urlname=WitchofAgnesi | title=Witch of Agnesi}}
*[http://demonstrations.wolfram.com/WitchOfAgnesi/ Witch of Agnesi] by Chris Boucher based on work by [[Eric W. Weisstein]], [[The Wolfram Demonstrations Project]].
*[http://www.mathcurve.com/courbes2d.gb/agnesi/agnesi.shtml "Witch of Agnesi" at "mathcurve"]
*{{citation|first=Evelyn|last=Lamb|publisher=[[Scientific American]]|url=https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-witch-of-agnesi/|title=A Few of My Favorite Spaces: The Witch of Agnesi|work=Roots of Unity|date=28 May 2018}}

[[Category:Algebraic curves]]