Witch of Agnesi

Witch of Agnesi

In mathematics, the witch of Agnesi (Italian pronunciation: [a.ˈɲe.zi]), sometimes called the witch of Maria Agnesi is the curve defined as follows.

The Witch of Agnesi with labeled points

Starting with a fixed circle, a point O on the circle is chosen. For any other point A on the circle, the secant line OA is drawn. The point M is diametrically opposite O. The line OA intersects the tangent of M at the point N. The line parallel to OM through N, and the line perpendicular to OM through A intersect at P. As the point A is varied, the path of P is the witch.

The curve is asymptotic to the line tangent to the fixed circle through the point O.

Contents

Equations

An animation showing the construction of the Witch of Agnesi

Suppose the point O is the origin, and that M is on the positive y-axis. Suppose the radius of the circle is a.

Then the curve has Cartesian equation

\!y = \frac{8a^3}{x^2+4a^2}.

Note that if a=1/2, then this equation becomes rather simple:

\!y = \frac{1}{x^2+1}.

Parametrically, if \theta\, is the angle between OM and OA, measured clockwise, then the curve is defined by the equations

\!x = 2a \tan \theta,\ y = 2a \cos ^2 \theta.\,

Another parameterization, with \theta\, being the angle between OA and the x-axis, increasing anti-clockwise is

\!x = 2a \cot \theta,\ y=2a\sin ^2 \theta.\,

Properties

The Witch of Agnesi with parameters a=1, a=2, a=4, and a=8
  • The area between the Witch and its asymptote is four times the area of the fixed circle (i.e., a2).
  • The volume of revolution of the Witch, about its asymptote, is 2a3.
\!\left(0,a\right),

which is the same as that of the generating circle (diameter = 2a).

History

The curve was studied by Pierre de Fermat in 1630. In 1703, Guido Grandi gave a construction for the curve. In 1718 Grandi suggested the name 'versoria' for the curve, the Latin term for sheet, the rope which turns (adjusts the trim of) the sail, and used the Italian word for it, 'versiera', a hint to sinus versus that appeared in his construction.[1]

In 1748, Maria Agnesi published her famous summation treatise Instituzioni analitiche ad uso della gioventù italiana, in which the curve was named according to Grandi, 'versiera'. [1] Coincidentally, the contemporary Italian word 'Aversiera'/'Versiera', derived from Latin 'Adversarius', a nickname for Devil, "Adversary of God", was synonymous with "witch".[2] Probably for this reason Cambridge professor John Colson mistranslated the name of the curve thusly. Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened. [3][4][5] Struik mentions that:

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 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 Newton's classification. ... The first to use the term 'witch' in this sense may have been B. Williamson, Integral calculus, 7 (1875), 173;[6] see Oxford English Dictionary.

On the other hand, Stephen Stigler suggests that Grandi himself "may have been indulging in a play on words".[7]

The Witch of Agnesi is also a fiction novel by Robert Spiller, in which a teacher gives a version of the history of the term.[8]

See also

References

  1. ^ a b C. Truesdell, "Correction and Additions for 'Maria Gaetana Agnesi'", Archive for History of Exact Science 43 (1991), 385-386. doi:10.1007/BF00374764
    • Per Grandi: "...nata da' seni versi, che da me suole chiamarsi la Versiera in latino pero Versoria..."
  2. ^ "Vocabolario dell'uso toscano", 1 By Pietro Fanfani, p. 334
  3. ^ Women in Mathematics By Lynn M. Osen (1975) p. 45
  4. ^ "Fermat's Enigma" by Simon Singh p. 100
  5. ^ The universal book of mathematics: from Abracadabra to Zeno's paradoxes By David J. Darling (2004) p. 8
  6. ^ "173 Find the area between the witch of Agnesi xy2 = 4a2(2ax) and its asymptote." (Oxford English Dictionary)
  7. ^ S.M.Stigler, "Cauchy and the witch of Agnesi: An historical note on the Cauchy distribution", Biometrika, 1974, vol. 61, no.2 p. 375-380
  8. ^ Spiller, Robert (2006). The Witch of Agnesi. Palm Beach, FL: Medallion Press. ISBN 9781932815726. OCLC 71259167. 

Sources

External links


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Look at other dictionaries:

  • witch of agnesi — änˈyāzē noun or witch Usage: usually capitalized A Etymology: after Maria Gaetana Agnesi died 1799 Italian mathematician; witch, translation of Italian versiera cubic curve (modification of New Latin versoria versed sine) influenced by versiera… …   Useful english dictionary

  • witch of Agnesi — noun Etymology: Maria Gaetana Agnesi died 1799 Italian mathematician; witch, translation of Italian versiera cubic curve (influenced by Italian versiera female demon) Date: 1875 a plane cubic curve that is symmetric about the y axis and… …   New Collegiate Dictionary

  • witch of Agnesi — /ah nyay zee/, Geom. a plane curve symmetrical about the y axis and asymptotic to the x axis, given by the equation x2y=4a2(2a y). Also called versiera. [1870 75; named after Maria Gaetana Agnesi (1718 99), Italian mathematician and philosopher]… …   Universalium

  • Agnesi — is a surname. People with this surname include:* Alberto Agnesi, Mexican telenovela actor * Maria Gaetana Agnesi (1718 1799), Italian linguist, mathematician and philosopher; sister of Maria Teresa * Maria Teresa Agnesi Pinottini (1720 1795),… …   Wikipedia

  • Agnesi-Kurve — Die Versiera der Agnesi, oder auch Versiera der Maria Agnesi, ist eine spezielle ebene Kurve, eine algebraische Kurve 3. Ordnung vom Geschlecht 0. Sie ist benannt nach der Mathematikerin Maria Agnesi, die sie 1748 veröffentlichte. Die Kurve wurde …   Deutsch Wikipedia

  • Agnesi, Maria Gaetana — ▪ Italian mathematician born May 16, 1718, Milan, Habsburg crown land [now in Italy] died January 9, 1799, Milan  Italian mathematician and philosopher, considered to be the first woman in the Western world to have achieved a reputation in… …   Universalium

  • Witch (disambiguation) — A witch is a practitioner of witchcraft. The word witch or its variants may also refer to:Performing arts* The Witch, a 1616 play by Thomas Middleton * The Witch (ballet), a ballet by John CrankoLiterature and comics* The Witch (fairy tale), a… …   Wikipedia

  • witch — I. noun Etymology: Middle English wicche, from Old English wicca, masculine, wizard & wicce, feminine, witch; akin to Middle High German wicken to bewitch, Old English wigle divination, and perhaps to Old High German wīh holy more at victim Date …   New Collegiate Dictionary

  • witch of A|gne|si — «ah NYAY zee», Mathematics. a plane cubic curve that is symmetrical about the y axis and asymptotic to the x axis, defined by the equation 4a2 (2a ― y) = x2y. ╂[translation of Italian versiera di Agnesi < versiera witch, versed sine + di of +… …   Useful english dictionary

  • Witch — Witch, n. [OE. wicche, AS. wicce, fem., wicca, masc.; perhaps the same word as AS. w[=i]tiga, w[=i]tga, a soothsayer (cf. {Wiseacre}); cf. Fries. wikke, a witch, LG. wikken to predict, Icel. vitki a wizard, vitka to bewitch.] [1913 Webster] 1.… …   The Collaborative International Dictionary of English

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