Witch of Agnesi
In mathematics, the witch of Agnesi (Italian pronunciation: [aɲˈɲeːzi, -eːsi; -ɛːzi]) is a cubic plane curve defined from two diametrically opposite points of a circle.
The curve was studied as early as 1653 by Pierre de Fermat, in 1703 by Guido Grandi, and by Isaac Newton.
It gets its name from Italian mathematician Maria Gaetana Agnesi who published it in 1748.
The Italian name la versiera di Agnesi is based on Latin versoria (sheet of sailing ships) and the sinus versus.
[1][2][3][4] The graph of the derivative of the arctangent function forms an example 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.
In its simplified form, this curve is the graph of the derivative of the arctangent function.
[8] The witch of Agnesi can also be described by parametric equations whose parameter θ is the angle between OM and OA, measured clockwise:[6][7]
[6] This is two times the volume of the torus formed by revolving the defining circle of the witch around the same line.
[9] The curve has a unique vertex at the point of tangency with its defining circle.
[9] The curve was studied by Pierre de Fermat in his 1659 treatise on 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].
[16] Paradís, Pla & Viader (2008) speculate that the geometer who suggested this curve to Fermat might have been Antoine de Laloubère.
[18] Grandi (1718) also suggested the name versiera (in Italian) or versoria (in Latin) for the curve.
[19] The Latin term is also used for a 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.
[9][18][20][21] In 1748, Maria Gaetana Agnesi published Instituzioni analitiche ad uso della gioventù italiana, an early textbook on calculus.
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.
[22] Maria Gaetana Agnesi named the curve according to Grandi, versiera.
[20][22] Coincidentally, at that time in Italy it was common to speak of the Devil through other words like aversiero or versiero, derived from Latin adversarius, the "adversary" of God.
[23] Because of this, Cambridge professor John Colson mistranslated the name of the curve as "witch".
[24] Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened.
[25][26] Struik mentions that:[22] 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 first to use the term 'witch' in this sense may have been B. Williamson, Integral calculus, 7 (1875), 173;[27] 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).
[18] A scaled version of the curve is the probability density function of the Cauchy distribution.
, another scaled version of the witch of Agnesi, when interpolating this function over the interval
[31] Curves with this shape have been used as the generic topographic obstacle in a flow in mathematical modeling.
[36] Witch of Agnesi is also the title of a music album by jazz quartet Radius.
